Reaching the Continuum Limit in Finite-Temperature Ab Initio Field-Theory Computations in Many-Fermion Systems

Abstract

Finite-temperature, grand-canonical computations based on field theory are widely applied in areas including condensed matter physics, ultracold atomic gas systems, and lattice gauge theory. However, these calculations have computational costs scaling as N 3 s with the size of the lattice or basis set, Ns. We report a new approach based on systematically controllable low-rank factorization which reduces the scaling of such computations to NsN 2 e , where Ne is the average number of fermions in the system. In any realistic calculations aiming to describe the continuum limit, Ns/Ne is large and needs to be extrapolated effectively to infinity for convergence. The method thus fundamentally changes the prospect for finite-temperature many-body computations in correlated fermion systems. Its application, in combination with frameworks to control the sign or phase problem as needed, will provide a powerful tool in ab initio quantum chemistry and correlated electron materials. We demonstrate the method by computing exact properties of the two-dimensional Fermi gas with zero-range attractive interaction, as a function of temperature in both the normal and superfluid states.Computations are playing an increasingly important role in addressing the fundamental challenges of understanding strong correlations in interacting quantum systems. Understandably, a major part of the development and application efforts in both physics and chemistry have focused on ground-state properties. However, experimental conditions are always at finite temperatures, where often rich and new properties can be revealed [1-4]. One example is the rapid development in the area of experiments with ultracold atoms, where temperature plays a crucial role and very precise measurements of properties are often possible with exquisite control over interaction strengths, environments, etc [5,6]. Accurate computations of thermodynamic properties allow direct comparisons with experiments, but are challenging because of the presence of strong coupling and thermal fluctuations. A second example is in strongly correlated materials, including high-temperature superconductors [7,8], where some of the outstanding and most interesting physics questions concern finite-temperature properties [9].A common finite-temperature formalism is based on field theory in which the thermodynamic properties are computed as path-integrals in field space. Approximations can be used to perform the path integrals, including the simplest, namely mean-field calculations. A more powerful approach is to evaluate the manydimensional integration by Monte Carlo methods [10]. This has become a key technique for many-body finitetemperature computations, widely applied in several fields of physics [11][12][13][14][15][16][17][18][19][20][21][22]. For example, many of the sign-problem-free computations in lattice models in condensed matter, and in Fermi gas and optical lattices of ultracold atoms, have been performed this way. For general Hamiltonians, for instance the doped Hubbard model or rea...

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