We conjecture that non-equilibrium boundary conditions generically trigger long range order in non-equilibrium steady states of locally interacting quantum chains. Our result is based on large scale density matrix renormalization group simulations of several models of quantum spin 1/2 chains which are driven far from equilibrium by coupling to a pair of unequal Lindblad reservoirs attached locally to the ends of the chain. In particular, we find a phase transition from exponentially decaying to long range spin-spin correlations in integrable Heisenberg XXZ chain by changing the anisotropy parameter. Long range order also typically emerges after breaking the integrability of the model.
Negative differential conductivity is reported for the far from equilibrium quantum spin transport in the insulating regime (J_x < J_z) of finite Heisenberg XXZ spin 1/2 chains. The phenomenon is reproduced analytically for small chains of N=4 spins and further analyzed numerically, for up to N=16, using an efficient pure-state simulation with stochastic spin baths.
Using quantization in the Fock space of operators we compute the non-equilibrium steady state in an open Heisenberg XY spin 1/2 chain of finite but large size coupled to Markovian baths at its ends. Numerical and theoretical evidence is given for a far from equilibrium quantum phase transition with spontaneous emergence of long-range order in spin-spin correlation functions, characterized by a transition from saturation to linear growth with the size of the entanglement entropy in operator space.
Time-dependent density matrix renormalization group method with a matrix product ansatz is employed for explicit computation of non-equilibrium steady state density operators of several integrable and non-integrable quantum spin chains, which are driven far from equilibrium by means of Markovian couplings to external baths at the two ends. It is argued that even though the time-evolution can not be simulated efficiently due to fast entanglement growth, the steady states in and out of equilibrium can be typically accurately approximated, so that chains of length of the order n ~ 100 are accessi…
We study the problem of adsorption and collapse transition of a linear polymer chain situated in a fractal container represented by a 4-simplex lattice and interacting with a surface adsorbed linear polymer chain. The adsorbed chain monomers act as pinning sites for the polymer chain. This problem has been solved exactly using real space renormalization group transformation. The resulting phase diagram and critical exponents are given.
The behavior of dynamical system interacting with non-equilibrium medium is investigated. Formally exact kinetic equations are derived for the statistical operator of the dynamical system and the macroscopic parameters of the medium. In the second order of perturbation with respect of interaction of the system with the medium, expression for the scattering integral is presented. The simplest applications of the obtained equations are considered.
We discuss that the thermodynamics of composite systems with non-additive entropies and additive energies can be equivalently derived considering additive entropies and non-additive energies. The general discussion is illustrated by a particular example.
In this note we study the approach to equilibrium of a chain of anharmonic oscillators. We find indications that a sufficiently large system always relaxes to the usual equilibrium distribution. There is no sign of an ergodicity threshold. The time however to arrive to equilibrium diverges when $g \to 0$, $g$ being the anharmonicity.
Long-range interacting Hamiltonian systems are believed to relax generically towards non-equilibrium states called "quasi-stationary" because they evolve towards thermodynamic equilibrium very slowly, on a time-scale diverging with particle number. We show here that, by applying a suitable perturbation operator for a finite time interval, we obtain, in a family of long-range systems, non-equilibrium states which appear to be strictly stationary. They exist even in the case of a harmonic potential, and are characterised by an ordered microscopic phase space structure. We give some simple heuris…
We show that, when a finite anisotropic Heisenberg spin-1/2 chain in the gapped regime is driven far from equilibrium, oppositely polarized ferromagnetic domains build up at the edges of the chain, thus suppressing quantum spin transport. As a consequence, a negative differential conductivity regime arises, where increasing the driving decreases the current. The above results are explained in terms of magnon localization and are shown to be structurally stable against breaking of integrability.
In Phys. Rev. Lett. 120, 200603 (2018), a segmented XXZ spin chain with zero anisotropy in one half and a large anisotropy on the other half gave rise to a spin current rectification which is perfect in the thermodynamic limit. Here we extend the previous study to segmented chains with interacting integrable as well as non-integrable halves, considering even cases in which no ballistic transport can emerge in either half. We demonstrate that, also in this more general case, it is possible to obtain giant rectification when the two interacting half chains are sufficiently different. We also sho…
The Motzkin spin chain is a spin-1 model introduced in \cite{shor} as an example of a system exhibiting a high degree of quantum fluctuations whose ground state can be mapped to Motzkin paths that are generated with local equivalence moves. This model is difficult to solve in general but keeping just the height preserving local equivalence moves we show that the model becomes integrable which when projected to certain subspaces of the full Hilbert space is isomorphic to the spin-$\frac{1}{2}$ XXX chain. In fact in the full Hilbert space the system is akin to two non-interacting spin-$\frac{1}{…
We investigate a simplified model of two fully connected magnetic systems maintained at different temperatures by virtue of being connected to two independent thermal baths while simultaneously being inter-connected with each other. Using generating functional analysis, commonly used in statistical mechanics, we find exactly soluble expressions for their individual magnetisations that define a two-dimensional non-linear map, the equations of which have the same form as those obtained for densely connected equilibrium systems. Steady states correspond to the fixed points of this map, separating…
An overarching action principle, the principle of minimal free action, exists for ergodic Markov chain dynamics. Using this principle and the Detailed Fluctuation Theorem, we construct a dynamic ensemble theory for non-equilibrium steady states (NESS) of Markov chains, which is in full analogy with equilibrium canonical ensemble theory. Concepts such as energy, free energy, Boltzmann macro-sates, entropy, and thermodynamic limit all have their dynamic counterparts. For reversible Markov chains, minimization of Boltzmann free action yields thermal equilibrium states, and hence provide a dynamic…
We characterise the equilibrium landscape, the entire manifold of local equilibrium states, of an interacting integrable quantum model. Focusing on the isotropic Heisenberg spin chain, we describe in full generality two complementary frameworks for addressing equilibrium ensembles: the functional integral Thermodynamic Bethe Ansatz approach, and the lattice regularisation transfer matrix approach. We demonstrate the equivalence between the two, and in doing so clarify several subtle features of generic equilibrium states. In particular we explain the breakdown of the canonical Y-system, which …
We derive an exact formula for the scaled cumulant generating function of the time-integrated current associated to an arbitrary ballistically transported conserved charge. Our results rely on the Euler-scale description of interacting, many-body, integrable models out of equilibrium given by the generalized hydrodynamics, and on the large deviation theory. Crucially, our findings extend previous studies by accounting for inhomogeneous and dynamical initial states in interacting systems. We present exact expressions for the first three cumulants of the time-integrated current. Considering the …
We study the effect of mergers in the force chain model describing the stress profile in static granular materials. Combining numerical and analytical calculations we show that granular materials do not generally behave in an elastic-like manner, however they may under specific conditions, which are elaborated. Non-elastic behavior resulting from the non-linearity of the full force chain model is discussed.
We argue that a particle language provides a conceptually simple framework for the description of anomalous equilibration in isolated quantum systems. We address this paradigm in the context of integrable models, which are those with particles that are stable against decay. In particular, we demonstrate that a complete description of equilibrium ensembles for interacting integrable models requires a formulation built from the mode occupation numbers of the underlying particle content, mirroring the case of non-interacting particles. This yields an intuitive physical interpretation of generaliz…
We use a Hamiltonian dynamics to discuss the statistical mechanics of long-lasting quasi-stationary states particularly relevant for long-range interacting systems. Despite the presence of an anomalous single-particle velocity distribution, we find that the Central Limit Theorem implies the Boltzmann expression in Gibbs' $Γ$-space. We identify the nonequilibrium sub-manifold of $Γ$-space characterizing the anomalous behavior and show that by restricting the Boltzmann-Gibbs approach to this sub-manifold we obtain the statistical mechanics of the quasi-stationary states.
After the rejection of their comment [arXiv:cond-mat/0609399v1] to our Phys. Rev. Lett. {\bf 97}, 100601 (2006), the Authors informed us that an extended version of their comment is going to be published in a different journal under the direct editorial responsibility of one of them. We then decided to make publicly available our formal reply, originally prepared for publication in Phys. Rev. Lett.
We consider a system of one-dimensional non-interacting fermions in external harmonic confinement. Using an efficient Green's function method we evaluate the exact profiles and the pair correlation function, showing a direct signature of the Fermi statistics and of the single quantum-level occupancy. We also study the dynamical properties of the gas, obtaining the spectrum both in the collisionless and in the collisional regime. Our results apply as well to describe a one-dimensional Bose gas with point-like hard-core interactions.
We study the transport process of interacting Brownian particles in a tube of varying cross section. To describe this process we introduce a modified Fick-Jacobs equation, considering particles that interact through a hard-core potential. We were able to solve the equation with numerical methods for the case of symmetric and asymmetric cavities. We focused in the concentration of particles along the direction of the tube. We also preformed Monte Carlo simulations to evaluate the accuracy of the results, obtaining good agreement between theory and simulations.
Entropy production is one of the most important characteristics of non-equilibrium steady states. We study here the steady-state entropy production, both at short times as well as in the long-time limit, of two important classes of non-equilibrium systems: transport systems and reaction-diffusion systems. The usefulness of the mean entropy production rate and of the large deviation function of the entropy production for characterizing non-equilibrium steady states of interacting many-body systems is discussed. We show that the large deviation function displays a kink-like feature at zero entro…
We consider two statistically independent systems described by the same entropy belonging to the two-parameter family of Sharma-Mittal. Assuming a weak interaction among the systems, allowing in this way an exchange of heat and work, we analyze, both in the entropy representation and in the energy representation, the evolution toward the equilibrium. The thermodynamics evolution is controlled by two scalar quantities identified with the temperature and the pressure of the system. The thermodynamical stability conditions of the equilibrium state are analyzed in both representations. Their relat…
We study the non-equilibrium time evolution of the average transverse magnetisation and end-to-end correlation functions of the random Ising quantum chain. Starting with fully magnetised states, either in the $x$ or $z$ direction, we compute numerically the average quantities. They show similar behaviour to the homogeneous chain, that is an algebraic decay in time toward a stationary state. During the time evolution, the spatial correlations, measured from one end to the other of the chain, are building up and finally at long time they reach a size-dependent constant depending on the distance …
We consider two quantum Ising chains initially prepared at thermal equilibrium but with different temperatures and coupled at a given time through one of their end points. In the long-time limit the system reaches a non-equilibrium steady state. We discuss properties of this non-equilibrium steady state, and characterize the convergence to the steady regime. We compute the mean energy flux through the chain and the large deviation function for the quantum and thermal fluctuations of this energy transfer.
The low temperature magnetization process of the ferromagnetic-antiferromagnetic Heisenberg chain is studied using the interacting boson approximation. In the low field regime and near the saturation field, the spin wave excitations are approximated by the $δ$ function boson gas for which the Bethe ansatz solution is available. The finite temperature properties are calculated by solving the integral equation numerically. The comparison is made with Monte Carlo calculation and the limit of the applicability of the present approximation is discussed.
We examine the long-term behaviour of non-integrable, energy-conserved, 1D systems of macroscopic grains interacting via a contact-only generalized Hertz potential and held between stationary walls. Existing dynamical studies showed the absence of energy equipartitioning in such systems, hence their long-term dynamics was described as quasi-equilibrium. Here we show that these systems do in fact reach thermal equilibrium at sufficiently long times, as indicated by the calculated heat capacity. This phase is described by equilibrium statistical mechanics, opening up the possibility that the mac…
A chain of kinetic equations for non-equilibrium one-particle, two-particle and $ s $-particle distribution functions of particles which take into account nonlinear hydrodynamic fluctuations is proposed. The method of Zubarev non-equilibrium statistical operator with projection is used. Nonlinear hydrodynamic fluctuations are described with non-equilibrium distribution function of collective variables that satisfies generalized Fokker-Planck equation. On the basis of the method of collective variables, a scheme of calculation of non-equilibrium structural distribution function of collective va…
In recent years statistical physicists have developed {\it discrete} "particle-hopping" models of vehicular traffic, usually formulated in terms of {\it cellular automata}, which are similar to the microscopic models of interacting charged particles in the presence of an external electric field. Concepts and techniques of non-equilibrium statistical mechanics are being used to understand the nature of the steady states and fluctuations in these so-called "microscopic" models. In this brief review we explain, primarily to the nonexperts, these models and the physical implications of the results…
We consider the exact solution of a model of correlated particles, which is presented as a system of interacting XX and Fateev-Zamolodchikov chains. This model can also be considered as a generalization of the multiband anisotropic $t-J$ model in the case we restrict the site occupations to at most two electrons. The exact solution is obtained for the eigenvalues and eigenvectors using the Bethe-ansatz method.
We investigate the non-equilibrium two-time correlation and response functions and the associated fluctuation-dissipation ratio for the ferromagnetic Ising chain with Glauber dynamics. The scaling behavior of these quantities at low temperature and large times is studied in detail. This analysis encompasses the self-similar domain-growth (aging) regime, the spatial and temporal Porod regimes, and the convergence toward equilibrium. The fluctuation-dissipation ratio admits a non-trivial limit value $X_\infty=1/2$ at zero temperature, and more generally in the aging regime.
We consider the XY spin chain with arbitrary time-dependent magnetic field and anisotropy. We argue that a certain subclass of Gaussian states, called Coherent Ensemble (CE) following [1], provides a natural and unified framework for out-of-equilibrium physics in this model. We show that $all$ correlation functions in the CE can be computed using form factor expansion and expressed in terms of Fredholm determinants. In particular, we present exact out-of-equilibrium expressions in the thermodynamic limit for the previously unknown order parameter one-point function, dynamical two-point functio…
Suitable Langevin thermostats are introduced which are able to control both the temperature and the chemical potential of a one-dimensional lattice of nonlinear Schrödinger oscillators. The resulting non-equilibrium stationary states are then investigated in two limit cases (low temperatures and large particle densities), where the dynamics can be mapped onto that of a coupled-rotor chain with an external torque. As a result, an effective kinetic definition of temperature can be introduced and compared with the general microcanonical (global) definition.
We study quantum chaos and spectral correlations in periodically driven (Floquet) fermionic chains with long-range two-particle interactions, in the presence and absence of particle number conservation ($U(1)$) symmetry. We analytically show that the spectral form factor precisely follows the prediction of random matrix theory in the regime of long chains, and for timescales that exceed the so-called Thouless/Ehrenfest time which scales with the size $L$ as ${\cal O}(L^2)$, or ${\cal O}(L^0)$, in the presence, or absence of $U(1)$ symmetry, respectively. Using random phase assumption which ess…
Continuum mechanics of entanglement in noisy interacting fermion chains
Rudolf Peierls Centre for Theoretical Physics, Clarendon Laboratory, Parks Road, Oxford OX1 3PU, UK（鲁道夫·皮尔斯理论物理中心，克拉伦登实验室）；Laboratoire de Physique de l’École Normale Supérieure, CNRS, ENS & Université PSL, Sorbonne Université, Université Paris Cité, 75005 Paris, France（巴黎高等师范学院物理实验室，法国国家科学研究中心，巴黎文理研究大学，索邦大学，巴黎西岱大学）
In this short note, we consider a system of two rotors, one of which interacts with a Langevin heat bath. We show that the system relaxes to its invariant measure (steady state) no faster than a stretched exponential $\exp(-c t^{1/2})$. This indicates that the exponent $1/2$ obtained earlier by the present authors and J.-P. Eckmann for short chains of rotors is optimal.
We examine the long-term behaviour of non-integrable, energy-conserved, 1D systems of macroscopic grains interacting via a contact-only generalized Hertz potential and held between stationary walls. We previously showed that in homogeneous configurations of such systems, energy is equipartitioned at sufficiently long times, thus these systems ultimately reach thermal equilibrium. Here we expand on our previous work to show that heterogeneous configurations of grains also reach thermal equilibrium at sufficiently long times, as indicated by the calculated heat capacity. We investigate the trans…
We study long-range interacting systems perturbed by external stochastic forces. Unlike the case of short-range systems, where stochastic forces usually act locally on each particle, here we consider perturbations by external stochastic fields. The system reaches stationary states where external forces balance dissipation on average. These states do not respect detailed balance and support non-vanishing fluxes of conserved quantities. We generalize the kinetic theory of isolated long-range systems to describe the dynamics of this non-equilibrium problem. The kinetic equation that we obtain app…
A Hamiltonian-based model of many harmonically interacting massive particles that are subject to linear friction and coupled to heat baths at different temperatures is used to study the dynamic approach to equilibrium and non-equilibrium stationary states. Based on the exactly calculated dynamic approach to the stationary distribution, the functional that governs this approach, which is called the free entropy, is constructed. For the stationary distribution the free entropy becomes maximal and its time derivative is minimal and vanishes. Thus, the free entropy characterizes equilibrium as wel…
Non-equilibrium steady states (NESS) describe particularly simple and stationary non-equilibrium situations. A possibility to obtain such states is to consider the asymptotic evolution of two infinite heat baths brought into thermal contact. In this work we generalise corresponding results of Doyon~et.~al. (J.\ Phys.\ A 18 (2015) no.9) for free Klein-Gordon fields in several directions. Our analysis is carried out directly at the level of correlation functions and in the algebraic approach to QFT. We discuss non-trivial chemical potentials, condensates, inhomogeneous linear models and homogene…
This paper discusses the elastic behavior of a very long crosslinked polyelectrolyte chain (Debye-Hückel chain), which is weakly charged. Therefore the response of the crosslinked chain (network) on an external constant force $f$ acting on the ends of the chain is considered. A selfconsistent variational computation of an effective field theory is employed. It is shown, that the modulus of the polyelectrolyte network has two parts: the first term represents the usual entropy elasticity of connected flexible chains and the second term takes into account the electrostatic interaction of the mono…
This work introduces a methodology for the statistical mechanical analysis of polymeric chains under tension controlled by optical or magnetic tweezers at thermal equilibrium with an embedding fluid medium. The response of single bonds between monomers or of entire groups of monomers to tension is governed by the activation of statistically interacting particles representing quanta of extension or contraction. This method of analysis is capable of describing thermal unbending of the freely jointed or wormlike chain kind, linear or nonlinear contour elasticity, and structural transformations in…
I explicitly construct a strong zero mode in the XYZ chain or, equivalently, Majorana wires coupled via a four-fermion interaction. The strong zero mode is an operator that pairs states in different symmetry sectors, resulting in identical spectra up to exponentially small finite-size corrections. Such pairing occurs in the Ising/Majorana fermion chain and possibly in parafermionic systems and strongly disordered many-body localized phases. The proof here shows that the strong zero mode occurs in a clean interacting system, and that it possesses some remarkable structure -- despite being a rat…
We conjecture exact and simple formulas for physical quantities in two quantum chains. A classic result of this type is Onsager, Kaufman and Yang's formula for the spontaneous magnetization in the Ising model, subsequently generalized to the chiral Potts models. We conjecture that analogous results occur in the XYZ chain when the couplings obey J_xJ_y + J_yJ_z + J_x J_z=0, and in a related fermion chain with strong interactions and supersymmetry. We find exact formulas for the magnetization and gap in the former, and the staggered density in the latter, by exploiting the fact that certain quan…
Using the density-matrix renormalization-group algorithm (DMRG) and a finite-size scaling analysis, we study the properties of the one-dimensional completely-anisotropic spin-1/2 XYZ model with Dzyaloshinsky-Moriya (DM) interactions. The model shows a rich phase diagram: depending on the value of the coupling constants, the system can display different kinds of ferromagnetic order and Luttinger-liquid behavior. Transitions from ferromagnetic to Luttinger-liquid phases are first order. We thoroughly discuss the transition between different ferromagnetic phases, which, in the absence of DM inter…
The relaxation dynamics of correlated electron transport (ET) along molecular chains is studied based on a substantially improved numerically exact path integral Monte Carlo (PIMC) approach. As archetypical model we consider a Hubbard chain containing two interacting electrons coupled to a bosonic bath. For this generalization of the ubiquitous spin-boson model, the intricate interdependence of correlations and dissipation leads to non-Boltzmann thermal equilibrium distributions for many-body states. By mapping the multi-particle dynamics onto an isomorphic single particle motion this phenomen…
We solve for the time-dependent finite-size scaling functions of the 1D transverse-field Ising chain during a linear-in-time ramp of the field through the quantum critical point. We then simulate Mott-insulating bosons in a tilted potential, an experimentally-studied system in the same equilibrium universality class, and demonstrate that universality holds for the dynamics as well. We find qualitatively athermal features of the scaling functions, such as negative spin correlations, and show that they should be robustly observable within present cold atom experiments.
The non-equilibrium dynamics of a gas of cold atoms in which Rydberg states are off-resonantly excited is studied in the presence of noise. The interplay between interaction and off-resonant excitation leads to an initial dynamics where aggregates of excited Rydberg atoms slowly nucleate and grow, eventually reaching long-lived meta-stable arrangements which then relax further on much longer timescales. This growth dynamics is governed by an effective Master equation which permits a transparent and largely analytical understanding of the underlying physics. By means of extensive numerical simu…
We obtain the large scale limit of the fluctuations around its hydrodynamic limit of the density of particles of a weakly asymmetric exclusion process in dimension up to three. The proof is based upon a sharp estimate on the relative entropy of the law of the process with respect to product reference measures associated to the hydrodynamic limit profile, which holds in any dimension and is of independent interest. As a corollary of this entropy estimate, we obtain some quantitative bounds on the speed of convergence of the aforementioned hydrodynamic limit.
We introduce and study a lattice fermion model in one dimension with explicit $\mathcal{N}=1$ supersymmetry (SUSY). The Hamiltonian of the model is defined by the square of a supercharge built from Majorana fermion operators. The model describes interacting Majorana fermions and its properties depend only on a single parameter $g$. When $g=1$, we find that SUSY is unbroken and the ground states are identical to those of the frustration-free Kitaev chains. We also find a parameter regime in which SUSY is restored in the infinite volume limit. For sufficiently large $g$, we prove that SUSY is sp…
Many experimentally relevant quantum spin chains are approximately integrable, and support long-lived quasiparticle excitations. A canonical example of integrable model of quantum magnetism is the XXZ spin chain, for which energy spreads ballistically, but, surprisingly, high-temperature spin transport can be diffusive or superdiffusive. We review the transport properties of this model using an intuitive quasiparticle picture that relies on the recently introduced framework of generalized hydrodynamics. We discuss how anomalous linear response properties emerge from hierarchies of quasiparticl…
We study the evolution in equilibrium of the fluctuations for the conserved quantities of a chain of anharmonic oscillators in the hyperbolic space-time scaling. Boundary conditions are determined by applying a constant tension at one side, while the position of the other side is kept fixed. The Hamiltonian dynamics is perturbed by random terms conservative of such quantities. We prove that these fluctuations evolve macroscopically following the linearized Euler equations with the corresponding boundary conditions, even in some time scales larger than the hyperbolic one.
We consider a model of heat conduction which consists of a finite nonlinear chain coupled to two heat reservoirs at different temperatures. We study the low temperature asymptotic behavior of the invariant measure. We show that, in this limit, the invariant measure is characterized by a variational principle. We relate the heat flow to the variational principle. The main technical ingredient is an extension of Freidlin-Wentzell theory to a class of degenerate diffusions.
