arXiv subjects

Wojciech De Roeck

Publications and source records attributed to Wojciech De Roeck.

At least 37 records · Page 2Linked to original sources

arXiv2022

A many-body Fredholm index for ground state spaces and Abelian anyons

We propose a many-body index that extends Fredholm index theory to many-body systems. The index is defined for any charge-conserving system with a topologically ordered $p$-dimensional ground state sector. The index is fractional with the denominator given by $p$. In particular, this yields a new short proof of the quantization of the Hall conductance and of Lieb-Schulz-Mattis theorem. In the case that the index is non-integer, the argument provides an explicit construction of Wilson loop operators exhibiting a non-trivial braiding and that can be used to create fractionally charged Abelian anyons.

math-ph↗

arXiv2020

A Rigorous Theory of Prethermalization without Temperature

Prethermalization refers to the physical phenomenon where a system evolves toward some long-lived non-equilibrium steady state before eventual thermalization sets in. One general scenario where this occurs is in driven systems with dynamics governed by an effective Hamiltonian (in some rotating frame), such that ergodicity of the latter is responsible for the approach to the prethermal state. This begs the question whether it is possible to have a prethermal state not associated to any effective Hamiltonian. Here, we answer this question in the affirmative. We exhibit a natural class of systems in which the prethermal state is defined by emergent, global symmetries, but where the dynamics that takes the system to this state has no additional conservation laws, in particular energy. We explain how novel prethermal phases of matter can nevertheless emerge under such settings, distinct from those previously discussed.

quant-ph↗

arXiv2020

Exactness of linear response in the quantum Hall effect

In general, linear response theory expresses the relation between a driving and a physical system's response only to first order in perturbation theory. In the context of charge transport, this is the linear relation between current and electromotive force expressed in Ohm's law. We show here that, in the case of the quantum Hall effect, all higher order corrections vanish. We prove this in a fully interacting setting and without flux averaging.

math-ph↗

arXiv2020

Subdiffusion in one-dimensional Hamiltonian chains with sparse interactions

We establish rigorously that transport is slower than diffusive for a class of disordered one-dimensional Hamiltonian chains. This is done by deriving quantitative bounds on the variance in equilibrium of the energy or particle current, as a function of time. The slow transport stems from the presence of rare insulating regions (Griffiths regions). In many-body disordered quantum chains, they correspond to regions of anomalously high disorder, where the system is in a localized phase. In contrast, we deal with quantum and classical disordered chains where the interactions, usually referred to as anharmonic couplings in classical systems, are sparse. The system hosts thus rare regions with no interactions and, since the chain is Anderson localized in the absence of interactions, the non-interacting rare regions are insulating. Part of the mathematical interest of our model is that it is one of the few non-integrable models where the diffusion constant can be rigorously proven not to be infinite.

math-ph↗

arXiv2019

Very slow heating for weakly driven quantum many-body systems

It is well understood that many-body systems driven at high frequency heat up only exponentially slowly and exhibit a long prethermalization regime. We prove rigorously that a certain relevant class of systems heat very slowly under weak periodic driving at intermediate frequency as well. This class of systems are those whose time-dependent, possibly translation-invariant, Hamiltonian is a weak perturbation of a sum of mutually commuting terms. This condition covers several periodically kicked systems that have been considered in the literature recently, in particular kicked Ising models. In contrast to the high-frequency regime, the prethermalization dynamics of our systems is in general not related to any time-independent effective Floquet Hamiltonian. Our results also have non-trivial implications for closed (time-independent) systems. We use the example of an Ising model with transversal and longitudinal field to show how they imply confinement of excitations. More generally, they show how "glassy" kinetically constrained models emerge naturally from simple many-body Hamiltonians, thus connecting to the topic of 'translation-invariant localization'.

cond-mat.stat-mech↗

arXiv2019

A many-body index for quantum charge transport

We propose an index for pairs of a unitary map and a clustering state on many-body quantum systems. We require the map to conserve an integer-valued charge and to leave the state, e.g. a gapped ground state, invariant. This index is integer-valued and stable under perturbations. In general, the index measures the charge transport across a fiducial line. We show that it reduces to (i) an index of projections in the case of non-interacting fermions, (ii) the charge density for translational invariant systems, and (iii) the quantum Hall conductance in the two-dimensional setting without any additional symmetry. Example (ii) recovers the Lieb-Schultz-Mattis theorem, and (iii) provides a new and short proof of quantization of Hall conductance in interacting many-body systems.

math-ph↗

arXiv2019

Prethermalization and Thermalization in Isolated Quantum Systems

Prethermalization has been extensively studied in systems close to integrability. We propose a more general, yet conceptually simpler, setup for this phenomenon. We consider a---possibly nonintegrable---reference dynamics, weakly perturbed so that the perturbation breaks at least one conservation law of the reference dynamics. We argue then that the evolution of the system proceeds via intermediate (generalized) equilibrium states of the reference dynamics. The motion on the manifold of equilibrium states is governed by an autonomous equation, flowing towards global equilibrium in a time of order 1/g^2, where g is the perturbation strength. We also describe the leading correction to the time-dependent reference equilibrium state, which is, in general, of order g. The theory is well confirmed in numerical calculations of model Hamiltonians, for which we use a numerical linked cluster expansion and full exact diagonalization.

cond-mat.stat-mech↗

arXiv2019

Glassy dynamics in strongly anharmonic chains of oscillators

We review the mechanism for transport in strongly anharmonic chains of oscillators near the atomic limit where all oscillators are decoupled. In this regime, the motion of most oscillators remains close to integrable, i.e. quasi-periodic, on very long time scales, while a few chaotic spots move very slowly and redistribute the energy across the system. The material acquires several characteristic properties of dynamical glasses: intermittency, jamming and a drastic reduction of the mobility as a function of the thermodynamical parameters. We consider both classical and quantum systems, though with more emphasis on the former, and we discuss also the connections with quenched disordered systems, which display a similar physics to a large extent.

cond-mat.stat-mech↗

arXiv2019

The adiabatic theorem in a quantum many-body setting

In these lecture notes, we review the adiabatic theorem in quantum mechanics, focusing on a recent extension to many-body systems. The role of locality is emphasized and the relation to the quasi-adiabatic flow discussed. An important application of these results to linear response theory is also reviewed.

math-ph↗

arXiv2019

Producing suprathermal tails in the stationary velocity distribution

We revisit effective scenarios for the origin of heavy tails in stationary velocity distributions. A first analysis combines localization with diffusive acceleration. That gets realized in space plasmas to find the so-called kappa-distributions having power-law decay at high speeds. There, localization at high energy already takes place for the reversible dynamics but becomes effective by an active diffusion in velocity space. A model for vibrating granular gases and giving rise to stretched exponential tails is also briefly discussed, where negative friction is the energizer. In all cases, the resulting nonMaxwellian velocity distributions are frenetically caused by the dependence on the speed of kinetic parameters.

cond-mat.stat-mech↗

arXiv2018

Quantization of conductance in gapped interacting systems

We provide a short proof of the quantisation of the Hall conductance for gapped interacting quantum lattice systems on the two-dimensional torus. This is not new and should be seen as an adaptation of the proof of [1], simplified by making the stronger assumption that the Hamiltonian remains gapped when threading the torus with fluxes. We argue why this assumption is very plausible. The conductance is given by Berry's curvature and our key auxiliary result is that the curvature is asymptotically constant across the torus of fluxes.

math-ph↗

arXiv2018

Note on linear response for interacting Hall insulators

We relate explicitly the adiabatic curvature -- in flux space -- of an interacting Hall insulator with nondegenerate ground state to various linear response coefficients, in particular the Kubo response and the adiabatic response. The flexibility of the setup, allowing for various driving terms and currents, reflects the topological nature of the adiabatic curvature. We also outline an abstract connection between Kubo response and adiabatic response, corresponding to the fact that electric fields can be generated both by electrostatic potentials and time-dependent magnetic fields. Our treatment fits in the framework of rigorous many-body theory, thanks to the gap assumption.

math-ph↗

arXiv2018

Many-body delocalization as a quantum avalanche

We propose a multi-scale diagonalization scheme to study disordered one-dimensional chains, in particular the transition between many-body localization (MBL) and the ergodic phase, expected to be governed by resonant spots. Our scheme focuses on the dichotomy MBL versus {ETH} (eigenstate thermalization hypothesis). We show that a few natural assumptions imply that the system is localized with probability one at criticality. On the ergodic side, delocalization is induced by a quantum avalanche seeded by large ergodic spots, whose size diverges at the transition. On the MBL side, the typical localization length tends to a finite universal value at the transition, but there is a divergent length scale related to the response to an inclusion of large ergodic spots. A mean field approximation analytically illustrates these results and predicts as a power-law distribution for thermal inclusions at criticality.

cond-mat.dis-nn↗

arXiv2017

A microscopically motivated renormalization scheme for the MBL/ETH transition

We introduce a multi-scale diagonalization scheme to study the transition between the many-body localized and the ergodic phase in disordered quantum chains. The scheme assumes a sharp dichotomy between subsystems that behave as localized and resonant spots that obey the Eigenstate Thermalization Hypothesis (ETH). We establish a set of microscopic principles defining the diagonalization scheme, and use them to numerically study the transition in very large systems. To a large extent the results are in agreement with an analytically tractable mean-field analysis of the scheme: We find that at the critical point the system is almost surely localized in the thermodynamic limit, hosting a set of thermal inclusions whose sizes are power-law distributed. On the localized side the {\em typical} localization length is bounded from above. The bound saturates upon approach to criticality, entailing that a finite ergodic inclusion thermalizes a region of diverging diameter. The dominant thermal inclusions have a fractal structure, implying that averaged correlators decay as stretched exponentials throughout the localized phase. Slightly on the ergodic side thermalization occurs through an avalanche instability of the nearly localized bulk, whereby rare, supercritically large ergodic spots eventually thermalize the entire sample. Their size diverges at the transition, while their density vanishes. The non-local, avalanche-like nature of this instability entails a breakdown of single parameter scaling and puts the delocalization transition outside the realm of standard critical phenomena.

cond-mat.stat-mech↗

arXiv2017

The adiabatic theorem and linear response theory for extended quantum systems

The adiabatic theorem refers to a setup where an evolution equation contains a time-dependent parameter whose change is very slow, measured by a vanishing parameter $ε$. Under suitable assumptions the solution of the time-inhomogenous equation stays close to an instantaneous fixpoint. In the present paper, we prove an adiabatic theorem with an error bound that is independent of the number of degrees of freedom. Our setup is that of quantum spin systems where the manifold of ground states is separated from the rest of the spectrum by a spectral gap. One important application is the proof of the validity of linear response theory for such extended, genuinely interacting systems. In general, this is a long-standing mathematical problem, which can be solved in the present particular case of a gapped system, relevant e.g.~for the integer quantum Hall effect.

math-ph↗

arXiv2017

Adiabatic Theorem for Quantum Spin Systems

The first proof of the quantum adiabatic theorem was given as early as 1928. Today, this theorem is increasingly applied in a many-body context, e.g. in quantum annealing and in studies of topological properties of matter. In this setup, the rate of variation $\varepsilon$ of local terms is indeed small compared to the gap, but the rate of variation of the total, extensive Hamiltonian, is not. Therefore, applications to many-body systems are not covered by the proofs and arguments in the literature. In this letter, we prove a version of the adiabatic theorem for gapped ground states of quantum spin systems, under assumptions that remain valid in the thermodynamic limit. As an application, we give a mathematical proof of Kubo linear response formula for a broad class of gapped interacting systems.

math-ph↗

arXiv2017

Many-Body Localization: Stability and Instability

Rare regions with weak disorder (Griffiths regions) have the potential to spoil localization. We describe a non-perturbative construction of local integrals of motion (LIOMs) for a weakly interacting spin chain in one dimension, under a physically reasonable assumption on the statistics of eigenvalues. We discuss ideas about the situation in higher dimensions, where one can no longer ensure that interactions involving the Griffiths regions are much smaller than the typical energy-level spacing for such regions. We argue that ergodicity is restored in dimension d > 1, although equilibration should be extremely slow, similar to the dynamics of glasses.

math-ph↗

arXiv2017

A Rigorous Theory of Many-Body Prethermalization for Periodically Driven and Closed Quantum Systems

Prethermalization refers to the transient phenomenon where a system thermalizes according to a Hamiltonian that is not the generator of its evolution. We provide here a rigorous framework for quantum spin systems where prethermalization is exhibited for very long times. First, we consider quantum spin systems under periodic driving at high frequency $ν$. We prove that up to a quasi-exponential time $τ_* \sim e^{c \fracν{\log^3 ν}}$, the system barely absorbs energy. Instead, there is an effective local Hamiltonian $\hat D$ that governs the time evolution up to $τ_*$, and hence this effective Hamiltonian is a conserved quantity up to $τ_*$. Next, we consider systems without driving, but with a separation of energy scales in the Hamiltonian. A prime example is the Fermi-Hubbard model where the interaction $U$ is much larger than the hopping $J$. Also here we prove the emergence of an effective conserved quantity, different from the Hamiltonian, up to a time $τ_*$ that is (almost) exponential in $U/J$.

math-ph↗