Quenched dynamics of classical isolated systems: the spherical spin model with two-body random interactions or the Neumann integrable model

Abstract

We study the Hamiltonian dynamics of the spherical spin model with fully-connected two-body interactions drawn from a zero-mean Gaussian probability distribution. In the statistical physics framework, the potential energy is of the so-called p = 2 spherical disordered kind, closely linked to the O(N ) scalar field theory. Most importantly for our setting, the energy conserving dynamics are equivalent to the ones of the Neumann integrable system. We take initial conditions from the Boltzmann equilibrium measure at a temperature that can be above or below the static phase transition, typical of a disordered (paramagnetic) or of an ordered (disguised ferromagnetic) equilibrium phase. We subsequently evolve the configurations with Newton dynamics dictated by a different Hamiltonian, obtained from an instantaneous global rescaling of the elements in the interaction random matrix. In the limit of infinitely many degrees of freedom, N → ∞, we identify three dynamical phases depending on the parameters that characterise the initial state and the final Hamiltonian. We obtain the global dynamical observables (energy density, self correlation function, linear response function, static susceptibility, etc.) with numerical and analytic methods and we show that, in most cases, they are out of thermal equilibrium. We note, however, that for shallow quenches from the condensed phase the dynamics are close to (though not at) thermal equilibrium à la Gibbs-Boltzmann. Surprisingly enough, in the N → ∞ limit and for a particular relation between parameters the global observables comply Gibbs-Boltzmann equilibrium. We next set the analysis of the system with finite number of degrees of freedom in terms of N non-linearly coupled modes. These are the projections of the vector spin (or particle's position on the sphere) on the eigenvectors of the interaction matrix, the most relevant being those linked to the eigenvalues at the edge of the spectrum. We argue that in a system with infinite size the modes decouple at long times. We evaluate the mode temperatures and we relate them to the frequency-dependent effective temperature measured with the fluctuation-dissipation relation in the frequency domain, similarly to what was recently proposed for quantum integrable cases. Finally, we analyse the N − 1 integrals of motion, notably, their scaling with N , and we use them to show that the system is out of equilibrium in all phases, even for parameters that show an apparent Gibbs-Boltzmann behaviour of global observables. We elaborate on the role played by these constants of motion in the post-quench dynamics and we briefly discuss the possible description of the asymptotic dynamics in terms of a Generalised Gibbs Ensemble.

Equilibration of quasi-integrable systems

“…Recently, an extensive study addressed GGEs in describing quantum integrable systems [26] (and references therein), for which, e.g., the response and correlation function provide a direct access to β k [27]. Analogous classical systems has been studied as well [28].…”

Universal late-time dynamics in isolated one-dimensional statistical systems with topological excitations

“…The subsequent coarsening dynamics, characterized by the formation of spatial domains of different ordered phases and diverging relaxation times 53,59 , has been shown to occur also for isolated Hamiltonian dynamics [60][61][62][63][64] and in certain cases it is accompanied by a novel behavior, with no counterpart in the presence of thermal baths [65][66][67][68] . Glassy features with similarities and differences from the ones found under dis-sipative dynamics 69 have also been exhibited in solvable models [70][71][72] .…”

Chaos in a classical limit of the Sachdev-Ye-Kitaev model

“…It has a paramagnetic phase for T > T sg and a spin glass phase for T < T sg , with the spinglass transition temperature given by T sg /J 0 = 1/ √ 2. It is important to emphasize, however, that the Poissonbracket dynamics we consider for this model is different from previously studied dynamical models involving pspin models 33,34 .…”