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arXiv:2012.15491 · Journal of Mathematical Physics

A classification of invertible phases of bosonic quantum lattice systems in one dimension

Anton Kapustin, Nikita Sopenko, Bowen Yang

Published Aug 1, 2021• 21 citations
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Abstract

We study invertible states of 1D bosonic quantum lattice systems. We show that every invertible 1D state is in a trivial phase: after tensoring with some unentangled ancillas, it can be disentangled by a fuzzy analog of a finite-depth quantum circuit. If an invertible state has symmetries, it may be impossible to disentangle it in a way that preserves the symmetries, even after adding unentagled ancillas. We show that in the case of a finite unitary symmetry G, the only obstruction is an index valued in degree-2 cohomology of G. We show that two invertible G-invariant states are in the same phase if and only if their indices coincide.

DOI
10.1063/5.0055996

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We study invertible states of 1D bosonic quantum lattice systems. We show that every invertible 1D state is in a trivial phase: after tensoring with some unentangled ancillas, it can be disentangled by a fuzzy analog of a finite-depth quantum circuit. If an invertible state has symmetries, it may be impossible to disentangle it in a way that preserves the symmetries, even after adding unentagled ancillas. We show that in the case of a finite unitary symmetry G, the only obstruction is an index valued in degree-2 cohomology of G. We show that two invertible G-invariant states are in the same phase if and only if their indices coincide.
We develop a lattice characterization of fermionic 't Hooft anomalies of finite internal symmetries in one and two spatial dimensions, formulated in terms of obstructions to symmetric short-range-entangled (SRE) states. We consider lattice systems formed by tensor product of onsite fermionic and bosonic local degrees of freedom, and finite internal symmetry given by a central extension Z2F→Gf→Gb\mathbb Z_2^F\to G_f\to G_bZ2F​→Gf​→Gb​. We extract a hierarchy of fermionic anomaly indices for a given symmetry operator in both (1+1)D and (2+1)D. In (1+1)D, an exact lattice symmetry is characterized by a pair of cohomological data (n2,ν3)(n_2,ν_3)(n2​,ν3​). For Gf=Gb×Z2FG_f=G_b\times\mathbb Z_2^FGf​=Gb​×Z2F​, we further show that a symmetry with trivial lattice anomaly indices (n2,ν3)(n_2,ν_3)(n2​,ν3​) is onsiteable and hence admits a symmetric SRE state, establishing that these indices faithfully detect the lattice anomaly. Comparing with continuum fermionic QFT, we find that exact lattice symmetries do not realize the additional H1(BGb,Z2)H^1(BG_b,\mathbb Z_2)H1(BGb​,Z2​) anomaly layer in continuum QFT. In particular, for Gb=Z2G_b=\mathbb Z_2Gb​=Z2​, exact lattice symmetries realize only the even Z4\mathbb Z_4Z4​ subgroup of the continuum Z8\mathbb Z_8Z8​ classification. In (2+1)D, we identify three successive anomaly layers of cohomological data (n2,n3,ν4)(n_2,n_3,ν_4)(n2​,n3​,ν4​). We show that a nontrivial value of any layer obstructs a symmetric SRE state. For Gf=Gb×Z2FG_f=G_b\times\mathbb Z_2^FGf​=Gb​×Z2F​, it also forbids a symmetric invertible state. We also construct a non-onsite Z4F\mathbb Z_4^FZ4F​ lattice symmetry in (2+1)D with nontrivial lattice anomaly index that forbids any symmetric invertible states with c−∈Zc_-\in\mathbb{Z}c−​∈Z, therefore forbids any invertible states compatible with onsite Z4F\mathbb{Z}_4^FZ4F​ symmetry. Our results motivate a systematic study of which continuum anomalies admit exact microscopic lattice realizations.
We define an index for invertible phases of two-dimensional fermionic and bosonic quantum spin systems without any additional symmetry. Conjecturally, it provides a microscopic definition of an invariant related to the chiral central charge of the boundary modes c− mod 24c_- \bmod 24c−​mod24 when the effective conformal field theory description is valid. Using this index, we prove that free fermionic systems with Chern number ν mod 48≠0ν\bmod 48 \neq 0νmod48=0 are in a non-trivial invertible phase.
We develop a framework for the classification of invertible translation-invariant stabilizer codes modulo condensation and stabilization with simple codes. Our stabilizer codes generalize the Pauli codes: they are extracted from local commuting projector Hamiltonians whose terms are made of local alphabets more general than the Pauli operators. In particular, not all such codes can be entangled/disentangled with Clifford Quantum Cellular Automata. We compute their Witt-equivalence classes in any spatial dimension in terms of relative L-theory groups. In particular, we show that the group of equivalence classes of such codes in three spatial dimensions is isomorphic to the Witt group of abelian topological orders in two spatial dimensions. In four spatial dimensions we obtain a group of order two, conjecturally corresponding to the non-trivial invertible phase. Additionally, we propose a representing spectrum for the generalized cohomology theory corresponding to the invertible stabilizer codes.
Bosonic invertible and symmetry-protected topological (SPT) phases are well-known to be described by ordinary cohomology groups in low dimensions, but `beyond-cohomology' phases appear in higher dimensions. We make a systematic study of them, and find that the first major non-triviality is a mod-3 phenomenon, and not a mod-2 phenomenon as in the case of fermionic phases. We also point out that this is a dual manifestation of the classic question of Steenrod, namely the issue of the existence of homology cycles without manifold representatives. Thom developed the theory of cobordisms to answer this question, and we explain how the same analysis leads to Dijkgraaf-Witten phases which are nontrivial on general simplicial complexes but become trivial on manifolds.
1+1D symmetry-protected topological (SPT) phases with invertible symmetry can be characterized by the symmetric entanglers: globally symmetric finite-depth local unitary circuits that generate representative ground states from a product state. For non-invertible symmetries, however, distinct SPT phases, such as the three Rep(D8)\mathrm{Rep}(D_8)Rep(D8​) SPT phases, may not be connected by any symmetric entangler. We show that the three Rep(D8)\mathrm{Rep}(D_8)Rep(D8​) SPT phases can nevertheless be connected by \emph{symmetry-permuting} finite-depth local unitary circuits, realizing dualities predicted by topological quantum field theory (TQFT). We construct an explicit matrix product unitary connecting two of the Rep(D8)\mathrm{Rep}(D_8)Rep(D8​) SPT phases, and show that its action on string order parameters reproduces the anyon permutation predicted by the TQFT duality.