Superfluidity in Condensed Matter

Superfluidity is not one measurement and not one order parameter. A material claim may concern equilibrium stiffness, a mass superfluid fraction, metastable flow, quantized circulation, a two-fluid sound mode, or vortex response. Those objects are related, but they are not synonyms. This page supplies the ledger needed to compare them and to decide what a combined experimental record actually licenses.

Required background. Bose–Einstein Condensation owns Penrose–Onsager occupation and dimensional limits. Off-Diagonal Long-Range Order owns number-conserving one- and two-body coherence.

Helpful background. Use Density and Current Operators and Order Parameters for source and response language; Weakly Interacting Bose Gas Preview for a controlled dilute-gas spectrum; and Low-Dimensional Quantum Gases for the phase-only and BKT branch. Thermodynamic Limit, Finite-Size Effects, Goldstone Modes, Collective Modes, and Hydrodynamics and Effective Theory Preview become useful on the corresponding branches. Lattice and path-integral questions route to the Bose–Hubbard Model and Quantum Monte Carlo Preview.

Which Neutral-Superfluid Claim Is Being Made?

Before using a formula, record this ten-field claim ledger. If a field is irrelevant, say why; a blank field is an untested assumption.

1. Platform and state: species, continuum or lattice, dimension, density, geometry, ensemble, temperature, pressure, rotation, confinement, disorder, substrate, preparation, and history.
2. Claimed object: occupation, ODLRO, algebraic order, stiffness, helicity modulus, number or mass superfluid density, normal density, critical velocity, persistent current, circulation, vortex, or sound pole.
3. Carrier and normalization: coherent mass $M$, atomic or pair number, bulk or areal units, tensor axes, phase convention, and per-particle, per-volume, or per-area normalization.
4. Symmetry and topology: physical global $U(1)$ symmetry, number-conserving or source-selected description, limit order, winding sector, contour, defects, and multicomponent identifications.
5. Scale hierarchy: interaction, chemical potential, $T$, stiffness, compressibility, healing length, core size, mean free path, system or pore size, flow, rotation, $(\omega,\mathbf q)$, damping, equilibration, drive, and resolution.
6. Framework and validity: Galilean continuum, lattice boson, dilute gas, phase-only, two-fluid, BKT, vortex, path-integral, porous-medium, or driven description, including its failure test.
7. Source, observable, and limits: twist, moving wall, rotation, heat or pressure pulse, imposed current, or spectral probe, with component, derivative, limit order, and equilibration protocol.
8. Forward model: cell, annulus, substrate, pore coupling, slip, roughness, weak links, loading, form factors, acceptance, backgrounds, heating, and resolution.
9. Evidence and alternatives: equilibrium, rotational, thermodynamic, flow, sound, spectral, interference, and vortex evidence versus ballistic flow, finite coherence, viscoelastic decoupling, percolation, heating, structural change, metastability, or instrumental artifacts.
10. Uncertainty and stopping rule: calibration, geometry, model, finite-size, finite-frequency, sample, and extrapolation errors, plus the observation that would falsify or narrow the claim.

The ledger prevents an attractive but invalid shortcut: a condensate peak is not automatically a mass-response measurement, and a low-dissipation flow record is not automatically an equilibrium stiffness measurement.

Condensation, ODLRO, and Stiffness Are Different

A condensate is defined by a macroscopic eigenvalue of a reduced density matrix. ODLRO describes the long-distance behavior of a correlation function. A helicity modulus is a derivative of the equilibrium free energy with respect to a phase twist. Robust flow additionally requires a decay barrier on the experimental timescale. These statements ask different questions.

Two standard counterexamples make the separation concrete:

• The homogeneous ideal Bose gas below its condensation temperature has macroscopic occupation, yet its quadratic spectrum gives a zero Landau velocity. Condensation alone does not guarantee metastable flow.
• A homogeneous two-dimensional BKT fluid has finite long-distance stiffness and algebraic order at nonzero temperature but no true one-body ODLRO in the thermodynamic limit.

Interacting bulk $^4$He gives the complementary material lesson: its low-temperature superfluid fraction can approach unity while its condensate fraction remains much smaller. The response of the whole liquid is not the count of particles in one orbital.

For a finite fixed-number system, $\langle\Psi\rangle$ need not be nonzero. Either declare the symmetry-breaking source and the order in which it and the volume limits are removed, or use number-conserving density matrices and response coefficients.

Static Twist Response and Superfluid-Density Conventions

Let a neutral coherent field be written locally as $\Psi=\sqrt n\,e^{i\theta}$, with dimensionless phase $\theta$ and coherent carrier mass $M$. Away from defects, rotation, and synthetic gauge sources, define

$$\mathbf v_s = \frac{\hbar}{M} \boldsymbol\nabla\theta .$$

Reversing the phase convention reverses phase-gradient and winding signs, not physical energies. Rotation and synthetic sources require separate signed definitions; they are not the electromagnetic vector potential of the charged London problem.

For a small, slowly varying twist about an equilibrated defect-free state, define the helicity-modulus tensor $\Upsilon_{ij}$ through the leading local quadratic response,

$$F[\theta]-F[0] = \frac12 \int d^d x\, \Upsilon_{ij} \partial_i\theta\, \partial_j\theta = \frac12 \int d^d x\, \rho^{\,s}_{ij} v_{s,i}v_{s,j}.$$

Higher powers of the phase gradient, higher spatial derivatives, and possible nonlocal response kernels are omitted from this long-wavelength expansion. They must be restored near defects, sharp boundaries, strong flow, or a nonlocal instability.

In a Galilean continuum with the declared mass,

$$\Upsilon_{ij} = \frac{\hbar^2}{M^2} \rho^{\,s}_{ij}, \qquad n^{\,s}_{ij} = \frac{\rho^{\,s}_{ij}}{M}.$$

This conversion is not automatic in a lattice, mixture, multiband system, porous host, or driven fluid. There, the directly defined object may be a stiffness rather than a Galilean mass fraction.

For a dimensionless twist $\varphi_i$ across a rectangular box of length $L_i$ and volume $V$,

$$F(\boldsymbol\varphi)-F(\mathbf0) = \frac{V}{2} \sum_{ij} \Upsilon_{ij} \frac{\varphi_i}{L_i} \frac{\varphi_j}{L_j} +o(\varphi^2),$$

so that

$$\Upsilon_{ij} = \left. \frac{L_iL_j}{V} \frac{\partial^2F} {\partial\varphi_i\partial\varphi_j} \right|_{\boldsymbol\varphi=\mathbf0}.$$

The displayed formulas use the Helmholtz free energy at fixed $(N,V,T)$. At fixed chemical potential use the grand potential, and for another ensemble differentiate the corresponding equilibrium potential while holding its natural controls fixed. At zero temperature use the matching constrained ground-state energy. An equilibrium stiffness first equilibrates the finite system at strictly zero frequency and fixed twist, then takes a declared thermodynamic and aspect-ratio limit. A current-response construction takes the static transverse long-wavelength limit with its source convention, contact term, and normalization declared; those details remain with the density, current, and susceptibility owners. This is not the uniform dynamic order $\mathbf q=0$ followed by $\omega\to0$ that defines a Drude-weight question.

For periodic equilibrium worldlines at zero imposed twist or current source, where equilibrium symmetry gives $\langle W_i\rangle=0$, a common rectangular-box convention gives

$$\frac{\rho^{\,s}_{ij}}{\rho} = \frac{M L_iL_j} {\beta\hbar^2N} \left\langle W_iW_j\right\rangle .$$

For a cubic isotropic average,

$$\frac{\rho_s}{\rho} = \frac{M L^2} {\beta\hbar^2Nd} \left\langle |\mathbf W|^2\right\rangle .$$

Here $N$ is the total number of identical coherent carriers of mass $M$, $\rho=MN/V$, $\beta=(k_BT)^{-1}$, and $W_i$ is the net dimensionless winding of all worldlines around direction $i$ in a periodic box of length $L_i$. At nonzero source, use the connected covariance $\langle W_iW_j\rangle-\langle W_i\rangle\langle W_j\rangle$. Sampling must converge across winding sectors. Production estimators and autocorrelation control remain with the Quantum Monte Carlo Preview.

Normal Density and the Controlled Two-Fluid Regime

Only in a nondissipative single-component Galilean two-fluid regime may one use

$$\rho=\rho_s+\rho_n, \qquad \mathbf j_{\mathrm{mass}} = \rho_s\mathbf v_s+ \rho_n\mathbf v_n, \qquad \mathbf j_S=s\mathbf v_n,$$

where $s$ is entropy density. Heat conduction, viscosity, mutual friction, and other dissipative terms correct the entropy current and produce entropy; $\mathbf j_S=s\mathbf v_n$ is only the ideal nondissipative identity. The superfluid and normal components are response components, not permanently labeled microscopic species. Mixtures, crystals, anisotropic media, and multicomponent condensates may require tensor densities and an entrainment matrix.

For a controlled equilibrium quasiparticle gas with the Galilean momentum-current relation,

$$\rho^{\,n}_{ij} = -\sum_a \int \frac{d^d p}{(2\pi\hbar)^d}\, p_i p_j \frac{\partial n_a[\varepsilon_a(\mathbf p)]} {\partial\varepsilon_a}, \qquad \rho_n = \frac1d \operatorname{tr}\rho^{\,n}.$$

This is a useful benchmark, not a universal definition for a lattice, a strongly damped continuum, or a driven fluid. For one isotropic linear phonon branch in three dimensions it yields

$$\rho_n^{\mathrm{ph}} = \frac{2\pi^2}{45} \frac{(k_BT)^4} {\hbar^3c^5}.$$

The $T^4$ law tests the assumed low-energy spectrum and dimensionality; it does not include rotons, boundaries, or additional branches.

In the negligible-thermal-expansion benchmark, with entropy per unit mass $\bar s$ and heat capacity per unit mass $c_p$,

$$c_1^2 \simeq \left( \frac{\partial P}{\partial\rho} \right)_{\bar s}, \qquad c_2^2 \simeq \frac{\rho_s}{\rho_n} \frac{T\bar s^{\,2}}{c_p}.$$

Outside that limit, density and entropy oscillations mix and one must solve the full two-fluid thermodynamic eigenproblem. Collective Modes owns pole, residue, damping, continuum, and probe-weight calculations.

The Landau Bound and Actual Critical Flow

For a Galilean-invariant homogeneous fluid, work in the fluid rest frame and let $\mathbf p$ be mechanical momentum. For homogeneous flow in direction $\widehat{\mathbf v}$, Landau’s energetic bound is

$$v_{\mathrm L}(\widehat{\mathbf v}) := \inf_{\mathbf p\cdot\widehat{\mathbf v}>0} \frac{\varepsilon(\mathbf p)} {\mathbf p\cdot\widehat{\mathbf v}}.$$

It gives $c$ for an ideal linear phonon branch. For a narrow roton-like minimum of gap $\Delta$ at $p_0$, the candidate is approximately $\Delta/p_0$; the full directional minimization still decides which branch controls.

On a lattice or in a synthetic gauge field, crystal or canonical momentum cannot simply replace that mechanical momentum. One must audit the excitation spectrum of the source-twisted, current-carrying state and separately test energetic and dynamical stability.

This result is an energetic instability bound for the declared spectrum. It is not a prediction of the measured threshold. Wall roughness, apertures, weak links, vortex nucleation, phase slips, heating, dynamical instabilities, disorder, and history can lower and broaden observed critical flow. A lower threshold therefore does not contradict the measured excitation spectrum.

Conversely, low viscosity, ballistic propagation, a narrowed momentum distribution, or one sharp dissipation onset does not by itself establish equilibrium stiffness. The strongest material claim comes from independent equilibrium, flow, rotational, sound, and spectral records with compatible normalizations and state points.

First Sound, Second Sound, and Spectral Evidence

A sound claim must identify a response operator and a pole, not merely a ridge in an intensity map. For each candidate mode record its dispersion, residue, linewidth, neighboring continuum, matrix element, background, and instrumental resolution. A peak that broadens into a continuum can remain a response feature even after a sharp-quasiparticle description fails.

In the small-thermal-expansion two-fluid limit, first sound is mainly a density wave and second sound is mainly an entropy–temperature counterflow. Outside that limit they are coupled eigenmodes. Their speeds, damping, and probe weights must be obtained from the full thermodynamic and dissipative response problem; labeling the lower branch “second sound” does not by itself measure $\rho_s$.

The excitation spectrum enters a different inference. Its low-momentum slope supplies a phonon candidate for the Landau bound, while roton-like structure can lower the directional minimum. The same spectrum contributes to equilibrium entropy and normal density only after its state dependence, degeneracy, linewidth, and integration window are controlled. Collective Modes owns the reusable pole calculation, and Neutron Scattering owns the cross section and resolution correction. This page owns the cross-probe material verdict.

Persistent Flow, Rotation, and Quantized Vortices

For a closed contour $\mathcal C$ in a single coherent component, avoiding cores and singular cuts,

$$\oint_{\mathcal C} \mathbf v_s\cdot d\boldsymbol\ell = \ell\frac{h}{M}, \qquad \ell\in\mathbb Z.$$

The quantum uses the mass of the coherent carrier. Ordinary paired $^3$He uses $M=2m_3$, subject to its multicomponent order-parameter and fractional-defect qualifications. A phase winding proves a circulation statement only after the carrier, contour, and order-parameter manifold are known.

For coarse-grained steady rigid rotation with ordinary singly quantized vortices,

$$n_v = \frac{2\Omega}{\kappa} = \frac{M\Omega}{\pi\hbar}, \qquad \kappa:=\frac{h}{M}.$$

Finite vessels, pinning, vortex sheets, multiply quantized defects, and nonequilibrium tangles lie outside this benchmark. Persistent flow is metastable, not eternally dissipationless: it decays when a phase slip or vortex crossing becomes allowed on the observation timescale.

Two-Dimensional Films and the BKT Regime

For a clean equilibrium two-dimensional single-component system, the long-distance renormalized stiffness obeys the Nelson–Kosterlitz jump

$$\Upsilon_R(T_{\mathrm{BKT}}^-) = \frac{2k_BT_{\mathrm{BKT}}}{\pi},$$

or, for areal mass density,

$$\rho_s^{\,2D}(T_{\mathrm{BKT}}^-) = \frac{2M^2k_BT_{\mathrm{BKT}}} {\pi\hbar^2}.$$

The second quantity is not a bulk density. For a diagonal anisotropic stiffness, the rescaled criterion uses $\sqrt{\Upsilon_x\Upsilon_y}$.

The jump is a statement about the renormalized, equilibrium, long-distance stiffness. A bare microscopic stiffness or finite-frequency fit cannot be inserted without a scale-flow and convergence audit. Finite size, finite frequency, trap profiles, film thickness, substrate coupling, disorder, and drive round or shift BKT-compatible features. Stop at a crossover claim unless size and frequency refinement supports a controlled long-distance extrapolation and the record also contains vortex-unbinding-compatible evidence.

Generic phase-only and universal-jump theory remains with Low-Dimensional Quantum Gases and the criticality owners. This page owns the material inference from those results.

Disorder, Porous Media, Lattices, and Finite Geometry

The Galilean liquid is a benchmark, not a universal template. A lattice breaks the direct conversion from phase stiffness to a moving-mass fraction. A porous host introduces pore-size distributions, dead layers, tortuosity, substrate drag, and disconnected coherent regions. A torsional oscillator measures the response of the loaded cell, not a superfluid fraction without a mechanical forward model. In mixtures, a response of one component can entrain another.

Three distinctions are especially important:

• Local coherence versus global stiffness. Coherent puddles do not establish a connected winding response. Percolation and weak links may control the sample-scale result.
• Equilibrium versus observation-time response. A winding sector can look stiff when phase slips are slower than the experiment. Frequency dependence is then part of the claim, not a nuisance to discard.
• Finite sample versus thermodynamic phase. A nonzero curvature, occupation eigenvalue, or sharp crossover in one finite system licenses a finite-instance result. Scaling is required for a thermodynamic statement.

For lattice bosons, the Bose–Hubbard Model owns the Hamiltonian and phase diagram; this page interprets a declared stiffness, winding estimator, flow record, or material probe. Open polariton, exciton, magnon, and photon fluids require a driven steady-state response and stability ledger. Equilibrium two-fluid and BKT formulas do not transfer automatically.

Canonical boundaries

• BEC and ODLRO pages own occupation, coherence, and their dimensional restrictions.
• Weak-gas, Gross–Pitaevskii Equation, and Bogoliubov Theory retain the dilute-gas equation of state, depletion, excitation spectrum, and controlled vortex cores.
• Finite-Temperature Phase Transitions, Universality, and Critical Exponents and Scaling retain generic BKT flow, essential singularities, universality, and scaling.
• Goldstone, collective-mode, and hydrodynamic owners retain reusable mode counting, pole calculations, conservation laws, and constitutive derivations.
• Numerical owners retain production winding estimators, sampling, autocorrelation, and convergence.
• Bose–Einstein Condensates owns trapped-gas preparation, expansion, interference, and platform-specific workflows.
• How Quantum Matter Is Measured and individual probe pages retain acquisition, calibration, cross sections, inversion, and resolution. This page owns the bounded neutral-material conclusion.
• Once the carrier is charged or the question concerns Meissner screening, penetration depth, fluxoid quantization, charged amplitude healing, or critical fields, use London Theory or Ginzburg–Landau Theory.

Worked Audit: Bulk Liquid Helium-4

Use an explicitly constructed cross-probe audit at saturated vapor pressure and $T=1.60\ \mathrm K$. The thermodynamic inputs are adopted from the state-resolved Donnelly–Barenghi property tables. Their temperature-unspecified reference dispersion is used only as a separate low-temperature spectral benchmark. The cell geometry, instrumental resolutions, and flow threshold are declared synthetic audit inputs, not measurements reported in that compilation.

1. Platform and state: homogeneous liquid $^4$He in a $20\ \mathrm{cm}$-long square channel of cross section $2.0\ \mathrm{cm}\times2.0\ \mathrm{cm}$, ending in a circular aperture of diameter $0.20\ \mathrm{mm}$. The wall roughness is $2\pm1\ \mathrm{\mu m}$, the liquid is at saturated vapor pressure, $\rho=145.21\ \mathrm{kg\,m^{-3}}$, and no rotation is imposed.
2. Claimed objects: equilibrium mass superfluid fraction, two sound modes, the excitation-based Landau bound, a constructed critical-flow threshold, and condensate occupation as a separate comparison.
3. Carrier and normalization: $M=m_4=6.646\times10^{-27}\ \mathrm{kg}$; all densities are bulk mass densities. The adopted values are $\rho_s=121.63\ \mathrm{kg\,m^{-3}}$, $\rho_n=23.58\ \mathrm{kg\,m^{-3}}$, and $\rho_s/\rho=0.838$.
4. Symmetry and topology: a number-conserving neutral liquid is used. No vortex is deliberately trapped, so a winding number is inapplicable to the initial equilibrium record.
5. Scale hierarchy: the same compilation gives $\bar s=281.5\ \mathrm{J\,kg^{-1}K^{-1}}$, $c_p=1.578\times10^3\ \mathrm{J\,kg^{-1}K^{-1}}$, $c_1=234.5\ \mathrm{m\,s^{-1}}$, and $c_2=20.33\ \mathrm{m\,s^{-1}}$. The separate reference dispersion covers the validated interval $0.0894\leq k\leq3.60\ \text{Å}^{-1}$, including the phonon, maxon, roton, and post-roton branches. It is not represented as a state-resolved $1.60\ \mathrm K$ spectrum.
6. Framework and validity: the compact scalar two-fluid formula is used only as a negligible-thermal-expansion benchmark. The reference spline is minimized only on its validated tabulated interval and compared separately with its physical linear-phonon continuation; a two-parameter phonon-plus-roton estimate is retained only as a check.
7. Source, observable, and limits: a slow rotational or inertial response supplies $\rho_s/\rho$; heat and pressure pulses supply sound speeds. For the constructed flow record, $v_{\mathrm{ap}}=Q/A_{\mathrm{ap}}$ and $v_{\mathrm{ch}}=Q/A_{\mathrm{ch}}$, with $A_{\mathrm{ap}}/A_{\mathrm{ch}}=7.85\times10^{-5}$. A declared aperture-profile factor $\eta=1.0\pm0.1$ converts the volume flux to the calibrated local relative velocity $v_{\mathrm{flow}}=\eta Q/A_{\mathrm{ap}}=12\pm2\ \mathrm{m\,s^{-1}}$; it is not the channel mean velocity.
8. Forward model: cell compliance, aperture enhancement, wall roughness, thermal boundary resistance, and transducer loading are retained. The synthetic probe record has representative resolutions $\delta E=0.10\ \mathrm{meV}$ and $\delta k=0.03\ \text{Å}^{-1}$; these are not convolved into the external reference spline, so that spline supplies a rounded central-value benchmark rather than a measurement uncertainty. Its cross section belongs to Neutron Scattering.
9. Evidence and alternatives: the inertial, sound, flow, and spectral results are compared. Vortex nucleation at rough walls and heating are credible alternatives to a bulk quasiparticle instability for the smaller flow threshold.
10. Uncertainty and stopping rule: the claim is bounded by cell coupling, temperature calibration, the thermal-expansion correction, the reference dispersion’s unspecified temperature, interpolation and synthetic resolution, aperture velocity calibration, and wall history. A strong frequency dependence of the inertial response or loss of a resolved second-sound pole would narrow the equilibrium claim.

The two-fluid benchmark gives

$$c_{2,\mathrm{bench}} = \left[ \frac{0.838}{0.162} \frac{(1.60\ \mathrm K) (281.5\ \mathrm{J\,kg^{-1}K^{-1}})^2} {1.578\times10^3\ \mathrm{J\,kg^{-1}K^{-1}}} \right]^{1/2} \simeq 20.39\ \mathrm{m\,s^{-1}}.$$

This agrees with the adopted $20.33\ \mathrm{m\,s^{-1}}$ value at the accuracy appropriate to the compact formula. For the Landau audit, do not extend the interpolation spline to $k=0$: its small negative interpolation offset would make $E_K(k)/k$ unphysical there. Use a physical linear-phonon continuation of slope $c_{\mathrm{ref}}\simeq238\ \mathrm{m\,s^{-1}}$ and minimize the spline only on its validated interval:

$$v_{\mathrm L}^{\mathrm{ref}} = \min\left\{ c_{\mathrm{ref}}, \min_{0.0894\leq k\leq3.60\,\text{Å}^{-1}} \frac{k_B E_K(k)}{\hbar k} \right\} \simeq 58\ \mathrm{m\,s^{-1}},$$

where $E_K(k)=\varepsilon(k)/k_B$ is the Donnelly–Barenghi spline. Its central-value minimum is $57.8\ \mathrm{m\,s^{-1}}$ near $k=2.00\ \text{Å}^{-1}$; reporting $58\ \mathrm{m\,s^{-1}}$ does not imply that the synthetic resolution was propagated. The familiar $\Delta/(\hbar k_0)\simeq58.6\ \mathrm{m\,s^{-1}}$ estimate is close but not identical. The constructed local $12\ \mathrm{m\,s^{-1}}$ threshold is compatible with aperture-enhanced boundary vortex nucleation and does not invalidate the spectrum or stiffness.

Azuah and collaborators report a neutron-derived condensate fraction $n_0=0.060\pm0.020$ at $1.60\ \mathrm K$ and saturated vapor pressure. It must not be substituted for $\rho_s/\rho=0.838$. The licensed conclusion is a cross-probe equilibrium-superfluid record in the declared state, with a constructed boundary-limited critical-flow test. If cell decoupling, heating, or the spectral normalization is unresolved, stop at that bounded statement.

Worked Audit: A Two-Dimensional Helium Film

Now consider a synthetic but fully calibrated $^4$He audit record near $0.70\ \mathrm K$.

1. Platform and state: a film on an exfoliated-graphite substrate with total areal coverage $n_{2D}=18.0\pm0.3\ \mathrm{nm^{-2}}$, measured in square cells of side $L=2$ and $8\ \mathrm{mm}$. The sample is equilibrated at each temperature with no imposed rotation.

2. Claimed object: finite-scale areal stiffness and a BKT-compatible crossover. A momentum-distribution peak is supporting coherence evidence, not the claimed thermodynamic object.

3. Carrier and normalization: $M=m_4$; $\rho_s^{2D}$ is mass per area and $\Upsilon_{\mathrm{eff}}(T;L,\omega)$ is the finite-size, finite-frequency stiffness. The calibrated observable is reported as $\Upsilon_{\mathrm{eff}}/k_B$ in kelvin, with phase convention $\Psi\propto e^{i\theta}$.

4. Symmetry and topology: physical global $U(1)$ symmetry, algebraic order below the crossover, and integer vortices in the single-component film are assumed. No multicomponent identification is used.

5. Scale hierarchy: for $T=(0.68,0.70,0.72)\ \mathrm K$, the calibrated synthetic $\Upsilon_{\mathrm{eff}}/k_B$ values are, respectively:

   • $L=2\ \mathrm{mm}$, $100\ \mathrm{Hz}$: $(0.472\pm0.010,\ 0.449\pm0.010,\ 0.420\pm0.011)\ \mathrm K$;
   • $L=8\ \mathrm{mm}$, $100\ \mathrm{Hz}$: $(0.463\pm0.008,\ 0.445\pm0.008,\ 0.407\pm0.009)\ \mathrm K$;
   • $L=2\ \mathrm{mm}$, $1000\ \mathrm{Hz}$: $(0.490\pm0.012,\ 0.471\pm0.012,\ 0.444\pm0.013)\ \mathrm K$;
   • $L=8\ \mathrm{mm}$, $1000\ \mathrm{Hz}$: $(0.478\pm0.010,\ 0.459\pm0.010,\ 0.431\pm0.011)\ \mathrm K$.

   Substrate thickness variations correspond to a coverage uncertainty of $0.3\ \mathrm{nm^{-2}}$; the drive is verified linear over a factor of three in amplitude.

6. Framework and validity: the equilibrium long-distance BKT criterion is the target. A finite-frequency response probes only the length reached during a cycle and must converge toward the low-frequency, large-size result.

7. Source, observable, and limits: a torsional response is converted to $\Upsilon_{\mathrm{eff}}(T;L,\omega)$ using the loaded-cell model. The desired limit is low frequency after equilibration, followed by controlled size extrapolation. Only that controlled limit may be called $\Upsilon_R$.

8. Forward model: graphite elasticity, viscoelastic loading, a $0.40\pm0.05$-layer inert dead layer, slip, thermometer lag, and empty-cell backgrounds are propagated through the stiffness calibration.

9. Evidence and alternatives: the $100\ \mathrm{Hz}$ dissipation peak occurs at $0.704\pm0.006\ \mathrm K$ and shifts to $0.716\pm0.008\ \mathrm K$ at $1000\ \mathrm{Hz}$. A simultaneous factor-of-five rise in the inferred phase-slip rate is vortex compatible. Viscoelastic decoupling, coverage gradients, and a finite-size crossover remain alternatives.

10. Uncertainty and stopping rule: linear interpolation against the universal-jump benchmark $2T/\pi$ gives these candidate crossings:

    • $T_\star(2\ \mathrm{mm},100\ \mathrm{Hz})=0.702\pm0.006\ \mathrm K$;
    • $T_\star(8\ \mathrm{mm},100\ \mathrm{Hz})=0.700\pm0.005\ \mathrm K$;
    • $T_\star(2\ \mathrm{mm},1000\ \mathrm{Hz})=0.713\pm0.008\ \mathrm K$;
    • $T_\star(8\ \mathrm{mm},1000\ \mathrm{Hz})=0.707\pm0.007\ \mathrm K$.

    The low-frequency size drift is small, while both frequencies show a common upward trend that is not individually resolved and has not converged. A third larger cell and a decade-lower frequency are the stopping tests before a thermodynamic transition is claimed.

At the representative candidate crossing $T_\star=0.70\ \mathrm K$, the universal-jump benchmark is

$$\Upsilon_{\mathrm{jump}}(T_\star) = \frac{2k_BT_\star}{\pi} \simeq 6.15\times10^{-24}\ \mathrm J.$$

Equivalently,

$$n_{s,\mathrm{jump}}^{2D} = \frac{\rho_{s,\mathrm{jump}}^{2D}}{M} = \frac{2Mk_BT_\star}{\pi\hbar^2} \simeq 3.68\ \mathrm{nm^{-2}},$$

or $\rho_{s,\mathrm{jump}}^{2D}\simeq2.45\times10^{-8}\ \mathrm{kg\,m^{-2}}$. These are comparison values on the universal line, not established thermodynamic densities for the finite records.

The universal line has rounded values $(0.433,0.446,0.458)\ \mathrm K$ at the three supplied temperatures; the interpolation uses the unrounded $2T/\pi$ values. Every quoted crossing can therefore be reconstructed rather than read from an unexplained fit. The strongest conclusion is finite-size and finite-frequency evidence for a BKT-compatible stiffness crossover, strengthened by vortex-sensitive dissipation. A thermodynamic claim still requires stable long-distance extrapolation and exclusion of substrate and viscoelastic alternatives.

Common Claim Failures

1. Condensate fraction equals superfluid fraction. It does not; the two are different density- matrix and response observables.
2. A finite $\langle\Psi\rangle$ is required. A number-conserving finite system can be audited through density matrices and twist response.
3. The Landau number is the measured critical velocity. It is an energetic bound; boundaries, vortices, heating, and drive determine the observed threshold.
4. Every second-sound fit measures $\rho_s$. The compact formula needs its thermodynamic and damping assumptions, and the measured pole needs a forward model.
5. One rounded BKT crossing proves a transition. The relevant stiffness is renormalized and long-distance; size, frequency, thickness, substrate, and disorder must be controlled.
6. Torsional decoupling is automatically a mass superfluid fraction. Viscoelasticity, loading, geometry, porous coupling, and observation time can mimic or reshape it.
7. A vortex image measures stiffness. It identifies a defect only after the carrier and order- parameter manifold are known; thermodynamic density requires a separate response measurement.
8. Persistent means permanent. Winding states are metastable and decay when a phase-slip path becomes accessible.

Exercises

Exercise 1 — Twist Curvature and Units

For a rectangular sample, suppose

$$F(\boldsymbol\varphi)-F(0) = \frac12 \sum_i C_i\varphi_i^2.$$

Find $\Upsilon_{ii}$, $\rho^s_{ii}$, and $n^s_{ii}$ for a Galilean continuum. State the units in three dimensions and explain what cannot be inferred on a lattice without an additional current response convention.

Solution

Comparison with the box-twist definition gives

$$\Upsilon_{ii} = \frac{L_i^2}{V}C_i, \qquad \rho^s_{ii} = \frac{M^2}{\hbar^2}\Upsilon_{ii}, \qquad n^s_{ii} = \frac{M}{\hbar^2}\Upsilon_{ii}.$$

In three dimensions, $\Upsilon$ has units of energy per length, $\rho^s$ has units $\mathrm{kg\,m^{-3}}$, and $n^s$ has units $\mathrm{m^{-3}}$. In two dimensions the same symbols refer to energy and areal densities instead. A lattice twist still defines $\Upsilon$, but Galilean boost symmetry is absent. Multiplying it by $M^2/\hbar^2$ does not automatically produce a literal moving-mass fraction; the lattice current and source convention must be declared.

Exercise 2 — Which Objects Exist?

Classify condensate occupation, one-body ODLRO, algebraic order, stiffness, and robust critical flow for (a) a three-dimensional homogeneous ideal Bose gas below $T_c$, (b) interacting bulk $^4$He at low temperature, and (c) an equilibrium homogeneous two-dimensional BKT fluid below its transition.

Solution

(a) The three-dimensional ideal gas has macroscopic occupation and one-body ODLRO below $T_c$; its correlations approach a constant rather than showing algebraic-only decay. With the canonical equilibrated boundary-twist limit, the condensate supplies a nonzero helicity modulus. Nevertheless the quadratic spectrum gives $v_{\mathrm L}=0$, so there is no robust metastable critical flow.

(b) Interacting bulk $^4$He has macroscopic occupation and true one-body ODLRO, rather than merely algebraic decay. It also has nonzero equilibrium mass stiffness and geometry-dependent metastable critical flow with quantized vortices. Its condensate fraction is much smaller than its low-temperature superfluid fraction.

(c) The equilibrium homogeneous two-dimensional BKT fluid has no extensive condensate occupation and no true finite-temperature one-body ODLRO in the thermodynamic limit. It does have algebraic correlations and nonzero renormalized long-distance stiffness. It can support metastable flow, but the observed critical velocity is controlled by geometry and vortex nucleation. Thus all five objects have been classified without forcing them into one scalar definition.

Exercise 3 — Phonon–Roton Landau Bound

An isotropic model has two excitation branches,

$$\varepsilon_{\mathrm{ph}}(k)=\hbar ck, \qquad \varepsilon_{\mathrm{rot}}(k) = \Delta+ \frac{\hbar^2(k-k_0)^2}{2\mu_*},$$

with $c=180\ \mathrm{m\,s^{-1}}$, $\Delta/k_B=7.5\ \mathrm K$, $k_0=1.70\ \text{Å}^{-1}$, and $\mu_*=0.18m_4$. Minimize $\varepsilon(k)/(\hbar k)$ over both branches and identify the controlling one. An aperture experiment finds dissipation at $15\ \mathrm{m\,s^{-1}}$. Is that flow result inconsistent with the supplied spectrum?

Solution

The phonon ratio is constant, so $v_{\mathrm L}^{\mathrm{ph}}=c=180\ \mathrm{m\,s^{-1}}$. For the roton branch, stationarity of $\varepsilon_{\mathrm{rot}}(k)/(\hbar k)$ gives

$$k_*^2 = k_0^2+ \frac{2\mu_*\Delta}{\hbar^2}, \qquad k_* \simeq 1.764\ \text{Å}^{-1}.$$

At that stationary point, the ratio equals the group velocity,

$$v_{\mathrm L}^{\mathrm{rot}} = \frac{\hbar(k_*-k_0)}{\mu_*} \simeq 56.7\ \mathrm{m\,s^{-1}}.$$

The stationary point is the positive minimum of the roton ratio, and comparison with the phonon value shows that the roton branch controls. The observed $15\ \mathrm{m\,s^{-1}}$ threshold is not inconsistent: the Landau value is an upper energetic bound within that spectrum. Vortex nucleation at the aperture, roughness, heating, or an inhomogeneous local velocity can trigger dissipation earlier. A quantitative audit would vary aperture geometry and history while checking the local temperature and vortex record.

Exercise 4 — Phonon Normal Density

Starting from the quasiparticle normal-density formula, derive the low-temperature normal density for one isotropic three-dimensional branch $\varepsilon(p)=cp$.

Solution

Isotropy gives

$$\rho_n = \frac{\beta}{3} \int \frac{d^3p}{(2\pi\hbar)^3} p^2 \frac{e^{\beta cp}} {(e^{\beta cp}-1)^2}.$$

With $x=\beta cp$,

$$\rho_n = \frac{1}{6\pi^2\hbar^3\beta^4c^5} \int_0^\infty dx\, x^4 \frac{e^x}{(e^x-1)^2}.$$

The integral is $4\pi^4/15$, hence

$$\rho_n = \frac{2\pi^2}{45} \frac{(k_BT)^4}{\hbar^3c^5}.$$

The units are mass per volume. The result assumes an equilibrium Galilean continuum, one sharp isotropic linear branch, negligible boundaries and other excitations, and a temperature low enough for the linear dispersion. A lattice, strong damping, rotons, extra polarizations, or dimensional crossover invalidates the unmodified formula.

Exercise 5 — Second-Sound Benchmark

At a stated temperature, let $\rho_s/\rho_n=4$, $\bar s=100\ \mathrm{J\,kg^{-1}K^{-1}}$, $c_p=1000\ \mathrm{J\,kg^{-1}K^{-1}}$, and $T=1.0\ \mathrm K$. Compute the compact second-sound benchmark and state when it is insufficient.

Solution

The negligible-thermal-expansion expression gives

$$c_2 = \left[ 4 \frac{(1.0\ \mathrm K) (100\ \mathrm{J\,kg^{-1}K^{-1}})^2} {1000\ \mathrm{J\,kg^{-1}K^{-1}}} \right]^{1/2} = \sqrt{40}\ \mathrm{m\,s^{-1}} \simeq 6.32\ \mathrm{m\,s^{-1}}.$$

The units follow because $T\bar s^2/c_p$ is energy per mass. If thermal expansion is appreciable, the density and entropy waves hybridize; one must solve the full coupled two-fluid eigenproblem. Strong damping, confinement, multiple components, and probe loading also require an extended forward model. Agreement of one fitted speed alone does not establish $\rho_s$.

Exercise 6 — The BKT Jump and Scale Drift

For a $^4$He film at $0.70\ \mathrm K$, calculate the universal-jump benchmark for the areal number and mass superfluid densities. Fits of $\Upsilon_{\mathrm{eff}}/k_B$ to $2T/\pi$ give the following candidate crossings:

• $0.702\pm0.006\ \mathrm K$ at $(L,f)=(2\ \mathrm{mm},100\ \mathrm{Hz})$;
• $0.700\pm0.005\ \mathrm K$ at $(8\ \mathrm{mm},100\ \mathrm{Hz})$;
• $0.713\pm0.008\ \mathrm K$ at $(2\ \mathrm{mm},1000\ \mathrm{Hz})$;
• $0.707\pm0.007\ \mathrm K$ at $(8\ \mathrm{mm},1000\ \mathrm{Hz})$.

Audit the size and frequency drift.

Solution

With $M=m_4=6.646\times10^{-27}\ \mathrm{kg}$,

$$n_{s,\mathrm{jump}}^{2D} = \frac{2Mk_BT}{\pi\hbar^2} \simeq 3.68\ \mathrm{nm^{-2}},$$

and

$$\rho_{s,\mathrm{jump}}^{2D} = Mn_{s,\mathrm{jump}}^{2D} \simeq 2.45\times10^{-8}\ \mathrm{kg\,m^{-2}}.$$

These are universal-line benchmark values, not an established thermodynamic density or the bare microscopic stiffness. At $100\ \mathrm{Hz}$ the size shift is $0.002\pm0.008\ \mathrm K$, while at $1000\ \mathrm{Hz}$ it is $0.006\pm0.011\ \mathrm K$; neither is individually resolved. For the $8\ \mathrm{mm}$ cell, the frequency shift is $0.007\pm0.009\ \mathrm K$, and for the $2\ \mathrm{mm}$ cell it is $0.011\pm0.010\ \mathrm K$. The common upward tendency at higher frequency is directionally consistent with probing a shorter dynamical scale, but it remains statistically unresolved. The data license a BKT-compatible finite-scale crossover. A thermodynamic transition requires lower-frequency and larger-size convergence plus vortex-compatible evidence.

Exercise 7 — Circulation, Annular Flow, and Rotation

A thin $^4$He annulus of radius $R=1.0\ \mathrm{cm}$ occupies winding sector $\ell=3$. Find the circulation, uniform flow speed, kinetic energy per coherent atom, and the vortex density expected under steady rotation at $\Omega=1.0\ \mathrm{s^{-1}}$.

Solution

The circulation is

$$\kappa_\ell = \ell\frac{h}{m_4} \simeq 2.99\times10^{-7}\ \mathrm{m^2\,s^{-1}}.$$

Uniform annular flow gives

$$v_\ell = \frac{\kappa_\ell}{2\pi R} = \frac{\ell\hbar}{m_4R} \simeq 4.76\times10^{-6}\ \mathrm{m\,s^{-1}}.$$

The kinetic energy per coherent atom is

$$\frac12m_4v_\ell^2 \simeq 7.5\times10^{-38}\ \mathrm J.$$

For ordinary singly quantized vortices,

$$n_v = \frac{m_4\Omega}{\pi\hbar} \simeq 2.0\times10^7\ \mathrm{m^{-2}}.$$

Quantization fixes the allowed circulation, not its lifetime. Surface roughness and weak links set the phase-slip barrier. For ordinary paired $^3$He the mass is $2m_3$, and its multicomponent order allows additional textures and defects beyond this scalar benchmark.

Exercise 8 — License the Strongest Claim

A neutral fluid shows a narrow occupation peak, a frequency-dependent torsional decoupling, a resolved second-sound-like pole, a phonon–roton spectrum, a low flow threshold, and occasional vortex events. What is the strongest immediate claim? Give two alternatives, one falsification test, and the canonical next owner.

Solution

The record supports a candidate neutral-superfluid response with mutually consistent coherence, collective-mode, spectral, flow, and defect evidence, but it does not yet establish a unique thermodynamic superfluid fraction. The occupation peak addresses coherence, the torsional signal needs a mechanical and frequency extrapolation, the sound pole needs thermodynamic and probe-weight identification, and the low flow threshold may be boundary limited.

Two credible alternatives are (1) viscoelastic or substrate decoupling combined with finite-size coherence and (2) a metastable, percolating response whose relaxation time exceeds the measurement period but whose equilibrium stiffness vanishes at larger scale. A decisive test is to demonstrate that the inferred static stiffness converges under lower frequency, larger size, and changed cell geometry while the same equilibrium state retains the sound and vortex record. Acquisition and inversion route to How Quantum Matter Is Measured; the relevant sound, neutron, finite-size, or hydrodynamic owner is then selected according to the failed field of the ten-field ledger.

References

• J. F. Allen and A. D. Misener, “Flow of Liquid Helium II,” Nature 141, 75 (1938), doi:10.1038/141075a0.
• E. L. Andronikashvili, “A Direct Observation of the Two Kinds of Motion in Helium II,” Journal of Physics USSR 10, 201–206 (1946).
• R. T. Azuah, W. G. Stirling, H. R. Glyde, M. Boninsegni, P. E. Sokol, and S. M. Bennington, “Condensate and Final-State Effects in Superfluid $^4$He,” Physical Review B 56, 14620–14630 (1997), doi:10.1103/PhysRevB.56.14620.
• D. J. Bishop and J. D. Reppy, “Study of the Superfluid Transition in Two-Dimensional $^4$He Films,” Physical Review Letters 40, 1727–1730 (1978), doi:10.1103/PhysRevLett.40.1727.
• R. J. Donnelly and C. F. Barenghi, “The Observed Properties of Liquid Helium at the Saturated Vapor Pressure,” Journal of Physical and Chemical Reference Data 27, 1217–1274 (1998), doi:10.1063/1.556028.
• R. J. Donnelly, Quantized Vortices in Helium II, Cambridge University Press (1991).
• R. P. Feynman, “Application of Quantum Mechanics to Liquid Helium,” in Progress in Low Temperature Physics, Vol. 1, pp. 17–53, North-Holland (1955).
• M. E. Fisher, M. N. Barber, and D. Jasnow, “Helicity Modulus, Superfluidity, and Scaling in Isotropic Systems,” Physical Review A 8, 1111–1124 (1973), doi:10.1103/PhysRevA.8.1111.
• H. E. Hall and W. F. Vinen, “The Rotation of Liquid Helium II. I. Experiments on the Propagation of Second Sound in Uniformly Rotating Helium II,” Proceedings of the Royal Society A 238, 204–214 (1956), doi:10.1098/rspa.1956.0212.
• G. B. Hess and W. M. Fairbank, “Measurements of Angular Momentum in Superfluid Helium,” Physical Review Letters 19, 216–218 (1967), doi:10.1103/PhysRevLett.19.216.
• P. C. Hohenberg and P. C. Martin, “Microscopic Theory of Superfluid Helium,” Annals of Physics 34, 291–359 (1965), doi:10.1016/0003-4916(65)90280-0.
• P. Kapitza, “Viscosity of Liquid Helium below the $\lambda$-Point,” Nature 141, 74 (1938), doi:10.1038/141074a0.
• I. M. Khalatnikov, An Introduction to the Theory of Superfluidity, W. A. Benjamin (1965).
• L. D. Landau, “The Theory of Superfluidity of Helium II,” Physical Review 60, 356–358 (1941), doi:10.1103/PhysRev.60.356.
• A. J. Leggett, “Superfluidity,” Reviews of Modern Physics 71, S318–S323 (1999), doi:10.1103/RevModPhys.71.S318.
• D. R. Nelson and J. M. Kosterlitz, “Universal Jump in the Superfluid Density of Two-Dimensional Superfluids,” Physical Review Letters 39, 1201–1205 (1977), doi:10.1103/PhysRevLett.39.1201.
• L. Onsager, “Statistical Hydrodynamics,” Il Nuovo Cimento Supplemento 6, 279–287 (1949), doi:10.1007/BF02780991.
• O. Penrose and L. Onsager, “Bose–Einstein Condensation and Liquid Helium,” Physical Review 104, 576–584 (1956), doi:10.1103/PhysRev.104.576.
• V. P. Peshkov, “‘Second Sound’ in Helium II,” Journal of Physics USSR 8, 381 (1944).
• L. Pitaevskii and S. Stringari, Bose–Einstein Condensation and Superfluidity, Oxford University Press (2016).
• E. L. Pollock and D. M. Ceperley, “Path-Integral Computation of Superfluid Densities,” Physical Review B 36, 8343–8352 (1987), doi:10.1103/PhysRevB.36.8343.
• D. Vollhardt and P. Wölfle, The Superfluid Phases of Helium 3, Dover (2013).