Fermionic sign problem

A general solution of the fermion sign problem is still unavailable, although interesting algorithms have been proposed [11]. A class of methods try to... 

A. Alavi and A.J.W. Thom Path Resummations and the Fermion Sign Problem, Lect. Notes Phys. 703, 685-704 (2006)... 

We do not mean to imply that QMC has rigorously been shown to have a more favorable complexity this is the crux of the infamous fermion sign problem of QMC that we discuss later. Rather, we argue... 

The nonlocality, however, is a problem for the DMC simulations because the matrix element for the evolution of the imaginary-time diffusion is not necessarily positive. For realistic pseudopotentials the matrix elements are indeed negative and thus create a sign problem (even for one electron), with consequences similar to those of the fermion sign problem (see, e.g., work of Bosin et al. [49]). 

It seems that QMC has all the requisites to become the method of choice in the future, as, we believe, it can fulfill all three of these requirements. Clearly, QMC s ability to scale up and treat the many-body effects directly is invaluable. But QMC also has many other attributes. For example, it is straightforward to include thermal, zero-point, or classical nuclear effects in PIMC. Certainly, until the fermion sign problem is solved, there is always a question mark hanging over the field Is the method a fundamental advance, or is it merely a candidate for the most accurate approximate scheme currently known ... 

For the sake of simplicity, let us consider the case where 4> = 1. (This would be directly applicable to the fermion sign problem [30], which arises in imaginary-time QMC sampling of many fermionic systems. Extension to general real-time QMC simulations where is a complex factor is straightforward.) In this case, = 1 always and the variance of the signal is controlled entirely by the size ( ). Under normal circumstances, ( ) can be very smalt, for the reasons that we have described earlier, and the simulation becomes unstable at long time. 

Fermion sign problem in quantum Monte Carlo... 

Although ET processes occur in systems consisting of many electrons, essentially only one of them moves during the reaction. Therefore, the many-fermion problem can be reduced to an effective one-electron problem which has no fermionic exchange and hence no fermion sign problem. However, because in ET studies dynamical quantities such as the rate are the objects of interesL one has to confront the dynamic sign problem. 

While Section 4.4 considers electron transport in the framework of a single-charge tunneling problem, such a theory would not explain the conductivity of polyacetylene. It turns out that the conductivity of many ID conductors often involves many-electron effects, and to study ID electronic systems with path integral simulations, one has to first solve the fermion sign problem. 

In ID, the fermion sign problem can actually be eliminated completely. In lattice models of electronic systems, this is accomplished by a trick called the checkerboard decomposition. This method is best illustrated using noninteracting... 

To eliminate the fermion sign problem, the checkerboard decomposition follows an unconventional method to obtain the path integral. First, the single-particle Hamiltonian is decomposed into two pieces hj = /i /t , where /t is given by the... 

Virtually all of the successful path integral simulations of 2-d models for electronic systems have been carried out by the auxiliary field MC method, sometimes called the determinantal method. The only thing that complicates the computation of the fermion partition in equation (8) is the interaction action 5i. As explained in Section 5.3, without 5i, the sum over exchanges can be performed analytically. Therefore, if the two-electron interaction term can be eliminated or at least decoupled, the fermion sign problem could be partially removed. This can be accomplished by a so-called Hubbard-Stratonovich transformation. The details can be found in the original paper. Briefly, two electrons (of opposite spin) on the same site i experience a repulsion of strength U and add a term —eUni ni to the action Si, where = 0, 1 is the occupation number of an f-spin electron on site i, and n, is the same for a -spin electron. To decouple the two-electron interaction, the following transformation (correct up to a multiplicative constant) can be used. 

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