Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Topological Order and Conformal Quantum Critical Points

Eddy Ardonne, Paul Fendley, Eduardo Fradkin

TL;DR

The paper develops a framework in which certain 2+1D quantum systems possess ground-state wave functionals that are local two-dimensional classical weights, yielding a time-independent conformal structure in space and a dynamical exponent z=2 (quantum Lifshitz) at conformal quantum critical points. It applies this to quantum dimer models and quantum eight-vertex models, deriving exact ground states and mapping equal-time correlators to 2D conformal field theories, including Gaussian/free-boson descriptions with continuously varying exponents along critical lines and to topologically ordered phases such as Z2 and non-Abelian gauge theories. The work constructs explicit RK-type and other lattice Hamiltonians whose ground states are exactly known, enabling precise analyses of confinement, deconfinement, and phase diagrams, and connects these 2D quantum criticalities to 2D classical and 2D WZW/Chern–Simons frameworks, including doubled theories to preserve gauge invariance. These insights illuminate the role of topological order in 2D quantum systems and have potential implications for topological quantum computation through explicit models with robust, gapped topological phases.

Abstract

We discuss a certain class of two-dimensional quantum systems which exhibit conventional order and topological order, as well as two-dimensional quantum critical points separating these phases. All of the ground-state equal-time correlators of these theories are equal to correlation functions of a local two-dimensional classical model. The critical points therefore exhibit a time-independent form of conformal invariance. These theories characterize the universality classes of two-dimensional quantum dimer models and of quantum generalizations of the eight-vertex model, as well as Z_2 and non-abelian gauge theories. The conformal quantum critical points are relatives of the Lifshitz points of three-dimensional anisotropic classical systems such as smectic liquid crystals. In particular, the ground-state wave functional of these quantum Lifshitz points is just the statistical (Gibbs) weight of the ordinary 2D free boson, the 2D Gaussian model. The full phase diagram for the quantum eight-vertex model exhibits quantum critical lines with continuously-varying critical exponents separating phases with long-range order from a Z_2 deconfined topologically-ordered liquid phase. We show how similar ideas also apply to a well-known field theory with non-abelian symmetry, the strong-coupling limit of 2+1-dimensional Yang-Mills gauge theory with a Chern-Simons term. The ground state of this theory is relevant for recent theories of topological quantum computation.

Topological Order and Conformal Quantum Critical Points

TL;DR

The paper develops a framework in which certain 2+1D quantum systems possess ground-state wave functionals that are local two-dimensional classical weights, yielding a time-independent conformal structure in space and a dynamical exponent z=2 (quantum Lifshitz) at conformal quantum critical points. It applies this to quantum dimer models and quantum eight-vertex models, deriving exact ground states and mapping equal-time correlators to 2D conformal field theories, including Gaussian/free-boson descriptions with continuously varying exponents along critical lines and to topologically ordered phases such as Z2 and non-Abelian gauge theories. The work constructs explicit RK-type and other lattice Hamiltonians whose ground states are exactly known, enabling precise analyses of confinement, deconfinement, and phase diagrams, and connects these 2D quantum criticalities to 2D classical and 2D WZW/Chern–Simons frameworks, including doubled theories to preserve gauge invariance. These insights illuminate the role of topological order in 2D quantum systems and have potential implications for topological quantum computation through explicit models with robust, gapped topological phases.

Abstract

We discuss a certain class of two-dimensional quantum systems which exhibit conventional order and topological order, as well as two-dimensional quantum critical points separating these phases. All of the ground-state equal-time correlators of these theories are equal to correlation functions of a local two-dimensional classical model. The critical points therefore exhibit a time-independent form of conformal invariance. These theories characterize the universality classes of two-dimensional quantum dimer models and of quantum generalizations of the eight-vertex model, as well as Z_2 and non-abelian gauge theories. The conformal quantum critical points are relatives of the Lifshitz points of three-dimensional anisotropic classical systems such as smectic liquid crystals. In particular, the ground-state wave functional of these quantum Lifshitz points is just the statistical (Gibbs) weight of the ordinary 2D free boson, the 2D Gaussian model. The full phase diagram for the quantum eight-vertex model exhibits quantum critical lines with continuously-varying critical exponents separating phases with long-range order from a Z_2 deconfined topologically-ordered liquid phase. We show how similar ideas also apply to a well-known field theory with non-abelian symmetry, the strong-coupling limit of 2+1-dimensional Yang-Mills gauge theory with a Chern-Simons term. The ground state of this theory is relevant for recent theories of topological quantum computation.

Paper Structure

This paper contains 20 sections, 126 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Scale-Invariant Wave Functions and Quantum Criticality
  3. Dimers, fermions and the quantum Lifshitz field theory
  4. From the square to the triangular lattice
  5. The critical field theory
  6. The off-critical field theory
  7. A quantum eight-vertex model
  8. Construction of the quantum eight-vertex Hamiltonian
  9. The phase diagram
  10. Operators
  11. Another quantum eight-vertex model
  12. Non-Abelian Topological States
  13. Field theories with continuous non-abelian symmetries
  14. Yang-Mills, Chern-Simons, and Wess-Zumino-Witten
  15. The doubled theory
  16. ...and 5 more sections

Figures (9)

  • Figure 1: Dimer flips on the triangular lattice
  • Figure 2: 4 height configurations
  • Figure 3: The eight vertices and their Boltzmann weights
  • Figure 4: A typical configuration in the eight-vertex model
  • Figure 5: Phase diagram of the quantum eight-vertex model: phases $I$ and $II$ are separated by a dual $6$-vertex transition (same with $I$ and $III$); $6$-vertex denotes the $6$-vertex model critical lines and KT are $2D$ Kosterlitz-Thouless transitions; the dotted line shows that the Kitaev point is smoothly connected to the critical regime of the eight-vertex model.
  • ...and 4 more figures