We numerically investigate one-dimensional Bose-Hubbard chains with on-site disorder by means of exact diagonalization. A primary focus of our work is on characterizing Fock-space localization in this model from the single-particle perspective. For this purpose, we compute the one-particle density matrix (OPDM) in many-body eigenstates. We show that the natural orbitals (the eigenstates of the OPDM) are extended in the ergodic phase and real-space localized when one enters the many-body localization (MBL) phase. Furthermore, the distributions of occupations of the natural orbitals can be used as measures of Fock-space localization in the respective basis. Consistent with previous studies, we observe signatures of a transition from the ergodic to the MBL regime when increasing the disorder strength. We further demonstrate that Fock-space localization, albeit weaker, is also evidently present in the distribution of the physical densities in the MBL regime, for both soft- and hard-core bosons. Moreover, the full distribution of the densities of the physical particles provides a one-particle measure for the detection of the ergodic-MBL transition which could be directly accessed in experiments with ultracold gases.
We study interacting fermions in one dimension subject to random, uncorrelated onsite disorder, a paradigmatic model of many‐body localization (MBL). This model realizes an interaction‐driven quantum phase transition between an ergodic and a many‐body localized phase, with the transition occurring in the many‐body eigenstates. We propose a single‐particle framework to characterize these phases by the eigenstates (the natural orbitals) and the eigenvalues (the occupation spectrum) of the one‐particle density matrix (OPDM) in individual many‐body eigenstates. As a main result, we find that the natural orbitals are localized in the MBL phase, but delocalized in the ergodic phase. This qualitative change in these single‐particle states is a many‐body effect, since without interactions the single‐particle energy eigenstates are all localized. The occupation spectrum in the ergodic phase is thermal in agreement with the eigenstate thermalization hypothesis, while in the MBL phase the occupations preserve a discontinuity at an emergent Fermi edge. This suggests that the MBL eigenstates are weakly dressed Slater determinants, with the eigenstates of the underlying Anderson problem as reference states. We discuss the statistical properties of the natural orbitals and of the occupation spectrum in the two phases and as the transition is approached. Our results are consistent with the existing picture of emergent integrability and localized integrals of motion, or quasiparticles, in the MBL phase. We emphasize the close analogy of the MBL phase to a zero‐temperature Fermi liquid: in the studied model, the MBL phase is adiabatically connected to the Anderson insulator and the occupation‐spectrum discontinuity directly indicates the presence of quasiparticles localized in real space. Finally, we show that the same picture emerges for interacting fermions in the presence of an experimentally‐relevant bichromatic lattice and thereby demonstrate that our findings are not limited to a specific model. image
Many-body localization (MBL) is well characterized in Fock space. To quantify the degree of this Fock-space localization, the multifractal dimension
Dq is employed; it has been claimed that
Dq shows a jump from the delocalized value
Dq=1 in the eigenstate thermalization hypothesis (ETH) phase to a smaller value
0<Dq<1 at the ETH-MBL transition, yet exhibiting a conspicuous discrepancy from the fully localized value
Dq=0, which indicates that multifractality remains inside the MBL phase. Here, to better quantify the situation, we employ, instead of the commonly used computational basis, the one-particle density matrix (OPDM) and use its eigenstates (natural orbitals) as a Fock state basis for representing many-body eigenstates $|\ensuremath{\psi}\ensuremath{\rangle}$ of the system. Using this basis, we compute
Dq and other indices quantifying the Fock-space localization, such as the local purity
S, which is derived from the occupation spectrum ${{n}_{\ensuremath{\alpha}}}$ (eigenvalues of the OPDM). We highlight the statistical distribution of Hamming distance ${x}_{\ensuremath{\mu}\ensuremath{\nu}}$ occurring in the pairwise coefficients $|{a}_{\ensuremath{\mu}}{|}^{2}{|{a}_{\ensuremath{\nu}}|}^{2}$ in
S, and compare this with a related quantity considered in the literature.
In the presence of strong enough disorder one-dimensional systems of interacting spinless fermions at nonzero filling factor are known to be in a many-body localized phase. When represented in `Fock space,' the Hamiltonian of such a system looks like that of a single `particle' hopping on a Fock lattice in the presence of a random disordered potential. The coordination number of the Fock lattice increases linearly with the system size
L in one dimension. Thus in the thermodynamic limit $L\ensuremath{\rightarrow}\ensuremath{\infty}$, the disordered interacting problem in one dimension maps on to an Anderson model with infinite coordination number. Despite this, this system displays localization which appears counterintuitive. A close observation of the on-site disorder potentials on the Fock lattice reveals a large degree of correlation among them as they are derived from an exponentially smaller number of on-site disorder potentials in real space. This indicates that the correlations between the on-site disorder potentials on a Fock lattice has a strong effect on the localization properties of the corresponding many-body system. This intuition is also consistent with studies of quantum random energy model where the typical mid-spectrum states are ergodic and the on-site potentials in Fock space are completely uncorrelated. In this work we perform a systematic quantitative exploration of the nature of correlations of the Fock space potential required for localization. We study different functional variations of the disorder correlation in Fock lattice by analyzing the eigenspectrum obtained through exact diagonalization. Without changing the typical strength of the on-site disorder potential in Fock lattice we show that changing the correlation strength can induce thermalization or localization in systems. From among the various forms of correlations we study, we find that only the linear variation of correlations with Hamming distance in Fock space is able to drive a thermal-MBL phase transition where the transition is driven by the correlation strength. Systems with the other forms of correlations we study are found to be ergodic.
This article reviews recent progress in understanding the physics of many-body localisation (MBL) in disordered and interacting quantum many-body systems, from the perspective of ergodicity breaking on the associated Fock space. This approach to MBL is underpinned by mapping the dynamics of the many-body system onto that of a fictitious single particle on the high-dimensional, correlated and disordered Fock-space graph; yet, as we elaborate, the problem is fundamentally different from that of conventional Anderson localisation on high-dimensional or hierarchical graphs. We discuss in detail the nature of eigenstate correlations on the Fock space, both static and dynamic, and in the ergodic and many-body localised phases as well as in the vicinity of the MBL transition. The latter in turn sheds light on the nature of the transition, and motivates a scaling theory for it in terms of Fock-space based quantities. We also illustrate how these quantities can be concretely connected to real-space observables. An overview is given of several analytical and numerical techniques which have proven important in developing a comprehensive picture. Finally, we comment on some open questions in the field of MBL where the Fock-space approach is likely to prove insightful.
Characterizing the delocalization transition in closed quantum systems with a many-body localized phase is a key open question in the field of nonequilibrium physics. We exploit the fact that localization of particles as realized in Anderson localization and standard many-body localization (MBL) implies Fock-space localization in single-particle basis sets characterized by a real-space index. Using a recently introduced quantitative measure for Fock-space localization computed from the density distributions, the occupation distance, we systematically study its scaling behavior across delocalization transitions and identify critical points from scaling collapses of numerical data. Excellent agreement with literature results is found for the critical disorder strengths of noninteracting fermions, such as the one-dimensional Aubry-Andr\'e model and the three-dimensional Anderson model. We observe a distinctively different scaling behavior in the case of interacting fermions with random disorder consistent with a Kosterlitz-Thouless transition. Finally, we use our measure to extract the transition point as a function of filling for interacting fermions.
We investigate a spatial subsystem entropy extracted from the one-particle density matrix (OPDM) of one-dimensional disordered interacting fermions that host a many-body localized (MBL) phase. Deep in the putative MBL regime, this OPDM entropy exhibits the salient scaling features of localization, even though it provides only an upper bound to the von Neumann entropy. First, we numerically show that the OPDM entropy of the eigenstates obeys an area law. Second, like the von Neumann entropy, the OPDM entropy grows logarithmically with time after a quantum quench, albeit with a different prefactor. Both these features survive at moderately large interactions and well toward the transition into the ergodic phase. We discuss prospects for calculating the OPDM entropy using approximate numerical methods and for its measurement in quantum gas experiments.