We study the applicability of the formula that factors the trace of the diagonal part of spin operator products in the case of a relatively small number of particles of an isolated spin system. The validity of this formula for a large number of particles follows from the basic principles of quantum statistical mechanics. The spin system under consideration includes dipole–dipole interaction and the Zeeman interaction with an external magnetic field. We establish that the accuracy of this formula monotonically increases as the magnetic field increases. At the same time, the dependence on the number of particles in the range $$2\div10$$ for various configurations turns out to be sharply nonmonotone.
We study the applicability of the formula that factors the trace of the diagonal part of spin operator products in the case of a relatively small number of particles of an isolated spin system. The validity of this formula for a large number of particles follows from the basic principles of quantum statistical mechanics. The spin system under consideration includes dipole–dipole interaction and the Zeeman interaction with an external magnetic field. We establish that the accuracy of this formula monotonically increases as the magnetic field increases. At the same time, the dependence on the number of particles in the range 2÷10 for various configurations turns out to be sharply nonmonotone.
The development of methods of quantum statistical mechanics is considered in light of their applications to quantum solid-state theory. We discuss fundamental problems of the physics of magnetic materials and the methods of the quantum theory of magnetism, including the method of two-time temperature Green’s functions, which is widely used in various physical problems of many-particle systems with interaction. Quantum cooperative effects and quasi-particle dynamics in the basic microscopic models of quantum theory of magnetism: the Heisenberg model, the Hubbard model, the Anderson Model, and the spinfermion model are considered in the framework of novel self-consistent-field approximation. We present a comparative analysis of these models; in particular, we compare their applicability for description of complex magnetic materials. The concepts of broken symmetry, quantum protectorate, and quasi-averages are analyzed in the context of quantum theory of magnetism and theory of superconductivity. The notion of broken symmetry is presented within the nonequilibrium statistical operator approach developed by D.N. Zubarev. In the framework of the latter approach we discuss the derivation of kinetic equations for a system in a thermal bath. Finally, the results of investigation of the dynamic behavior of a particle in an environment, taking into account dissipative effects, are presented.
Abstract While detailed information about the semiclassics for single-particle systems is available, much less is known about the connection between quantum and classical dynamics for many-body systems. As an example, we focus on interacting spins, which are of considerable conceptual and practical importance. We derive a trace formula for coupled spin j particles which relates the quantum energy levels to the classical dynamics. Our derivation is valid in the limit j → ∞ with j ħ = c o n s t . and applies to time-continuous as well as to periodically driven dynamics. We provide a simple explanation why the Solari–Kochetov phase can be omitted if the correct classical Hamiltonian is chosen.
The evolution operator of the spin–orbit coupled system in an external magnetic field is factorized according to the operator equation:[Formula: see text]where HSO is the spin–orbit coupled Hamiltonian and HZ is the Zeeman Hamiltonian. The preceding equation is expressed in matrix form using the theory of the O(4) group representations. For the special case of [Formula: see text], an explicit form of the matrix C(t) is found. The operator corresponding to the matrix C(t) is found in a closed form using the projection operators constructed in the Appendix. Two special cases, corresponding to a weak and a strong external field, are considered, and possible applications of the results obtained are indicated.
Quantum statistical mechanics of spin systems is formulated by the use of the representation of atomic coherent state of spin, which is specified by a complex number. After introducing a newly defined distribution function of spin, the Bloch equation for the equilibrium density matrix, the von Neumann equation for the nonequilibrium density matrix and the Heisenberg equation of motion are converted into partial differential equations of the type of separating variables respectively. Formal solutions of them are presented. Any operator which is composed of spins can be converted uniquely into a differential operator. Thus we succeed in formulating equilibrium and nonequilibrium quantum statistical mechanics in the complex variable space, which differs from the previous one in the diagonal representation of density matrix. We further introduce the interaction representation for the perturbational expansion. The linear response formula is written down in the present representation.
The rather new field of quantum thermodynamics intends to derive thermodynamics from quantum mechanics, as well as to study thermodynamical behavior of quantum systems. It turns out that, indeed, thermalization can be obtained even for very small quantum systems due to the interaction with an appropriate environment. The consideration of an ensemble, as in statistical mechanics, is not necessary. However, the total system - consisting of system and environment - in quantum thermodynamics is usually considered to be closed. In particular, there exists no interaction with an external observer. Since quantum measurements typically influence the measured system, one may wonder which influence such an observation will have in the quantum thermodynamical setting. In this thesis, we study this influence of measurements in quantum thermodynamical systems. It turns out that various concepts known from statistical mechanics arise from such a "quantum thermodynamics under observation", although they do not have any meaning in the isolated quantum thermodynamical setting. The model discussed in the first part consists of a two-level system coupled to a large spin environment, the magnetization of which is measured periodically. This leads to quasi-classical trajectories for the system and - as in statistical mechanics - thermalization and temperature only have a meaning in terms of an ensemble average or a long-time average. Calculating these averages, we prove the ergodicity of the system. Interestingly, the entropy of the obtained attractor state depends on the time between the measurements and can be shown to be connected with the subjective lack of knowledge of the observer. In case of optimal measurements, the ensemble averaged state of the system under periodic measurements equals the state of the system in the quantum thermodynamical model without measurements. Introducing a detuning between system and environment, we find that the system can be prepared in almost arbitrary thermal states due to the periodic measurements. Even the creation of inversion becomes possible. As shown in numerical simulations, our results, which were analytically obtained for large environments, are even applicable to describe the behavior of the system coupled to small environments consisting of only a few spins. Besides measurements of the environment, we also study direct projective measurements of the system, as well as measurements with finite duration, obtaining that the dynamics of the system is almost independent of the concrete realization of the measurements. In the second part of the thesis, we discuss the meaning of temperature fluctuations in quantum thermodynamical systems. In statistical mechanics, there exist different approaches to the problem of temperature fluctuations, which lead to the well-known temperature fluctuation formula. The nature and operational accessibility of these fluctuations, however, remains rather unclear. We argue that according to quantum thermodynamics a system relaxes into a quasi-stationary thermal state and therefore the temperature does not fluctuate. However, since temperature is not an observable, it can only be determined indirectly. For modular systems, we show that the fluctuations of the temperature estimate obtained from a single energy measurement, indeed, are given by the temperature fluctuation formula derived in statistical mechanics, but only within a certain temperature range. The validity range of this formula increases with increasing size of the system. Thus, one can conclude that these "temperature fluctuations" can be interpreted as representing a fundamental limit for the instantaneous determination of temperature.