Harmonic Chain Research Articles

Stationary state of harmonic chains driven by boundary resetting

We study the nonequilibrium steady state of an ordered harmonic chain of N oscillators connected to two walls which undergo diffusive motion with stochastic resetting. The intermittent resetting of the walls effectively emulates two nonequilibrium reservoirs that exert temporally correlated forces on the boundary oscillators. These reservoirs are characterized by the diffusion constants and resetting rates of the walls. We find that, for any finite N, the velocity distribution of the bulk oscillators remains non-Gaussian, as evidenced by a non-zero bulk kurtosis that decays . We calculate the spatio-temporal correlation of the velocity of the oscillators both analytically as well as using numerical simulation. The signature of the boundary resetting is present at the bulk in terms of the two-time velocity correlation of a single oscillator and the equal-time spatial velocity correlation. For the resetting driven chain, the two-time velocity correlation decay as at the large time, and there exists a non-zero equal-time spatial velocity correlation when . A non-zero average energy current will flow through the system when the boundary walls reset to their initial positions at different rates. This average energy current can be computed exactly in the thermodynamic limit. Numerically we show that the distribution of the instantaneous energy current at the boundary is independent of the system size. However, the distribution of the instantaneous energy current in the bulk approaches a stationary distribution in the thermodynamic limit.

Derivation of Anomalous Behavior from Interacting Oscillators in the High-Temperature Regime

AbstractA microscopic model of interacting oscillators, which admits two conserved quantities, volume, and energy, is investigated. We begin with a system driven by a general nonlinear potential under high-temperature regime by taking the inverse temperature of the system asymptotically small. As a consequence, one can extract a principal part (by a simple Taylor expansion argument), which is driven by the harmonic potential, and we show that previous results for the harmonic chain are covered with generality. We consider two fluctuation fields, which are defined as a linear combination of the fluctuation fields of the two conserved quantities, volume, and energy, and we show that the fluctuations of one field converge to a solution of an additive stochastic heat equation, which corresponds to the Ornstein–Uhlenbeck process, in a weak asymmetric regime, or to a solution of the stochastic Burgers equation, in a stronger asymmetric regime. On the other hand, the fluctuations of the other field cross from an additive stochastic heat equation to a fractional diffusion equation given by a skewed Lévy process.

Nonequilibrium steady states of long-range coupled harmonic chains.

We perform a numerical study of transport properties of a one-dimensional chain with couplings decaying as an inverse power r^{-(1+σ)} of the intersite distance r and open boundary conditions, interacting with two heat reservoirs. Despite its simplicity, the model displays highly nontrivial features in the strong long-range regime -1<σ<0. At weak coupling with the reservoirs, the energy flux departs from the predictions of perturbative theory and displays anomalous superdiffusive scaling of the heat current with the chain size. We trace this behavior back to the transmission spectrum of the chain, which displays a self-similar structure with a characteristic σ-dependent fractal dimension.

Effects of infinite-range interaction on the localization properties of one-dimensional disordered harmonic chains

Effects of infinite-range interaction on the localization properties of one-dimensional disordered harmonic chains

Acoustic transparency of the chain-chain interface.

We study propagation of wave packets through the interface between two dissimilar harmonic chains with on-site potentials (e.g., chains lying on elastic foundations). An expression for the transmission coefficient, relating energies of the incident and transmitted wave packets is derived using two different approaches. Without elastic foundation, the transmission coefficient monotonically decreases with increasing wave frequency. We show that by adding elastic foundations, one can qualitatively change this dependence and make it nonmonotonic or even increasing. Moreover, in some cases, the interface is totally transparent (the transmission coefficient is equal to unity at some frequency) if at least one of the chains has the elastic foundation. Presented results may serve for manipulation of the transmission coefficient and corresponding interfacial thermal resistance in low-dimensional nanosystems.

Statistical field theory for nonlinear elasticity of polymer networks with excluded volume interactions.

Polymer networks formed by cross linking flexible polymer chains are ubiquitous in many natural and synthetic soft-matter systems. Current micromechanics models generally do not account for excluded volume interactions except, for instance, through imposing a phenomenological incompressibility constraint at the continuum scale. This work aims to examine the role of excluded volume interactions on the mechanical response. The approach is based on the framework of the self-consistent statistical field theory of polymers, which provides an efficient mesoscale approach that enables the accounting of excluded volume effects without the expense of large-scale molecular modeling. A mesoscale representative volume element is populated with multiple interacting chains, and the macroscale nonlinear elastic deformation is imposed by mapping the end-to-end vectors of the chains by this deformation. In the absence of excluded volume interactions, it recovers the closed-form results of the classical theory of rubber elasticity. With excluded volume interactions, the model is solved numerically in three dimensions using a finite element method to obtain the energy, stresses, and linearized moduli under imposed macroscale deformation. Highlights of the numerical study include: (i) the linearized Poisson's ratio is very close to the incompressible limit without a phenomenological imposition of incompressibility; (ii) despite the harmonic Gaussian chain as a starting point, there is an emergent strain-softening and strain-stiffening response that is characteristic of real polymer networks, driven by the interplay between the entropy and the excluded volume interactions; and (iii) the emergence of a deformation-sensitive localization instability at large excluded volumes.

Quantum propagator for a general time-dependent quadratic Hamiltonian: Application to interacting oscillators in external fields

In this paper, we find the quantum propagator for a general time-dependent quadratic Hamiltonian. The method is based on the properties of the propagator and the fact that the quantum propagator fulfills two independent partial differential equations originating from Heisenberg equations for positions and momenta. As an application of the method, we find the quantum propagator for a linear chain of interacting oscillators for both periodic and Dirichlet boundary conditions. The state and excitation propagation along the harmonic chain in the absence and presence of an external classical source is studied and discussed. The location of the first maxima of the probability amplitude P(n, τ) is a straight line in the (n, τ)-plane, indicating a constant speed of excitation propagation along the chain.

Evolution of circuit complexity in a harmonic chain under multiple quenches

We study Nielsen’s circuit complexity in a periodic harmonic oscillator chain, under single and multiple quenches. This simple system is amenable to analytical computations and yet offers considerable insight into the dynamics of quantum information. For a single quench scenario, we highlight some important differences between our results that explicitly use the wavefunction, as compared to the recently proposed covariance matrix method and point out the consequences. In a multiple quench scenario, the complexity shows remarkably different behaviour compared to the other information theoretic measures, such as the entanglement entropy and the out of time ordered correlator. In particular, the latter is known to show signs of chaos in this integrable system, but our results indicate the contrary. We further show the presence of a ‘residual complexity,’ i.e. after two successive quenches, when the frequency returns to its initial value, the complexity has a non-zero lower limit. Further, applying a large number of successive quenches, the complexity of the time evolved state can be increased to a high value, which is not possible by applying a single quench. Finally, we show that this simplistic model exhibits the interesting phenomenon of ‘complexity crossover’ between two successive quenches performed at different times.

Heat Flow in a Periodically Forced, Thermostatted Chain II

We derive a macroscopic heat equation for the temperature of a pinned harmonic chain subject to a periodic force at its right side and in contact with a heat bath at its left side. The microscopic dynamics in the bulk is given by the Hamiltonian equation of motion plus a reversal of the velocity of a particle occurring independently for each particle at exponential times, with rate gamma . The latter produces a finite heat conductivity. Starting with an initial probability distribution for a chain of n particles we compute the current and the local temperature given by the expected value of the local energy. Scaling space and time diffusively yields, in the nrightarrow +infty limit, the heat equation for the macroscopic temperature profile T(t, u), t>0, u in [0,1]. It is to be solved for initial conditions T(0, u) and specified T(t,0)=T_-, the temperature of the left heat reservoir and a fixed heat flux J, entering the system at u=1. |J| equals the work done by the periodic force which is computed explicitly for each n.

Multidimensional entropic bound: Estimator of entropy production for Langevin dynamics with an arbitrary time-dependent protocol

Entropy production (EP) is a key quantity in thermodynamics, and yet measuring EP has remained a challenging task. Here we introduce an EP estimator, called multidimensional entropic bound (MEB), utilizing an ensemble of trajectories. The MEB can accurately estimate the EP of overdamped Langevin systems with an arbitrary time-dependent protocol. Moreover, it provides a unified platform to accurately estimate the EP of underdamped Langevin systems under certain conditions. In addition, the MEB is computationally efficient because optimization is unnecessary. We apply our developed estimator to three physical systems driven by time-dependent protocols pertaining to experiments using optical tweezers: A dragged Brownian particle, a pulling process of a harmonic chain, and an unfolding process of an RNA hairpin. Numerical simulations confirm the validity and efficiency of our method.

Harmonic representatives in homology over arbitrary fields

We introduce a notion of harmonic chain for chain complexes over fields of positive characteristic. A list of conditions for when a Hodge decomposition theorem holds in this setting is given and we apply this theory to finite CW complexes. An explicit construction of the harmonic chain within a homology class is described when applicable. We show how the coefficients of usual discrete harmonic chains due to Eckmann can be reduced to localizations of the integers, allowing us to compare classical harmonicity with the notion introduced here. We focus on applications throughout, including CW decompositions of orientable surfaces and examples of spaces arising from sampled data sets.

Localization properties of harmonic chains with correlated mass and spring disorder: Analytical approach.

We study the localization properties of normal modes in harmonic chains with mass and spring weak disorder. Using a perturbative approach, an expression for the localization length L_{loc} is obtained, which is valid for arbitrary correlations of the disorder (mass disorder correlations, spring disorder correlations, and mass-spring disorder correlations are allowed), and for practically the whole frequency band. In addition, we show how to generate effective mobility edges by the use of disorder with long range self-correlations and cross-correlations. The transport of phonons is also analyzed, showing effective transparent windows that can be manipulated through the disorder correlations even for relative short chain sizes. These results are connected to the problem of heat conduction in the harmonic chain; indeed, we discuss the size scaling of the thermal conductivity from the perturbative expression of L_{loc}. Our results may have applications in modulating thermal transport, particularly in the design of thermal filters or in manufacturing high-thermal-conductivity materials.

Heat Flow in a Periodically Forced, Thermostatted Chain

We investigate the properties of a harmonic chain in contact with a thermal bath at one end and subjected, at its other end, to a periodic force. The particles also undergo a random velocity reversal action, which results in a finite heat conductivity of the system. We prove the approach of the system to a time periodic state and compute the heat current, equal to the time averaged work done on the system, in that state. This work approaches a finite positive value as the length of the chain increases. Rescaling space, the strength and/or the period of the force leads to a macroscopic temperature profile corresponding to the stationary solution of a continuum heat equation with Dirichlet-Neumann boundary conditions.

Optical phonons of GeSbTe alloys: Influence of structural disorder

Phase-change materials such as pseudobinary GeSbTe (GST) alloys are widely used for fabrication of rewritable and nonvolatile memory devices and also reveal perspective thermoelectric parameters. Their electron spectrum demonstrates the topological insulator behavior. These promising properties are related to unique structural features of a lattice, study of which attracts much attention since full clarity was not achieved so far. Here, we study Raman spectra of monocrystalline GST alloys to examine influence of lattice structure on phonon spectrum. In the Raman spectra, the broad bands typical for the amorphous state and sharp lines of the crystalline one are observed. The use of a one-dimensional harmonic chain model of atoms composing the lattice unit cell allows us to explain the origin of these features. It is shown that the first component is related to dispersion of phonon frequencies in the different lattice unit cells, which is caused by Ge/Sb disorder. The sharp spectral lines are attributed to phonons originated from those of Sb2Te3 as the structural basis of GST. Their frequencies are invariants relative to Ge/Sb disorder, and GST alloy is an ordered crystal for these modes.

One-dimensional harmonic chain model of vibration-mode matching in solid-liquid interfacial thermal transport.

Understanding the atomistic mechanism of interfacial thermal transport at solid-liquid interfaces is a key challenge in thermal management at the nanoscale. A recent molecular-dynamics study demonstrated that interfacial thermal resistance (ITR) at the interface between a solid and a surfactant solution can be minimized by adjusting the molecular mass of the surfactant. In the present study, we explain the mechanism of this ITR minimization in view of vibration-mode matching using a one-dimensional (1D) harmonic chain model of a solid-liquid interface having an interfacial adsorption layer of surfactant molecules. The equation of motion for the 1D chain is described by a classical Langevin equation and is analytically solved by the nonequilibrium Green's function (NEGF) method. The resultant ITR is expressed in a form of vibrational matching, and its relationship to the overlap of the vibrational density of states is also discussed. The analysis leads to a conclusion that the damping coefficient η in the Langevin equation should be a finite and sufficiently large value to represent the rapid damping of vibration modes at solid-liquid interfaces. This conclusion provides a clue to seamlessly extend the conventional NEGF-phonon transmission picture of solid-solid interfacial thermal transport, which assumes η to be infinitesimal, to solid-liquid interfaces.

Stationary states of activity-driven harmonic chains.

We study the stationary state of a chain of harmonic oscillators driven by two active reservoirs at the two ends. These reservoirs exert correlated stochastic forces on the boundary oscillators which eventually leads to a nonequilibrium stationary state of the system. We consider three most well-known dynamics for the active force, namely, the active Ornstein-Uhlenbeck process, run-and-tumble process, and active Brownian process, all of which have exponentially decaying two-point temporal correlations but very different higher-order fluctuations. We show that, irrespective of the specific dynamics of the drive, the stationary velocity fluctuations are Gaussian in nature with a kinetic temperature which remains uniform in the bulk. Moreover, we find the emergence of an "equipartition of energy" in the bulk of the system-the bulk kinetic temperature equals the bulk potential temperature in the thermodynamic limit. We also calculate the stationary distribution of the instantaneous energy current in the bulk which always shows a logarithmic divergence near the origin and asymmetric exponential tails. The signatures of specific active driving become visible in the behavior of the oscillators near the boundary. This is most prominent for the RTP- and ABP-driven chains where the boundary velocity distributions become non-Gaussian and the current distribution has a finite cutoff.

Topological adiabatic dynamics in classical mass-spring chains with clamps

The path dependence of adiabatic evolution in classical harmonic chains with clamps is examined. It is shown that cutting and joining a chain may braid adiabatic normal mode frequencies. Accordingly, different adiabatic paths with the same endpoints may transport a normal mode to a different one, and an adiabatic cycle pumps action variables, i.e., the adiabatic invariants of integrable classical systems. Another adiabatic pump for artificial edge modes induced by clamps is shown as an application. Extensions to completely integrable systems and quantum systems are outlined.

Symmetry resolved entanglement of excited states in quantum field theory. Part II. Numerics, interacting theories and higher dimensions

In a recent paper we studied the entanglement content of zero-density excited states in complex free quantum field theories, focusing on the symmetry resolved entanglement entropy (SREE). By zero-density states we mean states consisting of a fixed, finite number of excitations above the ground state in an infinite-volume system. The SREE is defined for theories that possess an internal symmetry and provides a measure of the contribution to the total entanglement of each symmetry sector. In our work, we showed that the ratio of Fourier-transforms of the SREEs (i.e. the ratio of charged moments) takes a very simple and universal form for these states, which depends only on the number, statistics and symmetry charge of the excitations as well as the relative size of the entanglement region with respect to the whole system’s size. In this paper we provide numerical evidence for our formulae by computing functions of the charged moments in two free lattice theories: a 1D Fermi gas and a complex harmonic chain. We also extend our results in two directions: by showing that they apply also to excited states of interacting theories (i.e. magnon states) and by developing a higher dimensional generalisation of the branch point twist field picture, leading to results in (interacting) higher-dimensional models.

Rényi entropy and negativity for massless complex boson at conformal interfaces and junctions

We consider the ground state of a theory composed by M species of massless complex bosons in one dimension coupled together via a conformal interface. We compute both the Rényi entropy and the negativity of a generic partition of wires, generalizing the approach developed in a recent work for free fermions. These entanglement measures show a logarithmic growth with the system size, and the universal prefactor depends both on the details of the interface and the bipartition. We test our analytical predictions against exact numerical results for the harmonic chain.

Derivation of Euler equations from quantum and classical microscopic dynamics

We derive Euler equations from a Hamiltonian microscopic dynamics. The microscopic system is a one-dimensional disordered harmonic chain, and the dynamics is either quantum or classical. This chain is an Anderson insulator with a symmetry protected mode: thermal fluctuations are frozen while the low modes ensure the transport of elongation, momentum and mechanical energy, that evolve according to Euler equations in an hyperbolic scaling limit. In this paper, we strengthen considerably the results in Bernardin et al (2019 Commun. Math. Phys. 365 215–37); Hannani (2022 Commun. Math. Phys. 390 349–23), where we established a limit in mean starting from a local Gibbs state: we now control the second moment of the fluctuations around the average, yielding a limit in probability, and we enlarge the class of admissible initial states.