Low-energy theory for superfluid and solid matter and its application to the neutron star crust

We formulate a low-energy effective theory describing phases of matter that are both solid and superfluid. These systems simultaneously break translational symmetry and the phase symmetry associated with particle number. The symmetries restrict the combinations of terms that can appear in the effective action and the lowest order terms featuring equal number of derivatives and Goldstone fields are completely specified by the thermodynamic free energy or, equivalently, by the long-wavelength limit of static correlation functions in the ground state. We show that the underlying interaction between particles that constitute the lattice and the superfluid gives rise to entrainment, and mixing between the Goldstone modes. As a concrete example we discuss the low-energy theory for the inner crust of a neutron star, where a lattice of ionized nuclei coexists with a neutron superfluid.

Figure 1

The velocities of the two eigenmodes. The dotted lines are $v_l$ and $v_{\phi }$ with mixing ignored.