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arXiv:hep-lat/0212034 · International Journal of Modern Physics C

DIRECT SIMULATIONS OF SMALL MULTI-FERMION SYSTEMS

Michael J. Creutz

Published Oct 1, 2003• 0 citations
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Abstract

I explore computer simulations of the dynamics of small multi-fermion lattice systems. The method is more general, but I concentrate on Hubbard type models where the fermions hop between a small number of connected sites. I use the natural mapping of fermion occupation numbers onto computer bits. Signs from fermion interchange are reduced to bit counting. The technique inherently requires computer resources growing exponentially with the system volume; so, it restricted to modestly small systems. Large volume results would require combining these techniques with further approximations, perhaps in a recursive renormalization group manner.

DOI
10.1142/s0129183103005169

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I explore computer simulations of the dynamics of small multi-fermion lattice systems. The method is more general, but I concentrate on Hubbard type models where the fermions hop between a small number of connected sites. I use the natural mapping of fermion occupation numbers onto computer bits. Signs from fermion interchange are reduced to bit counting. The technique inherently requires computer resources growing exponentially with the system volume; so, it restricted to modestly small systems. Large volume results would require combining these techniques with further approximations, perhaps in a recursive renormalization group manner.
I explore computer simulations of the dynamics of small multi-fermion lattice systems. The method is more general, but I concentrate on Hubbard type models where the fermions hop between a small number of connected sites. I use the natural mapping of fermion occupation numbers onto computer bits. Signs from fermion interchange are reduced to bit counting. The technique inherently requires computer resources growing exponentially with the system volume; so, it restricted to modestly small systems. Large volume results would require combining these techniques with further approximations, perhaps in a recursive renormalization group manner.
The calculation time for the energy of atoms and molecules scales exponentially with system size on a classical computer but polynomially using quantum algorithms. We demonstrate that such algorithms can be applied to problems of chemical interest using modest numbers of quantum bits. Calculations of the water and lithium hydride molecular ground-state energies have been carried out on a quantum computer simulator using a recursive phase-estimation algorithm. The recursive algorithm reduces the number of quantum bits required for the readout register from about 20 to 4. Mappings of the molecular wave function to the quantum bits are described. An adiabatic method for the preparation of a good approximate ground-state wave function is described and demonstrated for a stretched hydrogen molecule. The number of quantum bits required scales linearly with the number of basis functions, and the number of gates required grows polynomially with the number of quantum bits.
In this paper, we propose a framework for studying the properties of the Lefschetz thimbles decomposition for lattice fermion models approaching the thermodynamic limit. The proposed set of algorithms includes the Schur complement solver and the exact computation of the derivatives of the fermion determinant. It allows us to solve the gradient flow equations taking into account the fermion determinant exactly, with high performance. Being able to do so, we can find both real and complex saddle points and describe the structure of the Lefschetz thimbles decomposition for large enough lattices which allows us to extrapolate our results to the thermodynamic limit. We describe the algorithms for a general lattice fermion model, with emphasis on two widely used types of lattice discretizations for relativistic fermions (staggered and Wilson fermions), as well as on interacting tight-binding models for condensed matter systems. As an example, we apply these algorithms to the Hubbard model on a hexagonal lattice. Several technical improvements allow us to deal with lattice volumes as large as $12\ifmmode\times\else\texttimes\fi{}12$ with ${N}_{\ensuremath{\tau}}=256$ steps in Euclidean time, in order to capture the properties of the thimbles decomposition as the thermodynamic, low-temperature, and continuum limits are approached. Different versions of the Hubbard-Stratonovich (HS) transformation were studied, and we show that the complexity of the thimbles decomposition is very dependent on its specific form. In particular, we provide evidence for the existence of an optimal regime for the hexagonal lattice Hubbard model, with a reduced number of thimbles becoming important in the overall sum. In order to check these findings, we have performed quantum Monte Carlo (QMC) simulations using the gradient flow to deform the integration contour into the complex plane. These calculations were made on small volumes (Ns=8{N}_{s}=8Ns​=8 sites in space), albeit still at low temperatures and with the chemical potential tuned to the van Hove singularity, thus entering into a regime where standard QMC techniques exhibit an exponential decay of the average sign. The results are compared versus exact diagonalization, and we demonstrate the importance of choosing an optimal form for the HS transformation for the Hubbard model to avoid issues associated with ergodicity. We compare the residual sign problem with the state-of-the-art BSS (Blankenbecler, Scalapino, and Sugar)-QMC and show that the average sign can be kept substantially higher using the Lefschetz thimbles approach.