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arXiv:2202.12880 · Physical review. B./Physical review. B

One-dimensional symmetric phases protected by frieze symmetries

Bram Vancraeynest-De Cuiper, Jacob C. Bridgeman, Nicolas Dewolf, Jutho Haegeman, Frank Verstraete

Published Mar 10, 2023• 2 citations
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Abstract

We make a systematic study of symmetry-protected topological gapped phases of quantum spin chains in the presence of the frieze space groups in one dimension using matrix product states. Here, the spatial symmetries of the one-dimensional lattice are considered together with an additional ``vertical reflection,'' which we take to be an on-site ${\mathbb{Z}}_{2}$ symmetry. We identify seventeen distinct non-trivial phases, define canonical forms, and compare the topological indices obtained from the MPS analysis with the group cohomological predictions. We furthermore construct explicit renormalization group fixed-point wave functions for symmetry-protected topological phases with global on-site symmetries, possibly combined with time reversal and parity symmetry. En route, we demonstrate how group cohomology can be computed using the Smith normal form.

DOI
10.1103/physrevb.107.115123

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We make a systematic study of symmetry-protected topological gapped phases of quantum spin chains in the presence of the frieze space groups in one dimension using matrix product states. Here, the spatial symmetries of the one-dimensional lattice are considered together with an additional ``vertical reflection,'' which we take to be an on-site Z2{\mathbb{Z}}_{2}Z2​ symmetry. We identify seventeen distinct non-trivial phases, define canonical forms, and compare the topological indices obtained from the MPS analysis with the group cohomological predictions. We furthermore construct explicit renormalization group fixed-point wave functions for symmetry-protected topological phases with global on-site symmetries, possibly combined with time reversal and parity symmetry. En route, we demonstrate how group cohomology can be computed using the Smith normal form.
Spatially modulated symmetries are one of the new types of symmetries the symmetry actions of which are position dependent. Yet exotic phases resulting from these spatially modulated symmetries are not fully understood and classified. In this work, we systematically classify one-dimensional bosonic symmetry-protected topological phases of multipole symmetries by employing the matrix product state formalism. The symmetry action induces projective representations at the ends of an open chain, which we identify via group cohomology. In particular, for rrr-pole symmetries---for instance, r=0r=0r=0 (global), 1 (dipole), and 2 (quadrupole)---the classification is determined by distinct components of second cohomology groups that encode the boundary projective representations.
We provide a classification of symmetry-protected topological (SPT) phases of many-body localized (MBL) spin and fermionic systems in one dimension. For spin systems, using tensor networks we show that all eigenstates of these phases have the same topological index as defined for SPT ground states. For unitary on-site symmetries, the MBL phases are thus labeled by the elements of the second cohomology group of the symmetry group. A similar classification is obtained for anti-unitary on-site symmetries, time-reversal symmetry being a special case with a [Formula: see text] classification (see [Wahl 2018 Phys. Rev. B 98 054204]). For the classification of fermionic MBL phases, we propose a fermionic tensor network diagrammatic formulation. We find that fermionic MBL systems with an (anti-)unitary symmetry are classified by the elements of the (generalized) second cohomology group if parity is included into the symmetry group. However, our approach misses a [Formula: see text] topological index expected from the classification of fermionic SPT ground states. Finally, we show that all found phases are stable to arbitrary symmetry-preserving local perturbations. Conversely, different topological phases must be separated by a transition marked by delocalized eigenstates. Finally, we demonstrate that the classification of spin systems is complete in the sense that there cannot be any additional topological indices pertaining to the properties of individual eigenstates, but there can be additional topological indices that further classify Hamiltonians.
We provide a classification of symmetry-protected topological (SPT) phases of many-body localized (MBL) spin and fermionic systems in one dimension. For spin systems, using tensor networks we show that all eigenstates of these phases have the same topological index as defined for SPT ground states. For unitary on-site symmetries, the MBL phases are thus labeled by the elements of the second cohomology group of the symmetry group. A similar classification is obtained for anti-unitary on-site symmetries, time-reversal symmetry being a special case with a [Formula: see text] classification (see [Wahl 2018 Phys. Rev. B 98 054204]). For the classification of fermionic MBL phases, we propose a fermionic tensor network diagrammatic formulation. We find that fermionic MBL systems with an (anti-)unitary symmetry are classified by the elements of the (generalized) second cohomology group if parity is included into the symmetry group. However, our approach misses a [Formula: see text] topological index expected from the classification of fermionic SPT ground states. Finally, we show that all found phases are stable to arbitrary symmetry-preserving local perturbations. Conversely, different topological phases must be separated by a transition marked by delocalized eigenstates. Finally, we demonstrate that the classification of spin systems is complete in the sense that there cannot be any additional topological indices pertaining to the properties of individual eigenstates, but there can be additional topological indices that further classify Hamiltonians.
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.
We provide a simple ansatz for finding integer spin SO(3)×Z2T×Z2RSO(3)\times\Z_2^T\times\Z_2^RSO(3)×Z2T​×Z2R​-invariant matrix product states representing a phase for each triple of Z2\Z_2Z2​ topological indices studied in the literature by Tasaki \cite{tasakiheistop}\cite{tasakitop} and Ogata (corresponding to time-reversal and reflection symmetries) \cite{ogatatime}\cite{ogatareflection}. Every combination of trivial and non-trivial indices has a spin-111 SU(2)SU(2)SU(2), time-reversal, and site-reflection symmetric MPS representative with injectivity length l≤8l\leq 8l≤8. We provide some exact examples for spin-111 and a Fortran program which will find representatives of any triple for any integer spin, as well as some data from random sampling for small spin. The existence of such states agrees with the classification of Chen, Gu, and Wen \cite{chen}, and with the recent results of Tasaki \cite{tasakinew} equating the hidden-order and one of Ogata's indices, wherein the Ogata index is that associated to the dihedral D2D_2D2​ symmetry.
The equivalence of topological indices for U(1)⋊Z2\mathrm{U}(1)\rtimes\mathbb{Z}_2U(1)⋊Z2​-symmetric quantum spin chains
Symmetry-protected topological (SPT) phases in quantum spin chains are distinguished by topological indices. Ogata's operator-algebraic indices apply to a broad range of symmetry classes without requiring continuous symmetry, whereas the elementary twist index of Tasaki requires U(1)\mathrm{U}(1)U(1) symmetry. We prove their equivalence in the common setting of integer-spin chains with on-site U(1)⋊Z2\mathrm{U}(1)\rtimes\mathbb{Z}_2U(1)⋊Z2​ symmetry and a symmetric locally-unique gapped ground state, without assuming translation invariance or a matrix product representation. A charge-fluctuation estimate yields a strong-limit construction of half-chain rotations with their phases fixed. An exact group commutator identity then identifies the Ogata index with the limit of local twist expectations and gives an explicit error bound. This provides a computable formula for the index when a lower bound on the gap and the relevant ground-state expectations are available. Combining the equivalence with a recent boundary-field gap theorem of OpenAI and our earlier twist-index calculation, we obtain a spin-rotation-invariant locally-unique gapped ground state of the infinite S=1S=1S=1 antiferromagnetic Heisenberg chain with both indices equal to −1-1−1, without assuming infinite-volume uniqueness. The author formulated the problem and proposed the basic strategy, while ChatGPT Astra made substantial contributions to the development of the proof.