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The Journal of Chemical Physics

Renormalization group studies of tunneling systems in the condensed phase

Gary A. Jongeward, Peter Guy Wolynes

Published Oct 1, 1983• 26 citations
Publisher

Abstract

In the path integral representation of the thermal density matrix, the sum over quantum paths is equivalent to the configurational partition function sum for a system composed of long polyatomic molecules. We give an iterative (renormalization group) approach which successively reduces the number of degrees of freedom in this isomorphic classical many-body problem to a more tractable few-body problem. We give as examples a two state tunneling system coupled to an adiabatic and nonadiabatic bath and an impurity with spin coupled to conduction band electrons.

DOI
10.1063/1.446205

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In the path integral representation of the thermal density matrix, the sum over quantum paths is equivalent to the configurational partition function sum for a system composed of long polyatomic molecules. We give an iterative (renormalization group) approach which successively reduces the number of degrees of freedom in this isomorphic classical many-body problem to a more tractable few-body problem. We give as examples a two state tunneling system coupled to an adiabatic and nonadiabatic bath and an impurity with spin coupled to conduction band electrons.
In the path integral representation of quantum theory, a few body quantum problem becomes a classical many body problem. To exploit this isomorphism, it becomes necessary to develop methods by which degrees of freedom can be explicitly removed from consideration. The interactions among the remaining relevant variables are described by effective interactions. In this paper, we present a general methodology to carry out the reduction in numbers of degrees of freedom. Certain path integral algorithms are shown to correspond to reference systems for the full isomorphic classical many body problem. The correspondence allows one to determine systematic corrections to the algorithms by low order perturbation approximations familiar in the theory of simple classical fluids. We show how to use discretized path integrals to compute rigorous upper and lower bounds to the free energy for nontrivial quantum systems, and we discuss how to optimize the upper bounds with variational theories. Several illustrative examples are provided including treatments of a tunneling system that is strongly coupled to a nonadiabatic heat bath. The examples demonstrate that even for strongly perturbed systems in which the quantum energy scales are as large as 40kBT, the methods discussed herein provide a practical route for obtaining equilibrium properties with both high accuracy and analytical simplicity.
This thesis develops mixed quantum-classical methods to model dynamical nonequilibrium quantum phenomena in condensed phase systems, and also explores properties of 2-d systems near a quantum critical point using field theoretic techniques and purely classical simulation. Mixed quantum-classical methods involve dividing the system into a quantum sub-system coupled to a classical bath, which is useful for examining the dynamics of systems in the condensed phase since the full quantum treatment of a large number of degrees of freedom is often computationally infeasible. Although the bath is treated classically, its coupling to the quantum system precludes a Newtonian description. We implement an approach that uses the quantum-classical Liouville equation to propagate the quantum subsystem density matrix, which results in an ensemble of surface-hopping trajectories. We test this method on model systems and compare with a linearized path-integral approach. We also develop a novel approach to propagating the reduced density matrix that incorporates ideas from both methods while
We introduce a new approach for calculating quantum time-correlation functions and time-dependent expectation values in many-body thermal systems; both electronically adiabatic and non-adiabatic cases can be treated. Our approach uses a path integral simulation to sample an initial thermal density matrix; subsequent evolution of this density matrix is equivalent to solution of the time-dependent Schrödinger equation, which we perform using a linear expansion of Gaussian wavepacket basis functions which evolve according to simple classical-like trajectories. Overall, this methodology represents a formally exact approach for calculating time-dependent quantum properties; by introducing approximations into both the imaginary-time and real-time propagations, this approach can be adapted for complex many-particle systems interacting through arbitrary potentials. We demonstrate this method for the spin Boson model, where we find good agreement with numerically exact calculations. We also discuss future directions of improvement for our approach with a view to improving accuracy and efficiency.
From a discretization of the path integral formulation of quantum mechanics, it is possible to relate equilibrium quantum many body theory to classical statistical mechanics. In this paper, we significantly extend and analyze this well known isomorphism in terms of the equilibrium theory of classical molecular fluids composed of flexible polyatomic species. We show how quantum influence functionals are isomorphic to classical cavity distribution functions. The former describe the influence of surrounding media on the dynamics of quantal degrees of freedom, and the latter describe environmental effects for classical models of flexible molecules and chemical equilibria. The connection allows the use of classical theories to perform nonperturbative calculations of influence functionals which treat the influence functionals and many body correlation functions in a self-consistent fashion. We illustrate the computational advantages of the method by studying its predictions for a hard sphere model of liquid helium above the l transition. The nature of quantum indistinguishability of identical particles (i.e., quantum exchange) is treated in our theory in terms of an exact isomorphism with chemical equilibria. This connection allows the treatment of exchange in condensed phases in terms of the classical law of mass action, and provides a computational advance over existing methods for interacting systems. By picturing exchange in terms of classical association equilibrium, we arrive at a view (hinted at long ago by Feynman and by Penrose and Onsager) in which the Bose condensation is related to an equilibrium polymeric sol–gel transition. Thus, below the l transition, the correlations in liquid helium are equivalent to those in a classical fluid containing a finite concentration of macroscopic polymers. We stress how the path integral aspect of the isomorphism leads to useful geometrical interpretations of quantum phenomena. For example, tunneling phenomena can be viewed in terms of solitonic (or instantonic) configurations or flexible chain molecules. With this picture, we show how the isomorphism can be employed to understand both adiabatic and nonadiabatic solvent effects on chemical bonding. For concreteness, we provide a detailed analysis of a particular model of the chemical bond for which a partial summation over intermediate quantum paths leads to an Ising model problem in the isomorphism. While applications like this are presented in the form of qualitative illustrations, a variety of methods can be employed to produce quantitative results. We sketch how these calculations can be performed for various problems, making connections with methods like the renormalization group (RG) technique. The classical isomorphism together with the modern theory of classical polyatomic systems provides a powerful framework for quantitative solutions of condensed matter quantum mechanical problems.
In the first three chapters we have considered problems related to the physical behaviour of one or a few particles (or some other physical objects which can be effectively described by one or a few (quasi)particles as, e.g., the random walk model in polymer physics considered in chapter 1). Though quantum field theory describes systems with an arbitrary number of degrees of freedom (an arbitrary number of (quasi)particles), in chapter 3 we actually applied it to systems with a restricted and small number of particles (e.g., for the description of the scattering process for a few particles). Another way to express this fact is to say that in chapter 3 we have considered field theories at zero temperature. However, the majority of realistic systems contain many identical (indistinguishable) particles such as atoms, electrons, photons etc. An attempt to describe these systems in terms of the individual trajectories of all particles is absolutely hopeless. Instead, we are interested in the collective behaviour of systems and describe them in terms of partition functions, mean values, correlation functions, etc. The methods of derivation, analysis and calculation of such collective characteristics constitute the subject of classical and quantum statistical mechanics.