arXiv subjects

David E. Logan

Publications and source records attributed to David E. Logan.

At least 19 recordsLinked to original sources

arXiv2026

Krylov-space anatomy and spread complexity of a disordered quantum spin chain

We investigate the anatomy and complexity of quantum states in Krylov space, in the ergodic and many-body localised (MBL) phases of a disordered, interacting spin chain. The Krylov basis generated by the Hamiltonian from an initial state provides a representation in which the spread of the time-evolving state constitutes a basis-optimised measure of complexity. We show that the long-time Krylov spread complexity sharply distinguishes the two phases. In the ergodic regime, the infinite-time complexity scales linearly with the Fock-space dimension, indicating that the state spreads over a finite fraction of the Krylov chain. By contrast, it grows sublinearly in the MBL regime, implying that the long-time state occupies only a vanishing fraction of the chain. Further, the profile of the infinite-time state along the Krylov chain exhibits a stretched-exponential decay in the MBL regime. This behaviour reflects a broad distribution of decay lengthscales, associated with different eigenstates contributing to the long-time state. Consistently, a large-deviation analysis of the statistics of eigenstate spread complexities shows that while the ergodic regime receives contributions from almost all eigenstates, the complexity in the MBL regime is dominated by a vanishing fraction of eigenstates, which have anomalously large complexity relative to the typical ones.

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arXiv2025

Multifractality in high-dimensional graphs induced by correlated radial disorder

We introduce a class of models containing robust and analytically demonstrable multifractality induced by disorder correlations. Specifically, we investigate the statistics of eigenstates of disordered tight-binding models on two classes of rooted, high-dimensional graphs -- trees and hypercubes -- with a form of strong disorder correlations we term `radial disorder'. In this model, site energies on all sites equidistant from a chosen root are identical, while those at different distances are independent random variables (or their analogue for a deterministic but incommensurate potential, a case of which is also considered). Analytical arguments, supplemented by numerical results, are used to establish that this setting hosts robust and unusual multifractal states. The distribution of multifractality, as encoded in the inverse participation ratios (IPRs), is shown to be exceptionally broad. This leads to a qualitative difference in scaling with system size between the mean and typical IPRs, with the latter the appropriate quantity to characterise the multifractality. The existence of this multifractality is shown to be underpinned by an emergent fragmentation of the graphs into effective one-dimensional chains, which themselves exhibit conventional Anderson localisation. The interplay between the exponential localisation of states on these chains, and the exponential growth of the number of sites with distance from the root, is the origin of the observed multifractality.

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arXiv2024

The Fock-space landscape of many-body localisation

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.

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arXiv2022

Probability transport on the Fock space of a disordered quantum spin chain

Within the broad theme of understanding the dynamics of disordered quantum many-body systems, one of the simplest questions one can ask is: given an initial state, how does it evolve in time on the associated Fock-space graph, in terms of the distribution of probabilities thereon? A detailed quantitative description of the temporal evolution of out-of-equilibrium disordered quantum states and probability transport on the Fock space, is our central aim here. We investigate it in the context of a disordered quantum spin chain which hosts a disorder-driven many-body localisation transition. Real-time dynamics/probability transport is shown to exhibit a rich phenomenology, which is markedly different between the ergodic and many-body localised phases. The dynamics is for example found to be strongly inhomogeneous at intermediate times in both phases, but while it gives way to homogeneity at long times in the ergodic phase, the dynamics remain inhomogeneous and multifractal in nature for arbitrarily long times in the localised phase. Similarly, we show that an appropriately defined dynamical lengthscale on the Fock-space graph is directly related to the local spin-autocorrelation, and as such sheds light on the (anomalous) decay of the autocorrelation in the ergodic phase, and lack of it in the localised phase.

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arXiv2022

Scaling of Fock-space propagator and multifractality across the many-body localization transition

We implement a recursive Green function method to extract the Fock space (FS) propagator and associated self-energy across the many-body localization (MBL) transition, for one-dimensional interacting fermions in a random onsite potential. We show that the typical value of the imaginary part of the local FS self-energy, Δ_t, related to the decay rate of an initially localized state, acts as a probabilistic order parameter for the thermal to MBL phase transition; and can be used to characterize critical properties of the transition as well as the multifractal nature of MBL states as a function of disorder strength W. In particular, we show that a fractal dimension D_s extracted from Δ_t jumps discontinuously across the transition, from D_s<1 in the MBL phase to D_s= 1 in the thermal phase. Moreover, Δ_t follows an asymmetrical finite-size scaling form across the thermal-MBL transition, where a non-ergodic volume in the thermal phase diverges with a Kosterlitz-Thouless like essential singularity at the critical point W_c, and controls the continuous vanishing of Δ_t as W_c is approached. In contrast, a correlation length (ξ) extracted from Δ_t exhibits a power-law divergence on approaching W_c from the MBL phase.

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arXiv2022

Anomalous multifractality in quantum chains with strongly correlated disorder

We demonstrate numerically that a robust and unusual multifractal regime can emerge in a one-dimensional quantum chain with maximally correlated disorder, above a threshold disorder strength. This regime is preceded by a mixed and an extended regime at weaker disorder strengths, with the former hosting both extended and multifractal eigenstates. The multifractal states we find are markedly different from conventional multifractal states in their structure, as they reside approximately uniformly over a continuous segment of the chain, and the lengths of these segments scale non-trivially with system size. This anomalous nature also leaves imprints on dynamics. An initially localised wavepacket shows ballistic transport, in contrast to the slow, generally sub-diffusive, transport commonly associated with multifractality. However, the timescale over which this ballistic transport persists again scales non-trivially with the system size.

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arXiv2021

Fock-space anatomy of eigenstates across the many-body localisation transition

We explore the Fock-space structure of eigenstates across the many-body localisation (MBL) transition in a disordered, interacting quantum spin-1/2 chain. Eigenstate expectation values of spatially local observables, which distinguish an MBL phase from an ergodic one, can be represented in terms of eigenstate amplitudes on the Fock space. Motivated by this, we introduce and study spatial correlations on the Fock space. From these, a correlation length emerges, which is found to vary discontinuously across the MBL transition; and is intimately connected to the discontinuous jump in the multifractal exponents characterising the Fock-space wavefunctions. Exploiting the direct connection between the local observables and Fock-space correlations, we show that the discontinuity in the lengthscale also implies discontinuous behaviour of the local observables across the transition. A scaling theory based on these Fock-space correlations is constructed, which is closely connected to that for the inverse participation ratio. It yields a volume-scale in the ergodic phase and a length-scale in the MBL phase, whose critical properties suggest a Kosterlitz-Thouless-like scenario for the MBL transition, as is predicted by recent phenomenological theories. Finally, we also show how correlation functions on the Fock space reveal the inhomogeneities in eigenstate amplitudes on the Fock space in the MBL phase.

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arXiv2021

Localisation in quasiperiodic chains: a theory based on convergence of local propagators

Quasiperiodic systems serve as fertile ground for studying localisation, due to their propensity already in one dimension to exhibit rich phase diagrams with mobility edges. The deterministic and strongly-correlated nature of the quasiperiodic potential nevertheless offers challenges distinct from disordered systems. Motivated by this, we present a theory of localisation in quasiperiodic chains with nearest-neighbour hoppings, based on the convergence of local propagators; exploiting the fact that the imaginary part of the associated self-energy acts as a probabilistic order parameter for localisation transitions and, importantly, admits a continued-fraction representation. Analysing the convergence of these continued fractions, localisation or its absence can be determined, yielding in turn the critical points and mobility edges. Interestingly, we find anomalous scalings of the order parameter with system size at the critical points, consistent with the fractal character of critical eigenstates. Self-consistent theories at high orders are also considered, shown to be conceptually connected to the theory based on continued fractions, and found in practice to converge to the same result. Results are exemplified by analysing the theory for three quasiperiodic models covering a range of behaviour.

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arXiv2020

Self-consistent theory of mobility edges in quasiperiodic chains

We introduce a self-consistent theory of mobility edges in nearest-neighbour tight-binding chains with quasiperiodic potentials. Demarcating boundaries between localised and extended states in the space of system parameters and energy, mobility edges are generic in quasiperiodic systems which lack the energy-independent self-duality of the commonly studied Aubry-André-Harper model. The potentials in such systems are strongly and infinite-range correlated, reflecting their deterministic nature and rendering the problem distinct from that of disordered systems. Importantly, the underlying theoretical framework introduced is model-independent, thus allowing analytical extraction of mobility edge trajectories for arbitrary quasiperiodic systems. We exemplify the theory using two families of models, and show the results to be in very good agreement with the exactly known mobility edges as well numerical results obtained from exact diagonalisation.

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arXiv2020

Localisation on certain graphs with strongly correlated disorder

Many-body localisation in interacting quantum systems can be cast as a disordered hopping problem on the underlying Fock-space graph. A crucial feature of the effective Fock-space disorder is that the Fock-space site energies are strongly correlated -- maximally so for sites separated by a finite distance on the graph. Motivated by this, and to understand the effect of such correlations more fundamentally, we study Anderson localisation on Cayley trees and random regular graphs, with maximally correlated disorder. Since such correlations suppress short distance fluctuations in the disorder potential, one might naively suppose they disfavour localisation. We find however that there exists an Anderson transition, and indeed that localisation is more robust in the sense that the critical disorder scales with graph connectivity $K$ as $\sqrt{K}$, in marked contrast to $K\ln K$ in the uncorrelated case. This scaling is argued to be intimately connected to the stability of many-body localisation. Our analysis centres on an exact recursive formulation for the local propagators as well as a self-consistent mean-field theory; with results corroborated using exact diagonalisation.

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arXiv2019

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

We consider the problem of many-body localisation on Fock space, focussing on the essential features of the Hamiltonian which stabilise a localised phase. Any many-body Hamiltonian has a canonical representation as a disordered tight-binding model on the Fock-space graph. The underlying physics is however fundamentally different from that of conventional Anderson localisation on high-dimensional graphs because the Fock-space graph possesses non-trivial correlations. These correlations are shown to lie at the heart of whether or not a stable many-body localised phase can be sustained in the thermodynamic limit, and a theory is presented for the conditions the correlations must satisfy for a localised phase to be stable. Our analysis is rooted in a probabilistic, self-consistent mean-field theory for the local Fock-space propagator and its associated self-energy; in which the Fock-space correlations, together with the extensive nature of the connectivity of Fock-space nodes, are key ingredients. The origins of many-body localisation in typical local Hamiltonians where the correlations are strong, as well as its absence in uncorrelated random energy-like models, emerge as predictions from the same overarching theory. To test these, we consider three specific microscopic models, first establishing in each case the nature of the associated Fock-space correlations. Numerical exact diagonalisation is then used to corroborate the theoretical predictions for the occurrence or otherwise of a stable many-body localised phase; with mutual agreement found in each case.

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arXiv2019

Self-consistent theory of many-body localisation in a quantum spin chain with long-range interactions

Many-body localisation is studied in a disordered quantum spin-1/2 chain with long-ranged power-law interactions, and distinct power-law exponents for interactions between longitudinal and transverse spin components. Using a self-consistent mean-field theory centring on the local propagator in Fock space and its associated self-energy, a localisation phase diagram is obtained as a function of the power-law exponents and the disorder strength of the random fields acting on longitudinal spin-components. Analytical results are corroborated using the well-studied and complementary numerical diagnostics of level statistics, entanglement entropy, and participation entropy, obtained via exact diagonalisation. We find that increasing the range of interactions between transverse spin components hinders localisation and enhances the critical disorder strength. In marked contrast, increasing the interaction range between longitudinal spin components is found to enhance localisation and lower the critical disorder.

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arXiv2019

Simple probability distributions on a Fock-space lattice

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 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 diagonalization 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.

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arXiv2019

Many-body localization in Fock-space: a local perspective

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 Lévy 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 Lévy-tail regime, indicative of the incipient MBL phase.

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arXiv2018

Percolation in Fock space as a proxy for many-body localisation

We study classical percolation models in Fock space as proxies for the quantum many-body localisation (MBL) transition. Percolation rules are defined for two models of disordered quantum spin-chains using their microscopic quantum Hamiltonians and the topologies of the associated Fock-space graphs. The percolation transition is revealed by the statistics of Fock-space cluster sizes, obtained by exact enumeration for finite-sized systems. As a function of disorder strength, the typical cluster size shows a transition from a volume law in Fock space to sub-volume law, directly analogous to the behaviour of eigenstate participation entropies across the MBL transition. Finite-size scaling analyses for several diagnostics of cluster size statistics yield mutually consistent critical properties. We show further that local observables averaged over Fock-space clusters also carry signatures of the transition, with their behaviour across it in direct analogy to that of corresponding eigenstate expectation values across the MBL transition. The Fock-space clusters can be explored under a mapping to kinetically constrained models. Dynamics within this framework likewise show the ergodicity-breaking transition via Monte Carlo averaged local observables, and yield critical properties consistent with those obtained from both exact cluster enumeration and analytic results derived in our recent work [arXiv:1812.05115]. This mapping allows access to system sizes two orders of magnitude larger than those accessible in exact enumerations. Simple physical pictures based on freezing of local real-space segments of spins are also presented, and shown to give values for the critical disorder strength and correlation length exponent $ν$ consistent with numerical studies.

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arXiv2018

Exact solution of a percolation analogue for the many-body localisation transition

We construct and solve a classical percolation model with a phase transition that we argue acts as a proxy for the quantum many-body localisation transition. The classical model is defined on a graph in the Fock space of a disordered, interacting quantum spin chain, using a convenient choice of basis. Edges of the graph represent matrix elements of the spin Hamiltonian between pairs of basis states that are expected to hybridise strongly. At weak disorder, all nodes are connected, forming a single cluster. Many separate clusters appear above a critical disorder strength, each typically having a size that is exponentially large in the number of spins but a vanishing fraction of the Fock-space dimension. We formulate a transfer matrix approach that yields an exact value $ν=2$ for the localisation length exponent, and also use complete enumeration of clusters to study the transition numerically in finite-sized systems.

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arXiv2016

Mott transitions in the Periodic Anderson Model

The periodic Anderson model (PAM) is studied within the framework of dynamical mean-field theory, with particular emphasis on the interaction-driven Mott transition it contains, and on resultant Mott insulators of both Mott-Hubbard and charge-transfer type. The form of the PAM phase diagram is first deduced on general grounds using two exact results, over the full range of model parameters and including metallic, Mott, Kondo and band insulator phases.The effective low-energy model which describes the PAM in the vicinity of a Mott transition is then shown to be a one-band Hubbard model, with effective hoppings that are not in general solely nearest neighbour, but decay exponentially with distance. This mapping is shown to have a range of implications for the physics of the problem, from phase boundaries to single-particle dynamics; all of which are confirmed and supplemented by NRG calculations. Finally we consider the locally degenerate, non-Fermi liquid Mott insulator, to describe which requires a two-self-energy description. This is shown to yield a number of exact results for the associated local moment, charge, and interaction-renormalised levels, together with a generalisation of Luttinger's theorem to the Mott insulator.

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arXiv2015

Mott insulators and the doping-induced Mott transition within DMFT: exact results for the one-band Hubbard model

The paramagnetic phase of the one-band Hubbard model is studied at zero-temperature, within the framework of dynamical mean-field theory, and for general particle-hole asymmetry where a doping-induced Mott transition occurs. Our primary focus is the Mott insulator (MI) phase, and our main aim to establish what can be shown exactly about it. To handle the locally doubly-degenerate MI requires two distinct self-energies, which reflect the broken symmetry nature of the phase and together determine the standard single self-energy. Exact results are obtained for the local charge, local magnetic moment and associated spin susceptibilities, the interaction-renormalised levels, and the low-energy behaviour of the self-energy in the MI phase. The metallic phase is also considered briefly, and shown to acquire an emergent particle-hole symmetry as the Mott transition is approached. Throughout the metal, Luttinger's theorem is reflected in the vanishing of the Luttinger integral; for the generic MI by contrast this is shown to be non-vanishing, but again to have a universal magnitude. Numerical results are also obtained using NRG, for the metal/MI phase boundary, the scaling behaviour of the charge as the Mott transition is aproached from the metal, and associated universal scaling of single-particle dynamics as the low-energy Kondo scale vanishes.

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