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Absence of nontrivial local conserved quantities in the quantum compass model on the square lattice

Mahiro Futami, Hal Tasaki

TL;DR

The paper proves that the $S= frac{1}{2}$ quantum compass model on the square lattice possesses no nontrivial local conserved quantities besides multiples of the Hamiltonian. It extends Shiraishi’s method through a diagonal-length framework and a Shiraishi shift to systematically eliminate all candidate local conserved quantities for diagonal lengths $\bar{k}$ in $[1,L/2]$, with a separate treatment for $\bar{k}=2$ showing the only allowed conserved piece is proportional to $\hat{H}$. The result holds both with and without a magnetic field, implying strong non-integrability for this model and highlighting its role as a simple, rigorous test case for non-integrable quantum spin systems. The discussion situates the finding relative to Kitaev-like models and notes potential generalizations via newer schemes (e.g., Hokkyo’s injectivity approach) to higher dimensions.

Abstract

By extending the method developed by Shiraishi, we prove that the quantum compass model on the square lattice does not possess any local conserved quantities except for the Hamiltonian itself.

Absence of nontrivial local conserved quantities in the quantum compass model on the square lattice

TL;DR

The paper proves that the quantum compass model on the square lattice possesses no nontrivial local conserved quantities besides multiples of the Hamiltonian. It extends Shiraishi’s method through a diagonal-length framework and a Shiraishi shift to systematically eliminate all candidate local conserved quantities for diagonal lengths in , with a separate treatment for showing the only allowed conserved piece is proportional to . The result holds both with and without a magnetic field, implying strong non-integrability for this model and highlighting its role as a simple, rigorous test case for non-integrable quantum spin systems. The discussion situates the finding relative to Kitaev-like models and notes potential generalizations via newer schemes (e.g., Hokkyo’s injectivity approach) to higher dimensions.

Abstract

By extending the method developed by Shiraishi, we prove that the quantum compass model on the square lattice does not possess any local conserved quantities except for the Hamiltonian itself.

Paper Structure

This paper contains 10 sections, 5 theorems, 13 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Definitions and the main theorem
  3. Proof
  4. Basic strategy
  5. First step: Shiraishi shift
  6. Second step for odd $\bar{k}$
  7. Second step for even $\bar{k}$
  8. The case $\bar{k}=2$
  9. The model with a magnetic field
  10. Discussion

Key Result

Theorem 2.1

The only local conserved quantities with diagonal length $\bar{k}$ such that $1\le\bar{k}\le L/2$ are constant multiples of the Hamiltonian $\hat{H}$.

Figures (2)

  • Figure 1: An example of the Shiraishi shift in the case $\bar{k}=5$. $\hat{\mathsf{A}}$ is a product with $\operatorname{Diag}\hat{\mathsf{A}}=5$ that satisfies conditions in Lemma \ref{['L:basic']} with (1a) and (2a). $\hat{\mathsf{B}}$ is defined by the commutator \ref{['e:SS1']}. Since $\hat{Y}$ at the top-right is appended, the product has $\operatorname{Diag}\hat{\mathsf{B}}=6$. We then truncated $\hat{X}$ at the bottom-left by taking the commutator \ref{['e:SS2']}. The resulting product $\hat{\mathsf{A}}'$ with $\operatorname{Diag}\hat{\mathsf{A}}'=5$ is the Shiraishi shift ${\cal S}(\hat{\mathsf{A}})$. Note that $\operatorname{Supp}{\cal S}(\hat{\mathsf{A}})$ has two bottom-left sites and does not satisfy condition (2). This means ${\cal S}^2(\hat{\mathsf{A}})$ does not exist and also $q_{{\cal S}(\hat{\mathsf{A}})}=0$. We also find $q_{\hat{\mathsf{A}}}=0$ from \ref{['e:q=q']}.
  • Figure 2: Products $\hat{\mathsf{C}}^{\bar{k}}_j$, $\hat{\mathsf{D}}_j$, and $\hat{\mathsf{E}}_j$ for $\bar{k}=5$. The dotted squares indicate $\hat{X}_{u_{\bar{k}-1}+e_\mathrm{x}}$ that are "appended" by the commutators in \ref{['e:Dj']} and \ref{['e:Ej']}.

Theorems & Definitions (5)

  • Theorem 2.1
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4