Sistema de temperatura

We begin with a brief introduction to group cohomology. In this paper, we will not present the most general definition of group cohomology.

discrete), one can consider an arbitrary function that maps n elements of 𝐺 to an ele- ment in 𝑀 ; 𝜔 : 𝐺𝑛_{→ 𝑀 or equivalently 𝜔(𝑔}

1, 𝑔2, ..., 𝑔𝑛) ∈ 𝑀 , ∀𝑔1, 𝑔2, ...𝑔𝑛∈ 𝐺. Such a group function is called an n-cochain. The set of all n-cochains, which is denoted as 𝐶𝑛_{(𝐺, 𝑀 ), forms an abelian group in the usual sense: (𝜔}

1 · 𝜔2)(𝑔1, 𝑔2, ..., 𝑔𝑛) = 𝜔1(𝑔1, 𝑔2, ..., 𝑔𝑛) · 𝜔2(𝑔1, 𝑔2, ..., 𝑔𝑛), in which the identity n-cochain is a group function whose value is always the identity in 𝑀 .

One can define a mapping 𝛿 from 𝐶𝑛_{(𝐺, 𝑀 ) to 𝐶}𝑛+1_{(𝐺, 𝑀 ): ∀𝜔 ∈ 𝐶}𝑛_{(𝐺, 𝑀 ),} define 𝛿𝜔 ∈ 𝐶𝑛+1_{(𝐺, 𝑀 ) as} 𝛿𝜔(𝑔1, ..., 𝑔𝑛+1) = 𝜔(𝑔2, ..., 𝑔𝑛+1) · 𝜔(−1) 𝑛+1 (𝑔1, ..., 𝑔𝑛) × 𝑛 ∏︁ 𝑖=1 𝜔(−1)𝑖(𝑔1, .., 𝑔𝑖−1, 𝑔𝑖· 𝑔𝑖+1, 𝑔𝑖+1, .., 𝑔𝑛+1). (2.47)

It is easy to show that the mapping 𝛿 is nilpotent: 𝛿2𝜔 = 1 (here 1 denotes the identity (n+2)-cochain). In addition, for two n-cochains 𝜔1, 𝜔2, obviously 𝛿 satisfies 𝛿(𝜔1· 𝜔2) = (𝛿𝜔1) · (𝛿𝜔2).

An n-cochain 𝜔(𝑔1, ...𝑔𝑛) is called an n-cocyle if and only if it satisfies the condition: 𝛿𝜔 = 1, where 1 is the identity element in 𝐶𝑛+1_{(𝐺, 𝑀 ). When this condition is} satisfied, we also say that 𝜔(𝑔1, ...𝑔𝑛) is an n-cocycle of group 𝐺 with coefficients in 𝑀 . The set of all n-cocycles, denoted by 𝑍𝑛_{(𝐺, 𝑀 ), forms a subgroup of 𝐶}𝑛_{(𝐺, 𝑀 ).}

Not all different cocyles are inequivalent. Below we define an equivalence relation in 𝑍𝑛_{(𝐺, 𝑀 ). Because 𝛿 is nilpotent, for any (n-1)-cochain 𝑐(𝑔}

1, ..., 𝑔𝑛−1), we can find the n-cocyle 𝛿𝑐. And if an n-cocyle 𝑏 can be represented as 𝑏 = 𝛿𝑐, for some 𝑐 ∈ 𝐶𝑛−1_{(𝐺, 𝑀 ), 𝑏 is called an n-coboundary. The set of all n-coboundaries, denoted} by 𝐵𝑛(𝐺, 𝑀 ), forms a subgroup of 𝑍𝑛(𝐺, 𝑀 ). Two n-cocycles 𝜔1, 𝜔2 are equivalent (denoted by 𝜔1 ∼ 𝜔2) if and only if they differ by an n-coboundary: 𝜔1 = 𝜔2· 𝑏, where 𝑏 ∈ 𝐵𝑛(𝐺, 𝑀 ).

The n-th cohomology group of group 𝐺 with coefficients in 𝑀 , 𝐻𝑛_{(𝐺, 𝑀 ), is}

formed by the equivalence classes in 𝑍𝑛(𝐵, 𝑀 ). More precisely: 𝐻𝑛(𝐺, 𝑀 ) = 𝑍𝑛(𝐺, 𝑀 )/𝐵𝑛(𝐺, 𝑀 ). In this paper we will make a lot of use of 4-cocycles 𝜔. We will always choose them

is equal to 1 (the identity element of group 𝐺). For any of the inequivalent cocy- cles mentioned above, it is always possible to choose a gauge such that 𝜔 becomes canonical [21].

Chapter 3

Symmetry enriched topological

phases and tensor network states

3.1

Introduction

In this chapter and next chapter, we develop a generic framework to write down general variational wavefunctions for a large class of symmetric topological phases using tensor network methods. In this chapter, we will mainly focus on symmetry enriched topological phases (SET phases). And in the next chapter, we will consider symmetry protected topological phases for both on-site symmetries as well as lattice symmetries.

In physical systems, one needs to consider both global symmetries and topological orders. In particular, it is very important to understand the interplay between global symmetries and the topological order. Here, we attempt to build a partial but sys- tematic understanding of gapped quantum phases with both global symmetries and topological orders, which have been termed as SET phases. In particular, we will focus on cases with toric code type topological orders (conventional discrete gauge theory) in 2+1D. And we will consider symmetries include both on-site symmetries and lattice symmetries. We focus on a particular type of tensor networks: Projected Entangled Pairs states (PEPS). We find, in the presence of topological orders and global symmetries, PEPS states are grouped to different classes, which are related,

but not limited to different SET types. For each class, we can write down general variational wavefunctions, which are very useful for numerical simulations.

This chapter is organized as follows. In Sec.3.2, we introduce some basics of PEPS. In particular, We discuss gauge redundancy as well as the implementation of symme- tries in PEPS. We introduce a special kind of gauge transformation named as invariant gauge group (𝐼𝐺𝐺). In phases with no symmetry breaking, 𝐼𝐺𝐺 leads to low-energy gauge dynamics. Further, for fractional filled systems, there are minimal required non- trivial 𝐼𝐺𝐺s for any symmetric PEPS under our basic assumption. This phenomenon is consistent with the Hastings-Oshikawa-Lieb-Schultz-Mattis theorem[61, 103, 90]. In Sec.3.3, we classify symmetric PEPS according to their distinct short-range physics, which is characterized by algebraic data Θ’s, 𝜒’s and 𝜂’s. Relations of the data 𝜒’s and 𝜂’s to second cohomology are discussed.

As a main example, we give the classification result for symmetric PEPS on the kagome lattice with a half-integer spin per site and 𝐼𝐺𝐺 = 𝑍2, and obtain the con- straints on the sub-Hilbert spaces for local tensors for each given class. The detailed calculation is presented in Ref.[76]. We give the physical interpretation of the al- gebraic data in Section 3.4. Particularly, we construct fractionalized symmetry op- erators to explicitly show that 𝜂’s are describing the symmetry fractionalization of spinons in the 𝑍2 QSL member phase. Detectable signatures of the data Θ’s, 𝜒’s and 𝜂’s are discussed. In Sec.4.5 we consider generalizations and limitations of our study, comment on relations with previous works, and conclude.