Three-dimensional symmetry-protected topological phase close to antiferromagnetic Néel order

Abstract

It is well known that the Haldane phase of a one-dimensional spin-1 chain is a symmetry-protected topological (SPT) phase, which is described by a nonlinear sigma model (NLSM) with a $\ensuremath{\Theta}$ term at $\ensuremath{\Theta}=2\ensuremath{\pi}$. In this work we study a three-dimensional (3d) SPT phase of a SU(2$N$) antiferromagnetic spin system with a self-conjugate representation on every site. The spin-ordered N\'eel phase of this system has a ground state manifold $\mathcal{M}=\frac{\mathrm{U}(2N)}{\mathrm{U}(N)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(N)}$, and this system is described by a NLSM defined with manifold $\mathcal{M}$. Since the homotopy group ${\ensuremath{\pi}}_{4}[\mathcal{M}]=\mathbb{Z}$ for $N>1$, this NLSM can naturally have a $\ensuremath{\Theta}$ term. We will argue that when $\ensuremath{\Theta}=2\ensuremath{\pi}$ this NLSM describes a SPT phase. This SPT phase is protected by the SU(2$N$) spin symmetry, or its subgroup SU($N$)$\ifmmode\times\else\texttimes\fi{}$SU$(N)\ensuremath{\rtimes}{Z}_{2}$, without assuming any other discrete symmetry. We will also construct a trial SU(2$N$) spin state on a 3d lattice; we argue that the long-wavelength physics of this state is precisely described by the aforementioned NLSM with $\ensuremath{\Theta}=2\ensuremath{\pi}$.