Elsevier

Annals of Physics

Volume 418, July 2020, 168164
Annals of Physics

Fusion rules from entanglement

https://doi.org/10.1016/j.aop.2020.168164Get rights and content

Highlights

  • A derivation of the anyon theory from the entanglement area law is initiated.

  • Basic concepts of the algebraic theory of anyon are defined from a quantum state.

  • The topological charges/superselection sectors of the anyon theory are defined.

  • The fusion rules of anyons are defined and shown to satisfy the expected properties.

  • The well-known formula of topological entanglement entropy is reproduced.

Abstract

We derive some of the axioms of the algebraic theory of anyon (Kitaev, 2006) from a conjectured form of entanglement area law for two-dimensional gapped systems. We derive the fusion rules of topological charges and show that the multiplicities of the fusion rules satisfy these axioms. Moreover, even though we make no assumption about the exact value of the constant sub-leading term of the entanglement entropy of a disk-like region, this term is shown to be equal to lnD, where D is the total quantum dimension of the underlying anyon theory. These derivations are rigorous and follow from the entanglement area law alone. More precisely, our framework starts from two local entropic constraints which are implied by the area law. From these constraints, we prove what we refer to as the “isomorphism theorem.” The existence of superselection sectors and fusion multiplicities follow from this theorem, even without assuming anything about the parent Hamiltonian. These objects and the axioms of the anyon theory are shown to emerge from the structure and the internal self-consistency relations of the information convex sets.

Introduction

One of the outstanding questions in modern physics concerns the classification of quantum phases. Many attempts have been already made to classify quantum phases over the past decade. For instance, gapped free-electron systems are completely classified [1], [2]. For more general short-range entangled states, an approach based on cobordism was proposed [3]. One-dimensional (1D) gapped systems are completely classified at this point [4], [5], [6], [7]. A general gapped two-dimensional (2D) systems are expected to be described within the framework topological quantum field theory; see [8], for example.

This whole slew of different approaches raises a natural question. Why are there so many different approaches, and how can we ever be sure that the classification is complete? The main difficulty lies in identifying the correct framework. In the presence of interaction, one often needs to make a nontrivial assumption. The only exception so far is the one-dimensional (1D) gapped system. Hastings’ theorem [9] implies that any gapped 1D system obeys an area law. This subsequently implies that a matrix product state can approximate the ground state with a moderate bond dimension. It is this result from which a classification of quantum phases of 1D gapped system [4], [5], [6] follows.

However, in higher dimensions, an analog of Hastings’ theorem is unknown. This is mainly because proving area law in 2D gapped systems remains challenging. Furthermore, even if area law turns out to be correct, the states that satisfy area law may not be well-approximated by an efficient tensor network [10]. These facts suggest that a classification program in 2D cannot merely mimic the classification program for 1D gapped systems. In fact, in any classification proposal based on tensor networks, there will always be a lingering question on whether we are not missing any unknown phases.

While it is widely believed at this point that topological quantum field theory (TQFT) describes all possible gapped phases in 2D, there is currently no rigorous argument that supports this belief. The existence of a three-dimensional (3D) gapped phase outside of the TQFT framework [11] shows that there may be gapped phases of matter that lie outside of the TQFT framework. Even if TQFT turns out to be the correct framework in 2D, understanding of where this framework comes from remains as an important fundamental problem.

Motivated by this state of affairs, we initiate a program in which a familiar set of axioms of TQFT can be derived from a seemingly innocuous assumption about entanglement. We show that some of the basic concepts of the algebraic theory of anyon [12], i.e., superselection sectors and fusion multiplicities, emerge from a familiar form of entanglement area law [13], [14]: S(A)=αγ,where S(A) is the von Neumann entropy of a simply connected region A, is the perimeter of A, and γ is a constant correction term1 that only depends on the topology of A. The sub-leading correction, which vanishes in the limit, is suppressed here.

We then show that our definition of the fusion multiplicities satisfies all the properties one would have expected from the algebraic theory of anyon. Again, these properties are derived from Eq. (1). Moreover, we further derive the following well-known formula: γ=lnD,where D is the total quantum dimension of the anyon theory we defined. Our derivation is rigorous under the assumption (Eq. (1)) and is completely independent from the previous approaches, i.e., an approach based on an effective field theory description [13] and explicit calculations in exactly solvable models [14].

While our assumption is not as rigorous as Hastings’ proof of the 1D area law, it is something that is widely accepted at this point. Therefore, we believe this would be a reasonable starting point to obtain a general understanding of gapped phases. A similar, but a markedly different starting point of our work would be the two axioms we have identified. These two axioms are entropic conditions on bounded-radius disks (Axioms A0 and A1 in Section 2.1). We can show that these two axioms follow from Eq. (1), but after that, we never use Eq. (1) explicitly. All of our results follow directly from the axioms.

In other words, the axioms of the anyon fusion theory can be derived from Eq. (1). The same conclusion follows from our axioms as well; see Axioms A0 and A1. While the conclusion would be the same either way, we would like to advocate for the use of the axioms over Eq. (1) for the following reasons. The first reason is that Axioms A0 and A1 are assumed to hold on patches whose size is independent of the system size. Therefore, in principle, one can verify these axioms in time that scales linearly with the system size. Under a promise that the state is translation-invariant, the time can be reduced to a constant. In contrast, Eq. (1) is defined over length scales that are comparable to the system size. Verifying this assumption will incur an exponential computational cost. Secondly, in the continuum limit, the leading term of Eq. (1) depends on the ultraviolet-cutoff. On the other hand, the axioms manifestly cancel out this divergent piece.

Our framework is completely Hamiltonian-independent, in the sense that we only require the existence of a global state on the system satisfying the two local entropic constraints. This work is motivated from a number of recent observations: that local reduced density matrices of topological quantum phases often have a quantum Markov chain structure [15], [16], [17], [18], [19], [20], [21], [22], [23]. The key overarching concept is a convex set of density matrices introduced in [18], which is later rediscovered and studied under the name “information convex (set)” [22], [23]. Roughly speaking, this is a set of density matrices which are locally indistinguishable from some reference state. In our context, this reference state would be the ground state of some local Hamiltonian. However, we do not use the fact that the state is a ground state.

Our framework opens up a concrete route to classify gapped quantum phases without resorting to ad-hoc assumptions. In addition, we believe our framework is capable of answering a long-standing open question about topological phases. The question is if a single ground state contains all the data necessary to define a topological phase. Given that we can define a notion of topological charges and fusion multiplicities from a single ground state, progress may be made by using our framework. Our approach can be generalized to a broader context, e.g., to higher dimensions and to setups in which a topological defect [24] or a boundary is present [25]. We will discuss these applications in our upcoming work.

The rest of this paper is organized as follows. In Section 2, we specify our formal setup and summarize our main results. In Section 3, we prove fundamental properties of the information convex sets, which are the key to obtaining some of the axioms of the algebraic theory of anyon. We shall refer to this part of the full algebraic theory of anyon as the anyon fusion theory from now on. In Section 4, we define the notion of superselection sectors and fusion multiplicities in our framework and prove that the definition satisfies all the axioms of the anyon fusion theory. In Section 5, we show that the constant term γ in the area law equals the logarithm of the total quantum dimension. In Section 6, we conclude with a discussion.

Section snippets

Setup and summary

Let us begin with a general setup and state our physical assumptions. Before we delve into the details, it will be instructive to discuss the physical system we have in mind. We are envisioning a gapped system in 2D, which consists of microscopic degrees of freedom, e.g., spins. We would like to coarse-grain these microscopic degrees of freedom so that we can view non-overlapping blocks of spins as gigantic “supersites”, see Fig. 1. We can consider the limit in which the length scale of each

Axiom extension, information convex set and isomorphism theorem

We have proclaimed in Section 2.2 that we can, among many things, establish a globally well-defined notion of topological charge. Because our axioms (Axioms A0 and A1) are assumed to hold only on bounded-radius disks, the fact that such a notion can even exist in the first place is not obvious at all.

In order to explain how this works, we choose the main theme of this section to be “local to global”. Starting from our local axioms (Axioms A0 and A1), we will see how we can infer some of the

Fusion data from information convex sets

The isomorphism theorem (Theorem 3.10) guarantees that the structure of the information convex set only depends on the topology, as long as the underlying subsystems can be smoothly deformed from one to another along some path.

We now focus on how to extract the information of the topological charges and the corresponding fusion rules from the information convex set. We do this by studying how the geometry of the information convex set depends on the topology of the underlying subsystem. We then

Topological entanglement entropy

In this section, we show that the sub-leading term γ of the area law (1) for a disk is given by the well-known formula γ=lnD,where D is the total quantum dimension defined from our definition of the fusion multiplicities {Nabc}. We show this result by calculating two different linear combinations of subsystem entropies7

Summary and discussions

In this paper, we have initiated a derivation of the axioms of the algebraic theory of anyon from a conjectured form of entanglement area law for the ground states of 2D gapped phases. Our framework is based on two entropic constraints (Axioms A0 and A1), which are implied by the area law formula. We have defined the superselection sectors and the fusion spaces through the geometry of the information convex sets. The axioms of the anyon fusion theory are derived from the internal

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

BS wishes to thank Yilong Wang for a discussion on possible obstruction of the categorification of fusion rings. BS is supported by the National Science Foundation, USA under Grant No. NSF DMR-1653769. IK’s research at Perimeter Institute was supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Economic Development and Innovation. IK also acknowledges support from IBM T. J. Watson research center as well as Simons foundation, USA.

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