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Journal of Sichuan Normal University

The Correlation Mode and Its Graphical Representation in Quantum Statistical Correlation Dynamics

Jun Yan

Published Jan 1, 2003• 0 citations
OpenAlex

Abstract

The correlation mode and its graphical representation in quantum statistical correlation dynamics are investigated in this paper. At first, the quantum correlation dynamics equations are given, then the matrices of correlation mode are calculated. These matrix elements are relative to creation, annihilation and change of correlation mode. Moreover, the twobody correlation dynamics is discussed, the possible application of this method for transport process in high temperature superconductive physics is also proposed.

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The correlation mode and its graphical representation in quantum statistical correlation dynamics are investigated in this paper. At first, the quantum correlation dynamics equations are given, then the matrices of correlation mode are calculated. These matrix elements are relative to creation, annihilation and change of correlation mode. Moreover, the twobody correlation dynamics is discussed, the possible application of this method for transport process in high temperature superconductive physics is also proposed.
The survey discusses the problem of measuring correlations in composite quantum and statistical systems. After briefly recalling the standard methods of describing correlations, the emphasis is on the method of correlation indices. The latter method for quantifying the strength of correlations in composite physical systems is sufficiently general allowing for measuring in a unique way the correlation strength in both quantum and statistical systems, for static as well as for dynamic processes. The correlation indices, can be defined for correlation operators, reduced density matrices, and other operators containing information on correlations in the studied composite systems. The correlation indices measure all types of correlations, quantum correlations, such as entanglement, as well as classical correlations. They are applicable for characterizing static as well as dynamical processes. Examples are given of correlation indices for several quantum states and for spin and pseudospin systems. The quantification of correlations in a nonequilibrium system is exemplified by calculating the correlation index for a trapped Bose-Einstein condensate subject to the action of an alternating field.
A new numerical approach to time correlation functions in the Wigner representation of quantum statistical mechanics has been developed. The time correlation functions have been presented in the form of the integral of the Weil symbol of operators and the Fourier transform of the product of matrix elements of the dynamic propagators. For the last function the integral Wigner-Liouville type equation has been derived. The initial condition for this equation has been obtained in the form of the Fourier transform of the Wiener path integral representation of the matrix elements of the propagators at initial time. The numerical procedure for solving this equation combining both molecular dynamics and Monte Carlo methods has been developed.
So far, we have proposed a method of expressing finite temperature in the form of ground state and pure state by changing the shape of the distribution function and using the step function. Then, in the second quantized system in this form, the Hamiltonian and the method of giving its eigenvalues are presented. We did this in the form that the creation and annihilation operators, whose matrix elements changed as the statistics changed, acted on the Hamiltonian in the ground state. We also discussed the basics of the parastatistics method used for that. In this paper, we apply this finite temperature expression method , express the eigen- values of the Hamiltonian of the interacting system, and try to express the energy given by the interaction as heat.
In the previous chapter we studied the kinematics of many-body systems, and the form of operators representing the physical properties of a system, all of which were embodied by the quantum field. In this chapter we shall study the quantum dynamics of many-body systems, which can also be embodied by the quantum fields. We shall employ the fact that the quantum dynamics of a system, instead of being described in terms of the dynamics of the states or the evolution operator, i.e. as previously done through the Schrödinger equation, can instead be carried by the quantum fields. The quantum dynamics is therefore expressed in terms of the correlation functions or Green's functions of the quantum fields evaluated with respect to some state of the system. In particular we shall consider the general case of quantum dynamics for arbitrary non-equilibrium states. After introducing various types of Green's functions and relating them to measurable quantities, we will discuss the simplifications reigning for the special case of equilibrium states. Quantum dynamics Quantum dynamics can be described in different ways since quantum mechanics is a linear theory and the dynamics described by a unitary transformation of states. This will come in handy in the next chapter when we study a quantum theory in terms of its perturbative expansion using the so-called interaction picture. Here we first discuss the Schrödinger and Heisenberg pictures. The Schrödinger picture Having the Hamiltonian on the multi-particle space at hand we can consider the dynamics described in the multi-particle space.
The graphical technique of evaluation in the second quantization representation of matrix elements of operators associated with the interactions in atoms is developed. The diagrams corresponding to operators and many-electron states given in terms of irreducible tensorial product of creation and annihilation operators are considered. The general rules for the construction of the diagrams for operators and matrix elements are presented. The procedures of reduction of the diagrams to obtain the algebraic expression are discussed.
We propose an improved finite temperature Lanczos method using the stochastic state selection method. In the finite temperature Lanczos method, we generate Lanczos states and calculate the eigenvalues. In addition we have to calculate matrix elements that are the values of an operator between two Lanczos states. In the calculations of the matrix elements we have to keep the set of Lanczos states on the computer memory. Therefore the memory limits the system size in the calculations. Here we propose an application of the stochastic state selection method in order to weaken this limitation. This method is to select some parts of basis states stochastically and to abandon other basis state. Only by the selected basis states we calculate the inner product. After making the statistical average, we can obtain the correct value of the inner product. By the stochastic state selection method we can reduce the number of the basis states for calculations. As a result we can relax the limitation on the computer memory. In order to study the Higgs mode at finite temperature, we calculate the dynamical correlations of the two spin operators in the spin-1/2 Heisenberg antiferromagnet on the square lattice using the improved finite temperature Lanczos method. Our results on the lattices of up to 32 sites show that the Higgs mode exists at low temperature and it disappears gradually when the temperature becomes large. At high temperature we do not find this mode in the dynamical correlations.