SYK模型的精确光谱密度矩至1/N^2

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"这篇研究论文详细探讨了Sachdev-Ye-Kitaev（SYK）模型的光谱密度矩，并提供了这些矩至1/N^2阶的精确校正。SYK模型是一种特殊的量子混沌系统，涉及到马里亚纳费米子，通常用于研究黑洞物理和量子信息理论等领域。在N个马里亚纳费米子的背景下，文章指出当阶数为1/N时，矩与Q-赫尔米特多项式的权重函数有关。" 正文： Sachdev-Ye-Kitaev (SYK) 模型是一个高度简化的量子多体系统，它在低维度下展现出强烈的局部相互作用和量子混沌特性。这个模型因其在理解量子引力、黑洞熵以及复杂性理论中的潜在应用而受到广泛关注。在本文中，作者Antonio García-García、Yiyang Ji和Jacobus J.M. Verbaarschot对q体SYK模型进行了深入研究，特别是关注其光谱密度的统计特性。为了得到1/N^2阶的校正，研究者使用了有根弦图来表示维克收缩，这是一种图形表示方法，用于量化量子场论中的关联函数。他们发现，每个弦图的1/N^2修正项与对应相交图的三角形环的数量成比例。对于q为奇数的情况，还会出现额外的分级因子。这一发现将寻找1/N^2校正的问题转化为一个纯粹的图论问题——三角计数。通过解决这个图论问题，研究者能够计算出q=1和q=2的SYK模型的精确矩，这些结果可以通过其他方法得到验证。此外，他们还利用这些矩来获取SYK模型的光谱密度，从而得到了至1/N^2阶的精确描述。这为理解和模拟SYK模型的复杂动力学提供了重要的数学工具。更进一步，研究者不仅限于矩的计算，他们还为所有贡献矩的收缩图得到了精确的解析结果，这些结果可被评估到八阶。这些结果证明，即使在相对较小的N值下，Q-赫尔米特多项式的方法也能提供准确的近似。这项工作深化了我们对SYK模型的理解，尤其是其精细结构和统计特性。它为探索量子混沌、量子引力和复杂系统中的非平凡物理现象提供了新的见解和计算工具。此外，由于其数学上的精巧性和物理上的相关性，这些成果可能对理论物理和量子信息科学的其他领域产生广泛的影响。

JHEP04(2018)146

be treated as being independent. However, we obtain an important approximation if we

nevertheless treat all crossings as independent: if a diagram has E crossings the result is

then simply given by [43]

≈ η

. (2.19)

This approximation expresses η

of an intersection graph G by a product of its edges.

This approximation is in fact at least 1/N-exact, as will be discussed in detail in section 3

and appendix A. It is exact for chord diagrams where multiple crossings are indeed in-

dependent (having tree graphs as intersection graphs, see appendix D). The approxima-

tion (2.19) allows us to use the Riordan-Touchard formula [58, 59] to sum over all inter-

section graphs [27, 42]:

(2p−1)!!

i=1

≈

(2p−1)!!

i=1

(1 −η)

k=−p

(−1)

k(k−1)/2



p + k



, (2.20)

where E

denotes the number of edges of G

. The unique spectral density that gives the

moments (2.20) is the weight function of Q-Hermite polynomials [60]:

(E) = c

1 −(E/E

)

∞

k=1



1 −4



2 + η

+ η

−k



, (2.21)

with c

a normalization constant and E

a scale factor that drops out of the ratio (2.9).

For this reason we refer to this approximation as the Q-Hermite approximation and also

introduce the Q-Hermite moments

(1 − η)

k=−p

(−1)

k(k−1)/2



p + k



. (2.22)

We take equations (2.19), (2.20) and (2.21) as the ﬁrst approximation to η

, the

moments and the spectral density respectively, and we use this as the starting point to

investigate further 1/N

corrections. We stress again that Q-Hermite approximation al-

ready contains many higher order terms in 1/N, although the approximation is only exact

to order 1/N.

3 1/N expansion

The goal of this paper is to understand better why the Q-Hermite result discussed in the

previous section is such a good approximation to the spectral density of the SYK model.

This section is a step in this direction as we show that indeed there are no 1/N corrections

to the Q-Hermite moments, and give an argument that the 1/N

corrections are determined

by the total number of triangles in an intersection graph. In appendix A, we rigorously

demonstrate this statement.

The scaled Q-Hermite moments M

only depend on η which has the 1/N

expansion

η = (−1)



1 −

(q − 1)



+ O





. (3.1)

– 6 –

JHEP04(2018)146

Keeping only the leading power in q at each order of 1/N, it can be shown that this

simpliﬁes to (see appendix B)

η = (−1)

∞

k=0



−2q



= (−1)

−2q

. (3.2)

The Q-Hermite moments thus have a nontrivial large N limit when q

/N is kept ﬁxed. For

q 

√

N we have η → 0 so that only the nested contractions contribute, which give the

moments of a semi-circle. For q 

√

N we have to distinguish even and odd q. For even

q we have η → 1 so that all contractions contribute equally which gives the moments of a

Gaussian distribution while in the case of odd q we obtain η → −1 corresponding to the

moments of the sum of two delta functions located symmetrically about zero. In the latter

case, all scaled moments are equal to one (see eq. (C.9)).

To understand the corrections to the Q-Hermite moments we evaluate 1/N corrections

at ﬁxed q and p,

− M

(p, q) +

(p, q) + ··· . (3.3)

The Q-Hermite result is obtained when all crossings between contraction lines are treated

independently with each crossing contributing a factor η. Corrections of order 1/N occur

when two crossed contraction lines have one common index. Since this involves a single

crossing, this correction is the same for the exact result and the Q-Hermite result and we

thus have that

(p, q) = 0. (3.4)

Corrections to the Q-Hermite result occur when the crossed contracting lines can no

longer be permuted independently. Generally, this happens when intersection graphs have

closed loops, and all vertices in the closed loop have at least pairwise common indices.

If a pair of vertices does not have any common indices, they can be commuted or anti-

commuted resulting in a loop that is no longer closed and is thus given by the Q-Hermite

results. A closed loop of length k, thus diﬀers by O(1/N

k−1

) from the Q-Hermite result.

Therefore, for the O(1/N

) correction we only need to consider the triangular closed loops.

For a closed loop of three crossed contraction lines, say α, β and γ, let us consider the

crossed pair βγ with one common index and let the crossed pair αβ also have a common

index. This is a 1/N correction that is part of the Q-Hermite result, and thus contributes

as q

/N to leading order in 1/N. Deviations from the Q-Hermite result to this closed loop

occur when also the crossed pair αγ has a common index. Choosing this index of γ to be

one from α gives a second factor q/N . The index can either be among the indices shared

with β or not. We thus conclude that the 1/N

corrections due to a triangular diagram

occur as

− η

∼

. (3.5)

The proportionality factor in (3.5) can be obtained from the simplest triangular intersection

graph, whose value is referred to as T

, see ﬁgure 1 (c). This graph ﬁrst occurs in the

calculation of the sixth moment and can be calculated by keeping track of the combinatorial

– 7 –

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