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Path integrals over Poisson trajectories : analytical and numerical estimates of ground state energies

Georg F. Bolz, Maurizio Serva

Published Jan 1, 1990• 0 citations
OpenAlex

Abstract

The aim of this contribution is to demonstrate both a path integral method based on birth and death processes and a related numerical technique by applying them to a simple (but nontrivial) model of an electron in interaction with a boson field mode.

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Hamiltonian semigroups associated with boson systems: a probabilistic approach

Maurizio Serva

In a previous paper the author introduced a path integral technique based on birth and death processes. The method provides an analogue of the Feynman-Kac formula which is associated to the Hamiltonian semigroups of some boson systems. The author briefly demonstrates this technique in a general form. Furthermore, the author studies a simple spin-boson model for which the author obtains some estimates on ground-state energy and spin-flip rate.

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In this thesis the path integral formalism is applied to the calculation of the dynamics of dissipative quantum systems. The time evolution of a system of bilinearly coupled bosonic modes is treated using the real-time path integral technique in coherent-state representation. This method is applied to a damped harmonic oscillator within the Caldeira-Leggett model. In order to get the stationary trajectories the corresponding Lagrangian function is diagonalized and then the path integrals are evaluated by means of the stationary-phase method. The time evolution of the reduced density matrix in the basis of coherent states is given in simple analytic form for weak system-bath coupling, i.e. the so-called rotating-wave terms can be evaluated exactly but the non-rotating-wave terms only in a perturbative manner. The validity range of the rotating-wave approximation is discussed from the viewpoint of spectral equations. In addition, it is shown that systems without initial system-bath correlations can exhibit initial jumps in the population dynamics even for rather weak dissipation. Only with initial correlations the classical trajectories for the system coordinate can be recovered. The path integral formalism in a combined phase-space and coherent-state representation is applied to the problem of curve-crossing dynamics. The system of interest is described by two coupled one-dimensional harmonic potential energy surfaces interacting with a heat bath. The mapping approach is used to rewrite the Lagrangian function of the electronic part of the system. Using the Feynman-Vernon influence-functional method the bath is eliminated whereas the non-Gaussian part of the path integral is treated using the perturbation theory in the small coordinate shift between potential energy surfaces. The vibrational and the population dynamics is considered in a lowest order of the perturbation. The dynamics of a Gaussian wave packet is analyzed along a one-dimensional reaction coordinate. Also the damping rate of coherence in the electronic part of the relevant system is evaluated within the ordinary and variational perturbation theory. The analytic expressions for the rate functions are obtained in the low and high temperature regimes.

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Non-static effects in ordered and disordered quantum spin systems I: theoretical formulation, energy gap and non-extensive terms of ground-state energy of the ferromagnetic Ising model in a transverse field

Yang Wei Koh

In the path integral formulation of the partition function of quantum spin models, most current treatments employ the so-called static approximation to simplify the process of summing over all possible paths. Although sufficient for studying the thermodynamic aspects of the system, static approximation ignores the contributions made by time-dependent, or non-static, fluctuations in the paths of the path integral. This non-static component is very small relative to the static part, and its careful treatment is necessary for the calculation of small non-extensive quantities such as the energy gap within the path integral framework. We propose a formalism for incorporating non-static effects into the path integral calculation of a class of spin models whose partition functions are reducible to the trace of a single spin (often known as the effective Hamiltonian). We first show that the time-dependent behavior of the single spin trace is governed by the Pauli equation. Time-dependent perturbation theory is used to obtain a perturbative expansion of the solution of the Pauli equation, and then for the single spin trace. This gives us a perturbative expansion of the path integral which can be integrated systematically using standard techniques. In this paper, we develop the theoretical framework outlined above in detail and apply it to a simple ordered spin model, the infinite-range ferromagnetic Ising model in a transverse field. We calculated two non-extensive quantites with this non-static approach: the $N^0$ and $N^{-1}$ terms of the ground-state energy ($N$=number of spins) and the energy gap between the ground and first-excited states. We checked our results by comparing with those of Holstein-Primakoff transform and numerical diagonalization of the Hamiltonian. The application of the method to quantum spin-glasses is briefly discussed.

arXiv:1802.09152•Feb 26, 2018•1 citation
In this thesis the path integral formalism is applied to the calculation of the dynamics of dissipative quantum systems. The time evolution of a system of bilinearly coupled bosonic modes is treated using the real-time path integral technique in coherent-state representation. This method is applied to a damped harmonic oscillator within the Caldeira-Leggett model. In order to get the stationary trajectories the corresponding Lagrangian function is diagonalized and then the path integrals are evaluated by means of the stationary-phase method. The time evolution of the reduced density matrix in the basis of coherent states is given in simple analytic form for weak system-bath coupling, i.e. the so-called rotating-wave terms can be evaluated exactly but the non-rotating-wave terms only in a perturbative manner. The validity range of the rotating-wave approximation is discussed from the viewpoint of spectral equations. In addition, it is shown that systems without initial system-bath correlations can exhibit initial jumps in the population dynamics even for rather weak dissipation. Only with initial correlations the classical trajectories for the system coordinate can be recovered. The path integral formalism in a combined phase-space and coherent-state representation is applied to the problem of curve-crossing dynamics. The system of interest is described by two coupled one-dimensional harmonic potential energy surfaces interacting with a heat bath. The mapping approach is used to rewrite the Lagrangian function of the electronic part of the system. Using the Feynman-Vernon influence-functional method the bath is eliminated whereas the non-Gaussian part of the path integral is treated using the perturbation theory in the small coordinate shift between potential energy surfaces. The vibrational and the population dynamics is considered in a lowest order of the perturbation. The dynamics of a Gaussian wave packet is analyzed along a one-dimensional reaction coordinate. Also the damping rate of coherence in the electronic part of the relevant system is evaluated within the ordinary and variational perturbation theory. The analytic expressions for the rate functions are obtained in the low and high temperature regimes.
In the path integral formulation of the partition function of quantum spin models, most current treatments employ the so-called static approximation to simplify the process of summing over all possible paths. Although sufficient for studying the thermodynamic aspects of the system, static approximation ignores the contributions made by time-dependent, or non-static, fluctuations in the paths of the path integral. This non-static component is very small relative to the static part, and its careful treatment is necessary for the calculation of small non-extensive quantities such as the energy gap within the path integral framework. We propose a formalism for incorporating non-static effects into the path integral calculation of a class of spin models whose partition functions are reducible to the trace of a single spin (often known as the effective Hamiltonian). We first show that the time-dependent behavior of the single spin trace is governed by the Pauli equation. Time-dependent perturbation theory is used to obtain a perturbative expansion of the solution of the Pauli equation, and then for the single spin trace. This gives us a perturbative expansion of the path integral which can be integrated systematically using standard techniques. In this paper, we develop the theoretical framework outlined above in detail and apply it to a simple ordered spin model, the infinite-range ferromagnetic Ising model in a transverse field. We calculated two non-extensive quantites with this non-static approach: the N0N^0N0 and N−1N^{-1}N−1 terms of the ground-state energy (NNN=number of spins) and the energy gap between the ground and first-excited states. We checked our results by comparing with those of Holstein-Primakoff transform and numerical diagonalization of the Hamiltonian. The application of the method to quantum spin-glasses is briefly discussed.