Upper critical dimension of the 3-state Potts model Shai M. Chester Jeerson Physical Laboratory Harvard University Cambridge MA 02138 USA

2025-05-06 0 0 703.66KB 8 页 10玖币

侵权投诉

Upper critical dimension of the 3-state Potts model

Shai M. Chester

Jeﬀerson Physical Laboratory, Harvard University, Cambridge, MA 02138, USA

Center of Mathematical Sciences and Applications,

Harvard University, Cambridge, MA 02138, USA

Ning Su

Department of Physics, University of Pisa, I-56127 Pisa, Italy

We consider the 3-state Potts model in d≥2 dimensions. For dless than the upper critical

dimension dcrit, the model has a critical and a tricritical ﬁxed point. In d= 2, these ﬁxed points

are described by minimal models, and so are exactly solvable. For d > 2, however, strong coupling

makes them diﬃcult to study and there is no consensus on the value of dcrit. We use the numerical

conformal bootstrap to compute critical exponents of both the critical and tricritical ﬁxed points

for general d. In d= 2 our results match the expected values, and as we increase dwe ﬁnd that the

critical exponents of each ﬁxed point get closer until they merge near dcrit .2.5.

I. INTRODUCTION

Lattice models in ddimensions [64] are useful descrip-

tions of many physical systems such as magnets and su-

perﬂuids. For a range of d, these models can be tuned

to undergo an interacting second order phase transition,

which is described by a unitary conformal ﬁeld theory

(CFT). For instance, the Ising model has a phase tran-

sition described by a CFT in 2 ≤d≤4, but in d≥4

the CFT becomes a free theory. The largest value of d

such that a theory ﬂows to an interacting unitary CFT

is called the upper critical dimension. Surprisingly, the

upper critical dimension is still unknown for the simplest

lattice model after the Ising model: the 3-state Potts

model. We will address this question using the modern

conformal bootstrap.

The q-state Potts model is deﬁned on a square lattice

of random spins siby the partition function [1]

Z=X

{si}

e−H[{si}], H[{si}] = βX

hiji

δsi,sj,(1)

where βis the inverse temperature. At large tempera-

ture this model has a disordered phase with one ground

state with Sqsymmetry, while at small temperature Sq

is broken and one spin value is preferred. We can tune

β=βcrit to get a phase transition called the critical Potts

model. If we allow some lattice sites to be vacant [65],

we can tune the chemical potential of these vacancies

along with βto get another transition called the tricriti-

cal Potts model. In d= 2, both the tricritical and critical

phase transitions are second order for q≤qcrit = 4 [2]

[66], and so are described by a conformal ﬁeld theory.

In fact, as q→qcrit the critical exponents of each theory

merge and then go oﬀ into the complex plane [3–6], which

is an example of the merger and annihilation scenario of

CFTs [7].

The q= 3 model is of special experimental and the-

oretical interest for general d. The critical model has

applications in d= 2 to 4He atoms on graphite [8–10].

(dcrit, qcrit) : (2,4) (2.32,2.85) (2.5,2.68) (3,2.45)

(3,2.21)

Ref. [2] [16] [17] [15]

TABLE I: Previous estimates for the upper critical dimension

dcrit for various q. For d= 2 the result is exact, while the

d > 2 cases come from lattice studies.

In d= 2, the critical and tricritical CFTs are described

by minimal models, and so are exactly solvable [11,12].

In d∼6 one can write down weakly coupled Landau-

Ginzburg Lagrangians for these theories using a complex

ﬁeld [13,14], but this description is very strongly coupled

in the 2 ≤d < 3 regime of interest [67], where we re-

quire non-perturbative methods. In d= 3, lattice Monte

Carlo simulations suggest that the q= 3 critical model

is ﬁrst order, so there is no CFT description, and in fact

qcrit ∼2.45 [15] [68]. For 2 < d < 3, the upper critical

dimension dcrit for q= 3 has not been determined, and

it is not even known if the critical and tricritical models

disappear via the merger and annihilation scenario [69].

Lattice simulations in fractional dimensions give predic-

tions for (dcrit, qcrit) that are listed in Table I, but there

is no consensus for dcrit when q= 3. It is also not clear if

the fractional dlattices used in these studies are analyt-

ically connected to the integer dﬁxed points, as indeed

some of the diﬀerent fractional dstudies in Table Igive

conﬂicting answers in d= 3.

In this paper we will use the numerical conformal boot-

strap [18] [70] to estimate dcrit for q= 3. This method

bounds the allowed values of critical exponents for any

CFT with a given symmetry, by applying a ﬁnite trun-

cation of the inﬁnite constraints of crossing equations for

a given set of four-point functions. Special features in

the space of the allowed region often correspond to phys-

ical theories. For instance, a kink in the space of allowed

critical exponents of O(N) invariant theories was found

to match Monte Carlo predictions for the critical O(N)

models [19,20], and this correspondence was later con-

arXiv:2210.09091v3 [hep-th] 7 Dec 2022

ﬁrmed by more sophisticated bootstrap studies that re-

stricted the allowed region to a tiny island around the

kink [21–24].

To bootstrap the 3-state critical Potts model, we con-

sider correlations functions of the two S3charged opera-

tors and the one singlet operator that were relevant in the

known d= 2 theory [71]. We ﬁnd that the allowed region

in the space of the critical exponents of these three oper-

ators is roughly approximated by a cone for each d. The

numerics are extremely intensive, as they require scan-

ning over not only the critical exponents, but also their

various three-point structures (i.e. OPE coeﬃcients). To

eﬃciently ﬁnd the tip of the cone, we use the recently

developed Navigator method [25] to minimize the lowest

lying charged relevant critical exponent, which is much

faster than exploring the full allowed region. In d= 2, the

cone tip precisely matches the known minimal model val-

ues, which motivates our conjecture that the tip in d > 2

continues to describe the physical theory. The bootstrap

setup for the tricritical theory is similar, except there is

now an extra relevant singlet operator, so we scan over a

four-dimensional space and again ﬁnd a four-dimensional

generalization of the tip in general dthat matches the

known minimal model in d= 2. As we increase dabove

2, we ﬁnd that the tips identiﬁed with the critical and

tricritical theories get closer to each other until around

d∼2.5, where the theories merge, and in particular the

extra singlet operator in the tricritical theory becomes

marginal. Our results are summarized by Figure 2, and

we will discuss two possibilities for dcrit .2.5 based on

this data.

The rest of this paper is organized as follows. In Sec-

tion II, we review properties of the conformal ﬁeld theo-

ries that describe the critical and tricritical ﬁxed points

in general d, starting with the exactly known d= 2 min-

imal model descriptions. In Section III, we describe our

bootstrap setup, how to numerically implement it using

the Navigator, and the resulting estimates for the criti-

cal exponents. We end with a discussion of our results in

Section IV.

II. CRITICAL AND TRICRITICAL CFTS

We begin by reviewing the CFTs that describe the crit-

ical and tricritical ﬁxed points of the 3-state Potts model.

Operators in these theories transform in representations

of the global conformal symmetry group SO(d, 2), as la-

beled by scaling dimension ∆ and spin `, as well as rep-

resentations rof the ﬂavor group S3. The S3symmetry

group has three irreducible representations: the dimen-

sion two fundamental 1(i.e. the charged), the dimension

1 singlet 0+, and the dimension 1 sign representation

0−. By deﬁnition, the critical theory has one relevant

spin zero singlet , while the tricritical has an extra rel-

evant spin zero singlet 0. As unitary local CFTs, they

must also contain a conserved stress tensor operator with

∆ = dand spin 2. A priori, this is all we know about

these CFTs in general d≥2.

For d= 2, the global conformal symmetry group

SO(d, 2) is enhanced to a pair of inﬁnite dimensional Vi-

rasoro groups Vl⊗ Vr. This enhanced symmetry group

was used in [11,12] to solve a set of theories known as Vi-

rasoro minimal models Mp,q labeled by coprime integers

q > p > 2 and central charge

cp,q = 1 −6(p−q)2

pq .(2)

The partition functions of these CFTs must also be mod-

ular invariant, and there are diﬀerent ways of imposing

this constraint for a given (p, q), which yield a diﬀerent

spectrum of operators and thus a diﬀerent theory. When

q=p+ 1 and p < 5, there is a unique invariant that

requires the weights hl,r under Vl,r to be identical. These

theories are the free theory for p= 2, and the p-critical

Ising model for p > 2, e.g. p= 2 is the critical Ising, p= 3

is the tricritical Ising, etc. These minimal models have

aZ2invariant Landau-Ginzburg Lagrangian in terms of

a real scalar ﬁeld φwith potential φ2(p−1). The upper

critical dimension of these theories can thus be trivially

determined by demanding that φ2(p−1) is marginal, i.e.

that the theory merge and annihilate with the free theory,

which ﬁxes [72]

crit = 2p−1

p−2.(3)

For instance, for the critical Ising model with p= 3, we

recover the expected d3

crit = 4.

For p≥5, there is now a non-diagonal modular invari-

ant where hl,r can be diﬀerent. For p= 5 this yields the

critical Potts model, with Virasoro primaries [73]

I = [0,0,0+], σ = [2/15,0,1], σ0= [4/3,0,1],

ε= [4/5,0,0+], ε0= [14/5,0,0+], ε2= [6,0,0+],

γ= [9/5,1,0−], W = [3,3,0−]

(4)

labelled as [∆, `, r], and these Virasoro primaries generate

an inﬁnite set of quasiprimaries under the global confor-

mal group SO(2,2), such as the stress tensor. Note that

there is only one relevant singlet scalar, ε, as expected

for a critical ﬁxed point, there are two relevant charged

scalars σand σ0, and no relevant scalars in the 0−. The

OPE coeﬃcients have also been computed and are given

in [26], but we will not make use of them. For p= 6,

the non-diagonal minimal model is the tricritical Potts

model, with Virasoro primaries [74]

I = [0,0,0+], σ = [2/21,0,1], σ0= [20/21,0,1],

σ2= [8/3,0,1], ε = [2/7,0,0], ε0= [10/7,0,0],

ε2= [24/7,0,0+], ε3= [44/7,0,0+], ε4= [10,0,0+],

γ= [17/7,1,0−], γ0= [23/7,3,0−], W = [5,5,0−].

(5)

There are now two relevant singlets, εand ε0, as we ex-

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

Uppercriticaldimensionofthe3-statePottsmodelShaiM.ChesterJeersonPhysicalLaboratory,HarvardUniversity,Cambridge,MA02138,USACenterofMathematicalSciencesandApplications,HarvardUniversity,Cambridge,MA02138,USANingSuDepartmentofPhysics,UniversityofPisa,I-56127Pisa,ItalyWeconsiderthe3-statePottsmodelind...

展开>> 收起<<

Upper critical dimension of the 3-state Potts model Shai M. Chester Jeerson Physical Laboratory Harvard University Cambridge MA 02138 USA.pdf

共8页,预览2页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Upper critical dimension of the 3-state Potts model Shai M. Chester Jeerson Physical Laboratory Harvard University Cambridge MA 02138 USA

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: