Each chapter of this thesis is based on separate pieces of work where all except one has been published in peer reviewed journals. The chapters has been kept as much as possible close to the original published work. Therefore each chapter is largely self-containted, have its own introduction and conclusion, and can in principle be read without reading other chapters.
In chapter 2 we propose a way, called Universal Wave Function Overlap, to extract universal topological data from generic ground states of gapped systems in any dimen- sions. Those extracted topological data should fully characterize the topological orders with gapped or gapless boundary. For non-chiral topological orders in 2+1D, this universal topological data consist of two matrices, S and T, which generate a projective representa- tion of SL(2, Z) on the degenerate ground state Hilbert space on a torus. For topological orders with gapped boundary in higher dimensions, this data constitutes a projective rep- resentation of the mapping class group MCG(Md) of closed spatial manifold Md. For a set of simple models and perturbations in two dimensions, we show that these quantities are protected to all orders in perturbation theory. We also propose a simple and effective numerical algorithm to compute these quantities using intrinsic gauge structure of local tensors in a Projected Entangled Pair States (PEPS) tensor network.
Based on the proposal in chapter 2, in chapter 3 we introduce a systematic numerical method based on tensor networks to calculate modular S and T matrices in 2+1D systems, which might fully identify topological order with gapped edges. Moreover, it is shown numerically that modular matrices, including S and T matrices, are robust characterization to describe phase transitions between topologically ordered states and trivial states, which can work as topological order parameters. This method only requires local information of one ground state in the form of a tensor network, and directly provides the universal data (S and T matrices), without any non-universal contributions. Furthermore it is generalizable to higher dimensions. Unlike calculating topological entanglement entropy by extrapolating, which numerical complexity is exponentially high, this method extracts a much more complete set of topological data (modular matrices) with much lower numerical cost.
I chapter 2 we conjectured that a certain set of universal topological quantities char- acterize topological order in any dimension. Those quantities can be extracted from the universal overlap of the ground state wave functions. For systems with gapped bound- aries, these quantities are representations of the mapping class group MCG(M ) of the space manifold M on which the systems lives. In chapter4 we consider simple examples in 3 + 1 dimensions and give physical interpretation of these quantities, related to fusion algebra and statistics of particle and string excitations. In particular, we will consider dimensional reduction from 3+1D to 2+1D, and show how the induced 2+1D topological data contains information on the fusion and the braiding of non-Abelian string excitations in 3D. These universal quantities generalize the well-known modular S and T matrices to any dimension. In a system with chiral topological order, there is a remarkable correspondence between the edge and entanglement spectra: the low-energy spectrum of the system in the presence of a physical edge coincides with the lowest part of the entanglement spectrum (ES) across a virtual cut of the system, up to rescaling and shifting. In chapter5, we explore whether the edge-ES correspondence extends to nonchiral topological phases. Specifically, we consider the Wen-plaquette model which has Z2 topological order. The unperturbed model displays an exact correspondence: both the edge and entanglement spectra within each topological sector a(a = 1, ..., 4) are flat and equally degenerate. Here, we show, through a detailed microscopic calculation, that in the presence of generic local perturbations: (i) the effective degrees of freedom for both the physical edge and the entanglement cut consist of a spin- 1/2 chain, with effective Hamiltonians Ha
edge and Hent.a , respectively, both of which have a Z2 symmetry enforced by the bulk topological order; (ii) there is in general no match between their low energy spectra, that is, there is no edge-ES correspondence. However, if supplement the Z2 topological order with a global symmetry (translational invariance along the edge/cut), i.e. by considering the Wen-plaquette model as a symmetry enriched topological phase (SET), then there is a finite domain in Hamiltonian space in which both Ha
edge and Hent.a realize the critical Ising model, whose low-energy effective theory is the c = 1/2 Ising CFT. This is achieved because the presence of the global symmetry implies that both Hamiltonians, in addition to being Z2 symmetric, are Kramers-Wannier self-dual. Thus, the bulk topological order and the global translational symmetry of the Wen-plaquette model as a SET imply an edge-ES correspondence at least in some finite domain in Hamiltonian space.
It is well known that the bulk physics of a topological phase constrains its possible edge physics through the bulk-edge correspondence. Therefore, the different types of edge theories that a topological phase can host constitute a universal piece of data which can be used to characterize topological order. In chapter 6, we argue that, beginning from only the fixed-point wave function (FPW) of a nonchiral topological phase and by locally
deforming it, all possible edge theories can be extracted from its entanglement Hamiltonian (EH). We give a general argument, and concretely illustrate our claim by deforming the FPW of the Wen-plaquette model, the quantum double of Z2. In that case, we show that the possible EHs of the deformed FPW reflect the known possible types of edge theories, which are generically gapped, but gapless if translational symmetry is preserved. We stress that our results do not require an underlying Hamiltonian and thus, this lends support to the notion that a topological phase is indeed characterized by only a set of quantum states and can be studied through its FPWs.
In chapter7we introduce the notion of fermionic gapped edges, a new kind of topological gapped boundary theory of a bosonic abelian topological state. These gapped edges exist naturally if the bosonic topological order is emergent from original, local fermionic degrees of freedom, so that domain walls between the bosonic topological state and a fermionic vacuum (such as a trivial band insulator) must be considered. Using the framework of La- grangian subgroups of [89] and [90, 91], we argue that the condition that the self-statistics of quasiparticles in a Lagrangian subgroup to be self-bosons should be removed. Physically, this implies that quasiparticles which are self-fermions can possibly condense on the bound- ary, leading to these fermionic gapped edges. We illustrate the presence of such a fermionic gapped edge in a system with bosonic Z2 topological order, and explicitly construct a mi- croscopic model, the Z2 Wen-plaquette model coupled to an array of Majorana fermions (which can be considered as the ferminonic vacuum). We explore the rich phase diagram of the edge theory of this model, which can be mapped to a variant of the Ashkin-Teller model, and show that there are critical lines of c = 1 Z2-orbifold boson CFTs separating the three gapped phases, including critical points with exotic symmetries such as twisted N = 2 supersymmetry. We also see that the notion of fermionic gapped edge leads to an enhancement of the anyonic symmetries, as far as boundary physics is concerned, giving rise to non-abelian Kramers-Wannier Dualities on the boundary, constraining the structure of the boundary phase diagram.
Chapter 2
Universal Wave Function Overlap
and Universal Topological Data from
Generic Gapped Ground States
This chapter was published in [1].