effect of hubbard interaction on quadratic band touching

Introduction Existing Results Our Results Conclusion

Effect of Hubbard Interaction on Quadratic BandTouching (QBT) semi-metal in Bernal-stacked

Honeycomb Bilayer

Sumiran Pujariin collaboration with T. C. Lang, G. Murthy and R. K. Kaul

IIT Bombay

September, 2016

1 / 63


Outline of the Talk

History, The Big Picture, etc.Problem StatementMotivationMean-Field TheoryPrevious Results on N = 2 : 1) Weak-coupling RG 2)det-QMCOur Results on general N : 1) Weak-coupling RG 2) det-QMCConclusion

2 / 63


The Big Picture : Condensed Matter Physics

Physics is HUGE (screenshot from Wiki)

Condensed Matter is one of biggest sub-fieldsAffects our lives in a big way ...Synergies with Materials Science, Applied and Engg. Physics,and allied fields

3 / 63


The Big Picture : Various States of Matter

One of the oldest philosophical questions

Matter Metals

InsulatorsMagnets

Superconductors

... Living

Today we focus on electronic states of quantum matter4 / 63


The Big Picture : Approaches to Condensed Matter

5 / 63


Going towards our Problem ...Some History

1920s : Classical Mechanics → Quantum Mechanics.Applied to electrons in solids immediately (Sommerfeld,Peierls, etc.)“Crystallized” into Free-Electron Theory or Band Theory

Bloch’s Theorem leads to Electronic band states fornon-interacting electronsPauli’s Exclusion leads to filling of a certain fraction of bandsFermi Surface with many low-energy excitations

Figure : 1) cartoon, 2) Silicon, 3) Copper

Basically explains many metals (Na, Al),Insulators/Semiconductors (Si) 6 / 63


Going towards our Problem ...Interacting Electrons

But what about electrons interactions ?Answer given by Landau .... Fermi Liquid TheoryLow energy quasiparticles “smoothly connected” tonon-interacting electronsThey are well-defined fermions with long lifetimesDue to Fermi surface and energy-momentum conservation, nophase space to scatter ...BUT .. there are electronic states of matter where this breaksdowne.g. Mott Insulators, lower dimensionsThis brings us to Interacting Electrons.. and our problem

7 / 63


Problem Statement

Hamiltonian :Lattice : Bernal Stack honeycomb bilayernearest neighbour hopping on :

∑〈i,j〉,σ c†

i,σcj,σ + h.c.On-site Hubbard Interaction :

∑i,σ 6=σ′ (ni,σ − 0.5)(ni,σ′ − 0.5)

σ is spin/flavour index for SU(N)Symmetries : Lattice, SU(N) flavour, Time Reversal

Figure : LEFT : Side view RIGHT : Top view

8 / 63


Motivation : The Quadratic Band Touching (QBT)semi-metal

Relevant to Graphene Material

G

QBT2QBT1

Figure : LEFT : Spectrum RIGHT : Brillouin Zone

SEMI-METAL : Fermi Points instead of Fermi SurfacesTwo Fermi Points in the Brillouin Zone

9 / 63


Motivation : Power Counting

(Engineering) Dimensional Analysis of Effect of InteractionsAlso poor man’s RGEffective Low-Energy Continuum Action of QBT

S0 ∼∑σ

∫dd~r dt Ψ†σ(~r , t)(τ0 ∂t+τ1(∂2

x−∂2y )/m+τ2∂x∂y/m)Ψσ(~r , t)

above S0 gives the QBT dispersionunder Space-time Rescaling : ~r → b~r , t → b2t with b & 1 andField-Rescaling : Ψσ → b−d/2Ψσ

S0 scale-invariant

10 / 63


Motivation : Power Counting

(Local) Interaction Term∫dd~r dt U

(Ψ†σ(~r , t)Ψσ(~r , t)Ψ†−σ(~r , t)Ψ−σ(~r , t)

)U → b2−dUIn d = 2, Local Interactions are marginalCan expect Instabilities for QBT semi-metal to LocalInteractionsWhat are possible ground states ??ASIDE Dirac semi-metal, ~r → b~r , t → bt → U → b1−dU →Interactions are irrelevant → StabilityHeuristically,

∼ 0 d.o.s. near Dirac Cones → Irrelevancefinite d.o.s. near QBT points → Possible Instabilities

11 / 63


Mean-Field Theory

Antiferromagnet (AFM) and Valence Bond (VBS) Ansatzes

Figure : Left : AFM ansatz. Right : VBS ansatz

AFM order at zero or Γ wave-vectorVBS order at QBT wave-vectorBoth open a gap → Not Semi-metallic

12 / 63


Mean-Field Theory

Kekule Current Ansatz

0

0

0

0

0

Current Order at QBT wave-vectorOpens a gap → Not Semi-metallic

13 / 63


Mean-Field Theory

Method : Self-consistent Hartree-Fock MFT

0

5

10

15

20

0 5 10 15 20

Neel

Neel

J

U

N = 2;L = 12 : Map of Neel order m1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

5

10

15

20

0 5 10 15 20

VBS

Neel

0

5

10

15

20

0 5 10 15 20

VBS

Neel

JU


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

JU


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

MFT → QBT unstable to AFM order for all N and UAFM order opens a gap at QBT → Insulating state

14 / 63


Existing Results : weak-coupling RG

Perturbative RG goes beyond MFTneeds a small parameter → useful in weak-couplingO. Vafek did the N = 2 calculation Phys. Rev. B 82, 205106 (2010)

zeroth-order effective Action S0 → quadratic

S0 =

∫ψ(r, τ)

(∂τ +

∂2x − ∂2

y2m I⊗ σx ⊗ IN −

∂x∂ym τz ⊗ σy ⊗ IN

)ψ(r, τ)

Internal Indices of field configurations :Valley : I2, τx , τy , τzLayer : I2, σx , σy , σzSpin/Flavour : IN above SU(N) symmetric

15 / 63


Existing Results : weak-coupling RG

general local interaction → quartic

Sint =12∑S,T

gST

∫dτd2r (ψ(r, τ) S ⊗ IN ψ(r, τ))×

(ψ(r, τ) T ⊗ IN ψ(r, τ))

S, T are 4× 4 matrices combining Valley and Layer indicesSU(N) spin/flavour symmetric → IN aboveLow energy projection of U Hubbard term → 3 non-zero gSTDo pert-RG calc to compute RG flows of gST s to lowest order

Integrate momentum shell high energy modes, apply Cumulant Expansion, Tree-level O(g) = 0 is

same as marginality from Power counting , One-loop O(g2) corrections. R. Shankar, Rev. Mod.

Phys. 66, 129 (1994)

6 new couplings generatedCompute RG flows of Susceptibility to quadratic order sources∝ ∆O ∫ dτd2r ψ†(r, τ)Oψ(r, τ)

16 / 63


Existing Results : weak-coupling RG : U → 0 instability

N=2

AFMCurrents

0 10 20 30 40 50ln s

2

4

6

8

10

12

14

d ln D

d ln s-2

Instability in Spin channel → magneticIdentity in Valley Indexdiagonal σz in layer index → AFM is leading instability

17 / 63


Existing Results : det-QMC

Det-QMC method for interacting fermionsSuzuki-Trotter, Hubbard-Stratonovich (HS) to decouple the Interaction term, MC sum over HS

variables, Importance sample using detailed Balance, Weight of HS configurations has

Determinantal structure, Measure Observables as a weighted-average in this ensemble, Assaad and

Evertz, Volume 739 of Lecture Notes in Physics pp 277-356 (2008)

N = 2 study by T. C. Lang et alAFM order for all USingle particle gap opens at QBT wavevectorZero spin gap at AFM wavevector due to Goldstone ModesAgreement with Vafek’s N = 2 RG prediction

18 / 63


Existing Results : det-QMC

One example : AFM order parameter vs 1/L

0.00

0.02

0.04

0.06

0.08

0.10

0 1 2 3 4 5

∆ σ/t

U/t

(a)

L = 3 L = 6 L = 9 L = 12

0.1 0.2 0.30.40

0.50

0.60

0.70

0.80

0.90

S AF,

3

1/L

(b)

U/t = 2.8

U/t = 2.5

U/t = 2.4

U/t = 2.0

Γ ← K → M

ε(k)

L = 12L = 6

Figure : from Lang et al, Phys. Rev. Lett. 109, 126402 (2012).

19 / 63


Our Results : weak-coupling RG : U → 0 instability

N=2

AFMCurrents

0 10 20 30 40 50ln s

2

4

6

8

10

12

14

d ln D

d ln s-2

N=4

AFM

Currents

0 10 20 30 40 50ln s

2

4

6

8

10

12

14

d ln D

d ln s-2

N ≥ 4 : Instability in Charge channel → Non-magneticN ≥ 4 : Off-diagonal τx in Valley index → Order at QBTwavevector which joins the two valleysN ≥ 4 : Off-diagonal σy in Layer index → Odd under TimeReversalConsistent with Kekule Currents → look for them in det-QMC

20 / 63


Our Results : det-QMC

Search for Order → Bragg Peaks in associated StructureFactorsQuantify (presence or absence of) Bragg Peaks using BinderRatiosLook at Single Particle and Spin Gap

21 / 63


Our Results : det-QMC : structure factors

N = 4 First look at AFM structure factor

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

ky

kx

N = 4 L = 12tpt = 1. U

t = 10.

Γ

QBT1 QBT2

0.28

0.3

0.32

0.34

0.36

0.38

0.4

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

ky

kx

N = 4 L = 12tpt = 1. U

t = 32.0

Γ

QBT1 QBT2

0

0.5

1

1.5

2

2.5

3

Figure : LEFT : Weak Coupling RIGHT : Strong Coupling

No Bragg peak in Weak CouplingConsistent with Expectation of Kekule CurrentsAFM in Strong coupling : Peak size scales with system sizeLook for Currents in weak-coupling ... 22 / 63



N = 4 Now look for Current Structure Factor Bragg peak atQBT wavevector in weak-coupling ..

−1.5

−1

−0.5

0

0.5

1

1.5

2

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

Γ

QBT1 QBT2

ky

kx

tp = 1. t = 1. N = 2 L = 12 U = 10.

4

6

8

10

12

14

16

−1.5

−1

−0.5

0

0.5

1

1.5

2

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

Γ

QBT1 QBT2

ky

kx

tp = 1. t = 1. N = 2 L = 12 U = 32.0

15

15.2

15.4

15.6

15.8

16

16.2

16.4


There is no Bragg peak at QBT !!Where are the currents ???

23 / 63



N = 6 AFM structure factor

−1.5

−1

−0.5

0

0.5

1

1.5

2

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

Γ

QBT1 QBT2

ky

kx

tp = 1. t = 1. N = 6 L = 12 U = 20.

0.27

0.28

0.29

0.3

0.31

0.32

0.33

0.34

−1.5

−1

−0.5

0

0.5

1

1.5

2

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

Γ

QBT1 QBT2

ky

kx

tp = 1. t = 1. N = 6 L = 12 U = 60.0

0.25

0.3

0.35

0.4

0.45

0.5


No AFM order in weak-coupling as expected ..No AFM order in strong-coupling unlike N = 4 ..What is the order in strong coupling ?

24 / 63



N = 6 Dimer VBS structure factor

−1.5

−1

−0.5

0

0.5

1

1.5

2

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

Γ

QBT1 QBT2

ky

kx

tp = 1. t = 1. N = 6 L = 12 U = 20.

2

4

6

8

10

12

14

−1.5

−1

−0.5

0

0.5

1

1.5

2

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

Γ

QBT1 QBT2

ky

kx

tp = 1. t = 1. N = 6 L = 12 U = 60.0

0

50

100

150

200

250

300

350

400

450


N = 6 strong coupling is VBS ordered.Are there Currents at weak coupling ?? ..

25 / 63



N = 6 Current structure factor

−1.5

−1

−0.5

0

0.5

1

1.5

2

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

Γ

QBT1 QBT2

ky

kx

tp = 1. t = 1. N = 6 L = 12 U = 20.

5

10

15

20

25

30

−1.5

−1

−0.5

0

0.5

1

1.5

2

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

Γ

QBT1 QBT2

ky

kx

tp = 1. t = 1. N = 6 L = 12 U = 60.0

26.5

27

27.5

28

28.5

29

29.5

30


No Bragg peak in weak coupling again !!What is this mysterious state ??

26 / 63



Revisit N = 2 AFM structure factor.. strange scaling of AFMorder parameter in weak-coupling remarked earlier on pg. 15 ..

−1.5

−1

−0.5

0

0.5

1

1.5

2

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

Γ

QBT1 QBT2

ky

kx

tp = 1. t = 1. N = 2 L = 12 U = 1.6

0.28

0.29

0.3

0.31

0.32

0.33

0.34

0.35

0.36

−1.5

−1

−0.5

0

0.5

1

1.5

2

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

Γ

QBT1 QBT2

ky

kx

tp = 1. t = 1. N = 2 L = 12 U = 3.8

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2


No AFM Bragg peak in weak coupling again !Is the mysterious state perhaps a stable QBT semi-metal ??

27 / 63


Our Results : det-QMC : Binder ratios

Construct Binder Ratios R to quantify Bragg peaks presenceConstruction : R → 1 ordered, while R → 0 not orderedN = 4 AFM Binder ratio

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

14 15 16 17 18 19

1−

〈S.S

(Qk1) 1

4〉

〈S.S

(QN

eel) 1

4〉

U

tp = 1. t = 1. N = 4

L = 6L = 9

L = 12L = 15

Figure : LEFT : Ratio Crossing RIGHT : Crossing Drifts

AFM in Strong Coupling as seen beforeNo AFM order in Weak Coupling as seen before 28 / 63



N = 4 VBS and Current Binder ratios

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30 35 40

1−

〈P.P

(Qnea

rQBT)〉/

〈P.P

(QQBT)〉

U

tp = 1. t = 1. N = 4

L = 6L = 9

L = 12L = 15

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30 35 40

1−

〈cI.cI(Q

nea

rQBT)〉/

〈cI.cI(Q

QBT)〉

U

tp = 1. t = 1. N = 4

L = 6L = 12L = 15

Figure : LEFT : VBS RIGHT : Current

No VBS order as expectedNo Current Order even in weak-coupling

29 / 63



N = 2 AFM Binder ratio

0

0.2

0.4

0.6

0.8

1

1.5 2 2.5 3 3.5 4

1−

〈S.S

(Qk1) 1

4〉

〈S.S

(QN

eel) 1

4〉

U

tp = 1. t = 1. N = 2

L = 6L = 9

L = 12L = 15

Figure : LEFT : Ratio Crossing RIGHT : Crossing Drifts

No AFM order in Weak Coupling evidenced here too !AFM in Strong Coupling as seen before

30 / 63



N = 2 VBS and Current Binder ratios

0

0.2

0.4

0.6

0.8

1

1.5 2 2.5 3 3.5 4

1−

〈P.P

(Qnea

rQBT)〉/

〈P.P

(QQBT)〉

U

tp = 1. t = 1. N = 2

L = 6L = 9

L = 12L = 15

0

0.2

0.4

0.6

0.8

1

1.5 2 2.5 3 3.5 4

1−

〈cI.cI(Q

nea

rQBT)〉/

〈cI.cI(Q

QBT)〉

U

tp = 1. t = 1. N = 2

L = 6L = 9

L = 12L = 15

Figure : LEFT : VBS RIGHT : Current

No VBS or Current Order in weak-coupling

31 / 63



N = 6 VBS Binder ratio

0

0.2

0.4

0.6

0.8

1

20 25 30 35 40 45 50 55 60

1−

〈P.P

(Qnea

rQBT)〉/

〈P.P

(QQBT)〉

U

tp = 1. t = 1. N = 6

L = 6L = 9

L = 12L = 15


VBS in Strong Coupling as seen beforeNo VBS in Weak Coupling as seen before

32 / 63



N = 6 AFM and Current Binder ratios

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50 60 70 80

1−

〈S.S

(Qk1) 1

1〉

〈S.S

(QN

eel) 1

1〉

U

tp = 1. t = 1. N = 6

L = 6L = 9

L = 12L = 15

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50 60 70 80

1−

〈cI.cI(Q

nea

rQBT)〉/

〈cI.cI(Q

QBT)〉

U

tp = 1. t = 1. N = 6

L = 6L = 9

L = 12L = 15

Figure : LEFT : AFM RIGHT : Current

No AFM order as expectedNo Current Order even in weak-couplingWhat is this mysterious state in weak-coupling ??

33 / 63


What about RG ? : Hypothesis of Dirac phase

RG argument :QBT has no linear term in dispersion by definitionIMP : Instead self-energy processes due to the interactiongenerate a Linear term in dispersion !Linear term not disallowed by symmetries of the problemOnce generated, they are RG relevant at the QBT fixed pointFlow to a Dirac semi-metallic fixed point in weak-couplingDirac semi-metal stable to weak local interactionsThus no weak-coupling instability of Bernal StackedHoneycomb Bilayer Hubbard modelInstead Dirac fermions form a stable weak-coupling phaseIMP : irrelevant cubic corrections to QBT are needed togenerate the linear term

34 / 63


What about RG ? : RG flows

Q

D

GN

α

gi

(a) (b)

Q

D

GN

α

gi

How to verify this RG picture ?Dynamical critical exponent : QBT z = 2 or Dirac z = 1(Gross-Neveu)Look into Gap data

35 / 63


Our Results : det-QMC : Spin Gaps

N = 4 Spin Gap at AFM wavevector

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

∆σat

QNeel

1/L

tp = 1. t = 1. N = 4

U = 10.U = 32.

Figure : Strong and Weak shown together

Strong Coupling : gapless spin excitations at AFM wavevectorConsistent with AFM order with gapless Goldstone modesWeak Coupling : no gapless modesConsistent with no AFM order

36 / 63




0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

∆σat

QNeel

1/L

tp = 1. t = 1. N = 2

U = 2.U = 4.


Strong U : gapless spin excitations at AFM wavevectorConsistent with AFM order with gapless Goldstone modesWeak U : no gapless modesConsistent with NO AFM order (IMP)

37 / 63




0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

∆σat

QNeel

1/L

tp = 1. t = 1. N = 6

U = 10.U = 32.U = 48.


Strong and Weak U : no gapless modesConsistent with no AFM (no SU(N) symm breaking)

38 / 63


Our Results : det-QMC : Single Particle Gaps

N = 4 Single Particle Gap at QBT wavevector

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

∆sp

atQBT

1/L

tp = 1. t = 1. N = 4

U = 10.U = 32.

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

∆sp

atQBT

1/L

tp = 1. t = 1. N = 4

U = 10.

Figure : LEFT : Strong and Weak RIGHT : Only Weak

Strong Coupling : QBT is gapped outConsistent with insulating AFM orderWeak Coupling : tending towards 0 (Note y-axis)Not Inconsistent with gapless state

39 / 63




0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

∆sp

atQBT

1/L

tp = 1. t = 1. N = 2

U = 2.U = 4.

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

∆sp

atQBT

1/L

tp = 1. t = 1. N = 2

U = 2.


Strong Coupling : QBT is gapped outConsistent with insulating AFM orderWeak Coupling : Tending towards 0 (Note y-axis again)Not Inconsistent with gapless state

40 / 63




0

0.5

1

1.5

2

2.5

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

∆sp

atQBT

1/L

tp = 1. t = 1. N = 6

U = 10.U = 32.U = 48.

0

0.02

0.04

0.06

0.08

0.1

0.12

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

∆sp

atQBT

1/L

tp = 1. t = 1. N = 6

U = 10.U = 32.


Strong Coupling : QBT is gapped outConsistent with insulating VBS orderWeak Coupling : Tending towards 0 (Note y-axis again)Not Inconsistent with gapless state

41 / 63


Single Particle Gap scaling

0

1

2

3

4

−4 −2 0 2 4

2 30

0.5

1

1.5

2

L∆

sp

L1/ν (U − Uc)/Uc

L = 6L = 9L = 12L = 15

U

Uc estimate of 2.65(10).Above scaling consistent with z = 1.Scaling form : ∆sp = L−zF∆((U − Uc)L1/ν)

42 / 63


Our Results : det-QMC : AFM Order Parameter Scalings

revisit strange Scaling of pg. 15

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

〈S.S(Q

Neel=

(0,0))

11〉/L2

1/L

tp = 1. t = 1. N = 2

U = 0.U = 1.6U = 2.U = 2.4U = 2.8U = 3.2U = 3.6U = 4.0

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

〈S.S(Q

Neel=

(0,0))

11〉/L2

1/L

tp = 1. t = 1. N = 4

U = 4.U = 10.U = 16.U = 20.U = 24.U = 32.U = 40.

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

〈S.S(Q

Neel=

(0,0))

11〉/L2

1/L

tp = 1. t = 1. N = 6

U = 4.U = 10.U = 16.U = 20.U = 24.U = 32.U = 40.U = 48.U = 60.

Figure : AFM Order parameter scalings

It is not strange; rather no AFM order in weak-coupling forN = 2

43 / 63


AFM Order Parameter Scalings

Smooth crossing in R indicates at continuous transitionUc ∼ 2.6

0

0.2

0.4

0.6

0.8

1

−8 −6 −4 −2 0 2 4 6 8

Rm2

m2L−a

L1/ν (U − Uc)/Uc

L = 9L = 12L = 15L = 18

Scaling Collapse Uc = 2.65(10) ν = 0.90(5) a = 0.22(15)

Uc consistent with Gap scalingR = FR((U − Uc)L1/ν) ; m2 = LaFm2((U − Uc)L1/ν)

44 / 63


Conclusion : A Dirac phase for all N = 2, 4, . . .

No AFM for N = 2 in weak-couplingCrossing in Binder RatioNo gapless Goldstone modes seen at AFM wavevector

QBT gives way to Dirac fermions for general NNone of the possible dominant weak-coupling instabilitiesobservedgapless single particle excitationsz = 1 scaling of single particle gap ... CRUCIALAlso IMP Gap opening and Magnetic Transition at same Uc

What about Experiments ?Experiments on suspended samples ... Amir Yacoby’s groupInsulating state is observed ..Our comment : can not be weak short-range interactionsPerhaps long-range part of Coulomb interactions IMPCan the Dirac phase be observed ??

45 / 63


Thank you !Details in PRL 117, 086404 (2016), arxiv:1604.03876Acknowledgements :

Lexington Cond-Matt Group : J. D’emidioO. VafekNSF for financial supportXSEDE for Computational Resources

46 / 63


R map

0

0.5

1

1.5

1 3 5

Non-Magnetic

Magnetic

t p→

U →

Map of R for L = 12

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

47 / 63


Self-Energy with Cubic Corrections to QBT

48 / 63


det-QMC Extra 2 : Lang et al Plots

0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4 5

∆ sp

U/t

QMC, fRG MFT, fRG

L = 3 L = 6 L = 9 L = 12TDL Λc

10-2

10-1

100

0.2 0.4 0.6t/U

0.5 1.0 1.5

10-6

10-3

100

t/U

L

0.00

0.02

0.04

0.06

0.08

0.10

0 1 2 3 4 5

∆ σ/t

U/t

(a)

L = 3 L = 6 L = 9 L = 12

0.1 0.2 0.30.40

0.50

0.60

0.70

0.80

0.90

S AF,

3

1/L

(b)

U/t = 2.8

U/t = 2.5

U/t = 2.4

U/t = 2.0

Γ ← K → M

ε(k)

L = 12L = 6

Figure : For N = 2 figs from Phys. Rev. Lett. 109, 126402 (2012).

Note in Weak Coupling :AFM Order parm goes towards zero unlike strong couplingSingle Particle gap goes towards zero unlike strong couplingSpin Gap increases unlike strong coupling.

49 / 63


det-QMC Extra 3 : 3D Structure Factor Plots N = 4

−2−1.5 −1−0.5 0 0.5 1 1.5 2 −1.5−1−0.500.5 1 1.5 2

00.050.10.150.20.250.30.350.4

tp = 1. t = 1. N = 2 L = 12 U = 10.

kxky

0.280.30.320.340.360.380.4

−2−1.5 −1−0.5 0 0.5 1 1.5 2 −1.5−1−0.500.5 1 1.5 2

0246810

121416

tp = 1. t = 1. N = 2 L = 12 U = 10.

kxky

46810121416

−2−1.5 −1−0.5 0 0.5 1 1.5 2 −1.5−1−0.500.5 1 1.5 2

00.511.522.533.544.55

tp = 1. t = 1. N = 2 L = 12 U = 10.

kxky

11.522.533.544.55

−2−1.5 −1−0.5 0 0.5 1 1.5 2 −1.5−1−0.500.5 1 1.5 2

00.51

1.52

2.53

tp = 1. t = 1. N = 2 L = 12 U = 32.0

kxky

00.511.522.53

−2−1.5 −1−0.5 0 0.5 1 1.5 2 −1.5−1−0.500.5 1 1.5 2

0246810

12141618

tp = 1. t = 1. N = 2 L = 12 U = 32.0

kxky

1515.215.415.615.81616.216.4

−2−1.5 −1−0.5 0 0.5 1 1.5 2 −1.5−1−0.500.5 1 1.5 2

012345678

tp = 1. t = 1. N = 2 L = 12 U = 32.0

kxky

12345678

Figure : Top : Weak U = 10. Bottom : Strong U = 32.Left : Spin Centre : Currents Right : VBS

50 / 63



−2−1.5 −1−0.5 0 0.5 1 1.5 2 −1.5−1−0.500.5 1 1.5 2

00.050.10.150.20.250.30.350.4

tp = 1. t = 1. N = 2 L = 12 U = 1.6

kxky

0.280.290.30.310.320.330.340.350.36

−2−1.5 −1−0.5 0 0.5 1 1.5 2 −1.5−1−0.500.5 1 1.5 2

0123456

tp = 1. t = 1. N = 2 L = 12 U = 1.6

kxky

11.522.533.544.555.5

−2−1.5 −1−0.5 0 0.5 1 1.5 2 −1.5−1−0.500.5 1 1.5 2

00.050.10.150.20.250.30.350.40.45

tp = 1. t = 1. N = 2 L = 12 U = 1.6

kxky

0.260.280.30.320.340.360.380.40.420.440.46

−2−1.5 −1−0.5 0 0.5 1 1.5 2 −1.5−1−0.500.5 1 1.5 2

00.51

1.52

2.5

tp = 1. t = 1. N = 2 L = 12 U = 3.8

kxky

0.20.40.60.811.21.41.61.822.2

−2−1.5 −1−0.5 0 0.5 1 1.5 2 −1.5−1−0.500.5 1 1.5 2

0123456

tp = 1. t = 1. N = 2 L = 12 U = 3.8

kxky

22.533.544.555.56

−2−1.5 −1−0.5 0 0.5 1 1.5 2 −1.5−1−0.500.5 1 1.5 2

00.10.20.30.40.50.60.7

tp = 1. t = 1. N = 2 L = 12 U = 3.8

kxky

0.30.350.40.450.50.550.60.650.7

Figure : Top : Weak U = 1.6 Bottom : Strong U = 3.8Left : Spin Centre : Currents Right : VBS

51 / 63



−2−1.5 −1−0.5 0 0.5 1 1.5 2 −1.5−1−0.500.5 1 1.5 2

00.050.10.150.20.250.30.35

tp = 1. t = 1. N = 6 L = 12 U = 20.

kxky

0.270.280.290.30.310.320.330.34

−2−1.5 −1−0.5 0 0.5 1 1.5 2 −1.5−1−0.500.5 1 1.5 2

05

1015202530

tp = 1. t = 1. N = 6 L = 12 U = 20.

kxky

51015202530

−2−1.5 −1−0.5 0 0.5 1 1.5 2 −1.5−1−0.500.5 1 1.5 2

0246810

1214

tp = 1. t = 1. N = 6 L = 12 U = 20.

kxky

2468101214

−2−1.5 −1−0.5 0 0.5 1 1.5 2 −1.5−1−0.500.5 1 1.5 2

00.050.10.150.20.250.30.350.40.450.5

tp = 1. t = 1. N = 6 L = 12 U = 60.0

kxky

0.250.30.350.40.450.5

−2−1.5 −1−0.5 0 0.5 1 1.5 2 −1.5−1−0.500.5 1 1.5 2

05

1015202530

tp = 1. t = 1. N = 6 L = 12 U = 60.0

kxky

26.52727.52828.52929.530

−2−1.5 −1−0.5 0 0.5 1 1.5 2 −1.5−1−0.500.5 1 1.5 2

050100150200250300350400450

tp = 1. t = 1. N = 6 L = 12 U = 60.0

kxky

050100150200250300350400450

Figure : Top : Weak U = 1.6 Bottom : Strong U = 3.8Left : Spin Centre : Currents Right : VBS

52 / 63


Technical : weak-coupling RG 1

RG goes beyond MFTneeds a small parameter → useful in weak-couplingO. Vafek did the N = 2 calculationSteps of RG :

zeroth-order effective quadratic Action S0

S0 =

∫dτd2r ψ(r, τ)

(∂τ +

∂2x − ∂2

y2m I⊗ σx −

∂x∂ym τz ⊗ σy

)ψ(r, τ)

Internal Indices of field configurations :Valley : τx , τy , τzLayer : σx , σy , σzSpin/Flavour

53 / 63



Steps of RG cont. :local spin SU(N) symmetric quartic interaction

Sint =12∑S,T

gST

∫dτd2r (ψ(r, τ) S ⊗ IN ψ(r, τ))×

(ψ(r, τ) T ⊗ IN ψ(r, τ))

Valley and Layer combine to give an SU(4) indexS and T above chosen as SU(4) generatorsNo. of coupling constants gST ? Naively, 16 + 16 ∗ 15/2 = 136Apply symmetries to reduce : a) Lattice, b) Time ReversalFinal no. of couplings : 9 gST s !!!Low energy projection of U Hubbard term3 out of 9 coupling constants are non-zero

54 / 63



Steps of RG cont. :RG step : Integrate out high energy modes in a thinmomentum shellapply Cumulant expansion perturbatively in gST

S< = S0< + 〈δS〉0> +12 (〈δS2〉0> − 〈δS〉20>) + . . .

Tree Level O(g) : 〈δS〉0> = 0 → MarginalityOne-Loop O(g2) : (〈δS2〉0> − 〈δS〉20>) → RG flows ofcoupling constantsGiven above, compute RG flows of infinitesimal sourcequadtratic “mass” terms ∝ ∆O ∫ dτd2r ψ†(r, τ)Oψ(r, τ)both in Charge and Spin channel to get RG prediction ...

55 / 63


Technical : det-QMC 1

QMC method for Interacting FermionsDo Suzuki-Trotter Decomposition (gives controllablesystematic errors)

Z = Tr[e−β(HU +Ht )

]= Tr

[(e−∆τHU e−∆τHt )m

]+ O(∆2

τ )

∆τ like an imaginary time; m = β/∆τ

Apply (discrete) Hubbard-Stratonovich (HS) Transformationto U Hubbard term

e−∆τU∑

i (ni,↑−1/2)(ni,↓−1/2) = C∑

{si}=±1eα∑

i si (ni,↑−ni,↓)

cosh(α) = exp(∆τU/2)

Decouples Interacting problem to a non-interacting one56 / 63



Have to sum over the HS variables {si} to compute for theinteracting problemDo this sum as a Monte-Carlo sumOne MC configuration corresponds to one realization of theHS variables {si}MC step : Importance sample over the configurations withweights W ({si})Usual way : Detailed Balance

W ({si}1) T ({si}1 → {si}2) = W ({si}2) T ({si}2 → {si}1)

57 / 63



Weight of configurations has determinantal structure

Z ∼Cm ∑{si}=±1

Tr[(eαc†V ({si})ce−∆τ c†Tc)β/∆τ

]= Cm ∑

{si}=±1det[I + eαV ({si}1)e−∆τT eαV ({si}2)e−∆τT . . .]

Above is the heavy-lifting stepEasier to see the determinant structure with Grassmanvariables

58 / 63



Measure various observables of interest

〈O〉 =TrOe−β(HU +Ht )

Tre−β(HU +Ht )

As a MC averageProjector version of det-QMC also there ...

〈O〉0 =〈Ψ0|O|Ψ0〉〈Ψ0|Ψ0〉

= limΘ→∞

〈Ψtrial|e−ΘHOe−ΘH |Ψtrial〉〈Ψtrial|e−2ΘH |Ψtrial〉

Sign-Problem free guaranteed by Particle-Hole Symmetrybipartite hoppings at half-fillingRepulsive Hubbard Interactions favouring half-filling

59 / 63


Time reversal

Defn : ΘSΘ† = −SRequires anti-unitarity of Θ time-reversal operator For spin-1/2fermionic operators, we get ... Θc↑ → −c†↓ Θc↓ → c†↑Current operators is I =

∑σ i(c†i ,σcj,σ − c†j,σci ,σ)

ΘIΘ† = −I

60 / 63


Our Results 13 : OLD Hypothesis of stable QBT

Can the QBT semi-metal be stable to Hubbard Interactions ?Hypothesis contrary to RG : let’s go on ...Non-magnetic =⇒ no gapless Goldstone spin modes at AFMwavector=⇒ gapless single particle excitations at QBT wavevector

61 / 63


Fermi-Liquid : Phase Space argument

62 / 63


det-QMC Extra 1 : Ratio Crossing Drifts

0

0.5

1

1.5

2

2.5

3

3.5

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

Uc

1/L

N = 2 11− ratio

0

2

4

6

8

10

12

14

16

18

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

Uc

1/L

N = 4 11− ratio

Figure : LEFT : N = 2 RIGHT : N = 4

Both N = 2 and N = 4 extrapolate to finite Uc

63 / 63

effect of hubbard interaction on quadratic band touching

Documents