Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1806.01193 · Journal of Physics Condensed Matter

Simple probability distributions on a Fock-space lattice

Staszek Welsh, David E Logan

Published Aug 28, 2018• 20 citations
arXivHTMLPDF

Abstract

We consider some aspects of a standard model employed in studies of many-body localization: interacting spinless fermions with quenched disorder, for non-zero filling fraction, here on d-dimensional hypercubic lattices. The model may be recast as an equivalent tight-binding model on a 'Fock-space (FS) lattice' with an extensive local connectivity. In the thermodynamic limit exact results are obtained for the distributions of local FS coordination numbers, FS site-energies, and the density of many-body states. All such distributions are well captured by exact diagonalisation on the modest system sizes amenable to numerics. Care is however required in choosing the appropriate variance for the eigenvalue distribution, which has implications for reliable identification of mobility edges.

DOI
10.1088/1361-648x/aadd35

Related papers

Many-body localization in Fock space: A local perspective

David E. Logan, Staszek Welsh

A canonical model for many-body localization (MBL) is studied, of interacting spinless fermions on a lattice with uncorrelated quenched site disorder. The model maps onto a tight-binding model on a `Fock-space (FS) lattice' of many-body states, with an extensive local connectivity. We seek to understand some aspects of MBL from this perspective, via local propagators for the FS lattice and their self-energies (SE's), focusing on the SE probability distributions, over disorder and FS sites. A probabilistic mean-field theory (MFT) is first developed, centered on self-consistent determination of the geometric mean of the distribution. Despite its simplicity this captures some key features of the problem, including recovery of an MBL transition, and predictions for the forms of the SE distributions. The problem is then studied numerically in $1d$ by exact diagonalization, free from MFT assumptions. The geometric mean indeed appears to act as a suitable order parameter for the transition. Throughout the MBL phase the appropriate SE distribution is confirmed to have a universal form, with long-tailed L\'evy behavior as predicted by MFT. In the delocalized phase for weak disorder, SE distributions are clearly log-normal, while on approaching the transition they acquire an intermediate L\'evy-tail regime, indicative of the incipient MBL phase.

arXiv:1806.01688•Physical review. B./Physical review. B•Jan 17, 2019•56 citations

Many-body localization due to correlated disorder in Fock space

Soumi Ghosh, Atithi Acharya, Subhayan Sahu +1 more

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.

arXiv:1901.04384•Physical review. B./Physical review. B•Apr 22, 2019•35 citations

Structure of typical states of a disordered Richardson model and many-body localization

Francesco Buccheri, Andrea De Luca, Antonello Scardicchio

We present a thorough numerical study of the Richardson model with quenched disorder (a fully connected XX model with longitudinal random fields). We find that for any value of the interaction the eigenstates occupy an exponential number of sites on the unperturbed Fock space but that single-spin observables do not thermalize, as tested by a microcanonical version of the Edwards-Anderson order parameter. We therefore do not observe many-body localization in this model. We find a relation between the inverse participation ratio and the average Hamming distance between spin configurations covered by a typical eigenstate for which we hypothesize a remarkably simple form for the thermodynamic limit. We also studied the random process defined by the spread of a typical eigenstate on configuration space, highlighting several similarities with hopping on percolated hypercube, a process used to mimic the slow relaxation of spin glasses. A nearby nonintegrable model is also considered where delocalization is instead observed, although the presence of a phase transition at infinite temperature is questionable.

arXiv:1103.3431•Physical Review B•Sep 19, 2011•50 citations

Many-body localization of Z3 Fock parafermions

M. S. Bahovadinov, Wouter Buijsman, Aleksey K. Fedorov +2 more

We study the effects of a random magnetic field on a one-dimensional (1D) spin-1 chain with correlated nearest-neighbor XY interaction. We show that this spin model can be exactly mapped onto the 1D disordered tight-binding model of ${\mathbb{Z}}_{3}$ Fock parafermions (FPFs), exotic anyonic quasiparticles that generalize usual spinless fermions. Thus, we have a peculiar case of a disordered Hamiltonian that despite being bilinear in the creation and annihilation operators, exhibits a many-body localization (MBL) transition owing to the nontrivial statistics of FPFs. This is in sharp contrast to conventional bosonic and fermionic quadratic disordered Hamiltonians that show single-particle (Anderson) localization. We perform finite-size exact diagonalization calculations of level-spacing statistics, fractal dimensions, and entanglement entropy, and provide convincing evidence for the MBL transition at finite disorder strength.

arXiv:2210.02947•Physical review. B./Physical review. B•Dec 15, 2022•3 citations

Role of Fock-space correlations in many-body localization

Thibault Scoquart, I. V. Gornyi, A. D. Mirlin

Models of many-body localization (MBL) can be represented as tight-binding models in the many-body Hilbert space (Fock space). We explore the role of correlations between matrix elements of the effective Fock-space Hamiltonians in the scaling of MBL critical disorder ${W}_{c}(n)$ with the size $n$ of the system. For this purpose, we consider five models, which all have the same distributions of diagonal (energy) and off-diagonal (``hopping'') Fock-space matrix elements but different Fock-space correlations. These include quantum-dot (QD) and one-dimensional (1D) MBL models, their modifications (uQD and u1D models) with removed correlations of off-diagonal matrix elements, as well a quantum random energy model (QREM) with no correlations at all. Our numerical results are in full consistency with analytical arguments predicting ${n}^{3/4}{(lnn)}^{\ensuremath{-}1/4}\ensuremath{\lesssim}{W}_{c}\ensuremath{\lesssim}nlnn$ for the scaling of ${W}_{c}(n)$ in the QD model (we find ${W}_{c}\ensuremath{\sim}n$ numerically), ${W}_{c}(n)\ensuremath{\sim}\text{const}$ for the 1D model, ${W}_{c}\ensuremath{\sim}nlnn$ for the uQD and u1D models without off-diagonal correlations, and ${W}_{c}\ensuremath{\sim}{n}^{1/2}lnn$ for QREM. The key difference between the QD and 1D models is in the structure of correlations of many-body energies. Removing off-diagonal Fock-space correlations makes both these models ``maximally chaotic''. Our findings demonstrate that the scaling of ${W}_{c}(n)$ for MBL transitions is governed by a combined effect of Fock-space correlations of diagonal and off-diagonal matrix elements.

arXiv:2402.10123•Physical review. B./Physical review. B•Jun 7, 2024•22 citations

Fock-space correlations and the origins of many-body localization

Sthitadhi Roy, David E. Logan

The authors approach the problem of many-body localization on the Fock space, with nodes and links representing many-body states and Hamiltonian matrix-elements. The problem maps exactly onto a tight-binding model on a complex high-dimensional graph with nontrivial correlated disorder. These correlations are shown to lie at the heart of many-body localization. The central result from this theory is that the correlations in the Fock-space disorder must be maximal for a many-body localized phase to be stable.

arXiv:1911.12370•Physical review. B./Physical review. B•Apr 23, 2020•69 citations
We consider some aspects of a standard model employed in studies of many-body localization: interacting spinless fermions with quenched disorder, for non-zero filling fraction, here on d-dimensional hypercubic lattices. The model may be recast as an equivalent tight-binding model on a 'Fock-space (FS) lattice' with an extensive local connectivity. In the thermodynamic limit exact results are obtained for the distributions of local FS coordination numbers, FS site-energies, and the density of many-body states. All such distributions are well captured by exact diagonalisation on the modest system sizes amenable to numerics. Care is however required in choosing the appropriate variance for the eigenvalue distribution, which has implications for reliable identification of mobility edges.
A canonical model for many-body localization (MBL) is studied, of interacting spinless fermions on a lattice with uncorrelated quenched site disorder. The model maps onto a tight-binding model on a `Fock-space (FS) lattice' of many-body states, with an extensive local connectivity. We seek to understand some aspects of MBL from this perspective, via local propagators for the FS lattice and their self-energies (SE's), focusing on the SE probability distributions, over disorder and FS sites. A probabilistic mean-field theory (MFT) is first developed, centered on self-consistent determination of the geometric mean of the distribution. Despite its simplicity this captures some key features of the problem, including recovery of an MBL transition, and predictions for the forms of the SE distributions. The problem is then studied numerically in 1d1d1d by exact diagonalization, free from MFT assumptions. The geometric mean indeed appears to act as a suitable order parameter for the transition. Throughout the MBL phase the appropriate SE distribution is confirmed to have a universal form, with long-tailed L\'evy behavior as predicted by MFT. In the delocalized phase for weak disorder, SE distributions are clearly log-normal, while on approaching the transition they acquire an intermediate L\'evy-tail regime, indicative of the incipient MBL phase.
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 LLL 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.
We present a thorough numerical study of the Richardson model with quenched disorder (a fully connected XX model with longitudinal random fields). We find that for any value of the interaction the eigenstates occupy an exponential number of sites on the unperturbed Fock space but that single-spin observables do not thermalize, as tested by a microcanonical version of the Edwards-Anderson order parameter. We therefore do not observe many-body localization in this model. We find a relation between the inverse participation ratio and the average Hamming distance between spin configurations covered by a typical eigenstate for which we hypothesize a remarkably simple form for the thermodynamic limit. We also studied the random process defined by the spread of a typical eigenstate on configuration space, highlighting several similarities with hopping on percolated hypercube, a process used to mimic the slow relaxation of spin glasses. A nearby nonintegrable model is also considered where delocalization is instead observed, although the presence of a phase transition at infinite temperature is questionable.
We study the effects of a random magnetic field on a one-dimensional (1D) spin-1 chain with correlated nearest-neighbor XY interaction. We show that this spin model can be exactly mapped onto the 1D disordered tight-binding model of Z3{\mathbb{Z}}_{3}Z3​ Fock parafermions (FPFs), exotic anyonic quasiparticles that generalize usual spinless fermions. Thus, we have a peculiar case of a disordered Hamiltonian that despite being bilinear in the creation and annihilation operators, exhibits a many-body localization (MBL) transition owing to the nontrivial statistics of FPFs. This is in sharp contrast to conventional bosonic and fermionic quadratic disordered Hamiltonians that show single-particle (Anderson) localization. We perform finite-size exact diagonalization calculations of level-spacing statistics, fractal dimensions, and entanglement entropy, and provide convincing evidence for the MBL transition at finite disorder strength.
Models of many-body localization (MBL) can be represented as tight-binding models in the many-body Hilbert space (Fock space). We explore the role of correlations between matrix elements of the effective Fock-space Hamiltonians in the scaling of MBL critical disorder Wc(n){W}_{c}(n)Wc​(n) with the size nnn of the system. For this purpose, we consider five models, which all have the same distributions of diagonal (energy) and off-diagonal (``hopping'') Fock-space matrix elements but different Fock-space correlations. These include quantum-dot (QD) and one-dimensional (1D) MBL models, their modifications (uQD and u1D models) with removed correlations of off-diagonal matrix elements, as well a quantum random energy model (QREM) with no correlations at all. Our numerical results are in full consistency with analytical arguments predicting ${n}^{3/4}{(lnn)}^{\ensuremath{-}1/4}\ensuremath{\lesssim}{W}_{c}\ensuremath{\lesssim}nlnn$ for the scaling of Wc(n){W}_{c}(n)Wc​(n) in the QD model (we find ${W}_{c}\ensuremath{\sim}n$ numerically), ${W}_{c}(n)\ensuremath{\sim}\text{const}$ for the 1D model, ${W}_{c}\ensuremath{\sim}nlnn$ for the uQD and u1D models without off-diagonal correlations, and ${W}_{c}\ensuremath{\sim}{n}^{1/2}lnn$ for QREM. The key difference between the QD and 1D models is in the structure of correlations of many-body energies. Removing off-diagonal Fock-space correlations makes both these models ``maximally chaotic''. Our findings demonstrate that the scaling of Wc(n){W}_{c}(n)Wc​(n) for MBL transitions is governed by a combined effect of Fock-space correlations of diagonal and off-diagonal matrix elements.
The authors approach the problem of many-body localization on the Fock space, with nodes and links representing many-body states and Hamiltonian matrix-elements. The problem maps exactly onto a tight-binding model on a complex high-dimensional graph with nontrivial correlated disorder. These correlations are shown to lie at the heart of many-body localization. The central result from this theory is that the correlations in the Fock-space disorder must be maximal for a many-body localized phase to be stable.