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arXiv:2303.00719 · Physical review. B./Physical review. B

Gapped boundary of ( 4 + 1 ) d beyond-cohomology bosonic SPT phase

Xinping Yang, Meng Cheng

Published Jul 19, 2024• 6 citations
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Abstract

In this paper we study gapped boundary states of ${\mathbb{Z}}_{N}$ bosonic symmetry-protected topological (SPT) phases in (4+1)d, which are characterized by mixed ${\mathbb{Z}}_{N}$-gravity response, and the closely related phases protected by ${C}_{N}$ rotation symmetry. We show that if $N\ensuremath{\notin}{2,4,8,16}$, any symmetry-preserving boundary theory is necessarily gapless for the root SPT state. We then propose a (3+1)d ${\mathbb{Z}}_{2}$ gauge theory coupled to fermionic matter as a candidate boundary theory for $N=2,4,8,16$, where the anomalous symmetry is implemented by invertible topological defects obtained from gauging (2+1)d chiral topological superconductors. For the ${C}_{N}$ case, we present an explicit construction for the boundary states for $N=2,4,8,16$, and argue that the construction fails for other values of $N$.

DOI
10.1103/physrevb.110.045137

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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 ${\mathcal{T}}^{2}={(\ensuremath{-}1)}^{\mathcal{F}}$.
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1d lattice models for the boundary of 2d "Majorana" fermion SPTs: Kramers-Wannier duality as an exact Z2Z_2Z2​ symmetry
It is well known that symmetry protected topological (SPT) phases host non-trivial 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 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+1D fermion SPTs. We focus on the root SPT phase for the symmetry group G=Z2×Z2fG =Z_2 \times Z^f_2G=Z2​×Z2f​. Starting with an exactly solvable model for the bulk of this phase constructed by Tarantino and Fidkowski, we derive an effective 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 Z2Z_2Z2​ symmetry on this Hilbert space and find a simple 3-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 CFT where the Z2Z_2Z2​ 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+1D topological superconductor protected by time-reversal symmetry T{\cal T}T with T2=(−1)F{\cal T}^2 = (-1)^{\cal F}T2=(−1)F.