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arXiv:2006.00159 · Physical review. B./Physical review. B

Commuting projector models for ( 3+1 )-dimensional topological superconductors via a string net of ( 1+1 )-dimensional topological superconductors

Ryohei Kobayashi

Published Aug 25, 2020• 3 citations
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Abstract

We discuss a way to construct a commuting projector Hamiltonian model for a ($3+1$)-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 ($T$) 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 $T$ domain walls, which makes it possible to define fluctuating Kitaev wires on them. Upon breaking the $T$ 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.

DOI
10.1103/physrevb.102.075135

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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.
We construct exactly solved commuting projector Hamiltonian lattice models for all known (2+1)-dimensional (2+1D) fermionic symmetry protected topological phases (SPTs) with on-site unitary symmetry group ${G}_{f}=G\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{f}$, where GGG is finite and Z2f{\mathbb{Z}}_{2}^{f}Z2f​ is the fermion parity symmetry. In particular, our models transcend the class of group supercohomology models, which realize some, but not all, fermionic SPTs in 2+1D. A natural ingredient in our construction is a discrete form of the spin structure of the 2D spatial surface MMM on which our model is defined, namely a ``Kasteleyn'' orientation of a certain graph associated with the lattice. As a special case, our construction yields commuting projector models for all eight members of the Z8{\mathbb{Z}}_{8}Z8​ classification of 2D fermionic SPTs with G=Z2G={\mathbb{Z}}_{2}G=Z2​.
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 ${T}^{2}=\ensuremath{-}1$ 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.
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.
We construct an exactly solvable commuting projector model for a (4+14+14+1)-dimensional Z2{\mathbb{Z}}_{2}Z2​-symmetry-protected topological phase (SPT) which is outside the cohomology classification of SPTs. The model is described by a decorated domain wall construction, with ``three-fermion'' Walker-Wang phases on the domain walls. We describe the anomalous nature of the phase in several ways. One interesting feature is that, in contrast to in-cohomology phases, the effective Z2{\mathbb{Z}}_{2}Z2​ symmetry on a (3+13+13+1)-dimensional boundary cannot be described by a quantum circuit and instead is a nontrivial quantum cellular automaton. A related property is that a codimension-two defect (for example, the termination of a Z2{\mathbb{Z}}_{2}Z2​ domain wall at a trivial boundary) will carry nontrivial chiral central charge 4 mod 8. We also construct a gapped symmetric topologically ordered boundary state for our model, which constitutes an anomalous symmetry-enriched topological phase outside of the classification of Chen and Hermele, and define a corresponding anomaly indicator.
Using the decorated domain wall procedure, we construct finite-depth local unitaries that realize fermionic symmetry-protected topological (SPT) phases. This results in explicit full commuting projector Hamiltonians, where ``full'' implies the fact that the ground state as well as all excited states of these Hamiltonians realize the nontrivial SPT phase. We begin by constructing explicit examples of 1+1D phases protected by the symmetry group $G={\mathbb{Z}}_{2}^{T}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{F}$, which also has a free fermion realization in class BDI, and by the symmetry group $G={\mathbb{Z}}_{4}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{4}^{F}$, which does not. We then turn to 2+1D, and construct the square roots of the Levin-Gu bosonic SPT phase, protected by ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{F}$ symmetry, in a concrete model of fermions and spins on the triangular lattice. Edge states and the anomalous symmetry action on them are explicitly derived. Although this phase has a free fermion representation as two copies of p+ipp+ipp+ip superconductors combined with their $p\ensuremath{-}ip$ counterparts with a different symmetry charge, the full set of commuting projectors is only realized in the strongly interacting version, which also implies that it admits a many-body localized realization.
Inspired by a recently constructed commuting-projector Hamiltonian for a two-dimensional (2D) time-reversal-invariant topological superconductor [Z. Wang et al., Phys. Rev. B 98, 094502 (2018)], we introduce a commuting-projector model that describes an interacting yet exactly solvable 2D topological insulator. We explicitly show that both the gapped and gapless boundaries of our model are consistent with those of band-theoretic, weakly interacting topological insulators. Interestingly, on certain lattices our time-reversal-symmetric models also enjoy CP\mathcal{CP}CP symmetry, leading to intuitive interpretations of the bulk invariant for a CP\mathcal{CP}CP-symmetric topological insulator upon putting the system on a Klein bottle. We also briefly discuss how these many-body invariants may be able to characterize models with only time-reversal symmetry.