**Membrane Inclusions**

Long-ranged forces due to thermal fluctuations predicted in many *correlated fluids* in soft matter:

Liquid Crystals, polyelectrolyte solutions, polymer melts, ...

Inclusions on a membrane

Helfrich energy of a membrane with surface tension and bending rigidity:

There is considerable literature on the fluctuation-induced interactions between objects on such membrane:

M. Goulian, R. Bruinsma and P. Pincus, Euro. Phys. Lett.

22, 145 (1993)- - -

E. Noruzifar, J. Wagner, R. Zandi, Phys. Rev. E

88, 042314 (2013)General form of the interaction for particles of dimension

ahas the form:

Composition fluctuations of a membrane

Lipid mixtures composing a membrane can exhibit composition fluctuations, possibly poised close to an Ising critical point.

The resulting fluctuation-induced interactions are determined (at large separation) by appropriate critical exponents:

B.B. Machta, S.L. Veatch, and J.P. Sethna, Phys. Rev. Lett.

109, 138101 (2012) (circles)O. A. Vasilyev, E. Eisenriegler, and S. Dietrich, Phys. Rev. E

88, 012137 (2013) (needles)

Appealing to conformal invariance at two dimensional criticality, interactions can be computed for arbitrary shapes:

G. Bimonte, T. Emig, and M. Kardar, EuroPhys. Lett.

104, 21001 (2013) (any shape)

The interaction between wedges is set by dimensional arguments, up to an overall coefficient:

For certain critical systems (e.g. the tricritical Ising model), and boundary conditions, it can change sign.

A stable fixed point can be obtained by blunting the wedge at its tip!