Abstract

Abstract In primary school, we were told that there are four phases of matter: solid, liquid, gas, and plasma. In college, we learned that there are much more than four phases of matter, such as hundreds of crystal phases, liquid crystal phases, ferromagnet, anti-ferromagnet, superfluid, etc. Those phases of matter are so rich, it is amazing that they can be understood systematically by the symmetry breaking theory of Landau. However, there are even more interesting phases of matter that are beyond Landau symmetry breaking theory. In this paper, we review new ‘topological’ phenomena, such as topological degeneracy, that reveal the existence of those new zero-temperature phase—topologically ordered phases. Microscopically, topologically orders are originated from the patterns of long-range entanglement in the ground states. As a truly new type of order and a truly new kind of phenomena, topological order and long-range entanglement require a new language and a new mathematical framework, such as unitary fusion category and modular tensor category to describe them. In this paper, we will describe a simple mathematical framework based on measurable quantities of topological orders (S, T, c) proposed around 1989. The framework allows us to systematically describe all 2+1D bosonic topological orders (i.e. topological orders in local bosonic/spin/qubit systems).

Highlights

  • Before we present the result from the numerical calculation, let us discuss a stacking operation,[63] denoted by

  • It can be realized by the following filling-fraction ν = 3/2 bosonic fractional quantum Hall (FQH) wave function described by the following pattern of zeros: {nl} = {n0, n1, n2, · · · }: Ψ4B 9/5 : {nl} = 30|30|30| · · ·, (65) i.e. neven = 3 and nodd = 0

  • We review the discovery and development of topological order – a new kind of order beyond Landau symmetry breaking theory in many-body systems

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Summary

INTRODUCTION

Condensed matter physics is a branch of science that study various properties of all kinds of materials, such as mechanical properties, hydrodynamic properties, electric properties, magnetic properties, optical properties, thermal properties, etc. The common theme is the principle of emergence, which states that the properties of a material are mainly determined by how particles are organized in the material. Rapid and exciting developments in FQH effect and in high Tc superconductivity resulted in many new ideas and new concepts Looking back at those new developments, it becomes more and more clear that, in last 25 years, we were witnessing an emergence of a new theme in condensed matter physics. The new theme is associated with new kinds of orders, new states of matter and new class of materials beyond Landau’s symmetry breaking theory. This is an exciting time for condensed matter physics. The new paradigm may even have an impact in our understanding of fundamental questions of nature – the emergence of elementary particles and the four fundamental interactions.[8,9,10,11,12,13]

The discovery of topological order
Topological ground state degeneracy
Topological order and phase transitions
The current systematic theories of topological orders
TOPOLOGICAL EXCITATIONS
Local excitations and topological excitations
Fusion space and internal degrees of freedom for the quasiparticles
Simple type and composite type
Fusion of quasiparticles
Quasiparticle intrinsic spin
Quasiparticle mutual statistics
S and T satisfy:
A numerical approach
The stacking operation of topological order
Non-Abelian type of topological order
Quantum dimensions as algebraic numbers
Other topological orders beyond parafermion non-Abelian type
Abelian topological orders
Non-Abelian topological orders of Zn-parafermion type
VIII. SUMMARY
The space-time world lines and quasiparticle tunneling process
Planar string configurations
The first type of linear relations: the F-move
The second type of linear relations: the O-move
The third type of linear relations: the Y-move iα j
A freedom of changing basis at each vertex
10. The fourth type of linear relations: the H-move
11. Summary of the conditions on the linear relations
12. A derivation of string fusion algebra
Amplitudes for loops
Amplitude for linked loops and Verlinde formula
The relation between SLnk and S
The spin si of topological excitations and T
Relation between Nkij and Sij

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