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Physical review. B, Condensed matter

Counting world lines for many-fermion systems

Michael J. Creutz

Published Mar 1, 1992• 10 citations
Publisher

Abstract

I explore a recursive enumeration of world-line diagrams as a numerical approach to simulating many-fermion lattice systems. Signs from fermion exchange are treated exactly. In addition, all values of the coupling constants are treated simultaneously. The method is computationally fast, although large-memory requirements restrict it to small spatial systems.

DOI
10.1103/physrevb.45.4650

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I explore a recursive enumeration of world-line diagrams as a numerical approach to simulating many-fermion lattice systems. Signs from fermion exchange are treated exactly. In addition, all values of the coupling constants are treated simultaneously. The method is computationally fast, although large-memory requirements restrict it to small spatial systems.
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.
I present a numerical algorithm for direct evaluation of multiple Grassmann integrals. The approach is exact and suffers no Fermion sign problems. Memory requirements grow exponentially with the interaction range and the transverse size of the system. Low dimensional systems of order a thousand Grassmann variables can be evaluated on a workstation. In quantum field theory fermions are usually treated via integrals over anti-commuting Grassmann variables [2], providing an elegant framework for the formal establishment of Feynman perturbation theory. With non-perturbative approaches, such as Monte Carlo studies on the lattice, these objects are more problematic. Essentially all approaches formally integrate the fermionic fields in terms of determinants depending only on bosonic fields. When a background
We present a general strategy to solve the notorious fermion sign problem using cluster algorithms. The method applies to various systems in the Hubbard model family as well as to relativistic fermions. Here it is illustrated for nonrelativistic lattice fermions. A configuration of fermion world lines is decomposed into clusters that contribute independently to the fermion permutation sign. A cluster whose flip changes the sign is referred to as a meron. Configurations containing meron clusters contribute 000 to the path integral, while all other configurations contribute 111. The cluster representation describes the partition function as a gas of clusters in the zero-meron sector.
An impurity model describes an atom or molecule embedded in some host or bath, with which it can exchange electrons. This exchange of electrons allows the impurity to make transitions between different quantum states, and leads to a non-trivial dynamics. Therefore, despite the zero dimensional nature (which makes impurity problems computationally much more tractable than fermionic lattice models), their numerical simulation remains a challenging task. Methods such as exact diagonalization or numerical RG, which explicitly treat a finite number of bath states, work well for single orbital models. However, because the number of bath states must be increased proportional to the number of orbitals, the computational effort grows exponentially with system size, and requires severe truncations of the bath already for two orbitals. Monte Carlo methods have the advantage that the bath is integrated out and thus the (infinite) size of the bath Hilbert space does not affect the simulation. While restricted to finite temperature, Monte Carlo methods are thus the method of choice for the solution of large multi-orbital or cluster impurity problems. Over the last few years, significant progress has been made (both in terms of efficiency and flexibility) with the development of diagrammatic Monte Carlo techniques. These notes provide an overview over two recently developed, complementary methods: (i) the weakcoupling approach, which scales favorably with system size and allows the efficient simulation of large impurity clusters, and (ii) the strong-coupling approach, which can handle impurity