Superfluid Order and Phase Stiffness

Superfluidity is phase rigidity measured by a free-energy cost or a transverse response, not merely the existence of an anomalous average. In a neutral paired fluid that average transforms under the physical global particle-number $U(1)$; in a charged theory it is gauge covariant. For the neutral system, the helicity modulus is the second derivative of the thermodynamic free energy with respect to a boundary twist; at long wavelengths it is the coefficient of $(\boldsymbol\nabla\theta)^2$. Its conversion to a “superfluid density” depends on whether $\theta$ is a particle or pair phase and on whether the system is continuum or lattice.

Required background. Phase–density hydrodynamics supplies the long-wavelength action. Symmetry realization and Goldstone poles distinguish an order parameter from a response. Kubo response fixes the order-of-limits problem.

Twist definition of phase rigidity

Put a system of linear size $L_i$ on a torus and impose

$$\theta(\mathbf x+L_i\hat{\mathbf e}_i) =\theta(\mathbf x)+\Phi_i.$$

For small twist, define the stiffness tensor $\rho_{s,ij}$ by

$$F(\boldsymbol\Phi)-F(0) =\frac{V}{2}\rho_{s,ij}\frac{\Phi_i}{L_i}\frac{\Phi_j}{L_j}+O(\Phi^4).$$

This definition is gauge invariant for a neutral system and applies on a lattice without invoking Galilean invariance. A normal metal can have a nonzero charge stiffness or Drude weight, but its equilibrium free energy is not rigid to a twist of a spontaneously selected pair phase. The two notions coincide only under additional assumptions.

If $\theta$ denotes the phase of a pair of particles of mass $m$, then the superflow velocity is $\mathbf v_s=\boldsymbol\nabla\theta/(2m)$ and

$$\frac12\rho_m\mathbf v_s^2 =\frac{n_s}{8m}(\boldsymbol\nabla\theta)^2.$$

Thus the coefficient called $\rho_s$ in a phase-only action may be $n_s/(4m)$, $n_s/m$, or a lattice helicity modulus depending on the phase normalization. Always define it through the free energy before translating symbols.

Static transverse response

Couple a neutral conserved current to a fictitious vector source. The stiffness is determined by the transverse, static kernel,

$$\rho_{s,ij}=K_{ij}^{\mathrm{dia}} +\lim_{\mathbf q\to0}K_{ij}^{\mathrm{para}}(\mathbf q,\omega=0) \quad (i,j\perp\mathbf q).$$

The sign shown assumes the full response kernel is the sum of diamagnetic and retarded paramagnetic pieces. Literature that defines the current correlator with an extra minus sign prints a subtraction. The invariant check is that the normal state’s uniform static transverse response cancels in a gauge-invariant continuum calculation, whereas the superfluid leaves a positive rigidity.

The order of limits matters. The equilibrium stiffness takes $\omega=0$ before $\mathbf q\to0$. The Drude weight probes the homogeneous dynamic limit, often $\mathbf q=0$ before $\omega\to0$. Translationally invariant normal matter can have the latter without the former. Scalapino, White, and Zhang 1993, §§II–III gives a clean lattice comparison.

Dimension, finite size, and vortices

At nonzero temperature in two dimensions, long-range order of a continuous phase is absent in the thermodynamic limit, yet the low-temperature phase can have algebraic correlations and nonzero helicity modulus. Vortex unbinding drives the Berezinskii–Kosterlitz–Thouless transition, with the thermodynamic-limit jump

$$\rho_s(T_{\mathrm{BKT}}^-)=\frac{2T_{\mathrm{BKT}}}{\pi}$$

when the phase action is $F=(\rho_s/2)\int\mathrm d^2x\,(\nabla\theta)^2$ and elementary vortices have $2\pi$ winding. Finite systems round the jump; extrapolation must use size dependence rather than a single nonzero value. Nelson and Kosterlitz 1977 derives the universal relation.

The chapter structure makes stiffness the bridge from a paired saddle to defect energetics and, after coupling charge, Meissner response.

Phase stiffness is an observable free-energy curvature. It is distinct from condensate fraction, a quasiparticle gap, the homogeneous Drude weight, and the charged Meissner kernel until the appropriate hypotheses connect them. Original schematic, not to scale.

The paired-matter claim test matrix records these definitions and their negative tests.

Exercise

Convert a pair-phase coefficient. A continuum paired fluid has mass density $mn_s$ and pair phase $\theta$. Express its kinetic energy in terms of $\nabla\theta$ and identify the helicity modulus in the convention $F=(\rho_s/2)\int(\nabla\theta)^2$.

Solution

Because each constituent has mass $m$ and the pair phase winds with charge two under particle number, $\mathbf v_s=\nabla\theta/(2m)$. Hence the energy density is $(mn_s/2)v_s^2=n_s(\nabla\theta)^2/(8m)$. Comparing with $\rho_s(\nabla\theta)^2/2$ gives $\rho_s=n_s/(4m)$. Using the single-particle phase instead would remove the factor four.

References

• Nelson, D. R., and Kosterlitz, J. M. (1977). “Universal jump in the superfluid density of two-dimensional superfluids.” Physical Review Letters 39, 1201–1205. doi:10.1103/PhysRevLett.39.1201.
• Scalapino, D. J., White, S. R., and Zhang, S. C. (1993). “Insulator, metal, or superconductor: The criteria.” Physical Review B 47, 7995–8007. doi:10.1103/PhysRevB.47.7995.