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Thermal Casimir interactions for higher derivative field Lagrangians: generalized Brazovskii models

We examine the Casimir effect for free statistical field theories which have Hamiltonians with second order derivative terms. Examples of such Hamiltonians arise from models of non-local electrostatics, membranes with non-zero bending rigidities and field theories of the Brazovskii type that arise for polymer systems. The presence of a second derivative term means that new types of boundary conditions can be imposed, leading to a richer phenomenology of interaction phenomena. In addition zero modes can be generated that are not present in standard first derivative models, and it is these zero modes which give rise to long range Casimir forces. Two physically distinct cases are considered: (i) unconfined fields, usually considered for finite size embedded inclusions in an infinite fluctuating medium, here in a two plate geometry the fluctuating field exists both inside and outside the plates, (ii) confined fields, where the field is absent outside the slab confined between the two plates. We show how these two physically distinct cases are mathematically related and discuss a wide range of commonly applied boundary conditions. We concentrate our analysis to the critical region where the underlying bulk Hamiltonian has zero modes and show that very exotic Casimir forces can arise, characterised by very long range effects and oscillatory behavior that can lead to strong metastability in the system.