Superfluid Stiffness Definition

Question

I am currently reading S. Sachdevs Book on Quantum Phase Transitions focusing on the Bose-Hubbard Model (Chapter 9) and especially the Dilute-Boson Field Theory (Chapter 16).

When describing the fluid phase of the one dimensional model Sachdev says that this phase has quasi-long range order in the superfluid order parameter, intermediate boson occupation number and a non-zero superfluid stiffness.

I could not find any definition of a superfluid stiffness in the entire book and also doing some research on the internet I was not able to find a clean definition of superfluid stiffness in this context (Most likely because of my incapability :D to find something).

Therefore my question:

• Could somebody provide a definition of a superfluid stiffness in the context of the Bose-Hubbard Model?
• Any further explaination of this quantity in the "quasi long-range ordered" phase of the XY-chain would also be very kind?

Thank you all in advance.

Answer 1

There are probably different conventions that lead to definitions that differ by some numerical factor or factors of the mass density, but essentially the superfluid stiffness is the coefficient $\alpha$ in the expressions for the energy density
$$ E[\theta]= \int d^3 x\frac 12 \alpha |\nabla \theta|^2, $$ where $\theta$ is the phase of the superfluid order parameter. A non-zero $\alpha$ means that it costs energy to have a space-varying phase, hence "stiffness". The superfluid particle-number current is then $$ \rho_s{\bf v}_s = \alpha \nabla \theta, $$ where $\rho_s$ is the superfluid (number) density. As $${\bf v_s}=\frac 1 m \nabla\theta $$ where mass is $m$ of the superfluid particle one often writes $$ E= \int d^3 x\frac {\rho_s}{2m} |\nabla \theta|^2, $$ so $\alpha= \rho_s/m$. At finite temperature, the "energy" should be understood to be a local free energy $F=E-TS$.