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Matrix Product States for Modulated Symmetries: SPT, LSM, and Beyond

Amogh Anakru, Sarvesh Srinivasan, Linhao Li, Zhen Bi

Abstract

Matrix product states (MPS) provide a powerful framework for characterizing one-dimensional symmetry-protected topological (SPT) phases of matter and for formulating Lieb-Schultz-Mattis (LSM)-type constraints. Here we generalize the MPS formalism to translationally invariant systems with general modulated symmetries. We show that the standard symmetry "push-through" condition for conventional global symmetry must be revised to account for symmetry modulation, and we derive the appropriate generalized condition. Using this generalized push-through structure, we classify one-dimensional SPT phases with modulated symmetries and formulate LSM-type constraints within the same MPS-based framework.

Matrix Product States for Modulated Symmetries: SPT, LSM, and Beyond

Abstract

Matrix product states (MPS) provide a powerful framework for characterizing one-dimensional symmetry-protected topological (SPT) phases of matter and for formulating Lieb-Schultz-Mattis (LSM)-type constraints. Here we generalize the MPS formalism to translationally invariant systems with general modulated symmetries. We show that the standard symmetry "push-through" condition for conventional global symmetry must be revised to account for symmetry modulation, and we derive the appropriate generalized condition. Using this generalized push-through structure, we classify one-dimensional SPT phases with modulated symmetries and formulate LSM-type constraints within the same MPS-based framework.
Paper Structure (15 sections, 5 theorems, 95 equations, 2 figures)

This paper contains 15 sections, 5 theorems, 95 equations, 2 figures.

Table of Contents

  1. Review of Matrix Product State Formalism
  2. Generalized push-through theorem
  3. Proof strategy
  4. Proof for generalized push-through condition
  5. Open Boundary Generalizations
  6. Examples of Symmetry Push-through for Various Modulated Symmetries
  7. General Modulated Symmetries
  8. Classification of Weak SPTs
  9. Exponential Symmetry
  10. Dipole Symmetry
  11. Tilted Exponential Symmetry
  12. Exponential symmetry with charge symmetry: Faithful and Unfaithful Unitary Representations
  13. LSM and SPT-LSM theorem
  14. Abelian exponential symmetries
  15. Dipole symmetries

Key Result

Lemma 1

Let $U$ be an on-site unitary, and consider the following diagrammatic equation: with normalization conditions $\Tr(v^\dag v \rho)= \Tr(w^\dag w \rho)=1$. Here $\rho$ is the unique dominant right eigenvector of the transfer operator associated with the injective MPS $A$. Then we have $\abs{\lambda}\leq 1$.

Figures (2)

  • Figure 1: Generalized push-through rule for modulated unitary symmetries.
  • Figure 2: Push-through rule for dipole symmetries.

Theorems & Definitions (14)

  • Lemma 1
  • Remark
  • proof : Proof:
  • Lemma 2
  • proof : Proof:
  • Remark
  • Lemma 3
  • proof : Proof:
  • Claim
  • proof : Proof:
  • ...and 4 more