We investigate the mixed-state entanglement between two spins embedded in the XXZ Heisenberg chain under thermal equilibrium. By deriving an analytical expression for the entanglement of two-spin thermal states and extending this analysis to larger spin chains, we demonstrate that mixed-state entanglement is profoundly shaped by both disorder and temperature. Our results reveal a sharp distinction between many-body localized (MBL) and ergodic phases, with entanglement vanishing above different finite temperature thresholds. Furthermore, by analyzing non-adjacent spins, we uncover an approximate exponential decay of entanglement with separation. This work advances the understanding of the quantum-to-classical transition by linking the entanglement properties of small subsystems to the broader thermal environment, offering a explanation for the absence of entanglement in macroscopic systems. These findings provide critical insights into quantum many-body physics, bridging concepts from thermalization, localization, and quantum information theory.
Abstract We investigate the mixed-state entanglement between two spins embedded in the XXZ Heisenberg chain under thermal equilibrium. By deriving an analytical expression for the entanglement of two-spin thermal states and extending this analysis to larger spin chains, we demonstrate that mixedstate entanglement is profoundly shaped by both disorder and temperature. Our results reveal a sharp distinction between many-body localized (MBL) and ergodic phases, with entanglement vanishing above different finite temperature thresholds. Furthermore, by analyzing non-adjacent spins, we uncover an approximate exponential decay of entanglement with separation. This work advances the understanding of the quantum-to-classical transition by linking the entanglement properties of small subsystems to the broader thermal environment, offering a explanation for the absence of entanglement in macroscopic systems. These findings provide critical insights into quantum manybody physics, bridging concepts from thermalization, localization, and quantum information theory.
Many-body localization is a profound phase of matter affecting the entire spectrum which emer ges in the presence of disorder in interacting many-body systems. Recently, the stability of many-body localization has been challenged by the avalanche mechanism, in which a small thermal region can spread, destabilizing localization and leading to global thermalization of the system. A key unresolved question is the critical competition between the thermal region's influence and the disorder strength required to trigger such an avalanche. Here, we numerically investigate many-body localization stability in an isolated Heisenberg spin chain of size
L subjected to a disordered magnetic field. By embedding a tunable thermal region of size
P, we analyze the system's behavior in both static and dynamical regimes using entanglement entropy and the gap ratio. Our study yields two main findings. Firstly, for strong disorder, the avalanche only occurs if the thermal region scales with system size, specifically when
P/L exceeds a threshold value. Secondly, at strong disorder, we identify an intermediate phase between many-body localization and ergodic behavior as
P increases. This intermediate phase leaves its fingerprint in both static and dynamic properties of the system and tends to vanish in the thermodynamic limit. Although our simulations are restricted to finite system sizes, the analysis suggests that these results hold in the thermodynamic limit for isolated many-body systems.
Disorder free many-body localization (MBL) can occur in interacting systems that can dynamically generate their own disorder. We address the thermal-MBL phase transition of two isotropic Heisenberg spin chains that are quasiperiodically coupled to each other. The spin chains are incommensurate and are coupled through a short-range exchange interaction of the X X Z type that decays exponentially with the distance. Using exact diagonalization, matrix product states, and a density matrix renormalization group, we calculate the time evolution of the entanglement entropy at long times and extract the inverse participation ratio in the thermodynamic limit. We show that this system has a robust MBL phase. We establish the phase diagram with the onset of MBL as a function of the interchain exchange coupling and of the incommensuration between the spin chains. The Ising limit of the interchain interaction optimizes the stability of the MBL phase over a broad range of incommensurations above a given critical exchange coupling. Incorporation of interchain spin flips significantly enhances entanglement between the spin chains and produces delocalization, favoring a prethermal phase whose entanglement entropy grows logarithmically with time. Published by the American Physical Society 2024
Abstract The connection between entanglement dynamics and non-equilibrium statistics in isolated many-body quantum systems has been established both theoretically and experimentally. Many-body localization (MBL), a phenomenon where interacting particles in disordered (i.e. random) chains fail to thermalize, exemplifies this connection. However, the systematic proof of critical phenomena such as MBL remains challenging due to the lack of robust methods for analyzing many-body entanglement dynamics. In this paper, we identify MBL through quasi-periodic dynamics in the entanglement evolution of subsystems in a disordered Heisenberg chain. This new form of characterizing MBL, through stable quasi-periodic dynamics of entanglement-where ‘stable’ means they persist in the thermodynamic limit—concretely distinguishes between two competing scenarios: fully localized behavior of subsystems or slowly, exponentially slow in disorder, thermalizing subsystems—a heated controversy in the literature. Utilizing perturbation theory, we derive the entanglement dynamics of single spins through an infinite perturbative series, while also modeling rare Griffiths regions (locally thermal inclusions). Our results prove that in regimes of sufficiently strong disorder, the entanglement evolution of individual subsystems remains quasi-periodic in the thermodynamic limit, thereby providing concrete evidence for the stability of MBL dynamics in disordered Heisenberg chains. This behavior contrasts with the widely reported logarithmic growth of subsystem entanglement in the MBL phase. We show that the logarithmic growth observed in prior studies arises from statistical ensemble averaging, which is prohibited due to the intrinsic non-ergodic dynamics characteristic of MBL systems, rooted in their quasi-periodic features.
Some interacting disordered many-body systems are unable to thermalize when the quenched disorder becomes larger than a threshold value. Although several properties of nonzero energy density eigenstates (in the middle of the many-body spectrum) exhibit a qualitative change across this many-body localization (MBL) transition, many of the commonly used diagnostics only do so over a broad transition regime. Here we provide evidence that the transition can be located precisely even at modest system sizes by sharply defined changes in the distribution of extremal eigenvalues of the reduced density matrix of subsystems. In particular, our results suggest that $p*={lim}_{{\ensuremath{\lambda}}_{2}\ensuremath{\rightarrow}ln{(2)}^{+}}{P}_{2}({\ensuremath{\lambda}}_{2})$, where ${P}_{2}({\ensuremath{\lambda}}_{2})$ is the probability distribution of the second lowest entanglement eigenvalue ${\ensuremath{\lambda}}_{2}$, behaves as an ``order parameter'' for the MBL phase:
p∗>0 in the MBL phase, while
p∗=0 in the ergodic phase with thermalization. Thus, in the MBL phase, there is a nonzero probability that a subsystem is entangled with the rest of the system only via the entanglement of one subsystem qubit with degrees of freedom outside the region. In contrast, this probability vanishes in the thermal phase.
We investigate the transition from the many-body localized phase to the ergodic thermalized phase at an infinite temperature in an
XY spin chain with
L spins, which experiences power-law decaying interactions in the form of ${V}_{ij}\ensuremath{\propto}1/{\left|i\ensuremath{-}j\right|}^{\ensuremath{\alpha}}$ (
i,j=1,...,L) and a random transverse field. By performing large-scale exact diagonalization for the chain size up to
L=18, we systematically analyze the energy gap statistics, the half-chain entanglement entropy, and the uncertainty of the entanglement entropy of the system at different interaction exponents $\ensuremath{\alpha}$. The finite-size critical scaling allows us to determine the critical disorder strength
Wc and the critical exponent $\ensuremath{\nu}$ at the many-body-localization phase transition, as a function of the interaction exponent $\ensuremath{\alpha}$ in the limit $L\ensuremath{\rightarrow}\ensuremath{\infty}$. We find that both
Wc and $\ensuremath{\nu}$ diverge when $\ensuremath{\alpha}$ decreases to a critical power of ${\ensuremath{\alpha}}_{c}\ensuremath{\simeq}1.16\ifmmode\pm\else\textpm\fi{}0.17$, indicating the absence of many-body localization for $\ensuremath{\alpha}<{\ensuremath{\alpha}}_{c}$. Our result differs from the analytical prediction of ${\ensuremath{\alpha}}_{c}=3/2$ found in a previous work based on a scaling argument [A. L. Burin, Phys. Rev. B 92, 104428 (2015)], but is close to the recent numerical result ${\ensuremath{\alpha}}_{c}\ensuremath{\approx}1$ obtained from quantum dynamics simulation [A. Safavi-Naini, M. L. Wall, O. L. Acevedo, A. M. Rey, and R. M. Nandkishore, Phys. Rev. A 99, 033610 (2019)].