二维张量Gross-Neveu模型：动态质量与对称性破缺

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"Tensorial Gross-Neveu模型的探索与分析" 在物理学领域，Gross-Neveu模型是一个在二维量子场论中被广泛研究的理论，它描述了具有四费米子相互作用的系统。这篇论文关注的是对这个模型的张量概括，即在二维空间中引入G 3 对称性的Tensorial Gross-Neveu模型。这里的G可以是单位ary群U(N)或反对称群O(N)。这些模型可以被视为Sachdev-Ye-Kitaev (SYK)模型的二维扩展，或者是Klebanov和Tarnopolsky提出的张量模型的二维版本。文章详细探讨了在大N极限下使用Schwinger-Dyson方程来研究模型的动力学特性，特别是动态质量产生的现象。动态质量产生是指在量子场论中，由于相互作用，无质量的粒子可以获得非零的质量。作者们分析了可能避免这种质量产生的耦合组合，并发现对于G = U(N)的情况，通过引入中间场表示来进行真空稳定性分析，唯一的明显可行的耦合组合导致不稳定的真空态。有趣的是，尽管Coleman-Mermin-Wagner定理指出，在二维系统中连续对称性不能自发破缺，但在大N扩张的框架内，这篇论文揭示了一种违反这一定理的现象。在Tensorial Gross-Neveu模型中，稳定的真空状态实际上会打破U(N) 3 对称性，这与手性Gross-Neveu模型中连续手性对称性的破坏相似，而这是在大N极限下特有的。此外，作者们来自不同的研究机构，包括法国巴黎萨克雷大学的理论物理实验室、加拿大Perimeter研究所、法国巴黎综合理工学院的理论物理中心和瑞士苏黎世联邦理工学院的理论物理研究所。他们的工作展示了国际跨学科合作在深入理解量子场论和高能物理中的复杂模型方面的重要性。论文的发表时间是2018年1月，表明这是近年来在该领域的最新研究成果。通过这样的张量概括，研究者们不仅拓宽了对Gross-Neveu模型的理解，也为二维量子系统的研究开辟了新的方向，特别是对于那些试图将低维量子系统与高维复杂模型联系起来的研究者来说，这项工作具有重要的参考价值。

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JHEP01(2018)003

Let us determine the A

-invariant interactions, which we choose to parametrize in

terms of the vertices I

2,+

. The completeness relation (A.10) can be used to prove that

S,`

2,+

= −

X=S,V,P

X,`−1

2,−

= −

S,`−1

2,+

V,`−1

2,+

P,`−1

2,+

, (2.17)

for any color `. By summing over `, this immediately implies:

= I

+ I

, (2.18)

where we have deﬁned I

X,`

2,+

. Therefore the A

-invariant sector contains at most

two independent coupling constants. Performing an odd permutation on the color indices

in (2.18) we ﬁnd that the left-hand side is invariant, while the right-hand side picks a minus

sign; therefore, both sides of the equation are zero. We are thus left with no invariants

under S

and at most one invariant under A

, which we could choose for example to be

. We are now going to show that this is in fact a non-trivial invariant, and that we can

choose to write it in such a way that we do not have to perform a sum over colors.

To go further, we can invoke the completeness relations (A.11) and (A.12) to ﬁnd:

V,`

2,+

= −I

S,`−1

2,−

+ I

P,`−1

2,−

= −I

S,`−1

2,+

− I

P,`−1

2,+

, (2.19)

P,`

2,+

= −

S,`−1

2,−

V,`−1

2,−

−

P,`−1

2,−

= −

S,`−1

2,+

−

V,`−1

2,+

P,`−1

2,+

. (2.20)

From now on, let us drop the ± index and simply denote I

X,`

2,+

by I

X,`

. For any given

`, the Fierz relations show that we are free to parametrize our action with the basis

S,`

, I

V,`

, I

P,`

). Let us denote by g

= (g

S,`

, g

V,`

, g

P,`

)

the coordinate vector in this basis,

i.e. the interaction action in the sector 2 is S

int,2

= −

X,`

. The Fierz identities

can be summarized in matrix form by

`−1

= F g

, F :=







−1/2 −1 −1/2

1/2 0 −1/2

1/2 −1 1/2







. (2.21)

We can furthermore introduce the A

averaging operator

A :=



+ F + F









0 0 0

0 1/3 −1/3

0 −2/3 2/3







. (2.22)

This matrix turns out to have a two-dimensional null eigenspace generated by (1, 0, 0)

and

(0, 1, 1)

, and a one-dimensional eigenspace with eigenvalue 1 generated by (0, −1, 2)

. We

recover in particular that two of the a priori allowed invariants identically vanish:

= 0 = I

+ I

. (2.23)

The addition operation on color labels is to be understood modulo 3.

– 8 –

JHEP01(2018)003

The A

-invariant space of I

interactions is therefore one-dimensional and generated by

the `-independent quantity (one can indeed check that (0, −1, 2)

is an eigenvector of F

with eigenvalue 1):

:= −I

V,`

+ 2I

P,`

. (2.24)

Summing over ` we ﬁnd I

(−I

+ 2I

) = I

In summary, we conclude that: 1) there is no I

interaction invariant under the full

permutation group S

; 2) the space of I

interactions invariant under the subgroup of even

permutations A

is one-dimensional, and generated by (2.24).

Remembering also that I

= I

= 0 for Majorana fermions, we write the A

-invariant

interacting action for the Majorana case as

int

= −





x I

X=S,V,P

x I

+ g

x I





. (2.25)

3 Large N : Schwinger-Dyson equation and mass generation

A powerful tool in the study of the large-N limit is provided by the Schwinger-Dyson (SD)

equations. In zero dimensions they typically allow to solve tensor models and ﬁnd their

critical behavior. In one dimension, they play a crucial role for the SYK model and its

tensorial generalizations, as they allow to identify a conformal IR regime and to solve the

theory in that limit. In this section we will study the SD equations for our Dirac and

Majorana two-dimensional tensorial ﬁeld theories, and in particular we will address the

question of dynamical mass generation.

3.1 Dirac case

In two dimensions we do not expect any breaking of continuous global symmetries. This

fact is known in the quantum ﬁeld theory literature as Coleman theorem [38] and in the

statistical mechanics literature as Mermin-Wagner theorem [37], and we will come back to

it in section 4. Therefore, we should be allowed to assume U(N)

-invariance of the theory,

which implies that the propagator is proportional to the identity in tensor space:

hψ

(x)

)i = G(x, x

) δ

. (3.1)

It is convenient to introduce a set of rescaled couplings deﬁned as

= N

, λ

= N

. (3.2)

Then, the large-N SD equations are:

−1

(x, x

) = G

−1

(x, x

) − Σ(x, x

) , (3.3)

Σ(x, x

) = − 2

X=S,V,P

(λ

+ 3λ

)Tr[G(x, x)Γ

]δ(x, x

) Γ

, (3.4)

– 9 –

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