effect of hubbard interaction on ``non-relativistic

Introduction Mean-Field Theory Existing Results Our Results Conclusion

Effect of Hubbard Interaction on “non-relativistic”Quadratic Band Touching (QBT) semi-metal in

Bernal-stacked Honeycomb Bilayer

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

University of Kentucky

January, 2016

1 / 51


Outline of the Talk

Problem Statement (1 slide)Motivation (3 slides)Mean-Field Theory (3 slides)Previous Results on N = 2 : 1) Weak-coupling RG (4 slides)2) det-QMC (2 slides)Our Results on general N : 1) Weak-coupling RG (1 slide) 2)det-QMC (21 slides)Conclusion (2 slides)

2 / 51


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

3 / 51


Motivation 1 : “Non-relativistic” semi-metal (QBT)

Relevant to Graphene Material

G

QBT2QBT1

Figure: LEFT : Spectrum RIGHT : Brillouin Zone

Two Fermi Points in the Brillouin ZoneA Different kind of Semi-metal dubbed “non-relativistic”

4 / 51


Motivation 2 : 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)(iτ0 ∂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

5 / 51


Motivation 2 : 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, U → b1−dU → Interactions areirrelevant → StabilityHeuristically,

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

6 / 51


Mean-Field Theory 1

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

7 / 51


Mean-Field Theory 2

Kekule Current Ansatz

0

0

0

0

0

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

8 / 51


Mean-Field Theory 3

Method : Self-consistent Hartree-Fock MFT

N = 2

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

Neel

Neel

J

U

N = 2;L = 12 : Map of columnar VBS order

−0.0002

0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0.0014

0.0016

0.0018

N = 4

0

5

10

15

20

0 5 10 15 20

VBS

Neel

0

5

10

15

20

0 5 10 15 20

VBS

Neel

J

U


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

J

U


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

J

U


−1

−0.8

−0.6

−0.4

−0.2

0

0.2

J

U


−1

−0.8

−0.6

−0.4

−0.2

0

0.2

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

9 / 51


Existing Results : weak-coupling RG 1

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

10 / 51



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)

11 / 51



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

12 / 51


Existing Results : det-QMC 1

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

13 / 51


Existing Results : det-QMC 2

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).

14 / 51


Our Results : weak-coupling RG

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

15 / 51


Our Results : det-QMC 0

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

16 / 51


Our Results : det-QMC 1 : 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

Γ

QBT1 QBT2

ky

kx

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

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

Γ

QBT1 QBT2

ky

kx

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

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 ... 17 / 51



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 ???

18 / 51



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 ?

19 / 51



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 ?? ..

20 / 51



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 ??

21 / 51



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 a stable QBT semi-metal ??

22 / 51


Our Results : det-QMC 7 : 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

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 14− ratio

Figure: LEFT : Ratio Crossing RIGHT : Crossing Drifts

AFM in Strong Coupling as seen beforeNo AFM order in Weak Coupling as seen before 23 / 51



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

24 / 51



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

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 14− ratio

Figure: LEFT : Ratio Crossing RIGHT : Crossing Drifts

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

25 / 51



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

26 / 51



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

27 / 51



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-coupling

28 / 51


Our Results 13 : 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

29 / 51


Our Results : det-QMC 14 : 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

30 / 51




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)

31 / 51




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)

32 / 51


Our Results : det-QMC 17 : 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 QBT

33 / 51




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 QBT

34 / 51




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 QBT

35 / 51


Our Results : det-QMC 20 : AFM Order ParameterScalings

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

36 / 51


Conclusion 1 : What about RG result ?

It is a one-loop O(g2) resultRunaway flow predictions will violate O(g2) eventuallyPerhaps QBT stable when O(g3) and higher orders taken into account ..

g1

g2

QBT

Exotic Scenario (has been seen by G. Murthy in differentcontext)Another possibility : An intermediate Interacting Fixed point

Strong Coup.QBTInt. FP gc

Another possibility : finite t⊥/t vs. t⊥/t →∞37 / 51


Conclusion 2 : A Stable QBT for N = 2, 4, . . .

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

QBT stable to Hubbard Interaction for general Ngapless single particle excitationsNone of the possible dominant weak-coupling instabilitiesobservedRemark : For longer-range interactions (forward scatteringlimit) a gapless nematic state possible (K. Yang and O. Vafek)

To do :Check for quadratic band in Single particle gapsQuantify order parameter scalings in QBT phaseQuantify Criticality in det-QMCLonger term : What theory describes the transition ?

38 / 51


Thank you !Acknowledgements :

Lexington Cond-Matt Group : J. D’emidio, G. MurthyO. VafekNSF for financial supportXSEDE for Computational Resources

39 / 51


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

40 / 51


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.

41 / 51


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

42 / 51



−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

43 / 51



−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

44 / 51


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

45 / 51



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

46 / 51



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 ...

47 / 51


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 one

48 / 51



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)

49 / 51



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

50 / 51



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

51 / 51

effect of hubbard interaction on ``non-relativistic

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