Abstract We study the one-dimensional spin-1/2 Heisenberg chain with Dzyaloshinskii-Moriya interaction in a random magnetic field using exact diagonalization. In order to obtain many-body mobility edge at infinite temperature, we employ a polynomial filtered Lanczos method that can avoid the fill-in problem when implementing the commonly used shift-and-invert transformation. In stark contrast to the original Heisenberg model, although the localized phase always conforms to Poisson statistics, the ergodic phase exhibits the Gaussian unitary ensemble rather than the Gaussian orthogonal ensemble statistics due to the lack of complex conjugation symmetry. The boundary between the ergodic and localized phases is determined by carefully performing finite-size scalings for the level statistics, entanglement entropy and its standard deviation, as well as fluctuations of the total spin of the system. The two phases are also well distinguished by the full delocalization or localization in the Hilbert space wherein the participation entropies present. To indicate the localized phase in experiment, we propose a scheme for realizing the out-of-time-order correlator on a modern nuclear magnetic resonance quantum simulator.
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)].
We analyze a disordered central spin model, where a central spin interacts equally with each spin in a periodic one-dimensional (1D) random-field Heisenberg chain. If the Heisenberg chain is initially in the many-body localized (MBL) phase, we find that the coupling to the central spin suffices to delocalize the chain for a substantial range of coupling strengths. We calculate the phase diagram of the model and identify the phase boundary between the MBL and ergodic phase. Within the localized phase, the central spin significantly enhances the rate of the logarithmic entanglement growth and its saturation value. We attribute the increase in entanglement entropy to a nonextensive enhancement of magnetization fluctuations induced by the central spin. Finally, we demonstrate that correlation functions of the central spin can be utilized to distinguish between MBL and ergodic phases of the 1D chain. Hence, we propose the use of a central spin as a possible experimental probe to identify the MBL phase.
We study the many-body localization of the random-field Heisenberg chain using the nested shift-invert Lanczos method with an iterative linear solver. We use the minimum residual method (MINRES) inside each Lanczos iteration. The memory consumption of the proposed method is only proportional to the dimension of the Hilbert space. We also introduce a preconditioner that takes into account the effects of disorder and interaction in the random-field Heisenberg chain. As a probe of many-body localization transition, we propose a unitary operator called the twist operator, which has a clear interpretation in the real space. We discuss its behavior for thermal and localized eigenstates. We demonstrate the efficiency of the nested iterative shift-invert diagonalization method with the proposed preconditioner for the many-body localization problem and estimate the transition point of the random-field Heisenberg chain more precisely based on the finite-size analysis of the expectation value of the twist operator.
Despite enormous efforts devoted to the study of the many-body localization (MBL) phenomenon, the nature of the high-energy behavior of the Heisenberg spin chain in a strong random magnetic field is lacking consensus. Here, we take a step back by exploring the weak interaction limit starting from the Anderson localized (AL) insulator. Through shift-invert diagonalization, we find that, below a certain disorder threshold h^{*}, weak interactions necessarily lead to an ergodic instability, whereas at strong disorder the AL insulator directly turns into MBL, in agreement with a simple interpretation of the avalanche theory for restoration of ergodicity. We further map the phase diagram for the generic XXZ model in the disorder h-interaction Δ plane. Taking advantage of the total magnetization conservation, our results unveil the remarkable behavior of the spin-spin correlation functions: in the regime indicated as MBL by standard observables, their exponential decay undergoes an inversion of orientation ξ_{z}>ξ_{x}. We find that the longitudinal length ξ_{z} is a key quantity for capturing ergodic instabilities, as it increases with system size near the thermal phase, in sharp contrast to its transverse counterpart.
We consider a one-dimensional spin-
21 chain with Heisenberg interaction in a disordered parallel magnetic field. This system is known to exhibit the many-body localization (MBL) transition at critical strength of disorder. We analyze the response of the chain when additional perpendicular magnetic field is applied to an individual spin and propose a method for accurate determination of the mobility edge via local spin measurements. We further demonstrate that the exponential decrease of the spin response with the distance between perturbed spin and measured spin can be used to characterize the localization length in the MBL phase. We also studied the effect of the quench on statistics of the entire energy spectrum. We demonstrate that the quench introduces level repulsion between energy states corresponding to different eigenvalues of the total spin in the delocalized regime, while level spacing is described by the Poisson statistics in the localized regime even at strong quench.
We examine the interplay of interaction and disorder for a Heisenberg spin-1/2 ladder system with random fields. We identify many-body localized states based on the entanglement entropy scaling, where delocalized and localized states have volume and area laws, respectively. We first establish the dynamic phase transition at a critical random field strength ${h}_{c}\ensuremath{\sim}8.5\ifmmode\pm\else\textpm\fi{}0.5$, where all energy eigenstates are localized beyond that value. Interestingly, the entanglement entropy and fluctuations of the bipartite magnetization show distinct probability distributions which characterize different phases. Furthermore, we show that for weaker
h, energy eigenstates with higher-energy density are delocalized while states at lower-energy density are localized, which defines a mobility edge separating these two phases. With increasing disorder strength, the mobility edge moves towards higher-energy density, which drives the system to the phase of the full many-body localization.