Hubbard-Stratonovich Transformation and Fluctuating Fields

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. 

In the following we introduce a minimal transformation to decouple the interactions between polymer chains. Two different schemes have to be employed for the thermal interactions between the monomers and the incompressibility constraint. For the thermal interactions which give rise to the term (0a - in the Hamiltonian, we use the Hubbard-Stratonovich formula 

To rewrite the incompressibility constraint, we use the Fourier representation of the 5-function  

The field W, which couples to the composition of the mixture, gives rise to a real contribution in the exponent, while the field 17, which enforces incompressibility, yields a complex contribution. 

We note that this complex term cannot be avoided easily. Replacing the incompressibility constraint by a finite compressibihty 

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