Transformation Hubbard—Stratonovich

This transformation is a generalization of a result for multivariate Gaussian integrals to functionals so that for any real, symmetric, positive-definite operator A(r, r ). 

Note that both of these identities are also vahd when positive sign in front of linear J(i) term is replaced by a negative sign. This is a generalization of the fact that for simple Gaussian integrals 

Furthermore, in these equations, inverse operator A (r, r ) is defined through the relation 

Similarly, taking into account the fact that the Poisson s equation must be satisfied even for position-dependent dielectric constant [78], it can be shown that inverse operator for A(r,r ) = l/ (r)[r-r [ is A-i(r,r ) = -6(r-r )V,-e(r )V,V4jt. 

In order to use this transformation for the Hamiltonian as represented by Eq. (6.68), microscopic density terms that are quadratic in nature need to be written in the form given on the left-hand side in Eqs. (6.77) and (6.78). Electrostatic terms in He are already in the appropriate form. It is only the terms in Hw that needs to be rewritten. This can be achieved by rewriting in terms of order parameters and total density. For an n component system, all microscopic densities can be described by n—1 independent order parameters (due to the incompressibility constraint serving as the nth relation among the densities). There are many different ways of defining these order parameters. One convenient definition, which makes mathematics simple, is the deviation of densities of solutes from the solvent density, that is, defining c )j(r) = Qj(r)—Qj(r) forj = 1,2. (n—1), wherej is the index for different solutes (monomers, counterions, and the salt ions). Using the transformation for each quadratic term in the Hamiltonian (cf. Eq. (6.68)), the partition function becomes 

---

Here a = a t - 1), t = T/T°P, and a, b, c are either phenomenological parameters or else they can be calculated from mean-field calculations [48] or better yet, by using the Hubbard-Stratonovich transformation to convert the partition function into a functional integral [39]. In the latter case, one obtains around t = 1 the values... 

The partition function Eq. 6 describes a system of mutually interacting chains. Introducing auxiliary fields, U and W, via a Hubbard-Stratonovich transformation, one can decouple the interaction between the chains and rewrite the Hamiltonian in terms of independent chains in fluctuating fields. Then, one can integrate over the chain conformations and obtain a Hamiltonian which only depends on the auxiliary fields. Thermodynamic averages like density or structure factors can be expressed as averages over the fields, U and W, without approximation. 

By the Hubbard-Stratonovich transformation we have rewritten the partition function of the interacting multi-chain systems in terms of noninteracting chains in complex fluctuating fields, il7 -i- W and il7 - W. hi field theoretical polymer simulations, one samples the fields U and W via computer simulation using the above Hamiltonian (cf. Sect. 4.4). 

Can we expect this to be the case here To estimate the range of the reweighting factor, we briefly re-inspect the Hubbard-Stratonovich transformation of the total density + < b that leads to the fluctuating field 17. For simplicity we consider a one-component system. In a compressible polymer solution or blend, the contribution of the repulsive interaction energy to the partition function can be written as... 

Hubbard-Stratonovich Transformation Field-Theoretic Reformulation of the Particle-Based Partition Function... 

A number of attempts have been made to incorporate the effect of composition fluctuations [86-88] in theories involving neutral polymers. Here, we present systematic one-loop expansion to go beyond the saddle-point approximation described in the previous section. In order to carry out the loop expansion, it is advantageous to use Hubbard-Stratonovich transformation to get rid of redundant functional integrals over collective density variables (q in Eq. (6.85)) and use Eq. (6.81) as the starting point for the partition function with the explicitly known normalization constants except A,. Saddle-point approximation within this formalism now requires taking functional derivatives with respect to fields only. 

Now, using the methods of collective variables (cf. Section 6.4.2.1) for decoupling all the interactions except the electrostatics and the Hubbard-Stratonovich transformation [14, 55] (cf. Section 6.4.2.2) for the electrostatic part in Eq. (6.124), the partition function can be written as integrals over the collective densities and corresponding fields so that Eq. (6.124) becomes... 

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. 

---