The purpose of the Concepts in Condensed Matter Journal Club is to create a forum for critical discussion of the ideas that form the foundations of our field.

**Pizza**, physics, and love. What more could you want?

Come Join us in 57 Birge, every other Wednesday at 4pm.

Topics so far have included:

- Symmetry breaking
- The Landau theory of second order phase transitions
- Landau’s theory of the Fermi Liquid
- The fluctuation-dissipation theorem
- Onsager’s Reciprocal relations
- The Integer Quantum Hall Effect
- The Wigner crystal
- Exchange, superexchange and magnetic order
- Luttinger’s theorem
- Luttinger-Tomonaga 1D liquids
- Goldstone Modes and Generalized rigidity
- Cooper Pairing
- Weyl Fermions

Today we looked at Goldstone modes and the idea of generalized rigidity. Reading included CH2 of P. W. Anderson’s “Basic Notions in Condensed Matter” and CH9 of James Sethna’s “Entropy, Order Parameters and complexity.”

I think there are two things missing in the reading that I couldn’t find in many places:

1) A general elaboration on the connection between the Landau free energy and the excitation spectrum of the unsymmetric ground state.

2) A broader discussion of the consequences of having unsymmetric ground states that are and are not eigenstates of the Hamiltonian (i.e. contrast a ferromagnet to an anti-ferromagnet)

The subtlety of (1) comes from the general fact that if a system is rigid, there will always be a $(\nabla \Phi)^2$ in the free energy, where $\Phi$ will be associated with the continuous symmetry. The reason why such a term appears is because the system is rigid in $\Phi$. For example, in a crystal the system is rigid in displacement $\Phi=u$, in a ferromagnet to the rotation of spin $\Phi=\theta$ and in a superfluid to a particle’s phase $\Phi=\phi$. Such rigidity means that the free energy cannot depend on the absolute value of the position of a crystal, magnetization direction of a ferromagnet or phase of a particle in a superfluid. But it does depend on

gradientsof these. So the free energies of these systems all end up having terms like $(\nabla \Phi)^2$ – the elastic energy in the case of a crystal, $(\nabla M)^2$ in a ferromagnet or phase stiffness in a superfluid. Each of these terms immediately imply that there is an energy cost to changes in displacement, magnetization or the phase as function of distance. This is of course how rigidity is constructed into the free energy, but it is also how the Goldstone modes arise – waves in any of these objects costs energy, leading to a dispersion for what we know as phonons in a crystal, spin waves in a ferromagnet and whatever superfluids have….–James