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arXiv:1409.1234 · Physical review. B./Physical review. B

Interaction-enabled topological crystalline phases

Matthew F. Lapa, Jeffrey C. Y. Teo, Taylor L. Hughes

Published Mar 18, 2016• 27 citations
arXivHTMLPDF

Abstract

In this article we provide a general mechanism for generating interaction-enabled fermionic topological phases. We illustrate the mechanism with crystalline symmetry-protected topological phases in one, two, and three spatial dimensions. These nontrivial phases require interactions for their existence, and in the cases we consider, the free-fermion classification yields only a trivial phase. For the one- and two-dimensional phases we consider, we provide explicit exactly solvable models which realize the interaction-enabled phases. Similar to the interpretation of the Kitaev Majorana wire as a mean-field $p$-wave superconductor Hamiltonian arising from an interacting model with quartic interactions, we show that our systems can be interpreted as ``mean-field'' charge-$4e$ superconductors arising, e.g., from an interacting model with eight-body interactions or through another physical mechanism. The quartet superconducting nature allows for the teleportation of full Cooper pairs and, in two dimensions, for interesting semiclassical crystalline defects with non-Abelian anyon bound states.

DOI
10.1103/physrevb.93.115131

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In this article we provide a general mechanism for generating interaction-enabled fermionic topological phases. We illustrate the mechanism with crystalline symmetry-protected topological phases in one, two, and three spatial dimensions. These nontrivial phases require interactions for their existence, and in the cases we consider, the free-fermion classification yields only a trivial phase. For the one- and two-dimensional phases we consider, we provide explicit exactly solvable models which realize the interaction-enabled phases. Similar to the interpretation of the Kitaev Majorana wire as a mean-field ppp-wave superconductor Hamiltonian arising from an interacting model with quartic interactions, we show that our systems can be interpreted as ``mean-field'' charge-4e4e4e superconductors arising, e.g., from an interacting model with eight-body interactions or through another physical mechanism. The quartet superconducting nature allows for the teleportation of full Cooper pairs and, in two dimensions, for interesting semiclassical crystalline defects with non-Abelian anyon bound states.
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.
Crystalline symmetry and time-reversal symmetry are commonly present in real superconducting materials. However, the topological classification of systems respecting these symmetries, particularly for interacting fermions, remains incomplete. In this work, we systematically classify time-reversal symmetry-protected crystalline topological superconductors in two-dimensional interacting fermionic systems using an explicit real-space construction. Among the resulting phases, we identify intrinsically interacting fermionic topological superconductors, i.e., phases that cannot be realized in either free-fermion or interacting bosonic systems. For spinless fermions with protecting symmetry group C4×Z2TC_4 \times Z_2^TC4​×Z2T​ or D4×Z2TD_4 \times Z_2^TD4​×Z2T​ (plus fermion parity), the intrinsic sector has a Z4Z_4Z4​ classification. The corresponding root phases generating this Z4Z_4Z4​ classification admit a transparent real-space construction in terms of decorated 1D blocks. These blocks are 1D fermionic symmetry-protected topological (FSPT) phases, realizable as double Majorana chains. We further find the corresponding Z4Z_4Z4​ spinless intrinsic phases for wallpaper groups p4p4p4, p4mp4mp4m, and p4gp4gp4g. We also find an additional Z2Z_2Z2​ intrinsically interacting phase for spinless fermions with wallpaper group pmpmpm, which is absent with the corresponding point-group symmetry alone. Moreover, these intrinsic phases naturally give rise to higher-order FSPT phases that support corner zero modes. Finally, we verify the crystalline equivalence principle for generic 2D interacting FSPT systems with both crystalline and internal symmetries.
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.
Although classification for free-fermion topological superconductors (TSC) is established, systematically understanding the classification of 3D interacting TSCs remains difficult, especially those protected by crystalline symmetries like the 230 space groups. We build up a general framework for systematically classifying 3D interacting TSCs protected by crystalline symmetries together with discrete internal symmetries. We first establish a complete classification for fermionic symmetry protected topological phases (FSPT) with purely discrete internal symmetries, which determines the crystalline case via the crystalline equivalence principle. Using domain wall decoration, we obtain classification data and formulas for generic FSPTs, what are suitable for systematic computation. The four layers of decoration data (n1,n2,n3,ν4)(n_1, n_2, n_3, ν_4)(n1​,n2​,n3​,ν4​) characterize a 3D FSPT with symmetry Gb×ω2Z2fG_b\times_{ω_2}Z_2^fGb​×ω2​​Z2f​, corresponding to p+ipp+ipp+ip, Kitaev chain, complex fermion, and bosonic SPT layers. Inspired by previous works, a crucial aspect is the p+ipp+ipp+ip layer, where classification involves two possibilities: anti-unitary and infinite-order symmetries (e.g., translation). We show the former maps to some mirror FSPT classification with the mirror plane decorated by a p+ipp+ipp+ip superconductor, while the latter is determined by the free part of H1(Gb,ZT)H^1(G_b, Z_T)H1(Gb​,ZT​), corresponding to weak TSCs. Another key point is the Kitaev chain decoration for the anti-unitary symmetries, which differs essentially from unitary ones. We explicitly obtain formulas for all three layers of decoration (n2,n3,ν4)(n_2, n_3, ν_4)(n2​,n3​,ν4​), which are amenable to automatic computation. As an application, we classify the 230 space-group topological crystalline superconductors in interacting electronic systems.
The construction and classification of crystalline symmetry protected topological (SPT) phases in interacting bosonic and fermionic systems have been intensively studied in the past few years. Crystalline SPT phases are not only of conceptual importance, but also provide us great opportunities towards experimental realization since space group symmetries naturally exist for any realistic material. In this paper, we systematically classify the crystalline topological superconductors (TSC) and topological insulators (TI) in 2D interacting fermionic systems by using an explicit real-space construction. In particular, we discover an intriguing fermionic crystalline topological superconductor that can only be realized in interacting fermionic systems (i.e., not in free-fermion or interacting bosonic systems). Moreover, we also verify the recently conjectured crystalline equivalence principle for generic 2D interacting fermionic systems.
We study the classification of interacting fermionic symmetry-protected topological (SPT) phases with both rotation symmetry and Abelian internal symmetries in one, two, and three dimensions. By working out this classification, on the one hand, we demonstrate the recently proposed correspondence principle between crystalline topological phases and those with internal symmetries through explicit block-state constructions. We find that for the precise correspondence to hold it is necessary to change the central extension structure of the symmetry group by the Z2{\mathbb{Z}}_{2}Z2​ fermion parity. On the other hand, we uncover new classes of intrinsically fermionic SPT phases that are only enabled by interactions, both in 2D and 3D. Several 3D Majorana-type fermionic SPT phases are identified. Moreover, several new instances of Lieb-Schultz-Mattis-type theorems for Majorana-type fermionic SPT phases are obtained and we discuss their interpretations from the perspective of bulk-boundary correspondence.