Abstract We perform a detailed numerical study of the very peculiar thermodynamic properties of the localized high-energy phase of the discrete non-linear Schrödinger equation (DNLSE). A numerical sampling of the microcanonical ensemble done by means of Hamiltonian dynamics reveals a new and subtle relation between the presence of the localized phase and a property of the system that we have called classical entanglement . Our main finding is that a quantity defined for our classical system in perfect analogy with the entanglement entropy of quantum ones, and that we have therefore called S ent , grows with the system size N in the localized phase as S ent ( N ) ∼ log ( N ) , therefore revealing the presence of subtle non-local correlations between any finite portion of the system and the rest of it. This manifestation of classical entanglement beautifully captures the lack of system separability in the DNLSE localized phase, revealing how statistical correlations specific to the microcanonical ensemble and non-reproducible in the canonical one, may concur to determine a property totally analogous to the one produced by non-local quantum correlations.
When constrained to high energies, the wave function of the non-linear Schrodinger equation on a lattice becomes localized. We demonstrate here that localization occours as a first-order transition in the microcanonical ensemble, in a region of phase space where the canonical and the microcanonical ensembles are not equivalent. Quite remarkably, the transition to the localized non-ergodic phase takes place in a region where the microcanonical entropy,
SN, grows sub-extensively with the system size
N, namely
SN∼N1/3. This peculiar feature signals a shrinking of the phase space and exemplifies quite a typical scenario of broken ergodicity phases. In particular, the localized phase is characterized by a condensation phenomenon, where a finite fraction of the total energy is localized on a few lattice sites. The detailed study of
SN(E) close to the transition value of the energy,
Ec , reveals that its first-order derivative,
∂E∂SN(E)∣E=Ec, exhibits a jump, thus naturally accounting for a first-order phase transition from a thermalized phase to a localized one, which is properly assessed by studying the participation ratio of the site energy as an order parameter. Moreover, we show that
SN predicts the existence of genuine negative temperatures in the localized phase. In a physical perspective this indicates that by adding energy to the system it condensates into increasingly ordered states.
Abstract A unique characteristic of quantum mechanics is entanglement describing correlations between particles irrespective of their locations. This property, called non-locality, has no classical analogue. Over the past few years, quantum physicists have reached a consensus that we lack a physical theory to account for a class of states whose non-local character exceeds the bounds allowed by quantum mechanics. Motivated by our observation that an extension of the Schrödinger equation with non-linear terms is directly linked to a relaxation of Born’s rule, an axiom of quantum mechanics, we derive a physical theory that accounts for such hyper-correlated states and modifies Born’s rule. We model correlated particles with a generalized probability theory whose dynamics are described with a non-linear version of Schrödinger’s equation and demonstrate how that deviates from the standard formulation of quantum mechanics in experimental probability-prediction. We show also that the violation of the Clauser-Horn-Shimony-Holt inequality, the amount of non-locality, is proportional to the degree of non-linearity, which can be experimentally tested.
The fundamental question of how information spreads in closed quantum many-body systems is often addressed through the lens of the bipartite entanglement entropy, a quantity that describes correlations in a comprehensive (nonlocal) way. Among the most striking features of the entanglement entropy are its unbounded linear growth in the thermodynamic limit, its asymptotic extensivity in finite-size systems, and the possibility of measurement-induced phase transitions, all of which have no obvious classical counterpart. Here, we show how these key qualitative features emerge naturally also in classical information spreading, as long as one treats the classical many-body problem on par with the quantum one, that is, by explicitly accounting for the exponentially large classical probability distribution. Our analysis is supported by extensive numerics on prototypical cellular automata and Hamiltonian systems, for which we focus on the classical mutual information and also introduce a ``classical entanglement entropy.'' Our study sheds light on the nature of information spreading in classical and quantum systems, and opens avenues for quantum-inspired classical approaches across physics, information theory, and statistics.
In this work we discuss the essential quantum origin of correlation energy as measured through the wave function regardless of any extrinsic Hamiltonian. The ambiguous physical meaning of correlation energy is clarified by identifying the non-dynamical correlations inherent in the system state of Slater rank different than one with the quantum phenomenon of nonlocality. This is achieved by relating correlation energy to entanglement as measured through the von Neumann and the linear entropies. Indeed, for helium-like systems with varying Z we observe one-to-one correspondence between entanglement and correlation energy. We present numerical evidence of the linear relation between the correlation energy and the quantum entanglement for various members of the helium isoelectronic series by use of highly correlated wave functions of configuration-interaction type.
We show that the one-dimensional discrete nonlinear Schrödinger chain (DNLS) at finite temperature has three different dynamical regimes (ultralow-, low-, and high-temperature regimes). This has been established via (i) one-point macroscopic thermodynamic observables (temperature T, energy density ε, and the relationship between them), (ii) emergence and disappearance of an additional almost conserved quantity (total phase difference), and (iii) classical out-of-time-ordered correlators and related quantities (butterfly speed and Lyapunov exponents). The crossover temperatures T_{l-ul} (between low- and ultra-low-temperature regimes) and T_{h-l} (between the high- and low-temperature regimes) extracted from these three different approaches are consistent with each other. The analysis presented here is an important step forward toward the understanding of DNLS which is ubiquitous in many fields and has a nonseparable Hamiltonian form. Our work also shows that the different methods used here can serve as important tools to identify dynamical regimes in other interacting many-body systems.
During the evolution of coupled nonlinear oscillators on a lattice, with dynamics dictated by the discrete nonlinear Schrödinger equation (DNLSE systems), two quantities are conserved: system energy (Hamiltonian) and system density (number of particles). If the number of system oscillators is large enough, a significant portion of the array can be considered to be an "open system", in intimate energy and density contact with a "bath" - the rest of the array. Thus, as indicated in previous works, the grand canonical formulation can be exploited in order to determine equilibrium statistical properties of thermalized DNLSE systems. In this work, given the values of the two conserved quantities, we have calculated the necessary values of the two Lagrange parameters (typically designated
β,μ) associated with the grand canonical partition function in two different ways. One is numerical and the other is analytic, based on a published approximate entropy expression. In addition we have accessed a purposely-derived approximate PDF expression of site-densities. Applying these mathematical tools we have generated maps of temperatures, chemical potentials, and field correlations for DNLSE systems over the entire thermalization zone of the DNLSE phase diagram, subjected to all system-nonlinearity levels. The end result is a rather complete picture, characterizing equilibrated large DNLSE systems.