柏拉图多面体标量场理论的重整化群固定点研究

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本文主要探讨的是柏拉图场论，这是一种基于标量场理论的量子场论，特别关注其在重归化组(RG)不变点上的性质。RG不变点是量子场论中的关键概念，它们描述了在改变能量尺度时，理论参数如何随时间演化。文章关注的是标量场模型，这些模型具有规则多面体的离散对称性，这在数学上对应于柏拉图多面体，如正六面体、四面体、十面体等。研究者利用了功能摄动重整化组(FPRG)方法，这是一个强大的工具，用于计算量子场论在低维度下的行为。FPRG通过构建一个有效平均动作函数，能够处理量子场论中的非线性效应，并在接近临界维度(d_c)时提供精确的描述。作者所考虑的临界维度(d_c)列表包括：d_c = 6, 4, 10/3, 3, 14/5, 8/3, 5/2, 12/5，这意味着理论在这些特定维度下展现出特殊的行为，可能是相变或理论简化的关键点。通过多成分β函数的导出，文章旨在揭示这些临界维度与理论参数之间的确切关系。β函数是RG流动的关键组成部分，它描述了参数随能量变化的速率。在d = d_c - ϵ的ε-expansion框架下，作者细致地探索了这些理论在接近临界维度时的稳定性和不稳定性，这对于理解理论的物理意义以及可能的物理现象至关重要。此外，本文还包含了研究的时间线，从提交到接受再到最终发表，展示了科学论文从构思到发表的严谨流程。值得注意的是，该研究成果是开放获取的，意味着研究结果可供全球科学家免费查阅，促进了学术交流和知识共享。这篇论文深入探讨了柏拉图场论的RG不变点，利用FPRG方法在不同临界维度下分析了标量场的性质，为理解量子场论中的对称性、相变和重整化过程提供了新的洞察。其研究对于理论物理学，特别是凝聚态物理和统计物理领域有着重要的理论价值。

JHEP04(2019)152

Not all the invariants I

(k)

are independent, as can be inferred from table 2. For each

polytope, we identify the basic N independent ones by increasing the polynomial degree k.

To this purpose it is useful to consider the Molien series which, for a given symmetry group

G, counts the number of homogeneous polynomials of a given degree k that are invariants

for G itself [26]. It is deﬁned as:

M(t) =

|G|

g∈G

det[I − t ρ(g)]

, (2.3)

where ρ is a linear representation of the group G on the underlying N dimensional vector

space. Once the series is expanded, the coeﬃcient of the monomial t

gives the number of

linearly independent homogeneous invariants of degree m; the Molien series furthermore

suggests which is the polynomial degree of the basic N independent invariant polynomials,

as it can be understood cross-checking tables 1 and 2. We always ﬁnd only one quadratic

independent invariant

which we call ρ := I

(2)

, while, independently of the order at which

they ﬁrst appear, we call τ the second and, when present, σ and ω respectively the third

and the fourth ones (see table 2). Let’s call B

the set given by the basic N indepen-

dent invariants of a given polytope P. In terms of the elements of B

we can consider

(k)

(ρ, τ, σ, ω) as the most general homogeneous G-invariant polynomial of degree k; in

general it can be expressed as

(k)

(ρ, τ, σ, ω) =

µ=1

(k)

(ρ, τ, σ, ω) , (2.4)

where M

(k)

(ρ, τ, σ, ω) are monomials given by powers and products of elements of B

such

that their overall polynomial degree is k, g

are some real coeﬃcients [22] and the number

r of homogeneous polynomials of degree k that are invariant under G

, is given in terms

of the Molien series as explained above. In the framework of the -expansion we are going

to renormalize PFTs in d = d

− , where the upper critical dimension d

is uniquely

determined by the degree of the homogeneous polynomials P

(k)

. Indeed we can express

the GL G-invariant potential V (φ

) ≡ U(ρ, τ, σ, ω) simply as

U(ρ, τ, σ, ω) =

k=2

(k)

(ρ, τ, σ, ω) , (2.5)

and we understand that the coeﬃcients g

play the role of coupling constants. By imposing

the GL potential U to be marginal (remember that φ has dimensions

d−2

as it can be

gleaned out inspecting the kinetic part of the action (2.1)) we obtain the upper critical

dimensions as

(k) =

k − 2

. (2.6)

In this paper, for any polytope P, we considered all the possible upper critical dimensions

corresponding to the allowed P

(k≤k

max

)

, where k

max

is the degree of the highest order

polynomial in B

. We exclude from the analysis those d

related to polynomials P

(k)

which

A single quadratic invariant guarantees that the underlying fundamental representation of O(N) remains

irreducible under G and that we have only one phase transition.

– 5 –

JHEP04(2019)152

are expressed as powers of ρ only, since they will simply describe the corresponding O(N)

symmetric theory.

Let us make all this more concrete and give an example for the Square polygon {4}.

First we construct the basic D

-symmetric invariant polynomials I

(k)

. To this purpose, we

ﬁx the versors {e

} choosing the four vertices of the Square to be the permutations of the

coordinates (±

√

2, ±

√

2). We then proceed performing the sum in eq. (2.2) which in this

case extends up to N = 2 and n = 4. Starting from k = 2 we ﬁnd

(2)

= 2



+ φ



, (2.7)

(3)

= 0 , (2.8)

(4)

= φ

+ 6φ

+ φ

, (2.9)

and therefore the two elements of B

{4}

are ρ

{4}

≡ I

(2)

and τ

{4}

≡ I

(4)

. Since the Square

interaction term is represented by the invariant polynomial τ

{4}

of degree k = 4, the only

interesting upper critical dimension in this case is d

= 4. The Molien Series for the Square

group D

is given by

M(t) = [(1 − t

)(1 − t

)]

−1

= 1 + t

+ 2t

+ O(t

) , (2.10)

from which we understand that the r = 2 monomials of degree 4 are M

(4)

= ρ

and

(4)

= τ, so that the corresponding marginal potential U(ρ, τ) is given by

U(ρ, τ) =

(ρ, τ) =



X ρ

{4}

+ Y τ

{4}



, (2.11)

where we named the coupling constants g

= X and g

= Y .

As a further example, consider the case of the dual pair {3, 4}, {4, 3} namely the

Octahedron and the Cube. We ﬁx the versors {e

} choosing the eight Cube vertices as

the permutations of the coordinates

/3(±1, ±1, ±1) so that, once we perform the sum

in eq. (2.2) which now extends up to N = 3 and n = 8, we ﬁnd that the three elements of

{4,3}

in the Cube basis are

{4,3}

≡ I

(2)



+ φ



, (2.12)

{4,3}

≡ I

(4)



+ 6



+ φ



+ φ

+ 6φ



, (2.13)

{4,3}

≡ I

(6)



+ 15



+ φ



+ 15



+ 6φ

+ φ



+ φ

+φ

+ 15φ



. (2.14)

In the Octahedron basis the independent invariants are given in appendix A. The duality

between the two Platonic solids is expressed as a map between the invariants (ρ, τ, σ) in

the two representations which, in the case of the Octahedron/Cube reads

{4,3}

{3,4}

{4,3}

{3,4}

−

{3,4}

{4,3}

{3,4}

−

{3,4}

. (2.15)

The map between invariants translates in a smooth map between couplings and thus their

RG properties are trivially the same.

– 6 –

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