We study the many body localization aspects a single-particle mobility edges in fermionic systems. We investigate incommensurate lattices and random disorder Anderson models. Many body localization and quantum nonergodic properties are studied by comparing entanglement and internal entropy and by calculating the scaling of subsystem particle number fluctuations respectively. Every crystalline structure shows some deviations or the other from the regular atomic arrangement, as prescribed by the symmetry and structure of the respective unit cell. These deviations from the ideal crystal structure are called imperfections. The translational symmetry in a crystal is consistent with laws of thermodynamics, since these laws are applied to describe the growth of crystals all of which have some imperfections. Any increase in the defect concentration raises the entropy, which in term lowers the free energy at a finite temperature. In the equilibrium state there is a finite concentration of imperfection in the crystal.
The concentration of a particular type of imperfection depends on the type of the crystal lattice, the binding energy of the lattice and structure of the imperfection itself. Entropy is the measure of the disorder of a system, the greater the disorder, the higher is the entropy. In the magnetic field the moments will be partially ordered, so that the entropy is lowered by the field. The relatively small heat capacity associated with the lattice vibrations of solids at temperature near and below 1oK makes this region interesting in connection with an evaluation of contribution of the conduction electrons to the heat capacity of the metals. In the present study we limit ourselves to find out expression of entropy for tunneling model. For this purpose we have developed first the defect contribution to the specific heat for tunneling model.
We study the many body localization aspects a single-particle mobility edges in fermionic systems. We investigate incommensurate lattices and random disorder Anderson models. Many body localization and quantum nonergodic properties are studied by comparing entanglement and internal entropy and by calculating the scaling of subsystem particle number fluctuations respectively. Every crystalline structure shows some deviations or the other from the regular atomic arrangement, as prescribed by the symmetry and structure of the respective unit cell. These deviations from the ideal crystal structure are called imperfections. The translational symmetry in a crystal is consistent with laws of thermodynamics, since these laws are applied to describe the growth of crystals all of which have some imperfections. Any increase in the defect concentration raises the entropy, which in term lowers the free energy at a finite temperature. In the equilibrium state there is a finite concentration of imperfection in the crystal.
The concentration of a particular type of imperfection depends on the type of the crystal lattice, the binding energy of the lattice and structure of the imperfection itself. Entropy is the measure of the disorder of a system, the greater the disorder, the higher is the entropy. In the magnetic field the moments will be partially ordered, so that the entropy is lowered by the field. The relatively small heat capacity associated with the lattice vibrations of solids at temperature near and below 1oK makes this region interesting in connection with an evaluation of contribution of the conduction electrons to the heat capacity of the metals. In the present study we limit ourselves to find out expression of entropy for tunneling model. For this purpose we have developed first the defect contribution to the specific heat for tunneling model.
We review the physics of many-body localization in models with incommensurate potentials. In particular, we consider one-dimensional quasiperiodic models with single-particle mobility edges. A conventional perspective suggests that delocalized states act as a thermalizing bath for the localized states in the presence of of interactions. However, contrary to this intuition there is evidence that such systems can display non-ergodicity. This is in part due to the fact that the delocalized states do not have any kind of protection due to symmetry or topology and are thus susceptible to localization. A study of such incommensurate models, in the non-interacting limit, shows that they admit extended, partially extended, and fully localized many-body states. Non-interacting incommensurate models cannot thermalize dynamically and remain localized upon the introduction of interactions. In particular, for a certain range of energy, the system can host a non-ergodic extended (i.e. metallic) phase in which the energy eigenstates violate the eigenstate thermalization hypothesis (ETH) but the entanglement entropy obeys volume-law scaling. The level statistics and entanglement growth also indicate the lack of ergodicity in these models. The phenomenon of localization and non-ergodicity in a system with interactions despite the presence of single-particle delocalized states is closely related to the so-called many-body proximity effect and can also be observed in models with disorder coupled to systems with delocalized degrees of freedom. Many-body localization in systems with incommensurate potentials (without single-particle mobility edges) have been realized experimentally, and we show how this can be modified to study the the effects of such mobility edges. Demonstrating the failure of thermalization in the presence of a single-particle mobility edge in the thermodynamic limit would indicate a more robust violation of the ETH.
Localization of particle wavefunctions in quasi-disordered one dimensional incommensurate lattices is studied both numerically and analytically. Through exact diagonalization, we show that energy dependent mobility edges can appear in the case of shallow lattices. We also show that these mobility edges can be studied with a tight-binding model (an extension of the Aubry-Andre model) that has energy dependent mobility edges that can be determined analytically. Topological aspects of the Aubry-Andre/Harper model are also studied by numerically calculating the Chern number. We first verify arguments by numerical calculations that variations in the Chern density decrease with increasing system size when the potential is incommensurate with the lattice. Next we introduce random disorder into the model and study the Chern number and the Chern density as a function of disorder strength by using the non-commutative Brillouin zone. We show that variations of the Chern density take on the same trends for both commensurate and incommensurate potentials after some critical disorder strength is reached. Strongly correlated quantum Hall states are also examined. We numerically examine the entanglement entropy and the entanglement spectrum of fractional quantum hall states as a function of the finite layer thickness d of the quasi-two-dimensional system for a number of filling fractions ν in the lowest and the second Landau levels: ν = 1/3, 7/3, 1/2, and 5/2. We observe that the entanglement measures are dependent on which Landau level the electrons fractionally occupy and are completely consistent with the results based on wavefunction overlap calculations. We also compare the ground state energies by variational Monte Carlo of the spin unpolarized Halperin 331 and the spin polarized Moore-Read (MR) Pfaffian fractional quantum Hall states at half filling of the lowest Landau level (LLL) and the second Landau level (SLL) as a function of small deviations around the Coulomb point. Our results suggest that even under moderate deviations in the interaction potential the MR Pfaffian description is more energetically favorable than the Halperin 331 state in the half filled SLL, consistent with recent experimental investigations.
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.
Quantum thermalization of isolated systems undergoing unitary time evolution is a fundamental problem in quantum statistical mechanics. Its study has been revived recently in the context of many-body Anderson localization. Previous works have focused on localization of many-body systems with all the single-particle states being localized. As a significant step forward, this work studies localization aspects of noninteracting many-particle systems in the presence of a single-particle mobility edge. By systemically investigating entanglement entropy scaling and nonthermal fluctuations in various lattice models, the authors establish a nonergodic extended phase as a generic intermediate phase (between purely ergodic extended and nonergodic localized phases) for the many-body localization transition of noninteracting fermions. This work also sheds light on the interacting transition scenario as well.
We use numerical linked cluster expansions (NLCEs) to study correlated lattice systems in one and two dimensions.For fermions in the honeycomb lattice, we study its finite-temperature properties and short-range spin correlations using NLCEs and determinantal quantum Monte Carlo (DQMC) simulations.For the homogeneous system, we analyze a number of thermodynamic quantities, including the entropy, the specific heat, uniform and staggered spin susceptibilities, short-range spin correlations, and the double occupancy at and away from half filling.Employing a local density approximation (LDA), we examine the viability of adiabatic cooling by increasing the interaction strength for homogeneous as well as for trapped systems.We also use NLCEs to study thermodynamic properties of the two-dimensional spin-1/2 Ising, XY, and Heisenberg models with bimodal random-bond disorder on the square and honeycomb lattices.In all cases, the nearest-neighbor coupling between the spins takes values ±J with equal probability.We obtain the disorder averaged (over all disorder configurations) energy, entropy, specific heat, and uniform magnetic susceptibility in each case.These results are compared with the corresponding ones in the clean models.Analytic expressions are obtained for low orders in the expansion of these thermodynamic quantities in inverse temperature.For many-body localization in disordered isolated systems, we study quantum quenches in the thermodynamic limit.By a quantum quench it is meant that the initial state is stationary with respect iii to an initial Hamiltonian, which is suddenly changed to a new (time-independent) Hamiltonian.The latter then drives the (unitary) dynamics of the system.We are interested in the time average of observables after relaxation following the quench, which can be computed under the so-called diagonal ensemble.
Impurities, defects, and other types of imperfections are ubiquitous in realistic quantum many-body systems and essentially unavoidable in solid state materials. Often, such random disorder is viewed purely negatively as it is believed to prevent interesting new quantum states of matter from forming and to smear out sharp features associated with the phase transitions between them. However, disorder is also responsible for a variety of interesting novel phenomena that do not have clean counterparts. These include Anderson localization of single-particle wave functions, many-body localization in isolated many-body systems, exotic quantum critical points, and glassy ground-state phases. This brief review focuses on two separate but related subtopics in this field. First, we review under what conditions different types of randomness affect the stability of symmetry-broken low-temperature phases in quantum many-body systems and the stability of the corresponding phase transitions. Second, we discuss the fate of quantum phase transitions that are destabilized by disorder as well as the unconventional quantum Griffiths phases that emerge in their vicinity.