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Path integral Monte Carlo for dissipative many-body systems

Luca Capriotti, R. VAIA, Alessandro Cuccoli, Andrea Fubini, V. Tognetti

Published Jan 1, 2003• 2 citations
OpenAlex

Abstract

We address the possibility of performing numerical Monte Carlo simulations for the thermodynamics of quantum dissipative systems. Dissipation is considered within the Caldeira-Leggett formulation, which describes the system in the path-integral formalism through the inclusion of an influence action that is bilocal and quadratic in the system's coordinates. At a first sight the usual direct approach of discretizing the path integral could seem feasible, but complications arise when one tries to introduce a physically meaningful dissipation kernel: in particular its imaginary-time dependence turns out to be severely singular and difficult to evaluate analytically, in spite of the simple expressions for its Matsubara components. We therefore propose to face the numerical problem using Fourier path-integral Monte Carlo, that can be formulated in two different ways: transforming the continuous paths and then truncating the high Fourier components (with possible improvements upon the truncation procedure), or performing the Fourier transformation upon the discretized paths. The latter choice leads to a simpler formulation and allows for a better control of the extrapolation to the limit of infinite Trotter number. The method is implemented for a single nonlinear particle with Ohmic dissipation and for a phi(4) chain with Drude-like dissipation.

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We address the possibility of performing numerical Monte Carlo simulations for the thermodynamics of quantum dissipative systems. Dissipation is considered within the Caldeira-Leggett formulation, which describes the system in the path-integral formalism through the inclusion of an influence action that is bilocal and quadratic in the system's coordinates. At a first sight the usual direct approach of discretizing the path integral could seem feasible, but complications arise when one tries to introduce a physically meaningful dissipation kernel: in particular its imaginary-time dependence turns out to be severely singular and difficult to evaluate analytically, in spite of the simple expressions for its Matsubara components. We therefore propose to face the numerical problem using Fourier path-integral Monte Carlo, that can be formulated in two different ways: transforming the continuous paths and then truncating the high Fourier components (with possible improvements upon the truncation procedure), or performing the Fourier transformation upon the discretized paths. The latter choice leads to a simpler formulation and allows for a better control of the extrapolation to the limit of infinite Trotter number. The method is implemented for a single nonlinear particle with Ohmic dissipation and for a phi(4) chain with Drude-like dissipation.
Abstract We address the possibility of performing numerical Monte Carlo simulations for the thermodynamics of quantum dissipative systems. Dissipation is considered within the Caldeira–Leggett formulation, which describes the system in the path‐integral formalism through the inclusion of an influence action that is bilocal and quadratic in the system's coordinates. At a first sight the usual direct approach of discretizing the path integral could seem feasible, but complications arise when one tries to introduce a physically meaningful dissipation kernel: in particular its imaginary‐time dependence turns out to be severely singular and difficult to evaluate analytically, in spite of the simple expressions for its Matsubara components. We therefore propose to face the numerical problem using Fourier path‐integral Monte Carlo, that can be formulated in two different ways: transforming the continuous paths and then truncating the high Fourier components (with possible improvements upon the truncation procedure), or performing the Fourier transformation upon the discretized paths. The latter choice leads to a simpler formulation and allows for a better control of the extrapolation to the limit of infinite Trotter number. The method is implemented for a single nonlinear particle with Ohmic dissipation and for a ϕ4 chain with Drude‐like dissipation.
An efficient Path Integral Monte Carlo procedure is proposed to simulate the behavior of quantum many-body dissipative systems described within the framework of the influence functional. Thermodynamic observables are obtained by Monte Carlo sampling of the partition function after discretization and Fourier transformation in imaginary time of the dynamical variables. The method is tested extensively for model systems, using realistic dissipative kernels. Results are also compared with the predictions of a recently proposed semiclassical approximation, thus testing the reliability of the latter approach for weak quantum coupling. Our numerical method opens the possibility to quantitatively describe real quantum dissipative systems as, e.g. , Josephson junction arrays.
We consider quantum nonlinear many-body systems with dissipation described within the Caldeira-Leggett model, i.e., by a nonlocal action in the path integral for the density matrix. Approximate classical-like formulas for thermodynamic quantities are derived for the case of many degrees of freedom, with general kinetic and dissipative quadratic forms. The underlying scheme is the pure-quantum self-consistent harmonic approximation (PQSCHA), equivalent to the variational approach by the Feynman-Jensen inequality with a suitable quadratic nonlocal trial action. A low-coupling approximation permits us to get manageable PQSCHA expressions for quantum thermal averages with a classical Boltzmann factor involving an effective potential and an inner Gaussian average that describes the fluctuations originating from the interplay of quanticity and dissipation. The application of the PQSCHA to a quantum phi(4) chain with Drude-like dissipation shows nontrivial effects of dissipation, depending upon its strength and bandwidth.
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.
The stochastic quantization of dissipative systems is discussed. It is shown that in order to stochastically quantize a system with dissipation, one has to restrict the Fourier transform of the space-time variable to the positive half domain in the complex plane. This breaks the time-reversal invariance, which manifests in the formulation through the resulting noninvariant forms for the propagators. The relation of the stochastic approach with the Caldeira and Leggett path-integral method is also analyzed. † tzani@vana.physto.se; address after September 1994: Imperial College, Physics Dept., Prince Consort Road, London SW72BZThe quantization of a classical system may, in general, be hindered by several obstacles, such as nonrenormalizability, topological obstructions, anomalies etc. There exists, moreover, a whole class of systems for which there is no available method of quantization, namely non-hamiltonian systems. The quantization of systems with no lagrangian or hamiltonian formulation, apart from the wellknown examples of systems with dissipation, includes several interesting physical problems, such as the Bargmann-Wigner higher spin field equations [1] and field theories of fundamental anyons [2]. For the Bargmann-Wigner system of equations there is no action which directly reproduces this set of equations, although the quantization for higher spin systems has been done using more complicated and indirect ways [3]. The case of anyonic systems is known to be unsolved. The most common example of a non-hamiltonian system is, nevertheless, the case of a particle on the line with friction. Its equation of motion is m¨x + γ ˙x + dV dx = 0 (1) where overdot denotes time derivative and V (x) is the potential of the particle. If we assume an action of the form S = f(x, ˙x, t) dt (2) for the above equation, the variation of the action gives δS δx = dt ( ∂f ∂ ∂f ∂ ∂f
The cavity method is a well-established technique for solving classical spin models on sparse random graphs (mean-field models with finite connectivity). Laumann et al. [Phys. Rev. B 78, 134424 (2008)] proposed recently an extension of this method to quantum spin-1/2 models in a transverse field, using a discretized Suzuki-Trotter imaginary-time formalism. Here we show how to take analytically the continuous imaginary-time limit. Our main technical contribution is an explicit procedure to generate the spin trajectories in a path-integral representation of the imaginary-time dynamics. As a side result we also show how this procedure can be used in simple heat bath Monte Carlo simulations of generic quantum spin models. The replica symmetric continuous-time quantum cavity method is formulated for a wide class of models and applied as a simple example on the Bethe lattice ferromagnet in a transverse field. The results of the methods are confronted with various approximation schemes in this particular case. On this system we performed quantum Monte Carlo simulations that confirm the exactness of the cavity method in the thermodynamic limit.