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

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 Θ term at Θ=2π. In this work we study a three-dimensional (3d) SPT phase of a SU(2N) 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 M=U(2N)U(N)×U(N), and this system is described by a NLSM defined with manifold M. Since the homotopy group π₄[M]=Z for $N>1$, this NLSM can naturally have a Θ term. We will argue that when Θ=2π this NLSM describes a SPT phase. This SPT phase is protected by the SU(2N) spin symmetry, or its subgroup SU(N)×SU(N)Z₂, 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 Θ=2π.