arXiv subjects

Daniele Toniolo

Publications and source records attributed to Daniele Toniolo.

11 recordsLinked to original sources

arXiv2026

Dynamics of many-body localized systems: logarithmic lightcones and $\log \, t$-law of $α$-Rényi entropies

In the context of the Many-Body-Localization phenomenology we consider arbitrarily large one-dimensional local spin systems, the XXZ model with random magnetic field is a prototypical example. Without assuming the existence of exponentially localized integrals of motion (LIOM), but assuming instead that the system's dynamics gives rise to a Lieb-Robinson bound (L-R) with a logarithmic lightcone, we rigorously evaluate the dynamical generation, starting from a generic product state, of $ α$-Rényi entropies, with $ α$ close to one, obtaining a $\log \, t$-law, that denotes a slow spread of entanglement. This is in sharp contrast with Anderson localized phases that show no dynamically generated entanglement. To prove this result we apply a general theory recently developed by us in arXiv:2408.00743 that quantitatively relates the L-R bounds of a local Hamiltonian with the dynamical generation of entanglement. Assuming instead the existence of LIOM we provide new independent proofs of the known facts that the L-R bound of the system's dynamics has a logarithmic lightcone and show that the dynamical generation of the von Neumann entropy has for large times a $ \log \, t$-shape. L-R bounds, that quantify the dynamical spreading of local operators, may be easier to measure in experiments in comparison to global quantities such as entanglement.

cond-mat.dis-nn↗

arXiv2025

Logarithmic lightcones in the multiparticle Anderson model with sparse interactions

We prove that the dynamics of the one-dimensional $ XY $ model with random magnetic field perturbed by a sparse set of $ ZZ $ terms with a large coupling constant $ Δ$ gives rise to Lieb-Robinson (L-R) bounds with a logarithmic lightcone and amplitude proportional to $ Δ^{-1} $. These spin systems are equivalent to a set of spinless lattice fermions subjected to a random on site potential and sparse density-density interactions. In the absence of the random magnetic field we also obtain a suppression of the L-R bounds as $ Δ^{-1} $. These results follow from the application of a general theorem about the L-R bound of a generic local time-dependent one-dimensional spin system with local time-dependent perturbations. Adopting the interaction picture of the dynamics, the large and sparse $ ZZ $ perturbations of the $ XY $ model, with or without disorder, are mapped into high-frequency periodic perturbations. All our results are non-perturbative.

math-ph↗

arXiv2025

The dynamical $α$-Rényi entropies of local Hamiltonians grow at most linearly in time

We consider a generic one dimensional spin system of length $ L $, arbitrarily large, with strictly local interactions, for example nearest neighbor, and prove that the dynamical $ α$-Rényi entropies, $ 0 < α\le 1 $, of an initial product state grow at most linearly in time. This result arises from a general relation among dynamical $ α$-Rényi entropies and Lieb-Robinson bounds. We extend our bound on the dynamical generation of entropy to systems with exponential decay of interactions, for values of $α$ close enough to $ 1 $, and moreover to initial pure states with low entanglement, of order $ \log L $, that are typically represented by critical states. We establish that low entanglement states have an efficient MPS representation that persists at least up to times of order $ \log L $. The main technical tools are the Lieb-Robinson bounds, to locally approximate the dynamics of the spin chain, a strict upper bound of Audenaert on $ α$-Rényi entropies and a bound on their concavity. Such a bound, that we provide in an appendix, can be of independent interest.

quant-ph↗

arXiv2024

Stability of slow Hamiltonian dynamics from Lieb-Robinson bounds

We rigorously show that a local spin system giving rise to a slow Hamiltonian dynamics is stable against generic, even time-dependent, local perturbations. The sum of these perturbations can cover a significant amount of the system's size. The stability of the slow dynamics follows from proving that the Lieb-Robinson bound for the dynamics of the total Hamiltonian is the sum of two contributions: the Lieb-Robinson bound of the unperturbed dynamics and an additional term coming from the Lieb-Robinson bound of the perturbations with respect to the unperturbed Hamiltonian. Our results are particularly relevant in the context of the study of the stability of Many-Body-Localized systems, implying that if a so called ergodic region is present in the system, to spread across a certain distance it takes a time proportional to the exponential of such distance. The non-perturbative nature of our result allows us to develop a dual description of the dynamics of a system. As a consequence we are able to prove that the presence of a region of disorder in a ergodic system implies the slowing down of the dynamics in the vicinity of that region.

quant-ph↗

arXiv2023

Absence of localization in two-dimensional Clifford circuits

We analyze a Floquet circuit with random Clifford gates in one and two spatial dimensions. By using random graphs and methods from percolation theory, we prove in the two dimensional setting that some local operators grow at ballistic rate, which implies the absence of localization. In contrast, the one-dimensional model displays a strong form of localization characterized by the emergence of left and right-blocking walls in random locations. We provide additional insights by complementing our analytical results with numerical simulations of operator spreading and entanglement growth, which show the absence (presence) of localization in two-dimension (one-dimension). Furthermore, we unveil that the spectral form factor of the Floquet unitary in two-dimensional circuits behaves like that of quasi-free fermions with chaotic single particle dynamics, with an exponential ramp that persists till times scaling linearly with the size of the system. Our work sheds light on the nature of disordered, Floquet Clifford dynamics and its relationship to fully chaotic quantum dynamics.

quant-ph↗

arXiv2022

On the Bott index of unitary matrices on a finite torus

This article reviews the foundations of the theory of the Bott index of a pair of unitary matrices in the context of condensed matter theory, as developed by Hastings and Loring in J. Math. Phys. (51), 015214, (2010) and Annals of Physics 326 (2011) 1699-1759, providing a novel proof of the equality with the Chern number. The Bott index is defined for a pair of unitary matrices, then extended to a pair of invertible matrices and homotopic invariance of the index is proven. An insulator defined on a lattice on a 2-torus, that is a rectangular lattice with periodic boundary conditions, is considered and a pair of quasi-unitary matrices associated to this physical system are introduced. It is shown that their Bott index is well defined and the connection with the transverse conductance, the Chern number, is established proving the equality of the two quantities.

cond-mat.mes-hall↗

arXiv2022

Mixing and localisation in random time-periodic quantum circuits of Clifford unitaries

How much does local and time-periodic dynamics resemble a random unitary? In the present work we address this question by using the Clifford formalism from quantum computation. We analyse a Floquet model with disorder, characterised by a family of local, time-periodic, random quantum circuits in one spatial dimension. We observe that the evolution operator enjoys an extra symmetry at times that are a half-integer multiple of the period. With this we prove that after the scrambling time, namely when any initial perturbation has propagated throughout the system, the evolution operator cannot be distinguished from a (Haar) random unitary when all qubits are measured with Pauli operators. This indistinguishability decreases as time goes on, which is in high contrast to the more studied case of (time-dependent) random circuits. We also prove that the evolution of Pauli operators displays a form of mixing. These results require the dimension of the local subsystem to be large. In the opposite regime our system displays a novel form of localisation, produced by the appearance of effective one-sided walls, which prevent perturbations from crossing the wall in one direction but not the other.

quant-ph↗

arXiv2021

The Bott index of two unitary operators and the integer quantum Hall effect

The Bott index of two unitary operators on an infinite dimensional Hilbert space is defined. homotopic invariance with respect to multiplicative unitary perturbations of the type identity plus trace class and the "logarithmic" law for the index are proven. The index and its properties are then extended to the case of a pair of invertible operators. An application to the physics of two dimensional quantum systems proves that the index is equal to the Chern number therefore showing that the transverse Hall conductance is an integer.

math-ph↗

arXiv2021

Dimensional reduction and scattering formulation for even topological invariants

Strong invariants of even-dimensional topological insulators of independent Fermions are expressed in terms of an invertible operator on the Hilbert space over the boundary. It is given by the Cayley transform of the boundary restriction of the half-space resolvent. This dimensional reduction is routed in new representation for the $K$-theoretic exponential map. It allows to express the invariants via the reflection matrix at the Fermi energy, for the scattering set-up of a wire coupled to the half-space insulator.

math-ph↗

arXiv2018

Time-dependent topological systems: A study of the Bott index

The Bott index is an index that discerns among pairs of unitary matrices that can or cannot be approximated by a pair of commuting unitary matrices. It has been successfully employed to describe the approximate integer quantization of the transverse conductance of a system described by a short-range, bounded and spectrally gapped Hamiltonian on a finite two dimensional lattice on a torus and to describe the invariant of the Bernevig- Hughes-Zhang model even with disorder. This paper shows the constancy in time of the Bott index and the Chern number related to the time-evolved Fermi projection of a thermodynamically large system described by a short-range and time-dependent Hamiltonian that is initially gapped. The general situation of a ramp of a time-dependent perturbation is considered, a section is dedicated to time-periodic perturbations.

cond-mat.mes-hall↗

arXiv2014

Superfluid Breakdown and Multiple Roton Gaps in Spin-Orbit Coupled Bose-Einstein Condensates on an Optical Lattice

We investigate the superfluid phases of a Rashba spin-orbit coupled Bose-Einstein condensate residing on a two dimensional square optical lattice in the presence of an effective Zeeman field $Ω$. At a critical value $Ω=Ω_c$, the single-particle spectrum $ E_k $ changes from having a set of four degenerate minima to a single minimum at $k=0$, corresponding to condensation at finite or zero momentum, respectively. We describe this quantum phase transition and the symmetry breaking of the condensate phases. We use the Bogoliubov theory to treat the superfluid phases and determine the phase diagram, the excitation spectrum and the sound velocity of the phonon excitations. A novel dynamically unstable superfluid regime occurring when $Ω$ is close to $Ω_c$ is analytically identified and the behavior of the condensate quantum depletion is discussed. Moreover, we show that there are two types of roton excitations occurring in the $Ω<Ω_c$ regime and obtain explicit values for the corresponding energy gaps.

cond-mat.quant-gas↗