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In 1966, William Russell Abel, Ansel Cochran Anderson and John Charles Wheatley sent acoustic waves through bulk liquid helium-3 at $0.32\mathrm{atm}$ and frequencies $15.4$ and $45.5\mathrm{MHz}$. A quartz delay line and receiving acoustic transducer bounded a $0.138\mathrm{mm}$ liquid gap; shielding reduced electrical pickup. Powdered cerium magnesium nitrate provided magnetic thermometry during a slow warm-up. Pulse amplitudes measured acoustic attenuation; phase difference comparison measured velocity. Calibration pulses checked gain; early received cycles avoided multiple reflections. On cooling through a maximum, acoustic attenuation changed from $\alpha \propto {\omega }^2/T^{\ast 2}$ to $\alpha \propto T^{\ast 2}$ with a velocity shift supporting zero sound; $T^{\ast }$ is their magnetic temperature scale. Weak acoustic attenuation was harder to resolve.^[ref]

This 1966 test followed Lev Landau's 1962 Nobel Prize in Physics.^[ref]

The experiment concerned normal liquid helium-3, distinct from the later discovery of superfluid helium-3.^[ref] Landau's original papers are theoretical calculations, not reports of an apparatus.

Anthony Leggett developed a theoretical account of the newly discovered superfluid phases of helium-3. Unlike helium-4 atoms, helium-3 atoms are fermions, so their superfluid state involves correlated pairing. The internal spin and orbital structure of the pairs allows several distinct ordered phases. His account connected this structure with unusual nuclear magnetic resonance observations, using magnetic response to distinguish possible forms of the condensate. The work illustrates how a small interaction can make the internal symmetry of a quantum fluid experimentally visible.^[ref]

A physicist who studied superfluidity and macroscopic quantum mechanics at Illinois.^[ref]

The condensate fraction is the population $N_0$ of a macroscopically occupied one-particle state divided by the total particle number $N$. For an interacting system, use a macroscopic eigenvalue of the one-particle density matrix, rather than assume that the relevant mode is an undeformed single-particle trap ground state. The condensate fraction is not generally the fraction measured as superfluid by a flow or rotation response.

Expanding the cosine gives a wave equation $I{\varphi }_{tt}=J{\varphi }_{xx}$ and a gapless phase mode with physical frequency $\sqrt{J/I}|q|$. The exact harmonic lattice frequency is $2\sqrt{J/I}|\sin (q/2)|$. It resembles superfluid sound. In one dimension, Gaussian phase fluctuations and possible compact phase slips require care before inferring true long-range order from this branch.

Helium is the chemical element with atomic number 2 and symbol $\mathrm{He}$. It is a noble gas. It has a closed-shell electronic structure and is exceptionally unreactive. At ordinary pressure it remains liquid to very low temperatures without freezing; helium-4 also provides the classic example of a superfluid at sufficiently low temperature.^[ref]

Pyotr Leonidovich Kapitza's 1938 test compared helium flow through optically flat glass disks. Suspension set the tube-to-bath level difference; a cathetometer read levels. The gap was estimated as $0.5\mu \mathrm{m}$. Levels barely moved over minutes above the transition, but equalized in seconds below. A laminar flow viscosity calculation gave only an upper bound.^[ref]

The essential control is to compare the same restriction and level readout at different temperatures. A changed pressure head is not itself evidence for a changed material response: the driving head must be included when interpreting the flow. For an idealized comparison at equal vapor pressure, negligible surface tension corrections and nearly constant mass density $\rho$, hydrostatic equilibrium givesHere $\Delta h$ is the difference in free-surface heights and $g$ is gravitational acceleration. This explains why following a liquid level provides both a driving pressure and a transported volume; it does not turn a finite-resolution observation into proof of zero viscosity.

Holographic D.C. conductivity describes the zero-frequency electric response of a boundary system using a bulk gauge field. In a clean normal state at nonzero charge density, the electric current generally overlaps with conserved momentum. A constant electric field then continuously accelerates that momentum, producing a zero-frequency delta function rather than an ordinary finite D.C. response.

Introducing relaxation of momentum can broaden this contribution and yield finite conductivity when no other protected dissipationless channel remains. That qualification matters: a superfluid can retain a delta function even when translations are broken. One must specify the phase, conserved quantities and frequency limit before interpreting a bulk calculation as a measured D.C. conductivity.

John McGreevy develops quantum field theory through connections with many-particle physics. His authored lectures construct coherent-state path integrals for both bosons and fermions, relating the choice of integration variables to their different quantum statistics. The same treatment derives low-energy phase dynamics for a superfluid. This gives a concrete route from microscopic quantum degrees of freedom to a collective field description, explaining why similar field-theory methods recur in particle and condensed-matter physics.^[ref]

Author of Physics 215C: Quantum Field Theory, Part 3 by John McGreevy. Course record: mcgreevy.physics.ucsd.edu/s22/lecture-notes.html

The lambda point is the temperature of the transition between ordinary liquid helium-4 and its superfluid phase, named after the sharp lambda-shaped heat capacity feature. At saturated vapor pressure it lies near $2.17\mathrm{K}$. Its value varies with pressure.

The Landau criterion asks whether creating an excitation can lower a moving superfluid's energy. For a weakly coupled, heavy obstacle moving through a uniform liquid initially at zero temperature, let $ϵ(p)>0$ be the excitation energy in the fluid rest frame, with momentum magnitude $p$. Transferring momentum $\mathbf{p}$ from an obstacle moving at velocity $\mathbf{v}$ supplies, to leading order in its negligible recoil, energy $\mathbf{v}\cdot \mathbf{p}$. Emission requiresFor an isotropic dispersion relation, the easiest direction is parallel to $\mathbf{v}$, giving the necessary kinematic conditionAn infimum includes cases in which the limiting slope occurs only as $p\to 0$. A finite-mass obstacle of mass $M$ instead loses $\mathbf{v}\cdot \mathbf{p}-p^2/(2M)$, so the corresponding emission inequality includes the extra positive recoil term $p^2/(2M)$ on the left. This is an energy and momentum balance, not a calculation of an emission rate.

The Landau criterion excludes spontaneous emission into the specified spectrum below its threshold. Above it, kinematic permission alone does not guarantee strong coupling or rapid decay. It is not a universal observed flow threshold: walls, channel geometry and quantum vortex nucleation can enable other decay routes, and finite temperature permits scattering from existing excitations. Using only bulk phonons and rotons leaves those processes outside the model.

Landau's 1941 energy and momentum argument used a roton approximation different from the finite-momentum minimum illustrated below.^[ref]

Landau received his Nobel Prize in Physics in 1962 while hospitalized in Moscow after a car accident.^[ref]

For a deliberately simple Bose-Einstein condensation comparison, stipulate an ideal free-particle excitation law $ϵ=p^2/(2m)$. Then $ϵ/p=p/(2m)$ has infimum zero. Macroscopic one-particle occupation therefore supplies no positive emission threshold by this argument alone. By contrast a linear phonon law $ϵ=cp$ yields $v_L=c$. The dispersion relation and the collective response contain information that the occupation criterion alone does not supply; Bose-Einstein condensation and thermodynamic superfluidity must be tested separately.

To check the picture algebraically, on the stipulated roton branch,Thus tangency occurs at $q_L=\sqrt{4.3}$, slightly above the roton energy minimum. At an energy minimum $e^{\prime }=0$, but a tangent through the origin needs $e^{\prime }=e/q>0$. Minimizing energy and minimizing the energy-to-momentum ratio are different operations. This geometric construction makes the criterion useful even when the full dispersion relation is supplied numerically rather than by a simple formula.

Lev Landau developed descriptions of interacting quantum fluids through their elementary excitations. In the Landau critical velocity argument, creating an excitation of energy $ϵ(\mathbf{p})$ in flowing superfluid changes the relevant energy by $ϵ(\mathbf{p})-\mathbf{v}\cdot \mathbf{p}$. Requiring this to remain positive yields the ideal bound $v_c={\min }_pϵ(p)/p$; additional processes can limit actual flow at lower speeds.^[ref] His Fermi liquid theory explains how strongly interacting fermions can retain long-lived low-energy quasiparticles, with interactions modifying their effective mass and collective response.^[ref]

A physicist.^[ref]

Awarded in equal thirds to Alexei Alexeyevich Abrikosov, Vitaly Lazarevich Ginzburg and Anthony J. Leggett for pioneering theoretical contributions to superconductors and superfluids.^[ref]

The normal component of a superfluid describes the momentum and entropy carried by excitations rather than the phase-gradient condensate current. Its velocity need not equal the superfluid velocity. In phenomenological condensate equations, prescribing its motion supplies a reference flow for dissipation; a full two-component fluid theory would also evolve its momentum and energy.

These are two responses of one liquid, not separable kinds of atoms. Their fractions need not equal the Bose-Einstein condensation fraction.^[ref]

A pairing gap is an energy scale for fermionic excitations above a paired superfluid or superconductor ground state. In a simple isotropic paired system the quasiparticle dispersion has the form $E_{\mathbf{k}}=\sqrt{{\xi }_{\mathbf{k}}^2+|\Delta {|}^2}$, where ${\xi }_{\mathbf{k}}$ measures the unpaired single-particle energy relative to the chemical potential. A nonzero minimum gap gives exponentially small fermionic thermal occupations when $k_{B}T\ll |\Delta |$. If the gap varies with direction and has nodes, gapless directions can instead produce power-law thermodynamics. Collective excitations must be counted separately from the gapped fermionic quasiparticles.

For smooth phase differences, $-J\cos ({\varphi }_{n+1}-{\varphi }_n)=-J+J({\varphi }_{n+1}-{\varphi }_n{)}^2/2+\cdots$. After integrating the angular momenta in the same inverse-energy time coordinate $\tau$, the coupled quantum rotor chain has the fluctuation actionUnit spacing replaces the lattice sum and difference by their continuum limits. The constant energy $-J$ per bond contributes an overall factor $e^{\beta JN}$, suppressed when describing fluctuations. The phase is compact; the Gaussian action describes its smooth sector.

There is an imaginary-time normalization for a coupled rotor chain issue in the printed formula. Factoring out $1/{\hslash }^2$ from the action above places ${\hslash }^{2}J$, not $J$, in its spatial bracket. The printed bracket is therefore correct in $\hslash =1$ units, or with its spatial coefficient redefined as $J_{\mathrm{bracket}}={\hslash }^2{J}_{\mathrm{Hamiltonian}}$. With the same dimensional $J$ as in the Hamiltonian, it lacks that factor.

In physical real time the wave equation is $I{\partial }_t^2\varphi =J{\partial }_x^2\varphi$. The harmonic phase mode of a quantum rotor chain consequently hasThe lattice expression is $\omega (q)=2\sqrt{J/I}|\sin (q/2)|$. The mode is a gapless phase wave analogous to sound in a superfluid, with rotor angular momentum conjugate to its phase. In one dimension this harmonic regime has strong long-distance phase fluctuations rather than true continuous-symmetry long-range order; compact phase slips can change the infrared phase outside the smooth-phase approximation.

Write $\theta =\pi /2-\vartheta$ and $\varphi =\phi$. To quadratic order,After dropping the total derivative $S{\phi }_t$, the quadratic Lagrangian density isIts Euler--Lagrange equations areFor modes proportional to $e^{i(kx-\omega t)}$, nontrivial amplitudes requireThusAt short wavelength the anisotropy is negligible and the quadratic dispersion is that of a conventional ferromagnetic magnon. At long wavelength, easy-plane anisotropy makes the out-of-plane fluctuation the conjugate density of the in-plane phase; eliminating it produces the linear Goldstone sound mode characteristic of a superfluid.

Entropy at absolute zero. The third law of thermodynamics concerns the low-temperature limit of equilibrium states. In the zero-entropy formulation, the entropy of a perfect crystal with a nondegenerate ground state tends to zero as $T\to 0$. The statistical interpretation is $S=k_B\ln \Omega$: a unique accessible ground state has $\Omega =1$. The entropy-difference formulation of the third law of thermodynamics states that entropy differences associated with regular changes of mechanical parameters tend to zero as absolute zero is approached. These statements require equilibrium and appropriate ground-state assumptions; a frozen disordered material or a degenerate ground state can carry residual entropy.

For a phase of fixed composition and pressure,A finite limiting entropy constrains this integral. Ordinary equilibrium heat capacities therefore vanish as temperature approaches zero, with familiar examples $C\propto T^3$ for three-dimensional acoustic phonons and $C\propto T$ for a normal degenerate Fermi liquid. A constant positive heat capacity extrapolated to zero would give a logarithmically divergent entropy difference and is impossible in this regime. Discontinuous transitions and anomalous limits need separate treatment rather than extending a high-temperature approximation indefinitely.

Gibbs free energy. For fixed particle numbers,Thus, when $S\to 0$, the Gibbs free energy approaches its zero-temperature value with a horizontal tangent:The equality $G(0,p)=H(0,p)$ uses the enthalpy $H=U+pV$; neither limiting value needs to be zero. The low-temperature Gibbs free-energy expansion makes the approach quantitative. For $C_p=a{T}^r$ with $r>0$ and $S(0)=0$,The second derivative is ${\partial }_T^{2}G=-C_p/T\le 0$, although it need not tend to zero. For a reaction between such equilibrium phases, ${\partial }_T\Delta G=-\Delta S\to 0$, and $\Delta G$ and $\Delta H$ approach one another. If a genuine residual entropy $S_0$ is retained instead, $G=G(0,p)-S_{0}T+\cdots$ has limiting slope $-S_0$. A horizontal tangent cannot be asserted without the zero-entropy condition.

Why helium remains a liquid. Helium atoms attract each other weakly and are exceptionally light. Their zero-point energy is large compared with the binding that would localize them in a solid. Both helium-4 and helium-3 can therefore remain liquid down to the lowest temperatures at low pressure; substantial applied pressure is needed to solidify them. This is not a violation of the third law. A quantum liquid can have a unique low-entropy ground state, and zero-point motion does not itself imply thermal entropy. The third law does not require every substance to crystallize at atmospheric pressure.

A helium-4 atom has integer spin zero and is a boson. Below the lambda point near $2.17\mathrm{K}$ at saturated vapor pressure, its liquid is superfluid. Its strong interactions mean that it is not quantitatively an ideal Bose gas, and the condensate fraction need not equal the superfluid fraction. The two-fluid model of superfluidity separates an entropy-carrying normal component of a superfluid from the superfluid component, whose entropy is zero. At sufficiently low temperature the dominant thermal excitations are acoustic phonons: their linear dispersion in three dimensions gives $U-U_0\propto V{T}^4$, $C_V\propto V{T}^3$ and $S\propto V{T}^3$. Higher-energy rotons are exponentially suppressed. Thus the normal component of a superfluid and entropy disappear as $T\to 0$, while superfluid flow and quantum motion can persist.

A helium-3 atom has spin $1/2$ and is a fermion. Well below its Fermi temperature, but above its superfluid transition, liquid helium-3 is a degenerate interacting Fermi liquid. Only quasiparticles close to the Fermi surface are thermally excited, giving $C_V\propto T$ and $S\propto T$. At temperatures of order a millikelvin, helium-3 also becomes superfluid through formation of Cooper pairs of fermions. This is not Bose-Einstein condensation of individual helium-3 atoms. Pairing reorganizes the low-energy excitation spectrum; in the fully gapped bulk B phase, the fermionic quasiparticle contribution to heat capacity is exponentially suppressed at temperatures well below the pairing gap. Gapless collective modes can still give thermal terms following a power law. The entropy nevertheless tends to zero. The normal Fermi-liquid law cannot be extrapolated through its pairing transition unchanged.^[ref]

Helium-3 also illustrates the importance of nuclear spin entropy. Over an intermediate low-temperature range, localized spins in the solid can retain approximately $k_B\ln 2$ of entropy per atom, while the degenerate liquid has much less. Reversible conversion of liquid to the higher-entropy solid then absorbs heat; adiabatic compression along the melting curve can cool the sample. This is Pomeranchuk cooling. At still lower temperature, interactions between the nuclear spins remove this nearly free-spin entropy, so a constant $k_B\ln 2$ cannot be extrapolated to absolute zero.^[ref]

Liquefaction and cooling methods. A gas must be below its liquid-gas critical point to form a distinct liquid phase by compression. Compression above the critical temperature alone cannot achieve ordinary liquid-vapour coexistence. Practical liquefaction removes the compression heat, cools the compressed gas in a heat exchanger and uses expansion to lower its temperature further.

A throttling valve operates approximately at constant specific enthalpy when kinetic-energy changes and heat leakage are negligible. The Joule-Thomson coefficient isusing consistent molar quantities, or consistent total quantities for a fixed amount. A pressure drop cools the gas if ${\mu }_{\mathrm{JT}}>0$. An ideal gas has zero Joule-Thomson coefficient; real intermolecular interactions are essential. In particular, room-temperature helium warms on throttling, so it must first be brought into a region of positive Joule-Thomson coefficient. Expansion through an engine or turbine can instead remove work and produce approximately isentropic cooling.

A regenerative liquefier sends the cold returning gas through a counterflow heat exchanger to precool the incoming compressed gas before its next expansion. Expansion engines and external precooling can supply the initial cooling; repeated operation then produces liquid. Modern helium liquefaction commonly combines work-producing expansion with a final throttle. ^[ref]

Once liquid exists, pumping away vapour lowers its saturation pressure and boiling temperature. The escaping vapour carries latent heat away, cooling the remaining liquid. At still lower temperatures, a dilution refrigerator exploits the different entropy carried by helium-3 in concentrated and dilute helium mixtures; adiabatic demagnetization and Pomeranchuk cooling provide other routes.

Approaching zero. The unattainability formulation of the third law states that regular cooling operations with finite resources cannot reach absolute zero in finitely many stages. As the entropy available to remove becomes small and thermal couplings weaken, each method loses its ability to take the final step. An isentropic cooling trajectory starting at nonzero entropy cannot reach a common zero-entropy endpoint through regular finite changes of parameters. A fully general finite-time claim also requires assumptions about available controls and heat-transfer rates. Quantum liquids and refrigeration methods can approach $0\mathrm{K}$ while remaining consistent with vanishing equilibrium entropy; they do not provide a finite-operation route to absolute zero.

The coupled quantum rotor chain has a compact phase and conjugate integer number $n_j={\ell }_j/\hslash$. Expanding the cosine givesIn the phase-space Euclidean path integral, using inverse-energy time, each rotor has the action term ${\ell }_j^2/(2I)-i{\ell }_j{\dot{\varphi }}_j/\hslash$. Completing its Gaussian square produces $I{\dot{\varphi }}_j^2/(2{\hslash }^2)$. Equivalently, apply the short-time free-rotor kernel locally and retain winding sectors. The smooth-sector continuum limit at unit spacing is consequentlyThe overall constant is omitted when the reference bond energy is subtracted. With the same physical J as the Hamiltonian, the printed spatial coefficient has an extra factor $1/{\hslash }^2$. The imaginary-time normalization for a coupled rotor chain requires ${\hslash }^{2}J/2$ inside a bracket with common prefactor $1/{\hslash }^2$. Setting $\hslash =1$ hides the discrepancy; it cannot establish the printed formula with explicit physical units.

Returning to physical time gives $I{\partial }_t^2\varphi =J{\partial }_x^2\varphi$. The harmonic phase mode of a quantum rotor chain therefore hasThis is a gapless sound-like collective phase mode, analogous to neutral superfluid sound. In rotor-chain phase-number hydrodynamics, number fluctuations supply the compressibility and the intersite coupling supplies phase stiffness. The mode is a phase-density oscillation, not a free single-rotor transition. The global phase zero mode of a finite rotor chain is a compact rotor with inertia NI, which should not be counted as an independent zero-frequency oscillator.

The compactness qualification of a harmonic rotor chain matters here. The smooth harmonic theory requires small neighbouring phase differences and cannot settle the entire strong-fluctuation phase diagram. In one dimension its zero-temperature phase variance grows logarithmically, so the gapless regime has algebraic phase correlations rather than a true condensate. Thermal fluctuations and compact phase slips require additional care; phase-slip proliferation can produce a gapped phase outside the Gaussian regime. ^[ref]

A neutral superfluid has a complex order parameter whose phase describes coherent particle transport. For a Bose condensate it is represented by a field amplitude; for a paired Fermi superfluid it is a pair amplitude. In a symmetry-breaking description, selecting its phase breaks the global particle-number $U(1)$ symmetry. A common phase shift leaves the energy unchanged, while a phase gradient costs energy and produces superfluid velocity. Long-wavelength phase oscillations give a gapless Goldstone mode, generally a sound wave when density fluctuations provide the conjugate dynamics. A finite isolated state with exact particle number need not have a nonzero field expectation: the phase-selected language describes the ordered limit and relative-phase coherence.

In a superconductor, the paired order parameter carries charge of magnitude $2e$ and couples to the electromagnetic field. The meaningful spatial phase combination is $\hslash \nabla \theta -q\mathbf{A}$, with signed pair charge $q$. Gauge symmetry is a redundancy, so “breaking local gauge symmetry” is not a literal breaking of a measurable physical symmetry. The gauge-invariant electromagnetic response includes the Meissner effect and a finite London penetration depth. Long-range Coulomb forces move the would-be neutral phase mode to a plasma scale through the Anderson-Higgs mechanism in a superconductor.

Additional symmetries can be broken by particular pairing states. Conventional isotropic spin-singlet pairing need not break spatial rotations or time reversal. A d-wave superconductor has a gap transforming nontrivially under crystal symmetries, and some triplet or multicomponent states can also break spin-rotation or time-reversal symmetries. These additional choices are not required by superconductivity itself. A nonzero pairing amplitude, phase coherence and electromagnetic screening describe related but distinct aspects of the ordered state.

A neutral superfluid can select a phase of its complex order parameter, breaking a global $U(1)$ symmetry. A spatially uniform change of that phase costs no energy. Slowly varying changes therefore produce a gapless Goldstone mode. For a dilute weakly repulsive Bose gas, the Bogoliubov quasiparticle dispersion isso $E_k\simeq \hslash c_{s}k$ at long wavelengths, with $c_s=\sqrt{g_B{n}_0/m_B}$. Density and phase fluctuations are canonically coupled components of this sound branch, rather than two independent positive-frequency modes. Superfluid helium and dilute atomic condensates supply neutral examples, although the dilute-gas formula is not quantitatively valid for strongly interacting liquid helium.

In a superconductor, the pair field is charged and its phase couples to the electromagnetic gauge field. Only combinations such as $\nabla \theta -q\mathbf{A}/\hslash$ are physically meaningful. The Anderson-Higgs mechanism in a superconductor produces electromagnetic screening and removes the neutral sound mode as an independent gapless charged-density oscillation. In a homogeneous three-dimensional continuum, long-range Coulomb forces give a longitudinal plasma gap, approximately ${\omega }^2={\omega }_p^2+c_s^2{k}^2$ at small $k$. The static transverse response gives the Meissner effect andwhen $n_s$ counts superconducting electrons of effective mass $m$. Counting pairs instead uses density $n_s/2$, charge $-2e$ and mass $2m$, giving the same stiffness. The condensate amplitude may also have a gapped Higgs mode, but that amplitude oscillation is distinct from the electromagnetic Higgs mechanism. Local gauge transformations remain a redundancy; this is not spontaneous breaking of an observable local symmetry in the same sense as the neutral global phase symmetry.