Path integral simulations sign problem

It is necessary to sum over these pemuitations in a path integral simulation. (The same sum is needed for bosons, without the sign factor.) For femiions, odd pemuitations contribute with negative weight. Near-cancelling positive and negative pemuitations constitute a major practical problem [196]. 

Path integral simulations of real-time dynamics suffer from another form of the sign problem - the dynamic sign problem . In equation (10), similarly to the fermionic problem, the paths carry non-positive-definite weights. 

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. 

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. 

This article offers a brief overview of applications of path integral simulations to various aspects of condensed-phase electronic systems. Despite the success of some of the methods described here, many challenges remain for path integral simulations. In particular, a general solution to the sign problem in fermion and real-time problems is of fundamental importance, and such a general method has yet to be found. This search will no doubt dominate research activities in the field of path integral simulations for the next few years. 

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