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arXiv:cond-mat/0001193 · Physical Review Letters

Exact Monte Carlo Method for Continuum Fermion Systems

Malvin H. Kalos, Francesco Pederiva

Published Oct 23, 2000• 39 citations
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Abstract

We offer a new proposal for the Monte Carlo treatment of many-fermion systems in continuous space. It is based upon diffusion Monte Carlo with significant modifications: correlated pairs of random walkers that carry opposite signs, different functions "guide" walkers of different signs, the Gaussians used for members of a pair are correlated, and walkers can cancel so as to conserve their expected future contributions. We report results for free-fermion systems and a fermion fluid with 14 3He atoms, where it proves stable and correct. Its computational complexity grows with particle number, but slowly enough to make interesting physics within the reach of contemporary computers.

DOI
10.1103/physrevlett.85.3547

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We offer a new proposal for the Monte Carlo treatment of many-fermion systems in continuous space. It is based upon diffusion Monte Carlo with significant modifications: correlated pairs of random walkers that carry opposite signs, different functions "guide" walkers of different signs, the Gaussians used for members of a pair are correlated, and walkers can cancel so as to conserve their expected future contributions. We report results for free-fermion systems and a fermion fluid with 14 3He atoms, where it proves stable and correct. Its computational complexity grows with particle number, but slowly enough to make interesting physics within the reach of contemporary computers.
We review the fundamental challenge of fermion Monte Carlo for continuous systems, the “sign problem”. We seek that eigenfunction of the many-body Schrödinger equation that is antisymmetric under interchange of the coordinates of pairs of particles. We describe methods that depend upon the use of correlated dynamics for pairs of correlated walkers that carry opposite signs. There is an algorithmic symmetry between such walkers that must be broken to create a method that is both exact and as effective as for symmetric functions. In our new method, it is broken by using different “guiding” functions for walkers of opposite signs, and a geometric correlation between steps of their walks. With a specific process of cancellation of the walkers, overlaps with antisymmetric test functions are preserved. Finally, we describe the progress in treating free-fermion systems and a fermion fluid with 14 3He atoms.
The issues that prevent the development of efficient and stable algorithms for fermion Monte Carlo calculations in continuum systems are reexamined with special reference to the implications of the ``plus-minus'' symmetry. This is a property of many algorithms that use signed walkers, namely, that the dynamics are unchanged when the signs of the walkers are interchanged. Algorithms that obey this symmetry cannot exhibit the necessary stability. Specifically, estimates of the overlap with any antisymmetric test function cannot be bounded away from zero in the limit of many iterations. Within the framework of a diffusion Monte Carlo treatment of the Schr\"odinger equation, it is shown that this symmetry is easily broken for pairs of walkers while at the same time preserving the correct marginal dynamics for each member of the pair. The key is to create different classes of correlations between members of pairs and to use (at least) two distinct correlations for a pair and for the same pair with signs exchanged. The ideas are applied successfully for a class of simple model problems in two dimensions. \textcopyright{} 1996 The American Physical Society.
A diffusion Monte Carlo algorithm is introduced that can determine the correct nodal structure of the wave function of a few-fermion system and its ground-state energy without an uncontrolled bias. This is achieved by confining signed random walkers to the points of a uniform infinite spatial grid, allowing them to meet and annihilate one another to establish the nodal structure without the fixed-node approximation. An imaginary-time propagator is derived rigorously from a discretized Hamiltonian, governing a non-Gaussian, sign-flipping, branching, and mutually annihilating random walk of particles. The accuracy of the resulting stochastic representations of a fermion wave function is limited only by the grid and imaginary-time resolutions and can be improved in a controlled manner. The method is tested for a series of model problems including fermions in a harmonic trap as well as the He atom in its singlet or triplet ground state. For the latter case, the energies approach from above with increasing grid resolution and converge within 0.015E_{h} of the exact basis-set-limit value for the grid spacing of 0.08 a.u. with a statistical uncertainty of 10^{-5}E_{h} without an importance sampling or Jastrow factor.
In this work we present a detailed study of the fermion Monte Carlo algorithm (FMC), a recently proposed stochastic method for calculating fermionic ground-state energies. A proof that the FMC method is an exact method is given. In this work the stability of the method is related to the difference between the lowest (bosonic-type) eigenvalue of the FMC diffusion operator and the exact Fermi energy. It is shown that within a FMC framework the lowest eigenvalue of the new diffusion operator is no longer the bosonic ground-state eigenvalue as in standard exact diffusion Monte Carlo (DMC) schemes but a modified value which is strictly greater. Accordingly, FMC can be viewed as an exact DMC method built from a correlated diffusion process having a reduced Bose–Fermi gap. As a consequence, the FMC method is more stable than any transient method (or nodal release-type approaches). It is shown that the most recent ingredient of the FMC approach (Kalos M H and Pederiva F 2000 Phys. Rev. Lett. 85 3547), namely the introduction of non-symmetric guiding functions, does not necessarily improve the stability of the algorithm. We argue that the stability observed with such guiding functions is in general a finite-size population effect disappearing for a very large population of walkers. The counterpart of this stability is a control population error which is different in nature from the standard diffusion Monte Carlo algorithm and which is at the origin of an uncontrolled approximation in FMC. We illustrate the various ideas presented in this work with calculations performed on a very simple model having only nine states but a full ‘sign problem’. Already for this toy model it is clearly seen that FMC calculations are inherently uncontrolled.
We propose an improved framework of diffusion Monte Carlo method for quantum many-fermion systems. This approach enables us to find the ground state of the system without any a priori information on the nodal structure of the wave function. The main idea in this approach is to use sign-carrying replicas which are the basis vectors in a restricted Hilbert space. For efficiency, we propose to reduce the number of replicas in a proper manner. We test our method on a spinless fermion system, and compare the results with the exact values. We also discuss this method in relation to exact diagonalization.
We have studied the dynamics of the correlated diffusion of pairs of random walkers of opposite signs. The use of populations of such pairs has been proposed for the Monte Carlo treatment of many-fermion systems, where the possibility of their cancellation might prevent the characteristic decay of the signal-to-noise ratio. For four model systems – free fermions, the harmonic oscillator, an N-body system of attractive and repulsive harmonic forces, and an extensive system interacting by Pöschl–Teller potentials – we have explored analytically and by computation the behavior of the time to cancellation as a function of initial conditions and, equally important, as a function of system size. We find that for these systems the computational efficiency does not decay either with large imaginary time or with large N.