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arXiv:1806.09709 · Physical review. B./Physical review. B

Full commuting projector Hamiltonians of interacting symmetry-protected topological phases of fermions

Nathanan Tantivasadakarn, Ashvin Vishwanath

Published Oct 2, 2018• 29 citations
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Abstract

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 $p+ip$ 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.

DOI
10.1103/physrevb.98.165104

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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=Z2T\ifmmode×\else\texttimes\fiZ2FG={\mathbb{Z}}_{2}^{T}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{F}G=Z2T​\ifmmode×\else\texttimes\fiZ2F​, which also has a free fermion realization in class BDI, and by the symmetry group G=Z4\ifmmode×\else\texttimes\fiZ4FG={\mathbb{Z}}_{4}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{4}^{F}G=Z4​\ifmmode×\else\texttimes\fiZ4F​, which does not. We then turn to 2+1D, and construct the square roots of the Levin-Gu bosonic SPT phase, protected by Z2\ifmmode×\else\texttimes\fiZ2F{\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{F}Z2​\ifmmode×\else\texttimes\fiZ2F​ 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−ipp\ensuremath{-}ipp\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.
Symmetry-protected topological (SPT) phases are gapped short-range-entangled quantum phases with a symmetry GGG, which can all be smoothly connected to the trivial product states if we break the symmetry. It has been shown that a large class of interacting bosonic SPT phases can be systematically described by group cohomology theory. In this paper, we introduce a (special) group supercohomology theory which is a generalization of the standard group cohomology theory. We show that a large class of short-range interacting fermionic SPT phases can be described by the group supercohomology theory. Using the data of supercocycles, we can obtain the ideal ground state wave function for the corresponding fermionic SPT phase. We can also obtain the bulk Hamiltonian that realizes the SPT phase, as well as the anomalous (i.e., non-onsite) symmetry for the boundary effective Hamiltonian. The anomalous symmetry on the boundary implies that the symmetric boundary must be gapless for (1+1)-dimensional [(1+1)D] boundary, and must be gapless or topologically ordered beyond (1+1)D. As an application of this general result, we construct a new SPT phase in three dimensions, for interacting fermionic superconductors with coplanar spin order (which have T2=1{T}^{2}=1T2=1 time-reversal Z2T{Z}_{2}^{T}Z2T​ and fermion-number-parity Z2f{Z}_{2}^{f}Z2f​ symmetries described by a full symmetry group ${Z}_{2}^{T}\ifmmode\times\else\texttimes\fi{}{Z}_{2}^{f}$). Such a fermionic SPT state can neither be realized by free fermions nor by interacting bosons (formed by fermion pairs), and thus are not included in the KKK-theory classification for free fermions or group cohomology description for interacting bosons. We also construct three interacting fermionic SPT phases in two dimensions (2D) with a full symmetry group ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}^{f}$. Those 2D fermionic SPT phases all have central-charge c=1c=1c=1 gapless edge excitations, if the symmetry is not broken.
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​.
The classification and lattice model construction of symmetry protected topological (SPT) phases in interacting fermion systems are very interesting but challenging. In this paper, we give a systematic fixed point wave function construction of fermionic SPT (FSPT) states for generic fermionic symmetry group Gf=Z2f×ω2GbG_f=\mathbb{Z}_2^f \times_{\omega_2} G_bGf​=Z2f​×ω2​​Gb​ which is a central extension of bosonic symmetry group GbG_bGb​ (may contain time reversal symmetry) by the fermion parity symmetry group Z2f={1,Pf}\mathbb{Z}_2^f = \{1,P_f\}Z2f​={1,Pf​}. Our construction is based on the concept of equivalence class of finite depth fermionic symmetric local unitary (FSLU) transformations and decorating symmetry domain wall picture, subjected to certain obstructions. We will also discuss the systematical construction and classification of boundary anomalous SPT (ASPT) states which leads to a trivialization of the corresponding bulk FSPT states. Thus, we conjecture that the obstruction-free and trivialization-free constructions naturally lead to a classification of FSPT phases. Each fixed-point wave function admits an exactly solvable commuting-projector Hamiltonian. We believe that our classification scheme can be generalized to point/space group symmetry as well as continuum Lie group symmetry.
In closed quantum systems, strong randomness can localize many-body excitations, preventing ergodicity. An interesting consequence is that high energy excited states can exhibit quantum coherent properties, such as symmetry protected topological (SPT) order, that otherwise only occur in equilibrium ground states. Here, we ask: which types of SPT orders can be realized in highly excited states of a many-body (MB) localized system? We argue that this question is equivalent to whether an SPT order can be realized in an exactly solvable lattice model of commuting projectors. This perspective enables a sharp definition of MB localizability. Using this criterion, it is straightforward to establish that whereas all bosonic SPTs in spatial dimensions d=1, 2, 3d=1,\,2,\,3d=1,2,3 are MB localizable, chiral phases (e.g. quantum Hall fluids) are not. We also show that free fermion SPTs in d>1d >1d>1 (including topological insulators and superconductors) cannot be localized if interactions are weak. A key question is whether strong interactions can render them MB localizable, which we study in the context of a class of d=2d=2d=2 topological superconductors. Using a decorated domain wall (DDW) approach we show that some phases in this class are MB localizable, when they correspond to bosonic SPT orders. However, a similar DDW approach faces a fatal obstruction to realizing certain intrinsically fermionic SPT orders, an issue we argue may persist beyond this specific construction.
We use higher-dimensional bosonization and fermion decoration to construct exactly soluble interacting fermion models to realize fermionic symmetry-protected trivial (SPT) orders (which are also known as symmetry-protected topological orders) in any dimensions and for generic fermion symmetries Gf{G}_{f}Gf​, which can be a nontrivial Z2f{Z}_{2}^{f}Z2f​ extension ${Z}_{2}^{f}\ensuremath{\leftthreetimes}{G}_{b}$ (where Z2f{Z}_{2}^{f}Z2f​ is the fermion-number-parity symmetry and Gb{G}_{b}Gb​ is the bosonic symmetry). This generalizes the previous results from group supercohomology of Gu and Wen (arXiv:1201.2648), where Gf{G}_{f}Gf​ is assumed to be ${Z}_{2}^{f}\ifmmode\times\else\texttimes\fi{}{G}_{b}$. We find that the (d+1)(d+1)(d+1)-dimensional [(d+1)D][(d+1)\mathrm{D}][(d+1)D] fermionic SPT phases with bosonic symmetry Gb{G}_{b}Gb​ and from fermion decoration construction can be described in a compact way using higher group homomorphism: $\mathcal{B}{G}_{b}\stackrel{\ensuremath{\varphi}}{\ensuremath{\rightarrow}}\mathcal{B}({Z}_{2},2;{Z}_{2},d)$. In fact, the fermion symmetry is more precisely described by the structure ${Z}_{2}^{f}\ensuremath{\leftthreetimes}{G}_{b}\ensuremath{\leftthreetimes}S{O}_{\ensuremath{\infty}}$ (or ${Z}_{2}^{f}\ensuremath{\leftthreetimes}{G}_{b}\ensuremath{\leftthreetimes}{O}_{\ensuremath{\infty}}$ with time-reversal symmetry). In this case the (d+1)D(d+1)\mathrm{D}(d+1)D fermionic SPT phases are better described by $\mathcal{B}({Z}_{2}^{f}\ensuremath{\leftthreetimes}{G}_{b}\ensuremath{\leftthreetimes}S{O}_{\ensuremath{\infty}})\stackrel{\ensuremath{\varphi}}{\ensuremath{\rightarrow}}\mathcal{B}(S{O}_{\ensuremath{\infty}},1;{Z}_{2},d)$ [or $\mathcal{B}({Z}_{2}^{f}\ensuremath{\leftthreetimes}{G}_{b}\ensuremath{\leftthreetimes}{O}_{\ensuremath{\infty}})\stackrel{\ensuremath{\varphi}}{\ensuremath{\rightarrow}}\mathcal{B}({O}_{\ensuremath{\infty}},1;{Z}_{2},d)]$.
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.