arXiv subjects

Wojciech De Roeck

Publications and source records attributed to Wojciech De Roeck.

At least 55 records · Page 3Linked to original sources

arXiv2016

Asymptotic Localization in the Bose-Hubbard Model

We consider the Bose-Hubbard model. Our focus is on many-body localization, which was described by many authors in such models, even in the absence of disorder. Since our work is rigorous, and since we believe that the localization in this type of models is not strictly valid in the infinite-time limit, we necessarily restrict our study to 'asymptotic localization': We prove that transport and thermalization are small beyond perturbation theory in the limit of large particle density. Our theorem takes the form of a many-body Nekhoroshev estimate. An interesting and new aspect of this model is the following: even though our analysis proceeds by perturbation in the hopping, the localization can not be inferred from a lack of hybridization between zero-hopping eigenstates.

math-ph↗

arXiv2016

An exponentially local spectral flow for possibly non-self-adjoint perturbations of non-interacting quantum spins, inspired by KAM theory

Since its introduction by Hastings in [10], the technique of quasi-adiabatic continuation has become a central tool in the discussion and classification of ground state phases. It connects the ground states of self-adjoint Hamiltonians in the same phase by a unitary quasi-local transformation. This paper takes a step towards extending this result to non- self adjoint perturbations, though, for technical reason, we restrict ourselves here to weak perturbations of non-interacting spins. The extension to non-self adjoint perturbation is important for potential applications to Glauber dynamics (and its quantum analogues). In contrast to the standard quasi-adiabatic transformation, the transformation constructed here is exponentially local. Our scheme is inspired by KAM theory, with frustration-free operators playing the role of integrable Hamiltonians.

math-ph↗

arXiv2016

Invariance principle for `push' tagged particles for a Toom Interface

In many interacting particle systems, tagged particles move diffusively upon subtracting a drift. General techniques to prove such `invariance principles' are available for reversible processes (Kipnis-Varadhan) and for non-reversible processes in dimension $d>2$. The interest of our paper is that it considers a non-reversible one-dimensional process: the Toom model. The reason that we can prove the invariance principle is that in this model, push-tagged particles move manifestly slower than second-class particles.

math.PR↗

arXiv2016

Step Heat Profile in Localized Chains

We consider two types of strongly disordered one-dimensional Hamiltonian systems coupled to baths (energy or particle reservoirs) at the boundaries: strongly disordered quantum spin chains and disordered classical harmonic oscillators. These systems are believed to exhibit localization, implying in particular that the conductivity decays exponentially in the chain length $L$. We ask however for the profile of the (very slowly) transported quantity in the steady state. We find that this profile is a step-function, jumping in the middle of the chain from the value set by the left bath to the value set by the right bath. The width of the step grows not faster than $\sqrt{L}$. This is confirmed by numerics on a disordered quantum spin chain of 9 spins and on much longer chains of harmonic oscillators. In the case of harmonic oscillators, we also observe a drastic breakdown of local equilibrium at the step, resulting in a chaotic temperature profile.

cond-mat.stat-mech↗

arXiv2016

Stability and instability towards delocalization in MBL systems

We propose a theory that describes quantitatively the (in)stability of fully MBL systems due to ergodic, i.e. delocalized, grains, that can be for example due to disorder fluctuations. The theory is based on the ETH hypothesis and elementary notions of perturbation theory. The main idea is that we assume as much chaoticity as is consistent with conservation laws. The theory describes correctly -even without relying on the theory of local integrals of motion (LIOM)- the MBL phase in 1 dimension at strong disorder. It yields an explicit and quantitative picture of the spatial boundary between localized and ergodic systems. We provide numerical evidence for this picture. When the theory is taken to its extreme logical consequences, it predicts that the MBL phase is destabilised in the long time limit whenever 1) interactions decay slower than exponentially in $d=1$ and 2) always in $d>1$. Finer numerics is required to assess these predictions.

cond-mat.dis-nn↗

arXiv2015

Uniqueness regime for Markov dynamics on quantum lattice spin systems

We consider a lattice of weakly interacting quantum Markov processes. Without interaction, the dynamics at each site is relaxing exponentially to a unique stationary state. With interaction, we show that there remains a unique stationary state in the thermodynamic limit, i.e. absence of phase coexistence, and the relaxation towards it is exponentially fast for local observables. We do not assume that the quantum Markov process is reversible (detailed balance) w.r.t. a local Hamiltonian.

math-ph↗

arXiv2015

A theory of many-body localization in periodically driven systems

We present a theory of periodically driven, many-body localized (MBL) systems. We argue that MBL persists under periodic driving at high enough driving frequency: The Floquet operator (evolution operator over one driving period) can be represented as an exponential of an effective time-independent Hamiltonian, which is a sum of quasi-local terms and is itself fully MBL. We derive this result by constructing a sequence of canonical transformations to remove the time-dependence from the original Hamiltonian. When the driving evolves smoothly in time, the theory can be sharpened by estimating the probability of adiabatic Landau-Zener transitions at many-body level crossings. In all cases, we argue that there is delocalization at sufficiently low frequency. We propose a phase diagram of driven MBL systems.

cond-mat.dis-nn↗

arXiv2015

Exponentially slow heating in periodically driven many-body systems

We derive general bounds on the linear response energy absorption rates of periodically driven many-body systems of spins or fermions on a lattice. We show that for systems with local interactions, energy absorption rate decays exponentially as a function of driving frequency in any number of spatial dimensions. These results imply that topological many-body states in periodically driven systems, although generally metastable, can have very long lifetimes. We discuss applications to other problems, including decay of highly energetic excitations in cold atomic and solid-state systems.

cond-mat.stat-mech↗

arXiv2015

Local Perturbations Perturb -Exponentially- Locally

We elaborate on the principle that for gapped quantum spin systems with local interaction "local perturbations [in the Hamiltonian] perturb locally [the ground state]". This principle was established in [Bachmann et al. 2012], relying on the `spectral flow technique' or `quasi-adiabatic continuation' [Hastings 2004] to obtain locality estimates with sub-exponential decay in the distance to the spatial support of the perturbation. We use ideas of [Hamza et Al. 2009] to obtain similarly a transformation between gapped eigenvectors and their perturbations that is local with exponential decay. This allows to improve locality bounds on the effect of perturbations on the low lying states in certain gapped models with a unique `bulk ground state' or `topological quantum order'. We also give some estimate on the exponential decay of correlations in models with impurities where some relevant correlations decay faster than one would naively infer from the global gap of the system, as one also expects in disordered systems with a localized ground state.

math-ph↗

arXiv2015

Locality and nonlocality of classical restrictions of quantum spin systems with applications to quantum large deviations and entanglement

We study the projection on classical spins starting from quantum equilibria. We show Gibbsianness or quasi-locality of the resulting classical spin system for a class of gapped quantum systems at low temperatures including quantum ground states. A consequence of Gibbsianness is the validity of a large deviation principle in the quantum system which is known and here recovered in regimes of high temperature or for thermal states in one dimension. On the other hand we give an example of a quantum ground state with strong nonlocality in the classical restriction, giving rise to what we call measurement induced entanglement, and still satisfying a large deviation principle.

math-ph↗

arXiv2014

Asymptotic localization of energy in non-disordered oscillator chains

We study two popular one-dimensional chains of classical anharmonic oscillators: the rotor chain and a version of the discrete non-linear Schrödinger chain. We assume that the interaction between neighboring oscillators, controlled by the parameter $ε>0$, is small. We rigorously establish that the thermal conductivity of the chains has a non-perturbative origin, with respect to the coupling constant $ε$, and we provide strong evidence that it decays faster than any power law in $ε$ as $ε\rightarrow 0$. The weak coupling regime also translates into a high temperature regime, suggesting that the conductivity vanishes faster than any power of the inverse temperature.

cond-mat.stat-mech↗

arXiv2014

Scenario for delocalization in translation invariant systems

We investigate the possibility of Many-Body Localization in translation invariant Hamiltonian systems, which was recently brought up by several authors. A key feature of Many-Body Localized disordered systems is recovered, namely the fact that resonant spots are rare and far-between. However, we point out that resonant spots are mobile, unlike in models with strong quenched disorder, and that these mobile spots constitute a possible mechanism for delocalization, albeit possibly only on very long timescales. In some models, this argument for delocalization can be made very explicit in first order of perturbation theory in the hopping. For models where this does not work, we present instead a non-perturbative argument that relies solely on ergodicity inside the resonant spots.

cond-mat.stat-mech↗

arXiv2014

Can translation invariant systems exhibit a Many-Body Localized phase?

This note is based on a talk by one of us, F. H., at the conference PSPDE II, Minho 2013. We review some of our recent works related to (the possibility of) Many-Body Localization in the absence of quenched disorder (in particular arXiv:1305.5127,arXiv:1308.6263,arXiv:1405.3279). In these works, we provide arguments why systems without quenched disorder can exhibit `asymptotic' localization, but not genuine localization.

cond-mat.stat-mech↗

arXiv2014

Asymptotic quantum many-body localization from thermal disorder

We consider a quantum lattice system with infinite-dimensional on-site Hilbert space, very similar to the Bose-Hubbard model. We investigate many-body localization in this model, induced by thermal fluctuations rather than disorder in the Hamiltonian. We provide evidence that the Green-Kubo conductivity $κ(β)$, defined as the time-integrated current autocorrelation function, decays faster than any polynomial in the inverse temperature $β$ as $β\to 0$. More precisely, we define approximations $κ_τ(β)$ to $κ(β)$ by integrating the current-current autocorrelation function up to a large but finite time $τ$ and we rigorously show that $β^{-n}κ_{β^{-m}}(β)$ vanishes as $β\to 0$, for any $n,m \in \mathbb{N}$ such that $m-n$ is sufficiently large.

cond-mat.stat-mech↗

arXiv2013

Quantum Diffusion with Drift and the Einstein Relation I

We study the dynamics of a quantum particle hopping on a simple cubic lattice and driven by a constant external force. It is coupled to an array of identical, independent thermal reservoirs consisting of free, massless Bose fields, one at each site of the lattice. When the particle visits a site x of the lattice it can emit or absorb field quanta of the reservoir at x. Under the assumption that the coupling between the particle and the reservoirs and the driving force are sufficiently small, we establish the following results: The ergodic average over time of the state of the particle approaches a non-equilibrium steady state (NESS) describing a non-zero mean drift of the particle. Its motion around the mean drift is diffusive, and the diffusion constant and the drift velocity are related to one another by the Einstein relation.

math-ph↗

arXiv2013

Quantum Diffusion with Drift and the Einstein Relation II

This paper is a companion to 'Quantum Diffusion with Drift and the Einstein Relation I' (jointly submitted to arXiv). Its purpose is to describe and prove a certain number of technical results used in 'Quantum Diffusion with Drift and the Einstein Relation I', but not proven there. Both papers concern long-time properties (diffusion, drift) of the motion of a driven quantum particle coupled to an array of thermal reservoirs. The main technical results derived in the present paper are $(1)$ an asymptotic perturbation theory applicable for small driving, and, $(2)$ the construction of time-dependent correlation functions of particle observables.

math-ph↗

arXiv2013

Derivation of some translation-invariant Lindblad equations for a quantum Brownian particle

We study the dynamics of a Brownian quantum particle hopping on an infinite lattice with a spin degree of freedom. This particle is coupled to free boson gases via a translation-invariant Hamiltonian which is linear in the creation and annihilation operators of the bosons. We derive the time evolution of the reduced density matrix of the particle in the van Hove limit in which we also rescale the hopping rate. This corresponds to a situation in which both the system-bath interactions and the hopping between neighboring sites are small and they are effective on the same time scale. The reduced evolution is given by a translation-invariant Lindblad master equation which is derived explicitly.

math-ph↗

arXiv2012

Disordered quantum wires: microscopic origins of the DMPK theory and Ohm's law

We study the electronic transport properties of the Anderson model on a strip, modeling a quasi one-dimensional disordered quantum wire. In the literature, the standard description of such wires is via random matrix theory (RMT). Our objective is to firmly relate this theory to a microscopic model. We correct and extend previous work (arXiv:0912.1574) on the same topic. In particular, we obtain through a physically motivated scaling limit an ensemble of random matrices that is close to, but not identical to the standard transfer matrix ensembles (sometimes called TOE, TUE), corresponding to the Dyson symmetry classes β=1,2. In the β=2 class, the resulting conductance is the same as the one from the ideal ensemble, i.e.\ from TUE. In the β=1 class, we find a deviation from TOE. It remains to be seen whether or not this deviation vanishes in a thick-wire limit, which is the experimentally relevant regime. For the ideal ensembles, we also prove Ohm's law for all symmetry classes, making mathematically precise a moment expansion by Mello and Stone. This proof bypasses the explicit but intricate solution methods that underlie most previous results.

math-ph↗