Many‐body Green's function theory for thin ferromagnetic films: exact treatment of the single‐ion anisotropy

A theory for the magnetization of ferromagnetic films is formulated within the framework of many‐body Green's function theory which considers all components of the magnetization. The model Hamiltonian includes a Heisenberg term, an external magnetic field, a second‐ and fourth‐order uniaxial single‐ion anisotropy, and the magnetic dipole‐dipole coupling. The single‐ion anisotropy terms can be treated exactlyby introducing higher‐order Green's functions and subsequently taking advantage of relations between products of spin operators which leads to an automatic closure of the hierarchy of the equations of motion for the Green's functions with respect to the anisotropy terms. This is an improvement on the method of our previous work, which treated the corresponding terms only approximately by decoupling them at the level of the lowest‐order Green's functions. RPA‐like approximations are used to decouple the exchange interaction terms in both the low‐order and higher‐order Green's functions. As a first numerical example we apply the theory to a monolayer for spin S = 1 in order to demonstrate the superiority of the present treatment of the anisotropy terms over the previous approximate decouplings.     

Microscopic description of heavy-ion collisions in terms of interacting RPA modes

Starting from the TDHF equation, the equations of motion for the single-particle density matrices refering to projectile and target are derived in an approximative way in a translating and rotating reference frame. An expansion of the fluctuating part of the transformed density in terms of generalized RPA modes in the ph as well as the pp(hh) channel leads to a system of non-linear coupled equations which simultaneously determine the time development of the collective coordinates and momenta of the relative motion, of the intrinsic angular momentum, and of the amplitudes of coherent surface excitations of the fragments. In a schematic calculation the formalism is applied to the angular momentum dissipation in the excitation of damped giant dipole modes.     

Second quantized reduced Bloch equations and the exact solutions for pairing Hamiltonian

In this article, we present a set of hierarchy Bloch equations for the reduced density operators in either canonical or grand canonical ensembles in the occupation number representation. They provide a convenient tool for studying the equilibrium quantum statistical mechanics for some model systems. As an example of their applications, we solve the equations for the model system with a pairing Hamiltonian. With the aid of its symplectic group symmetry, we obtain the statistical reduced density matrices with different orders. As a special instance for the solutions, we also get the reduced density matrices of the ground state for a superconductor.     

Hierarchy of equations for reduced density matrices in the case of thermodynamic equilibrium

A hierarchy of equations for s-particle density matrices at thermodynamic equilibrium is obtained, with the equation for the nonequilibrium density matrix as the starting point. When deducing the hierarchy the hypothesis of maximum statistical independence for the density matrices is used. The hierarchy obtained is an analogue of the classical equilibrium BBGKY hierarchy and goes over into it when . It is shown that thermodynamic quantities can be expressed in terms of functions that enter only into the first hierarchy equations. The hierarchy is analysed in detail in the case of a uniform fluid. As an example in which the equations can be solved easily enough, a hard-sphere system wherein triplet correlations are neglected is considered. Different approximations that can be used when solving the equations derived are discussed. Comparisons are made with the results of other theoretical treatments.     

Truncation of time-dependent many-body theories

We investigate truncation schemes for the many-body dynamics of finite fermion systems beyond the time-dependent Hartree-Fock (TDHF) approximation. One approach starts from the quantum Bogolyubov-Born-Green-Kirkwood-Yvon (BBGKY-)hierarchy of equations of motion for density matrices. It is shown that simple truncation of higher correlations within this scheme is inconsistent because essential exchange correlations are lost. As an alternative, we study the (extended) exp(S) or coupled-cluster formalism. It provides a wider range of applicability. But it also runs into problems with non-unitary propagation after truncation.     

Linear integral transformations and hierarchies of integrable nonlinear evolution equations

Integrable hierarchies of nonlinear evolution equations are investigated on the basis of linear integral equations. These are (Riemann-Hilbert type of) integral transformations which leave invariant an infinite sequence of ordinary differential matrix equations of increasing order in an (indefinite) parameter k. The potential matrices in these equations obey a set of nonlinear recursion relations, leading to a heirarchy of nonlinear partial differential equations. In decreasing order the same equations give rise to a “reciprocal” hierarchy, associated with Heisenberg ferromagnet type of equations.Central in the treatment is an embedding of the hierarchy into an infinite-matrix structure, which is constructed on the basis of the integral equations. In terms of this infinite-matrix structure the equations governing the hierarchies become quite simple. Furthermore, it leads in a straightforward way to various generalizations, such as to other types of linear spectral problems, multicomponent system and lattice equations. Generalizations to equations associated with noncommuting flows follow as a direct consequence of the treatment. Finally, some results on conserved densities and the Hamiltonian structure are briefly discussed.     

From sum rules to RPA: 1. Nuclei

We take up the flexible formulation of RPA in subspaces of coordinate-like and momentum-like 1 ph operators which was developed in the preceeding paper. We draw the connection to many of the known collective approximations to RPA by proper choice of the subspace of operators. We finally apply the newly developed scheme to comparative studies of a hierarchy of approaches for the RPA spectra of doubly magic nuclei described with Skyrme forces.     

Many-body Green's function theory for the magnetic reorientation of thin ferromagnetic films

The field-induced reorientation of the magnetization of ferromagnetic films is treated within the framework of many-body Green's function theory by considering all components of the magnetization. We present a new method for the calculation of expectation values in terms of the eigenvalues and eigenvectors of the equations of motion matrix for the set of Green's functions. This formulation allows a straightforward extension of the monolayer case to thin films with many layers and for arbitrary spin and moreover provides a practicable procedure for numerical computation. The model Hamiltonian includes a Heisenberg term, an external magnetic field, a second-order uniaxial single-ion anisotropy, and the magnetic dipole-dipole coupling. We utilize the Tyablikov (RPA) decoupling for the exchange interaction terms and the Anderson-Callen decoupling for the anisotropy terms. The dipole coupling is treated in the mean-field approximation, a procedure which we demonstrate to be a sufficiently good approximation for realistic coupling strengths. We apply the new method to monolayers with spin and to multilayer systems with S=1. We compare some of our results to those where mean-field theory (MFT) is applied to all interactions, pointing out some significant differences. Received 19 June 2000 and Received in final form 2 August 2000     

Equations of motion for a (non‐linear) scalar field model as derived from the field equations

The problem of derivation of the equations of motion from the field equations is considered. Einstein's field equations have a specific analytical form: They are linear in the second order derivatives and quadratic in the first order derivatives of the field variables. We utilize this particular form and propose a novel algorithm for the derivation of the equations of motion from the field equations. It is based on the condition of the balance between the singular terms of the field equation. We apply the algorithm to a non‐linear Lorentz invariant scalar field model. We show that it results in the Newton law of attraction between the singularities of the field moved on approximately geodesic curves. The algorithm is applicable to the N‐body problem of the Lorentz invariant field equations.     

Infinite dimensional symmetric spaces and Lax equations compatible with the infinite Toda chain

In this paper we present a natural embedding of the infinite Toda chain in a set of Lax equations in the algebra $LT$ consisting of $\mathbb{Z}\times \mathbb{Z}$-matrices that possess only a finite number of nonzero diagonals above the main central diagonal. This hierarchy of Lax equations describes the evolution of deformations of a set of commuting anti-symmetric matrices and corresponds to splitting this algebra into its anti-symmetric part and the subalgebra of matrices in $LT$ that have no component above the main diagonal. We show that the projections of these deformations satisfy a set of zero curvature relations, which demonstrates the compatibility of the system. Further we introduce a suitable $LT$-module in which we can distinguish elements, the so-called wave matrices, that will lead you to solutions of the hierarchy. We conclude by showing how wave matrices of the infinite Toda chain hierarchy can be constructed starting from an infinite dimensional symmetric space.     

Form Factors of Branch-Point Twist Fields in Quantum Integrable Models and Entanglement Entropy

In this paper we compute the leading correction to the bipartite entanglement entropy at large sub-system size, in integrable quantum field theories with diagonal scattering matrices. We find a remarkably universal result, depending only on the particle spectrum of the theory and not on the details of the scattering matrix. We employ the “replica trick” whereby the entropy is obtained as the derivative with respect to n of the trace of the nth power of the reduced density matrix of the sub-system, evaluated at n=1. The main novelty of our work is the introduction of a particular type of twist fields in quantum field theory that are naturally related to branch points in an n-sheeted Riemann surface. Their two-point function directly gives the scaling limit of the trace of the nth power of the reduced density matrix. Taking advantage of integrability, we use the expansion of this two-point function in terms of form factors of the twist fields, in order to evaluate it at large distances in the two-particle approximation. Although this is a well-known technique, the new geometry of the problem implies a modification of the form factor equations satisfied by standard local fields of integrable quantum field theory. We derive the new form factor equations and provide solutions, which we specialize both to the Ising and sinh-Gordon models.     

Classical and quantum kinetic equations with exact conservation laws

Stationary and dynamic properties of reduced density matrices can be determined from formal or approximate closures of an infinite hierarchy of equations. The local macroscopic conservation laws place weak but important constraints on the reduced density matrices which should be respected by any closure. For pairwise additive forces conditions on the closure of the one- and two-particle equations are obtained that preserve the exact functional dependence of the conserved densities and their fluxes on the reduced density matrices. To illustrate the nature of these conditions, a closure approximation suitable for a quantum gas is given, yielding an extension of the time-dependent Hartree-Fock equations for the dynamics of a nuclear fluid to include collisions.     

Quantum Theory of Solid State Plasma Dielectric Response

We review the quantum mechanical derivation of the random phase approximation (RPA) for solid state plasmas, starting from the Hamilton equations for canonically paired “second quantized” creation and annhilation field operators of interacting quantum many‐body systems. Discussing variational differentiation, the coupled equations of motion for the quantum field operators are derived. The concept of Green's functions is reviewed and interpreted, first for retarded Green's functions, and their equations of motion are developed from the equations of motion for the field operators. Thermodynamic Green's functions are discussed, and their periodicity/antiperiodicity properties in imaginary time are carefully examined with discussion of Matsubara Fourier series and representation in terms of a spectral weight function. The analytic continuation from imaginary time to real time is treated. Finally, we define nonequilibrium Green's functions and discuss the linearized timedependent Hartree approximation leading to the random phase approximation. An interesting application to the case of Graphene in a perpendicular magnetic field is discussed in detail, along with applications to normal systems, in terms of attendant phenomenology involving electron‐hole pair excitations and plasmons (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Emission spectra and intrinsic optical bistability in a two-level medium

Scattering of resonant radiation in a dense two-level medium is studied theoretically accounting for local field effects and renormalisation of the resonance frequency. Intrinsic optical bistability is viewed as switching between different spectral patterns of fluorescent light controlled by the incident field strength. Response spectra are calculated analytically for the entire hysteresis loop of atomic excitation. The equations to describe the non-linear interaction of an atomic ensemble with light are derived from the Bogolubov-Born-Green-Kirkwood-Yvon hierarchy for reduced single particle density matrices of atoms and quantised field modes and their correlation operators. The spectral power of scattered light with separated coherent and incoherent constituents is obtained straightforwardly within the hierarchy. The formula obtained for emission spectra can be used to distinguish between possible mechanisms suggested to produce intrinsic bistability in experiments.     

Dispersive optical bistability in a nonideal Fabry-Pérot cavity

A stability analysis is performed for optical bistability in a Fabry-Pérot cavity with mirrors of arbitrary transmission coefficient. The mixed absorptive and dispersive régime is covered. In order to describe the system we use the Maxwell0Bloch equations formulated in terms of slowly varying envelopes. Standing-wave effects are completely taken into account by refraining from a truncation of the harmonic expansions for the polarization and the inversion density. We represent the solutions of the linearized Bloch hierarchy in terms of Chebyshev polynomials depending on the stationary electric field envelopes. In this way, we reduce the stability problem to a four-dimensional set of linear differential equations. Together with a couple of boundary conditions these equations govern the spatial behaviour of the deviations of the forward and the backward electric field envelopes. Our final stability problem becomes much simpler in the uniform-field limit and in the adiabatic limit. If we choose the stationary backward electric field equal to zero we recover results that were derived earlier for the case of a ring cavity.     

First order formalism off(R) gravity

The first order formalism is applied to study the field equations of a general Lagrangian density for gravity of the form . These field equations correspond to theories which are a subclass of conformally metric theories in which the derivative of the metric is proportional to the metric by a Weyl vector field. The resulting geometrical structure is unique, except whenf(R)=aR ², in the sense that the Weyl field is identifiable in terms of the trace of the energy-momentum tensor and its derivatives. In the casef(R)=aR ² the metric is only defined up to a conformai factor. We discuss the matter conservation equations which are implied by the invariance of the theories under diffeomorphisms. We apply the results to the case of dust and obtain that in general the dust particles will not follow geodesic Unes. We consider the linearized field equations and apply them to obtain the weak field slow motion limit. It is found that the gravitational potential acquires a new term which depends linearly on the mass density. The importance of these new equations is briefly discussed.     

Renormalized Kinetic Theory of Classical Fluids in and out of Equilibrium

We present a theory for the construction of renormalized kinetic equations to describe the dynamics of classical systems of particles in or out of equilibrium. A closed, self-consistent set of evolution equations is derived for the single-particle phase-space distribution function f, the correlation function C=〈δfδf〉, the retarded and advanced density response functions χ ^R,A =δf/δφ to an external potential φ, and the associated memory functions Σ ^R,A,C . The basis of the theory is an effective action functional Ω of external potentials φ that contains all information about the dynamical properties of the system. In particular, its functional derivatives generate successively the single-particle phase-space density f and all the correlation and density response functions, which are coupled through an infinite hierarchy of evolution equations. Traditional renormalization techniques (involving Legendre transform and vertex functions) are then used to perform the closure of the hierarchy through memory functions. The latter satisfy functional equations that can be used to devise systematic approximations that automatically imply the conservation laws of mass, momentum and energy. The present formulation can be equally regarded as (i) a generalization to dynamical problems of the density functional theory of fluids in equilibrium and (ii) as the classical mechanical counterpart of the theory of non-equilibrium Green’s functions in quantum field theory. It unifies and encompasses previous results for classical Hamiltonian systems with any initial conditions. For equilibrium states, the theory reduces to the equilibrium memory function approach used in the kinetic theory of fluids in thermal equilibrium. For non-equilibrium fluids, popular closures of the BBGKY hierarchy (e.g. Landau, Boltzmann, Lenard-Balescu-Guernsey) are simply recovered and we discuss the correspondence with the seminal approaches of Martin-Siggia-Rose and of Rose and we discuss the correspondence with the seminal approaches of Martin-Siggia-Rose and of Rose.     

Discrete stochastic evolution rules and continuum evolution equations

The continuum evolution equations are derived from updating rules for three classes of stochastic models. The first class corresponds to models whose stochastic continuum equations are of the Langevin type obtained after carrying out a “local average” known as coarse-graining. The second class consists of a hierarchy of continuum equations for the correlations of the dynamical variables obtained after making an average over realizations. This average generates a hierarchy of deterministic partial differential equations except when the dynamical variables do not depend on the values of the neighboring dynamical variables, in which case a hierarchy of ordinary differential equations is obtained. The third class of evolution equations for the correlations of the dynamical variable constitutes another hierarchy after calculating an average over both realizations and all the sites of the lattice. This double average generates a hierarchy of deterministic ordinary differential equations. The second and third classes of equations are truncated using a mean field (m,n)-closure approximation in order to obtain a finite set of equations. Illustrative examples of every class are given.