大N下O(N)标量理论的二维多临界点解析

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本文主要探讨了二维到四维（2<d<4）范围内的开放获取研究，针对的是大N极限下的O(N)对称标量理论。作者A. Katsis和N. Tetradis来自希腊国家和卡波迪斯特里安大学的物理系，他们在最新的一项研究中，对O(N)标量理论的重归一化群方程进行了深入分析，目标是寻找之前数值上观察到的非平凡不动点。在他们的研究中，关键焦点在于理论的潜在函数，即势能函数。他们通过解析方法解决了在大N极限条件下的重归一化群方程，这是一种在量子场论中用于描述系统随尺度变化行为的重要工具。他们特别关注的是那些可能导致物理性质发生显著变化的非线性结构，如极值点和其附近可能存在的奇异性。值得注意的是，他们发现了两种新的解：一种是实解，这些解在最小潜在值处的高阶导数表现出奇异性。这意味着在这些点，势能函数的行为与常规预期有所不同，可能暗示着理论中的新型相变或非平凡物理现象。另一种解是复数解，它们带有沿着负实轴的分支切口。这类解的出现可能与理论的对称性破裂或多重临界点有关，这些临界点是系统参数变化时，物理性质发生跃迁的特殊点。此外，文章还提到了版权信息，指出该研究是开放获取的，遵循Creative Commons Attribution (CC BY) 许可协议，并得到了SCOAP3项目的资助。这表明研究成果可以被广泛共享和引用，促进了学术界的交流与合作。这篇论文的重要性在于它不仅深化了对大N极限下O(N)标量理论的理解，还可能推动理论物理学领域中临界现象、重归一化群方法以及多临界点研究的进一步探索。对于数值模拟和实验验证这些理论预言的物理学家来说，这篇文章提供了宝贵的理论指导和新的思考方向。

Physics Letters B 780 (2018) 491–494

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Multicritical points of the O (N) scalar theory in 2 < d < 4for large N

A. Katsis, N. Tetradis

∗

Department of Physics, National and Kapodistrian University of Athens, Zographou 157 84, Greece

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

Article history:

Received

4 February 2018

Accepted

14 March 2018

Available

online 16 March 2018

Editor:

N. Lambert

Keywords:

Critical

phenomena

Renormalization

group

Scalar

theories

Large

We solve analytically the renormalization-group equation for the potential of the O (N)-symmetric scalar

theory in the large-N limit and in dimensions 2 < d < 4, in order to look for nonperturbative ﬁxed points

that were found numerically in a recent study. We ﬁnd new real solutions with singularities in the higher

derivatives of the potential at its minimum, and complex solutions with branch cuts along the negative

real axis.

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

1. Introduction

The O (N)-symmetric scalar theories have served for decades as

the testing ground of techniques developed for the investigation

of the critical behaviour of ﬁeld theories and statistical models. It

comes, therefore, as a surprise that a recent study [1] has found

that their phase structure may be much more complicated that

what had been found previously. In particular, it is suggested that,

in dimensions 2 < d < 4, several nonperturbative ﬁxed points ex-

ist,

which had not been identiﬁed until now. The large-N limit

[2–7]offers the possibility to identify such ﬁxed points analyt-

ically,

without resorting to perturbation theory. We shall con-

sider

the theory in this limit through the Wilsonian approach to

the renormalization group (RG) [8]. Its various realizations [9–13]

give

consistent descriptions of the ﬁxed-point structure of the

three-dimensional theory [14], in agreement with known results

for the Wilson–Fisher (WF) ﬁxed point [15] and the Bardeen–

Moshe–Bander

(BMB) endpoint of the line of tricritical ﬁxed points

[16–18].

shall employ the formalism of ref. [11], leading to the ex-

act

Wetterich equation for the functional RG ﬂow of the action.

For N →∞ the anomalous dimension of the ﬁeld vanishes and

higher-derivative terms in the action are expected to play a mi-

nor

role. This implies that the derivative expansion of the action

[19–21]can be truncated at the lowest order, resulting in the

Corresponding author.

E-mail

addresses: ariskatsis @phys .uoa .gr (A. Katsis), ntetrad @phys .uoa .gr

(N. Tetradis).

local potential approximation (LPA) [9,13,14,22]. The resulting evo-

lution

equation for the potential is exact in the sense explained in

ref. [14]. It has been analysed in refs. [23,24]in three dimensions.

In this work, we extend the analysis over the range 2 < d < 4, in

an attempt to identify new ﬁxed points.

2. Evolution equation for the potential

We consider the theory of an N-component scalar ﬁeld φ

with

O (N) symmetry in d dimensions. We are interested in the func-

tional

RG evolution of the action as a function of a sharp infrared

cutoff k. We work within the LPA approximation, neglecting the

anomalous dimension of the ﬁeld and higher-derivative terms in

the action. We deﬁne ρ =

, a =1...N, as well as the rescaled

ﬁeld

ρ = k

2−d

ρ. We denote derivatives with respect to

ρ with

primes. We focus on the potential U

(ρ) and its dimensionless ver-

sion

(

ρ) =k

−d

(ρ). In the large-N limit and for a sharp cutoff,

the evolution equation for the potential can be written as [23]

∂u



∂t

=−2u



+(d −2)

∂u



∂

−

1 + u



∂u



∂

, (1)

with t = ln(k/) and C

−1

= 2

d/2

(d/2). This equation can be

considered as exact, as explained in ref. [14]. The crucial assump-

tion

is that, for N →∞, the contribution from the radial mode

is negligible compared to the contribution from the N Goldstone

modes.

The

most general solution of eq. (1)can be derived with the

method of characteristics, generalizing the results of ref. [23]. It is

given by the implicit relation

https://doi.org/10.1016/j.physletb.2018.03.038

0370-2693/

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