The subject of this paper is the phase transition between symmetry protected topological states (SPTs). We consider spatial dimension d and symmetry group G so that the cohomology group, Hd+1(G,U(1)), contains at least one Z2n or Z factor. We show that the phase transition between the trivial SPT and the root states that generate the Z2n or Z groups can be induced on the boundary of a (d+1)-dimensional G×Z2T-symmetric SPT by a Z2T symmetry breaking field. Moreover we show these boundary phase transitions can be “transplanted” to d dimensions and realized in lattice models as a function of a tuning parameter. The price one pays is for the critical value of the tuning parameter there is an extra non-local (duality-like) symmetry. In the case where the phase transition is continuous, our theory predicts the presence of unusual (sometimes fractionalized) excitations corresponding to delocalized boundary excitations of the non-trivial SPT on one side of the transition. This theory also predicts other phase transition scenarios including first order transition and transition via an intermediate symmetry breaking phase.
The study of continuous phase transitions triggered by spontaneous symmetry breaking has brought revolutionary ideas to physics. Recently, through the discovery of symmetry protected topological phases, it is realized that continuous quantum phase transition can also occur between states with the same symmetry but different topology. Here we study a specific class of such phase transitions in 1 + 1 dimensions – the phase transition between bosonic topological phases protected by Z n × Z n . We find in all cases the critical point possesses two gap opening relevant operators: one leads to a Landau-forbidden symmetry breaking phase transition and the other to the topological phase transition. We also obtained a constraint on the central charge for general phase transitions between symmetry protected bosonic topological phases in 1 + 1 D .
Symmetry protected topological (SPT) phases are gapped short-range-entangled quantum phases with a symmetry
G. They can all be smoothly connected to the same trivial product state if we break the symmetry. The Haldane phase of spin-1 chain is the first example of SPT phases which is protected by
SO(3) spin rotation symmetry. The topological insulator is another example of SPT phases which are protected by
U(1) and time-reversal symmetries. In this paper, we show that interacting bosonic SPT phases can be systematically described by group cohomology theory: Distinct
d-dimensional bosonic SPT phases with on-site symmetry
G (which may contain antiunitary time-reversal symmetry) can be labeled by the elements in
H1+d[G,UT(1)], the Borel
(1+d)-group-cohomology classes of
G over the
G module
UT(1). Our theory, which leads to explicit ground-state wave functions and commuting projector Hamiltonians, is based on a new type of topological term that generalizes the topological $\ensuremath{\theta}$ term in continuous nonlinear $\ensuremath{\sigma}$ models to lattice nonlinear $\ensuremath{\sigma}$ models. The boundary excitations of the nontrivial SPT phases are described by lattice nonlinear $\ensuremath{\sigma}$ models with a nonlocal Lagrangian term that generalizes the Wess-Zumino-Witten term for continuous nonlinear $\ensuremath{\sigma}$ models. As a result, the symmetry
G must be realized as a non-on-site symmetry for the low-energy boundary excitations, and those boundary states must be gapless or degenerate. As an application of our result, we can use ${\mathcal{H}}^{1+d}[U(1)\ensuremath{\rtimes}{Z}_{2}^{T},{U}_{T}(1)]$ to obtain interacting bosonic topological insulators (protected by time reversal
Z2T and boson number conservation), which contain one nontrivial phase in one-dimensional (1D) or 2D and three in 3D. We also obtain interacting bosonic topological superconductors (protected by time-reversal symmetry only), in term of
H1+d[Z2T,UT(1)], which contain one nontrivial phase in odd spatial dimensions and none for even dimensions. Our result is much more general than the above two examples, since it is for any symmetry group. For example, we can use ${\mathcal{H}}^{1+d}[U(1)\ifmmode\times\else\texttimes\fi{}{Z}_{2}^{T},{U}_{T}(1)]$ to construct the SPT phases of integer spin systems with time-reversal and
U(1) spin rotation symmetry, which contain three nontrivial SPT phases in 1D, none in 2D, and seven in 3D. Even more generally, we find that the different bosonic symmetry breaking short-range-entangled phases are labeled by the following three mathematical objects: $({G}_{H},{G}_{\ensuremath{\Psi}},{\mathcal{H}}^{1+d}[{G}_{\ensuremath{\Psi}},{U}_{T}(1)])$, where
GH is the symmetry group of the Hamiltonian and ${G}_{\ensuremath{\Psi}}$ the symmetry group of the ground states.
Decades of research have revealed a deep understanding of topological quantum matter with protected edge modes. We report that even richer physics emerges when tuning between two topological phases of matter whose respective edge modes are incompatible. The frustration at the edge leads to novel boundary physics, such as symmetry-breaking phases with exotic non-Landau transitions---even when the edge is zero-dimensional. As a minimal case study, we consider spin chains with ${\mathbb{Z}}_{3}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{3}$ symmetry, exhibiting two nontrivial symmetry-protected topological (SPT) phases. At the bulk 1+1D critical transition between these SPT phases, we find two stable 0+1D boundary phases, each spontaneously breaking one of the
Z3 symmetries. Furthermore, we find that a single boundary parameter tunes a non-Landau boundary critical transition between these two phases. This constitutes a 0+1D version of an exotic phenomenon driven by charged vortex condensation known as deconfined quantum criticality. This work highlights the rich unexplored physics of criticality between nontrivial topological phases and provides insights into the burgeoning field of gapless topological phases.
We propose a series of simple two-dimensional (2D) lattice interacting fermion models that we demonstrate at low energy describe bosonic symmetry-protected topological (SPT) states and quantum phase transitions between them. This is because due to interaction, the fermions are gapped both at the boundary of the SPT states and at the bulk quantum phase transition, thus these models at low energy can be described completely by bosonic degrees of freedom. We show that the bulk of these models is described by a
Sp(N) principal chiral model with a topological $\mathrm{\ensuremath{\Theta}}$ term, whose boundary is described by a
Sp(N) principal chiral model with a Wess-Zumino-Witten term at level 1. The quantum phase transition between SPT states in the bulk is tuned by a particular interaction term, which corresponds to tuning $\mathrm{\ensuremath{\Theta}}$ in the field theory, and the phase transition occurs at $\mathrm{\ensuremath{\Theta}}=\ensuremath{\pi}$. The simplest version of these models with
N=1 is equivalent to the familiar O(4) nonlinear sigma model (NLSM) with a topological term, whose boundary is a
(1+1)D conformal field theory with central charge
c=1. After breaking the O(4) symmetry to its subgroups, this model can be viewed as bosonic SPT states with U(1), or
Z2 symmetries, etc. All of these fermion models, including the bulk quantum phase transitions, can be simulated with the determinant quantum Monte Carlo method without the sign problem. Recent numerical results strongly suggest that the quantum disordered phase of the O(4) NLSM with precisely $\mathrm{\ensuremath{\Theta}}=\ensuremath{\pi}$ is a stable
(2+1)D conformal field theory with gapless bosonic modes.