Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Competing orders, non-linear sigma models, and topological terms in quantum magnets

T. Senthil, Matthew P. A. Fisher

TL;DR

The paper argues that two-dimensional quantum magnets exhibiting non-LGW behavior can be described by field theories built from slow order fluctuations augmented with topological terms, notably an $O(4)$ nonlinear sigma model at $\theta=\pi$ and related WZW terms. It systematically connects quasi-one-dimensional weakly coupled chains, deconfined criticality on the square lattice, and massless QED$_3$ to a unified sigma-model framework, offering an alternative to gauge-theory approaches. Key results include the mapping of Neel–VBS competition to $O(4)$-type models with topological terms, the equivalence to NCCP$^1$ in easy-plane limits, and the emergence of topological terms (Hopf and theta) that govern phase structure and criticality. The findings suggest a broader role for sigma-model descriptions in capturing algebraic spin liquids and deconfined critical points, with implications for both theoretical understanding and experimental interpretation of quantum magnets.

Abstract

A number of examples have demonstrated the failure of the Landau-Ginzburg-Wilson(LGW) paradigm in describing the competing phases and phase transitions of two dimensional quantum magnets. In this paper we argue that such magnets possess field theoretic descriptions in terms of their slow fluctuating orders provided certain topological terms are included in the action. These topological terms may thus be viewed as what goes wrong within the conventional LGW thinking. The field theoretic descriptions we develop are possible alternates to the popular gauge theories of such non-LGW behavior. Examples that are studied include weakly coupled quasi-one dimensional spin chains, deconfined critical points in fully two dimensional magnets, and two component massless $QED_3$. A prominent role is played by an anisotropic O(4) non-linear sigma model in three space-time dimensions with a topological theta term. Some properties of this model are discussed. We suggest that similar sigma model descriptions might exist for fermionic algebraic spin liquid phases.

Competing orders, non-linear sigma models, and topological terms in quantum magnets

TL;DR

The paper argues that two-dimensional quantum magnets exhibiting non-LGW behavior can be described by field theories built from slow order fluctuations augmented with topological terms, notably an nonlinear sigma model at and related WZW terms. It systematically connects quasi-one-dimensional weakly coupled chains, deconfined criticality on the square lattice, and massless QED to a unified sigma-model framework, offering an alternative to gauge-theory approaches. Key results include the mapping of Neel–VBS competition to -type models with topological terms, the equivalence to NCCP in easy-plane limits, and the emergence of topological terms (Hopf and theta) that govern phase structure and criticality. The findings suggest a broader role for sigma-model descriptions in capturing algebraic spin liquids and deconfined critical points, with implications for both theoretical understanding and experimental interpretation of quantum magnets.

Abstract

A number of examples have demonstrated the failure of the Landau-Ginzburg-Wilson(LGW) paradigm in describing the competing phases and phase transitions of two dimensional quantum magnets. In this paper we argue that such magnets possess field theoretic descriptions in terms of their slow fluctuating orders provided certain topological terms are included in the action. These topological terms may thus be viewed as what goes wrong within the conventional LGW thinking. The field theoretic descriptions we develop are possible alternates to the popular gauge theories of such non-LGW behavior. Examples that are studied include weakly coupled quasi-one dimensional spin chains, deconfined critical points in fully two dimensional magnets, and two component massless . A prominent role is played by an anisotropic O(4) non-linear sigma model in three space-time dimensions with a topological theta term. Some properties of this model are discussed. We suggest that similar sigma model descriptions might exist for fermionic algebraic spin liquid phases.

Paper Structure

This paper contains 10 sections, 56 equations, 1 figure.

Table of Contents

  1. Algebraic spin liquid in one dimension
  2. Weakly coupled spin chains
  3. Ordered phases and transitions
  4. Deconfined criticality in the two dimensional square lattice
  5. Tanaka-Hu superspin field theory
  6. Equivalence to NCCP$^1$ field theory
  7. Sigma model description of massless $QED_3$
  8. Isotropic $O(4)$ model with a $\theta$ term
  9. Discussion and prospects
  10. Duality of the $\theta = \pi$ $O(2) \times O(2)$ model to NCCP$^1$

Figures (1)

  • Figure 1: Configuration with $Q = 1$ showing the location of the vortices in $z_1$ and $z_2$. The curve A is the vortex in $z_2$ and B is the vortex in $z_1$.