Wojciech De Roeck

Universität Heidelberg, Heidelburg, Baden-Württemberg, Germany

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Publications (47)47.58 Total impact

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    Nick Crawford, Wojciech de Roeck
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    ABSTRACT: We consider a one dimensional interacting particle system which describes the effective interface dynamics of the two dimensional Toom model at low temperature and noise. We prove a number of basic properties of this model. First we consider the dynamics on a half open finite interval $[1, N)$, bounding the mixing time from above by $2N$. Then we consider the model defined on the integers. Due to infinite range interaction, this is a non-Feller process that we can define starting from product Bernoulli measures with density $p \in (0, 1)$, but not from arbitrary measures. We show, under a modest technical condition, that the only possible invariant measures are those product Bernoulli measures. We further show that the unique stationary measure on $[-k, \infty)$ converges weakly to a product Bernoulli measure on $\Z$ as $k \rightarrow \infty$.
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    Wojciech De Roeck, Marius Schütz
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    ABSTRACT: 'Quasi-adiabatic continuation', introduced first in the seminal work [Hastings 2004], has proven a successful technique for studying locality properties in gapped quantum lattice systems. For example, when a gapped Hamiltonian is perturbed locally in a region $S$, then the quasi-adiabatic continuation describes the resulting change in the ground state by a quasi-local transformation whose strength decays 'sub-exponentially' with distance to $S$. Here we address the natural question whether this decay can be made exponential, with characteristic decay length roughly the correlation length $\xi$ of the ground states. We point out that (a slight generalization of) [Hamza et al. 2009] allows to do this in some cases. As applications, we improve the locality estimate in some results on topological order and we prove some exponential decay estimates 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 many-body localization.
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    Wojciech De Roeck, François Huveneers
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    ABSTRACT: We study two popular one-dimensional chains of classical anharmonic oscillators: the rotor chain and a version of the discrete non-linear Schr\"odinger chain. We assume that the interaction between neighboring oscillators, controlled by the parameter $\epsilon >0$, is small. We rigorously establish that the thermal conductivity of the chains has a non-perturbative origin, with respect to the coupling constant $\epsilon$, and we provide strong evidence that it decays faster than any power law in $\epsilon$ as $\epsilon \rightarrow 0$. The weak coupling regime also translates into a high temperature regime, suggesting that the conductivity vanishes faster than any power of the inverse temperature.
    Communications on Pure and Applied Mathematics 12/2014; DOI:10.1002/cpa.21550 · 3.08 Impact Factor
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    Wojciech De Roeck, François Huveneers
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    ABSTRACT: 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.
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    Wojciech De Roeck, Francois Huveneers
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    ABSTRACT: We investigate the possibility of many-body localization in translation invariant Hamiltonian systems. 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 quenched disorder, and that they could therefore in principle delocalize the system. The motion of these resonant spots is reminiscent of motion in kinetically constrained models, the frustration effect being caused by a mismatch of energies rather than by geometric constraints. We provide examples where, in first order in the hopping, the resonant spots are effectively trapped (analogous to a jammed phase) and therefore localization is not ruled out, but we also have examples where, despite the rareness of resonant spots, the resonant spots are not trapped and hence this obvious scenario for many-body localization is not realized. We comment on the analysis of higher orders, but we did not reach a final conclusion yet.
  • Wojciech De Roeck, Francois Huveneers
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    ABSTRACT: We investigate the possibility of many-body localization in translation invariant Hamiltonian systems. 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 quenched disorder, and that they could therefore in principle delocalize the system. The motion of these resonant spots is reminiscent of motion in kinetically constrained models, the frustration effect being caused by a mismatch of energies rather than by geometric constraints. We provide examples where, in first order in the hopping, the resonant spots are effectively trapped (analogous to a jammed phase) and therefore localization is not ruled out, but we also have examples where, despite the rareness of resonant spots, the resonant spots are not trapped and hence this obvious scenario for many-body localization is not realized. We comment on the analysis of higher orders, but we did not reach a final conclusion yet.
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    Roland Bauerschmidt, Wojciech de Roeck, Jürg Fröhlich
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    ABSTRACT: We consider a linear Boltzmann equation that arises in a model for quantum friction. It describes a particle that is slowed down by the emission of bosons. We study the stochastic process generated by this Boltzmann equation and we show convergence of its spatial trajectory to a multiple of Brownian motion with exponential scaling. The asymptotic position of the particle is finite in mean, even though its absolute value is typically infinite. This is contrasted to an approximation that neglects the influence of fluctuations, where the mean asymptotic position is infinite.
    Journal of Physics A Mathematical and Theoretical 03/2014; 47(27). DOI:10.1088/1751-8113/47/27/275003 · 1.69 Impact Factor
  • Wojciech De Roeck, Antti Kupiainen
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    ABSTRACT: We consider generalised versions of the spin-boson model at small coupling. We assume the spin (or atom) to sit at the origin and the propagation speed v (p) of free bosons to be constant, i.e. independent of momentum. In particular, the bosons are massless. We prove detailed bounds on the mean number of bosons contained in the ball . In particular, we prove that, as , this number tends to an asymptotic value that can be naturally identified as the mean number of bosons bound to the atom in the ground state. Physically, this means that bosons, that are not bound to the atom, are travelling outwards at a speed that is not lower than v (p) , hence the term 'minimal velocity estimate'. Additionally, we prove bounds on the number of emitted bosons with low momentum (soft mode bounds). This paper is an extension of our earlier work in De Roeck and Kupiainen (Annales Henri Poincar, 14:253-311, 2013). Together with the results in De Roeck and Kupiainen (Annales Henri Poincar, 14:253-311, 2013), the bounds of the present paper suffice to prove asymptotic completeness, as we describe in De Roeck et al. (Asymptotic completeness in the massless spin-boson model, 2012).
    Annales Henri Poincare 01/2014; 16(2). DOI:10.1007/s00023-014-0323-7 · 1.37 Impact Factor
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    ABSTRACT: 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.
    Journal of Mathematical Physics 12/2013; 56(2). DOI:10.1063/1.4906767 · 1.18 Impact Factor
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    Wojciech De Roeck, François Huveneers
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    ABSTRACT: 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 $\kappa(\beta)$, defined as the time-integrated current autocorrelation function, decays faster than any polynomial in the inverse temperature $\beta$ as $\beta \to 0$. More precisely, we define approximations $\kappa_{\tau}(\beta)$ to $\kappa(\beta)$ by integrating the current-current autocorrelation function up to a large but finite time $\tau$ and we rigorously show that $\be^{-n}\kappa_{\be^{-m}}(\beta)$ vanishes as $\be \to 0$, for any $n,m \in \bbN$ such that $m-n$ is sufficiently large.
    Communications in Mathematical Physics 08/2013; 332(3). DOI:10.1007/s00220-014-2116-8 · 1.90 Impact Factor
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    Wojciech De Roeck, Juerg Froehlich, Kevin Schnelli
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    ABSTRACT: 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.
    Journal of Mathematical Physics 06/2013; 55(7). DOI:10.1063/1.4881532 · 1.18 Impact Factor
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    Wojciech De Roeck, Juerg Froehlich, Kevin Schnelli
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    ABSTRACT: 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.
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    W. De Roeck, M. Griesemer, A. Kupiainen
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    ABSTRACT: We consider generalized versions of the massless spin-boson model. Building on the recent work in 'Approach to ground state and time-independent photon bound for massless spin-boson models' (arXiv:1109.5582, Annales H. Poincare, 2012) and 'Propagation bounds and soft photon bounds for the massless spin-boson model' (submitted to arXiv jointly with the present paper), we prove asymptotic completeness.
    Advances in Mathematics 01/2013; 268. DOI:10.1016/j.aim.2014.09.012 · 1.35 Impact Factor
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    W. De Roeck, A. Kupiainen
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    ABSTRACT: We consider generalized versions of the massless spin-boson model. We prove detailed bounds on the number of bosons in certain spatial regions (propagation bounds) and on the number of bosons with low momentum (soft photon bounds). This work is an extension of our earlier work in 'Approach to ground state and time-independent photon bound for massless spin-boson models' (arXiv:1109.5582, Ann. H. Poincare, 2012). Together with the results in arXiv:1109.5582, the bounds of the present paper suffice to prove asymptotic completeness, as we describe in a joint submission to arXiv: 'Asymptotic completeness for the massless spin-boson model'.
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    Wojciech De Roeck, Dominique Spehner
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    ABSTRACT: 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.
    Journal of Statistical Physics 08/2012; 150(2). DOI:10.1007/s10955-012-0649-9 · 1.28 Impact Factor
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    Sven Bachmann, Maximilian Butz, Wojciech De Roeck
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    ABSTRACT: 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. Our objective is to firmly relate this theory to a microscopic model. We correct and extend previous work (Bachmann and De Roeck in J. Stat. Phys. 139:541–564, 2010) 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 appropriate regime for metals. For the ideal ensembles, we also prove Ohm’s law for all symmetry classes, making mathematically precise a moment expansion by Mello and Stone in Phys. Rev. B 44:3559–3576, 1991. This proof bypasses the explicit but intricate solution methods that underlie most previous results.
    Journal of Statistical Physics 07/2012; 148(1). DOI:10.1007/s10955-012-0517-7 · 1.28 Impact Factor
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    Wojciech De Roeck, Antti Kupiainen
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    ABSTRACT: It is widely believed that an atom interacting with the electromagnetic field (with total initial energy well-below the ionization threshold) relaxes to its ground state while its excess energy is emitted as radiation. Hence, for large times, the state of the atom+field system should consist of the atom in its ground state, and a few free photons that travel off to spatial infinity. Mathematically, this picture is captured by the notion of asymptotic completeness. Despite some recent progress on the spectral theory of such systems, a proof of relaxation to the ground state and asymptotic completeness was/is still missing, except in some special cases (massive photons, small perturbations of harmonic potentials). In this paper, we partially fill this gap by proving relaxation to an invariant state in the case where the atom is modelled by a finite-level system. If the coupling to the field is sufficiently infrared-regular so that the coupled system admits a ground state, then this invariant state necessarily corresponds to the ground state. Assuming slightly more infrared regularity, we show that the number of emitted photons remains bounded in time. We hope that these results bring a proof of asymptotic completeness within reach.
    Annales Henri Poincare 09/2011; 14(2). DOI:10.1007/s00023-012-0190-z · 1.37 Impact Factor
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    W. De Roeck, A. Kupiainen
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    ABSTRACT: We prove diffusion for a quantum particle coupled to a field of bosons (phonons or photons). The importance of this result lies in the fact that our model is fully Hamiltonian and randomness enters only via the initial (thermal) state of the bosons. This model is closely related to the one considered in [De Roeck, Fr\"ohlich 2011], but various restrictive assumptions of the latter have been eliminated. In particular, depending on the dispersion relation of the bosons, the present result holds in dimension $d \geq 3$.
    Communications in Mathematical Physics 07/2011; 323(3). DOI:10.1007/s00220-013-1794-y · 1.90 Impact Factor
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    Wojciech De Roeck, Jürg Fröhlich, Alessandro Pizzo
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    ABSTRACT: We show that, in a model where a non-relativistic particle is coupled to a quantized relativistic scalar Bose field, the embedded mass shell of the particle dissolves in the continuum when the interaction is turned on, provided the coupling constant is sufficiently small. More precisely, under the assumption that the fiber eigenvectors corresponding to the putative mass shell are differentiable as functions of the total momentum of the system, we show that a mass shell could exist only at a strictly positive distance from the unperturbed embedded mass shell near the boundary of the energy–momentum spectrum.
    Annales Henri Poincare 12/2010; 11(8):1545-1589. DOI:10.1007/s00023-010-0066-z · 1.37 Impact Factor
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    W. De Roeck, A. Kupiainen
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    ABSTRACT: Recently, several authors studied small quantum systems weakly coupled to free boson fields at positive density. All the approaches we are aware of employ complex deformations of Liouvillians or Mourre theory (the infinitesimal version of the former). We present an approach based on polymer expansions of statistical mechanics. Despite the fact that our approach is elementary, our results are slightly sharper than those contained in the literature up to now. Essentially, we show that, whenever the small quantum system is known to admit a Markov approximation (Pauli master equation \emph{aka} Lindblad equation) in the weak coupling limit, and the Markov approximation is exponentially mixing, then the weakly coupled system approaches a unique invariant state that is perturbatively close to its Markov approximation. Comment: 23 pages, v1-->v2: inconsequential error in Section 3 corrected
    Communications in Mathematical Physics 05/2010; 305(3). DOI:10.1007/s00220-011-1247-4 · 1.90 Impact Factor

Publication Stats

330 Citations
47.58 Total Impact Points

Institutions

  • 2008–2014
    • Universität Heidelberg
      • Institute of Theoretical Physics
      Heidelburg, Baden-Württemberg, Germany
    • University of Leuven
      Louvain, Flemish, Belgium
  • 2012
    • University of Cologne
      • Institute of Physics
      Köln, North Rhine-Westphalia, Germany
  • 2009
    • ETH Zurich
      • Institute for Theoretical Physics
      Zürich, Zurich, Switzerland
    • University of Helsinki
      • Department of Mathematics and Statistics
      Helsinki, Uusimaa, Finland
  • 2004
    • University of Antwerp
      Antwerpen, Flanders, Belgium