The conditions for both the stability and the breakdown of the topological classification of gapped ground states of noninteracting fermions, the tenfold way, in the presence of quartic fermion-fermion interactions are given for any dimension of space. This is achieved by encoding the effects of interactions on the boundary gapless modes in terms of boundary dynamical masses. Breakdown of the noninteracting topological classification occurs when the quantum nonlinear $\ensuremath{\sigma}$ models for the boundary dynamical masses favor quantum disordered phases. For the tenfold way, we find that (i) the noninteracting topological classification ${\mathbb{Z}}_{2}^{\phantom{\rule{0.16em}{0ex}}}$ is always stable, (ii) the noninteracting topological classification $\mathbb{Z}$ in even dimensions is always stable, and (iii) the noninteracting topological classification $\mathbb{Z}$ in odd dimensions is unstable and reduces to ${\mathbb{Z}}_{N}^{\phantom{\rule{0.16em}{0ex}}}$ that can be identified explicitly for any dimension and any defining symmetries. We also apply our method to the three-dimensional topological crystalline insulator SnTe from the symmetry class $\mathrm{AII}+R$ , for which we establish the reduction $\mathbb{Z}\ensuremath{\rightarrow}{\mathbb{Z}}_{8}^{\phantom{\rule{0.16em}{0ex}}}$ of the noninteracting topological classification.The conditions for both the stability and the breakdown of the topological classification of gapped ground states of noninteracting fermions, the tenfold way, in the presence of quartic fermion-fermion interactions are given for any dimension of space. This is achieved by encoding the effects of interactions on the boundary gapless modes in terms of boundary dynamical masses. Breakdown of the noninteracting topological classification occurs when the quantum nonlinear $\ensuremath{\sigma}$ models for the boundary dynamical masses favor quantum disordered phases. For the tenfold way, we find that (i) the noninteracting topological classification ${\mathbb{Z}}_{2}^{\phantom{\rule{0.16em}{0ex}}}$ is always stable, (ii) the noninteracting topological classification $\mathbb{Z}$ in even dimensions is always stable, and (iii) the noninteracting topological classification $\mathbb{Z}$ in odd dimensions is unstable and reduces to ${\mathbb{Z}}_{N}^{\phantom{\rule{0.16em}{0ex}}}$ that can be identified explicitly for any dimension and any defining symmetries. We also apply our method to the three-dimensional topological crystalline insulator SnTe from the symmetry class $\mathrm{AII}+R$, for which we establish the reduction $\mathbb{Z}\ensuremath{\rightarrow}{\mathbb{Z}}_{8}^{\phantom{\rule{0.16em}{0ex}}}$ of the noninteracting topological classification.

arXiv:1505.06341Physical Review BSep 1, 2015103 citations

Classification of topological insulators and superconductors out of equilibrium

Max McGinley, Nigel R. Cooper

The authors have discovered a topological classification of quantum many-body systems undergoing dynamical evolution, which can be applied to insulators and superconductors driven out of equilibrium. This classification is analogous to the equilibrium classification of topological insulators, which explains the remarkably quantized experimental signatures that these phases are known for. The nonequilibrium classification successfully predicts whether topological edge modes remain stable under external time-dependent fluctuations. This, in turn, has ramifications for the use of such systems in quantum technologies.The authors have discovered a topological classification of quantum many-body systems undergoing dynamical evolution, which can be applied to insulators and superconductors driven out of equilibrium. This classification is analogous to the equilibrium classification of topological insulators, which explains the remarkably quantized experimental signatures that these phases are known for. The nonequilibrium classification successfully predicts whether topological edge modes remain stable under external time-dependent fluctuations. This, in turn, has ramifications for the use of such systems in quantum technologies.

arXiv:1811.00889Physical review. B./Physical review. BFeb 25, 201963 citations

Topological Phases of Quantum Matter

Wei-Feng Tsai, Hsin Lin, Arun Bansil

Springer series in solid-state sciencesJan 1, 20181 citation

Topological Phases in the Quantum Theory.

Tom D. Imbo

Jan 1, 198839 citations

Interacting symmetry-protected topological phases out of equilibrium

Max McGinley, Nigel R. Cooper

The authors make use of concepts and methods from the theory of topological phases to understand the dynamics of generic quantum many-body systems undergoing unitary time-evolution. They develop a topological classification scheme for wavefunctions far from equilibrium, and show that this classification can be used to predict a number of universal phenomena in certain non-equilibrium regimes. This classification is explicitly derived for strongly interacting bosonic systems in all spatial dimensions.The authors make use of concepts and methods from the theory of topological phases to understand the dynamics of generic quantum many-body systems undergoing unitary time-evolution. They develop a topological classification scheme for wavefunctions far from equilibrium, and show that this classification can be used to predict a number of universal phenomena in certain non-equilibrium regimes. This classification is explicitly derived for strongly interacting bosonic systems in all spatial dimensions.

arXiv:1908.06875Physical Review ResearchDec 26, 201919 citations

{\mathbb{Z}}_{2}^{\phantom{\rule{0.16em}{0ex}}}

is always stable, (ii) the noninteracting topological classification

\mathbb{Z}

in even dimensions is always stable, and (iii) the noninteracting topological classification

\mathbb{Z}

in odd dimensions is unstable and reduces to

{\mathbb{Z}}_{N}^{\phantom{\rule{0.16em}{0ex}}}

that can be identified explicitly for any dimension and any defining symmetries. We also apply our method to the three-dimensional topological crystalline insulator SnTe from the symmetry class

\mathrm{AII}+R

, for which we establish the reduction $\mathbb{Z}\ensuremath{\rightarrow}{\mathbb{Z}}_{8}^{\phantom{\rule{0.16em}{0ex}}}$ of the noninteracting topological classification.