O(N)理论中的各向同性Lifshitz临界点研究

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"这篇研究论文探讨了在O（N）标量理论中，各向同性的Lifshitz点的存在性，特别是在大N情况下。利用功能重整化群方程（Functional Renormalization Group equations，FRG）和改进的局部势近似（Improved Local Potential Approximation，LPA'），作者Dario Zappalà分析了在不同维度d下的Lifshitz点现象。文章指出，在维度d位于4与8之间时，存在非平凡的Lifshitz点，并且确定了相关本征方向的本征值谱。此外，通过LPA'的1/N阶计算，发现在d=4和d=8时，异常尺寸ηN消失。文中还讨论了研究结果对红外调节器选择的依赖性。" O（N）理论是一种在统计物理和量子场论中常见的模型，通常用于研究多组分系统的行为。在这个理论中，N是场的自由度数量，当N变得很大时，可以通过1/N展开进行近似处理，这被称为1/N扩展。在本文的研究中，作者关注的是Lifshitz点，这是一种特殊的相变点，其动力学具有非传统的时间-空间标度不变性，即时间尺度与空间尺度的比例不是常数。 Lifshitz点通常出现在具有多个竞争相互作用的系统中，它们导致相图中的复杂结构。在各向同性的Lifshitz点，这种非平凡的动力学特性并不依赖于空间的方向。功能重整化群（FRG）是一个强大的工具，它可以系统地研究量子场论或统计物理模型的连续相变和临界行为。通过FRG方程，可以追踪系统的标度行为随调节参数变化的过程。在LPA'中，局部势被考虑到了梯度项的修正，使得对系统动力学的描述更为精确。作者发现，在4维至8维的区间内，存在非平凡的Lifshitz点，这意味着在这个维度范围内，系统可能出现新的相变模式。同时，本征值谱的计算揭示了这些相变的稳定性特征和系统动态的复杂性。异常尺寸ηN是衡量标度不变性破坏程度的一个量，它的消失表明在d=4和d=8时，系统的标度行为更加规范，这可能是由于在这些维度下系统的特殊性质。最后，红外调节器的选择对研究结果有影响，表明在使用FRG方法时，需要谨慎选择合适的调节器以获得稳健的物理预测。这篇论文对理解高维度复杂系统的相变行为和动力学特性提供了深入见解，特别是对于那些可能存在Lifshitz点的系统，它提供了关键的理论工具和计算方法。

Physics Letters B 773 (2017) 213–218

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Isotropic Lifshitz point in the O (N) theory

Dario Zappalà

INFN, Sezione di Catania, Via Santa Soﬁa 64, 95123 Catania, Italy

a r t i c l e i n f o a b s t r a c t

Article history:

Received

14 March 2017

Received

in revised form 13 August 2017

Accepted

22 August 2017

Available

online 26 August 2017

Editor:

A. Ringwald

Keywords:

Functional

Renormalization Group

Lifshitz

point

1/N

expansion

The presence of an isotropic tricritical Lifshitz point for the O (N) scalar theory is investigated at large N

the improved Local Potential Approximation (LPA



) by means of the Functional Renormalization Group

equations. At leading order, the non-trivial Lifshitz point is observed if the number of dimensions d is

taken between d = 4and d = 8, and the eigenvalue spectrum of the associated eigendirections is derived.

At order 1/N of the LPA



the anomalous dimension η

is computed and it is found to vanish both in

d = 4and d = 8. The dependence of our ﬁndings on the infrared regulator is discussed.

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

1. Introduction

The description of a tricritical Lifshitz point by a Landau–

Ginzburg

model, where the derivatives of the ﬁeld with respect

to the coordinates of a m-dimensional subset of a d-dimensional

space and those of the complementary (d − m)-dimensional sub-

space

possess different scaling laws, was ﬁrst presented in [1].

More speciﬁcally, in [1] the kinetic term with square gradient of

the ﬁeld, O (∂

), is kept ﬁnite only for the second subset of coordi-

nates,

while the corresponding term of the m-dimensional subset

is suppressed, so that the term with four powers of the gradi-

ent,

O (∂

), becomes the leading kinetic term of the m-dimensional

subspace and this induces drastic changes in the scaling properties

of the theory.

The

Lifshitz points, which are related to the coexistence on the

phase diagram of three phases, one with vanishing order param-

eter,

another with ﬁnite constant order parameter and the third

characterized by a modulated order parameter with ﬁnite wave

vector, ﬁnd application in various ﬁelds such as magnetic systems

as well as polymer mixtures or high T

superconductors (for re-

views

see [2,3]), but, recently, also in different contexts such as

Lorentz symmetry violation, [4,5], or emergent gravity theories

[6–10]. In addition, an oscillating phase has been predicted for a

very wide class of systems [11–15], and it is conceivable to ex-

pect

that a Lifshitz point could be associated to these modulated

phases. In this sense, a more complete understanding of the prop-

erties

of the Lifshitz point is certainly desirable.

E-mail address: dario.zappala@ct.infn.it.

Rather than considering the general case with 0 < m < d, where

the different scaling properties in the two separate subspaces lead

to a peculiar critical behaviour that involves two different anoma-

lous

dimensions and correlation lengths, we shall focus on the

isotropic case with m = d. In fact, if m < d, due to the different

behaviour of the two sets of coordinates, the isotropy of the prob-

lem

is lost while, when m = d, all the space coordinates have the

same critical behaviour and spatial isotropy is preserved. Clearly,

in this latter case the critical scaling remains different from the

standard one because, as explained before, the kinetic term in the

action is quartic, rather than quadratic, in the ﬁeld derivatives.

The

critical properties of the Lifshitz point were studied in the

ε-expansion [1] as well as in the O(1/N) expansion [16]. The

isotropic case m = d was considered within an expansion around

d = 8 and ε = 8 − d [17] while, recently, a numerical Monte-

Carlo

study indicated a possible disappearance of the Lifshitz point,

when ﬂuctuations are properly included [18].

Furthermore,

another non-perturbative technique already em-

ployed

to study this problem is the Functional Renormalization

Group (FRG) [19–21] which consists of a set of differential ﬂow

equations either for various operators entering the effective ac-

tion

of the theory, or for one or more n-point Green functions

derived from the effective action. Fixed points correspond to sta-

tionary

points of these equations and the critical exponents, that

classify relevant, marginal and irrelevant operators, are extracted

by determining the eigenvalue spectrum of the linear reduction of

the differential equations around the ﬁxed point solutions. Coming

to the Lifshitz point, the FRG was applied to study this problem

for a one component scalar theory, N = 1, [22], and for the N = 3

theory,

[23], both in the uniaxial (m =1) case. Finally, the isotropic

http://dx.doi.org/10.1016/j.physletb.2017.08.051

0370-2693/

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