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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.

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1 Answer 1

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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 $F=E-TS$.

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    $\begingroup$ Just a curiosity: the concept is similar to the one of "spin rigidity", e.g. en.wikipedia.org/wiki/Spin_stiffness "The spin stiffness or spin rigidity or helicity modulus or the "superfluid density" (for bosons the superfluid density is proportional to the spin stiffness) is a constant which represents the change in the ground state energy of a spin system as a result of introducing a slow in plane twist of the spins." $\endgroup$
    – Quillo
    Commented Oct 17, 2020 at 15:35
  • $\begingroup$ Could you clarify $\nabla \theta$, is this the spatial change of the twist angle? Could you maybe also clarify the origin of the second and third equation? I'm sorry, but I don't see it directly why the current and the velocity should be related to the twist angle in the following way? Thank you for your answer so far :). $\endgroup$
    – jan0155
    Commented Oct 17, 2020 at 15:39
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    $\begingroup$ The current/phase relation follows from imagining that the particles have a unit U(1) charge and coupling in a gauge field ${\bf A}$ as a probe. The gradient $\nabla\theta$ is therefore replaced by the gauge covarant derivative $\nabla\theta -{\bf A}$. The particle number current asociated with the gauge field is then given by the functional derivative of the energy/action wrt ${\bf A}$. This leads to my formula for the current. $\endgroup$
    – mike stone
    Commented Oct 17, 2020 at 15:51
  • $\begingroup$ @Quillo Yes they are the same concept, which is likely why Sachdev used the term "superfluid stiffness" rather than "superfluid density" or "helicity modulus" as he does earlier in the book. $\endgroup$ Commented Oct 17, 2020 at 20:25

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