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Matrix product states and equivariant topological field theories for bosonic symmetry-protected topological phases in (1+1) dimensions

Ken Shiozaki, Shinsei Ryu

Abstract

Matrix Product States (MPSs) provide a powerful framework to study and classify gapped quantum phases --symmetry-protected topological (SPT) phases in particular--defined in one dimensional lattices. On the other hand, it is natural to expect that gapped quantum phases in the limit of zero correlation length are described by topological quantum field theories (TFTs or TQFTs). In this paper, for (1+1)-dimensional bosonic SPT phases protected by symmetry $G$, we bridge their descriptions in terms of MPSs, and those in terms of $G$-equivariant TFTs. In particular, for various topological invariants (SPT invariants) constructed previously using MPSs, we provide derivations from the point of view of (1+1) TFTs. We also discuss the connection between boundary degrees of freedom, which appear when one introduces a physical boundary in SPT phases, and "open" TFTs, which are TFTs defined on spacetimes with boundaries.

Matrix product states and equivariant topological field theories for bosonic symmetry-protected topological phases in (1+1) dimensions

Abstract

Matrix Product States (MPSs) provide a powerful framework to study and classify gapped quantum phases --symmetry-protected topological (SPT) phases in particular--defined in one dimensional lattices. On the other hand, it is natural to expect that gapped quantum phases in the limit of zero correlation length are described by topological quantum field theories (TFTs or TQFTs). In this paper, for (1+1)-dimensional bosonic SPT phases protected by symmetry , we bridge their descriptions in terms of MPSs, and those in terms of -equivariant TFTs. In particular, for various topological invariants (SPT invariants) constructed previously using MPSs, we provide derivations from the point of view of (1+1) TFTs. We also discuss the connection between boundary degrees of freedom, which appear when one introduces a physical boundary in SPT phases, and "open" TFTs, which are TFTs defined on spacetimes with boundaries.

Paper Structure

This paper contains 45 sections, 138 equations, 15 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Classification and topological invariants of SPT phases using MPSs
  3. Symmetry and group cohomology classification
  4. Edge degrees of freedom
  5. Simple and fixed point MPSs
  6. Topological invariants
  7. Topological invariants in terms of ground state wave functions
  8. Discrete torsion phase (1): symmetry action on twisted ground state
  9. Discrete torsion phase (2): partial symmetry action and swapping
  10. Crosscap from inversion symmetry: "partial inversion"
  11. Crosscap from time-reversal symmetry: "adjacent partial transposition"
  12. Klein bottle partition function from inversion symmetry
  13. Klein bottle partition function from time-reversal symmetry: "disjoint partial transposition with intermediate twist"
  14. $G$-equivariant topological field theories and MPSs
  15. Some basics of TFTs
  16. ...and 30 more sections

Figures (15)

  • Figure 1: The symmetry property of $A_m$ under [a] on-site unitary, [b] time-reversal, and [c] inversion symmetries, respectively.
  • Figure 2: [a] A fixed point MPS. The blue bond represents the singlet representation in $V^* \otimes V$. [b] $A$ matrix. [c] The transfer matrix.
  • Figure 3: [a] MPS expression of the symmetry action on the twisted ground state. [b] The equivalent path integral on the 2-torus $T^2$. The blue and red lines express the symmetry defect lines.
  • Figure 4: MPS expression of the partial symmetry action with swapping defined in (\ref{['eq:symmetry_action_and_swap']}).
  • Figure 5: The geometry of path integral for the partial symmetry action with swapping operator defined in (\ref{['eq:symmetry_action_and_swap']}). The red and blue lines express the symmetry defect lines. The intervals with arrows are identified with ones having the same number of arrows, which results in the 2-torus.
  • ...and 10 more figures