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Three dimensional Symmetry Protected Topological Phase close to Antiferromagnetic Neel order

Cenke Xu

TL;DR

This work extends the concept of symmetry-protected topological (SPT) phases from 1D Haldane systems to a 3D bosonic spin system with self-conjugate SU(2N) representations, showing that a $3+1$-D nonlinear sigma model on the manifold $\mathcal{M}=\mathrm{U}(2N)/[\mathrm{U}(N)\times \mathrm{U}(N)]$ with a theta-term $\Theta$ yields a nontrivial SPT at $\Theta=2\pi$. The authors derive a boundary theory—a $2+1$-D NLSM with a Wess-Zumino-Witten term at level $k=1$—and demonstrate, via symmetry reduction to $\mathrm{SU}(N)\times\mathrm{SU}(N)\rtimes \mathbb{Z}_2$, a corresponding boundary $\mathrm{SU}(N)$ principal chiral model with $\Theta'=\pi$, which cannot be realized in a strictly 2D system with the same symmetry. A concrete lattice realization using slave fermions on a diamond lattice reproduces the field-theory predictions and reveals domain-wall edge CFTs, with the color-singlet constraint essential to preserve the SPT phase. The study argues that the phase is protected by $\mathrm{SU}(2N)$ or its subgroup and may generalize to other odd dimensions, suggesting a path toward PSU$(2N)$-based classifications in higher-dimensional SPTs.

Abstract

It is well-known that the Haldane phase of one-dimensional spin-1 chain is a symmetry protected topological (SPT) phase, which is described by a nonlinear Sigma model (NLSM) with a Theta-term at Theta = 2Pi. In this work we study a three dimensional SPT phase of SU(2N) antiferromagnetic spin system with a self-conjugate representation on every site. The spin ordered Neel phase of this system has a ground state manifold M = U(2N)/[U(N)xU(N)], and this system is described by a NLSM defined with manifold M. Since the homotopy group Pi4[M] = Z for N > 1, this NLSM can naturally have a Theta-term. We will argue that when Theta = 2Pi this NLSM describes a SPT phase. This SPT phase is protected by the SU(2N) spin symmetry, or its subgroup SU(N)xSU(N)xZ2, without assuming any other discrete symmetry. We will also construct a trial SU(2N) spin state on a 3d lattice, we argue that the long wavelength physics of this state is precisely described by the aforementioned NLSM with Theta = 2Pi.

Three dimensional Symmetry Protected Topological Phase close to Antiferromagnetic Neel order

TL;DR

This work extends the concept of symmetry-protected topological (SPT) phases from 1D Haldane systems to a 3D bosonic spin system with self-conjugate SU(2N) representations, showing that a -D nonlinear sigma model on the manifold with a theta-term yields a nontrivial SPT at . The authors derive a boundary theory—a -D NLSM with a Wess-Zumino-Witten term at level —and demonstrate, via symmetry reduction to , a corresponding boundary principal chiral model with , which cannot be realized in a strictly 2D system with the same symmetry. A concrete lattice realization using slave fermions on a diamond lattice reproduces the field-theory predictions and reveals domain-wall edge CFTs, with the color-singlet constraint essential to preserve the SPT phase. The study argues that the phase is protected by or its subgroup and may generalize to other odd dimensions, suggesting a path toward PSU-based classifications in higher-dimensional SPTs.

Abstract

It is well-known that the Haldane phase of one-dimensional spin-1 chain is a symmetry protected topological (SPT) phase, which is described by a nonlinear Sigma model (NLSM) with a Theta-term at Theta = 2Pi. In this work we study a three dimensional SPT phase of SU(2N) antiferromagnetic spin system with a self-conjugate representation on every site. The spin ordered Neel phase of this system has a ground state manifold M = U(2N)/[U(N)xU(N)], and this system is described by a NLSM defined with manifold M. Since the homotopy group Pi4[M] = Z for N > 1, this NLSM can naturally have a Theta-term. We will argue that when Theta = 2Pi this NLSM describes a SPT phase. This SPT phase is protected by the SU(2N) spin symmetry, or its subgroup SU(N)xSU(N)xZ2, without assuming any other discrete symmetry. We will also construct a trial SU(2N) spin state on a 3d lattice, we argue that the long wavelength physics of this state is precisely described by the aforementioned NLSM with Theta = 2Pi.

Paper Structure

This paper contains 6 sections, 24 equations, 1 figure.

Table of Contents

  1. 1. Introduction
  2. 2. Haldane phase
  3. 3. 3d SPT phase of SU(2$N$) spin system
  4. 3.1 Field Theory Description
  5. 3.2 Lattice Construction
  6. 4. Discussion and Summary

Figures (1)

  • Figure 1: ($a$). The pictorial representation of the Haldane phase, where the blue and red links stand for the valence bonds of slave fermions with color $A = 1$ and $2$ respectively. ($b$) A domain wall of $\Theta^\prime$ on the boundary of our 3$d$ SPT. ($c$) The Young diagram of the self-conjugate representation of the SU(2$N$) spin system that we are considering.