One-dimensional symmetric phases protected by frieze symmetries

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.

arXiv comment13 + 2 pages, submitted to Physical Review B

References in corpus (17)

• Persistent entanglement in arrays of interacting particles
• Entanglement spectrum of a topological phase in one dimension
• Symmetry protected topological orders and the group cohomology of their symmetry group
• Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Protected Topological Order
• Classification of Gapped Symmetric Phases in 1D Spin Systems
• Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order
• Symmetry protection of topological order in one-dimensional quantum spin systems
• Classifying quantum phases using Matrix Product States and PEPS
• Matrix product states represent ground states faithfully
• Complete classification of 1D gapped quantum phases in interacting spin systems
• Valence Bond Solids for Quantum Computation
• Renormalization group transformations on quantum states
• Gauging spatial symmetries and the classification of topological crystalline phases
• Anyon condensation and a generic tensor-network construction for symmetry protected topological phases
• Symmetric tensor networks and practical simulation algorithms to sharply identify classes of quantum phases distinguishable by short-range physics
• Absolute anomalies in (2+1)D symmetry-enriched topological states and exact (3+1)D constructions
• Irreducible Projective Representations and Their Physical Applications

Cited by in corpus (2)

• Les Houches Lecture Notes on Tensor Networks
• From gauging to duality in one-dimensional quantum lattice models