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Connecting Entanglement Entropy Growth and Local Integrals of Motion in the Disordered Fermi-Hubbard Model

Ahad Nokhostin Helm

Published Jan 1, 2025• 0 citations
OpenAlex

Abstract

We study a one-dimensional Fermi-Hubbard model with disorder in charge and spin degrees of freedom. We calculate the time dependence of the entanglement entropy. While previous research on disordered interacting systems has typically focused on systems with either charge or spin, our model enables us to explore the interplay between charge and spin in shaping the behavior of entanglement. We use a method that identifies optimally local charge- and spin-specific integrals of motion. We ask how the locality level of these integrals of motion influences the capacity of low-order terms in the l-bit Hamiltonian to capture the entanglement entropy. Our results show that increasing the locality level improves the accuracy of low-order terms in capturing entanglement entropy dynamics. With equally strong charge and spin disorder, the behavior of the entanglement entropy closely resembles that observed in single-degree-of-freedom systems, and the l-bit Hamiltonian truncated at second order accurately captures this behavior. Author Keywords: Entanglement Entropy, Fermi-Hubbard Model, Hungarian Algorithm, l-bit Hamiltonian, Local Integrals of Motion, Many-Body Localization

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•Sep 22, 2026•0 citations
We study a one-dimensional Fermi-Hubbard model with disorder in charge and spin degrees of freedom. We calculate the time dependence of the entanglement entropy. While previous research on disordered interacting systems has typically focused on systems with either charge or spin, our model enables us to explore the interplay between charge and spin in shaping the behavior of entanglement. We use a method that identifies optimally local charge- and spin-specific integrals of motion. We ask how the locality level of these integrals of motion influences the capacity of low-order terms in the l-bit Hamiltonian to capture the entanglement entropy. Our results show that increasing the locality level improves the accuracy of low-order terms in capturing entanglement entropy dynamics. With equally strong charge and spin disorder, the behavior of the entanglement entropy closely resembles that observed in single-degree-of-freedom systems, and the l-bit Hamiltonian truncated at second order accurately captures this behavior. Author Keywords: Entanglement Entropy, Fermi-Hubbard Model, Hungarian Algorithm, l-bit Hamiltonian, Local Integrals of Motion, Many-Body Localization
Generically a quantum system initialized in an unentangled state will, under unitary dynamics, rapidly become entangled, a process closely related to information transport and to thermalization. Disorder can suppress the growth of entanglement and result in memory of initial conditions. In non-interacting systems this arises from localization of single-particle states, the occupancy of which is fixed by the initial condition. In interacting systems similar localized conserved quantities persist, but with the added feature that they are coupled, resulting in entanglement growth which is distinct from both non-interacting localized systems and from generic ergodic systems. The Fermi-Hubbard model has two degrees of freedom per site -- charge and spin -- and disorder may be present in both of these. We study the growth of entanglement in two scenarios -- disorder in charge equal and unequal to that in spin, and determine the distinct contributions of charge and spin degrees of freedom by expanding the Hamiltonian in terms of a set of optimally localized conserved quantities with separate charge and spin character. We find that coupling between charge and spin is significantly weaker than charge-charge and spin-spin coupling. While this decoupling is present in all our results, it is only apparent when the strength of the disorder in the two sectors is different such that there is a separation between the characteristic timescales of the contributions to entanglement made by charge and by spin.
We investigate the non-equilibrium dynamics of the disordered one-dimensional Fermi-Hubbard model with a focus on many-body localization. The system is initialized in a charge-density-wave-state, and its time evolution is analyzed through sublattice imbalance (spin and charge), and bipartite entanglement entropy. A clear crossover from ergodic to non-ergodic behavior is observed with increasing disorder strength. In the weak disorder regime, rapid decay of imbalance and the fast growth of entanglement indicate efficient thermalization. In contrast, a strong disorder leads to persistent imbalance and slow dynamics, signaling the breakdown of ergodicity. The charge and spin sectors exhibit distinct relaxation behavior, providing evidence for partial decoupling between these degrees of freedom. Furthermore, in the interacting regime, the entanglement entropy shows slow logarithmic growth, reflecting the dephasing-driven dynamics characteristic of the many-body localized phase. These results highlight the interplay between disorder and interactions in determining the dynamical properties of the system and establish robust signatures of many-body localization in the Fermi-Hubbard model.
Although many-body localization has primarily been studied in systems with a single local degree of freedom, experimental studies of many-body localization in cold atom systems motivate exploration of the disordered Hubbard model. With two coupled local degrees of freedom, it is natural to ask how localization in spin relates to disorder in charge and vice versa. Most prior work has addressed disorder in only one of these sectors and often has not used measures of localization which distinguish between charge and spin. Here, we explore localization in the Hubbard model with a wide range of independent values of charge and spin disorder using measures of localization based on charge and spin-specific integrals of motion. Our results demonstrate symmetry between the response of the spin to charge disorder and vice versa, and we find very weak disorder in one channel, so long as the disorder in the other channel is sufficiently strong, results in localization in both channels. The strength of disorder required in the less-disordered channel declines as the system size increases. Furthermore, the weaker the disorder in the less-disordered channel, the longer the timescale at which localization appears in the dynamics of this degree of freedom.
We study many-body localization for a disordered chain of spin-1/2 fermions. When both down and up components are exposed to the same strong disorder, a power-law growth of the entanglement entropy was observed [P. Prelov\ifmmode \check{s}\else \v{s}\fi{}ek et al., Phys. Rev. B 94, 241104 (2016)] that suggests that many-body localization is not complete; the density (charge) degree of freedom is localized, while the spin degree of freedom is apparently delocalized. We show that this powerlike behavior is only a transient effect and that, for longer times, the growth of the entanglement entropy is logarithmic in time. At the same time, the decay of the spin correlation slows down, showing that the spin transport is strongly reduced at long times. The dynamics of the system is quite similar to the one of many-body localized systems. Because of the limited duration of the numerical experiment, it is not possible to determine whether the system is truly many-body localized or diffusing very slowly at long times.
Many-body localization is believed to be generically unstable in quantum systems with continuous non-Abelian symmetries, even in the presence of strong disorder. Breaking these symmetries can stabilize the localized phase, leading to the emergence of an extensive number of quasilocally conserved quantities known as local integrals of motion, or lll bits. Using a sophisticated nonperturbative technique based on continuous unitary transforms, we investigate the one-dimensional Hubbard model subject to both spin and charge disorder, compute the associated lll bits and demonstrate that the disorder gives rise to a novel form of spin-charge separation. We examine the role of symmetries in delocalizing the spin and charge degrees of freedom, and show that while symmetries generally lead to delocalization through multiparticle resonant processes, certain subsets of states appear stable.
We present a theoretical and numerical study of the many-body localization (MBL) transition and entanglement dynamics in disordered spin-fermion chains with long-range interactions. Moving beyond standard single-particle pictures, we investigate the breakdown of ergodicity and thermalization by constructing an effective Hamiltonian via iterative Schrieffer-Wolff transformations and utilizing the local integrals of motion (l-bits) formalism. Our analysis examines entanglement entropy growth, level spacing statistics, and the transition from Wigner-Dyson to Poisson statistics. Furthermore, we derive analytical estimates for the dephasing rate scaling as a function of long-range potential parameters and validate the theoretical findings using exact diagonalization (ED) numerical protocols with finite-size scaling.