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String and Membrane condensation on 3D lattices

Alioscia Hamma, Paolo Zanardi, Xiao Gang Wen

TL;DR

This work analyzes three exactly solvable 3D lattice models that realize $Z_2$ string condensation and distinct topological orders. By constructing and comparing string and membrane operators, the authors classify how closed-string and closed-membrane condensations coexist and how the statistics of string ends and membrane edges shape the ground-state structure. The models reveal a spectrum of topological orders: a conventional $Z_2$ gauge theory with both string and membrane condensation, a string-condensation-only phase with fermionic endpoints, and a novel phase where string ends are bosonic but membrane edges are fermionic. The results illuminate how higher-dimensional condensates encode robust ground-state degeneracy and nontrivial operator algebras, contributing new examples of topological order and guiding principles for designing exactly solvable 3D quantum spin liquids.

Abstract

In this paper, we investigate the general properties of lattice spin models that have string and/or membrane condensed ground states. We discuss the properties needed to define a string or membrane operator. We study three 3D spin models which lead to Z_2 gauge theory at low energies. All the three models are exactly soluble and produce topologically ordered ground states. The first model contains both closed-string and closed-membrane condensations. The second model contains closed-string condensation only. The ends of open-strings behave like fermionic particles. The third model also has condensations of closed membranes and closed strings. The ends of open strings are bosonic while the edges of open membranes are fermionic. The third model contains a new type of topological order.

String and Membrane condensation on 3D lattices

TL;DR

This work analyzes three exactly solvable 3D lattice models that realize string condensation and distinct topological orders. By constructing and comparing string and membrane operators, the authors classify how closed-string and closed-membrane condensations coexist and how the statistics of string ends and membrane edges shape the ground-state structure. The models reveal a spectrum of topological orders: a conventional gauge theory with both string and membrane condensation, a string-condensation-only phase with fermionic endpoints, and a novel phase where string ends are bosonic but membrane edges are fermionic. The results illuminate how higher-dimensional condensates encode robust ground-state degeneracy and nontrivial operator algebras, contributing new examples of topological order and guiding principles for designing exactly solvable 3D quantum spin liquids.

Abstract

In this paper, we investigate the general properties of lattice spin models that have string and/or membrane condensed ground states. We discuss the properties needed to define a string or membrane operator. We study three 3D spin models which lead to Z_2 gauge theory at low energies. All the three models are exactly soluble and produce topologically ordered ground states. The first model contains both closed-string and closed-membrane condensations. The second model contains closed-string condensation only. The ends of open-strings behave like fermionic particles. The third model also has condensations of closed membranes and closed strings. The ends of open strings are bosonic while the edges of open membranes are fermionic. The third model contains a new type of topological order.

Paper Structure

This paper contains 6 sections, 54 equations, 7 figures.

Table of Contents

  1. Introduction
  2. $Z_2$ lattice gauge theory -- a model with string and membrane condensation
  3. A model with string condensation only
  4. Exactly solvable model with membrane condensation
  5. Edges of open membranes
  6. Conclusions

Figures (7)

  • Figure 1: A three dimensional model with Majorana fermions on the sites. The six Majorana fermions label the six vectors from a site in the following way: $x\mapsto +\hat{x}, \overline{x}\mapsto -\hat{x},y\mapsto \hat{x}, \overline{y}\mapsto -\hat{y},z\mapsto \hat{z}, \overline{z}\mapsto -\hat{z}.$ The plaquettes in the three planes are shown. $\hat{F}_{p_{xy}},\hat{F}_{p_{yz}},\hat{F}_{p_{zx}}$ are respectively the plaquettes $\hat{F}_{{\bf i}_1{\bf i}_2{\bf i}_3{\bf i}_4},\hat{F}_{{\bf i}_2{\bf i}_3{\bf i}_6{\bf i}_7},\hat{F}_{{\bf i}_4{\bf i}_3{\bf i}_6{\bf i}_5}$ where ${\bf i}_1={\bf i};{\bf i}_2={\bf i+\hat{x}};{\bf i}_3={\bf i+\hat{x}+\hat{y}};{\bf i}_4={\bf i+\hat{y}};{\bf i}_5={\bf i+\hat{y}+\hat{z}};{\bf i}_6={\bf i+\hat{x}+\hat{y}+\hat{z}};{\bf i}_7={\bf i+\hat{x}+\hat{z}}.$
  • Figure 2: The dash-lines represent the cube in the cubic lattice. The solid lines represent the octahedron.
  • Figure 3: The Cube operator and the Corner Loop operator with the nomenclature of the Majorana fermions forming a corner loop.
  • Figure 4: A triple of not commuting corner loops. The corner operator above does not commute with the two below. Each corner term affects three cubes.
  • Figure 5: The Cube operator and four Corner Loop operators. The cube is made of the product of only four such loops. These four operators all commute with each other because they have no Majorana fermion in common. The four Corner Loops shown in the figure are chosen on the odd sites.
  • ...and 2 more figures