A prime characterization of many-body localized (MBL) systems is the entanglement of their eigenstates; in contrast to the typical ergodic phase whose eigenstates are volume law, MBL eigenstates obey an area law. In this paper, we show that a spin-disordered Hubbard model has both a large number of area-law eigenstates as well as a large number of eigenstates whose entanglement scales logarithmically with system size (log-law). This model provides a microscopic Hamiltonian which is neither ergodic nor MBL. We establish these results through a combination of analytic arguments based on the eta-pairing operators combined with a numerical analysis of eigenstates. In addition, we describe and simulate a dynamic time evolution approach starting from product states through which one can separately probe the area-law and log-law eigenstates in this system.
A prime characterization of many-body localized (MBL) systems is the entanglement of their eigenstates; in contrast to the typical ergodic phase whose eigenstates are volume law, MBL eigenstates obey an area law. In this paper, we show that a spin-disordered Hubbard model has both a large number of area-law eigenstates as well as a large number of eigenstates whose entanglement scales logarithmically with system size (log-law). This model provides a microscopic Hamiltonian which is neither ergodic nor MBL. We establish these results through a combination of analytic arguments based on the eta-pairing operators combined with a numerical analysis of eigenstates. In addition, we describe and simulate a dynamic time evolution approach starting from product states through which one can separately probe the area-law and log-law eigenstates in this system.
We consider disordered many-body systems with periodic time-dependent Hamiltonians in one spatial dimension. By studying the properties of the Floquet eigenstates, we identify two distinct phases: (i) a many-body localized (MBL) phase, in which almost all eigenstates have area-law entanglement entropy, and the eigenstate thermalization hypothesis (ETH) is violated, and (ii) a delocalized phase, in which eigenstates have volume-law entanglement and obey the ETH. The MBL phase exhibits logarithmic in time growth of entanglement entropy when the system is initially prepared in a product state, which distinguishes it from the delocalized phase. We propose an effective model of the MBL phase in terms of an extensive number of emergent local integrals of motion, which naturally explains the spectral and dynamical properties of this phase. Numerical data, obtained by exact diagonalization and time-evolving block decimation methods, suggest a direct transition between the two phases.
We study the properties of excited states in one-dimensional many-body localized (MBL) systems using a matrix product state algorithm. First, the method is tested for a large disordered noninteracting system, where for comparison we compute a quasiexact reference solution via a Monte Carlo sampling of the single-particle levels. Thereafter, we present extensive data obtained for large interacting systems of $L\ensuremath{\sim}100$ sites and large bond dimensions $\ensuremath{\chi}\ensuremath{\sim}1700$, which allows us to quantitatively analyze the scaling behavior of the entanglement S in the system. The MBL phase is characterized by a logarithmic growth $S(L)\ensuremath{\sim}log(L)$ over a large scale separating the regimes where volume and area laws hold. We check the validity of the eigenstate thermalization hypothesis. Our results are consistent with the existence of a mobility edge.
Many-body localization of a disorder interacting boson system in one dimension is studied numerically at the filling factor being one-half, in terms of level statistics, local compressibility, correlation function and entanglement entropies. The von Neumann entanglement entropy is decoupled into a particle number entropy and a configuration entropy. An anomalous volume-law behavior is found for the configuration entanglement entropy to confirm a recent experimental observation [A. Lukin, M. Rispoli, R. Schittko, et al., Science 364, 256 (2019)] for sufficient strong disorder, while the particle number entropy fulfills an area-law corresponding to the total entropy for disordered spin chain. The localization length are extracted from a two-body correlation function for many-body localization states and corresponding time-evolutions states as well. A phase diagrams is established with consisting of an ergodic thermalized region and a many-body-localization region in a parameter space of the disorder strength and the energy density. Two regions are separated by a many-body mobility edge deducted from the standard deviation of the particle-number entanglement entropy, which appears consistent with that based on the localization length. Slow dynamics characterized by a logarithmic time-dependence is explicitly shown for both the particle number entropy and the configuration entropy in an intermediate regime of their time-evolutions, which does not show up in the Anderson localization case, i.e. non-interacting disorder systems.
In recent years, physicists have made great strides toward better understanding isolated many-body quantum systems. Already, two distinct phases are well known: systems that are ergodic and thermalize completely, and, by contrast, systems that exhibit many-body localization (MBL) due to a strong disorder potential, which prevents thermalization. It is tempting to wonder whether a phase of matter could exist between the extremes of full thermalization and MBL in a generic, i.e., nonintegrable, isolated many-body quantum system. Here, the authors provide evidence that such a nonergodic phase can be realized in a translationally invariant itinerant electron model, with both charge and spin degrees of freedom. Exact diagonalization calculations on finite system sizes reveal that there exists a ``spin band'' of nonthermal eigenstates. These states have entanglement entropy that scales as an overall ``volume law,'' but after a partial measurement of the spin on each site the scaling is modified to an ``area law,'' thus suggesting the states are not ergodic. These properties and others suggest that this model realizes a new nonthermal phase of matter, known as a quantum disentangled liquid (QDL). The putative existence of this phase has striking implications for the foundations of quantum statistical mechanics.
An interacting quantum system can transition from an ergodic to a many-body localized (MBL) phase under the presence of sufficiently large disorder. Both phases are radically different in their dynamical properties, which are characterized by highly excited eigenstates of the Hamiltonian. Each eigenstate can be characterized by the set of quantum numbers over the set of (local, in the MBL phase) integrals of motion of the system. In this work we study the evolution of the eigenstates of the disordered Heisenberg model as the disorder strength, W, is varied adiabatically. We focus on the probability that two `colliding' eigenstates hybridize as a function of both the range R at which they differ as well as the strength of their hybridization. We find, in the MBL phase, that the probability of a colliding eigenstate hybridizing strongly at range R decays as Pr(R)∝exp[−R/η], with a length scale η(W)=1/(Blog(W/Wc)) which diverges at the critical disorder strength Wc. This leads to range-invariance at the transition, suggesting the formation of resonating cat states at all ranges. This range invariance does not survive to the ergodic phase, where hybridization is exponentially more likely at large range, a fact that can be understood with simple combinatorial arguments. In fact, compensating for these combinatorial effects allows us to define an additional correlation length ξ in the MBL phase which is in excellent agreement with previous works and which takes the critical value 1/log(2) at the transition, found in previous works to destabilize the MBL phase. Finally, we show that deep in the MBL phase hybridization is dominated by two-level collisions of eigenstates close in energy.
Eigenstates of many-body localized (MBL) systems are characterized by area-law bipartite entanglement along with multifractal statistics of their amplitudes on Hilbert space. At the same time, sparse random pure states with fractal statistics are not compatible with area law and necessarily exhibit volume-law entanglement. This raises the question regarding what correlation functions of Hilbert-space amplitudes MBL eigenstates must possess for their area-law entanglement to be compatible with their multifractality. In this Letter, we identify and compute such appropriate Hilbert-space correlations which carry information of entanglement. We find that, for MBL eigenstates, these correlations are qualitatively different not only from those of ergodic states but also of sparse random states with fractal statistics. This enables us to show that indeed the said correlations lie at the heart of the coexistence of area-law entanglement and multifractality for MBL eigenstates.