Mori Formalism Research Articles

Estimation of equilibration time scales from nested fraction approximations.

We consider an autocorrelation function of a quantum mechanical system through the lens of the so-called recursive method, by iteratively evaluating Lanczos coefficients or solving a system of coupled differential equationsin the Mori formalism. We first show that both methods are mathematically equivalent, each offering certain practical advantages. We then propose an approximation scheme to evaluate the autocorrelation function and use it to estimate the equilibration time τ for the observable in question. With only a handful of Lanczos coefficients as the input, this scheme yields an accurate order of magnitude estimate of τ, matching state-of-the-art numerical approaches. We develop a simple numerical scheme to estimate the precision of our method. We test our approach using several numerical examples exhibiting different relaxation dynamics. Our findings provide a practical way to quantify the equilibration time of isolated quantum systems, a question which is both crucial and notoriously difficult.

Quantum phase transition of many interacting spins coupled to a bosonic bath: Static and dynamical properties

By using worldline and diagrammatic quantum Monte Carlo techniques, matrix product state, and a variational approach \`a la Feynman, we investigate the equilibrium properties and relaxation features of a quantum system of $N$ spins antiferromagnetically interacting with each other, with strength $J$, and coupled to a common bath of bosonic oscillators, with strength $\ensuremath{\alpha}$. We show that, in the Ohmic regime, a Beretzinski-Thouless-Kosterlitz quantum phase transition occurs. While for $J=0$ the critical value of $\ensuremath{\alpha}$ decreases asymptotically with $1/N$ by increasing $N$, for nonvanishing $J$ it turns out to be practically independent on $N$, allowing to identify a finite range of values of $\ensuremath{\alpha}$ where spin phase coherence is preserved also for large $N$. Then, by using matrix product state simulations, and the Mori formalism and the variational approach \`a la Feynman jointly, we unveil the features of the relaxation, that, in particular, exhibits a nonmonotonic dependence on the temperature reminiscent of the Kondo effect. For the observed quantum phase transition we also establish a criterion analogous to that of the metal-insulator transition in solids.

Theoretical Study of Gilbert Damping Constants in Magnetic Multilayer Films

A theory is presented, describing Gilbert damping of a ferromagnetic layer by treating the outflow of the spin angular momentum to lattice and nonmagnetic systems on the same footing. The Mori formalism is applied to the entire system, including at the interface between different bulk (or film) systems. The Gilbert damping constant is determined from both Kamersky’s spin-orbit-related torque correlation (TC) term and the hopping-related TC term. In addition, we confirm that the hopping-related TC term corresponds to conventional Gilbert damping enhanced by spin pumping.

Approximate but accurate quantum dynamics from the Mori formalism. II. Equilibrium time correlation functions.

The ability to efficiently and accurately calculate equilibrium time correlation functions of many-body condensed phase quantum systems is one of the outstanding problems in theoretical chemistry. The Nakajima-Zwanzig-Mori formalism coupled to the self-consistent solution of the memory kernel has recently proven to be highly successful for the computation of nonequilibrium dynamical averages. Here, we extend this formalism to treat symmetrized equilibrium time correlation functions for the spin-boson model. Following the first paper in this series [A. Montoya-Castillo and D. R. Reichman, J. Chem. Phys. 144, 184104 (2016)], we use a Dyson-type expansion of the projected propagator to obtain a self-consistent solution for the memory kernel that requires only the calculation of normally evolved auxiliary kernels. We employ the approximate mean-field Ehrenfest method to demonstrate the feasibility of this approach. Via comparison with numerically exact results for the correlation function Czz(t)=Re⟨σz(0)σz(t)⟩, we show that the current scheme affords remarkable boosts in accuracy and efficiency over bare Ehrenfest dynamics. We further explore the sensitivity of the resulting dynamics to the choice of kernel closures and the accuracy of the initial canonical density operator.

Approximate but accurate quantum dynamics from the Mori formalism: I. Nonequilibrium dynamics.

We present a formalism that explicitly unifies the commonly used Nakajima-Zwanzig approach for reduced density matrix dynamics with the more versatile Mori theory in the context of nonequilibrium dynamics. Employing a Dyson-type expansion to circumvent the difficulty of projected dynamics, we obtain a self-consistent equation for the memory kernel which requires only knowledge of normally evolved auxiliary kernels. To illustrate the properties of the current approach, we focus on the spin-boson model and limit our attention to the use of a simple and inexpensive quasi-classical dynamics, given by the Ehrenfest method, for the calculation of the auxiliary kernels. For the first time, we provide a detailed analysis of the dependence of the properties of the memory kernels obtained via different projection operators, namely, the thermal (Redfield-type) and population based (NIBA-type) projection operators. We further elucidate the conditions that lead to short-lived memory kernels and the regions of parameter space to which this program is best suited. Via a thorough analysis of the different closures available for the auxiliary kernels and the convergence properties of the self-consistently extracted memory kernel, we identify the mechanisms whereby the current approach leads to a significant improvement over the direct usage of standard semi- and quasi-classical dynamics.

Convolutionless Nakajima-Zwanzig equations for stochastic analysis in nonlinear dynamical systems.

Determining the statistical properties of stochastic nonlinear systems is of major interest across many disciplines. Currently, there are no general efficient methods to deal with this challenging problem that involves high dimensionality, low regularity and random frequencies. We propose a framework for stochastic analysis in nonlinear dynamical systems based on goal-oriented probability density function (PDF) methods. The key idea stems from techniques of irreversible statistical mechanics, and it relies on deriving evolution equations for the PDF of quantities of interest, e.g. functionals of the solution to systems of stochastic ordinary and partial differential equations. Such quantities could be low-dimensional objects in infinite dimensional phase spaces. We develop the goal-oriented PDF method in the context of the time-convolutionless Nakajima-Zwanzig-Mori formalism. We address the question of approximation of reduced-order density equations by multi-level coarse graining, perturbation series and operator cumulant resummation. Numerical examples are presented for stochastic resonance and stochastic advection-reaction problems.

Markovian dissipative coarse grained molecular dynamics for a simple 2D graphene model

Based upon a finite-element "coarse-grained molecular dynamics" (CGMD) procedure, as applied to a simple atomistic 2D model of graphene, we formulate a new coarse-grained model for graphene mechanics explicitly accounting for dissipative effects. It is shown that, within the Mori-projection operator formalism, the reversible part of the dynamics is equivalent to the finite temperature CGMD-equations of motion, and that dissipative contributions to CGMD can also be included within the Mori formalism. The CGMD nodal momenta in the present graphene model display clear non-Markovian behavior, a property that can be ascribed to the fact that the CGMD-weighting function suppresses high-frequency modes more effectively than, e.g., a simple center of mass (COM) based CG procedure. The present coarse-grained graphene model is also shown to reproduce the short time behavior of the momentum correlation functions more accurately than COM-variables and it is less dissipative than COM-CG. Finally, we find that, while the intermediate time scale represented directly by the CGMD variables shows a clear non-Markovian dynamics, the macroscopic dynamics of normal modes can be approximated by a Markovian dissipation, with friction coefficients scaling like the square of the wave vector. This opens the way to the development of a CGMD model capable of describing the correct long time behavior of such macroscopic normal modes.

DISSIPATIVE PROCESSES IN LOW DENSITY STRONGLY INTERACTING 2D ELECTRON SYSTEMS

A glassy phase in disordered two dimensional (2D) electron systems may exist at low temperatures for electron densities lying intermediate between the Fermi liquid and Wigner crystal limits. The glassy phase is generated by the combined effects of disorder and the strong electron-electron correlations arising from the repulsive Coulomb interactions. Our approach here is motivated by the observation that at low electron densities the electron pair correlation function, as numerically determined for a non-disordered 2D system from Monte Carlo simulations, is very similar to the pair correlation function for a 2D classical system of hard discs. This suggests that theoretical approaches to 2D classical systems of hard discs may be of use in studying the disordered, low density electron problem. We use this picture to study its dynamics on the electron-liquid side of a glass transition. At long times the major relaxation process in the electron-liquid will be a rearrangement of increasingly large groups of the discs, rather than the movement of the discs separately. Such systems have been studied numerically and they display all the characteristics of glassy behaviour. There is a slowing down of the dynamics and a limiting value of the retarded spatial correlations. Motivated by the success of mode-coupling theories for hard spheres and discs in reproducing experimental results in classical fluids, we use the Mori formalism within a mode-coupling theory to obtain semi-quantitative insight into the role of electron correlations as they affect the time response of the weakly disordered 2D electron system at low densities.

Comparing the Mori formalism and the green function methods

Using a simple model described by a Hamiltonian of fermions coupled to bosons, we show that the relaxation function calculated via a low temperature approximation to the Mori memory function is similar, at least to lowest order, to the relaxation function calculated using a Green function formalism.

Derivation of memory function from its equation of motion

A new form of Memory Function (MF) appearing in the Mori's formalism has been derived using plausible approximations. In addition to the fact that present form of MF satisfies sum-rules upto sixth order, it has special characteristic of presence of one more adjustable parameter. It is also found that the present form of MF behaves as sech v (bt) under suitable conditions. The utility of the present MF is exemplified by studying time evolution of velocity auto correlation function and transport coefficients of Lennard–Jones fluids.

Electron paramagnetic resonance in weakly anisotropic Heisenberg magnets with a symmetric anisotropy

Motivated by the recent results on new oxides, we revise the theory of the electron paramagnetic resonance (EPR) in weakly anisotropic Heisenberg magnets with a symmetric (or dipole-dipole) anisotropy. In the framework of Mori's formalism, we calculate the EPR absorption and obtain a self-consistent perturbative expression for ${\ensuremath{\chi}}^{\mathrm{xx}}(\ensuremath{\omega})$ at finite temperatures. We show that generally the EPR signal is a mixture of the absorption and the dispersion. In the case where the self-energy can be estimated within the RPA decoupling scheme, expressions for the line shape, the linewidth, and the signal position are also obtained. In particular, we show that the position of the EPR signal is defined by three contributions: the anisotropy of the static susceptibility, the spin dynamics, and the asymmetric part of the line shape.

Hierarchy of dielectric relaxation times in water

This work presents a simple model, which allows us to obtain the relation between macroscopic and microscopic dielectric relaxation times in water on the basis of the Zwanzig–Mori formalism for the time correlation functions. In the model it is supposed that water consists of areas, so-called clusters, the size of which are defined by the dipoles spatial correlation radius. The macroscopic dielectric relaxation time of a sample is the relaxation time of the cluster polarization. The obtained relation between macroscopic and microscopic dielectric relaxation times contains the parameters depending on the cluster shape and structure. Such an approach is valid for the associated liquids having one macroscopic relaxation time only. As to systems with the wide distribution of relaxation times this approach demands generalization.

Autoregressive vibrational-dephasing analysis of the ν 2 band of liquid methyl iodide in nanoporous glass

The spontaneous Raman spectrum of the ν 2 methyl-deformation mode of liquid methyl iodide confined in 50-Å-diameter pores of silica sol–gel glass was measured at room temperature. The ν 2 band was broader for the microconfined liquid than for the bulk liquid. Vibrational correlation functions obtained by Fourier transforming the isotropic bands were modeled by using a memory-function procedure based on the Zwanzig–Mori formalism. The memory functions were analyzed by stochastic time series using an autoregressive model. Based on this analysis, the increase in the line width is largely attributed to inhomogeneous broadening due to site inhomogeneities at the pore surface.

Shear viscosity of expanded rubidium

The zeroth, second and fourth frequency sum rules of the transverse stress auto-correlation function of Rb have been evaluated for six thermodynamic states along the liquid - vapour co-existence curve by using the Ashcroft pseudo-potential and the corresponding pair correlation function obtained by molecular dynamics simulation. The numerical results for the sum rules thus obtained and a model for the memory function appearing in the Mori formalism are used to calculate the shear viscosity. The results obtained have been used to compare with the experimental results and with those of Kahl and Kambayashi. It is found that our method provides almost quantitative explanation for the density and temperature dependence of the shear viscosity of expanded rubidium.

Regime Interpretation of Anomalous Vortex Dynamics in 2D Superconductors

Low-frequency dynamic impedance $[{\ensuremath{\sigma}}^{\ensuremath{-}1}(\ensuremath{\omega},T)\ensuremath{\equiv}({\ensuremath{\sigma}}_{1}+i{\ensuremath{\sigma}}_{2}{)}^{\ensuremath{-}1}]$ measurements on Josephson junction arrays found that ${\ensuremath{\sigma}}_{1}\ensuremath{\sim}|\mathrm{ln}\ensuremath{\omega}|$, ${\ensuremath{\sigma}}_{2}\ensuremath{\sim}\mathrm{const}$. This implies anomalously sluggish vortex mobilities ${\ensuremath{\mu}}_{V}(\ensuremath{\omega})\ensuremath{\sim}{\ensuremath{\sigma}}_{1}^{\ensuremath{-}1}$, and is in conflict with general dynamical scaling expressions. We calculate (a) $\ensuremath{\sigma}(\ensuremath{\omega},T)$ by real-space vortex scaling and (b) ${\ensuremath{\mu}}_{V}(\ensuremath{\omega})$ using Mori's formalism for a screened Coulomb gas. We find, in addition to the usual critical (large-\ensuremath{\omega}) and hydrodynamic (low-\ensuremath{\omega}) regimes, a new intermediate-frequency scaling regime into which the experimental data fall. This resolves the above mentioned conflict and makes explicit predictions for the scaling form of $\ensuremath{\sigma}(\ensuremath{\omega},T)$.

Reversible diffusion-controlled reactions between two species in a fluid

A model suspension of diffusing spherical particles with an internal degree of freedom taking two discrete values, or equivalently a two-component suspension with reactions between species, is studied. The internal degrees of freedom of neighboring particles are coupled by an Ising type interaction of short range. The dynamics of the Ising variables is described by a master operator of Glauber form. On a macroscopic level small deviations of the two densities from their equilibrium averages obey transport equations with memory character. The memory kernel is derived from the Mori formalism. The two-body contribution to the cluster expansion of the memory function is obtained analytically at zero wavevector. The spectrum of the memory function shows a marked dependence on the presence or absence of an external field, the ratio of time scales for transitions between states and for translational diffusion, and on the antiferromagnetic or ferromagnetic character of the Ising interaction.

Longitudinal and bulk viscosities of Lennard-Jones fluids

Expressions for the longitudinal and bulk viscosities have been derived using Green Kubo formulae involving the time integral of the longitudinal and bulk stress autocorrelation functions. The time evolution of stress autocorrelation functions are determined using the Mori formalism and a memory function which is obtained from the Mori equation of motion. The memory function is of hyperbolic secant form and involves two parameters which are related to the microscopic sum rules of the respective autocorrelation function. We have derived expressions for the zeroth-, second-and fourth- order sum rules of the longitudinal and bulk stress autocorrelation functions. These involve static correlation functions up to four particles. The final expressions for these have been put in a form suitable for numerical calculations using low- order decoupling approximations. The numerical results have been obtained for the sum rules of longitudinal and bulk stress autocorrelation functions. These have been used to calculate the longitudinal and bulk viscosities and time evolution of the longitudinal stress autocorrelation function of the Lennard-Jones fluids over wide ranges of densities and temperatures. We have compared our results with the available computer simulation data and found reasonable agreement.

Analysis on Hierarchical Equations of Mori and Tri-Diagonal Recurrence Relations

A mathematical correspondence is established between a set of equations due to Mori formalism and those of generalized tri-diagonal recurrence relations. Thus, once a set of coupled equations describing physical phenomena are given, the corresponding Langevin equations can be obtained directly. This enables us to construct a hierarchical structure of physical variables and time scales

Generalized negative bulk viscosity in liquids

Expressions for the wave number dependent shear viscosity and longitudinal viscosity are obtained using generalized hydrodynamics and Mori formalism. The results obtained for shear viscosity and longitudinal viscosity have been used to calculate the wave number dependent bulk viscosity for the Lennard-Jones potential near its triple point. It has been found that the bulk viscosity becomes negative for wave numbers greater than 0.8 AA-1.

A recursive solution of Heisenberg's equation and its interpretation

We present the generalization of the recursion method of Haydock and co-workers to systems of many interacting particles. This new method has close similarities to the memory function or Mori formalism, but with some important differences. Heisenberg's equation for the time evolution of a microscopic operator is recursively transformed into a tridiagonal matrix equation. This equation resolves the operator into components corresponding to transitions of different energies. The projected spectrum of transitions has a continued fraction expansion given by the elements of the tridiagonal matrix. We show that for an appropriate choice of inner product this density of transitions obeys a generalization of the black body theorem of electromagnetism, in that it is exponentially insensitive to distant parts of the system. This implies that the projected density of transitions is computationally stable and can be calculated even in macroscopic many-body systems. We argue that the physical content of the density of transitions is determined by the nature of its singular points, such as discrete transitions, continuous spectrum, band edges and van Hove singularities.