qcp/bh duality and quantum matters.hepth.hanyang.ac.kr/~sjs/mytalks/2018/2018.06.apctp.pdf ·...

Sang-JinSin(Hanyang)

2018.06@apctp

QCP/BHDualityandquantummatters.

118.06.apctp

Introduction:physicsisunification

Unification:Regardingtwodifferentthingsasthesame1.  QM:particle=wave2.  SR:time=space3.  GR:gravity=geometry

Today:QMmatter=geometry Everything=geometry,perhaps

(Mydef.ofstringtheoryifthereis.)

*Quantummatter=matterwithSlowelectrons218.06.apctp

Materialwithslowelectrons

3

Transition metal oxides

25

• Atomic quantum numbers: (n, l, m, ms)• Partially filled d-shell: strongly correlated, multi-orbital• O2-: anion

Spin

-orb

it co

uplin

gC

oulomb interaction

V3+: V2O3, d2

V4+: VO2, d1TransitionmetalOxideà3d

HiTcSCmaterials

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1.  Loss of particle à Loss of calculability:

Effect of strong Interaction i)

418.06.apctp

2. Rapid Thermalization à enables Hydro-dynamic description

Effect of strong Interaction (ii)

518.06.apctp

Planckian dissipation !

Universal entropy production time in QC system: !

€

τ = τ ! ≈!kBT

€

ρ ∝1τ !

∝ kBT

Observed in Quark gluon plasma (heavy ion colliders RIHC, LHC) and cold atom unitary fermi gas:!

Since early 1990s recognized as responsible for strange metal properties, also linear resistivity high Tc metals ??:!

s= T ~ =

1

4

~kB

PlankianDissipation:Arriving at universality instantly.

Planckian dissipation !

Universal entropy production time in QC system: !

€

τ = τ ! ≈!kBT

€

ρ ∝1τ !

∝ kBT

Observed in Quark gluon plasma (heavy ion colliders RIHC, LHC) and cold atom unitary fermi gas:!

Since early 1990s recognized as responsible for strange metal properties, also linear resistivity high Tc metals ??:!

s= T ~ =

1

4

~kB

LinearRegistivityinStrangeMetal

18.06.apctp

4

0 100 2000

5

10

15

20

25

Temperature (K)

L /

L0

B

1 10 100 1000109

1010

1011

Temperature (K)

nm

in (

cm

-2)

DisorderLimited

ThermallyLimited

S3S2S1

A

−6 −4 −2 0 2 4 60

4

8

12

16

20

n (1010 cm−2)

L/L

0

C

40 60 80 1000

2

4

6

8

10

T (K)

H

(eV

/µm

2)

CHe

h

-V+Ve

h

∆Vg = 0

FIG. 3. Disorder in the Dirac fluid. (A) Minimum car-rier density as a function of temperature for all three sam-ples. At low temperature each sample is limited by disorder.At high temperature all samples become limited by thermalexcitations. Dashed lines are a guide to the eye. (B) TheLorentz ratio of all three samples as a function of bath tem-perature. The largest WF violation is seen in the cleanestsample. (C) The gate dependence of the Lorentz ratio is wellfit to hydrodynamic theory of Ref. [5, 6]. Fits of all threesamples are shown at 60 K. All samples return to the Fermiliquid value (black dashed line) at high density. Inset showsthe fitted enthalpy density as a function of temperature andthe theoretical value in clean graphene (black dashed line).Schematic inset illustrates the di↵erence between heat andcharge current in the neutral Dirac plasma.

more pronounced peak but also a narrower density de-pendence, as predicted [5, 6].

More quantitative analysis of L(n) in our experimentcan be done by employing a quasi-relativistic hydrody-namic theory of the DF incorporating the e↵ects of weakimpurity scattering [5, 6, 39].

L =LDF

(1 + (n/n0)2)2 (2)

where

LDF =HvFlmT 2min

and n20 =

Hmin

e2vFlm. (3)

Here vF is the Fermi velocity in graphene, min is the elec-trical conductivity at the CNP, H is the fluid enthalpydensity, and lm is the momentum relaxation length from

impurities. Two parameters in Eqn. (2) are undeter-mined for any given sample: lm and H. For simplic-ity, we assume we are well within the DF limit wherelm and H are approximately independent of n. We fitEqn. (2) to the experimentally measured L(n) for alltemperatures and densities in the Dirac fluid regime toobtain lm and H for each sample. Fig 3C shows threerepresentative fits to Eqn. (2) taken at 60 K. lm is esti-mated to be 1.5, 0.6, and 0.034 µm for samples S1, S2,and S3, respectively. For the system to be well describedby hydrodynamics, lm should be long compared to theelectron-electron scattering length of 0.1 µm expectedfor the Dirac fluid at 60 K [18]. This is consistent withthe pronounced signatures of hydrodynamics in S1 andS2, but not in S3, where only a glimpse of the DF appearsin this more disordered sample. Our analysis also allowsus to estimate the thermodynamic quantity H(T ) for theDF. The Fig. 3C inset shows the fitted enthalpy densityas a function of temperature compared to that expectedin clean graphene (dashed line) [18], excluding renormal-ization of the Fermi velocity. In the cleanest sample H

varies from 1.1-2.3 eV/µm2 for Tdis < T < Telph. Thisenthalpy density corresponds to 20 meV or 4kBTper charge carrier — about a factor of 2 larger than themodel calculation without disorder [18].

To fully incorporate the e↵ects of disorder, a hydrody-namic theory treating inhomogeneity non-perturbativelyis necessary [40, 41]. The enthalpy densities reportedhere are larger than the theoretical estimation obtainedfor disorder free graphene, consistent with the picturethat chemical potential fluctuations prevent the samplefrom reaching the Dirac point. While we find thermalconductivity well described by Ref. [5, 6], electrical con-ductivity increases slower than expected away from theCNP, a result consistent with hydrodynamic transport ina viscous fluid with charge puddles [41].

In a hydrodynamic system, the ratio of shear viscos-ity to entropy density s is an indicator of the strengthof the interactions between constituent particles. It issuggested that the DF can behave as a nearly perfectfluid [18]: /s approaches a conjecture by Kovtun-Son-Starinets: (/s)/(~/kB) & 1/4 for a strongly inter-acting system [42]. A non-perturbative hydrodynamicframework can be employed to estimate , as we discusselsewhere [41]. A direct measurement of is of greatinterest.

We have experimentally discovered the breakdown ofthe WF law and provided evidence for the hydrodynamicbehavior of the Dirac fermions in graphene. This pro-vides an experimentally realizable Dirac fluid and opensthe way for future studies of strongly interacting rela-tivistic many-body systems. Beyond a diverging thermalconductivity and an ultra-low viscosity, other peculiarphenomena are expected to arise in this plasma. Themassless nature of the Dirac fermions is expected to re-sult in a large kinematic viscosity, despite a small shear

6

Other Effect of strong Interaction

Violation of Wiedemann-Franz law

PseudoGap

ForSIS,Noreliablecalculationmethodforsuchmatter. Condensed matter theory

= Structure (UV scale) à functionality (IR scale) Physics with reductionism is Not applicable to SIS.

Problem and Implications

718.06.apctp

keV meV

8

Black Hole has i) Universality /no hair / information loss ii)Thermodynamics(1stlawßàEinsteineq.)iii)Transport

So is the QCP. Absence of scaleàno structural dependence à info. Lost à Universality

idea?:SimilarityofBHandQCP

18.06.apctp

QuantumCriticalpoint

GoodObservablesforthenewtheory?

9

Z,θ:ω=kz,[s]=D-θ

Mostuniversalquantity:spectralfunctionandTransportnearQCP

Classifying QCP : dynamical exponent

18.06.apctp

10

Postulate: the dictionary for ads/cft is true in general.

QCP(z,θ) Geometry (z,θ)

1 Introduction

The AdS/CFT correspondence, sometimes called as gauge/gravity duality, is a conjectured relationship

between quantum field theory and gravity. Precisely, in this correspondence, quantum physics of strongly

correlated many-body systems is related to the classical dynamics of gravity which lives in one higher

dimension. On the other hand according to the AdS/CFT dictionary, an AdS geometry at the gravity

side could only address the conformal symmetry of the dual field theory. However, the generalization of

gauge/gravity correspondence to geometries which are not asymptotically AdS seems to be important

as long as such extension may be related to the invariance under a certain scaling of dual field theory

which does not even have conformal symmetry. Such generalization of AdS is actually motivated by

consideration of gravity toy models in condensed matter physics (the application of such generalization

can be found, for example, in [2]). A prototype of this generalization is a theory with the Lifshitz fixed

point in which the spatial and time coordinates of a field theory have been scaled as

t → ζzt, x → ζx, r → ζr, (1.1)

where z is the critical dynamical exponent. From the holographic duality point of view, for a (D + 1)-

dimensional theory, the corresponding (D+ 2)-dimensional gravity dual can be defined by the following

metric

ds2D+2 =−dt2

r2z+

dr2

r2+

1

r2

D!

i=1

dx2i , (1.2)

where in this paper the AdS radius is set to be one. Due to the anisotropy between space and time, it is

clear that this metric can not be an ordinary solution of the Einstein equation, in fact one needs some

sorts of matter fields to break the isotropy, e.g., by adding a massive vector field or a gauge field coupled

to a scalar field [3–6]. In general by adding a dilaton with non trivial potential and an abelian gauge

field to Einstein-Hilbert action (Einstein-Maxwell-Dilaton theory), one can find even more interesting

metrics, in particular the following metric has been used frequently [7]

ds2D+2 = r2θD

"−dt2

r2z+

dr2

r2+

1

r2

D!

i=1

dx2i

#

, (1.3)

where θ is hyperscaling violation exponent. This metric under the scale-transformation (1.1) transforms

as ds → ζθD ds. In a theory with hyperscaling violation, the thermodynamic parameters behave in such

a way that they are stated in D− θ dimensions; More precisely, in a (D+1)-dimensional theories which

are dual to background (1.2), the entropy scales with temperature as TD/z, however, in the presence of

θ namely dual to (1.3), it scales as T (D−θ)/z [7, 8]. Therefore one may associate an effective dimension

to the theory and this becomes important in studying the log behavior of the entanglement entropy of

system with Fermi surface in condense matter physics, explicitly it was shown that for θ = D − 1 for

any z, the entanglement entropy exhibits a logarithmic violation of the area law [9]. On the other hand

for such backgrounds time-dependency can be achieved by Vaidya metric with a hyperscaling violating

factor. It is the main aim of this paper to investigate how entanglement (mutual information) spreads

in time-dependent hyperscaling violating backgrounds.

Basically, the AdS-Vaidya metric is used to describe a gravitational collapse of a thin shell of matter

in formation of the black hole. This metric in D + 2 dimensions is given by

ds2 =1

ρ2

$

− f(ρ, v)dv2 − 2dρdv +D!

i=1

dx2i

%

, f(ρ, v) = 1−m(v)ρD+1, (1.4)

where ρ is the radial coordinate, xi’s (i = 1, .., D) are spatial boundary coordinates and, here, the

2

Dual?

Transport coefficients

Make a unique rule Exp. data

18.06.apctp

Analogy:Freeparticlewaveeq.àSchrodingereq.underforce.

holographymeetstheexperiment

11

SimplestQCPisz=1:graphene

Q: strong coupling?….10yearsofspeculation 1. 2. near Dirac Point : Tiny FS à No (insufficient) screening

g2 =e2 · c

4~c · vF 1

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4March2016

12

REFERENCES AND NOTES

1. L. D. Landau, E. M. Lifshitz, Fluid Mechanics (Pergamon Press,1987).

2. G. K. Batchelor, An Introduction to Fluid Dynamics (CambridgeUniv. Press, 1967).

3. B. V. Jacak, B. Müller, Science 337, 310–314 (2012).4. C. Cao et al., Science 331, 58–61 (2011).5. E. Elliott, J. A. Joseph, J. E. Thomas, Phys. Rev. Lett. 113,

020406 (2014).6. T. Schäfer, D. Teaney, Rep. Prog. Phys. 72, 126001 (2009).7. R. N. Gurzhi, Sov. Phys. Usp. 11, 255–270 (1968).8. A. O. Govorov, J. J. Heremans, Phys. Rev. Lett. 92, 026803

(2004).9. R. Bistritzer, A. H. MacDonald, Phys. Rev. B 80, 085109

(2009).10. M. Müller, J. Schmalian, L. Fritz, Phys. Rev. Lett. 103, 025301

(2009).11. A. V. Andreev, S. A. Kivelson, B. Spivak, Phys. Rev. Lett. 106,

256804 (2011).12. A. Tomadin, G. Vignale, M. Polini, Phys. Rev. Lett. 113, 235901

(2014).

13. B. N. Narozhny, I. V. Gornyi, M. Titov, M. Schütt, A. D. Mirlin,Phys. Rev. B 91, 035414 (2015).

14. G. F. Giuliani, G. Vignale, Quantum Theory of the Electron Liquid(Cambridge Univ. Press, 2005).

15. M. J. M. de Jong, L. W. Molenkamp, Phys. Rev. B 51,13389–13402 (1995).

16. J. C. W. Song, L. S. Levitov, Phys. Rev. Lett. 109, 236602(2012).

17. R. V. Gorbachev et al., Nat. Phys. 8, 896–901 (2012).18. Supplementary materials are available on Science Online.19. A. S. Mayorov et al., Nano Lett. 11, 2396–2399

(2011).20. L. Wang et al., Science 342, 614–617 (2013).21. T. Taychatanapat, K. Watanabe, T. Taniguchi, P. Jarillo-Herrero,

Nat. Phys. 9, 225–229 (2013).22. Q. Li, S. Das Sarma, Phys. Rev. B 87, 085406 (2013).23. M. Polini, G. Vignale, http://arxiv.org/abs/1404.5728

(2014).24. D. A. Abanin et al., Science 332, 328–330 (2011).25. A. Principi, G. Vignale, M. Carrega, M. Polini, http://arxiv.org/

abs/1506.06030 (2015).

ACKNOWLEDGMENTS

This work was supported by the European Research Council,the Royal Society, Lloyd’s Register Foundation, the GrapheneFlagship, and the Italian Ministry of Education, University andResearch through the program Progetti Premiali 2012 (projectABNANOTECH). D.A.B. and I.V.G. acknowledge support from theMarie Curie program SPINOGRAPH (Spintronics in Graphene).A.P. received support from the Nederlandse WetenschappelijkOrganisatie. R.K.K. received support from the Engineering andPhysical Sciences Research Council.

SUPPLEMENTARY MATERIALS

www.sciencemag.org/content/351/6277/1055/suppl/DC1Supplementary TextFigs. S1 to S14References (26–37)

14 July 2015; accepted 23 December 2015Published online 11 February 201610.1126/science.aad0201

ELECTRON TRANSPORT

Observation of the Dirac fluid and thebreakdown of the Wiedemann-Franzlaw in grapheneJesse Crossno,1,2 Jing K. Shi,1 Ke Wang,1 Xiaomeng Liu,1 Achim Harzheim,1

Andrew Lucas,1 Subir Sachdev,1,3 Philip Kim,1,2* Takashi Taniguchi,4 Kenji Watanabe,4

Thomas A. Ohki,5 Kin Chung Fong5*

Interactions between particles in quantum many-body systems can lead to collective behaviordescribed by hydrodynamics. One such system is the electron-hole plasma in graphene nearthe charge-neutrality point, which can form a strongly coupled Dirac fluid.This charge-neutralplasma of quasi-relativistic fermions is expected to exhibit a substantial enhancement of thethermal conductivity, thanks to decoupling of charge and heat currents within hydrodynamics.Employing high-sensitivity Johnson noise thermometry, we report an order of magnitudeincrease in the thermal conductivity and the breakdown of the Wiedemann-Franz law in thethermally populated charge-neutral plasma in graphene.This result is a signature of the Diracfluid and constitutes direct evidence of collective motion in a quantum electronic fluid.

Understanding the dynamics of many inter-acting particles is a formidable task in phys-ics. For electronic transport inmatter, stronginteractions can lead to a breakdown of theFermi liquid (FL) paradigm of coherent

quasi-particles scattering off of impurities. Insuch situations, provided that certain conditionsare met, the complex microscopic dynamics canbe coarse-grained to a hydrodynamic descriptionof momentum, energy, and charge transport onlong length and time scales (1). Hydrodynamicshas been successfully applied to a diverse array of

interacting quantum systems, fromhigh-mobilityelectrons in conductors (2) to cold atoms (3) andquark-gluon plasmas (4). Hydrodynamic effectsare expected to greatly modify transport coef-ficients compared with their FL counterparts, ashas been argued for strongly interactingmasslessDirac fermions in graphene at the charge-neutralitypoint (CNP) (5–8).Many-body physics in graphene is interesting

because of electron-hole symmetry and a lineardispersion relation at the CNP (9, 10). Togetherwith the vanishing Fermi surface, the ultra-relativistic spectrum leads to ineffective screening(11) and the formation of a strongly interactingquasi-relativistic electron-hole plasma known asa Dirac fluid (DF) (12). The DF shares many fea-tures with quantum critical systems (13): mostimportantly, the electron-electron scattering timeis fast (14–17) and well suited to a hydrodynamicdescription. Because of the quasi-relativistic na-ture of the DF, this hydrodynamic limit is de-scribed by equations (18) quite different from

those applicable to its nonrelativistic counter-parts. A number of unusual properties have beenpredicted, including nearly perfect (inviscid) flow(19) and a diverging thermal conductivity, whichresults in the breakdown of theWiedemann-Franz(WF) law at finite temperature (5, 6).Away from theCNP, graphenehas a sharpFermi

surface, and the standardFLphenomenologyholds.By tuning the chemical potential, we are able tomeasure thermal and electrical conductivity inboth the DF and the FL in the same sample. In aFL, the relaxation of heat and charge currents isclosely related, as they are carried by the samequasi-particles. The WF law (20) states that theelectronic contribution to a metal’s thermal con-ductivity ke is proportional to its electrical con-ductivity s and temperature T, such that theLorenz ratio L satisfies

L ≡kesT

¼ p2

3kBe

! "2

≡ L0 ð1Þ

where e is the electron charge, kB is the Boltz-mann constant, and L0 is the Sommerfeld valuederived from FL theory. L0 depends only onfundamental constants, not specific details of thesystem such as carrier density or effective mass.As a robust prediction of FL theory, the WF lawhas been verified in numerous metals (20). Athigh temperatures, the WF law can be violateddue to inelastic electron-phonon scattering orbipolar diffusion in semiconductors, even whenelectron-electron interactions are negligible (21).In recent years, several nontrivial violations oftheWF law—all of which are related to the emer-gence of non-FL behavior—have been reportedin strongly interacting systems such as Luttingerliquids (22), metallic ferromagnets (23), heavy fer-mionmetals (24), and underdoped cuprates (25).Owing to the strong Coulomb interactions be-

tween thermally excited charge carriers, the WFlaw is expected to be violated at the CNP in a DF.An electric field drives electrons and holes in op-posite directions; collisions between them intro-duce a frictional dissipation, resulting in a finiteconductivity even in the absence of disorder (26).In contrast, a temperature gradient causes elec-trons and holes to move in the same direction,

1058 4 MARCH 2016 • VOL 351 ISSUE 6277 sciencemag.org SCIENCE

1Department of Physics, Harvard University, Cambridge, MA02138, USA. 2John A. Paulson School of Engineering andApplied Sciences, Harvard University, Cambridge, MA 02138,USA. 3Perimeter Institute for Theoretical Physics, Waterloo,Ontario N2L 2Y5, Canada. 4National Institute for MaterialsScience, Namiki 1-1, Tsukuba, Ibaraki 305-0044, Japan.5Quantum Information Processing Group, Raytheon BBNTechnologies, Cambridge, MA 02138, USA.*Corresponding author. E-mail: [email protected] (P.K.);[email protected] (K.C.F.)

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originally published online February 11, 2016 (6277), 1055-1058. [doi: 10.1126/science.aad0201]351Science

M. Polini (February 11, 2016) Novoselov, I. V. Grigorieva, L. A. Ponomarenko, A. K. Geim andTomadin, A. Principi, G. H. Auton, E. Khestanova, K. S. D. A. Bandurin, I. Torre, R. Krishna Kumar, M. Ben Shalom, A.grapheneNegative local resistance caused by viscous electron backflow in

Editor's Summary

, this issue p. 1061, 1055, 1058; see also p. 1026Sciencetransport in graphene, a signature of so-called Dirac fluids.

observed a huge increase of thermalet al.fluid flowing through a small opening. Finally, Crossno found evidence in graphene of electron whirlpools similar to those formed by viscouset al.Bandurin

had a major effect on the flow, much like what happens in regular fluids.2fluid in thin wires of PdCoO found that the viscosity of the electronet al.counterexamples (see the Perspective by Zaanen). Moll

flow rarely resembles anything like the familiar flow of water through a pipe, but three groups describe Electrons inside a conductor are often described as flowing in response to an electric field. This

Electrons that flow like a fluid

This copy is for your personal, non-commercial use only.

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is a registered trademark of AAAS. ScienceAdvancement of Science; all rights reserved. The title Avenue NW, Washington, DC 20005. Copyright 2016 by the American Association for thein December, by the American Association for the Advancement of Science, 1200 New York

(print ISSN 0036-8075; online ISSN 1095-9203) is published weekly, except the last weekScience

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thus inducing an energy current that grows un-impeded by interparticle collisions as the totalmomentum is conserved. The thermal conduc-tivity is therefore limited by the rate at whichmomentum is relaxed by residual impurities.Realization of the DF in graphene requires

that the thermal energy be larger than the localchemical potential m(r), defined at position r:kBT ≳ jmðrÞj. Impurities cause spatial variationsin the local chemical potential, and even whenthe sample is globally neutral, it is locally dopedto form electron-hole puddles with finite m(r)(27–30). At high temperatures, formation of theDF is complicated by phonon scattering, whichcan relax momentum by creating additional in-elastic scattering channels. This high-temperaturelimit occurs when the electron-phonon scatteringrate becomes comparable to the electron-electronscattering rate. These two temperatures set theexperimental window in which the DF and thebreakdown of the WF law can be observed.Tominimize disorder, we usedmonolayer gra-

phene samples encapsulated in hexagonal boronnitride (31). All devices used in this study havetwo terminals, so as to keep awell-defined temper-ature profile (32), with contacts fabricated usingthe one-dimensional edge technique (33) to mini-mize contact resistance. We employed a back-gatevoltage Vg applied to the silicon substrate to tune

the charge carrier density n = ne – nh, where neand nh are the electron and hole densities, re-spectively (21). All measurements were performedin a cryostat to control the bath temperatureTbath.Figure 1A shows the resistance R versus Vg mea-sured at various fixed temperatures for a repre-sentative device [see (21) for all samples]. Fromthis, we used the known sample dimensions toestimate the electrical conductivity s (Fig. 1B). Atthe CNP, the residual charge carrier density nmin

can be estimated by extrapolating a linear fit oflog(s) as a function of log(n) out to theminimumconductivity (34). At the lowest temperatures,nmin

saturates to ~8 × 109 cm–2. Extraction of nmin bythis method prompts overestimation of the charge-puddle energy, consistentwith previous reports (31).Above the disorder temperature scale Tdis ~ 40 K,nmin increases as Tbath is raised, which suggeststhat thermal excitations begin to dominate andthe sample enters the nondegenerate regime nearthe CNP.Electronic thermal conductivity wasmeasured

using high-sensitivity Johnson noise thermometry(JNT) (32, 35). We applied a small bias currentthrough the sample, thus injecting a Joule heatingpower P directly into the electronic system andinducing a small difference between the temper-ature of the graphene electrons and that of thebath: DT≡Te−Tbath. The electron temperature

Te was monitored independently of the latticetemperature through the Johnson noise poweremitted at 100 MHz, with a 20-MHz bandwidthdefined by an inductor-capacitor matching net-work. We designed our JNT setup to be operatedover a wide temperature range, from 3 to 300 K(35). With a precision of ~10 mK, we measuredsmall deviations of Te from Tbath (i.e., DT≪Tbath).In this limit, the temperature of the graphenelattice is well thermalized to the bath (32), andour JNT setup allows us to sensitively measurethe electronic cooling pathways in graphene.When the temperature is low enough, electronand lattice interactions are weak (35, 36), andmost of the Joule heat generated in grapheneescapes via direct diffusion to the contacts (21).As the temperature increases, electron-phononscattering becomesappreciable, and thermal tran-sport becomes limited by the electron-phononcoupling strength (36–38). The onset temperatureof appreciable electron-phonon scattering, Tel-ph,depends on the sample disorder and device geo-metry:Tel-ph ~ 80K (35, 36, 39, 40) for our samples.Below this temperature, the electronic contribu-tion of the thermal conductivity can be obtainedfrom P and DT using the device dimensions (21).Figure 1C shows ke(Vg) plotted alongside the

simultaneously measured s(Vg) at various fixedbath temperatures. Here, for a direct quantitativecomparison based on the WF law, we plot thescaled electrical conductivity as sTL0 in the sameunits as ke. In a FL, these two values will coincide,in accordancewith Eq. 1. At low temperatures (T <Tdis ~ 40 K), where the puddle-induced densityfluctuations dominate, we find that ke ≈ sTL0,monotonically increasing as a function of carrierdensity with a minimum at the neutrality point,confirming theWF law in the disordered regime.As T increases (T > Tdis), however, we begin toobserve violation of theWF law. This violation ap-pears only close to the CNP, with the measuredthermal conductivitymaximized at n = 0 (Fig. 1C).The deviation is the largest at 75 K, where ke ismore than an order of magnitude larger than thevalue expected for a FL. The nonmonotonicity ofke(T) is consistent with acoustic phonons relaxingmomentum more efficiently than impurities as Tincreases (41). For T ≳ 100K in our samples, activ-ation of optical phonons introduces an additionalelectron-phonon cooling pathway (35), and themeasured thermal conductivity is larger than ke.This non-FL behavior quickly disappears as jnj in-creases; ke returns to the FL value and restores theWF law. In fact, away from the CNP, the WF lawholds for a wide temperature range, consistentwith previous reports (35, 36, 39) (Fig. 1E). For thisFL regime, we verify the WF law up to T ~ 80 K.Our observation of the breakdown of the WF

law in graphene is consistent with the emer-gence of the DF. Figure 2 shows the full densityand temperature dependence of the experimen-tally measured Lorenz ratio, highlighting thepresence of the DF. The blue region denotesL ∼ L0, suggesting that the carriers in grapheneexhibit FL behavior. The WF law is violated inthe DF (yellow-red region), with a peak Lorenzratio 22 times larger than L0. The green dotted

SCIENCE sciencemag.org 4 MARCH 2016 • VOL 351 ISSUE 6277 1059

Fig. 1. Temperature- and density-dependent electrical and thermal conductivity. (A) Resistance (R)versus gate voltage (Vg) at various temperatures. kW, kilohm. (B) Electrical conductivity (blue) as afunction of the charge density set by the back gate for different bath temperatures. The residual carrierdensity at the neutrality point (green) is estimated by the intersection of the minimum conductivity witha linear fit to log(s), away from neutrality (dashed gray lines). Curves have been offset vertically such thatthe minimum density (green) aligns with the temperature axis to the right. Solid black lines correspond to4e2/h. At low temperatures, the minimum density is limited by disorder (charge puddles). Above Tdis ~ 40 K(yellow shaded area), thermal excitations begin to dominate, and the sample enters the nondegenerateregime near the neutrality point. (C to E) Thermal conductivity (red points) as a function of (C) gate voltageand [(D) and (E)] bath temperature, compared to with the WF law, sTL0 (blue lines). At low temperatureand/or high doping ðjmj ≫ kBTÞ, we find the WF law to hold.This is a nontrivial check on the quality of ourmeasurement. In the nondegenerate regime ðjmj < kBTÞ, the thermal conductivity is enhanced and theWF law is violated. Above T ~ 100 K, electron-phonon coupling becomes appreciable and begins to dominatethermal transport at all measured gate voltages. At this temperature, the yellow shaded background ends. Alldata from this figure are taken from sample S2 [inset in (E)].

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0 100 2000

5

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Temperature (K)L

/ L

0

B

1 10 100 1000109

1010

1011

Temperature (K)

nm

in (

cm

-2)

DisorderLimited

ThermallyLimited

S3S2S1

A

−6 −4 −2 0 2 4 60

4

8

12

16

20

n (1010 cm−2)

L/L

0

C

40 60 80 1000

2

4

6

8

10

T (K)

H

(eV

/µm

2)

CHe

h

-V+Ve

h

∆Vg = 0

FIG. 3. Disorder in the Dirac fluid. (A) Minimum car-rier density as a function of temperature for all three sam-ples. At low temperature each sample is limited by disorder.At high temperature all samples become limited by thermalexcitations. Dashed lines are a guide to the eye. (B) TheLorentz ratio of all three samples as a function of bath tem-perature. The largest WF violation is seen in the cleanestsample. (C) The gate dependence of the Lorentz ratio is wellfit to hydrodynamic theory of Ref. [5, 6]. Fits of all threesamples are shown at 60 K. All samples return to the Fermiliquid value (black dashed line) at high density. Inset showsthe fitted enthalpy density as a function of temperature andthe theoretical value in clean graphene (black dashed line).Schematic inset illustrates the di↵erence between heat andcharge current in the neutral Dirac plasma.

more pronounced peak but also a narrower density de-pendence, as predicted [5, 6].

More quantitative analysis of L(n) in our experimentcan be done by employing a quasi-relativistic hydrody-namic theory of the DF incorporating the e↵ects of weakimpurity scattering [5, 6, 39].

L =LDF

(1 + (n/n0)2)2 (2)

where

LDF =HvFlmT 2min

and n20 =

Hmin

e2vFlm. (3)

Here vF is the Fermi velocity in graphene, min is the elec-trical conductivity at the CNP, H is the fluid enthalpydensity, and lm is the momentum relaxation length from

impurities. Two parameters in Eqn. (2) are undeter-mined for any given sample: lm and H. For simplic-ity, we assume we are well within the DF limit wherelm and H are approximately independent of n. We fitEqn. (2) to the experimentally measured L(n) for alltemperatures and densities in the Dirac fluid regime toobtain lm and H for each sample. Fig 3C shows threerepresentative fits to Eqn. (2) taken at 60 K. lm is esti-mated to be 1.5, 0.6, and 0.034 µm for samples S1, S2,and S3, respectively. For the system to be well describedby hydrodynamics, lm should be long compared to theelectron-electron scattering length of 0.1 µm expectedfor the Dirac fluid at 60 K [18]. This is consistent withthe pronounced signatures of hydrodynamics in S1 andS2, but not in S3, where only a glimpse of the DF appearsin this more disordered sample. Our analysis also allowsus to estimate the thermodynamic quantity H(T ) for theDF. The Fig. 3C inset shows the fitted enthalpy densityas a function of temperature compared to that expectedin clean graphene (dashed line) [18], excluding renormal-ization of the Fermi velocity. In the cleanest sample H

varies from 1.1-2.3 eV/µm2 for Tdis < T < Telph. Thisenthalpy density corresponds to 20 meV or 4kBTper charge carrier — about a factor of 2 larger than themodel calculation without disorder [18].

To fully incorporate the e↵ects of disorder, a hydrody-namic theory treating inhomogeneity non-perturbativelyis necessary [40, 41]. The enthalpy densities reportedhere are larger than the theoretical estimation obtainedfor disorder free graphene, consistent with the picturethat chemical potential fluctuations prevent the samplefrom reaching the Dirac point. While we find thermalconductivity well described by Ref. [5, 6], electrical con-ductivity increases slower than expected away from theCNP, a result consistent with hydrodynamic transport ina viscous fluid with charge puddles [41].

In a hydrodynamic system, the ratio of shear viscos-ity to entropy density s is an indicator of the strengthof the interactions between constituent particles. It issuggested that the DF can behave as a nearly perfectfluid [18]: /s approaches a conjecture by Kovtun-Son-Starinets: (/s)/(~/kB) & 1/4 for a strongly inter-acting system [42]. A non-perturbative hydrodynamicframework can be employed to estimate , as we discusselsewhere [41]. A direct measurement of is of greatinterest.

We have experimentally discovered the breakdown ofthe WF law and provided evidence for the hydrodynamicbehavior of the Dirac fermions in graphene. This pro-vides an experimentally realizable Dirac fluid and opensthe way for future studies of strongly interacting rela-tivistic many-body systems. Beyond a diverging thermalconductivity and an ultra-low viscosity, other peculiarphenomena are expected to arise in this plasma. Themassless nature of the Dirac fermions is expected to re-sult in a large kinematic viscosity, despite a small shear

SimpleàPureàHard!

experimentalevidence

18.06.apctp

Hydrodynamics vs quantum Holography in data fitting

13

LUCAS, CROSSNO, FONG, KIM, AND SACHDEV PHYSICAL REVIEW B 93, 075426 (2016)

(albeit with negligible quantum entanglement) insofar as theydo not admit a controllable description via kinetic theory.Furthermore, it has been shown [27] that strongly interactingquantum critical fluids have a somewhat different hydrody-namic description than the canonical Fermi liquids describedabove, and this can lead to very different hydrodynamicproperties, including in transport [20,21,27– 31], as we willreview in this paper.

Using novel techniques to measure thermal transport[32– 34], the Dirac fluid has finally been observed inmonolayer graphene, and evidence for its hydrodynamicbehavior has emerged [35], as we will detail. However,existing theories of hydrodynamic transport are not consistentwith the simultaneous density dependence in experimentallymeasured thermal and electrical conductivities. In this paper,we improve upon the hydrodynamic theory of Ref. [27],describe carefully effects of finite density, and develop anonperturbative relativistic hydrodynamic theory of transportin electron fluids near a quantum critical point. Undercertain assumptions about the equations of state of theDirac fluid, our theory is quantitatively consistent withexperimental observations. The techniques we employ areincluded in the framework of Ref. [36], which developed ahydrodynamic description of transport in relativistic fluidswith long-wavelength disorder in the chemical potential [36]was itself inspired by recent progress employing the AdS/CFTcorrespondence to understand quantum critical transport instrange metals [31,37– 44], but as we will discuss, this theoryis also well suited to describe the physics of graphene.

A. Summary of results

The recent experiment [35] reported order-of-magnitudeviolations of the Wiedemann-Franz law. The results werecompared with the standard theory of hydrodynamic transportin quantum critical systems [27], which predicts that

σ (n) = σQ + e2v2Fn

2τ

H, (2a)

κ(n) = v2FHτ

T

σQ

σ (n), (2b)

where e is the electron charge, s is the entropy density, n is thecharge density (in units of length−2), H is the enthalpy density,τ is a momentum relaxation time, and σQ is a quantum criticaleffect, whose existence is a new effect in the hydrodynamicgradient expansion of a relativistic fluid. Note that up to σQ,σ (n) is simply described by Drude physics. The Lorenz ratiothen takes the general form

L(n) = LDF

(1 + (n/n0)2)2, (3)

where

LDF = v2FHτ

T 2σQ

, (4a)

n20 = HσQ

e2v2Fτ

. (4b)

L(n) can be parametrically larger than LWF (as τ → ∞and n ≪ n0), and much smaller (n ≫ n0). Both of thesepredictions were observed in the recent experiment, and fits ofthe measuredL to (3) were quantitatively consistent, until largeenough n where Fermi liquid behavior was restored. However,the experiment also found that the conductivity did not growrapidly away from n = 0 as predicted in (2), despite a largepeak in κ(n) near n = 0, as we show in Fig. 1. Furthermore,the theory of Ref. [27] does not make clear predictions for thetemperature dependence of τ , which determines κ(T ).

In this paper, we argue that there are two related reasonsfor the breakdown of (2). One is that the dominant source ofdisorder in graphene—fluctuations in the local charge density,commonly referred to as charge puddles [45– 48]—are notperturbatively weak, and therefore a nonperturbative treatmentof their effects is necessary [49]. The second is that theparameter τ , even when it is sharply defined, is intimatelyrelated to both the viscosity and to n, and this n dependence isneglected when performing the fit to (2) in Fig. 1. We develop anonperturbative hydrodynamic theory of transport which relieson neither of the above assumptions, and gives us an explicitformula for τ in the limit of weak disorder. The key assumptionfor the validity of our theory is that the size of the chargepuddles is comparable to or larger than the electron interaction

hole FL elec. FLDirac fluid

−400 −200 0 200 4000

0.5

1

1.5

2

n (µm−2)

σ(k

Ω−

1)

hole FL elec. FLDirac fluid

−400 −200 0 200 4000

2

4

6

8

10

n (µm−2)

κ(n

W/K

)

FIG. 1. A comparison of our hydrodynamic theory of transport with the experimental results of Ref. [35] in clean samples of graphene atT = 75 K. We study the electrical and thermal conductances at various charge densities n near the charge neutrality point. Experimental dataare shown as circular red data markers, and numerical results of our theory, averaged over 30 disorder realizations, are shown as the solid blueline. Our theory assumes the equations of state described in (27) with the parameters C0 ≈11, C2 ≈9, C4 ≈200, η0 ≈110, σ0 ≈1.7, and(28) with u0 ≈0.13. The yellow shaded region shows where Fermi liquid behavior is observed and the Wiedemann-Franz law is restored, andour hydrodynamic theory is not valid in or near this regime. We also show the predictions of (2) as dashed purple lines, and have chosen thethree-parameter fit to be optimized for κ(n).

075426-2

3

which request the co-linearity of all momentum vectors~q1, · · · , ~q4. Therefore available phase space is greatly re-duced. Such kinematical constraints maintains the non-equilibrium states and as a consequence, the two currentsJe, Jh behave independently for a long time comparedwith the Planck time ~/kT , which is the time for hy-drodynamics to work.

The net electric current J and total number currentJn which become neutral at Dirac point, are defined byJ = Je + Jh, Jn = Je Jh, respectively and their elec-tric charge densities and number densities are relatedby Q1 = en1 and Q2 = en2. The total electric con-ductivity = @J

@E and can be expressed in terms ofQ = Q1 + Q2 and Qn = Q1 Q2 together with theproportionality constant gn of Qn = gnQ:

= 0(1 + (Q/Q0)2), =

1 + (1 + g2n)(Q/Q0)2, (16)

where

0 =e2

~ 2Z0, =4kB~

sT

k2, Q2

0 =~0

4kBsk2. (17)

To fix the parameters, we used four measured values ofref. [11] at 75K, 0 = 0.338/k, = 7.7nW/K , Q0 =e ·320/(µm)2, together with the curvature of density plotof to fix gn = 3.2. and assumed charge conjugationsymmetry to set W0 = Z0. Using these, the basic pa-rameters of the theory as well as the entropy density canbe determined: 2Z0 = 1.387, k2 = 454

(µm)2 , s = 2044 kB(µm)2 .

We replace all r0 dependence by s, the entropy densityby s = 4kBr20. Cosmological constant is not determineddue to the inherent scale symmetry.

Now, why we can set the proportionality of the two

charges as given in eq. (13). To avoid the issues involvedin the transport by puddle, we simply assume that thesystem is homogeneous. Then the number densities ofelectrons and holes created by thermal excitation is pro-portional to the net charge density: for the fermi liquidcase, out of total degree of freedom (d.o.f) n k2F µ2,excitable d.o.f is kT · µ, because the excitable shellwidth is kT . But in hydrodynamic regime, kT >> µ,therefore entire non-degenerate charge distribution re-gion is excitable. In fact this is a typical situation offermion dynamics described by AdS black hole [27, 28].In summary, in case of the hydrodynamic regime, thecharge carrier density created is proportional to total de-gree of freedom, Q, which is the volume of the Dirac coneabove the Dirac point.

We remark that due to strong Coulomb interaction,the created electron hole pairs can form the bound state,exciton. Such excitons in homogeneous graphene satisfiesthe linear relations between the electric charge and theexciton number. Although exciton in graphene has beendiscussed extensively [29, 30], mosts are only for bi-layergraphene. However, we expect that strong coulomb in-teraction in Dirac Fluid regime of single layer graphenealso should be able to make bound state.

(a)

(b)

FIG. 1. Theory vs. Data: (a)density plot of , (b)thatof . Red circles are for data used in [11, 13], dashed linesare for one current model and real lines are for two currentmodel. The parameters are fixed such that plot is well fit.The white color is the Dirac fluid regime in which our theoryworks, and the blue shaded is for the fermi liquid one.

Discussions: In the presence of an extra current thatcarries mainly heat, the violation of WFL is not directevidence of a Dirac fluid. However, the fact that sucha phenomenon is quantitatively well described by hydro-dynamics and gauge/gravity duality, indicates that thesystem is strongly correlated.Disorder and the nature of the scalar field: The scalar

field provides momentum dissipation only when both itsgradient and the vacuum expectation value of its dualoperator, hOIi, are nonzero. The latter is the analogueof charge density in electric field as one can see from theWard identity,

rTµ = hOIirµ

0I + F 0

µ hJi . (18)

The role of the source field 0I = kxI is the chemical po-

tential of impurity and that of hOIi is the density of im-purity, whose presence gives the momentum dissipation.It is identified as the subleading order term of the fluctu-ation of the scalar field near the boundary and nonzerodue to the presence of curvature in AdS spacetime. k2

can be understood as the density of the uniform impurity.

3

which request the co-linearity of all momentum vectors~q1, · · · , ~q4. Therefore available phase space is greatly re-duced. Such kinematical constraints maintains the non-equilibrium states and as a consequence, the two currentsJe, Jh behave independently for a long time comparedwith the Planck time ~/kT , which is the time for hy-drodynamics to work.



= 0(1 + (Q/Q0)2), =

1 + (1 + g2n)(Q/Q0)2, (16)

where

0 =e2

~ 2Z0, =4kB~

sT

k2, Q2

0 =~0

4kBsk2. (17)


(µm)2 , s = 2044 kB(µm)2 .



charges as given in eq. (13). To avoid the issues involvedin the transport by puddle, we simply assume that thesystem is homogeneous. Then the number densities ofelectrons and holes created by thermal excitation is pro-portional to the net charge density: for the fermi liquidcase, out of total degree of freedom (d.o.f) n k2F µ2,excitable d.o.f is kT · µ, because the excitable shellwidth is kT . But in hydrodynamic regime, kT >> µ,therefore entire non-degenerate charge distribution re-gion is excitable. In fact this is a typical situation offermion dynamics described by AdS black hole [27, 28].In summary, in case of the hydrodynamic regime, thecharge carrier density created is proportional to total de-gree of freedom, Q, which is the volume of the Dirac coneabove the Dirac point.

We remark that due to strong Coulomb interaction,the created electron hole pairs can form the bound state,exciton. Such excitons in homogeneous graphene satisfiesthe linear relations between the electric charge and theexciton number. Although exciton in graphene has beendiscussed extensively [29, 30], mosts are only for bi-layergraphene. However, we expect that strong coulomb in-teraction in Dirac Fluid regime of single layer graphenealso should be able to make bound state.

(a)

(b)

FIG. 1. Theory vs. Data: (a)density plot of , (b)thatof . Red circles are for data used in [11, 13], dashed linesare for one current model and real lines are for two currentmodel. The parameters are fixed such that plot is well fit.The white color is the Dirac fluid regime in which our theoryworks, and the blue shaded is for the fermi liquid one.

Discussions: In the presence of an extra current thatcarries mainly heat, the violation of WFL is not directevidence of a Dirac fluid. However, the fact that sucha phenomenon is quantitatively well described by hydro-dynamics and gauge/gravity duality, indicates that thesystem is strongly correlated.Disorder and the nature of the scalar field: The scalar


rTµ = hOIirµ

0I + F 0

µ hJi . (18)

The role of the source field 0I = kxI is the chemical po-

tential of impurity and that of hOIi is the density of im-purity, whose presence gives the momentum dissipation.It is identified as the subleading order term of the fluctu-ation of the scalar field near the boundary and nonzerodue to the presence of curvature in AdS spacetime. k2

can be understood as the density of the uniform impurity.

18.06.apctp

14

3

with the Planck time ~/kT , which is the time for hy-drodynamics to work.



= 0(1 + (Q/Q0)2), =

1 + (1 + g2n)(Q/Q0)2, (16)

where

0 =e2

~ 2Z0, =4kB~

sT

k2, Q2

0 =~0

4kBsk2. (17)


(µm)2 , s = 2044 kB(µm)2 .



charges as given in eq. (13). To avoid the issues involvedin the transport by puddle, we simply assume that welllocalized puddles do not contribute transport or simplyassume that the system is homogeneous. Under suchassumption, the number densities of electrons and holescreated by thermal excitation is proportional to the netcharge density: for the fermi liquid case, out of totaldegree of freedom (d.o.f) n k2F µ2, excitable d.o.f is kT · µ, because the excitable shell width is kT . Butin hydrodynamic regime, kT >> µ, therefore entire non-degenerate charge distribution region is excitable. In factthis is a typical situation of fermion dynamics describedby AdS black hole [27, 28]. In summary, in case of thehydrodynamic regime, the charge carrier density createdis proportional to total degree of freedom, Q, which isthe volume of the Dirac cone above the Dirac point.

We remark that due to strong Coulomb interaction,the created electron hole pairs can form the bound state,exciton. Such excitons in homogeneous graphene satisfiesthe linear relations between the electric charge and theexciton number. Although exciton in graphene has beendiscussed extensively [29, 30], mosts are only for bi-layergraphene. However, we expect that strong coulomb in-teraction in Dirac Fluid regime of single layer grapheneshould be able to make bound state. The abundance ofsuch excitons are remained to be verified experimentally.

Discussions: In the presence of an extra current thatcarries mainly heat, the violation of WFL is not directevidence of a Dirac fluid. However, the fact that such

(a)

(b)

FIG. 1. Holography v.s the experimental data: (a) densityplot of . (b) density plot of . Red circles are for data usedin [11, 13], dashed lines are for one current model and reallines are for two current model. The white region is the Diracfluid regime in which our theory works, while the shaded areais the fermi liquid regime.

a phenomenon is quantitatively well described by hydro-dynamics and gauge/gravity duality, indicates that thesystem is strongly correlated.Disorder and the nature of the scalar field: The scalar


rTµ = hOIirµ

0I + F 0

µ hJi . (18)

The role of the source field 0I = kxI is the chemical

potential of impurity and that of hOIi is the density ofimpurity. Therefore, k2 can be understood as the densityof the uniform impurity.The puddle e↵ect on the transport: One important

source of the disorder in graphene is known to be thethe charge density inhomogeneity, [11, 13, 31], which iscompletely neglected here. Then why the theory couldmatch the experiment so well? A partial answer is thatwhile the DC conductivities directly depends only on the

3




= 0(1 + (Q/Q0)2), =

1 + (1 + g2n)(Q/Q0)2, (16)

where

0 =e2

~ 2Z0, =4kB~

sT

k2, Q2

0 =~0

4kBsk2. (17)


(µm)2 , s = 2044 kB(µm)2 .






(a)

(b)




rTµ = hOIirµ

0I + F 0

µ hJi . (18)




3




= 0(1 + (Q/Q0)2), =

1 + (1 + g2n)(Q/Q0)2, (16)

where

0 =e2

~ 2Z0, =4kB~

sT

k2, Q2

0 =~0

4kBsk2. (17)


(µm)2 , s = 2044 kB(µm)2 .






(a)

(b)




rTµ = hOIirµ

0I + F 0

µ hJi . (18)




3




= 0(1 + (Q/Q0)2), =

1 + (1 + g2n)(Q/Q0)2, (16)

where

0 =e2

~ 2Z0, =4kB~

sT

k2, Q2

0 =~0

4kBsk2. (17)


(µm)2 , s = 2044 kB(µm)2 .






(a)

(b)




rTµ = hOIirµ

0I + F 0

µ hJi . (18)




3




= 0(1 + (Q/Q0)2), =

1 + (1 + g2n)(Q/Q0)2, (16)

where

0 =e2

~ 2Z0, =4kB~

sT

k2, Q2

0 =~0

4kBsk2. (17)


(µm)2 , s = 2044 kB(µm)2 .






(a)

(b)




rTµ = hOIirµ

0I + F 0

µ hJi . (18)




4basicparametersfromthemeasuredvalue.

T=

(4)2

2r0, (7)

where the o↵-diagonal components of each transport coecient are zero in this case. The

other point is that the thermo-electric coecient ↵ and ↵ is linear in the charge density q.

The thermal conductivity is defined by

ij = ij T ↵ik1kl ↵lj. (8)

Together with (7) and (8), we get simple relation between the electric conductivity and the

thermal conductivity;

=

=

(4)2

2· r0T

1 +q2

r202

. (9)

2.2 Momentum relaxation with a finite magnetization

The thermo-electric transport coecients with q is given by

xx =(F H

2)(F + G2

)

(F2 +H2G2)

xy =HG(2F + G2 H

2) + (F2

+H2G2

)

(F2 +H2G2)

↵xx = ↵xx =sG(F H

2)

F2 +H2G2

↵xy = ↵xy =sH(F + G2

)

F2 +H2G2

xx =s2T F

(F2 +H2G2)

yx =s2T HG

(F2 +H2G2),

(10)

where

F = r20

2+H

21 +

2 qH ,

G = q H, s = 4r20 .

(11)

2.3 Momentum relaxation with two U(1) field

In this section, we start from the action with two gauge fields;

S =

Zd4xpg

"R 1

2

(@)

2+ 1()(@1)

2+ 2()(@2)

2 V () Z()

4F

2 W ()

4G

2

#,

(12)

2

holographicmodel

Idea:neutralcurrentàEnhancetheheatconductivity

18.06.apctp

Transport:

Remark:allanalytical

15

2

One can see that if ev r2 in asymptotic region, Qi

corresponds to the charge density of the boundary fieldtheory. To compute the transport coecients, we turnon small fluctuations around the background solution asgiven in equation (A3, A4). From the A field fluctuationequations, the currents are defined by [18],

J1 =pgZ()F xr, J2 =

pgW ()Gxr

Q = U(r)2d

dr

gtx(r)

U(r)

A(r)J1 B(r)J2. (4)

Notice that near the boundary, the heat current becomesQ = T tx

µ1J1µ2J2. Moreover, these currents are con-served along radial direction r. Therefore their boundaryvalues are related to that of horizon data.

Finally, we get the boundary current in terms of theexternal sources:

J1 =

Z0 +

ev0Q21

k20

E1 +

ev0Q1Q2

k2E2 +

4TQ1

k20,

J2 =

W0 +

ev0Q22

k20

E2 +

ev0Q1Q2

k2E1 +

4TQ2

k20

Q =4TQ1

k20E1 +

4TQ2

k20E2 +

(4T )2ev0

k20.

(5)

The eq. (5) can be written in matrix form, Ji = ijEj ,with J3 = Q and E3 = . The transport coecients canbe read o↵ from the eq. (5) and the definition

0

@1 ↵1T 2 ↵2T

↵1T ↵2T T

1

A := . (6)

Notice that the matrix is real and symmetric, so that theOnsager relations hold:

↵i = ↵i, = . (7)

The heat conductivity is defined by the response of thetemperature gradient to the heat current in the absenceof other currents: setting J1 and J2 to be zero in (5), wecan express E1 and E2 in terms of . Substituting theseto the first line of (5), we get

= T ↵1(↵12 ↵2)

12

T ↵2(↵21 ↵1)

12 . (8)

To discuss more explicitly, we consider a black hole solu-tion with two charges:

U(r) = r2 m0

r

k2

2+

1

4r2

Q2

1

Z0+

Q22

W0

, (9)

where m0 is given by U(r0) = 0 and the temperature is

T =r04

3

k2

2r20

Q21

4Z0r40

Q22

4W0r40

. (10)

The solutions of U(1) gauge fields are a(r) = µ1 q1r ,

b(r) = µ2 q2r . Notice qi = Qi/Zi with Z1, Z2 being

Z0,W0 respectively. For the finite vector norm gµAµA

at the horizon r = r0, we need µi = qi/r0.The conductivities for any number of conserved cur-

rents can be calculated explicitly:

i = Zi +Q2

i

r20k2, ij =

QiQj

r20k2, =

1 +P

i 4Q2i /sk

2Zi,

with = 4sT/k2, s = 4r20 and Zi is the coupling ofAi. If we identify the total electric current as J =

Pi Ji

and thermo-electric force Ei = E Tr(µi/T ), we cancalculate the electric conductivity to give

=@J

@E=

X

i

i +X

i,j

ij = Z + 4Q2/sk2, (11)

where Q =P

i Qi and Z =P

i Zi, showing the additivityof density independent part of the electric conductivity.If we define the heat conductivity due to the i-th currentby 1/i = 1/ + Q2

i /Zis2T , then the heat conductivityformula leads us to additivity of density dependent part

of the inverse heat conductivity. Therefore

D[1/] =X

i

D[1/i], D[] =X

i

D[i], (12)

where D[f ], D[f ] denote the density dependent and in-dependent part of f , respectively.Finally we claim that the experimental data of

graphene will be well fit with two current theory if weassume the proportionality of two charges

Q2 = gQ1, (13)

whose justification will be discussed later. This assump-tion together with the second additivity eq.(12) is whatmakes our two current model work.Origin of two Currents in Graphene: What is

the nature and the origin of the extra current in thegraphene. There are a few attractive candidates. Thefirst idea is the e↵ect of imbalance [4] between the elec-trons and holes due to the kinematical constraints of theDirac cone. When there is such an deviation of electronand hole density from their equilibrium value, then thesystem has tendency to reduce the di↵erence by creat-ing/absorbing electron-hole pair:

e $ e + h+ + e, h+$ h+ + h+ + e (14)

In such processes, energy and momemtum conservationsmust hold. The point is that, for the graphene, the lin-ear dispersion relation severely reduces the kinematicallyavailable states [4]: If we define ~q as a momentum mea-sured from a Dirac point,

~q1 = ~q2 + ~q3 + ~q4, |~q1| = |~q2|+ |~q2|+ |~q3|, (15)

2



J1 =pgZ()F xr, J2 =

pgW ()Gxr

Q = U(r)2d

dr

gtx(r)

U(r)

A(r)J1 B(r)J2. (4)




J1 =

Z0 +

ev0Q21

k20

E1 +

ev0Q1Q2

k2E2 +

4TQ1

k20,

J2 =

W0 +

ev0Q22

k20

E2 +

ev0Q1Q2

k2E1 +

4TQ2

k20

Q =4TQ1

k20E1 +

4TQ2

k20E2 +

(4T )2ev0

k20.

(5)


0

@1 ↵1T 2 ↵2T

↵1T ↵2T T

1

A := . (6)


↵i = ↵i, = . (7)


= T ↵1(↵12 ↵2)

12

T ↵2(↵21 ↵1)

12 . (8)


U(r) = r2 m0

r

k2

2+

1

4r2

Q2

1

Z0+

Q22

W0

, (9)


T =r04

3

k2

2r20

Q21

4Z0r40

Q22

4W0r40

. (10)






i = Zi +Q2

i

r20k2, ij =

QiQj

r20k2, =

1 +P

i 4Q2i /sk

2Zi,


Pi Ji


=@J

@E=

X

i

i +X

i,j

ij = Z + 4Q2/sk2, (11)

where Q =P

i Qi and Z =P




D[1/] =X

i

D[1/i], D[] =X

i

D[i], (12)



Q2 = gQ1, (13)



e $ e + h+ + e, h+$ h+ + h+ + e (14)


~q1 = ~q2 + ~q3 + ~q4, |~q1| = |~q2|+ |~q2|+ |~q3|, (15)

2



J1 =pgZ()F xr, J2 =

pgW ()Gxr

Q = U(r)2d

dr

gtx(r)

U(r)

A(r)J1 B(r)J2. (4)




J1 =

Z0 +

ev0Q21

k20

E1 +

ev0Q1Q2

k2E2 +

4TQ1

k20,

J2 =

W0 +

ev0Q22

k20

E2 +

ev0Q1Q2

k2E1 +

4TQ2

k20

Q =4TQ1

k20E1 +

4TQ2

k20E2 +

(4T )2ev0

k20.

(5)


0

@1 ↵1T 2 ↵2T

↵1T ↵2T T

1

A := . (6)


↵i = ↵i, = . (7)


= T ↵1(↵12 ↵2)

12

T ↵2(↵21 ↵1)

12 . (8)


U(r) = r2 m0

r

k2

2+

1

4r2

Q2

1

Z0+

Q22

W0

, (9)


T =r04

3

k2

2r20

Q21

4Z0r40

Q22

4W0r40

. (10)






i = Zi +Q2

i

r20k2, ij =

QiQj

r20k2, =

1 +P

i 4Q2i /sk

2Zi,


Pi Ji


=@J

@E=

X

i

i +X

i,j

ij = Z + 4Q2/sk2, (11)

where Q =P

i Qi and Z =P




D[1/] =X

i

D[1/i], D[] =X

i

D[i], (12)



Q2 = gQ1, (13)



e $ e + h+ + e, h+$ h+ + h+ + e (14)


~q1 = ~q2 + ~q3 + ~q4, |~q1| = |~q2|+ |~q2|+ |~q3|, (15)

2



J1 =pgZ()F xr, J2 =

pgW ()Gxr

Q = U(r)2d

dr

gtx(r)

U(r)

A(r)J1 B(r)J2. (4)




J1 =

Z0 +

ev0Q21

k20

E1 +

ev0Q1Q2

k2E2 +

4TQ1

k20,

J2 =

W0 +

ev0Q22

k20

E2 +

ev0Q1Q2

k2E1 +

4TQ2

k20

Q =4TQ1

k20E1 +

4TQ2

k20E2 +

(4T )2ev0

k20.

(5)


0

@1 ↵1T 2 ↵2T

↵1T ↵2T T

1

A := . (6)


↵i = ↵i, = . (7)


= T ↵1(↵12 ↵2)

12

T ↵2(↵21 ↵1)

12 . (8)


U(r) = r2 m0

r

k2

2+

1

4r2

Q2

1

Z0+

Q22

W0

, (9)


T =r04

3

k2

2r20

Q21

4Z0r40

Q22

4W0r40

. (10)






i = Zi +Q2

i

r20k2, ij =

QiQj

r20k2, =

1 +P

i 4Q2i /sk

2Zi,


Pi Ji


=@J

@E=

X

i

i +X

i,j

ij = Z + 4Q2/sk2, (11)

where Q =P

i Qi and Z =P




D[1/] =X

i

D[1/i], D[] =X

i

D[i], (12)



Q2 = gQ1, (13)



e $ e + h+ + e, h+$ h+ + h+ + e (14)


~q1 = ~q2 + ~q3 + ~q4, |~q1| = |~q2|+ |~q2|+ |~q3|, (15)

2



J1 =pgZ()F xr, J2 =

pgW ()Gxr

Q = U(r)2d

dr

gtx(r)

U(r)

A(r)J1 B(r)J2. (4)




J1 =

Z0 +

ev0Q21

k20

E1 +

ev0Q1Q2

k2E2 +

4TQ1

k20,

J2 =

W0 +

ev0Q22

k20

E2 +

ev0Q1Q2

k2E1 +

4TQ2

k20

Q =4TQ1

k20E1 +

4TQ2

k20E2 +

(4T )2ev0

k20.

(5)


0

@1 ↵1T 2 ↵2T

↵1T ↵2T T

1

A := . (6)


↵i = ↵i, = . (7)


= T ↵1(↵12 ↵2)

12

T ↵2(↵21 ↵1)

12 . (8)


U(r) = r2 m0

r

k2

2+

1

4r2

Q2

1

Z0+

Q22

W0

, (9)


T =r04

3

k2

2r20

Q21

4Z0r40

Q22

4W0r40

. (10)






i = Zi +Q2

i

r20k2, ij =

QiQj

r20k2, =

1 +P

i 4Q2i /sk

2Zi,


Pi Ji


=@J

@E=

X

i

i +X

i,j

ij = Z + 4Q2/sk2, (11)

where Q =P

i Qi and Z =P




D[1/] =X

i

D[1/i], D[] =X

i

D[i], (12)



Q2 = gQ1, (13)



e $ e + h+ + e, h+$ h+ + h+ + e (14)


~q1 = ~q2 + ~q3 + ~q4, |~q1| = |~q2|+ |~q2|+ |~q3|, (15)

2



J1 =pgZ()F xr, J2 =

pgW ()Gxr

Q = U(r)2d

dr

gtx(r)

U(r)

A(r)J1 B(r)J2. (4)




J1 =

Z0 +

ev0Q21

k20

E1 +

ev0Q1Q2

k2E2 +

4TQ1

k20,

J2 =

W0 +

ev0Q22

k20

E2 +

ev0Q1Q2

k2E1 +

4TQ2

k20

Q =4TQ1

k20E1 +

4TQ2

k20E2 +

(4T )2ev0

k20.

(5)


0

@1 ↵1T 2 ↵2T

↵1T ↵2T T

1

A := . (6)


↵i = ↵i, = . (7)


= T ↵1(↵12 ↵2)

12

T ↵2(↵21 ↵1)

12 . (8)


U(r) = r2 m0

r

k2

2+

1

4r2

Q2

1

Z0+

Q22

W0

, (9)


T =r04

3

k2

2r20

Q21

4Z0r40

Q22

4W0r40

. (10)






i = Zi +Q2

i

r20k2, ij =

QiQj

r20k2, =

1 +P

i 4Q2i /sk

2Zi,


Pi Ji


=@J

@E=

X

i

i +X

i,j

ij = Z + 4Q2/sk2, (11)

where Q =P

i Qi and Z =P




D[1/] =X

i

D[1/i], D[] =X

i

D[i], (12)



Q2 = gQ1, (13)



e $ e + h+ + e, h+$ h+ + h+ + e (14)


~q1 = ~q2 + ~q3 + ~q4, |~q1| = |~q2|+ |~q2|+ |~q3|, (15)

18.06.apctp

Thestorywaspublishedas

16

Phys.Rev.Lett.118(2017)no.3,036601Editors'Suggestion

Holography of the Dirac Fluid in Graphene with Two Currents

Yunseok Seo,1 Geunho Song,1 Philip Kim,2,3 Subir Sachdev,2,4 and Sang-Jin Sin11Department of Physics, Hanyang University, Seoul 133-791, Korea

2Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA3Department of Physics and Astronomy, Seoul National University, Seoul 151-747, Korea

4Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada(Received 17 September 2016; published 18 January 2017)

Recent experiments have uncovered evidence of the strongly coupled nature of graphene: theWiedemann-Franz law is violated by up to a factor of 20 near the charge neutral point. We describethis strongly coupled plasma by a holographic model in which there are two distinct conserved U(1)currents. We find that our analytic results for the transport coefficients for the two current model have asignificantly improved match to the density dependence of the experimental data than the models with onlyone current. The additive structure in the transport coefficients plays an important role. We also suggest theorigin of the two currents.

DOI: 10.1103/PhysRevLett.118.036601

Introduction.—It has been argued that graphene nearcharge neutrality forms a strongly interacting plasma, theDirac fluid. It does not have well-defined quasiparticleexcitations, and is amenable to a hydrodynamic description[1–10]. Evidence for such a Dirac fluid has appeared inrecent experiments [11] on the violation of the Wiedemann-Franz law (WFL) in extremely clean graphene near thecharge neutral point; the ratio of heat conductivity andelectric conductivity, L ¼ κ=Tσ, was found to be up to20 times the Fermi liquid value.The simplest hydrodynamic model [12], with pointlike

and uncorrelated disorder and a single conserved U(1)current, agrees with the overall experimental trends, buthas difficulty capturing the density dependencies of boththe electrical (σ) and thermal (κ) conductivities [13]. Analternative hydrodynamic model, the “puddle” model,with long-wavelength disorder in the chemical potentialand a single conserved U(1) current, led to a betteragreement with observations [13], but still left room forimprovement.In this Letter, we explore a model with two conserved

U(1) currents. The idea is that introducing a new neutralcurrent can enhance the transport of the heat relative tothat of the charge. Our model is formulated in holographicterms [14,15], to utilize the recent progress in the develop-ment of transport calculation in gauge-gravity duality[16–27]. The Dirac fluid in our model is described by ananti–de Sitter (AdS) black hole in 3þ 1 dimensions, theholographic dual of a 2þ 1-dimensional system at finitetemperature. The momentum dissipation is treated usingscalar fields, which corresponds to weak pointlike dis-order. We calculate electric, thermoelectric power, andthermal conductivities analytically. We find that, under the

assumption that the conserved charges Q1, Q2 are propor-tional to each other, the theoretical results for the densitydependencies of the electric and heat conductivities cannow satisfactorily match the experimental data in theDirac fluid regime.One possible mechanism for the extra current is the

kinematic constraints of energy-momentum conservationon the Dirac cone, which reduce the phase space ofelectron and hole scattering significantly [4], allowingelectrons and holes to form independent currents as longas the relaxation time for mixing between the currents ispresumed to be much longer than the Planckian relax-ation time ℏ=kBT, the time required for the hydro-dynamic regime to work. It should be noted, however,that the estimates of electron and hole equilibrationtimes are made in a quasiparticle framework [4], whosevalidity in hydrodynamic regime is just assumed here.We see that the kinematics on the Dirac cone alsoprovide a reason why the two charge densities can beproportional.dc transport with two Uð1Þ fields.—We start from the

action S¼Rd4x

ffiffiffiffiffiffi−gpL with two gauge fields Aμ, Bμ, a

dilaton field ϕ, and the scalar fields χ1, χ2 for momentumdissipation,

L ¼ R −1

2½ð∂ϕÞ2 þ Φ1ðϕÞð∂χ1Þ2 þ Φ2ðϕÞð∂χ2Þ2&

− VðϕÞ − ZðϕÞ4

F2 −WðϕÞ4

G2; ð1Þ

where F ¼ dA,G ¼ dB, F2 ¼ FμνFμν, etc. We also requirepositivity of ΦiðϕÞ, ZðϕÞ, and WðϕÞ. The action (1) yieldsequations of motion,

PRL 118, 036601 (2017) P HY S I CA L R EV I EW LE T T ER Sweek ending

20 JANUARY 2017

0031-9007=17=118(3)=036601(5) 036601-1 © 2017 American Physical Society

18.06.apctp

Otherz=1material?Diracmaterialisaclass(1405.5774):우리 예측:soistheanomaloustransport.

SurfaceofTISimilar,butdifferbystrongspin-orbitinteraction

17

May 23, 2014 Advances in Physics DiracMat˙AdvPhys

Material Pseudospin Energy scale (eV) References

Graphene, Silicene, Germanene Sublattice 1–3 eV [5, 6, 17, 19, 36, 37]Artificial Graphenes Sublattice 108–0.1 eV [28, 29, 38–40]Hexagonal layered heterostructures Emergent 0.01–0.1 eV [41–47]Hofstadter butterfly systems Energent 0.01 eV [46]Graphene-hBN heterostructures in high magnetic fields

Band inversion interfaces Spin-orbit ang. mom. 0.3 eV [48–50]SnTe/PbTe, CdTe/HgTe, PbTe

2D Topological Insulators Spin-orbit ang. mom. < 0.1eV [7, 8, 22, 24, 51, 52]HgTe/CdTe, InAs/GaSb, Bi bilayer, ...

3D Topological Insulators Spin-orbit ang. mom. . 0.3eV [7, 8, 23, 52–55]Bi1xSbx, Bi2Se3, strained HgTe, Heusler alloys, ...

Topological crystalline insulators orbital . 0.3eV [56–59]SnTe, Pb1xSnxSe

d-wave cuprate superconductors Nambu pseudospin . 0.05eV [60, 61]3He Nambu pseudospin 0.3 µeV [2, 3]3D Weyl and Dirac semimetals Energy bands Unclear [32–34]Cd3As2, Na3Bi

Table 1. Table of Dirac materials indicated by material family, pseudospin realization in the Dirac Hamiltonian,and the energy scale for which the Dirac spectrum is present without any other states.

Recent ARPES measurements on Na3Bi [32] and Cd3As2 [33, 34] have found evidencefor a three-dimensional Dirac semimetal state in these materials.

At first (microscopic) sight, a material like graphene does hardly display any similaritywith typical d-wave superconductors or superfluids. There are important materials spe-cific properties making all these materials distinct: some are superconductors and someare (bulk) insulators; some are crystalline with honeycomb lattice (graphene or silicene),others have a more complicated Perovskite crystal structure (cuprate superconductors)or do not exhibit any crystalline order (3He-A phase). Also the physical realizations ofthe Dirac pseudospin di↵er between these materials (c.f. Table 1) and the list of di↵er-ences can be continued. But again, it is the universal properties related to the existenceof the low-energy Dirac excitations that justify the concept of Dirac materials. As a uni-fying principle, the presence of nodes leads to a sharp reduction of the phase space forlow-energy excitations in Dirac materials. More precisely, the dimensionality of the setof points in momentum space where we have zero-energy excitations is reduced in Diracmaterials as compared to normal metals. For example, nodes for a three-dimensionalDirac material mean the e↵ective Fermi surface is shrunk from a two-dimensional ob-ject to a point. Lines of Dirac nodes in three dimensions would mean that the Fermisurface has shrunk from a two-dimensional surface to a one-dimensional line. In eithercase there is a reduction of dimensionality for the zero-energy states. This reduction ofphase space controlled by additional symmetry in the system is an indicator for Diracmaterials. Reduced phase space and controlling symmetries are important for applica-tions. First of all, it is possible to lift the protected symmetry of the Dirac node andtherefore destroy the nodes and open an energy gap. This modification of the spectrumof quasiparticles drastically changes the response of the Dirac material, as for exampleis the case for topological insulator in a magnetic field [35]. Second, Dirac nodes andthe resultant reduction of phase space suppress dissipation and are thus attractive forapplications exploiting the coherence of low-energy states in the nodes.

4

18.06.apctp

SurfacephenomenaofTI:WALàWLtransition

1818.06.apctp

1.Bi2Te3withCrdoping:Baoet.al,SREP02391

directions and apBerry’s phase is accumulated10,11,44. The destructiveinterference due to pBerry’s phase leads to an enhancement of MC.Applying an external magnetic field suppresses the destructiveinterference, giving rise to a negative MC11,44,45. One interestingquestion to ask is what if the magnetic impurities are incorporatedinto the TI materials? Theoretical predictions suggested that whendoping with magnetic impurities, a competing WL effect will beintroduced and the localization behavior is a result of thecompetition between WAL and WL27,29,46,47. For the slightly dopedsample (x 5 0.08, Fig. 2b), at T 5 1.9 K the sharp upward cuspfeature of WAL is gone and the magnetoconductance exhibits anon-monotonic increase with the increase of magnetic field, wherea sharp downward cusp is developed at small magnetic fields. Whenthe temperature warms up to 3.1 K, the non-monotonic behaviordisappears and the WAL shows up again. Further increasing thetemperature to 3.7 K flattens the cusp feature, indicating that theWAL is weakened and it can only survive in a small temperaturerange. For the sample doped with x 5 0.10 (Fig. 2c), WAL iscompletely suppressed and a non-monotonic behavior is presentedup to 3.1 K before a classical parabolic dependence of the magneticfield (,B2) of the MC appears around 4 K (Supplementary materialsFig. S1). Further increasing the doping concentration to x 5 0.14(Fig. 2d), downward cusp feature is persistent up to 10 K, indicativeof a WL dominated behavior.

The quantum corrections to the 2D MC can be described bythe Hikami-Larkin-Nagaoka (HLN) model48 and is given analytically

by the equation Dsxx:sxx Bð Þsxx 0ð Þ~ae2

ph½y 12z

Bw

B

! "

lnBw

B

! "

$, where e is the electron charge, h is Planck’s constant, B

is the magnetic field, y is the digamma function, and a is a coefficientwhose value is determined by the nature of the corrections being WL

or WAL, or having contributions from both effects. Additionally, wehave Bw~

.4el2

w in which the coherence length is characterized bylw~

ffiffiffiffiffiffiffiffiDtw

p, D is the diffusion coefficient and tw is the dephasing

time. The undoped samples show a WAL behavior (Fig. 2a) and canbe fitted well to the HLN model (Figs. 2e and 2f). The resultant avalue ranges from 20.65 to 20.75 (black squares in Fig. 2h) withincreasing temperatures, consistent with the typical values of WALoriginated from 2D surface states of TI11,49–51. And for the heavilydoped samples with x 5 0.14, the MC has an excellent fit to the HLNmodel (Fig. 2g) with a values from 0.25 to 0.09 (blue triangles inFig. 2h) suggesting a typical WL behavior28,51–53. However, the fitbecomes challenging for the lightly (x 5 0.08, Fig. 2b) and inter-mediate doping (x 5 0.10, Fig. 2c) samples, primarily because of thecompetition between WAL and WL. Under these circumstances, theweight ratios of competing terms of WAL and WL are difficult to beextracted. Nevertheless, in low magnetic fields (20.3 T , B ,0.3 T), the sample with intermediate doping yields avalues rangingfrom 1.0 to 0.37 with increasing temperatures (T # 2.8 K) asopposed to a large deviation from the HLN model at high fields(for B . 0.3 T, supplementary Fig. S1)27.

It has been proposed that the opening of the surface energy gapfrom the TRS breaking is responsible for this crossover from WAL toWL27,28. Experimental observation in CrxBi2-xTe3 thin films showedthat with x 5 0.23, the surface states were completely suppressed.Correspondingly the system became a dilute magnetic semi-conductor (DMS)28. It is well known that the incorporation of mag-netic impurities leads to the increased disorder in the films causinglocalization in the electronic states, known as WL, which is stronglyrelated to field-induced magnetization29. In our scenario, with amuch lower Cr doping of x 5 0.14, the MC is completely governedby the WL effect as opposed to the crossover behavior from WL tounitary parabola with x 5 0.10. This suggests that a long-range

Figure 2 | Crossover of quantum corrections of magnetoconductance (MC) with increasing Cr content in CrxBi2-xTe3 thin films (x # 0.14).(a) MC curves of pure Bi2Te3 thin films, showing the negative MC features of WAL. (b) MC curves of Cr0.08Bi1.92Te3 thin film, indicating a non-monotonic behavior with sharp downward cusp at low temperatures (, 3.1 K) and re-presence of WAL at higher temperatures (3.1 and 3.7 K). (c) MCcurves of Cr0.10Bi1.90Te3 thin film, showing a crossover from downward cusp feature to parabolic dependence with increasing temperatures. (d) MCcurves of Cr 0.14Bi1.86Te3 thin film shows a WL dominated behavior. (e) HLN model fitting of MC curves with different Cr content at temperature of1.9 K. (f) and (g) HLN model fitting of MC curves of pure Bi2Te3 and heavily doped Cr0.14Bi1.86Te3 thin films, showing that both WAL and WL can befitted well to the HLN model. (h) Pre-factor of a in HLN model of thin films with different Cr concentrations.

www.nature.com/scientificreports

SCIENTIFIC REPORTS | 3 : 2391 | DOI: 10.1038/srep02391 3

2..Bi2Se3withMndoping:Zhanget.al,prB86,205127(2012)DUMING ZHANG et al. PHYSICAL REVIEW B 86, 205127 (2012)

0.3

0.2

0.1

0.0

-0.1

-0.2

∆σxx

(e

2/h

)

-2 -1 0 1 2µ0H (T)

0.15

0.00

-0.15∆σxx

(e

2/ h

)

1050T (K)

Bi/Mn = 8.3

0.5 K

1.0 K

1.5 K

2.7 K

3.0 K

4.0 K

10 K

1.2 T

-0.2

-0.1

0.0

0.1

∆σxx

(e

2/h

)

-2 -1 0 1 2µ0H (T)

0.05

0.00

-0.05∆σxx

(e

2/h

)

1050T (K)

Bi/Mn = 10.3

0.5 K

1.0 K

1.5 K

4.0 K

8.0 K

0.85 T

-0.6

-0.4

-0.2

0.0

0.2

0.4

∆σxx

(e

2/h

)

-1.0 -0.5 0.0 0.5 1.0µ0H (T)

-0.4

0.0

0.4

∆σxx

(e

2/ h

)

1050T (K)

Bi/Mn = 12.5

0.5 K1.0 K

3.0 K4.0 K

5.0 K

6.0 K

10 K

0.7 T-1.0

-0.5

0.0

0.5

∆σxx

(e

2/h

)

-1.0 -0.5 0.0 0.5 1.0µ0H (T)

-0.4

0.0

0.4

∆σxx

(e

2/h

)

1050T (K)

Bi/Mn = 23.6

0.5 K1.5 K3.0 K

4.0 K5.0 K

7.0 K

10 K

0.55 T

(a)

(d)(c)

(b)

-0.4

-0.2

0.0

0.2

0.4

∆σxx

(e

2/ h

)

-0.8 -0.4 0.0 0.4 0.8µ0H (T)

0.5 K

Bi/Mn = 12.5

23.6

10.3

8.3

undoped

(e) (f)

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

∆σxx

(e

2/ h

)

-0.4 -0.2 0.0 0.2 0.4µ0H (T)

∆ /2EF = 0.99

0.9

0.80.50.2

0.1

0.0

FIG. 6. (Color online) MC of Mn-Bi2Se3 in field perpendicular to the sample plane. (a) MC of the most lightly doped sample (Bi/Mn = 23.6)shows a crossover from weak antilocalization to weak localization at T ∼ 3.0 K, consistent with the TC obtained from SQUID measurements.(b) MC of the Bi/Mn = 12.5 sample shows larger positive MC, which survives up to ∼5 K. (c) MC of the Bi/Mn = 10.3 sample. (d) MC ofthe most highly doped sample (Bi/Mn = 8.3). The insets in panels (a)–(d) plot temperature-dependent !σxx at fixed perpendicular magneticfield. (e) MC of four Mn-Bi2Se3 and one undoped Bi2Se3 sample. (f) Simulation using Eq. (2) showing crossover from weak antilocalizationto weak localization as parameter !/(2EF ) changes.

the bulk states dominate the conductivity because of the largeFermi energy: a simple estimate using the surface-state energydispersion and a Fermi energy 300 meV above the Dirac pointshows that the surface carrier density (∼1.3 × 1013 cm−2) isabout an order of magnitude smaller than that of the bulk inour samples. Thus, an important question to address is the

coexisting surface and bulk conduction in our samples. Theclassical contribution to the MC from bulk channels is wellknown to result in a parabolic positive magnetoresistance or anegative MC. According to diagrammatic calculations,28 thequantum corrections to the conductivity of the lowest 2D bulkquantum well states in Bi2Se3 are purely in the orthogonal

205127-6

DUMING ZHANG et al. PHYSICAL REVIEW B 86, 205127 (2012)

0.3

0.2

0.1

0.0

-0.1

-0.2

∆σxx

(e

2/h

)

-2 -1 0 1 2µ0H (T)

0.15

0.00

-0.15∆σxx

(e

2/ h

)

1050T (K)

Bi/Mn = 8.3

0.5 K

1.0 K

1.5 K

2.7 K

3.0 K

4.0 K

10 K

1.2 T

-0.2

-0.1

0.0

0.1

∆σxx

(e

2/h

)

-2 -1 0 1 2µ0H (T)

0.05

0.00

-0.05∆σxx

(e

2/h

)

1050T (K)

Bi/Mn = 10.3

0.5 K

1.0 K

1.5 K

4.0 K

8.0 K

0.85 T

-0.6

-0.4

-0.2

0.0

0.2

0.4

∆σxx

(e

2/h

)

-1.0 -0.5 0.0 0.5 1.0µ0H (T)

-0.4

0.0

0.4

∆σxx

(e

2/ h

)

1050T (K)

Bi/Mn = 12.5

0.5 K1.0 K

3.0 K4.0 K

5.0 K

6.0 K

10 K

0.7 T-1.0

-0.5

0.0

0.5

∆σxx

(e

2/h

)

-1.0 -0.5 0.0 0.5 1.0µ0H (T)

-0.4

0.0

0.4

∆σxx

(e

2/h

)

1050T (K)

Bi/Mn = 23.6

0.5 K1.5 K3.0 K

4.0 K5.0 K

7.0 K

10 K

0.55 T

(a)

(d)(c)

(b)

-0.4

-0.2

0.0

0.2

0.4

∆σxx

(e

2/ h

)

-0.8 -0.4 0.0 0.4 0.8µ0H (T)

0.5 K

Bi/Mn = 12.5

23.6

10.3

8.3

undoped

(e) (f)

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

∆σxx

(e

2/ h

)

-0.4 -0.2 0.0 0.2 0.4µ0H (T)

∆ /2EF = 0.99

0.9

0.80.50.2

0.1

0.0

FIG. 6. (Color online) MC of Mn-Bi2Se3 in field perpendicular to the sample plane. (a) MC of the most lightly doped sample (Bi/Mn = 23.6)shows a crossover from weak antilocalization to weak localization at T ∼ 3.0 K, consistent with the TC obtained from SQUID measurements.(b) MC of the Bi/Mn = 12.5 sample shows larger positive MC, which survives up to ∼5 K. (c) MC of the Bi/Mn = 10.3 sample. (d) MC ofthe most highly doped sample (Bi/Mn = 8.3). The insets in panels (a)–(d) plot temperature-dependent !σxx at fixed perpendicular magneticfield. (e) MC of four Mn-Bi2Se3 and one undoped Bi2Se3 sample. (f) Simulation using Eq. (2) showing crossover from weak antilocalizationto weak localization as parameter !/(2EF ) changes.

the bulk states dominate the conductivity because of the largeFermi energy: a simple estimate using the surface-state energydispersion and a Fermi energy 300 meV above the Dirac pointshows that the surface carrier density (∼1.3 × 1013 cm−2) isabout an order of magnitude smaller than that of the bulk inour samples. Thus, an important question to address is the

coexisting surface and bulk conduction in our samples. Theclassical contribution to the MC from bulk channels is wellknown to result in a parabolic positive magnetoresistance or anegative MC. According to diagrammatic calculations,28 thequantum corrections to the conductivity of the lowest 2D bulkquantum well states in Bi2Se3 are purely in the orthogonal

205127-6

SurfacestatesofTI[1703.07361]

19

3

(a)




ph½y 12z

Bw

B

! "

lnBw

B

! "




.4el2









(b)

(c)

2.5K

6.0K3.1K

-0.4 -0.2 0.0 0.2 0.4

-0.6

-0.4

-0.2

0.0

0.2

H(T)

Δσ(e2 ℏ)

(d)

(e)




ph½y 12z

Bw

B

! "

lnBw

B

! "




.4el2









(f)

FIG. 1. Evolution of MC curve as temperature increases.(a), (c), (e) are theoretical results and (b), (d) and (f) areexperimental data of reference [21]. For ((a) and (b)) = 0;for ((c) and (d)) = 0.1; for ((e) and (f)) = 0.14. (d) isdrawn to compare the theory and data on top of each other.We used 2 = 1717

(µm)2, vF = 7.5 104m/s, q = 7.12.

Magneto-conductance: Now we focus on themagneto-conductance (MC) of non-ferromagnetic state.To compare our results with the data for the non-ferromagnetic material, we take µ = 0 so that M0 = 0. Thelongitudinal conductivity in this limit is

xx =(F + G

2)(F H2)

F2 +H2G2. (15)

And the MC is defined by

xx(H) xx(0). (16)

Fig.1 is the colloection of plots of MC as a functionof magnetic field and temperature. Left three figures(a,c,e) are plots of theoretical results for three di↵erentmagnetic doping parameters = 0, 0.1, 0.14 and the

right three figures (b,d,e) are corresponding experimen-tal data. Since our results are analytic we can easily draw3D graphs while experimental results are 2D curves for afew discrete temperatures. From Fig.1, we can say thatthe overall features of theoretical results are overwhelm-ingly consistent with the experimental data apart fromthe horns in Fig 1(b). Especially for the medium dopingcase shown in Fig.1(c,d), MC changes nontrivially fromone temperature slice to the other and this has been theregion of hard-to-fit by any theory so far. Remarkably,our theoretical result fits the details of data as one cansee in figure 1(d), where the theoretical curves are drawnon top of the experimental data.The non-trivial behavior of magneto-conductivity in

crossover regime can be understood by the competitionbetween the enhancement in conductivity by Witten ef-fect and the suppression by external magnetic fields. Theinteraction term gives Witten e↵ect: external magneticfield generates extra charge carriers in the presence of qgiving enhancement of conductivity. The result of thecompetition is the sign change in the curvature of MCcurve near H = 0, where

2(1 42/9)

r20

2H

2 +O(H4). (17)

and = q2/r

20. It also explains why crossover from

WAL to WL appears only in relatively low but not verylow temperature region, because r0 T for high tem-perature and becomes small so that 1 2/3 cannotchange the sign. The fundamental reason for the appear-ance of the Witten e↵ect is the Berry potential which inturn is due to the spin-orbit interaction. That our theoryis good only for the strongly interacting systems suggeststhat the electrons in this regime are strongly interacting.In fact, as the gap increases, the density of states belowFermi energy decreases and the condition µ/kT << 1holds, providing the validity of hydrodynamic descrip-tion.On the other hand, for the very low or very high dop-

ing region, where conventional theory can explain thedata well, our results agree with data only qualitatively.The most problematic feature is the horns of the Fig.1(b), which is missing in our result Fig 1(a,e). Actu-ally such sharp peak in MC curve is the hall-mark ofweak anti-localizaion(WAL) in usual (weakly interacting)topological insulators. Here we suggest that the horns aresmoothed out by strong interaction: the suppression ofbackward scattering is due to the relative Berry phaseof between the localization amplitudes of a trajectoryand its time reversal pair [31]. Trajectories are fuzzy ifquasi-particles disappear by strong interaction, then sois the cancellation argument. Finally we show our MCformula as a function of magnetic field and the dopingparameter as a 3D graph in FIG.2, from which we cansee the evolution of the magneto-conductivity along thechange of magnetic doping, which agrees with the dataof ref. [21]. Our result predict how the MC curve shouldbehave outside the experimented region.

4

FIG. 2. Evolution of MC curve from WAL to WL as oneincreases doping parameter .

Future directions: In this letter, we only examinedthe Magneto-conductance in non-ferromagnetic phase,That is, we need to investigate the regime for nonzerocharge parameter q. Other transport coecients like

thermal conductivities and Seeback coecients with orwithout magnetic fields are also important aspects thatrequest future investigations. The graphene has evennumber of Dirac cones, weak spin-orbit interaction anddi↵erent mechanism for WL/WAL. Because of such dif-ferences, we need to find other interaction term in holo-graphic model for graphene. It is also interesting to clas-sify all possible pattern of interaction that provides thefermion surface gap in the presence of strong e-e correla-tion in our context.

ACKNOWLEDGMENTS

This work is supported by Mid-career Researcher Pro-gram through the National Research Foundation of Ko-rea grant No. NRF-2016R1A2B3007687. YS is alsosupported in part by Basic Science Research Programthrough NRF funded by the Ministry of Education grantNo. NRF-2016R1D1A1B03931443.

[1] S. Sachdev, Quantum Phase Transitions. (CambridgeUniversity Press. (2nd ed.), 2011).

[2] J. Zaanen, Y. Liu, Y.-W. Sun, and K. Schalm, Holo-

graphic duality in condensed matter physics (CambridgeUniversity Press, 2015).

[3] E. Oh and S.-J. Sin, Phys. Lett. B726, 456 (2013),arXiv:1302.1277 [hep-th].

[4] S.-J. Sin, J. Korean Phys. Soc. 66, 151 (2015),arXiv:1310.7179 [hep-th].

[5] S. A. Hartnoll, P. K. Kovtun, M. Muller, and S. Sachdev,Phys. Rev. B 76, 144502 (2007), arXiv:0706.3215.

[6] A. Lucas, J. Crossno, K. C. Fong, P. Kim,and S. Sachdev, Phys. Rev. B93, 075426 (2016),arXiv:1510.01738 [cond-mat.str-el].

[7] J. M. Maldacena, Int.J.Theor.Phys. 38, 1113 (1999),arXiv:hep-th/9711200 [hep-th].

[8] E. Witten, Adv. Theor. Math. Phys. 2, 253 (1998),arXiv:hep-th/9802150.

[9] S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, Phys.Lett. B428, 105 (1998), arXiv:hep-th/9802109 [hep-th].

[10] J. Crossno, J. K. Shi, K. Wang, X. Liu, A. Harzheim,A. Lucas, S. Sachdev, P. Kim, T. Taniguchi, K. Watan-abe, T. A. Ohki, and K. C. Fong, Science 351, 1058(2016), arXiv:1509.04713 [cond-mat.mes-hall].

[11] Y. Seo, G. Song, P. Kim, S. Sachdev, and S.-J. Sin, Phys.Rev. Lett. 118, 036601 (2017), arXiv:1609.03582.

[12] M. Z. Hasan and C. L. Kane, Reviews of Modern Physics82, 3045 (2010).

[13] X.-L. Qi and S.-C. Zhang, Reviews of Modern Physics83, 1057 (2011).

[14] G. Bergmann, Solid State Commun. 42, 815 (1982).[15] R. Yu, W. Zhang, H.-J. Zhang, S.-C. Zhang, X. Dai, and

Z. Fang, Science 329, 61 (2010).[16] L. Fu and C. Kane, Phys.Rev.Lett. 102, 216403 (2009).[17] X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Phys. Rev. B

78, 195424 (2008).[18] F. Tikhonenko, A. Kozikov, A. Savchenko, and R. Gor-

bachev, Physical Review Letters 103, 226801 (2009).[19] M. Liu, J. Zhang, C.-Z. Chang, Z. Zhang, X. Feng, K. Li,

K. He, L.-l. Wang, X. Chen, X. Dai, et al., Physical re-view letters 108, 036805 (2012).

[20] D. Zhang et al., Physical Review B 86, 205127 (2012).[21] L. Bao, W. Wang, N. Meyer, Y. Liu, C. Zhang, K. Wang,

P. Ai, and F. Xiu, Scientific reports 3 (2013).[22] S. Hikami, A. I. Larkin, and Y. Nagaoka, Progress of

Theoretical Physics 63, 707 (1980).[23] H.-Z. Lu, J. Shi, and S.-Q. Shen, Physical review letters

107, 076801 (2011).[24] Y. Seo, K.-Y. Kim, K. K. Kim, and S.-J. Sin, Phys. Lett.

B759, 104 (2016), arXiv:1512.08916 [hep-th].[25] N. Nagaosa, J. Sinova, S. Onoda, A. MacDonald, and

N. Ong, Reviews of modern physics 82, 1539 (2010).[26] A. M. Essin, J. E. Moore, and D. Vanderbilt, Physical

review letters 102, 146805 (2009).[27] A. Donos and J. P. Gauntlett, JHEP 1411, 081 (2014),

arXiv:1406.4742 [hep-th].[28] A. Amoretti and D. Musso, JHEP 09, 094 (2015),

arXiv:1502.02631 [hep-th].[29] M. Blake, A. Donos, and N. Lohitsiri, JHEP 08, 124

(2015), arXiv:1502.03789 [hep-th].[30] K.-Y. Kim, K. K. Kim, Y. Seo, and S.-J. Sin, JHEP 07,

027 (2015), arXiv:1502.05386 [hep-th].[31] T. Ando, T. Nakanishi, and R. Saito, Journal of the

Physical Society of Japan 67, 2857 (1998).

1.9 K2.5 K3.1 K6.0 K

-0.4 -0.2 0.0 0.2 0.4

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

H(T)

Δσ(e

2 ℏ)

18.06.apctp

Universalityintransitionduetosurfacegap

2018.06.apctp

TheoryfitsnotonlyforCrdopedBi2Te3butalsoMndopedBi2Se3

T=3T=4T=5T=7

0.0 0.2 0.4 0.6 0.8

-0.8

-0.6

-0.4

-0.2

0.0

0.2

H(T)Δσ

T=2.5KT=3.1KT=6K

-0.4 -0.2 0.0 0.2 0.4

-0.6

-0.4

-0.2

0.0

0.2

H(T)

Δσ

results

21

Strong Correlation E↵ects on Surfaces of Topological Insulators via Holography

Yunseok Seo, Geunho Song and Sang-Jin SinDepartment of Physics, Hanyang University, Seoul 04763, Korea.

(Dated: March 21, 2017)

We investigate e↵ects of strong correlation on the surface state of topological insulator (TI). Weargue that electrons in the regime of crossover from weak anti-localization to weak localization, arestrongly correlated and calculate magneto-transport coecients of TI using gauge gravity principle.Then, we examine, magneto-conductivity (MC) formula and find excellent agreement with the dataof chrome doped Bi2Te3 in the crossover regime. We also find that cusp-like peak in MC at lowdoping is absent, which is natural since quasi-particles disappear due to the strong correlation.

PACS numbers: 11.25.Tq, 71.10.-d, 72.15.Rn

Introduction: Understanding strongly correlatedelectron systems has been a theoretical challenge for sev-eral decades. Typically, such systems lose quasi-particlesand show mysteriously rapid thermalization [1–4], whichprovide the hydrodynamic description [5, 6] of them nearquantum critical point (QCP). Recently, the principle ofgauge-gravity duality [7–9] attracted much interest as apossibility of the paradigm for strongly interacting sys-tems, where the system near QCP is mapped to a blackhole. More recently, large violation of Widermann-Frantzlaw was observed in graphene near charge neutral point,indicating that it is a strongly interacting system [10] ina window of temperature, and the gauge gravity prin-ciple applied to it exhibited remarkable agreement withthe experimental data [11].

The fundamental reason for the appearance of thestrong interaction in graphene is the smallness of thefermi sea: in the presence of the Dirac cone, when fermisurface is near the tip of the cone, electron hole pair cre-ation from such a small fermi sea is insucient to screenthe Coulomb interaction. Because this is so simple anduniversal, one can expects that for any Dirac material,there should be a regime of parameters where electronsare strongly correlated. Dirac cone also provides the rea-son why it is a quantum critical system with Lorentzinvariance. The most well known Dirac material otherthan the graphene is the surface of a topological insula-tor (TI) [12, 13]. The latter has an unpaired Dirac coneand strong spin-orbit coupling, and as a consequence, ithas a variety of interesting physics[14–16] including weakanti-localization (WAL) [17].

Magnetic doping in TI can open a gap in the surfacestate by breaking the time reversal symmetry [18–20],and it is responsible for the transition from WAL to weaklocalization(WL). For extreme low doping, the sharphorn of the magneto-conductivity curve near zero mag-netic field can be described by Hikami-Larkin-Nagaoka(HLN) function [21]. However, for intermediate dopingwhere the tendency of WAL and weak localization (WL)compete, a satisfactory theory is still wanted [18, 20, 22]although there is a phenomenological description [23]Even in the case the fermi surface is large at low dop-ing so that the system is a fermi liquid, increasing thesurface gap pushes up the dispersion curve, which makes

the fermi sea small. Then, the logic for strong cou-pling in graphene works for transition region in surface ofTI. Therefore electron system near the transition regionshould be strongly correlated.In this paper, we investigate magneto-conductivity

(MC) for the surface of a topological insulator with cor-related electrons using gauge gravity principle. We willgive analytic formulae of all the magneto-transports onthe surface of TI with strong correlation as a functionof magnetic field, temperature and impurity density andcompare the result with Bi2Te3 data of [20]. Most inter-estingly, in the doping regime with crossover from WALto WL, our theory agrees with experimental data nicelyin a window of temperature justifying our suggestion thatelectrons in the experimented material are strongly cor-related in this regime. Our results also show that thecusp-like peak in MC curve at fixed temperature, whichis the hall-mark of WAL in the weakly interacting sys-tem, is absent, which can be argued to be a consequenceof strong correlation.Idea of the model: Our system is the surface of topo-

logical insulator which is a 2+1 dimensional system withodd number of Dirac cones. Our question is what hap-pens if such system has strong correlation as well andthe recipe for strong electron-electron interaction is touse gauge gravity principle or holography. For TI, spe-cial care is necessary to encode strong spin-orbit coupling(SOC). Our holographic model is defined on a manifoldM which is asymptotically AdS4. With these setup, ourmodel is defined by the action,

22S =

Z

Md4xpg

2

4R+6

L2

1

4F

2

X

I,a=1,2

1

2(@(a)

I)2

3

5

q

16

Z

M

X

I=1,2

(@(2)I

)2F ^ F (1)

where q is the coupling and 2 = 8G and L is the

AdS radius. From now on, we set 22 = L = 1. Theaction contains two pairs of bosons, one for the magneticimpurities and the other for the non-magnetic ones. Toencode the e↵ect of SOC in the presence of the magneticdoping, we introduced the last term which is a couplingbetween the impurity density and the instanton density.

18.06.apctp

+SJS

PublishedinPhys.Rev.B96(2017)no.4,041104(rapidcommunications)

SmallFermiSurfacesandStrongCorrelationEffectsinDiracMaterialswithHolography Y.Seo,G.Song,C.Park+SJS PublishedinJHEP1710(2017)204

Sofar,transport,Whataboutspectrum?

18.06.apctp 22

•  Spectraldata-ARPES

•  Itcomesfromfundamentalfermion’stwopointfunction.

•  Motttransitionisfirstcandidatetounderstand

•  DMFTissuccessfultoanextendsowecancompare

MottTransition

2318.06.apctp

Hubbardmodel

24

Coupling=U/t

대부분의 실험은t를 조절한다.

e:Tomoveornotto,thatistheproblemRealproblemtohuman:unsolvable!

18.06.apctp

25

Theme

18.06.apctp

Hubbardmodel(HM)àNotsolvableHolographyiseffectivetool.CanwereplacetheHubbardModelbyacalculableh-model?

Spectralfunction

2618.06.apctp

2

(p,m) in the holographic model that gives qualitativelythe same spectral density. As a result, we can consider pand m as functions of the Hubbard model parameter U .As U change the trajectory (p(U),m(U)) in the phasespace defines a Embedding of the Hubbard model intothe holographic model. Any line connecting a point infermi-liquid phase and a point in the gapped phase wouldgive an holographic model describing a Mott transition.Therefore we may say that one embedding defines oneholographic Hubbard model. We will give a few simpleembeddings and give the evolution of the model along theincreasing U/t. Comparing with DMFT results for Hub-bard model, two features agree: the appearance of ‘threepeaks’ and ‘transfer’ of the density of state(DOS) to theshoulders. However, the way gap disappears is di↵erentfrom each other after the critical strength of interaction,where both DMFT and holography calculations can notbe trusted completely. Comparing with the experiment,the vanadium oxides data seem to fit quite well to outtheory.

The holographic model : Our starting point is thefermion action with non-minimal dipole interaction,

SD =

Zd4xpgi

MDM m ipMN

FMN

+ Sbd,

(1)

where the subscript D denotes the Dirac fermion and thecovariant derivative is

DM = @M +1

4!abMab iqAM . (2)

For fermions, the equation of motions are first orderand we can not fix the values of all the componentat the boundary, which make it necessary to introduce‘Gibbons-Hawking term’ Sbd to guarantee the equationof motion which defined as

Sbd =±1

2

Zd3x

ph =

±1

2

Zd3x

ph( + + + ),

(3)

where h = ggrr, ± are the spin-up and down compo-

nents of the bulk spinors. The sign is to be chosen suchthat, when we fix the value of + at the boundary, Sbd

cancel the terms including that comes from the to-tal derivative of SD. Similar story is true when we fix . The former defines the standard quantization andthe latter does the alternative quantization. The back-ground solution we will use is Reisner-Nordstrom blackhole in asymptotic AdS4 spacetime,

ds2 = r

2f(r)

L2dt

2 +L2

r2f(r)dr

2 +r2

L2d~x

2

f(r) = 1 +Q

2

r4 M

r3, A = µ

1 r0

r

, (4)

where L is AdS radius, r0 is the radius of the black holeand Q = r0 µ,M = r0(r20 + µ

2). The temperature of the

boundary theory is given by T = f0(r0)/4 and it can

be solved for r0 to give r0 = (2T +p

(2T )2 + 3µ2)/3.In this paper we treat the fermion as a probe and donot consider its back reaction and further mathematicaldetails of getting the spectral density is described in theappendix.

It was pointed out [21] that the high frequency behav-ior of the spectral function diverges like !2m so that thesum of the degree of freedom over frequency is infiniteif m is positive. Therefore we need to take the negativebulk mass only in the standard quantization. For the easeof discussion we want to maintain the positivity of themass which can be done simply by going to the alterna-tive quantization. Summarizing, we work in alternativequantization with positive mass.

The Phases of holographic fermion : We studythe phases of holographic fermion as function of dipolecoupling p and the Dirac bulk mass m. Six phasesappear: these are Fermi Liquid(FL), Bad metal, BadMetal prime(BM’), Pseudo-gap(PG), Gapped(G) andSemi-metal(SM) phases. The phase boundaries are fuzzysince transitions are all smooth. FL, BM, BM’ and SMare metalic phases, G is insulating and PG is betweenbad-metal and insulator. The character of each phaseis illustrated in Figure 2. In this paper, we use the thesymmetrized version A(!) + A(!) of the spectral den-sity, to meet the general tendency in the literature. Thedi↵erence between the bad metal and its primed phase iswhether they have shoulder or not. See Figure 2(d) and(b). Similarly, the di↵erence between the Fermi-liquidand semi-metal is also whether or not there is a shoul-der. See Figure 2(a) and (c). The di↵erence of Bad metalprime (BM’) and Pseudo gap is whether or not the cen-tral peak is higher or lower than the shoulder. See Figure2(b) and (e). The di↵erence of semi-metal and Gappedphase is whether they have central peak or not. See Fig-ure 2(c) and (f).

(a)fermi liquid (FL) (b)bad metal prime

(BM’)

(c)semi-metal (SM)

(d)bad metal (BM) (e)pseudogap (PG) (f)gapped (G)

FIG. 2. (a)-(e):Typical Fermion Phases and their Tempera-ture evolutions. a) FL with (p=0.5, m=0.2), (b) BM’ withshoulder (p=2, m=0.1,0.4), (c) SM (p=6, m=0.45), (d) BMwithout shoulder (p=0.5, m=0.1), (e) PG (p=2.5,m=0.15),(f) G (p=6, m=0.15)

2



SD =

Zd4xpgi

MDM m ipMN

FMN

+ Sbd,

(1)


DM = @M +1

4!abMab iqAM . (2)


Sbd =±1

2

Zd3x

ph =

±1

2

Zd3x

ph( + + + ),

(3)




ds2 = r

2f(r)

L2dt

2 +L2

r2f(r)dr

2 +r2

L2d~x

2

f(r) = 1 +Q

2

r4 M

r3, A = µ

1 r0

r

, (4)









(BM’)

(c)semi-metal (SM)



2



SD =

Zd4xpgi

MDM m ipMN

FMN

+ Sbd,

(1)


DM = @M +1

4!abMab iqAM . (2)


Sbd =±1

2

Zd3x

ph =

±1

2

Zd3x

ph( + + + ),

(3)




ds2 = r

2f(r)

L2dt

2 +L2

r2f(r)dr

2 +r2

L2d~x

2

f(r) = 1 +Q

2

r4 M

r3, A = µ

1 r0

r

, (4)









(BM’)

(c)semi-metal (SM)



2



SD =

Zd4xpgi

MDM m ipMN

FMN

+ Sbd,

(1)


DM = @M +1

4!abMab iqAM . (2)


Sbd =±1

2

Zd3x

ph =

±1

2

Zd3x

ph( + + + ),

(3)




ds2 = r

2f(r)

L2dt

2 +L2

r2f(r)dr

2 +r2

L2d~x

2

f(r) = 1 +Q

2

r4 M

r3, A = µ

1 r0

r

, (4)









(BM’)

(c)semi-metal (SM)



2



SD =

Zd4xpgi

MDM m ipMN

FMN

+ Sbd,

(1)


DM = @M +1

4!abMab iqAM . (2)


Sbd =±1

2

Zd3x

ph =

±1

2

Zd3x

ph( + + + ),

(3)




ds2 = r

2f(r)

L2dt

2 +L2

r2f(r)dr

2 +r2

L2d~x

2

f(r) = 1 +Q

2

r4 M

r3, A = µ

1 r0

r

, (4)









(BM’)

(c)semi-metal (SM)



2



SD =

Zd4xpgi

MDM m ipMN

FMN

+ Sbd,

(1)


DM = @M +1

4!abMab iqAM . (2)


Sbd =±1

2

Zd3x

ph =

±1

2

Zd3x

ph( + + + ),

(3)




ds2 = r

2f(r)

L2dt

2 +L2

r2f(r)dr

2 +r2

L2d~x

2

f(r) = 1 +Q

2

r4 M

r3, A = µ

1 r0

r

, (4)









(BM’)

(c)semi-metal (SM)



PhaseDiagramhas6phases

3

The result of the detailed study of phases are sum-marized by the phase diagram given in Figure 3(a). Thedashed line represents the free fermion wall, the FL phaseis located at the upper-left corner and gapped phase is atthe lower-right corner. All other phases are in betweenthe two and can be understood as their proximity andcompetition e↵ects. Notice also that the phase diagramis divided by the line of m ' 0.35: the lower half regionis where gap-generation phenomena is observed as we ex-pected from the presence of the dipole term. However,in the upper region, a new metalic phase appears insteadof gapped one. We call it semi-metal phase, because sig-nificant fraction of density of state is depleted from thequasi-particle peak near the fermi level and moved tothe shoulder region which is called Hubbard side peak.The emergence of this new metalic phase in the stronglycoupled system was unexpected, which might be as sur-prising as when DMFT calculation discovered a narrowquasiparticle peak between the two Hubbard bands [22–25]. The appearance of each phase are explained in detailin the appendix for interested reader.

1 2 3 4 5 6 7p

0.1

0.2

0.3

0.4

0.5

m

(FL) (SM)

(G)(PG)

(BM’)

(BM)

(a)Phase diagram

(FL) (SM)

(G)(PG)

(BM’)

(BM)

(b)Embedding of Hubbard

models

FIG. 3. (a) Phase diagram in (p,m) space. Phase boundariesare smooth crossover except SM-G transition. (b)An embed-ding defines a holographic Hubbard model: ↵ = 1/50 (1/20)for yellow (red) line in straight line embedding. =8/9 (4/3)for blue (green) curve in hyperbolic embedding.

Holographic Hubbard Models : Traditionally, thephysics of Mott transition is usually discussed using Hub-bard model which describe the transition as the competi-tion of hopping and onsite coulomb repulsion. So one ofthe test of the usefulness of holographic method is to seeif it can encode the physics of Hubbard model at leastqualitatively. It is given by the Hamiltonian

H = t(X

c†icj +H.c.) + U

Xnini- (5)

+V

X

<ij>

ninj + V2 + V3+

where and is for spin index, t is the hopping rate, Urepresents the on-site Coulomb repulsion and V is forthe nearest o↵-site repulsion and Vn is next to nearestrepulsion. These can be set to be zero in leading order.The purpose of this section is to discuss how to realize thephysics of the Hubbard model in terms of the simplestholographic model and to compare the result with DMFTcalculations and experiment.

Here, the question is, for any Hubbard model withgiven values of t, U, V, Vi, · · · , can we find m and p thatproduce the same or at least qualitatively the same spec-tral density. The answer seems to be yes, because we al-ready have seen that our phase diagram contains all sixphases that are available for the Hubbard model, there-fore if we consider a line in the phase diagram connectingtwo points, one in Fermi liquid phase and the other ingapped phase, then it will give a mapping of an Hubbardmodel into a holographic model and each such a path in(p,m) space defines a “holographic Hubbard model”.Here we consider the simplest embedding for U V

and Vi = 0 for i 2. Since both p and U are gap gen-eration mechanism in each side, it is natural to identifythese: p = U/t. To identify the m in terms of the Hub-bard model parameter, we need to observe following two:i) m = 1/2 line has singularly strong e↵ect of restoringthe free fermion nature in the sense that there is a met-alic spectrum around fermi sea with dispersion relation! + µ = ±k, whatever large value p may have, as wecan see in Figure 8. Therefore we expect that t ! 1when m ! 1/2 0+; ii) m, by which we mean mL withL being the radius of AdS, is related to the long rangecoulomb repulsion V . Then it is reasonable to identifyV/t = (1/2 m) and V = ↵U with ↵ 1. EliminatingV/t, we get

m+ ↵p = 1/2, p = U/t. (6)

Notice that it passes the free fermion point at (p,m) =(0, 1/2) automatically. This embedding is analogousto the original t-U model where all o↵-site repulsionsare negligible compared to on-site one. So we call itstandard embedding. We take two di↵erent values of↵ = 1/50, 1/20 and give the resulting evolution as U in-creases in Figure 3(b).

(a) (b) (c) (d)

FIG. 4. U -evolutions for Straight line Embedding ( V U ).(a)-(d) are at marked point of embedding diagram for Yellowline

When the line passes through the SM phase, the evolu-tion is similar to that of single-site one band DMFT resultin the sense that the MIT progresses with narrowing thequasi-particle peak by transferring the DOS to the shoul-der region, i.e, Hubbard band. However, the way Mottgap open is di↵erent. In DMFT, the quasiparticle peakdisappear by making its residue vanish at critical Uc. Onthe other hand, in holography, the gap open by movingo↵ the dispersion curve from the fermi level, which can bedone either by lowering m or by increasing p. For exam-ple, when the embedding line crosses the phase boundaryat large p as in the yellow line of embedding diagram,

18.06.apctp 27

Freefermion

Gap

6phases

18.06.apctp 28

understandingphasediagram

2918.06.apctp

3


1 2 3 4 5 6 7p

0.1

0.2

0.3

0.4

0.5

m

(FL) (SM)

(G)(PG)

(BM’)

(BM)

(a)Phase diagram

(FL) (SM)

(G)(PG)

(BM’)

(BM)


models



H = t(X

c†icj +H.c.) + U

Xnini- (5)

+V

X

<ij>

ninj + V2 + V3+




m+ ↵p = 1/2, p = U/t. (6)


(a) (b) (c) (d)



9

without and with bulk mass. which is contrasted with thenonzero mass case where the fermi surface is remained atsubstantially high temperature. See the Figure 11(c).That is, the fermi liquid behavior is enhanced by thepresence of the bulk mass. Here again, we can see therole of mass is the stabilizer of the quasi-particle naturein holographic matter.

(a)Fast decay of FS by T (b)Mass as FS stabilizer

FIG. 11. Role of mass in stabilizing the fermi surface(FS):(a)T = 0, 0.1, 0.4 at m = p = 0. (b)m = 0, 0.224, 0.448 atT = 0.1 and p = 0.

As m ! 1/2, such ‘quasi-particle stabilizing tendency’increases infinitely so that the system is a fermi liquidwhatever strength of the dipole term. In fact, the spec-tral function shows that the dispersion curve is straightline just as if we study a free fermions. Therefore it isnatural to think the bulk fermions with mass near 1/2describe a weakly interacting system, i.e, quasi-particles.For applications to the realistic material, having such adial to make the system fermi liquid in a limit is very use-ful and in fact essential because in the real experiments,one can tune the coupling by applying pressure or dop-ing rate. In fact, for p = 0, it can be understood if wenotice that is the dual of the operator with dimension = d/2 m which is dimension of free fermion whenm = 1/2 so that the fermion with m = 1/2 in alternatequantization is dual to the free fermion [7, 10]. In thepresence of the dipole coupling whose role is to introducea gap which break the conformal symmetry dynamically,there is no guarantee that such trend continue to hold.Our observation is that, nevertheless, such free fermionnature at m = 1/2 persists even in the presence of thedipole interaction regardless of its strength. We call itFree fermion Wall in m-p phase diagram. After care-ful examination of full range of phase diagram, we foundthat at mass region m > 0.35, some metalic phases existalways.

5. Hyperboic Embeddings

Following embedding is interesting since it illuminatemany hidden aspects of the theory including the physicalorigin of the bulk mass. We set V 6= 0 but fixed insteadof being related to U .

p = (U + V )/t, and 2m = V/U 1. (13)

The constraint V/U 1 comes since the o↵-site repul-sion is weaker than the on-site one. The correspondingholographic Hubbard model is a hyperbola

p = (1

2m+ 1), with =

V

t= fixed. (14)

This embedding has following consistency properties.

• In large U limit, both t-U -V model and its gravitydual m-p model are in Gapped phase and even inthe smallest U limit, it does not pass the obviousfree fermion point (m, p) = (1/2, 0) because U V .

• The first embedding with = 4/3 (Blue curve)passes SM ! BM

0 ! PG ! G and the sec-ond one with = 8/9 (Green curve) passes FL !BM

0 ! PG ! G phases as U increases, The lat-ter is also qualitatively consistent with evolution ofHubbard model calculated with multi-site DMFT.The graph is given below.

(a) (b) (c) (d)

FIG. 12. U -evolutions for Straight line Embedding ( V << U). (a)-(d) are at marked point on Red line in the Figure 3(b).

(a) (b) (c) (d)

(e) (f) (g) (h)

FIG. 13. U -evolutions at kc for V 6= 0 cases (HyperbolicEmbedding) . (a)-(d) for upper (Blue) curve, (e)-(h) for lower(Green) curve, Spectral functions at kc for V 6= 0 (GreenCurve) in the Embedding diagram Figure 3(b).

Although we know the reason for the free fermion na-ture of m = 1/2 by counting the conformal weight, itis still somewhat mysterious from the interaction pointof view: How such freeness is achieved in a theory ofstrong interaction? The bulk mass enter into dynamicsonly in a combination mL with L being the AdS radiuswhich encodes the interaction strength of the boundarytheory through L

1/4. In fact, 1/2 is the maximalvalue of mL within the unitarity bound. The most natu-ral way to identify the mL’s gauge coupling dependanceis the o↵-site coulomb interaction V , and following ar-gument shows a plausible way to understand the free

FF = (d 1)/2

m=1/2istheFreefermionicregardlessofdim.

RoleofpinBM’andSMcreation

3018.06.apctp

3


1 2 3 4 5 6 7p

0.1

0.2

0.3

0.4

0.5

m

(FL) (SM)

(G)(PG)

(BM’)

(BM)

(a)Phase diagram

(FL) (SM)

(G)(PG)

(BM’)

(BM)


models



H = t(X

c†icj +H.c.) + U

Xnini- (5)

+V

X

<ij>

ninj + V2 + V3+




m+ ↵p = 1/2, p = U/t. (6)


(a) (b) (c) (d)



(a) p = 1 (b) p = 2 (c) p = 3 (d) p = 6

-3 -2 -1 0 1 2 3 ω

0.5

1.0

1.5

2.0A(ω)

(e) p = 1

-3 -2 -1 0 1 2 3 ω

0.2

0.4

0.6

0.8A(ω)

(f) p = 2

-3 -2 -1 0 1 2 3 ω

0.5

1.0

1.5

A(ω)

(g) p = 3

-3 -2 -1 0 1 2 3 ω

0.5

1.0

1.5

A(ω)

(h) p = 6

Figure 9. (a)-(d):p-evolution of the DOS at m = 0.45 shows appearance of and bad metal prime andsemi-metal; (e)-(h) spectral function along vertical red line in each figure (a)-(d).

The figure 11(a) shows that in the absence of the bulk mass, the DOS peak in !-plot goes

away very rapidly as soon as we turn on temperature. That is, quasi-particles are fragile at

finite temperature, which is the character of non-fermi liquids. On the other hand, if we turn on

the mass, the fermi surface peaks becomes sharper as we can see in the Figure 11(b). The same

phenomena can be seen in the k-plot: in Figure 11(c) and (d), we contrasted the temperature

evolution of the fermi surface without and with bulk mass. which is contrasted with the nonzero

mass case where the fermi surface is remained at substantially high temperature. See the Figure

11(c). That is, the fermi liquid behavior is enhanced by the presence of the bulk mass. Here

again, we can see the role of mass is the stabilizer of the quasi-particle nature in holographic

matter.

As m ! 1/2, such ’quasi-particle stabilizing tendency’ increases infinitely so that the system

is a fermi liquid whatever strength is the dipole term. In fact, the spectral function shows that

the dispersion curve is straight line just as if we study a free fermions. Therefore it is natural

to think the bulk fermions with mass near 1/2 describe a weakly interacting system, i.e, quasi-

particles. For applications to the realistic material, having such a dial to make the system fermi

liquid in a limit is very useful and in fact essential because in the real experiments, one can tune

the coupling by applying pressure or doping rate. In fact, for p = 0, it can be understood if we

notice that is the dual of the operator with dimension = d/2 m which is dimension of

free fermion when m = 1/2 so that the fermion with m = 1/2 in alternate quantization is dual

to the free fermion [8, 11]. In the presence of the dipole coupling whose role is to introduce a

gap which break the conformal symmetry dynamically, there is no guarantee that such trend

continue to hold. Our observation is that, nevertheless, such free fermion nature at m = 1/2

persists even in the presence of the dipole interaction regardless of its strength. We call it Free

– 14 –

RoleofpinPGandGapcreation

3118.06.apctp

3


1 2 3 4 5 6 7p

0.1

0.2

0.3

0.4

0.5

m

(FL) (SM)

(G)(PG)

(BM’)

(BM)

(a)Phase diagram

(FL) (SM)

(G)(PG)

(BM’)

(BM)


models



H = t(X

c†icj +H.c.) + U

Xnini- (5)

+V

X

<ij>

ninj + V2 + V3+




m+ ↵p = 1/2, p = U/t. (6)


(a) (b) (c) (d)



(a) p = 1 (b) p = 2 (c) p = 6 (d) p = 8

-4 -2 0 2 40.0

0.5

1.0

1.5

2.0

ω

A(ω

)

(e) p = 1

-4 -2 0 2 40.0

0.5

1.0

1.5

2.0

ω

A(ω

)

(f) p = 2

-4 -2 0 2 40

2

4

6

8

10

ω

A(ω

)

(g) p = 6

-4 -2 0 2 40

2

4

6

8

10

ω

A(ω

)

(h) p = 8

Figure 8. (a)-(d) on p-evolution of the DOS along the line m = 0.1 show appearance of and pseudo-gapand gapped phase; (e)-(h) spectral function along vertical red line in each figure (a)-(d).

The overall feature of the evolution is from metalic to the insulating phase with pseudo gap

phase in the middle and it agrees with our expectation.

We now show the evolution along the line of m = 0.45 with increasing p to demonstrate the

changes of phase in the upper half part of the phase diagram. The !-plots in Figures 9(e)-(h)

are is along the red vertical redlines in Figure 9(a)-(d) respectively. We can see three di↵erent

phases:

1. Gapless metalic phase with linear dispersion: it is a fermi liquid (FL) regime.

2. The bad metal phase due to development of incomplete generation of conduction band.

3. New metalic phase which we call semi-metal due to the development of the conduction

band. Semi-metal because half of the DOS at the fermi sea is depleted and moved to

shoulder region.

A.3 Role of the bulk mass

We now study the role of mass more systematically by calculating the evolution of the DOS

at two nonzero fixed values of p, that is along two vertical lines p = 2.5 and p = 6.0 in phase

dragram. In the Figure 10(a)-(h), the m-evolution along p = 0.2 line is drawn, where a few

physically interesting phases appear. From the figures 10(i)-(p), we can see that the bulk mass

sharpens the peak at the fermi surface consistently regardless of the value of p. One can see that

increasing m pushes up the lower middle band so that gap is reduced. When the middle band

crosses the fermi level, central peak appears signaling the creation of the semi-metalic phase.

For both cases the final stage is SM phase.

– 13 –

RoleofbulkmassinGapcreation

3218.06.apctp

(a) m = 0 p = 2.5 (b) m = 0.15 p = 2.5 (c) m = 0.3 p = 2.5 (d) m = 0.45 p = 2.5

(e) m = 0 p = 2.5 (f) m = 0.15 p = 2.5 (g) m = 0.3 p = 2.5 (h) m = 0.45 p = 2.5

(i) m = 0 p = 6 (j) m = 0.15 p = 6 (k) m = 0.3 p = 6 (l) m = 0.45 p = 6

(m) m = 0 p = 6 (n) m = 0.15 p = 6 (o) m = 0.3 p = 6 (p) m = 0.45 p = 6

Figure 10. (a)-(h): m- evolution of the spectral density with increasingm with p = 2.5. (i)-(p):Evolutionof the spectral density at p = 6.0. For both cases the final stage is SM phase.

fermion Wall in m-p phase diagram. After careful examination of full range of phase diagram,

we found that at mass region m > 0.35, some metalic phases exist always.

A.4 Hyperboic Embeddings

Following embedding is interesting since it illuminate many hidden aspects of the theory includ-

ing the physical origin of the bulk mass. We set V 6= 0 but fixed instead of being related to

U .

p = (U + V )/t, and 2m = V/U 1. (A.15)

– 15 –

Nowaswedecreasesmfurther,theloweronegoesdownupperonegoesupàgapcreation,i.e,As(½-m)increases,gapiscreated.

3


1 2 3 4 5 6 7p

0.1

0.2

0.3

0.4

0.5

m

(FL) (SM)

(G)(PG)

(BM’)

(BM)

(a)Phase diagram

(FL) (SM)

(G)(PG)

(BM’)

(BM)


models



H = t(X

c†icj +H.c.) + U

Xnini- (5)

+V

X

<ij>

ninj + V2 + V3+




m+ ↵p = 1/2, p = U/t. (6)


(a) (b) (c) (d)



Roleofbulkmassinpsuedo-Gapgeneration

3318.06.apctp

(a) m = 0 p = 2.5 (b) m = 0.15 p = 2.5 (c) m = 0.3 p = 2.5 (d) m = 0.45 p = 2.5

-4 -2 2 4ω

0.2

0.4

0.6

0.8

1.0

1.2

1.4

A(ω)

(e) m = 0 p = 2.5

-1.5 -1.0 -0.5 0.5 1.0 1.5ω

0.2

0.4

0.6

0.8

1.0

A(ω)

(f) m = 0.15 p = 2.5

-1.5 -1.0 -0.5 0.5 1.0 1.5ω

0.2

0.4

0.6

0.8

1.0

1.2

A(ω)

(g) m = 0.3 p = 2.5

-2 -1 1 2ω

0.2

0.4

0.6

0.8

1.0

1.2

1.4

A(ω)

(h) m = 0.45 p = 2.5

(i) m = 0 p = 6 (j) m = 0.15 p = 6 (k) m = 0.3 p = 6 (l) m = 0.45 p = 6

(m) m = 0 p = 6 (n) m = 0.15 p = 6 (o) m = 0.3 p = 6 (p) m = 0.45 p = 6

Figure 10. (a)-(h): m- evolution of the spectral density with increasingm with p = 2.5. (i)-(p):Evolutionof the spectral density at p = 6.0. For both cases the final stage is SM phase.

fermion Wall in m-p phase diagram. After careful examination of full range of phase diagram,

we found that at mass region m > 0.35, some metalic phases exist always.

A.4 Hyperboic Embeddings

Following embedding is interesting since it illuminate many hidden aspects of the theory includ-

ing the physical origin of the bulk mass. We set V 6= 0 but fixed instead of being related to

U .

p = (U + V )/t, and 2m = V/U 1. (A.15)

– 15 –

3


1 2 3 4 5 6 7p

0.1

0.2

0.3

0.4

0.5

m

(FL) (SM)

(G)(PG)

(BM’)

(BM)

(a)Phase diagram

(FL) (SM)

(G)(PG)

(BM’)

(BM)


models



H = t(X

c†icj +H.c.) + U

Xnini- (5)

+V

X

<ij>

ninj + V2 + V3+




m+ ↵p = 1/2, p = U/t. (6)


(a) (b) (c) (d)



DataforTransitionMetalOxide

18.06.apctp 34


25


Spin

-orb

it co

uplin

gC

oulomb interaction

V3+: V2O3, d2

V4+: VO2, d1


25


Spin

-orb

it co

uplin

g

Coulom

b interaction

V3+: V2O3, d2

V4+: VO2, d1

.pptfromAraGo

18.06.apctp 35

Cf:DMFTvsExperiment//HolographyvsExp1.22 Dieter Vollhardt

-3 -2 -1 0

Energy(eV)

Inte

nsi

ty(a

rbit

rary

un

its)

SrVO3,LDA+DMFT(QMC)

CaVO3,LDA+DMFT(QMC)

SrVO3(Sekiyamaetal.)

CaVO3(Sekiyamaetal.)

0 1 2 3 4

Energy(eV)

SrVO3,LDA+DMFT(QMC)

CaVO3,LDA+DMFT(QMC)

SrVO3(Inoueetal.)

Ca0.9

Sr0.1

VO3(Inoueetal.)

Fig. 12: Comparison of the calculated, parameter-free LDA+DMFT(QMC) spectra of SrVO3

(solid line) and CaVO3 (dashed line) with experiment. Left: Bulk-sensitive high-resolution PES

(SrVO3: circles; CaVO3: rectangles). Right: XAS for SrVO3 (diamonds) and Ca0.9Sr0.1VO3

(triangles) [86]. Horizontal line: experimental subtraction of the background intensity; after

Ref. [85].

7 Summary and outlook

Due to the intensive international research over the last two decades the DMFT has quickly

developed into a powerful method for the investigation of electronic systems with strong cor-

relations. It provides a comprehensive, non-perturbative and thermodynamically consistent ap-

proximation scheme for the investigation of finite-dimensional systems (in particular for dimen-

sion d = 3), and is particularly useful for the study of problems where perturbative approaches

are inapplicable. For this reason the DMFT has now become the standard mean-field theory

for fermionic correlation problems, including cold atoms in optical lattices [88–92]. The study

of models in non-equilibrium within a suitable generalization of the DMFT has become yet

another fascinating new research area [93–101].

Until a few years ago research into correlated electron systems concentrated on homogeneous

bulk systems. DMFT investigations of systems with internal or external inhomogeneities such

as thin films and multi-layered nanostructures are still very new [102–107]. They are par-

ticularly important in view of the novel types of functionalities of such systems, which may

have important applications in electronic devices. Here the DMFT and its generalizations will

certainly be very useful.

In particular, the development of the ab initio band-structure calculation technique referred to

as LDA+DMFT has proved to be a breakthrough in the investigation of electronically correlated

DMFTschool(2011)DieterVollhardt

4

very thin but high peak moves away from the fermi levelinstead of disappearing. In the symmetrized density ofstates in Figure 4(c,d), it appears as if the quasi-particlepeak is splitted into two peaks with a small gap betweenthem. In this figure, one should notice the sharpness oftransition between (c) and (d). Note also (d) is close tothe phase boundary. The peak does not disappear butmove away from the fermi sea to give an insulator, andthis is a sharp di↵erence between the DMFT result andholographic one.

When the embedding line do not pass the SM phase(Red line), the U -evolution resembles multi-site DMFTresult where the transition is smooth and it passesthrough the Bad metal and Pseudo gap phases. Herealso it is manifest that during the U -evolution the degreeof freedom is being moved from from the head (quasi-particle peak) to shoulder region which is consistent withDMFT result [20]. See Figure 4. For more example seeFigure 12 in the appendix, where hyperbolic type embed-dings are also considered.

Comparing with experiment : Our study on m, p

0-1-2-3

ω

A(ω)

(a)PES data vs theory

0 1 2 3 4

ω

A(ω)

(b)XAS data vs theory

FIG. 5. Experimenal data vs holographic theory: (a) PESdata, (b) XAS data ; In both case (color red) is for SrVO3

and (color blue) is for CaVO3. The data for SrVO3 is from[26], and that for CaVO3 is from [25].

holographic model will be useful for the interpretationof experiments in Vanadium Oxides and other transitionmetal Oxides. We take Photoemission (PES) data andX-ray absorpsion spectroscopy data for SrVO3 (red cir-cles and diamonds) [26] and Ca0.9Sr0.1VO3 (blue boxesand triangles) [25] following the lecture note of Vollhardtin [27] and DMFT study in [28, 29] and fit those with

our theory. The result is given in Figure 5. The pa-rameters values of (m, p, kc) we used for the holographictheory are as follows : (a) (0.47, 2.2,2.08) for red lineand (0.47, 2.05,2.04) for blue line, (b) (0.47, 1.9,2.05)for red line and (0.455, 1.7,2.07) for blue line. We seethat the agreements between the theory and experimentare remarkable. We recommend the readers to see theFIG. 6 of ref. [29] to compare the same data with theLDA+DMFT QMC spectra.

Our study shows that for any holographic Hubbardmodel, the transfer of the degree of freedom fromthe head(quasi-particle peak) to shoulder under the U -evolution is manifest and that is consistent with DMFTresult [20]. For the order of phase transition our resultis smooth almost everywhere except the transition fromsemi-metal to Gapped phase, where there are extremelynarrow channels of PG and BM’ phases which can behardly seen in the phase diagram so that the transitionis mathematically smooth but so rapid so that it can beregarded as a first order. So the order of phase tran-sition of a holographic Hubbard model depends on theembedding trajectory.

The single site result of DMFT result shows that thetransition from metal to insulator is by narrowing of thequasi particle peak (the central peak) and it remains un-til it become an insulator. This is very similar to theparticular trajectory that passes through the SM phase.However, we believe that the presence of the SM phaseis due to lack of back reaction, which is much more dif-ficult to consider and this is remained as a future work.The cluster DMFT study of the Hubbard model showsthe appearance of pseudo gap and bad metals but if oneincreases the frustration, first order phase transition canalso happen. The authors of ref. [30] reported the coexis-tence of insulating and metallic state in the intermediateregion of U/t, which is qualitatively similar to our situa-tion to the case where embedding trajectory pass throughthe phase boundary of SM to Gapped phase.

ACKNOWLEDGMENTS

We thank Ki-Seok Kim, Myeong-Joon Han and Hun-pyo Lee for useful discussions. This work is supportedby Mid-career Researcher Program through the Na-tional Research Foundation of Korea grant No. NRF-2016R1A2B3007687. YS is also supported in part byBasic Science Research Program through NRF grant No.NRF-2016R1D1A1B03931443.

[1] N. F. MOTT, Reviews of Modern Physics 40, 677 (1968).[2] In this paper we consider PG in the original meaning

introduced in ref. [1] rather than that of High Tc super-conductivity.

[3] J. Zaanen, Y.-W. Sun, Y. Liu, and K. Schalm, Holo-

graphic Duality in Condensed Matter Physics (Cam-

bridge Univ. Press, 2015).[4] S. A. Hartnoll, A. Lucas, and S. Sachdev, (2016),

arXiv:1612.07324 [hep-th].[5] S.-S. Lee, Phys. Rev. D79, 086006 (2009),

arXiv:0809.3402 [hep-th].

3


1 2 3 4 5 6 7p

0.1

0.2

0.3

0.4

0.5

m

(FL) (SM)

(G)(PG)

(BM’)

(BM)

(a)Phase diagram

(FL) (SM)

(G)(PG)

(BM’)

(BM)


models



H = t(X

c†icj +H.c.) + U

Xnini- (5)

+V

X

<ij>

ninj + V2 + V3+




m+ ↵p = 1/2, p = U/t. (6)


(a) (b) (c) (d)



O

ComparingwithHubbardModel?

TherearesomeessentialdifferencebetweenHubbardmodelandthepresentholographicmodelwithdipoleinteraction.Hubbardmodelathalffillingcanshowthephasetransition.However,DipoleTermatzerochemicalpotentialdoesnotgenerateaGAP.WeneedOthergapgeneratingmechanism.WefoundafewothersandinfactclassifiedallYukawaint.….Staytuned.

Halffillingßàzerochemicalpotential.

Conclusion

•  Newfieldtheorybasedonholography

•  AppliedtoDiracmaterial,Transitionmetal•  21centuryphysics=highlyinterestingtheorywithexp.

3718.06.apctp

Thanksyou.

18.06.apctp 38

MottTransitionFirstorder?

18.06.apctp 39

Mott transition in single-site DMFT

29

G Kotliar and D Vollhardt, Physics Today 57, 53 (2004).

Method:quantumholography

2dSIS(nearQCP)hologram = 3dimBlackholeàquantumblackhole

40

3dim.ClassicalBH

2dim.QuantumMatter

18.06.apctp

quantumholography

QCP:dynamicalexponentButBH:

ßà equilibrium,fluiddynamicbehavior ßà transport(transportisinputintraditionalfluiddynamics)

4118.06.apctp

method

Figure 3: !e sketch of the AdS/CFT correspondence.

Supersymmetric D-brane

Susy gauge theory Susy AdS gravity

42

TheoryofQuantumMatter

ClassicalGeometry

General case without SUSY i.Findexampleoutsidestringtheory.ii.Asumeàcaculateàcomparewithexp..

18.06.apctp

Ryu & Takayanagi (2006)

1. entanglement entropy calculation in 2d

i) Evidence outside string theory

43

2. Tensor network :(MultiscaleEntanglementRenormalizationAnsatz)

[Swingle]

18.06.apctp

Ryu & Takayanagi (2006) by product. Presence of dual space time=

presence of high entanglement Raamsdonk : classical . Space is sewn by entanglement. Entanglement first law à Linearized gravity equation.

Comment : Entanglement and Holography

44

CompleteEinsteinequationfromthegeneralizedFirstLawofEntanglementEunseokOh(HanyangU.),I.Y.Park(PhilanderSmithColl.),Sang-JinSin(HanyangU.):arXiv:1709.05752[hep-th]|PDF

A B

A~

Figure 2: According to [9], the entanglement entropy S(A) between regions A and B inthe field theory is related to the area of the minimal surface A in the dual geometry suchthat the boundary of A coincides with the boundary of A: S(A) = Area(A)/(4GN).In the diagram, spatial geometry in the gravity picture is represented by the interiorof the ball, while the geometry on which the field theory lives is identified with theboundary sphere.

gravity duality are built up by entangling degrees of freedom in the non-perturbativedescription.

A disentangling experiment

Let us return to the simpler case of a single CFT on Sd. We would like to do a thoughtexperiment in which we start with the vacuum state of the field theory, dual to gravityon pure global AdS spacetime, and see what happens to the dual geometry when wegradually change the state to disentangle some of the degrees of freedom. To be specific,we divide the sphere into two parts (e.g. hemispheres) which we label A and B.

Since the CFT is a local quantum field theory, there are specific degrees of freedomassociated with specific spatial regions, so we can decompose the Hilbert space H =HA⊗HB. A simple quantitative measure of the entanglement between A and B is theentanglement entropy [8], defined to be the von Neumann entropy

S(A) = −Tr(ρA log ρA)

of the density matrix for the subsystem A,

ρA = TrB(|Ψ⟩⟨Ψ|) .

This is typically a divergent quantity, but we can consider a field theory defined witha cutoff (e.g. on a lattice), such that the entanglement entropy is finite. Now, startingwith the vacuum state, we can ask what happens to the dual spacetime when wevary the quantum state in such a way that the entanglement entropy S(A) decreases.Using the recent proposal of Ryu and Takayanagi [9], we can make a very precisestatement about what happens: the area of the minimal surface A in the dual spacetimewhich separates the spherical boundary into its two components A and B decreases, indirect proportionality to the decrease in entanglement entropy (see figure 2). Since thesurface A is a dividing surface between two regions of the dual space, we see that as

3

A B

A B

Figure 4: Effect on geometry of decreasing entanglement between holographic degreesof freedom corresponding to A and B: area separating corresponding spatial regionsdecreases while distance between points increases. The boundary geometry remainsfixed (despite appearances in the diagram).

largerβ

Figure 5: Spatial section of eternal black hole for two different temperatures (corre-sponding to a horizontal line through the middle of the Penrose diagram of figure 1).For low temperature (large β), where entanglement between the two CFTs is smaller,the asymptotic regions are further apart and separated by a surface of smaller area.

Combining (3) and (2), we see that as the entanglement between degrees of freedomin region A and region B (and therefore the mutual information I(C,D)) drops tozero, the length of the shortest bulk path between the points xC and xD must goto infinity (figure 3). Together with the result of the previous subsection, we obtainthe following picture. As the entanglement between two sets of degrees of freedom ina nonperturbative description of quantum gravity drops to zero, the proper distancebetween the corresponding spacetime regions goes to infinity, while the area of theminimal surface separating the regions decreases to zero. Roughly speaking, the tworegions of spacetime pull apart and pinch off from each other, as shown in figure 4. Asseen in figure 5, these quantitative features can be seen explicitly in the example ofthe eternal AdS black hole, where we can decrease the entanglement between the twoCFTs by increasing the inverse temperature parameter β.

Conclusions

We have seen that we can connect up spacetimes by entangling degrees of freedom and

5

18.06.apctp

45

Def. of AdS/CMT

QCP(z,θ) Geometry (z,θ)

1 Introduction

The AdS/CFT correspondence, sometimes called as gauge/gravity duality, is a conjectured relationship

between quantum field theory and gravity. Precisely, in this correspondence, quantum physics of strongly

correlated many-body systems is related to the classical dynamics of gravity which lives in one higher

dimension. On the other hand according to the AdS/CFT dictionary, an AdS geometry at the gravity

side could only address the conformal symmetry of the dual field theory. However, the generalization of

gauge/gravity correspondence to geometries which are not asymptotically AdS seems to be important

as long as such extension may be related to the invariance under a certain scaling of dual field theory

which does not even have conformal symmetry. Such generalization of AdS is actually motivated by

consideration of gravity toy models in condensed matter physics (the application of such generalization

can be found, for example, in [2]). A prototype of this generalization is a theory with the Lifshitz fixed

point in which the spatial and time coordinates of a field theory have been scaled as

t → ζzt, x → ζx, r → ζr, (1.1)

where z is the critical dynamical exponent. From the holographic duality point of view, for a (D + 1)-

dimensional theory, the corresponding (D+ 2)-dimensional gravity dual can be defined by the following

metric

ds2D+2 =−dt2

r2z+

dr2

r2+

1

r2

D!

i=1

dx2i , (1.2)

where in this paper the AdS radius is set to be one. Due to the anisotropy between space and time, it is

clear that this metric can not be an ordinary solution of the Einstein equation, in fact one needs some

sorts of matter fields to break the isotropy, e.g., by adding a massive vector field or a gauge field coupled

to a scalar field [3–6]. In general by adding a dilaton with non trivial potential and an abelian gauge

field to Einstein-Hilbert action (Einstein-Maxwell-Dilaton theory), one can find even more interesting

metrics, in particular the following metric has been used frequently [7]

ds2D+2 = r2θD

"−dt2

r2z+

dr2

r2+

1

r2

D!

i=1

dx2i

#

, (1.3)

where θ is hyperscaling violation exponent. This metric under the scale-transformation (1.1) transforms

as ds → ζθD ds. In a theory with hyperscaling violation, the thermodynamic parameters behave in such

a way that they are stated in D− θ dimensions; More precisely, in a (D+1)-dimensional theories which

are dual to background (1.2), the entropy scales with temperature as TD/z, however, in the presence of

θ namely dual to (1.3), it scales as T (D−θ)/z [7, 8]. Therefore one may associate an effective dimension

to the theory and this becomes important in studying the log behavior of the entanglement entropy of

system with Fermi surface in condense matter physics, explicitly it was shown that for θ = D − 1 for

any z, the entanglement entropy exhibits a logarithmic violation of the area law [9]. On the other hand

for such backgrounds time-dependency can be achieved by Vaidya metric with a hyperscaling violating

factor. It is the main aim of this paper to investigate how entanglement (mutual information) spreads

in time-dependent hyperscaling violating backgrounds.

Basically, the AdS-Vaidya metric is used to describe a gravitational collapse of a thin shell of matter

in formation of the black hole. This metric in D + 2 dimensions is given by

ds2 =1

ρ2

$

− f(ρ, v)dv2 − 2dρdv +D!

i=1

dx2i

%

, f(ρ, v) = 1−m(v)ρD+1, (1.4)

where ρ is the radial coordinate, xi’s (i = 1, .., D) are spatial boundary coordinates and, here, the

2

Dual?

Transport coefficients

Make a unique rule Exp. data

18.06.apctp

AdS/CFT:anexactdualitywheredictionaryisgiven HydrogenatomofHolographicDuality

Why3d?whyOxide?

46

Transitionmetaloxide3d1-104s1-2O18.06.apctp

qcp/bh duality and quantum matters.hepth.hanyang.ac.kr/~sjs/mytalks/2018/2018.06.apctp.pdf ·...

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