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 ${\mathrm{QED}}_{3}$ . A prominent role is played by an anisotropic O(4) nonlinear sigma model in three space-time dimensions with a topological theta term. Some properties of this model are discussed. We speculate that similar sigma model descriptions might exist for fermionic algebraic spin liquid phases.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 ${\mathrm{QED}}_{3}$. A prominent role is played by an anisotropic O(4) nonlinear sigma model in three space-time dimensions with a topological theta term. Some properties of this model are discussed. We speculate that similar sigma model descriptions might exist for fermionic algebraic spin liquid phases.

arXiv:cond-mat/0510459Physical Review BAug 8, 2006226 citations

Topological Terms, Quantum Magnets and Deconfined Criticality

Adam Nahum

Springer thesesOct 1, 20140 citations

A short guide to topological terms in the effective theories of condensed matter

Akihiro Tanaka, Shintaro Takayoshi

This article is meant as a gentle introduction to the topological terms that often play a decisive role in effective theories describing topological quantum effects in condensed matter systems. We first take up several prominent examples, mainly from the area of quantum magnetism and superfluids/superconductors. We then briefly discuss how these ideas are now finding incarnations in the studies of symmetry-protected topological phases, which are in a sense a generalization of the concept of topological insulators to a wider range of materials, including magnets and cold atoms.This article is meant as a gentle introduction to the topological terms that often play a decisive role in effective theories describing topological quantum effects in condensed matter systems. We first take up several prominent examples, mainly from the area of quantum magnetism and superfluids/superconductors. We then briefly discuss how these ideas are now finding incarnations in the studies of symmetry-protected topological phases, which are in a sense a generalization of the concept of topological insulators to a wider range of materials, including magnets and cold atoms.

arXiv:1502.05464Science and Technology of Advanced MaterialsFeb 18, 201511 citations

Variational model for one-dimensional quantum magnets

Yu. B. Kudasov, R. V. Kozabaranov

arXiv:1801.10602Physics Letters AFeb 27, 20184 citations

Effective Field Theories for Topological States of Matter

T. H. Hansson, Thomas Klein Kvorning

Springer proceedings in physicsJan 1, 20200 citations

Geometric phases and the magnetization process in quantum antiferromagnets

Akihiro Tanaka, Keisuke Totsuka, Xiao Hu

The physics underlying the magnetization process of quantum antiferromagnets is revisited from the viewpoint of geometric phases. A continuum variant of the Lieb-Schultz-Mattis-type approach to the problem is put forth, where the commensurability condition of Oshikawa et al. derives from a Berry connection formulation of the system's crystal momentum. We then go on to formulate an effective field theory which can deal with higher dimensional cases as well. We find that a topological term, whose principle function is to assign Berry phase factors to space-time vortex objects, ultimately controls the magnetic behavior of the system. We further show how our effective action maps into a ${\mathbf{Z}}_{2}$ gauge theory under certain conditions, which in turn allows for the occurrence of a fractionalized phase with topological order.The physics underlying the magnetization process of quantum antiferromagnets is revisited from the viewpoint of geometric phases. A continuum variant of the Lieb-Schultz-Mattis-type approach to the problem is put forth, where the commensurability condition of Oshikawa et al. derives from a Berry connection formulation of the system's crystal momentum. We then go on to formulate an effective field theory which can deal with higher dimensional cases as well. We find that a topological term, whose principle function is to assign Berry phase factors to space-time vortex objects, ultimately controls the magnetic behavior of the system. We further show how our effective action maps into a ${\mathbf{Z}}_{2}$ gauge theory under certain conditions, which in turn allows for the occurrence of a fractionalized phase with topological order.

arXiv:0712.4316Physical Review BFeb 17, 200931 citations

{\mathrm{QED}}_{3}

. A prominent role is played by an anisotropic O(4) nonlinear sigma model in three space-time dimensions with a topological theta term. Some properties of this model are discussed. We speculate that similar sigma model descriptions might exist for fermionic algebraic spin liquid phases.

{\mathbf{Z}}_{2}

gauge theory under certain conditions, which in turn allows for the occurrence of a fractionalized phase with topological order.