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Twisted locality-preserving automorphisms, anomaly index, and generalized Lieb-Schultz-Mattis theorems with anti-unitary symmetries

Ruizhi Liu, Jinmin Yi, Liujun Zou

Abstract

Symmetries and their anomalies are powerful tools to understand quantum matter. In this work, for quantum spin chains, we define twisted locality-preserving automorphisms and their Gross-Nesme-Vogts-Werner indices, which provide a unified framework to describe both unitary and anti-unitary symmetries, on-site and non-on-site symmetries, and internal and translation symmetries. For a symmetry $G$ with actions given by twisted locality-preserving automorphisms, we give a microscopic definition of its anomaly index, which is an element in $H^3_\varphi(G; U(1))$, where the subscript $\varphi$ means that anti-unitary elements of $G$ act on $U(1)$ by complex conjugation. We show that an anomalous symmetry leads to multiple Lieb-Schultz-Matttis-type theorems. In particular, any state with an anomalous symmetry must either have long-range correlation or violate the entanglement area law. Based on this theorem, we further deduce that any state with an anomalous symmetry must have long-range entanglement, and any Hamiltonian that has an anomalous symmetry cannot have a unique gapped symmetric ground state, as long as the interactions in the Hamiltonian decay fast enough as the range of the interaction increases. For Hamiltonians with only two-spin interactions, the last theorem holds if the interactions decay faster than $1/r^2$, where $r$ is the distance between the two interacting spins. We demonstrate these general theorems in various concrete examples.

Twisted locality-preserving automorphisms, anomaly index, and generalized Lieb-Schultz-Mattis theorems with anti-unitary symmetries

Abstract

Symmetries and their anomalies are powerful tools to understand quantum matter. In this work, for quantum spin chains, we define twisted locality-preserving automorphisms and their Gross-Nesme-Vogts-Werner indices, which provide a unified framework to describe both unitary and anti-unitary symmetries, on-site and non-on-site symmetries, and internal and translation symmetries. For a symmetry with actions given by twisted locality-preserving automorphisms, we give a microscopic definition of its anomaly index, which is an element in , where the subscript means that anti-unitary elements of act on by complex conjugation. We show that an anomalous symmetry leads to multiple Lieb-Schultz-Matttis-type theorems. In particular, any state with an anomalous symmetry must either have long-range correlation or violate the entanglement area law. Based on this theorem, we further deduce that any state with an anomalous symmetry must have long-range entanglement, and any Hamiltonian that has an anomalous symmetry cannot have a unique gapped symmetric ground state, as long as the interactions in the Hamiltonian decay fast enough as the range of the interaction increases. For Hamiltonians with only two-spin interactions, the last theorem holds if the interactions decay faster than , where is the distance between the two interacting spins. We demonstrate these general theorems in various concrete examples.

Paper Structure

This paper contains 37 sections, 37 theorems, 152 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Review of the operator algebra formalism
  3. Algebras of local and quasi-local operators
  4. Quantum cellular automata and locality-preserving automorphisms
  5. States, GNS construction and factors
  6. Split property
  7. Anti-unitary equivalence
  8. Hamiltonians and ground states
  9. Twisted locality-preserving automorphisms, their GNVW index, and symmetry actions
  10. Anomaly index
  11. General definition
  12. Example: $G=G_0\times\mathbb{Z}$
  13. Generalized Lieb-Schultz-Mattis theorems for infinite systems with anti-unitary symmetry
  14. The root lemma and its corollaries
  15. The more standard LSM theorems: Constraints on the spectrum and symmetry-enforced long-range entanglement
  16. ...and 22 more sections

Key Result

Theorem 2.1

A state $\psi$ on $\mathscr{A}^{ql}$ is clustering iff it is a factor state.

Figures (4)

  • Figure 1: The block-partitioned unitary $S$ which swaps the original system (yellow) and its copy (blue). This figure only shows 6 sites around the origin, but the lattice actually extends from $-\infty$ to $\infty$.
  • Figure 2: The swap operator with a shift. This figure only shows 6 sites around the origin, but the lattice actually extends from $-\infty$ to $\infty$.
  • Figure 3: The dashed lines, which separate the figure into 3 parts, perform a decomposition of $\tilde{\alpha}'(g, n)$ as in Proposition \ref{['prop:decomp_twisted']}, i.e., $\tilde{\alpha}'_{\geqslant 0}(g, n)$ is the right most part, $\tilde{\alpha}'(g, n)_{<0}$ is the left most part, and $\alpha'_0(g, n)$ can be obtained from the region between the two dashed lines, $\tilde{\alpha}'_{\geqslant 0}(g, n)$ and $\tilde{\alpha}'(g, n)_{<0}$. This figure only shows 6 sites around the origin, but the lattice actually extends from $-\infty$ to $\infty$. Also, $n=2$ in this figure, but the decomposition of $\tilde{\alpha}'(g, n)$ for other values of $n$ can be similarly obtained.
  • Figure 4: The circuit $\tilde{S}_{+}S_{+}$. The black arrows are implemented first. This figure only shows 6 sites around the origin, but the lattice actually extends from $-\infty$ to $\infty$.

Theorems & Definitions (90)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Example 2.1
  • Definition 2.6
  • Definition 2.7
  • Theorem 2.1: Theorem 2.6.10 of Ref. bratteli2013operator1
  • Definition 2.8
  • ...and 80 more