Strongly correlated

systems

T. Giamarchi

http://dpmc.unige.ch/gr_giamarchi/

Details in:

TG, cond-mat/0605472 (Salerno

lectures)

TG, Quantum physics in one

dimension, Oxford (2004)

Questions

• Understood: free electrons

• Real systems : Coulomb interaction

• Properties of realistic systems

• Free electron theory works quite well

Effect of interactions

Landau Fermi liquid

Individual fermionic excitations

exist (quasiparticles)

Fermi liquid theory

• Shown perturbatively in U

• Much more general and robust

Element m*/m cc

Nb 2 1

3He 6 20

Heavy fermion 100-1000 100

Photoemission

Measures Spectral function A(k,)

4 3 2 1 0

binding energy

0°

60°

em

issio

n a

ng

le

E(k)

wave vector [Å-1]

bin

din

g e

nerg

y [eV

]

EF

-4

-3

-2

-1

0

1.51.00.50.0

zoomed

Bandmapping

[001]

[111]

[110]

X

L

W

.

..

Cu Fermi surface (Ashcroft/Mermin)

Fermi surface mapping

k(EF)

(M. Grioni)

T. Valla et al. PRL 83 2085 (1999)

Mo(110) surface

• Transistor 1956

• Supraconductivité

• Magnétorésistance

géante

(1913),1972,

1973,1987,2003

2007

Need to undestand the effects

of interactions ?

Doped SrTiO3

CMR-manganites

Organics, fullerenes,

novel compounds

largest polarization

reached by the

field effect in oxides

FM-insulator

FM-M

High-Tc cupratesSC metalAF-insulator

SC

AF-M

Silicon, GaAssemiconductor (Wigner, FQHE)

n 2D ( e / cm2 )10 1510 7 10 9 10 1310 1110 5

AF-insulator

FM-I

insulator / semiconductor / metal metal

Densities

Oxydes: do not follow band

theory

Leo Kouwenhoven et al

Quantum Dots

Nanoscopic Systems

b Moty Heiblum et al.

Fractional quantum Hall effect

I

V

two dimensional electron gas

High Tc

Bednorz Muller

Ferroelectrics, High Tc, Manganites, Oxydes,

Organics,, nanotechnologies....

Interactions: tomorrow’s

materials

Superfluid

Mott

Insulator

1 μm

Pb

Cold atoms

Atoms in a lattice

Tunnelling

Short range

interaction

Proposal: D. Jaksch et al PRL81 3108 (98)P. Zoller

Effect of interactions in CaG

M. Greiner, O. Mandel, T. Esslinger, T. W. Hansch, I. Bloch, Nature 415 39 (2002)

Cold atomic gases

• Extreme

control and

versatility

• Novel ``materials’’ to address condensed matter

issues

Quantum simulators !

Interactions

StatisticsDimensionality

ENS, ETH, LENS, Mainz, MIT,

NIST, Penn State,

Hubbard model

Reduced dimensionality ?

• Effect of interactions at their strongest

• Many realizations (CM and CA)

• Qualitatively Novel physics !

Plastic electronic1973: F.Wudl, D.Cowan, A.Garito, A.Heeger

TTF° + TCNQ°----->TTF+TCNQ-

2 TMTSF° + X- -----> TMTSF2X + e-

Charge transfer compounds and organic salts

+

+

Superconductivity ?

NO !!

Sliding CDW !!

Bechgaard salts

1979: K. Bechgaard

TMTTF : atoms S (Fabre)

TMTSF : atoms Se (Bechgaard)

TMTSF

(TMTSF)2X, X=PF6-,…. ClO4

-,.

c

a

c

b

• Salt

• Quarter filled

band

ta 3000K

tb 300K

tc 30K

Quasi 1D

Superconductivity !

1981: D. Jerome

D. Jerome, M,Ribault,J.Mazaud and K.Bechgaard (1980)

D. Jaccard et al., J. Phys. C, 13 L89 (2001)

O.M Ausslander et al., Science 298 1354 (2001)Y. Tserkovnyak et al., PRL 89 136805 (2002) Y. Tserkovnyak et al., PRB 68 125312 (2003)

Quantum wires

Cees Dekker

CARBONNANOTUBES

Nanotubes

Yao et al., Nature 402 273 (1999)

Spin chains and ladders

Ladder systemsSpin chains

Spin dimer systemsB. C. Watson et al., PRL 86 5168 (2001)

M. Klanjsek et al.,

PRL 101 137207 (2008)

B. Thielemann et al.,

arXiv:0809.0440 (2008)

Cold atoms: 1D systems

S. Hofferberth et al. Nat. Phys

(2008)

One dimension is different

• No individual excitation can exist (only

collective ones)

• Strong quantum fluctuations

Continuous symmetry

How to make a theory

60’: A. Larkin, I. Dzialoshinskii, L. Gorkov, Bichkov

E. Lieb, D. Mathis

70’: A. Luther, V.J. Emery

80’: F.D.M. Haldane

Good excitations: collective onesTomonaga Luttinger

Details in:

TG, cond-mat/0605472 (Salerno

lectures)

TG, Quantum physics in one

dimension, Oxford (2004)

So now let us start......

Labelling the particles

1D: unique way

of labelling

(x) varies slowly

q » 0 q » 20

CDW

: superfluid phase

Quantum

fluctuations

All short distance properties: form of

the operators. , : smooth fields

Hamiltonian

Luttinger liquid theory

Low energy effective description

All effects described by two parameters:

u and K (Luttinger liquid parameters)

Power law correlation functions

Luttinger parameters

u: velocity of collective excitations

K: dimensionless parameter

U0 1

K1 1

Tonks gas

Luttinger parameters

Correlations

1/2

Difference with ``hydrodynamics’’

(quasicondensates)

à = Ã0 Exp[i µ(x,t)]

S = s [ ( )2 + (x )2]

but !!!

½ = ½0

Condensate ?

No true condensate

(K 0)

Finite temperature

Conformal theory

b

c

Spins

Powerlaw correlation functions

Non universal exponents K(h,J)

1D : excitations are more like fermions, not bosons !!

Fermions

Right (+kF) and left (-kF) particles

Observed ?

Hint in spin systems

Tennent et al. PRB 52 13368 (1995)

H. J. Schulz PRB 34 6372 (1986)

Lineshape:

FT of power law better

than two

gaussian/Lorentzian peaks

Organic conductors

c

b

A. Schwartz et al. PRB 58

1261 (1998)First observation of LL !!

¾(!) » !º

TG Physica B 230 975

(1997)

Nanotubes

Z. Yao et al. Nature 402

273 (1999)

Tonks limit

U = 1 : spinless fermions

Not for n(k) : F B

B. Paredes et al Nature (2004)

T. Kinoshita et al. Science (2004)

M. Kohl et al. PRL (2004)

Interferences

s0L h Ãy(r) Ã(0) i

but K large !! ½ » ½0 hydrodynamic !!

S. Hofferberth et al. Nat. Phys

(2008)

Some aspects of LL but......

Exponent is an adjustable parameter

Universality ? Different correlation

functions ?

Variation with a control parameter ?

can one do a quantitative test ????

BPCB-HPIPB. C. Watson et al., PRL 86 5168 (2001)

M. Klanjsek et al.,

PRL 101 137207 (2008)

B. Thielemann et al.,

PRB 79, 020408(R) (2009)

A. Tsvelik (BNL)

R. Chitra (Jussieu)

E. Orignac (ENS-Lyon)

R. Citro (Salerno U.)

P. Bouillot (Geneva)

C. Kollath (Polytechnique)

M. Zvonarev (Geneva)

Collaborations:M. Klanjsek +

group C. Berthier

(Grenoble)

B. Thielemann +

group C. Ruegg

(LCN/PSI)

+LANL Group

D. Poilblanc ,

S Capponi (Toulouse)

A. Laüchli (Dresden)

B. Normand

Spin Ladders

J

J’

Jr

3 scales :

Jr : rungs

J : legs

J’ : interladder exchange

Triplons:

hard core bosons

1D : spinless fermions

Effect of magnetic field

E

H

S

T

Nature of the transition ?

Properties of massless phase ?

Magnetic field: like a chemical potential (hc1, hc2)

(TG and A. M. Tsvelik PRB 59 11398 (1999))

h

m

Gapped

hc1 hc2

A way to study interacting spinless fermions !

hc1 » Jr ; hc2 » Jr + J

All microscopic parameters are known !

Expected phase diagram

T

H

hc1 hc2

LL

3D

Thermal

gapped

J

J’

How to compute

• Analytical calculations (Luttinger liquid +

BA)

• Numerical calculations (DMRG+ MC)

DMRG

• Zero temperature DMRG

(LL parameters)

• Zero temperature DMRG

Mean field theory

+ (-1)i hx

Finite temperature DMRG

Specific heat Spin chain

P. Bouillot M. Zvonarev

C. Kollath

Magnetization

M. Klanjsek et al., PRL 101 137207 (2008)

Fixes:

Jr = 12.9 K

J = 3.6 K

Specific heat

Ch. Rüegg et al.,

PRL 101, 247202 (2008)

Peak : end of LLLuttinger liquid: Cv / ° T

Specific heat

Ch. Rüegg et al., PRL

101, 247202 (2008)

Can one control LL ?

Compute u(h) and K(h) from (Jr,J,h)

No adjustable parameters !!

Get all (several) correlation functions

Allows to test for Luttinger Liquid !

Luttinger parameters

M. Klanjsek et al., PRL 101 137207 (2008)

Red : Ladder (DMRG)

Green: Strong coupling (Jr 1 ) (BA)

Correlation functions

NMR relaxation rate:

Tc to ordered phase: 1/J’ = Â1D(Tc)

M. Klanjsek et al., PRL 101 137207 (2008)

R. Chitra, TG PRB 55 5816 (97); TG, AM Tsvelik PRB 59 11398 (99)

Ordered phase

Order parameter at T=0

M. Klanjsek et al.,

PRL 101 137207 (2008)

NMR

Neutrons

J’ = 27 mKB. Thielemann et al.,

PRB 79, 020408(R) (2009)

Systems with « spins »

Luttinger liquid

Same treatment

More convenient

1 1( ) ( )

2 2

kinH H H H H

int ( )( )

( )

i

H U U

U

H H H

( , )u K Charge excitations

( , )u K Spin excitations

Charge-spin separation

Spin

Spin-Charge Separation

Charge

Spinon Holon

Spin

Spin-Charge Separation

higher D ?

Charge

Energy increases with spin-charge separation

Confinement of spin-charge: « quasiparticle »

Can one observe spin-charge

separation ?

• Condensed matter: difficult

• One serious experiment: Yacoby et al.

O.M Ausslander et al., Science

298 1354 (2001)

Y. Tserkovnyak et al., PRL 89

136805 (2002)

Y. Tserkovnyak et al., PRB 68

125312 (2003)

Bosons with “spins”

[L.E. Sadler et. al., Nature 443, 164 (2006)]

Spin 1 system[J.M. McGuirk et. al., PRL 89, 090402 (2002)]

[J.M. Higbie et. al., PRL 95, 050401 (2005)]

87Rb atoms, F=1 states

|F=1,mF=-1i and |F=2, mF = 1i “spin” ½ system

[M.Erhard et. al., PRA 69, 032705 (2004)] [A. Widera et. al., PRL 92, 160406 (2004)]

Proposal

• 87Rb |F=2,mF=-1i and |F=2,mF=1i good

potential candidates to observe spin-charge

separation

• No problem of temperature ( fermions)

• Phase separation

A. Kleine, C. Kollath et al. PRA 77 013607 (2008);

NJP 10 045025 (2008)

(Partial) Conclusions

Highly non trivial 1D physics due to interactions

Luttinger liquid theory provides a framework to

study this physics

Testable in many different microscopic systems

(organics, nantubes, quantum wires, cold atoms,

spin systs)

Luttinger liquid provides a firm footing to go

beyond

Luttinger liquids as a cornerstone

to study additional effects

Effect of a lattice:

Mott transition

How to treat?

• Incommensurate: Q 2 0

• Commensutate: Q 2 0

Competition

2 21[ ( ( , )) ( ( , )) ]2

xu

dxdS x u x

K

Beresinskii-

Kosterlitz-Thouless

transition

K=2

Mott insulator:

is locked

Density is fixed

Gap in the excitation spectrumn(k)

T. Kuhner et al. PRB 61

12474 (2000) TG, Physica B 230 975 (97)

Important for 1D

J 0 ``trivial’’ MI

J À V but K < 2

MI !!

T. Stoferle et al.PRL 92 130403 (2004)

Cold atoms

Effect of disorder

Disorder: Bose Glass

TG + H. J. Schulz EPL 3 1287 (1987); PRB 37 325 (1988)

``Two’’ fourier components of disorder

Backward scattering (q » 2 0)

Responsible for localization (Bose Glass)

Pinning of a CDW of bosons

Tonks limit: Anderson localization of free

fermions

Bose glass phase

1D : TG + H. J. Schulz PRB 37 325 (1988)

Superfluid – Localized (Bose glass) transition

for K < 3/2

BKT like transition

Higher dimensions: M.P.A. Fisher et al. PRB 40 546 (1989)

Bose glass also exists (scaling theory)

continuous transition

Phase diagram

K

Db

1 3/2

Bose Glass

Superfluid

Important points

Weak disorder enough : localized for

K < 3/2 even if V ¿

Quantum effect: destructive interferences

Short correlations of disorder: rf » d

Probing: tilting of lattice (``transport’’)

Numerics

S. Rapsch, U.

Schollwoeck, W.

Zwerger EPL 46

559 (1999)

Experiments with interactions !

L. Fallani et al. PRL 98, 130404 (2007)

Quasiperiodics

V(x) = Cos(Q1 x) + Cos(Q2 x)

[Harper potential]

Commensurate (Mott transition): K=2 ??

Disorder (Bose Glass): K=3/2 ??

Other ??

Quasiperiodic

J. Vidal, D. Mouhanna, TG PRL 83 3908 (1999); PRB 65 014201 (2001)

G. Roux et al. PRA 78, 023628 2008

Expansion

Beyond Luttinger

liquids

• Coupling between 1D chains (deconfinement

transition)

[M. A. Cazalilla et al. N. J. Phys. 8 158 (2006) ]

• Coupling of the LL to an external bath

[A. M. Lobos et al. arxiv/0906.5570; E.G. Dalla

Torre et al. (2009)]

• Systems with non linear spectrum (e.g. ! » k2 –

ferromagnet)

[M. B. Zvonarev et al. PRL 99, 240404 (2007)]

Coupled chains: deconfinement

and higher dimensions

Spin systems

E

k

Weakly coupled 1D ladders

J’

Quasi-1D ; Luttinger liquid approach

¹ = h – hc1

Close to QCP:

Dimensional crossover

Always 3D close to QCP !

E

kJ’

¹ = h – hc1

J’ > ¹:

3D

behavior

Bose Einstein Condensation

(TG and A. M. Tsvelik PRB 59 11398 (1999))

triplon = hard core boson

h hc dilute limit: « free » bosons

BEC vs BEC

Cold atoms Dimers/Spins

TG, Ch. Rüegg,

O. Tchernyshyov,

Nat. Phys. 4 198 (08)

Crossover LL (``fermions’’) –

BEC (``bosons’’)

E. Orignac, R. Citro, TG, PRB 75, 140403R 2007

NMR,

1/T1

Local spin

correlations

Bosons

[87Rb]

A. F. Ho, M. A. Cazalilla, TG PRL 92 130405 (2003)

M. A. Cazalilla, A. F. Ho, TG, NJP 8 158 (2006)

T. Stoferle et al.

PRL 92 130403 (2004)

1D physics (Luttinger Liquids)

Cold atoms

1D Mott insulator

10-6

10-5

10-4

10-3

10-2

10-1

1 1.5 2 2.5

J/

K

1D MI2D MI

Anisotropic 3D SF (BEC)

( ) ( ) ( )( )

Phase diagram

Array of atomic

‘quantum dots’

transverse

hopping

repulsion

T. Stoferle et al.PRL 92 130403 (2004)

Experiments

Fermions

D. Jaccard et al., J. Phys. C, 13 L89 (2001)

Deconfinement

0 5 10 15 20 25

60

80

100

200

400

600 TMTTF2PF

6

a

T*

IC

SDW

C

SDWSpin-Peierls

c

Act

ivat

ion

en

ergy,

T* (

K)

Pressure (kbar)

P. Auban-Senzier, D. Jérome, C. Carcel and J.M. Fabre J de Physique IV, (2004)

TG Chemical

Review 104 5037

(2004)

Coupled 1D chains

C. Berthod, TG, S. Biermann, A. Georges, PRL 97 136401 (06)

Coupled tubes with Spin gap

b

M.A. Cazalilla, A. F. Ho, TG,

PRL 95 226402 (2005); arxiv

0604525

AF

Order

1

2

Triplet superconductivity

(repulsive interactions)

Non luttinger liquids

Itinerant ferromagnet

M. B. Zvonarev, V. V. Cheianov, TG, PRL 99 240404 (2007)

One dimensional two bosonic species

G?(x; t) ' t¡®"¯ ln

Ãt

tF

!+

it¹h

2m¤

#¡1=2exp

(im¤x2

2t¹h¡ 4i¯m¤ ln(t=tF)

)

Trapped/open

regimes

Light cone of

spinless bosons

Transverse spin excitations

Conclusions

Tour of one dimensional physics

Luttinger liquid theory provides a framework to

study this physics, and to go beyond

Many effects: lattice, disorder, long range forces,

competition between orders, out of equilibrium

physics, .......

Many experimental realizations: organics, spin

systems, nanowire, nanotubes, adatoms,

FRQHE, Joseph. junctions, cold atoms, ......