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arXiv:2011.13921 · Physical review. B./Physical review. B

Unwinding fermionic symmetry-protected topological phases: Supersymmetry extension

Abhishodh Prakash, Juven Wang

Published Feb 22, 2021• 18 citations
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Abstract

We show how $1+1$-dimensional fermionic symmetry-protected topological states (SPTs, i.e., nontrivial short-range entangled gapped phases of quantum matter whose boundary exhibits 't Hooft anomaly and whose bulk cannot be deformed into a trivial tensor product state under finite-depth local unitary transformations only in the presence of global symmetries), indeed can be unwound to a trivial state by enlarging the Hilbert space via adding extra degrees of freedom and suitably extending the global symmetries. The extended projective global symmetry on the boundary can become supersymmetric in a specific sense, i.e., it contains group elements that do not commute with the fermion number parity ${(\ensuremath{-}1)}^{F}$, while the antiunitary time-reversal symmetry becomes fractionalized. This also means we can uplift and remove certain exotic fermionic anomalies (e.g., ``parity'' anomaly in time-reversal or reflection symmetry) via appropriate supersymmetry extensions in terms of group extensions. We work out explicit examples for multilayers of $1+1\mathrm{d}$ Majorana fermion chains, then comment on models with Sachdev-Ye-Kitaev (SYK) interactions, intrinsic fermionic gapless SPTs protected by supersymmetry, and generalizations to higher space-time dimensions via a cobordism theory.

DOI
10.1103/physrevb.103.085130

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We show how 1+11+11+1-dimensional fermionic symmetry-protected topological states (SPTs, i.e., nontrivial short-range entangled gapped phases of quantum matter whose boundary exhibits 't Hooft anomaly and whose bulk cannot be deformed into a trivial tensor product state under finite-depth local unitary transformations only in the presence of global symmetries), indeed can be unwound to a trivial state by enlarging the Hilbert space via adding extra degrees of freedom and suitably extending the global symmetries. The extended projective global symmetry on the boundary can become supersymmetric in a specific sense, i.e., it contains group elements that do not commute with the fermion number parity (\ensuremath−1)F{(\ensuremath{-}1)}^{F}(\ensuremath−1)F, while the antiunitary time-reversal symmetry becomes fractionalized. This also means we can uplift and remove certain exotic fermionic anomalies (e.g., ``parity'' anomaly in time-reversal or reflection symmetry) via appropriate supersymmetry extensions in terms of group extensions. We work out explicit examples for multilayers of 1+1d1+1\mathrm{d}1+1d Majorana fermion chains, then comment on models with Sachdev-Ye-Kitaev (SYK) interactions, intrinsic fermionic gapless SPTs protected by supersymmetry, and generalizations to higher space-time dimensions via a cobordism theory.
Symmetry-protected topological (SPT) phases are gapped phases of quantum matter protected by global symmetries that cannot be adiabatically deformed to a trivial phase without breaking symmetry. In this work, we show that, for several SPT phases that are short-range entangled (SRE), enlarging symmetries may effectively achieve the consequences of explicitly breaking symmetries. In other words, we demonstrate that nontrivial SPT phases can be unwound to trivial ones by symmetry extension---through a path where the Hilbert space is enlarged and the Hamiltonian is invariant under an extended symmetry group applying the idea of Wang, Wen, and Witten, Phys. Rev. X 8, 031048 (2018). We show examples of both bosonic and fermionic SPT phases in 1+1 dimensions (1+1 D), including Haldane's bosonic spin chain and layers of Kitaev's fermionic Majorana chains. By adding degrees of freedom into the boundary/bulk, we can lift the zero mode degeneracy, or unwind the whole system. Furthermore, based on properties of Schur cover, we sketch a general picture of unwinding applicable to any (1+1)-D bosonic SPT phase protected by on-site finite symmetry. Altogether we show that SRE states can be unwound by symmetry breaking, inversion, and symmetry extension.
We prove that the boundaries of all nontrivial (1+1)-dimensional intrinsically fermionic symmetry-protected-topological phases, protected by finite on-site symmetries (unitary or antiunitary), are supersymmetric quantum mechanical systems. This supersymmetry does not require any fine-tuning of the underlying Hamiltonian, arises entirely as a consequence of the boundary 't Hooft anomaly that classifies the phase, and is related to a "Bose-Fermi" degeneracy different in nature from other well known degeneracies such as Kramers doublets.
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 ${\mathcal{T}}^{2}={(\ensuremath{-}1)}^{\mathcal{F}}$.
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.
Symmetry-protected topological (SPT) states have boundary 't Hooft anomalies that obstruct the effective boundary theory realized in its own dimension with UV completion and with an on-site G-symmetry.In this work, yet we show that a certain anomalous non-on-site G-symmetry along the boundary becomes on-site when viewed as an extended H-symmetry, via a suitable group extension 1 → K → H → G → 1. Namely, a nonperturbative global (gauge or gravitational) anomaly in G becomes anomaly free in H.This guides us to construct an exactly soluble lattice path integral and Hamiltonian of symmetric gapped boundaries applicable to any SPT state of any finite symmetry group, including on-site unitary and antiunitary time-reversal symmetries.The resulting symmetric gapped boundary can be described either by an H-symmetry extended boundary in any spacetime dimension or, more naturally, by a topological emergent K-gauge theory with a global symmetry G on a 3 þ 1D bulk or above.The excitations on such a symmetric topologically ordered boundary can carry fractional quantum numbers of the symmetry G, described by representations of H. (Applying our approach to a 1 þ 1D boundary of 2 þ 1D bulk, we find that a deconfined gauge boundary indeed has spontaneous symmetry breaking with long-range order.The deconfined symmetry-breaking phase crosses over smoothly to a confined phase without a phase transition.)In contrast to known gapped boundaries or interfaces obtained via symmetry breaking (either global symmetry breaking or the Anderson-Higgs mechanism for gauge theory), our approach is based on symmetry extension.More generally, applying our approach to SPT states, topologically ordered gauge theories, and symmetry enriched topologically ordered (SET) states leads to generic boundaries or interfaces constructed with a mixture of symmetry breaking, symmetry extension, and dynamical gauging.