算子展开揭示热奇数维共形场论的非平凡动力学与大N真空结构

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本文主要探讨了"算子乘积展开反演公式在有限温度下奇数维度玻色子和费米子共形场论中的动态作用"。作者Anastasios C. Petkou和Andreas Stergiou，分别来自希腊雅典的亚里士多德大学理论物理研究所和瑞士日内瓦的欧洲核子研究组织，他们在2018年6月提交并发表了一项关键成果。他们利用算子乘积展开（Operator Product Expansion, OPE）的反演公式，对这些高维量子场论的热两点函数进行了深入研究。OPE是量子场论中一种重要的工具，它用于近似表示两个算子在邻近位置的行为，通过将它们视为无穷乘积的形式。在有限温度环境下，这个技术被用来分析共形场论的动力学性质，特别是其算子谱，即理论中可能存在的基本物理量。研究发现，当热质量（即温度依赖的尺度参数）满足特定的代数超越方程时，会形成非平凡的热共形场论。这种超越方程确保了算子谱中不存在无限多个算子，这是理论的稳定性标志，避免了过于复杂的物理行为。对于任意奇数维度，这些间隙方程的解并非简单，而是通常表现为复数形式，并且展现出了独特的模式。作者认为，这种模式揭示了在零温度下，对应于奇数维共形场论的大型N（N代表某个量子数或自由度数量）真空结构。大型N极限在某些理论中是一个强大的简化工具，它允许在一定程度上控制复杂数值计算，从而更好地理解量子场论的行为。总结来说，这篇论文不仅提供了关于有限温度共形场论的新见解，还可能为后续的研究提供有价值的基础，特别是在探索量子场论在不同维度和温度条件下的行为，以及大型N极限在理论分析中的应用。这一工作对于理解复杂量子系统的动力学和结构具有重要意义。

Dynamics of Finite-Temperature Conformal Field Theories

from Operator Product Expansion Inversion Formulas

Anastasios C. Petkou

1,2

and Andreas Stergiou

Institute of Theoretical Physics, Aristotle University of Thessaloniki, Thessaloniki 54124, Greece

Theoretical Physics Department, CERN, 1211 Geneva 23, Switzerland

(Received 26 June 2018; published 17 August 2018)

We apply the operator product expansion inversion formula to thermal two-point functions of bosonic

and fermionic conformal field theories in general odd dimensions. This allows us to analyze in detail the

operator spectrum of these theories. We find that nontrivial thermal conformal field theories arise when the

thermal mass satisfies an algebraic transcendental equation that ensures the absence of an infinite set of

operators from the spectrum. The solutions of these gap equations for general odd dimensions are in general

complex numbers and follow a particular pattern. We argue that this pattern unveils the large-N vacuum

structure of the corresponding theories at zero temperature.

DOI: 10.1103/PhysRevLett.121.071602

Introduction.—The description of critical systems in

nontrivial backgrounds requires data not present in the

plane geometry. Perhaps the simplest example is that of

conformal field theories (CFTs) on S

× R

d−1

, with β the

radius of the circle, that describe finite-size or finite-

temperature critical systems. In such a case, the two-point

function of a scalar operator ϕðxÞ will in principle depend

on the one-point functions of all operators that appear in its

operator product expansion (OPE) with itself, since the

latter can be nonzero. In particular, for an operator OðxÞ

with dimension Δ

we schematically have hOðxÞi

×R

d−1

∝

=β

, where b

is a dimensionless parameter.

In d ¼ 2 the plane is conformally related to the cylinder

and, although one-point functions of conformal primaries

vanish on the latter, there exist operators such as the energy-

momentum tensor which transform anomalously under a

conformal map. This fixes their one-point functions on the

cylinder, and therefore the CFT data on R

determine the

finite-size or finite-temperature corrections to correlation

functions on S

× R [1,2].

For d>2 there is no conformal transformation between

and S

× R

d−1

, and generically one needs to find other

ways to determine the additional data b

. A first step in this

direction was described in Ref. [3], where the leading

anisotropic finite-size corrections to the two-point function

of scalars in R

were connected to the ratio of the thermal

free-energy density of the system and the normalization C

of the energy-momentum tensor two-point function. An

extension of these ideas to the nontrivial 3D OðNÞ vector

model was performed in Refs. [4,5], where the relevance of

the planar OPE to the description of the finite-size or finite-

temperature CFTs was demonstrated. In the latter works the

crucial point was that parameters such as b

were inde-

pendently determined by the gap equation of the vector

model. In particular, once the bosonic thermal mass was

determined, all one-point functions could be evaluated and

hence the full finite-temperature two-point function could

be reconstructed.

In more recent developments, the improved understand-

ing of CFTs on R

using numerical and analytic bootstrap

methods (see Ref. [6] for a recent review) calls for an

extension of these advances to finite-size or finite-temper-

ature critical systems. In this context an interesting work

has recently appeared [7], whose main result is a Lorentzian

inversion formula for the thermal two-point function of a

scalar ϕðxÞ with dimension Δ

. Using the OPE one can

show that the Euclidean position-space [8] thermal two-

point function takes the generic form

hϕðxÞϕð0Þi

≡ gðr; cos θÞ¼





ðcos θÞ

2Δ

;

ð1Þ

where x

¼ðτ; xÞ are coordinates on S

× R

d−1

with

period τ ∼ τ þ β, r ¼jxj, and θ ∈ ½0; π is a polar angle

when R

d−1

is written in spherical coordinates. C

ðcos θÞ are

Gegenbauer polynomials with ν ¼ d=2 − 1. The sum in

Eq. (1) runs over all operators O

in the OPE ϕ × ϕ with

spin s and dimension Δ

. The coefficients a

are given

by (following the conventions of Ref. [7])

Published by the American Physical Society under the terms of

the Creative Commons Attribution 4.0 International license.

Further distribution of this work must maintain attribution to

the author(s) and the published article’s title, journal citation,

and DOI. Funded by SCOAP

PHYSICAL REVIEW LETTERS 121, 071602 (2018)

0031-9007=18=121(7)=071602(6) 071602-1 Published by the American Physical Society

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