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arXiv:1505.06341 · Physical Review B

Breakdown of the topological classification Z for gapped phases of noninteracting fermions by quartic interactions

Takahiro Morimoto, Akira Furusaki, Christopher Mudry

Published Sep 1, 2015• 103 citations
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Abstract

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.

DOI
10.1103/physrevb.92.125104

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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 \ensuremathσ\ensuremath{\sigma}\ensuremathσ models for the boundary dynamical masses favor quantum disordered phases. For the tenfold way, we find that (i) the noninteracting topological classification Z2{\mathbb{Z}}_{2}^{\phantom{\rule{0.16em}{0ex}}}Z2​ is always stable, (ii) the noninteracting topological classification Z\mathbb{Z}Z in even dimensions is always stable, and (iii) the noninteracting topological classification Z\mathbb{Z}Z in odd dimensions is unstable and reduces to ZN{\mathbb{Z}}_{N}^{\phantom{\rule{0.16em}{0ex}}}ZN​ 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 AII+R\mathrm{AII}+RAII+R, for which we establish the reduction Z\ensuremath→Z8\mathbb{Z}\ensuremath{\rightarrow}{\mathbb{Z}}_{8}^{\phantom{\rule{0.16em}{0ex}}}Z\ensuremath→Z8​ of the noninteracting topological classification.
We classify interacting topological insulators and superconductors with order-two crystal symmetries (reflection or twofold rotation), focusing on the case where interactions reduce the noninteracting classification. We find that the free-fermion Z2{\mathbb{Z}}_{2}Z2​ classifications are stable against quartic contact interactions, whereas the Z\mathbb{Z}Z classifications reduce to ZN{\mathbb{Z}}_{N}ZN​, where NNN depends on the symmetry class and the dimension ddd. These results are derived using a quantum nonlinear $\ensuremath{\sigma}$ model (QNLSM) that describes the effects of the quartic interactions on the boundary modes of the crystalline topological phases. We use Clifford algebra extensions to derive the target spaces of these QNLSMs in a unified way. The reduction pattern of the free-fermion classification then follows from the presence or absence of topological terms in the QNLSMs, which is determined by the homotopy group of the target spaces. We show that this derivation can be performed using either a complex fermion or a real Majorana representation of the crystalline topological phases and demonstrate that these two representations give consistent results. To illustrate the breakdown of the noninteracting classification, we present examples of crystalline topological insulators and superconductors in one, two, and three dimensions whose surface modes are unstable against interactions. For the three-dimensional example, we show that the reduction pattern obtained by the QNLSM method agrees with the one inferred from the stability analysis of the boundary modes using bosonization.
We present a dimensional reduction argument to derive the classification reduction of fermionic symmetry protected topological phases in the presence of interactions. The dimensional reduction proceeds by relating the topological character of a d-dimensional system to the number of zero-energy bound states localized at zero-dimensional topological defects present at its surface. This correspondence leads to a general condition for symmetry preserving interactions that render the system topologically trivial, and allows us to explicitly write a quartic interaction to this end. Our reduction shows that all phases with topological invariant smaller than n are topologically distinct, thereby reducing the noninteracting Z classification to Z_{n}.
In this paper, we study five-dimensional Dirac fermions of which the extra-dimension is compactified on quantum graphs. We find that there is a nontrivial correspondence between matrices specifying boundary conditions at the vertex of the quantum graphs and zero-dimensional Hamiltonians in gapped free-fermion systems. Based on the correspondence, we provide a complete topological classification of the boundary conditions in terms of noninteracting fermionic topological phases. The ten symmetry classes of topological phases are fully identified in the language of five-dimensional Dirac fermions, and topological numbers of the boundary conditions are given. In analogy with the bulk-boundary correspondence in noninteracting fermionic topological phases, the boundary condition topological numbers predict four-dimensional massless fermions localized at the vertex of the quantum graphs and thus govern the low energy physics in four dimensions.
Abstract Guided by the many-particle quantum theory of interacting systems, we develop a uniform classification scheme for topological phases of disordered gapped free fermions, encompassing all symmetry classes of the Tenfold Way. We apply this scheme to give a mathematically rigorous proof of bulk-boundary correspondence. To that end, we construct real C ∗^*∗ ∗ -algebras harbouring the bulk and boundary data of disordered free-fermion ground states. These we connect by a natural bulk-to-boundary short exact sequence, realising the bulk system as a quotient of the half-space theory modulo boundary contributions. To every ground state, we attach two classes in different pictures of real operator KKK K -theory (or KRKRKR KR -theory): a bulk class, using Van Daele’s picture, along with a boundary class, using Kasparov’s Fredholm picture. We then show that the connecting map for the bulk-to-boundary sequence maps these KRKRKR KR -theory classes to each other. This implies bulk-boundary correspondence, in the presence of disorder, for both the “strong” and the “weak” invariants.
We show that topology can protect exponentially localized, zero energy edge modes at critical points between one-dimensional symmetry-protected topological phases. This is possible even without gapped degrees of freedom in the bulk-in contrast to recent work on edge modes in gapless chains. We present an intuitive picture for the existence of these edge modes in the case of noninteracting spinless fermions with time-reversal symmetry (BDI class of the tenfold way). The stability of this phenomenon relies on a topological invariant defined in terms of a complex function, counting its zeros and poles inside the unit circle. This invariant can prevent two models described by the same conformal field theory (CFT) from being smoothly connected. A full classification of critical phases in the noninteracting BDI class is obtained: Each phase is labeled by the central charge of the CFT, c∈1/2N, and the topological invariant, ω∈Z. Moreover, c is determined by the difference in the number of edge modes between the phases neighboring the transition. Numerical simulations show that the topological edge modes of critical chains can be stable in the presence of interactions and disorder.
We study the effect of electron interactions in topological crystalline insulators (TCIs) protected by mirror symmetry, which are realized in the SnTe material class and host multivalley Dirac fermion surface states. We find that interactions reduce the integer classification of noninteracting TCIs in three dimensions, indexed by the mirror Chern number, to a finite group Z8{Z}_{8}Z8​. In particular, we explicitly construct a microscopic interaction Hamiltonian to gap eight flavors of Dirac fermions on the TCI surface, while preserving the mirror symmetry. Our construction builds on interacting edge states of $U(1)\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ symmetry-protected topological phases of fermions in two dimensions, which we classify. Our work reveals a deep connection between three-dimensional topological phases protected by spatial symmetries and two-dimensional topological phases protected by internal symmetries.