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arXiv:cond-mat/0202283

Permutation zones and the fermion sign problem

Dean Lee

Published Feb 17, 2002• 0 citations
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Abstract

We present a new approach to the problem of alternating signs for fermionic many body Monte Carlo simulations. We demonstrate that the exchange of identical fermions is typically short-ranged even when the underlying physics is dominated by long distance correlations. We show that the exchange process has a maximum characteristic range of sqrt[2*(1-f)*beta*h] lattice sites, where beta is the inverse temperature, h is the hopping parameter, and f is the filling fraction. We introduce the notion of permutation zones, special regions of the lattice where identical fermions may interchange and outside of which they may not. Using successively larger permutation zones, one can extrapolate to obtain thermodynamic observables in regimes where direct simulation is impossible.

DOI
10.48550/arxiv.cond-mat/0202283

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We present a new approach to the problem of alternating signs for fermionic many body Monte Carlo simulations. We demonstrate that the exchange of identical fermions is typically short-ranged even when the underlying physics is dominated by long distance correlations. We show that the exchange process has a maximum characteristic range of sqrt[2*(1-f)betah] lattice sites, where beta is the inverse temperature, h is the hopping parameter, and f is the filling fraction. We introduce the notion of permutation zones, special regions of the lattice where identical fermions may interchange and outside of which they may not. Using successively larger permutation zones, one can extrapolate to obtain thermodynamic observables in regimes where direct simulation is impossible.
Starting from a \emph{probabilistic numerics} approach to the Fermion sign problem in path integral Monte Carlo, we recast the arithmetic calculation of a Fermionic observable as a statistical inference problem. We develop approaches that learn the behaviour of Fermion exchange cycles binned by the conjugacy class of the permutation group (which we term `permutation family'). This extends the work of DuBois, Brown and Alder\cite{dubois2017overcoming} to inhomogeneous and more complex systems. Monte Carlo samples are used to train models for both the probability of a permutation family and the energy of this set of exchange permutations. The overall Fermionic energy is then directly inferred from these models, without using a direct ratio estimator on the Monte Carlo samples. By imposing physical understanding as inductive priors, we produce accurate and useful fits that remain robust even in regimes with severe sign problems. We generalise the linear (ideal-gas style) models of DuBois et al. with Bayesian priors that enforce the intuitive models of Feynman\cite{Feynman1953A} at their asymptotic limits. These linear models serve as the baseline for a Long Short-Term Memory (LSTM) neural network, which is tasked with learning only the residual many-body \emph{correlations} on top of the physical model. We develop active important sampling methods driven by these models, which direct the Monte Carlo chains toward undersampled permutation regions, to efficiently reduce the variance in the observable. We apply this framework to small experiments on benchmark systems: the spin-polarised uniform electron gas, and electrons in a 2D harmonic confining potential. In both cases we demonstrate that this inference-based framework can extract stable energies in regimes where direct Monte Carlo sampling fails due to the sign problem.
Determinant quantum Monte Carlo (QMC) simulations of fermion models normally face the sign problem, which has a strong dependence of the interaction strength, electron filling, temperature, lattice geometry and lattice size. It is useful to set up the data set of such dependence, to make it clear which parameter region is accessible or not. To supplement the previous study, here we report the behavior of the average sign in the weak interaction region, as functions of lattice geometry, lattice size, electron filling and temperature. Furthermore, we find that in the weak interaction region, for several interesting lattice geometries (except for the kagome lattice) the average sign gets improved when the lattice sizes increase, which is in contrast to the general understanding of the lattice size dependence of the average sign. This finding points out that when the sign problem is present in the weak interaction region, the simulation of a larger lattice size can be better under control.
Initial characterizations of the fermion sign problem focused on its evolution with spatial lattice size LLL and inverse temperature $\ensuremath{\beta}$, emphasizing the implications of the exponential nature of the decay of the average sign $\ensuremath{\langle}\mathcal{S}\ensuremath{\rangle}$ for the complexity of its solution and associated limitations of quantum Monte Carlo studies of strongly correlated materials. Early interest was also on the evolution of $\ensuremath{\langle}\mathcal{S}\ensuremath{\rangle}$ with density $\ensuremath{\rho}$, either because commensurate filling is often associated with special symmetries for which the sign problem is absent, or because particular fillings are often primary targets, e.g., those densities, which maximize superconducting transition temperature (the top of the ``dome'' of cuprate systems). Here we describe an analysis of the sign problem, which demonstrates that the spin-resolved sign $\ensuremath{\langle}{\mathcal{S}}_{\ensuremath{\sigma}}\ensuremath{\rangle}$ already possesses signatures of universal behavior traditionally associated with order parameters, even in the absence of symmetry protection that makes $\ensuremath{\langle}\mathcal{S}\ensuremath{\rangle}=1$. When appropriately scaled, $\ensuremath{\langle}{\mathcal{S}}_{\ensuremath{\sigma}}\ensuremath{\rangle}$ exhibits universal crossings and data collapse. Moreover, we show these behaviors occur in the vicinity of quantum critical points of three well-understood models, exhibiting either second-order or Kosterlitz-Thouless phase transitions. Our results pave the way for using the average sign as a minimal correlator that can potentially describe quantum criticality in a variety of fermionic many-body problems.
Practically all the known Monte Carlo algorithms for identical fermions require the subtraction of separate contributions arising from, respectively, even and odd permutations of the particles. At low temperatures this becomes a serious problem since the two contributions nearly cancel. A new method is proposed for obtaining this difference directly. This amounts to a constraint that limits configuration space to a region bounded by the nodal surface of the fermionic ground-state wave function. A detailed description of the method is given for the general real Hamiltonian in the continuum where the partition function paths are approximated by ``polymers'' of finite length. The only departure from the usual bosonic treatment is the use of a nonlocal constraint which prohibits the polymer from simultaneously occupying regions of configuration space related by an odd permutation. With this constraint, the respective even- and odd-permutation images of the polymer are viewed as the two components in a phase-separated binary mixture. The condition for equilibrium of the interface formed between the two phases can be shown to be consistent with the boundary condition of the fermionic wave function at the nodal surface. Extension of the method to lattice systems and nonzero temperature is not straightforward and is not discussed.
Explicit treatment of many-body Fermi statistics in path integral Monte Carlo (PIMC) results in exponentially scaling computational cost due to the near cancellation of contributions to observables from even and odd permutations. Through direct analysis of exchange statistics we find that individual exchange probabilities in homogeneous systems are, except for finite size effects, independent of the configuration of other permutations present. For two representative systems, 3-He and the homogeneous electron gas, we show that this allows the entire antisymmetrized density matrix to be generated from a simple model depending on only a few parameters obtainable directly from a standard PIMC simulation. The result is a polynomial scaling algorithm and up to a 10 order of magnitude increase in efficiency in measuring fermionic observables for the systems considered.
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.