Finding local integrals of motion in quantum lattice models in the thermodynamic limit

TL;DR

The paper addresses the challenge of identifying local integrals of motion (LIOMs) in many-body quantum systems without relying on full Hamiltonian diagonalization. It introduces a thermodynamic-limit, Hilbert-Schmidt-based framework that builds a local operator basis and solves a linear eigenproblem to extract LIOMs and associated Mazur bounds. The authors demonstrate the method on the XXZ chain, an integrable unitary circuit, and nearly integrable spin ladders, validating both LIOMs and quasilocal conserved quantities, as well as slow relaxation modes. This approach enables model-independent analysis of long-time dynamics and conserved quantities in large or higher-dimensional systems, with practical estimates of relaxation rates and spin stiffness without exhaustive diagonalization.

Abstract

Local integrals of motion (LIOMs) play a key role in understanding the long-time properties of closed macroscopic systems. They were found for selected integrable systems via complex analytical calculations. The existence of LIOMs and their structure can also be studied via numerical methods, which, however, involve exact diagonalization of Hamiltonians, posing a bottleneck for such studies. We show that finding LIOMs in translationally invariant lattice models or unitary quantum circuits can be reduced to a problem for which one may numerically find an exact solution in the thermodynamic limit. We develop a simple algorithm and demonstrate its efficiency by calculating LIOMs and bounds on correlations (the Mazur bounds) for infinite integrable spin chains and unitary circuits. Finally, we demonstrate that this approach identifies slow modes in nearly integrable spin models and estimates their relaxation times.

Figures (9)

• Figure 1: Lowest eigenvalues obtained from \ref{['eig']} for the XXZ model with $\Delta=3/4$ and different operator bases: (a) Pauli string basis and (b) symmetry-resolved basis.
• Figure 2: (a) and (b) shows the Mazur bounds $B_s$ corresponding to LIOMs identified in Fig. \ref{['fig:xxz_lioms']}(a) and \ref{['fig:xxz_lioms']}(b), respectively. Index $s$ enumerates local operators, $\hat{O}_s$, sorted accordingly to their support $m$.
• Figure 3: Three smallest eigenvalues, $\lambda_{\alpha}$, and eigenvectors, $V_{\alpha,1}$, versus maximal support, $M$, for XXZ model within symmetry resolved basis with (a),(b): $\Delta=3/4$ and (c),(d),$\Delta=0.1$.
• Figure 4: Projection of QLIOM, $\hat{A}^1$, on the spin current for the same case as in Fig. \ref{['fig:quasilocal']}. (a) shows results for various $M$ and extrapolated results for $M\to \infty$. (b) compares extrapolated results with Mazur bounds for spin current, $D_Z$ and $D_K$, obtained analytically in Refs. Prosen2011a and Prosen2013a, respectively.
• Figure 5: Solution of \ref{['eig']} for the XXX quantum circuit defined via elementary gate in \ref{['eq:gate']} with $\delta=0.5$. (a) shows the eigenvalues and (b) shows the structure of LIOMs visualized via the Mazur bound from \ref{['mazur']}, $B_s=\sum_{\alpha=1}^{N_L} V^2_{\alpha s}$. The index $s$ enumerates local operators, $\hat{O}_s$, sorted accordingly to their support $m$.