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Journal of Mathematical Physics

Functional integrals for spin-Bose systems

G. J. Papadopoulos

Published Jan 1, 1986• 6 citations
Publisher

Abstract

The problem of obtaining the propagator of a spin-Bose system consisting of spins- 1/2 and several radiation modes in interaction is reduced to developing a bosonized Hamiltonian for the system from the propagator of which, by appropriate projection, the propagator relating to the original Hamiltonian is extracted. The analysis proceeds via functional integration involving the evaluation of two auxiliary propagators of Schrödinger equations whose Hamiltonians depend separately on radiation operators, and bosonized spin operators. These propagators are coupled through a complex field and the functional average of their product against a Gaussian measure of the field leads to the bosonized system’s propagator. The procedure is applied to the propagator for a single spin-Bose system which is obtained as a power series in the spin flipping energy. The analysis is also applied for an explicit evaluation in the case of a simplified spin-Bose Hamiltonian.

DOI
10.1063/1.527365

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The problem of obtaining the propagator of a spin-Bose system consisting of spins- 1/2 and several radiation modes in interaction is reduced to developing a bosonized Hamiltonian for the system from the propagator of which, by appropriate projection, the propagator relating to the original Hamiltonian is extracted. The analysis proceeds via functional integration involving the evaluation of two auxiliary propagators of Schrödinger equations whose Hamiltonians depend separately on radiation operators, and bosonized spin operators. These propagators are coupled through a complex field and the functional average of their product against a Gaussian measure of the field leads to the bosonized system’s propagator. The procedure is applied to the propagator for a single spin-Bose system which is obtained as a power series in the spin flipping energy. The analysis is also applied for an explicit evaluation in the case of a simplified spin-Bose Hamiltonian.
We consider the general problem of a single two-level atom interacting with a multimode radiation field (without the rotating-wave approximation), and additionally take the field to be coupled to a thermal reservoir. Using the method of bosonization of the spin operators in the Hamiltonian, and working in the Bargmann representation for all the boson operators, we obtain the propagator for the composite system using the techniques of functional integration, under a reasonable approximation scheme. The propagator is explicitly evaluated for a simplified version of the system with one spin and a dynamically coupled single-mode field. The results are also checked on the known problem of quantum Brownian motion.
A number of problems in statistical physics can be reformulated in terms of a two-state system evolving in a random field. The corresponding evolution operator can be written in the form of time-ordered operator exponential. Functional formalism allows us to rewrite the latter as a product of usual matrix exponentials using a nonlinear change of functional integration variables. In this review I present this formalism applied to two physical systems the quantum Heisenberg magnet and one-dimensional quantum mechanics in a spatially random potential. First, I derive a representation of the partition function of a quantum Heisenberg ferromagnet as a functional integral over number valued fields (a real one and a complex one) free of constraints. The fields of integration as functions of time obey initial conditions instead of the usual periodic boundary conditions. This is a manifestation of the finite-dimensionality of the space of spin states. In the subsequent sections I study the one-dimensional localization problem. The change of functional integration variables gives simultaneously explicit expressions for the averaging weight and for the Green function of the stationary Schrödinger equation. It allows to compute density correlators of arbitrary orders. The generalization to the case of different energy correlators (Berezinskii-Gor’kov equations) is considered too. In the present review such technical points as regularizations of functional integrals and transformations are discussed in more details than in the original papers.
We test a new approach to bosonization in relativistic field theories and many-body systems, based on the use of fermionic composites as integration variables in the Berezin integral defining the partition function of the system. The method appears promising since at zeroth order it correctly describes the propagators of the composites, which can be evaluated in a number of significant cases. Still to be established is a general procedure for deriving the free action of the composites starting from the one of the constituents. To shed light on this problem and to explore further features of the method we study a simplified version of the BCS model. In this simple case the action of the composites can indeed be obtained: whether this result can be generalized it remains hewever to be seen. Yet an interesting property of the wave operators appearing in the free actions of bilinear composites already emerges from the simple problem we have treated: here the wave operators do not describe any time evolution, even though they generate the right propagators. This outcome relates to the basic properties of the integrals over the even elements of a Grassmann algebra where the composites live, which entails that the propagators are no longer the inverse of the wave operators.
The path integral single spin partition function in the basis of boson coherent states, for a general normal ordered two-mode Schwinger boson Hamiltonian, has been computed. Within the initial phase of the calculational process, by a specific approach, the Schwinger boson constraint by means of the projection operator has been implemented.\nAfter, the case of the Zeeman Hamiltonian has been taken on. Using the expression for the partition function obtained for the general case, the appropriate single spin partition function has been produced. Additionally, partition functions have been computed for a few specific spin quantum numbers and then compared with those calculated by the straightforward means, which brought about great confidence in the projection operator implementation; both results match perfectly. \nFinally, the Heisenberg Model has been tackled, and the appropriate expressions for the partition functions have been computed. Their validity has not been verified for any specific cases, but the Zeeman Hamiltonian case dealt with earlier appears to be a strong indicator that these are indeed the correct expressions. \nMany open questions remain, however, for future research to address. All in all, it has been discovered that using the projection operator as a means to enforce the Schwinger boson constraint, veritably works for a Zeeman Hamiltonian problem for specific spin quantum numbers.
Numerical evaluation of functional integrals usually involves a finite ( L -slice) discretization of the imaginary-time axis. In the auxiliary-field method, the L -slice approximant to the density matrix can be evaluated as a function of inverse temperature at any finite L as L (β) = [ 1 (β/ L )] L , if the density matrix 1 (β) in the static approximation is known. We investigate the convergence of the partition function Z L (β) ≡ Tr L (β), the internal energy and the density of states g L ( E ) (the inverse Laplace transform of Z L ), as L → ∞ . For the simple harmonic oscillator, g L ( E ) is a normalized truncated Fourier series for the exact density of states. When the auxiliary-field approach is applied to spin systems, approximants to the density of states and heat capacity can be negative. Approximants to the density matrix for a spin-½ dimer are found in closed form for all L by appending a self-interaction to the divergent Gaussian integral and analytically continuing to zero self-interaction. Because of this continuation, the coefficient of the singlet projector in the approximate density matrix can be negative. For a spin dimer, Z L is an even function of the coupling constant for L < 3: ferromagnetic and antiferromagnetic coupling can be distinguished only for L ⩾3, where a Berry phase appears in the functional integral. At any non-zero temperature, the exact partition function is recovered as L → ∞ .
Quantum evolution was recently shown to be equivalent to the mathematical expectation of the stochastic motion described by the Schr\"odinger equation with momentum-coupled white noise [J. Shao, J. Chem. Phys. 159, 024128 (2023)]. The stochastic formulation provides a convenient means of evaluating quantum propagators because the involved kinetic and potential contributions are disentangled. With the help of the auxiliary functional averaging technique, we apply this scheme to the derivation of the quantum propagators for systems of exponential, linear, harmonic, and $\ensuremath{\delta}$-function potentials. Notably, our work presents a truly workable expression of the quantum propagator for the exponential potential problem. To demonstrate the features and advantages of this propagator, we use it to calculate the wave-packet dynamics and compare it to the split-operator method.