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arXiv:1708.01684 · Physical review. B./Physical review. B

Exactly solvable model for two-dimensional topological superconductors

Zitao Wang, Shang-Qiang Ning, Xie Chen

Published Sep 4, 2018• 17 citations
arXivHTMLPDF

Abstract

In this paper, we present an exactly solvable model for two-dimensional topological superconductors with helical Majorana edge modes protected by time-reversal symmetry. Our construction is based on the idea of decorated domain walls and makes use of the Kasteleyn orientation on a two-dimensional lattice, which was used for the construction of the symmetry protected fermion phase with ${Z}_{2}$ symmetry by Tarantino et al. and Ware et al. By decorating the time-reversal domain walls with spinful Majorana chains, we are able to construct a commuting projector Hamiltonian with zero correlation length ground state wave function that realizes a strongly interacting version of the two-dimensional topological superconductor. From our construction, it can be seen that the ${T}^{2}=\ensuremath{-}1$ transformation rule for the fermions is crucial for the existence of such a nontrivial phase; with ${T}^{2}=1$, our construction does not work.

DOI
10.1103/physrevb.98.094502

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In this paper, we present an exactly solvable model for two-dimensional topological superconductors with helical Majorana edge modes protected by time-reversal symmetry. Our construction is based on the idea of decorated domain walls and makes use of the Kasteleyn orientation on a two-dimensional lattice, which was used for the construction of the symmetry protected fermion phase with Z2{Z}_{2}Z2​ symmetry by Tarantino et al. and Ware et al. By decorating the time-reversal domain walls with spinful Majorana chains, we are able to construct a commuting projector Hamiltonian with zero correlation length ground state wave function that realizes a strongly interacting version of the two-dimensional topological superconductor. From our construction, it can be seen that the T2=\ensuremath−1{T}^{2}=\ensuremath{-}1T2=\ensuremath−1 transformation rule for the fermions is crucial for the existence of such a nontrivial phase; with T2=1{T}^{2}=1T2=1, our construction does not work.
A time-reversal invariant Kitaev-type model is introduced in which spins (Dirac matrices) on the square lattice interact via anisotropic nearest-neighbor and next-nearest-neighbor exchange interactions. The model is exactly solved by mapping it onto a tight-binding model of free Majorana fermions coupled with static Z2{\mathbb{Z}}_{2}Z2​ gauge fields. The Majorana fermion model can be viewed as a model of time-reversal-invariant superconductor and is classified as a member of symmetry class DIII in the Altland-Zirnbauer classification. The ground-state phase diagram has two topologically distinct gapped phases which are distinguished by a Z2{\mathbb{Z}}_{2}Z2​ topological invariant. The topologically nontrivial phase supports both a Kramers' pair of gapless Majorana edge modes at the boundary and a Kramers' pair of zero-energy Majorana states bound to a 0-flux vortex in the $\ensuremath{\pi}$-flux background. Power-law decaying correlation functions of spins along the edge are obtained by taking the gapless Majorana edge modes into account. The model is also defined on the one-dimension ladder, in which case again the ground-state phase diagram has Z2{\mathbb{Z}}_{2}Z2​ trivial and nontrivial phases.
It is well known that symmetry-protected topological (SPT) phases host nontrivial boundaries that cannot be mimicked in a lower-dimensional system with a conventional realization of symmetry. However, for SPT phases of bosons (fermions) within the cohomology (supercohomology) classification the boundary can be recreated without the bulk at the cost of a non-onsite-symmetry action. This raises the following question: Can one also mimic the boundaries of SPT phases which lie outside the (super)cohomology classification? In this paper, we study this question in the context of (2+1)(2+1)(2+1)-dimensional [(2+1)D][(2+1)\mathrm{D}][(2+1)D] fermion SPTs. We focus on the root SPT phase for the symmetry group $G={Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}^{f}$. Starting with an exactly solvable model for the bulk of this phase constructed by Tarantino and Fidkowski, we derive an effective one-dimensional (1D) lattice model for the boundary. Crucially, the Hilbert space of this 1D model does not have a local tensor product structure, but rather is obtained by placing a local constraint on a local tensor product Hilbert space. We derive the action of the Z2{Z}_{2}Z2​ symmetry on this Hilbert space and find a simple three-site Hamiltonian that respects this symmetry. We study this Hamiltonian numerically using exact diagonalization and DMRG and find strong evidence that it realizes an Ising conformal field theory where the Z2{Z}_{2}Z2​ symmetry acts as the Kramers-Wannier duality; this is the expected stable gapless boundary state of the present SPT. A simple modification of our construction realizes the boundary of the (2+1)D(2+1)\mathrm{D}(2+1)D topological superconductor protected by time-reversal symmetry T\mathcal{T}T with ${\mathcal{T}}^{2}={(\ensuremath{-}1)}^{\mathcal{F}}$.
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.
We construct an exactly sovable commuting projector Hamiltonian for (2+1)D bosonic topological insulator which is one of symmetry-protected topological (SPT) phases protected by U(1) and time-reversal Z2T\mathbb{Z}_2^TZ2T​ symmetry, where the symmetry group is U(1)⋊Z2T\rtimes\mathbb{Z}_2^T⋊Z2T​. The model construction is based on the decorated domain-wall interpretation of the E∞E_{\infty}E∞​-page of a spectral sequence of a cobordism group that classifies the SPT phases in question. We demonstrate nontriviality of the model by showing an emergence of a Kramers doublet when the system is put on a semi-infinite cylinder (−∞,0]×S1(-\infty,0]\times S^1(−∞,0]×S1 with an inserted πππ-flux. The surface anomaly manifests itself as a non-onsite representation of the U(1)⋊Z2T\rtimes\mathbb{Z}_2^T⋊Z2T​ symmetry. Anomaly matching on a boundary is discussed within a simple boundary theory.
Two-dimensional time-reversal-invariant topological superconductors host helical Majorana fermions at their boundary. We study the fate of these edge states under the combined influence of strong interactions and disorder, using the effective one-dimensional (1D) lattice model for the edge introduced by Jones and Metlitski [Phys. Rev. B 104, 245130 (2021)]. We specifically develop a strong-disorder renormalization-group analysis of the lattice model and identify a regime in which time-reversal is broken spontaneously, creating random magnetic domains; Majorana fermions localize to domain walls and form an infinite-randomness fixed point, identical to that appearing in the random transverse-field Ising model. While this infinite-randomness fixed point describes a fine-tuned critical point in a purely 1D system, in our edge context there is no obvious time-reversal-preserving perturbation that destabilizes the fixed point. Our analysis thus suggests that the infinite-randomness fixed point emerges as a stable phase on the edge of two-dimensional topological superconductors when strong disorder and interactions are present.
We discuss a way to construct a commuting projector Hamiltonian model for a (3+13+13+1)-dimensional topological superconductor in class DIII. The wave function is given by a sort of string net of the Kitaev wire, decorated on the time-reversal (TTT) domain wall. Our Hamiltonian is provided on a generic three-dimensional (3D) manifold equipped with a discrete form of the spin structure. We will see how the 3D spin structure induces a 2D spin structure (called a ``Kasteleyn'' direction on a 2D lattice) on TTT domain walls, which makes it possible to define fluctuating Kitaev wires on them. Upon breaking the TTT symmetry in our model, we find the unbroken remnant of the symmetry, which is defined on the time-reversal domain wall. The domain wall supports the 2D nontrivial symmetry-protected topological (SPT) phase protected by the unbroken symmetry, which allows us to determine the SPT classification of our model, based on the recent quantum field theory argument by Hason, Komargodski, and Thorngren.