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arXiv:1609.07469 · Physical review. B./Physical review. B

Interaction effects on the classification of crystalline topological insulators and superconductors

Xue-Yang Song, Andreas P. Schnyder

Published May 4, 2017• 34 citations
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Abstract

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 ${\mathbb{Z}}_{2}$ classifications are stable against quartic contact interactions, whereas the $\mathbb{Z}$ classifications reduce to ${\mathbb{Z}}_{N}$, where $N$ depends on the symmetry class and the dimension $d$. 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.

DOI
10.1103/physrevb.95.195108

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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 \ensuremathσ\ensuremath{\sigma}\ensuremathσ 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.
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 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 $\mathbb{Z}\ensuremath{\rightarrow}{\mathbb{Z}}_{8}^{\phantom{\rule{0.16em}{0ex}}}$ of the noninteracting topological classification.
We study the stability of topological crystalline superconductors in the symmetry class DIIIR and in two-dimensional space when perturbed by quartic contact interactions. It is known that no less than eight copies of helical pairs of Majorana edge modes can be gapped out by an appropriate interaction without breaking spontaneously any one of the protecting symmetries. Hence, the noninteracting classification Z\mathbb{Z}Z reduces to Z8 \mathbb{Z}^{\,}_{8}Z8​ when these interactions are present. It is also known that the stability when there are less than eight modes can be understood in terms of the presence of topological obstructions in the low-energy bosonic effective theories, which prevent opening of a gap. Here, we investigate the stability of the edge theories with four, two, and one edge modes, respectively. We give an analytical derivation of the topological term for the first case, because of which the edge theory remains gapless. For two edge modes we employ bosonization methods to derive an effective bosonic action. When gapped, this bosonic theory is necessarily associated to the spontaneous symmetry breaking of one of the discrete protecting symmetries whenever translation symmetry is imposed on the boundary. For the last case, we show the existence of two topological sectors measured by a certain Z2 \mathbb{Z}^{\,}_{2}Z2​-invariant.
The natural existence of crystalline symmetry in real materials manifests the importance of understanding crystalline symmetry-protected topological phases, especially for interacting systems. In this paper, we systematically construct and classify all the point-group crystalline topological superconductors and insulators in three-dimensional (3D) interacting fermion systems using the novel concept of a topological crystal. The corresponding higher-order topological surface theory can also be systematically studied via higher-order bulk-boundary correspondence. In particular, we discover several intrinsically interacting topological phases that cannot be realized in any free-fermion systems. Moreover, the crystalline equivalence principle for 3D interacting fermionic systems is also verified, with an additional subtle “twist” on the spin of fermions.
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.
In free fermion systems with given symmetry and dimension, the possible topological phases are labeled by elements of only three types of Abelian groups, 0, Z2, or Z. For example, noninteracting one-dimensional fermionic superconducting phases with S(z) spin rotation and time-reversal symmetries are classified by Z. We show that with weak interactions, this classification reduces to Z4. Using group cohomology, one can additionally show that there are only four distinct phases for such one-dimensional superconductors even with strong interactions. Comparing their projective representations, we find that all these four symmetry-protected topological phases can be realized with free fermions. Further, we show that one-dimensional fermionic superconducting phases with Z(n) discrete S(z) spin rotation and time-reversal symmetries are classified by Z4 when n is even and Z2 when n is odd; again, all these strongly interacting topological phases can be realized by noninteracting fermions. Our approach can be applied to systems with other symmetries to see which one-dimensional topological phases can be realized with free fermions.
While free fermion topological crystalline insulators have been largely classified, the analogous problem in the strongly interacting case has been only partially solved, and the relationship between the free and interacting classifications is not well understood. In this paper, we develop a characterization and classification of interacting, invertible fermionic topological phases in (2+12+12+1) dimensions with charge conservation, discrete magnetic translation, and MMM-fold point group rotation symmetries, which form the group ${G}_{f}=\text{U}{(1)}^{f}{\ifmmode\times\else\texttimes\fi{}}_{\ensuremath{\phi}}[{\mathbb{Z}}^{2}\ensuremath{\rtimes}{\mathbb{Z}}_{M}]$ for M=1,2,3,4,and6M=1,2,3,4,\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}6M=1,2,3,4,and6. $\ensuremath{\phi}$ is the magnetic flux per unit cell. We derive a topological response theory in terms of background crystalline gauge fields, which gives a complete classification of different phases and a physical characterization in terms of quantized response to symmetry defects. We then derive the same classification in terms of a set of real-space invariants ${{\mathrm{\ensuremath{\Theta}}}_{\text{o}}^{\ifmmode\pm\else\textpm\fi{}}}$ that can be obtained from ground state expectation values of suitable partial rotation operators. We explicitly relate these real-space invariants to the quantized coefficients in the topological response theory, and find the dependence of the invariants on the chiral central charge ${c}_{\ensuremath{-}}$ of the invertible phase. Finally, when $\ensuremath{\phi}=0,$ we derive an explicit map between the free and interacting classifications.