Caution on Gross-Neveu criticality with a single Dirac cone Violation of locality and its consequence of unexpected finite-temperature transition Yuan Da Liao12Xiao Yan Xu3Zi Yang Meng4and Yang Qi125

2025-04-27 0 0 4.75MB 8 页 10玖币

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Caution on Gross-Neveu criticality with a single Dirac cone:

Violation of locality and its consequence of unexpected ﬁnite-temperature transition

Yuan Da Liao,1, 2 Xiao Yan Xu,3Zi Yang Meng,4, ∗and Yang Qi1, 2, 5, †

1State Key Laboratory of Surface Physics, Fudan University, Shanghai 200438, China

2Center for Field Theory and Particle Physics, Department of Physics, Fudan University, Shanghai 200433, China

3Key Laboratory of Artiﬁcial Structures and Quantum Control (Ministry of Education),

School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China

4Department of Physics and HKU-UCAS Joint Institute of Theoretical and Computational Physics,

The University of Hong Kong, Pokfulam Road, Hong Kong SAR, China

5Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China

(Dated: November 9, 2023)

Lately there are many SLAC fermion investigations on the (2+1)D Gross-Neveu criticality of a single Dirac

cone. While the SLAC fermion construction indeed gives rise to the linear energy-momentum relation for all

lattice momenta at the non-interacting limit, the long-range hopping and its consequent violation of locality on the

Gross-Neveu quantum critical point (GN-QCP) – which a priori requires short-range interaction – has not been

veriﬁed. Here we show, by means of large-scale quantum Monte Carlo simulations, that the interaction-driven

antiferromagnetic insulator in this case is fundamentally diﬀerent from that on a purely local 𝜋-ﬂux Hubbard

model on the square lattice. In particular, the antiferromagnetic long-range order has a ﬁnite temperature

continuous phase transition, which appears to violate the Mermin-Wagner theorem, and smoothly connects to

the previously determined GN-QCP. The magnetic excitations inside the antiferromagnetic insulator are gapped

without Goldstone mode, even though the state spontaneously breaks continuous 𝑆𝑈 (2)symmetry. These

unusual results point out the fundamental diﬀerence between the QCP in SLAC fermion and that of GN-QCP

with short-range interaction.

I. INTRODUCTION

Massless Dirac fermions are ubiquitously present as the

low-energy description of many condensed matter systems in-

cluding graphene [1], twisted bilayer graphene [2–5], d-wave

superconductors [6–9], algebraic spin liquid [6,7,10–17] and

the deconﬁned quantum criticality [18–30]; in high-energy

physics, the dynamical massless Dirac fermions in quantum

chromodynamics and the existence of a deconﬁned phase in

compact quantum electrodynamics have attracted great atten-

tions and remains unsolved [10,15,31–36]. Nonetheless, it is

generally believed that strong local interactions can generate

a ﬁnite mass for the Dirac fermions and spontaneously result

in a quantum phase transition [37–42]. The corresponding

quantum critical point (QCP) are typically described by the

Gross-Neveu (GN) university classes [43,44]. In particular,

a single Dirac cone, realized in the the SLAC fermion model

with long-range hopping in (2+1)D [45,46], was found to

give rise to an Ising-type ferromagnetic order that generates a

𝑍2symmetry-breaking mass gap [47], or an antiferromagnetic

Mott insulator that breaks the 𝑆𝑈 (2)spin rotational symme-

try [48]. The associated QCPs from Dirac semimetal (DSM)

to insulators are believed to belong to the (2+1)D chiral Ising

or Heisenberg GN universality classes.

The SLAC fermion construction gives rise to a linear

energy-momentum relation for all lattice momenta at the non-

interacting limit (shown in Fig. 1(a)), therefore reduces the

ﬁnite-size eﬀect suﬀered by other local cousins such as the

∗zymeng@hku.hk

†qiyang@fudan.edu.cn

honeycomb and 𝜋-ﬂux models where only a small region of

the Brillouin zone (BZ) displays the relativistic behavior at

low-energy. The fundamental diﬀerence of the SLAC fermion

model compared with its local cousins, i.e., the necessity of

avoiding the Nielsen-Ninomiya theorem [49–51] by violat-

ing locality on ﬁnite size lattices and the assumption that the

locality of the Dirac operator is recovered in the thermody-

namic limit (TDL), has not be investigated. This means, with

the long-range interactions in the SLAC fermion models (the

bare interaction is on-site but the long-range hopping medi-

ates long-range interaction), whether the GN transition and

the symmetry-breaking phases obtained thereafter can be dis-

cussed as if they were from a purely local model in the origin

sense of GN-QCP [43,44], are questionable.

This is the problem solved in this article. Here we show, by

means of large-scale QMC simulations, that the phase diagram

of the SLAC fermion model is fundamentally diﬀerent from

that of a purely local 𝜋-ﬂux Hubbard model on the square

lattice. In particular, we ﬁnd the antiferromagnetic insula-

tor (AFMI) phase in the SLAC fermion model exists at ﬁnite

temperatures, which appears to violate the Mermin-Wagner

theorem [52–54]. The AFMI phase emerges from the high-

temperature paramagnetic (PM) phase via a ﬁnite-temperature

continuous phase transition, and this continuous transition line

smoothly connects to the previously determined GN-QCP at

the ground state [48]. Contrary to the picture of the Mermin-

Wagner theorem, where the low-energy ﬂuctuation of the gap-

less Goldstone mode destroys the long-range order at any ﬁnite

temperature, we ﬁnd that the magnetic excitations inside the

AFMI are gapped without Goldstone mode, although the state

spontaneously breaks continuous 𝑆𝑈(2)symmetry.

Our results suggest that the long-range interaction in the

SLAC fermion model has altered the low-energy eﬀective

arXiv:2210.04272v2 [cond-mat.str-el] 7 Nov 2023

theory of the interacting Dirac fermions, the QCP of SLAC

fermion model is fundamentally diﬀerent from that of the

local-interaction ones in this way. We note that examples of

nonlocal interaction stabilizing ﬁnite-temperature symmetry-

breaking phases and giving rise to gapped Goldstone modes at

zero-temperature, have also been seen in 1D Ising and SLAC

fermion model [55,56] and 2D Heisenberg model [57–59], and

in dissipative systems such as 1D Ohmic spin chain [60,61]

and 2D dissipative quantum XY models [62,63].

(a)

10 15 20 25 30

0.1

0.2

0.3

0.4

0 5

0.5

(b)

10 15 20 25 30

0.1

0.2

0.3

0.4

0 5

0.5

(d)

(c)

DSM

AFMI

PM PM

AFMI

FIG. 1. The dispersion of (a) SLAC fermion and (c) free 𝜋-ﬂux model

in the ﬁrst BZ. The 𝑈-𝑇phase diagram of (b) SLAC fermion and (d)

𝜋-ﬂux Hubbard model obtained from QMC simulation. In panel (b),

the red squares are obtained from the cross of 𝑅for diﬀerent 𝐿when

scanning 𝑇at ﬁx 𝑈=7.5,8,9,10,12,14,16,20,22 and 24. The blue

square is obtained from the cross of 𝑅for diﬀerent 𝐿when scanning

𝑈at ﬁx 𝑇=1/3. The black star denotes the position of QCP in

Ref. [48]. The red dash line is a guide to the eye. In panel (d), the

black diamond denotes the position of GN-QCP in Ref. [40].

II. MODEL AND METHOD

We consider the spin-1/2 SLAC fermion and the 𝜋-ﬂux Hub-

bard model on the square lattice at half-ﬁlling for comparison.

The SLAC fermion Hubbard model has the Hamiltonian

𝐻SLAC =−𝑡

𝑖 𝑗 𝜎

(𝐴𝑖 𝑗 𝑐†

𝑖𝑎 𝜎 𝑐𝑗 𝑏 𝜎 +h.c.)+𝑈

2

𝑖

𝜆=𝑎,𝑏

(𝑛𝑖𝜆 −1)2,

(1)

where we set 𝑡=1as the energy unit, 𝑐†

𝑖𝑎 𝜎 and 𝑐𝑖𝑏 𝜎 are the

creation and annihilation operators for an electron at unit cell

𝑖on sublattices 𝑎, 𝑏 with spin 𝜎=↑,↓;𝑛𝑖𝜆 =Í𝜎𝑐†

𝑖𝜆 𝜎 𝑐𝑖𝜆𝜎

denotes the local particle number operator at sublattice 𝜆of

unit cell 𝑖;𝐴𝑖 𝑗 =𝑖(−1)𝑥𝜋

𝐿sin(𝑥 𝜋/𝐿)𝛿𝑦,0+(−1)𝑦𝜋

𝐿sin(𝑦 𝜋/𝐿)𝛿𝑥,0denotes the

electron hopping amplitude with r≡ (𝑥, 𝑦)=r𝑖−r𝑗standing

for the relative distance between two diﬀerent unit cells 𝑖and

𝑗,𝑥=1,· · · , 𝐿 −1with 𝐿the linear system size. The kinetic

term of 𝐻SLAC is known as the SLAC fermion [45], and the

corresponding single particle spectrum is 𝜀(k)=±|k|, which

results in a single linearly dispersing Dirac cone at momentum

𝚪=(0,0)point, as shown in Fig. 1(a). We observe that on

ﬁnite-size lattices, the corresponding Fermi velocity is reduced

to ±1in the BZ. However, the fermion velocity changes sign at

the BZ boundary, resulting in a sigularity. The violation of the

locality of SLAC fermion represents itself as singular values

at the BZ boundary. And previous works [47,48] assume the

locality of the Dirac operator is recovered at the TDL.

To make a proper comparison with the local model, we also

simulate the 𝜋-ﬂux Hubbard model with the Hamiltonian

𝐻𝜋-Flux =−𝑡

⟨𝑖 𝑗 ⟩, 𝜎

(𝐵𝑖 𝑗 𝑐†

𝑖 𝜎 𝑐𝑗 𝜎 +h.c.) + 𝑈

2

𝑖

(𝑛𝑖−1)2,(2)

where hopping amplitudes 𝐵𝑖,𝑖+ ®𝑒𝑥=1and 𝐵𝑖,𝑖+®𝑒𝑦=(−1)𝑖𝑥,

the position of site 𝑖is given as r𝑖=𝑖𝑥®𝑒𝑥+𝑖𝑦®𝑒𝑦, such arrange-

ment bestows a 𝜋-ﬂux penetrating each square plaquette (the

dispersion is given in Fig. 1(c)). It is known that the 𝜋-ﬂux

model has a chiral Heisenberg GN-QCP at 𝑈𝑐=5.65(5)[39–

41,48], and the AFMI at 𝑈 > 𝑈𝑐spontaneously breaking

the spin 𝑆𝑈 (2)symmetry with Goldstone mode located at

M=(𝜋, 𝜋)point (see Fig. 1(d)).

We employ the projection QMC (PQMC) [64] method to

study the ground-state and dynamical spin correlation func-

tions and the ﬁnite temperature QMC (FTQMC) [65,66]

method to study the temperature dependence of the phys-

ical observables. These results give rise to a consistent

and complementary picture. For PQMC method, we can

measure a physical observable ⟨𝑂⟩according to ⟨𝑂⟩=

limΘ→∞ Ψ𝑇

𝑒−Θ

2𝐻𝑂𝑒−Θ

2𝐻

Ψ𝑇

⟨Ψ𝑇|𝑒−Θ𝐻|Ψ𝑇⟩, where Θis the projection length;

|Ψ𝑇⟩is the trial wave function; and |Ψ0⟩=limΘ→∞ 𝑒−𝜃

2𝐻|Ψ𝑇⟩

is the ground state wave function. For FTQMC method,

⟨𝑂⟩can be measured according to ⟨𝑂⟩=Tr[e−𝛽 𝐻 𝑂]

Tr[e−𝛽 𝐻 ], where

𝛽=1/𝑇is the inverse of temperature. We use discrete

Θ = 𝑀Δ𝜏(𝛽=𝑀Δ𝜏) and perform a Trotter decomposi-

tion for PQMC (FTQMC) method, and set Δ𝜏=0.1and

projection time Θ = 2𝐿+10 for 𝐻SLAC and Θ = 𝐿+10

for 𝐻𝜋-Flux when measuring imaginary-time physical quanti-

ties, and, in FTQMC method, we set Δ𝜏=0.01 for measure-

ment. With the aid of particle-hole symmetry, the PQMC

and FTQMC for 𝐻SLAC and 𝐻𝜋-Flux models are all sign-

problem free [40,48,64,67,68]. We have simulated the

square lattice system with 𝑁=2𝐿2sites and the linear size

𝐿=5,7,· · · ,19 for 𝐻SLAC, while 𝑁=𝐿2sites and the linear

size 𝐿=4,8,· · · ,32 for 𝐻𝜋-Flux.

III. RESULTS

We ﬁrst reveal the ﬁnite temperature continuous phase tran-

sition of the AFMI phase in 𝐻SLAC, with the phase boundary

determined as shown in Fig. 1(b). Here we use one vertical

scan with ﬁxed 𝑈=16 and varying 𝑇and one horizontal scan

with ﬁxed 𝑇=1/3and varying 𝑈, to demonstrate the generic

behavior. Fig. 1(d) are the 𝑈−𝑡phase diagram of 𝜋-ﬂux Hub-

bard model [40], we notice that there is no ﬁnite temperature

phase transition.

Ref. [48] investigated the ground state phase diagram of

𝐻SLAC. Following their approach, we deﬁne the AFMI spin

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CautiononGross-NeveucriticalitywithasingleDiraccone:Violationoflocalityanditsconsequenceofunexpectedfinite-temperaturetransitionYuanDaLiao,1,2XiaoYanXu,3ZiYangMeng,4,∗andYangQi1,2,5,†1StateKeyLaboratoryofSurfacePhysics,FudanUniversity,Shanghai200438,China2CenterforFieldTheoryandParticlePhysics,Depar...

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Caution on Gross-Neveu criticality with a single Dirac cone Violation of locality and its consequence of unexpected finite-temperature transition Yuan Da Liao12Xiao Yan Xu3Zi Yang Meng4and Yang Qi125

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