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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
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Introduction Existing Results Our Results Conclusion
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
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Introduction Existing Results Our Results Conclusion
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
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Introduction Existing Results Our Results Conclusion
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
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Introduction Existing Results Our Results Conclusion
The Big Picture : Approaches to Condensed Matter
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Introduction Existing Results Our Results Conclusion
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
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Introduction Existing Results Our Results Conclusion
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
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Introduction 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 Existing Results Our Results Conclusion
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
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Introduction Existing Results Our Results Conclusion
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
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Introduction Existing Results Our Results Conclusion
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
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Introduction Existing Results Our Results Conclusion
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
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Introduction Existing Results Our Results Conclusion
Mean-Field Theory
Kekule Current Ansatz
0
0
0
0
0
Current Order at QBT wave-vectorOpens a gap → Not Semi-metallic
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Introduction Existing Results Our Results Conclusion
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
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
JU
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
MFT → QBT unstable to AFM order for all N and UAFM order opens a gap at QBT → Insulating state
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Introduction Existing Results Our Results Conclusion
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
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Introduction Existing Results Our Results Conclusion
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)
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Introduction Existing Results Our Results Conclusion
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
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Introduction Existing Results Our Results Conclusion
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
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Introduction Existing Results Our Results Conclusion
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).
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Introduction Existing Results Our Results Conclusion
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
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Introduction Existing Results Our Results Conclusion
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
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Introduction Existing Results Our Results Conclusion
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
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Introduction Existing Results Our Results Conclusion
Our Results : det-QMC : 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 Existing Results Our Results Conclusion
Our Results : det-QMC : 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 Existing Results Our Results Conclusion
Our Results : det-QMC : 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 Existing Results Our Results Conclusion
Our Results : det-QMC : 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 Existing Results Our Results Conclusion
Our Results : det-QMC : 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 perhaps a stable QBT semi-metal ??
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Introduction Existing Results Our Results Conclusion
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
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Introduction Existing Results Our Results Conclusion
Our Results : det-QMC : 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 Existing Results Our Results Conclusion
Our Results : det-QMC : 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
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 Existing Results Our Results Conclusion
Our Results : det-QMC : 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 Existing Results Our Results Conclusion
Our Results : det-QMC : 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 Existing Results Our Results Conclusion
Our Results : det-QMC : 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-couplingWhat is this mysterious state in weak-coupling ??
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Introduction Existing Results Our Results Conclusion
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
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Introduction Existing Results Our Results Conclusion
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
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Introduction Existing Results Our Results Conclusion
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
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Introduction Existing Results Our Results Conclusion
Our Results : det-QMC : 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 Existing Results Our Results Conclusion
Our Results : det-QMC : 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 Existing Results Our Results Conclusion
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
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Introduction Existing Results Our Results Conclusion
Our Results : det-QMC : 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 state
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Introduction Existing Results Our Results Conclusion
Our Results : det-QMC : 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 state
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Introduction Existing Results Our Results Conclusion
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)
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Introduction Existing Results Our Results Conclusion
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
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Introduction Existing Results Our Results Conclusion
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)
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Introduction Existing Results Our Results Conclusion
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 ??
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Introduction Existing Results Our Results Conclusion
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
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Introduction Existing Results Our Results Conclusion
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
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Introduction Existing Results Our Results Conclusion
Self-Energy with Cubic Corrections to QBT
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Introduction 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 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 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 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 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 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 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 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 one56 / 63
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Introduction 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 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 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
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Introduction Existing Results Our Results Conclusion
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
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Introduction Existing Results Our Results Conclusion
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
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Introduction Existing Results Our Results Conclusion
Fermi-Liquid : Phase Space argument
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Introduction 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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