arXiv subjects

Guido Giachetti

Publications and source records attributed to Guido Giachetti.

29 records · Page 2Linked to original sources

arXiv2022

Spreading of a local excitation in a Quantum Hierarchical Model

We study the dynamics of the quantum Dyson hierarchical model in its paramagnetic phase. An initial state made by a local excitation of the paramagnetic ground state is considered. We provide analytical predictions for its time evolution, solving the single-particle dynamics on a hierarchical network. A localization mechanism is found and the excitation remains close to its initial position at arbitrary times. Furthermore, a universal scaling among space and time is found related to the algebraic decay of the interactions as $r^{-1-σ}$. We compare our predictions to numerics, employing tensor network techniques, for large magnetic fields, discussing the robustness of the mechanism in the full many-body dynamics.

quant-ph↗

arXiv2022

Villain model with long-range couplings

The nearest-neighbor Villain, or periodic Gaussian, model is a useful tool to understand the physics of the topological defects of the two-dimensional nearest-neighbor $XY$ model, as the two models share the same symmetries and are in the same universality class. The long-range counterpart of the two-dimensional $XY$ model has been recently shown to exhibit a non-trivial critical behavior, with a complex phase diagram including a range of values of the power-law exponent of the couplings decay, $σ$, in which there are a magnetized, a disordered and a critical phase (arXiv:2104.13217). Here we address the issue of whether the critical behavior of the two-dimensional $XY$ model with long-range couplings can be described by the Villain counterpart of the model. After introducing a suitable generalization of the Villain model with long-range couplings, we derive a set of renormalization-group equations for the vortex-vortex potential, which differs from the one of the long-range $XY$ model, signaling that the decoupling of spin-waves and topological defects is no longer justified in this regime. The main results are that for $σ<2$ the two models no longer share the same universality class. Remarkably, within a large region of its phase diagram, the Villain model is found to behave similarly to the one-dimensional Ising model with $1/r^2$ interactions.

cond-mat.stat-mech↗

arXiv2022

Symplectic coarse graining approach to the dynamics of spherical self-gravitating systems

We investigate the evolution of the phase-space distribution function around slightly perturbed stationary states and the process of violent relaxation in the context of the dissipationless collapse of an isolated spherical self-gravitating system. By means of the recently introduced symplectic coarse graining technique, we obtain an effective evolution equation that allows us to compute the scaling of the frequencies around a stationary state, as well as the damping times of Fourier modes of the distribution function, with the magnitude of the Fourier $k-$vectors themselves. We compare our analytical results with $N$-body simulations.

astro-ph.GA↗

arXiv2022

Energy fluctuation relations and repeated quantum measurements

In this review paper, we discuss the statistical description in non-equilibrium regimes of energy fluctuations originated by the interaction between a quantum system and a measurement apparatus applying a sequence of repeated quantum measurements. To properly quantify the information about energy fluctuations, both the exchanged heat probability density function and the corresponding characteristic function are derived and interpreted. Then, we discuss the conditions allowing for the validity of the fluctuation theorem in Jarzynski form $\langle e^{-βQ}\rangle = 1$, thus showing that the fluctuation relation is robust against the presence of randomness in the time intervals between measurements. Moreover, also the late-time, asymptotic properties of the heat characteristic function are analyzed, in the thermodynamic limit of many intermediate quantum measurements. In such a limit, the quantum system tends to the maximally mixed state (thus corresponding to a thermal state with infinite temperature) unless the system's Hamiltonian and the intermediate measurement observable share a common invariant subspace. Then, in this context, we also discuss how energy fluctuation relations change when the system operates in the quantum Zeno regime. Finally, the theoretical results are illustrated for the special cases of two- and three-levels quantum systems, now ubiquitous for quantum applications and technologies.

quant-ph↗

arXiv2021

Berezinskii-Kosterlitz-Thouless phase transitions with long-range couplings

The Berezinskii-Kostelitz-Thouless (BKT) transition is the paradigmatic example of a topological phase transition without symmetry-breaking, where a quasi-ordered phase, characterized by a power law scaling of the correlation functions at low temperature, is disrupted by the proliferation of topological excitations above the critical temperature $T_{\rm BKT}$. In this letter, we consider the effect of long-range decaying couplings $\sim r^{-2-σ}$ on this phenomenon. After pointing out the relevance of this non trivial problem, we discuss the phase diagram, which is far richer than the corresponding short-range one. It features -- for $7/4<σ<2$ -- a quasi ordered phase in a finite temperature range $T_c < T < T_{\rm BKT}$, which occurs between a symmetry broken phase for $T T_{\rm BKT}$. The transition temperature $T_c$ displays unique universal features quite different from those of the traditional, short-range XY model. Given the universal nature of our findings, they may be observed in current experimental realizations in $2D$ atomic, molecular and optical quantum systems.

cond-mat.stat-mech↗

arXiv2021

Thermalization processes induced by quantum monitoring in multi-level systems

We study the heat statistics of a multi-level $N$-dimensional quantum system monitored by a sequence of projective measurements. The late-time, asymptotic properties of the heat characteristic function are analyzed in the thermodynamic limit of a high, ideally infinite, number $M$ of measurements $(M \to \infty)$. In this context, the conditions allowing for an Infinite-Temperature Thermalization (ITT), induced by the repeated monitoring of the quantum system, are discussed. We show that ITT is identified by the fixed point of a symmetric random matrix that models the stochastic process originated by the sequence of measurements. Such fixed point is independent on the non-equilibrium evolution of the system and its initial state. Exceptions to ITT, to which we refer to as partial thermalization, take place when the observable of the intermediate measurements is commuting (or quasi-commuting) with the Hamiltonian of the quantum system, or when the time interval between measurements is smaller or comparable with the system energy scale (quantum Zeno regime). Results on the limit of infinite-dimensional Hilbert spaces ($N \to \infty$), describing continuous systems with a discrete spectrum, are also presented. We show that the order of the limits $M\to\infty$ and $N\to\infty$ matters: when $N$ is fixed and $M$ diverges, then ITT occurs. In the opposite case, the system becomes classical, so that the measurements are no longer effective in changing the state of the system. A non trivial result is obtained fixing $M/N^2$ where instead partial ITT occurs. Finally, an example of partial thermalization applicable to rotating two-dimensional gases is presented.

quant-ph↗

arXiv2021

Violent relaxation in the Hamiltonian mean field model: II. Non-equilibrium phase diagrams

A classical long-range-interacting $N$-particle system relaxes to thermal equilibrium on time scales growing with $N$; in the limit $N\to \infty$ such a relaxation time diverges. However, a completely non-collisional relaxation process, known as violent relaxation, takes place on a much shorter time scale independent of $N$ and brings the system towards a non-thermal quasi-stationary state. A finite system will eventually reach thermal equilibrium, while an infinite system will remain trapped in the quasi-stationary state forever. For times smaller than the relaxation time the distribution function of the system obeys the collisionless Boltzmann equation, also known as the Vlasov equation. The Vlasov dynamics is invariant under time reversal so that it does not "naturally" describe a relaxational dynamics. However, as time grows the dynamics affects smaller and smaller scales in phase space, so that observables not depending upon small-scale details appear as relaxed after a short time. Herewith we present an approximation scheme able to describe violent relaxation in a one-dimensional toy-model, the Hamiltonian Mean Field (HMF). The approach described here generalizes the one proposed in G. Giachetti and L. Casetti, J. Stat. Mech.: Theory Exp. 2019, 043201 (2019), that was limited to "cold" initial conditions, to generic initial conditions, allowing us to to predict non-equilibrium phase diagrams that turn out to be in good agreement with those obtained from the numerical integration of the Vlasov equation.

cond-mat.stat-mech↗

arXiv2020

Self-consistent harmonic approximation with non-local couplings

We derive the self-consistent harmonic approximation for the $2D$ XY model with non-local interactions. The resulting equation for the variational couplings holds for any form of the spin-spin coupling as well as for any dimension. Our analysis is then specialized to power-law couplings decaying with the distance $r$ as $\propto 1/r^{2+σ}$ in order to investigate the robustness, at finite $σ$, of the Berezinskii-Kosterlitz-Thouless (BKT) transition, which occurs in the short-range limit $σ\to \infty$. We propose an ansatz for the functional form of the variational couplings and show that for any $σ>2$ the BKT mechanism occurs. The present investigation provides an upper bound for the lower critical threshold $σ^\ast=2$, above which the traditional BKT transition persists in spite of the LR couplings.

cond-mat.stat-mech↗

arXiv2020

Coarse-grained collisionless dynamics with long-range interactions

We present an effective evolution equation for a coarse-grained distribution function of a long-range-interacting system preserving the symplectic structure of the non-collisional Boltzmann, or Vlasov, equation. We first derive a general form of such an equation based on symmetry considerations only. Then, we explicitly derive the equation for one-dimensional systems, finding that it has the form predicted on general grounds. Finally, we use such an equation to predict the dependence of the damping times on the coarse-graining scale and numerically check it for some one-dimensional models, including the Hamiltonian Mean Field (HMF) model, a scalar field with quartic interaction, a 1-d self-gravitating system, and the Self-Gravitating Ring (SGR).

cond-mat.stat-mech↗

arXiv2020

Quantum-heat fluctuation relations in $3$-level systems under projective measurements

We study the statistics of energy fluctuations in a three-level quantum system subject to a sequence of projective quantum measurements. We check that, as expected, the quantum Jarzynski equality holds provided that the initial state is thermal. The latter condition is trivially satisfied for two-level systems, while this is generally no longer true for $N$-level systems, with $N > 2$. Focusing on three-level systems, we discuss the occurrence of a unique energy scale factor $β_{\rm eff}$ that formally plays the role of an effective inverse temperature in the Jarzynski equality. To this aim, we introduce a suitable parametrization of the initial state in terms of a thermal and a non-thermal component. We determine the value of $β_{\rm eff}$ for a large number of measurements and study its dependence on the initial state. Our predictions could be checked experimentally in quantum optics.

quant-ph↗

arXiv2019

Violent relaxation in the Hamiltonian Mean Field model: I. Cold collapse and effective dissipation

In $N$-body systems with long-range interactions mean-field effects dominate over binary interactions (collisions), so that relaxation to thermal equilibrium occurs on time scales that grow with $N$, diverging in the $N\to\infty$ limit. However, a faster and non-collisional relaxation process, referred to as violent relaxation, sets in when starting from generic initial conditions: collective oscillations (referred to as virial oscillations) develop and damp out on timescales not depending on the system's size. After the damping of such oscillations the system is found in a quasi-stationary state that survives virtually forever when the system is very large. During violent relaxation the distribution function obeys the collisionless Boltzmann (or Vlasov) equation, that, being invariant under time reversal, does not "naturally" describe a relaxation process. Indeed, the dynamics is moved to smaller and smaller scales in phase space as time goes on, so that observables that do not depend on small-scale details appear as relaxed after a short time. We propose an approximation scheme to describe collisionless relaxation, based on the introduction of moments of the distribution function, and apply it to the Hamiltonian Mean Field (HMF) model. To the leading order, virial oscillations are equivalent to the motion of a particle in a one-dimensional potential. Inserting higher-order contributions in an effective way, inspired by the Caldeira-Leggett model of quantum dissipation, we derive a dissipative equation describing the damping of the oscillations, including a renormalization of the effective potential and yielding predictions for collective properties of the system after the damping in very good agreement with numerical simulations. Here we restrict ourselves to "cold" initial conditions; generic initial conditions will be considered in a forthcoming paper.

cond-mat.stat-mech↗