This paper presents two perturbation approaches to calculating the quantum statistics of multidimensional nonlinear oscillators in a simple, economic way. These are the power series expansion formalism and the system-specific split operator method. The former involves a harmonic reference system, while the latter allows for the use of physically motivated (anharmonic) zeroth-order representations. Both approaches are outlined with some improvements, and their possible limitations are discussed. The relative efficacy of the approaches is tested on two typical models that are often used by researchers as benchmarks for different numerical methods. Although no system-specific reference system is involved in the power series expansion of the Boltzmann operator, it quite accurately describes the quantum statistics in the entire temperature range, even though the coupling strength is rather large. Yet another important advantage of this approach is that it is essentially analytical, and therefore its numerical implementation does not require any computational effort. This makes the power series expansion technique particularly attractive for treating many-body problems. In contrast, the split operator method is found to be efficient only in the limit of separable motion. Otherwise, this approach results in a less than optimally efficient approximation for the density matrix, or even one which is worse than the standard Trotter approximation.
This paper compares the efficiency and accuracy of different second-order approximations for the single-step propagator obtained by means of the power series expansion formalism and the split operator method. The former is based on a harmonic reference system, while the latter allows for the use of physically motivated (anharmonic) zeroth-order representations. Three typical examples---a system-bath Hamiltonian, a H\'enon-Heiles anharmonic resonating system, and a Fokker-Planck chaotic model---are considered in the present testing. The examples cover a variety of situations with strongly anharmonic coupling. Although no system-specific reference system is involved in the power series representation of the propagator, it quite accurately describes the dynamics of very anharmonic processes in the entire time domain even though the coupling is strong. Another appealing feature of the approach is that it is essentially analytical and therefore does not require any computational effort to implement. This makes the power series expansion technique particularly attractive for efficiently treating many-body problems. In contrast, numerical implementation of the split operator method can be arduous as a general multidimensional calculation, while its utility is in general restricted to the separable limit, when the coupling is almost turned off. In that limit the method indeed provides accurate results over a broad range of t. With increasing coupling, however, the efficiency of the method deteriorates very rapidly regardless of the particular choice of the zeroth-order representation.
An effective and flexible numerical scheme is proposed to calculate the quantum and statistical mechanics of multidimensional systems in a simple economic way. The basic idea is to split the Hamiltonian operator into a reference separable part, whose solution can be obtained by a combination of analytic and numerical techniques, and a term containing nonseparable interactions and then to employ a symmetric decomposition of the time evolution operator, which is exact up to a high order in the time step. The method is applicable to a wide range of coupling potentials and requires numerical effort that scales only linearly with the number of degrees of freedom involved. To verify the utility of the present approach, two model systems with strongly anharmonic mode coupling are considered. The applications show that the method accurately describes the dynamics for fairly long times with moderate coupling strengths and is still much less arduous than a general numerically exact calculation.
We introduce a non perturbative general approximation scheme (NGAS) that can handle interactions of any strength in quantum theory. This approach starts with an input Hamiltonian that can be solved exactly. The interaction effects are then built into this Hamiltonian through nonlinear feedback enforced by self consistency conditions. While the method itself is nonperturbative it can be systematically improved using a new perturbation method called 'mean field perturbation theory' which does not involve power series expansion in any small parameter. We put this scheme to the test on one dimensional anharmonic interactions using the harmonic approximation. The results are consistently accurate across various cases including quartic, sextic, and octic anharmonic oscillators, as well as the quartic double well oscillator (QDWO) even when the coupling strength varies widely. The flexibility of the method is demonstrated when we swap the input Hamiltonian for that of an infinite square well and still achieve comparable accuracy. When applied to the λϕ4 quantum field theory this approach aligns with the Gaussian effective potential method under the harmonic approximation. Beyond that it reveals the condensate structure of the effective vacuum and highlights the instability of the perturbative ground state. Notably, our ground-state energy results for the QDWO stand in stark contrast to those from standard perturbation theory where Borel summation fails regardless of coupling strength.
A simple framework for accurate solution of a general class of one-dimensional Fokker-Planck and/or Schr\"odinger equations is presented. The main idea is representing the propagator in the form
P(x,t∣x0)=P0(x,t∣x0)exp[W(x,t∣x0)] and expanding the exponent
W in a power series in a given function of
t, where
P0 is the exact solution of a reference harmonic-oscillator problem. The expansion coefficients are analytically evaluated from recursive relations. This approach is shown to be a dramatic improvement over the standard Taylor series expansion for the propagator in that just a few terms of the present expansion are sufficient to attain a very accurate description in the whole time domain.
This is the first in a series of papers. It deals with the linked-cluster summation of the perturbation series in quantum mechanics and quantum statistical physics. In particular, a formalism is developed for calculating the matrix element of the operator
exp(+sH), where
H is the quantum-mechanical Hamiltonian operator. It is found that the matrix element may be written as an ordinary
c-number integral of a
c-number function over the phase space. This function is found by doing a linked-cluster summation of the usual perturbation series. It is in the form of an exponential of an effective Hamiltonian,
Heff, which is a function of the parameter
s and the
c-number momentum and position coordinates. We have assumed that the Hamiltonian has no spin-dependent terms. The matrix element is written in such a form that it may be used for calculating the transition matrix elements in time-dependent perturbation theory as well as for calculating the canonical partition function
ZN in quantum statistical mechanics. The former is obtained by substituting $s=\ensuremath{-}\frac{\mathrm{it}}{\ensuremath{\hbar}}$ and the latter by substituting $s=\ensuremath{-}\ensuremath{\beta}$. The series for
Heff depends only on the potential and is independent of the statistics. The effects of the statistics (Bose-Einstein, Fermi-Dirac, or Maxwell-Boltzmann) are all included in a certain factor multiplying
exp(sHeff). Thus we have succeeded in separating the statistical and quantum corrections. These latter arise from the noncommutativity of the position and momentum operators. The effective Hamiltonian is an infinite series. A diagrammatic technique is given for evaluating this series. It is an expansion in powers of the coupling constant and can also be expanded in powers of Planck's constant. In general, it is not possible to give a closed-form expression for
Heff. But for most practical cases it is sufficient to keep the first few powers of Planck's constant $\ensuremath{\hbar}$, because it is very small. We therefore have an entirely new method of approximation. The statistical effect may always be included exactly. No numerical estimates are made in this paper.
Numerous examples in condensed matter physics possess an effective Hamiltonian structure involving two-body anharmonic interactions between elementary excitations. The effective second-quantised Hamiltonian used to model these systems is essentially of the same form except for the particulars of the interaction parameters and the statistics of the particles which are interacting. The author investigates such systems in the limit of strongly anharmonic interactions which may possibly lead to symmetry breaking. He introduces quantum field operators and calculates their equations of motion. The interaction coefficients are expanded in a Taylor series about the critical mode to second order to illustrate the method, but he gives the expansion to infinite order for completeness. It is demonstrated that both in first and second order the equations of motion are reduced to a system of highly nonlinear coupled partial differential equations (PDE) for the field operator. He shows that the form of the second-order equation is exact due to renormalisation theory, in the case where the Hamiltonian coefficients represent operators in real space which are isotropic, and incorporates symmetries and topologies which would appear in an expansion to infinite order. This special situation arises in a very wide range of physical examples. These equations can be solved exactly using a particular ansatz in conjunction with recent mathematical developments in the field of non-linear analysis. In particular, the method of symmetry reduction for PDE provides a complete set of symmetry variables for the type of equations used in the study.