effect of hubbard interaction on ``non-relativistic
TRANSCRIPT
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
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Introduction Mean-Field Theory Existing Results Our Results Conclusion
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)
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Introduction Mean-Field Theory Existing Results Our Results Conclusion
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
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Introduction Mean-Field Theory Existing Results Our Results Conclusion
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”
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Introduction Mean-Field Theory Existing Results Our Results Conclusion
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
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Introduction Mean-Field Theory Existing Results Our Results Conclusion
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
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Introduction Mean-Field Theory Existing Results Our Results Conclusion
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
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Introduction Mean-Field Theory Existing Results Our Results Conclusion
Mean-Field Theory 2
Kekule Current Ansatz
0
0
0
0
0
Current Order at QBT wave-vectorOpens a gap → Not Semi-metallic
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Introduction Mean-Field Theory Existing Results Our Results Conclusion
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
N = 4;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
J
U
N = 4;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
J
U
N = 4;L = 12 : Map of columnar VBS order
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
J
U
N = 4;L = 12 : Map of columnar VBS order
−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
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Introduction Mean-Field Theory Existing Results Our Results Conclusion
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
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Introduction Mean-Field Theory Existing Results Our Results Conclusion
Existing Results : weak-coupling RG 2
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)
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Introduction Mean-Field Theory Existing Results Our Results Conclusion
Existing Results : weak-coupling RG 3
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
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Introduction Mean-Field Theory Existing Results Our Results Conclusion
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
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Introduction Mean-Field Theory Existing Results Our Results Conclusion
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).
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Introduction Mean-Field Theory Existing Results Our Results Conclusion
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
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Introduction Mean-Field Theory Existing Results Our Results Conclusion
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
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Introduction Mean-Field Theory Existing Results Our Results Conclusion
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
Introduction Mean-Field Theory Existing Results Our Results Conclusion
Our Results : det-QMC 2 : structure factors
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
Figure: LEFT : Weak Coupling RIGHT : Strong Coupling
There is no Bragg peak at QBT !!Where are the currents ???
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Introduction Mean-Field Theory Existing Results Our Results Conclusion
Our Results : det-QMC 3 : structure factors
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
Figure: LEFT : Weak Coupling RIGHT : Strong Coupling
No AFM order in weak-coupling as expected ..No AFM order in strong-coupling unlike N = 4 ..What is the order in strong coupling ?
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Introduction Mean-Field Theory Existing Results Our Results Conclusion
Our Results : det-QMC 4 : structure factors
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
Figure: LEFT : Weak Coupling RIGHT : Strong Coupling
N = 6 strong coupling is VBS ordered.Are there Currents at weak coupling ?? ..
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Introduction Mean-Field Theory Existing Results Our Results Conclusion
Our Results : det-QMC 5 : structure factors
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
Figure: LEFT : Weak Coupling RIGHT : Strong Coupling
No Bragg peak in weak coupling again !!What is this mysterious state ??
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Introduction Mean-Field Theory Existing Results Our Results Conclusion
Our Results : det-QMC 6 : structure factors
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
Figure: LEFT : Weak Coupling RIGHT : Strong Coupling
No AFM Bragg peak in weak coupling again !Is the mysterious state a stable QBT semi-metal ??
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Introduction Mean-Field Theory Existing Results Our Results Conclusion
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
Introduction Mean-Field Theory Existing Results Our Results Conclusion
Our Results : det-QMC 8 : Binder ratios
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
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Introduction Mean-Field Theory Existing Results Our Results Conclusion
Our Results : det-QMC 9 : Binder ratios
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
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Introduction Mean-Field Theory Existing Results Our Results Conclusion
Our Results : det-QMC 10 : Binder ratios
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
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Introduction Mean-Field Theory Existing Results Our Results Conclusion
Our Results : det-QMC 11 : Binder ratios
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
Figure: LEFT : Weak Coupling RIGHT : Strong Coupling
VBS in Strong Coupling as seen beforeNo VBS in Weak Coupling as seen before
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Introduction Mean-Field Theory Existing Results Our Results Conclusion
Our Results : det-QMC 12 : Binder ratios
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
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Introduction Mean-Field Theory Existing Results Our Results Conclusion
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
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Introduction Mean-Field Theory Existing Results Our Results Conclusion
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
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Introduction Mean-Field Theory Existing Results Our Results Conclusion
Our Results : det-QMC 15 : Spin Gaps
N = 2 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 = 2
U = 2.U = 4.
Figure: Strong and Weak shown together
Strong U : gapless spin excitations at AFM wavevectorConsistent with AFM order with gapless Goldstone modesWeak U : no gapless modesConsistent with NO AFM order (IMP)
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Introduction Mean-Field Theory Existing Results Our Results Conclusion
Our Results : det-QMC 16 : Spin Gaps
N = 6 Spin Gap at AFM wavevector
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.
Figure: Strong and Weak shown together
Strong and Weak U : no gapless modesConsistent with no AFM (no SU(N) symm breaking)
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Introduction Mean-Field Theory Existing Results Our Results Conclusion
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
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Introduction Mean-Field Theory Existing Results Our Results Conclusion
Our Results : det-QMC 18 : Single Particle Gaps
N = 2 Single Particle Gap at QBT wavevector
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.
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 again)Not Inconsistent with gapless QBT
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Introduction Mean-Field Theory Existing Results Our Results Conclusion
Our Results : det-QMC 19 : Single Particle Gaps
N = 6 Single Particle Gap at QBT wavevector
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.
Figure: LEFT : Strong and Weak RIGHT : Only Weak
Strong Coupling : QBT is gapped outConsistent with insulating VBS orderWeak Coupling : Tending towards 0 (Note y-axis again)Not Inconsistent with gapless QBT
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Introduction Mean-Field Theory Existing Results Our Results Conclusion
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
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Introduction Mean-Field Theory Existing Results Our Results Conclusion
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
Introduction Mean-Field Theory Existing Results Our Results Conclusion
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 ?
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Introduction Mean-Field Theory Existing Results Our Results Conclusion
Thank you !Acknowledgements :
Lexington Cond-Matt Group : J. D’emidio, G. MurthyO. VafekNSF for financial supportXSEDE for Computational Resources
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Introduction Mean-Field Theory Existing Results Our Results Conclusion
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
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Introduction Mean-Field Theory Existing Results Our Results Conclusion
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.
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Introduction Mean-Field Theory Existing Results Our Results Conclusion
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
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Introduction Mean-Field Theory Existing Results Our Results Conclusion
det-QMC Extra 4 : 3D Structure Factor Plots N = 2
−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
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Introduction Mean-Field Theory Existing Results Our Results Conclusion
det-QMC Extra 5 : 3D Structure Factor Plots N = 6
−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
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Introduction Mean-Field Theory Existing Results Our Results Conclusion
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
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Introduction Mean-Field Theory Existing Results Our Results Conclusion
Technical : weak-coupling RG 2
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
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Introduction Mean-Field Theory Existing Results Our Results Conclusion
Technical : weak-coupling RG 3
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 ...
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Introduction Mean-Field Theory Existing Results Our Results Conclusion
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
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Introduction Mean-Field Theory Existing Results Our Results Conclusion
Technical : det-QMC 2
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)
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Introduction Mean-Field Theory Existing Results Our Results Conclusion
Technical : det-QMC 3
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
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Introduction Mean-Field Theory Existing Results Our Results Conclusion
Technical : det-QMC 4
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