Condensation, Off-Diagonal Order, and Superfluidity

Condensation, off-diagonal long-range order, spontaneous breaking of particle-number $U(1)$, and superfluid response are related in common three-dimensional weak fluids, but they are not definitions of one another. Their diagnostics are respectively a macroscopic one-body eigenvalue, a long-distance correlator, an order-of-limits statement with a symmetry-breaking source, and a free-energy or current response.

Required background. Use the ideal-gas condensation criterion and the general distinction between symmetry realization and order parameters. Helpful background. Superfluid hydrodynamics supplies the response-language continuation.

Condensation, broken symmetry, stiffness, and flow

For a bosonic field, define the one-body density matrix

$$\rho_1(\mathbf x,\mathbf y)=\langle\psi^\dagger(\mathbf x)\psi(\mathbf y)\rangle.$$

Condensation means that its largest eigenvalue $N_0$ is extensive. In a stationary uniform system without imposed flow, off-diagonal long-range order (ODLRO) means $\rho_1(\mathbf r,0)\to n_0$ as $r\to\infty$. A global condensate phase cancels from this gauge-invariant two-point function; with a phase gradient, the asymptotic factor records the phase difference rather than an absolute phase. The eigenvalue and long-distance formulations agree under the usual homogeneous clustering assumptions, but algebraic decay is only quasi-long-range order.

At finite volume, a number eigenstate has $\langle\psi\rangle=0$. Spontaneous breaking is therefore an ordered limit:

$$\lim_{h\to0^+}\lim_{V\to\infty}\langle\psi\rangle_h\neq0,$$

with $h$ an explicit source. Reversing the limits returns zero. The phase of $\langle\psi\rangle_h$ labels a broken-symmetry representation; it is not a finite-system observable by itself.

Superfluidity is a response property. Impose a slow phase twist $\boldsymbol\varphi$ across a box. For volume $V$,

$$F(\boldsymbol\varphi)-F(0) =\frac{V}{2}\rho_s\left(\frac{\boldsymbol\varphi}{mL}\right)^2+o(\varphi^2),$$

which defines the superfluid mass density $\rho_s$ in the continuum convention. Equivalent transverse-current and winding-number definitions require their own limit and normalization. Leggett’s discussion makes clear why the response definition, rather than an anomalous average alone, carries the physical content Leggett 1999, pp. 318–323.

Counterexamples that separate them

The three-dimensional ideal gas condenses, yet its lack of interactions makes metastable flow and the usual finite critical-velocity argument singular. A two-dimensional BKT fluid at nonzero temperature has finite stiffness and algebraically decaying $\rho_1$, but no ODLRO. A one-dimensional zero-temperature Luttinger liquid likewise has algebraic order without an extensive zero-momentum eigenvalue. A Bose glass is compressible and may show local coherence, but its thermodynamic stiffness vanishes.

These counterexamples also show why a narrow momentum peak is insufficient: finite imaging resolution, trapping, and quasi-long-range order can all produce one without a nonzero thermodynamic helicity modulus.

A phase-classification protocol

For any proposed Bose phase, state the spatial dimension, temperature, geometry, disorder, ensemble, and order of $V$, $h$, $q$, and $\omega$ limits. Then report separately:

1. the scaling of the largest eigenvalue of $\rho_1$;
2. the asymptotic form of $\rho_1(r)$;
3. the stiffness or transverse-current response;
4. the compressibility and excitation gap; and
5. finite-size, trap, and disorder extrapolations.

Only the conjunction warranted by these measurements should appear in the conclusion. The distinction between BEC and superfluidity in dimensions and geometries of interest is treated systematically in Pitaevskii and Stringari 2016, chs. 2 and 9.

Exercises

Explain why an algebraic correlator $\rho_1(r)\sim r^{-\eta}$ in a square of side $L$ does not yield an extensive condensate eigenvalue.

Solution

The zero-momentum occupation scales as $N_0\sim\int_{L^2}\mathrm d^2r\,\rho_1(r)\sim L^{2-\eta}$. Since $N\sim L^2$, the condensate fraction scales as $N_0/N\sim L^{-\eta}\to0$ for every $\eta>0$, even though coherence decays only algebraically.

References

• Anthony J. Leggett, “Superfluidity,” Reviews of Modern Physics 71 (1999) S318–S323, doi:10.1103/RevModPhys.71.S318.
• Lev P. Pitaevskii and Sandro Stringari, Bose–Einstein Condensation and Superfluidity, Oxford University Press (2016), chs. 2 and 9, doi:10.1093/acprof:oso/9780198758884.001.0001.