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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

The t expansion: A nonperturbative analytic tool for Hamiltonian systems

D. Horn, Marvin Weinstein

Published Sep 15, 1984• 158 citations
Publisher

Abstract

A systematic nonperturbative scheme is developed to calculate the ground-state expectation values of arbitrary operators for any Hamiltonian system. Quantities computed in this way converge rapidly to their true expectation values. The method is based upon the use of the operator ${e}^{\ensuremath{-}tH}$ to contract any trial state onto the true ground state of the Hamiltonian $H$. We express all expectation values in the contracted state as a power series in $t$, and reconstruct $t\ensuremath{\rightarrow}\ensuremath{\infty}$ behavior by means of Pad\'e approximants. The problem associated with factors of spatial volume is taken care of by developing a connected graph expansion for matrix elements of arbitrary operators taken between arbitrary states. We investigate Pad\'e methods for the $t$ series and discuss the merits of various procedures. As examples of the power of this technique we present results obtained for the Heisenberg and Ising models in 1+1 dimensions starting from simple mean-field wave functions. The improvement upon mean-field results is remarkable for the amount of effort required. The connection between our method and conventional perturbation theory is established, and a generalization of the technique which allows us to exploit off-diagonal matrix elements is introduced. The bistate procedure is used to develop a $t$ expansion for the ground-state energy of the Ising model which is, term by term, self-dual.

DOI
10.1103/physrevd.30.1256

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arXiv:1909.12508•Sep 27, 2019•0 citations
A systematic nonperturbative scheme is developed to calculate the ground-state expectation values of arbitrary operators for any Hamiltonian system. Quantities computed in this way converge rapidly to their true expectation values. The method is based upon the use of the operator e\ensuremath−tH{e}^{\ensuremath{-}tH}e\ensuremath−tH to contract any trial state onto the true ground state of the Hamiltonian HHH. We express all expectation values in the contracted state as a power series in ttt, and reconstruct t\ensuremath→\ensuremath∞t\ensuremath{\rightarrow}\ensuremath{\infty}t\ensuremath→\ensuremath∞ behavior by means of Pad'e approximants. The problem associated with factors of spatial volume is taken care of by developing a connected graph expansion for matrix elements of arbitrary operators taken between arbitrary states. We investigate Pad'e methods for the ttt series and discuss the merits of various procedures. As examples of the power of this technique we present results obtained for the Heisenberg and Ising models in 1+1 dimensions starting from simple mean-field wave functions. The improvement upon mean-field results is remarkable for the amount of effort required. The connection between our method and conventional perturbation theory is established, and a generalization of the technique which allows us to exploit off-diagonal matrix elements is introduced. The bistate procedure is used to develop a ttt expansion for the ground-state energy of the Ising model which is, term by term, self-dual.
Employing a theorem on lower bounds on the zeros of orthogonal polynomials, the plaquette expansion to order 1/Np{\mathit{N}}_{\mathit{p}}Np​ of the tridiagonal Lanczos matrix elements is solved for the ground-state energy density in the infinite lattice limit. The resulting nonperturbative expression for the estimate of the energy density in terms of the connected coefficients to order 〈H4{\mathit{H}}^{4}H4〉c{\mathrm{〉}}_{\mathit{c}}〉c​ is completely general. This expression is applied to various Hamiltonian systems---the Heisenberg model in D dimensions and SU(2) and SU(3) lattice gauge theory in 3+1 dimensions. In all cases the analytic estimate to the energy density is not only a significant improvement on the trial state, but is typically accurate to a few percent. The energy density of the D-dimensional Heisenberg model is predicted to approach scrE0{\mathit{scrE}}_{0}scrE0​(N\'eel)-1/8 for large D. In the case of SU(2) and SU(3) the specific heat derived from the energy density peaks at the correct strong- to weak-coupling transition.
We propose a method to construct the ground state \ensuremath{\psi}(\ensuremath{\lambda}) of local lattice Hamiltonians with the generic form H0{\mathit{H}}_{0}H0​+\ensuremath{\lambda}H1{\mathit{H}}_{1}H1​, where \ensuremath{\lambda} is a coupling constant and H0{\mathit{H}}_{0}H0​ is a Hamiltonian with a nondegenerate ground state ${\mathrm{\ensuremath{\psi}}}_{0}$. The method is based on the choice of an exponential ansatz \ensuremath{\psi}(\ensuremath{\lambda})=exp[U(\ensuremath{\lambda})]${\mathrm{\ensuremath{\psi}}}_{0}$, which is a sort of generalized lattice version of a Jastrow wave function. We combine perturbative and variational techniques to get successive approximations of the operator U(\ensuremath{\lambda}). Perturbation theory is used to set up a variational method which in turn produces nonperturbative results. The computation with this kind of ansatz leads us to associate to the original quantum-mechanical problem a statistical-mechanical system defined in the same spatial dimension. In some cases these statistical-mechanical systems turn out to be integrable, which allows us to obtain exact upper bounds to the energy. The general ideas of our method are illustrated in the example of the Ising model in a transverse field.
New nonperturbative methods for dealing with Hamiltonian systems are introduced. The derivation of these methods requires identities for rewriting exponentials of sums of operators which are different from the usual Campbell-Hausdorff formula. These identities allow one to derive approximations to ${e}^{\ensuremath{-}\ensuremath{\delta}H}$ which are correct to higher order in $\ensuremath{\delta}$ and which contain fewer terms than the Campbell-Hausdorff formula. This allows one to generate path-integral actions which are more accurate for finite-size steps in time and which can be exploited to improve the rate of convergence of Monte Carlo calculations. To show that these methods allow one to include effects which show up in the stationary-phase approximation to the path integral, e.g., instantons, solitons, etc., I not only derive the connection between the Hamiltonian and path-integral formalism but the relationship between a specific stationary-phase approximation and the corresponding Hamiltonian calculation. My focus, however, is the direct application of these new identities to the study of nonperturbative Hamiltonian dynamics. I show that these methods are easier to apply and give better results than those based upon the naive ttt-expansion, block mean-field, or real-space renormalization ideas. Comparison of these older methods with the computational tools introduced in this paper are discussed in the context of simple examples. It is shown that the new methods allow one to extrapolate answers to finite ttt without the use of Pad\'e approximants.
An analysis of a general non-perturbative technique for calculating ground-state properties of extensive lattice many-body systems is presented, in order to extract accurate numerical values characterizing the ground-state spectrum. This technique, the plaquette expansion, employs an expansion about the thermodynamic limit of the coefficients that are generated by the Lanczos process. For the ground-state energy this error analysis, using theorems on the error bounds for the Lanczos method and the truncation in the plaquette expansion, allows for an accurate estimate when the approximation is taken to a given order. As an example we analyse the one-dimensional antiferromagnetic Heisenberg model, and find that the best ground-state energy density is within of the exact value, although the systematic error is . We also find, for this model, systematic improvement with each new order included in the expansion and have not observed any asymptotic tendencies. At equivalent orders of truncation we achieve far better results than for the other moment methods, such as the t-expansion or the connected-moment expansion.
Abstract It is well known that quantum-mechanical perturbation theory often gives rise to divergent series that require proper resummation. Here I discuss simple ways in which these divergences can be avoided in the first place. Using the elementary technique of relaxed fixed-point iteration, I obtain convergent expressions for various challenging ground-states wave functions, including quartic, sextic and octic anharmonic oscillators, the hydrogenic Zeeman problem, and the Herbst-Simon Hamiltonian (with finite energy but vanishing Rayleigh-Schrödinger coefficients), all at arbitarily strong coupling. These results challenge the notion that non-analytic functions of coupling constants are intrinsically “non-perturbative”. A possible application to exact diagonalization is briefly discussed.
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.