Contributor info: Dr Laura Foini

Guy Bunin, Laura Foini, Jorge Kurchan

SciPost Phys. 17, 114 (2024) · published 15 October 2024 | Toggle abstract · pdf

The spectral form factor of quantum chaotic systems has the familiar 'ramp $+$ plateau' form. Techniques to determine its form in the semiclassical or the thermodynamic limit have been devised, in both cases based on the average over an energy range or an ensemble of systems. For a single instance, fluctuations are large, do not go away in the limit, and depend on the element of the ensemble itself, thus seeming to question the whole procedure. Considered as the modulus of a partition function in complex inverse temperature $\beta_R+i\beta_I$ ($\beta_I \equiv \tau$ the time), the spectral form factor has regions of Fisher zeroes, the analogue of Yang-Lee zeroes for the complex temperature plane. The large spikes in the spectral form factor are in fact a consequence of near-misses of the line parametrized by $\beta_I$ to these zeroes. The largest spikes are indeed extensive and extremely sensitive to details, but we show that they are both exponentially rare and exponentially thin. Motivated by this, and inspired by the work of Derrida on the Random Energy Model, we study here a modified model of random energy levels in which we introduce level repulsion. We also check that the mechanism giving rise to spikes is the same in the SYK model.

Paola Ruggiero, Pasquale Calabrese, Thierry Giamarchi, Laura Foini

SciPost Phys. 13, 111 (2022) · published 18 November 2022 | Toggle abstract · pdf

The study of non-equilibrium dynamics of many-body systems after a quantum quench received a considerable boost and a deep theoretical understanding from the path integral formulation in imaginary time. However, the celebrated problem of a quench in the Luttinger parameter of a one dimensional quantum critical system (massless quench) has so far only been solved in the real-time Heisenberg picture. In order to bridge this theoretical gap and to understand on the same ground massive and massless quenches, we study the problem of a gaussian field characterized by a coupling parameter $K$ within a strip and a different one $K_0$ in the remaining two semi-infinite planes. We give a fully analytical solution using the electrostatic analogy with the problem of a dielectric material within a strip surrounded by an infinite medium of different dielectric constant, and exploiting the method of charge images. After analytic continuation, this solution allows us to obtain all the correlation functions after the quench within a path integral approach in imaginary time, thus recovering and generalizing the results in real time. Furthermore, this imaginary-time approach establishes a remarkable connection between the quench and the famous problem of the conductivity of a Tomonaga-Luttinger liquid coupled to two semi-infinite leads: the two are in fact related by a rotation of the spacetime coordinates.

Silvia Pappalardi, Laura Foini, Jorge Kurchan

SciPost Phys. 12, 130 (2022) · published 14 April 2022 | Toggle abstract · pdf

In recent years, there has been intense attention on the constraints posed by quantum mechanics on the dynamics of the correlation at low temperatures, triggered by the postulation and derivation of quantum bounds on the transport coefficients or on the chaos rate. However, the physical meaning and the mechanism enforcing such bounds is still an open question. Here, we discuss the quantum fluctuation-dissipation theorem (the KMS conditions) as the principle underlying bounds on correlation time scales. By restating the problem in a replicated space, we show that the quantum bound to chaos is a direct consequence of the KMS condition, as applied to a particular pair of two-time correlation and response functions. Encouraged by this, we describe how quantum fluctuation-dissipation relations act in general as a blurring of the time-dependence of correlations, which can imply bounds on their decay rates. Thinking in terms of fluctuation-dissipation opens a direct connection between bounds and other thermodynamic properties.

Laura Foini, Jorge Kurchan

SciPost Phys. 12, 080 (2022) · published 3 March 2022 | Toggle abstract · pdf

A disordered system is denominated 'annealed' when the interactions themselves may evolve and adjust their values to lower the free energy. The opposite ('quenched') situation when disorder is fixed, is the one relevant for physical spin-glasses, and has received vastly more attention. Other problems however are more natural in the annealed situation: in this work we discuss examples where annealed averages are interesting, in the context of matrix models. We first discuss how in practice, when system and disorder adapt together, annealed systems develop `planted' solutions spontaneously, as the ones found in the study of inference problems. In the second part, we study the probability distribution of elements of a matrix derived from a rotationally invariant (not necessarily Gaussian) ensemble, a problem that maps into the annealed average of a spin glass model.

Paola Ruggiero, Pasquale Calabrese, Laura Foini, Thierry Giamarchi

SciPost Phys. 11, 055 (2021) · published 13 September 2021 | Toggle abstract · pdf

We study the quantum quench in two coupled Tomonaga-Luttinger Liquids (TLLs), from the off-critical to the critical regime, relying on the conformal field theory approach and the known solutions for single TLLs. We consider a squeezed form of the initial state, whose low energy limit is fixed in a way to describe a massive and a massless mode, and we encode the non-equilibrium dynamics in a proper rescaling of the time. In this way, we compute several correlation functions, which at leading order factorize into multipoint functions evaluated at different times for the two modes. Depending on the observable, the contribution from the massive or from the massless mode can be the dominant one, giving rise to exponential or power-law decay in time, respectively. Our results find a direct application in all the quench problems where, in the scaling limit, there are two independent massless fields: these include the Hubbard model, the Gaudin-Yang gas, and tunnel-coupled tubes in cold atoms experiments.

Jacopo De Nardis, Miłosz Panfil, Andrea Gambassi, Leticia F. Cugliandolo, Robert Konik, Laura Foini

SciPost Phys. 3, 023 (2017) · published 27 September 2017 | Toggle abstract · pdf

Quantum integrable models display a rich variety of non-thermal excited states with unusual properties. The most common way to probe them is by performing a quantum quench, i.e., by letting a many-body initial state unitarily evolve with an integrable Hamiltonian. At late times, these systems are locally described by a generalized Gibbs ensemble with as many effective temperatures as their local conserved quantities. The experimental measurement of this macroscopic number of temperatures remains elusive. Here we show that they can be obtained by probing the dynamical structure factor of the system after the quench and by employing a generalized fluctuation-dissipation theorem that we provide. Our procedure allows us to completely reconstruct the stationary state of a quantum integrable system from state-of-the-art experimental observations.