Unified theory of local integrals of motion

Abstract

Many-body localization (MBL) is understood theoretically through the existence of an extensive number of local integrals of motion (LIOMs). These conserved quantities are related to the microscopic quantum degrees of freedom that are spatially localized. Here, we present a general framework for constructing exact LIOMs with the desired locality and quantum numbers supplied as input rather than arising as emergent properties. We show that one can express the task of finding LIOMs as an optimization problem. In simple cases, solving this problem amounts to matrix diagonalization, while in more complex settings, it connects to the question of finding classical ground states of spin-glass models. We illustrate our theory using paradigmatic examples of single-particle Anderson localization and MBL in interacting spin chains. These developments unify previous results and reveal intriguing connections among many-body localization, spin-glass physics and constrained optimization problems.

Figures (3)

• Figure 1: Support probabilities for the LIOMs maximized on a single site (black triangles), and on three adjacent sites (grey circles). Results are averaged over 1000 disorder realizations. Left panel: total probability for all operators at a given distance $d$ from the central site. Right panel: average probability per operator. Inset: ratio of $p_d$ for the LIOMs optimized on three sites to $p_d$ for the single site optimization.
• Figure 2: Same as in Fig. \ref{['fig:od']}, but in the presence of additional constraints $v_n=\pm1$ and $\sum_n v_n=0$.
• Figure 3: Support probability (log scale) of the integrals of motion in the Anderson chain \ref{['andham']} with 1000 sites and disorder $W=5$, as a function of distance from the central site $i=500$, averaged over 1000 realizations. The curves correspond to integrals of motion with maximized support on $k$ sites around the center, with varying $k$. Inset: Total probability of the local set of sites $(i-k,\dots i+k)$ for each integral of motion.