随机张量网络揭示全息对偶：新模型的AdS/CFT特性

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"来自随机张量网络的全息对偶"是一篇发表在《Journal of High Energy Physics》(JHEP)第11期的开放获取论文，由Patrick Hayden、Sepehr Nezami、Xiao-Liang Qi、Nathaniel Thomas、Michael Walter和Zhao Yang等作者合作完成。论文发表于2016年，探讨了张量网络在探索全息对偶性中的作用。张量网络因其满足纠缠面积定律而成为研究全息对偶的理想工具，这在阿贝尔-卡斯蒂利亚诺斯/规范场理论(AdS/CFT)对偶中的结构特性有着重要意义。在文中，作者利用随机张量网络构建模型，这些模型展现出与AdS/CFT对偶相似的特性。他们发现，当张量的键维度增大时，模型中的边界区域纠缠熵遵循 Ryu-Takayanagi 熵公式，这是一个关键的全息性质，它与辅助多部分纠缠密切相关。Rényi熵在模型中的行为也得到了深入研究，并与AdS/CFT中的对应现象进行了对比。此外，随机张量网络模型揭示了一个重要的概念，即每个边界区域编码了与其对应的“块状纠缠楔”，也就是边界区域及其最小表面所包围的区域的物理信息。作者通过将随机张量的平均值关联到经典铁磁伊辛模型的分配函数，使得最小表面在模型中表现为畴壁，类似于热力学相变中的界面。进一步地，引入体场后，模型再现了对Ryu-Takayanagi公式的修正，体最小表面不再是简单地对应于边界上的最小面积，而是受到体场纠缠的影响，熵也随之增加。这种体场的纠缠增长最终在拓扑上类似于形成黑洞，改变了最小表面的行为。通过体相关函数的扩展，作者能够计算出更精细的物理量。这篇论文不仅深化了对随机张量网络在全息对偶性中的理解，还为理论物理学家提供了一个新颖且直观的框架来研究量子引力和量子信息之间的深层次联系。通过这些模型，研究者们可以探索更复杂的全息现象，以及边界理论和引力理论之间的相互作用。"

JHEP11(2016)009

Figure 2. (a) Graphic representation of the single site density operator |V

ihV

| for a vertex in

the tensor network shown in ﬁgure 1. (b) The average over the state |V

ihV

| ⊗ |V

ihV

| in the

Hilbert space (see eq. (2.11)). On the right side of the equality, the dashed line connected to the

black dot stands for a sum over an Ising variable s

= ±1. When s

= 1 (s

= −1), each green

rectangle represents an operator I

), respectively. (c) The state average of Z

in eq. (2.10) for

the simple tensor network shown in ﬁgure 1. We consider a region A consisting of a single site,

and the green rectangle with X represents the swap operator F

. After contracting the doubled

line loops one obtains the partition function of an Ising model, with the blue arrows representing

the Ising variables. The dashed lines in the right of last equality represent three diﬀerent terms

in the Ising model contributed by the links, the bulk state (middle triangle) and the choice of

boundary region A.

That is, the typical second R´enyi entropy is given by the diﬀerence of the two free energies,

i.e., the “energy cost” induced by ﬂipping the boundary pinning ﬁeld to down (−1) in

region A, while keeping the remainder of the system with a pinning ﬁeld up (+1).

In summary, what we have achieved is that the second R´enyi entropy is related to

the partition function of a classical Ising model deﬁned on the same graph as the tensor

network. Besides the standard two-spin interaction term, the Ising model also has an

additional term in its energy contributed by the second R´enyi entropy of the bulk state

, and the Ising spins at the boundary vertices are coupled to a boundary “pinning ﬁeld”

determined by the boundary region A. If the bulk contribution from ρ

is small (which

means major part of quantum entanglement of the boundary states is contributed by the

tensor network itself), one can see that the parameters log D

and log D

x∂

determine

the eﬀective temperature of the Ising model. For simplicity, in the following we assume

= D

x∂

= D for all internal legs and boundary dangling legs. In this case we can take

β =

log D as the inverse temperature of the classical Ising model.

3 Ryu-Takayanagi formula

Once the mapping to the classical Ising model is established, it is easy to see how the

Ryu-Takayanagi formula emerges. In the large D limit, the Ising model is in the low-

temperature long-range ordered phase (as long as the bulk has spatial dimension ≥ 2), so

that the Ising action can be estimated by the lowest energy conﬁguration. The boundary

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JHEP11(2016)009

pinning ﬁeld h

leads to the existence of an Ising domain wall bounding the boundary

region A, and in the absence of a bulk contribution the minimal energy condition of the

domain wall is exactly the RT formula. In this section we will analyze this emergence of

the Ryu-Takayanagi formula and corrections due to bulk entanglement in more detail.

3.1 Ryu-Takayanagi formula for a bulk direct-product state

We ﬁrst consider the simplest situation with the bulk state a pure direct-product state

|φ

ihφ

|. In this case one can contract the bulk state at each site with the

tensor of that site, which leads to a new tensor with one fewer legs. Since each tensor is

a random tensor, the new tensor obtained from contraction with the bulk state is also a

random tensor. Therefore the holographic mapping with a pure direct-product state in

the bulk is equivalent to a purely in-plane random tensor network, similar to a MERA,

or a “holographic state” deﬁned in ref. [6]. The second R´enyi entropy of such a tensor

network state is given by the partition function of Ising model in eq. (2.13) without the ρ

term. Omitting the constant terms that appears in both Z

and Z

, the Ising action can

be written as

A[{s

}] = −

log D





hxyi

− 1) +

x∈∂

− 1)





. (3.1)

In the large D limit, the Ising model is in the low temperature limit, and the partition

function is dominated by the lowest energy conﬁguration. As illustrated in ﬁgure 3 (a),

the “energy” of an Ising conﬁguration is determined by the number of links crossed by

the domain wall between spin-up and spin-down domains, with the boundary condition

of the domain wall set by the boundary ﬁeld h

. For the calculation of denominator Z

= +1 everywhere, so that the lowest energy conﬁguration is obviously s

= +1 for all

x, with energy F

= 0. For F

, the nontrivial boundary ﬁeld h

= −1 for x ∈ A requires

the existence of a spin-down domain. Each link hxyi with spins anti-parallel leads to an

energy cost of log D. Therefore the R´enyi entropy in large D limit is

(A) = F

− F

' log D min

Σ bound A

|Σ| ≡ log D |γ

|. (3.2)

The minimization is over surfaces Σ such that Σ ∪ A form the boundary of a spin-down

domain, and |Σ| denotes the area of Σ, i.e., the number of edges that cross the surface.

Therefore the minimal area surface, denoted by γ

, is the geodesic surface bounding A

region. Here we have assumed that the geodesic surface is unique. More generally, if there

are k degenerate minimal surfaces (as will be the case for a regular lattice in ﬂat space),

is modiﬁed by −log k.

With this discussion, we have proved that Ryu-Takayanagi formula applies to the

second R´enyi entropy of a large dimensional random tensor network, with the area of

geodesic surface given by the graph metric of the network. As will be discussed later in

section 5, the higher R´enyi entropies take the same value in the large D limit, and it

can also be extended to the von Neumann entropy, at least if the minimal geodesics are

unique (see section 7). However, the triumph that the second R´enyi entropy is equal to

– 10 –

JHEP11(2016)009

the area of the minimal surface in the graph metric is in fact a signature that the random

tensor construction deviates from the holographic theory. The holographic calculation

of the second R´enyi entropy amounts to evaluating the Euclidean action of the two-fold

replica geometry, which satisﬁes the Einstein equation everywhere in the bulk. Thus, in

general, the second R´enyi entropy does not exactly correspond to the area of the minimal

surface in the original geometry. Due to the back-reaction of the gravity theory, the n-fold

replica geometry is in general diﬀerent from the geometry constructed by simply gluing

n copies of the original geometry around the minimal surfaces, the discrepancy between

which can be seen manifestly from the n-dependence of the holographic R´enyi-n entropy.

We will see in section 5 that our random tensor model can reproduce the correct R´enyi

entropies for a single boundary region if we replace the bond states |xyi by appropriate

short-range entangled states with non-trivial entanglement spectrum. However, this does

not resolve the problem for multiple boundary regions, for which we will have a more

detailed discussion in section 5.

To compare with the RT formula deﬁned on a continuous manifold, one can consider a

triangulation of a given spatial manifold and deﬁne a random tensor network on the graph

of the triangulation. (See [22, appendix A] for further discussion of the construction of

the triangulation graph.) Denoting by l

the length scale of the triangulation (the average

distance between neighboring triangles), the area |γ

| in our formula is dimensionless and

the area |γ

| deﬁned on the continuous Riemann manifold is given by |γ

| = l

d−1

|γ

(when the spatial dimension of bulk is d). Therefore S(A) = l

1−d

log D |γ

|, and we see

that l

1−d

log D corresponds to the gravitational coupling constant

Compared to previous results about the RT formula in tensor networks [6, 8], our proof

of RT formula has the following advantages: ﬁrstly, our result does not require the boundary

region A to be a single connected region on the boundary. Since the entropy in the large D

limit is always given by the Ising spin conﬁguration with minimal energy, the result applies

to multiple boundary regions. Secondly, our result does not rely on any property of the

graph structure, except for the uniqueness of the geodesic surface (if this is not satisﬁed

then the entropy formula acquires corrections as discussed above; cf. section 9). If we

obtain a graph by triangulation of a manifold, our formula applies to manifolds with zero

or positive curvature, even when the standard AdS/CFT correspondence does not apply.

In addition to these two points, we will also see in later discussions that our approach

allows us to study corrections to the RT formula systematically. Notice that we are not

limited to two-dimensional manifolds. One can consider a higher dimensional manifold and

construct a graph approximating its geometry. It follows from our results that the entropy

of a subregion of the boundary state is given by the size of the minimum cut on the graph,

i.e., the area of the minimal surface in the bulk homologous to the boundary region.

3.2 Ryu-Takayanagi formula with bulk state correction

If we do not assume the bulk state to be a pure direct-product state, the bulk entropy term

in eq. (2.13) is nonzero. If we still take the D → ∞ limit, the Ising model free energy is

still determined by the minimal energy spin conﬁguration, which is now determined by a

balance between the area law energy log D |Σ| for a domain wall Σ, and the energy cost from

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