Fermi liquid

Why can a strongly interacting liquid resemble a gas? Its low-energy excitations can be long-lived Landau quasiparticles near a Fermi surface. Normal liquid helium-3 motivates this Fermi liquid description: interactions survive in effective excitation energies and interactions.^[ref]

Interacting one-dimensional fermion systems often form Luttinger liquids; non-Fermi liquids also occur in higher dimensions. A Cooper instability can produce a paired phase.^[ref]

The Fermi-surface renormalization-group scaling below organizes the low-energy interactions; zero sound tests whether their collective dynamics survives even when collisions are rare.

For noninteracting fermions at zero temperature, the Fermi surface is the momentum-space boundary $ϵ(\mathbf{k})=\mu$ between occupied and unoccupied states. Its shape can be a circle, several pockets or a more complicated surface, depending on the dispersion and filling.

In an interacting Fermi liquid, occupation is not simply zero or one on either side. Instead the momentum distribution has a discontinuity with quasiparticle weight $Z$, and low-energy Landau quasiparticles define the Fermi surface. This distinguishes a sharp quasiparticle boundary from an ideal-gas step distribution. The geometry controls low-energy phase space, transport and collective response. Quantum oscillations constrain extremal orbit areas, while momentum-resolved spectral measurements can probe excitations near the chemical potential; the interpretation requires the relevant interaction and measurement model.

For a two-dimensional free electron gas with two spin states and $z$ electrons per square cell of side $a$, counting occupied wavevectors gives $z/a^2=2\pi k_F^2/(2\pi {)}^2$. The square first Brillouin zone has half-width $\pi /a$. A monovalent circle lies inside its faces; a divalent circle crosses its faces but not its corners. A real periodic potential can deform and reconnect the crossings, as in Fermi-surface reconstruction in a weak square potential; valence alone is insufficient to prove insulation.

For a free circle of radius $R$ and square reciprocal period $Q$ with $\sqrt{2}R<Q<2R$, the circle cuts the faces but not the corners of the first zone. Weak Bragg gaps reconnect the folded branches into two inequivalent edge-centred electron pockets and one corner-centred hole pocket. The zero-potential limit retains band crossings instead of avoided crossings.

For $s=Q/(2R)$, each face cap has area $C=R^2[\arccos s-s\sqrt{1-s^2}]$. The total second-band electron area is $4C$; the first-band hole area is $Q^2-\pi R^2+4C$. Multiplying either area by the spin degeneracy two and dividing by $(2\pi {)}^2$ gives the carrier densities. This is an infinitesimal-potential count, not an exact finite-gap formula.

A hole pocket surrounds unoccupied states near a band maximum. Its hole carrier density counts missing states in the nearly filled band, not all occupied electrons. In two dimensions the count is $g{A}_h/(2\pi {)}^2$. Four corner pieces of a square-zone plot can be one pocket on the periodically identified zone.

An electron pocket surrounds occupied states near a conduction-band minimum. In two dimensions its carrier density is $gA/(2\pi {)}^2$, where $A$ is its total reciprocal-space area and $g$ the unresolved degeneracy. Portions drawn at opposite zone faces can belong to the same pocket after periodic identification.

For an isotropic three-dimensional free electron gas at zero temperature, occupied wavevectors fill a sphere of radius $k_F$. Including two spin states, $n=k_F^3/(3{\pi }^2)$. The volume in reciprocal space counts occupied states; folding an extended-zone sphere into a first Brillouin zone does not remove states outside that first zone.

A Fermi point is a point of the zero-temperature Fermi surface in one-dimensional momentum space. For a spin-degenerate band with one occupied interval, its two boundaries are usually $-k_F$ and $+k_F$. Low-energy excitations near them form left- and right-moving branches. A perturbation at wavevector $2k_F$ can mix those branches, as in a Peierls distortion.

For a two-dimensional energy band, a Fermi contour is the set of wavevectors at the Fermi energy. It is the two-dimensional counterpart of a Fermi surface. At a regular value it consists locally of curves; at a band extremum or a crossing it can acquire isolated points or intersecting branches. At zero temperature, each band's occupied region satisfies $E_n(\mathbf{k})<E_F$, so the contours must be labelled by band when determining occupations. The Fermi contours of an isotropic Rashba band illustrate why two contours need not belong to two different bands.

The circular zero-temperature occupation boundary of an isotropic two-dimensional electron gas. The occupied disk has area $\pi k_F^2$; converting its area to particle number requires the reciprocal-space state density and a separately specified spin degeneracy.

Translating a Fermi circle of radius $p_F$ by $p_0\hat{\mathbf{x}}$ preserves its area. For $p_0<p_F$, its radial displacement from the original centre is the displayed expression. Its Taylor expansion is $\nu =p_0\cos \theta -p_0^2{\sin }^2\theta /(2p_F)+O(p_0^4/p_F^3)$. The second-order term cancels the apparent area increase from the first-order shape: ${\int }_0^{2\pi }(p_F\nu +{\nu }^2/2)d\theta =0$. This distinction matters when converting a grand-potential variation into a fixed-particle number energy change.

Fermi-surface renormalization-group scaling sends frequency and normal displacement from the Fermi surface to zero while leaving tangential momentum fixed. Generic interactions are irrelevant, while forward-scattering and opposite-momentum Cooper channels can remain marginal.

BCS theory describes a superconducting instability caused by attractive interactions between opposite-momentum fermions near a Fermi surface.

John Bardeen, Leon Neil Cooper and John Robert Schrieffer developed BCS theory in 1957, using attraction mediated by phonons.^[ref]

Weak-coupling Cooper pairs overlap. The Cooper instability reorganizes an unpaired Fermi surface; it is not Bose-Einstein condensation of compact, pre-existing molecules. The original paper explicitly distinguishes these pictures.^[ref]

Two complementary clues made the microscopic problem concrete. The superconducting isotope effect connected the transition to phonons.^[ref]

A 1957 transmission experiment used thin lead and tin films as samples in far-infrared and millimeter-microwave measurements. The measured transmission was converted to complex electrical conductivity, normalized to the normal-state value. Comparing film thickness, resistance, annealing and the two metals tested the response's reproducibility. Weak low-photon-energy absorption followed by an onset supported a superconducting energy gap; its extraction required a thin-film electrodynamic model.^[ref]

The later Giaever superconducting tunnelling experiment provided another way to resolve the superconducting energy gap through quantum tunnelling.^[ref] The derivations below explain what the model predicts, including the distinction between a single quasiparticle's minimum energy and the threshold for making two of them.

Eliashberg theory retains the frequency dependence of the electron–phonon interaction and the normal and anomalous electronic self-energies in a strong-coupling treatment of conventional superconductivity. The phonon spectral function ${\alpha }^{2}F(\omega )$ and a screened Coulomb pseudopotential enter the pairing equations. High phonon frequencies alone do not specify a transition temperature without the coupling strength and competing repulsion.

A fermionic excitation obtained by a canonical particle–hole Bogoliubov transformation of a paired mean-field Hamiltonian. The uniform spin-singlet problem has the displayed positive energy and weights $|u{|}^2=(1+\xi /E)/2$, $|v{|}^2=(1-\xi /E)/2$. It interpolates between particle-like and hole-like excitations across the Fermi surface. These single-particle excitations differ from collective phase and amplitude modes of the order parameter.

The continuous evolution, in suitable attractive Fermi systems, from weakly paired overlapping Cooper pairs to tightly bound composite bosons undergoing Bose-Einstein condensation. Pair formation, phase coherence and the chemical potential must be determined consistently across the crossover. A weak-coupling gap-temperature formula with fixed density of states is not uniformly valid through the molecular limit.

For a constant effective density of states and attractive coupling $\lambda$, let the linearized BCS theory gap equation integrate $\tanh (\xi /(2k_B{T}_c))/(2\xi )$ over $-\sigma <\xi <\theta$, with positive cutoffs. Its even integrand makes the result the mean of two positive-cutoff logarithms. When $k_B{T}_c\ll \sigma ,\theta$, $1/\lambda \simeq \frac{1}{2}\ln (1.13\theta /(k_B{T}_c))+\frac{1}{2}\ln (1.13\sigma /(k_B{T}_c))$, giving the displayed geometric mean of cutoffs.

If a band bottom truncates the lower side, $\sigma =\min (\mu ,\theta )$ for positive chemical potential $\mu$. The weak-coupling limit at fixed $\mu /\theta >0$ must not be interchanged with an uncontrolled $\mu \to 0$ extrapolation: the lower logarithm requires $\mu \gg k_B{T}_c$ when $\mu <\theta$.

Set the chemical potential equal to the degenerate three-dimensional spin-degenerate Fermi energy, proportional to $n^{2/3}$, and keep the effective coupling and upper cutoff fixed. The asymmetric-cutoff BCS transition temperature scales as $n^{1/3}$ below $n_{\theta }$ and saturates above it. At low density the predicted $k_B{T}_c/\mu$ instead grows as $n^{-1/3}$, eventually invalidating the degenerate and logarithmic approximations. This is a prescribed cutoff model, not a self-consistent treatment of the BCS-BEC crossover.

An attractive local pairing interaction can be decoupled with a complex Hubbard–Stratonovich transformation. Integrating the fermionic Grassmann variables gives an effective action for the pair field. Its uniform saddle is the BCS gap equation, and its slow-field expansion near the transition gives the Ginzburg-Landau theory of superconductivity. A trace over a fully doubled particle-hole basis requires the corresponding factor of one half; the displayed formula uses one independent pairing block.

Subtracting the normal-state grand potential fixes the reference constant in the uniform BCS pairing-field effective action. Here ${\omega }_n=(2n+1)\pi /\beta$ has energy units and $E_k^2={\xi }_k^2+|\Delta {|}^2$. The ratio cancels a field-independent determinant phase and produces a convergent high-frequency sum under the same momentum regulator. Differentiation gives the BCS gap equation. For a fixed attractive pairing shell, this is a finite regulated model; an unrestricted continuum contact interaction additionally needs ultraviolet renormalization. Its weak-coupling condensation density $-{\nu }_F{\Delta }^2/2$ uses a single-spin density of states per volume.

A two-component Nambu spinor $({\psi }_{k\uparrow },{\bar{\psi }}_{-k\downarrow })$ for every k accounts for both spin species once after relabelling. Its Grassmann integral contributes one functional determinant. A fully doubled spin-particle-hole basis instead has redundancy and requires one half in the trace logarithm or a corresponding Pfaffian convention. A reliable check is the zero-gap limit: the grand potential must become $-2{\beta }^{-1}{\sum }_k\ln (1+e^{-\beta {\xi }_k})$, with exactly two spin states and no invented vacuum offset.

The isotropic weak-coupling BCS theory predicts the gap ratio approximately $3.528$ and the electronic heat-capacity jump approximately $1.426$ in units of the normal-state heat capacity at the transition. These compare distinct measurements of the same paired state. They are conditional predictions of the weak-coupling fully gapped model, rather than universal numbers for all superconductors.

The gap feature in a superconducting single-particle excitation spectrum. In a simple BCS model, excitations have $E_{\mathbf{k}}=\sqrt{{\xi }_{\mathbf{k}}^2+|\Delta (\mathbf{k}){|}^2}$. The superconducting gap function specifies the pairing amplitude and its phase; its magnitude need not be constant, and nodes can leave gapless excitations.

A complex momentum-dependent pairing amplitude. Its crystal-symmetry transformation, nodes and relative signs distinguish candidate superconducting states. Magnitude-sensitive spectra alone generally do not establish all relative signs; phase-sensitive superconducting gap experiments test them through interference.

An experiment converting the relative phase of a superconducting pairing amplitude into a measurable interference or flux signal. Orthogonal-face Josephson junctions and tricrystal rings are examples. Junction asymmetry, trapped flux, loop inductance and comparison geometries are important controls.

Compare superconducting phase information from different crystal directions rather than only measuring the gap magnitude. Orthogonal-face junctions and a tricrystal half-flux experiment can test sign changes of a candidate d-wave gap. Junction properties, trapped flux and loop inductance must be controlled; a measured half-flux signal needs its actual device model.

A ring crossing three appropriately oriented grain-boundary junctions tests a superconducting pairing sign pattern. An odd total intrinsic $\pi$ offset frustrates integer phase matching and can favor a half-flux state. Actual enclosed flux also depends on loop inductance and junction currents; the measurement requires controls against ordinary trapped flux.^[ref]

A Cooper pair is a correlated pair of fermions associated with superconductivity. The conventional BCS ground state uses opposite-momentum, opposite-spin pair creation operators $c_{k\uparrow }^{†}{c}_{-k\downarrow }^{†}$. Their total momentum is zero for every relative momentum $k$; zero total momentum does not restrict $k$ to zero.

Two added electrons move above a frozen filled Fermi sea and attract within a thin energy shell. The Pauli-blocked continuum has a logarithmic pair susceptibility, so an arbitrarily weak net attractive constant interaction produces a state below the two-particle threshold at zero temperature. This two-added-particle result is not by itself a full many-electron superconducting state.

For shell width $D$, attraction $g>0$ and constant single-spin density of states $\rho$, $1=(g\rho /2)\log [(2D+B)/B]$ gives $B=2D/[\exp (2/(g\rho ))-1]$. The pair energy relative to the free threshold is $-B$. At weak coupling this scale is nonanalytic in $g$.

The pairing Hamiltonian is projected onto empty states in a specified shell above the inert Fermi sea. A state restricted to that shell is an eigenstate of this projected model, not necessarily of a contact interaction that remains unrestricted at all momenta.

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.

The BCS ground state is a product of empty and occupied pair states:The modulus squares are needed when the amplitudes are complex. Each pair block is a normalized two-state quantum superposition; multiplying independent blocks preserves normalization.

The positive-energy Bogoliubov quasiparticle annihilation operators annihilate this paired mean-field state, so it is their vacuum even though physical electron occupations are nonzero. The pair momentum is zero, but $k$ labels all relative momenta in the chosen pairing space; it is not restricted to $k=0$. ^[ref]

The state superposes different total particle numbers. A fixed-number approximation can project it onto one number sector. The utility of the unprojected state is that independent pair blocks make the BCS anomalous average, excitation spectrum and BCS gap equation directly calculable.

The empty and doubly occupied states of a paired momentum block form a two-level space. Define $S_k^{+}=a_{k\uparrow }^{†}{a}_{-k\downarrow }^{†}$, $S_k^{-}=a_{-k\downarrow }{a}_{k\uparrow }$ and $S_k^z=(n_{k\uparrow }+n_{-k\downarrow }-1)/2$. They obey the spin algebra on that space. The normalized factor $u_k|0⟩+v_k|\mathrm{pair}⟩$ is a spin-one-half coherent state; the product over blocks is the BCS ground state. Pauli exclusion prevents interpreting a block as an unrestricted harmonic-oscillator pair mode.

For real gap and the particle-hole block $\left(\begin{array}{cc}{\xi }_k& -{\Delta }_k\\ -{\Delta }_k& -{\xi }_k\end{array}\right)$, the BCS coherence factors obey $u_k^2=(1+{\xi }_k/E_k)/2$, $v_k^2=(1-{\xi }_k/E_k)/2$ and $u_k{v}_k={\Delta }_k/(2E_k)$, where $E_k=\sqrt{{\xi }_k^2+{\Delta }_k^2}$. They diagonalize the Bogoliubov--de Gennes Hamiltonian and give the empty/pair amplitudes of the BCS ground state. Their relative sign depends on the pairing convention.

For an original check, write $x={\xi }_k/\Delta$ with constant real $\Delta >0$. At $x=0$, the BCS coherence factors give $u_k^2=v_k^2=1/2$. At $x=1$, they give $(u_k^2,v_k^2)=(0.853553,0.146447)$; at $x=-1$ these values exchange places. Thus the zero-temperature occupation $v_k^2$ rounds the normal Fermi sea's step over an energy interval of order $\Delta$. This mixing is strongest at the Fermi surface, where the BCS quasiparticle spectrum has its minimum.

For a uniform isotropic weak-gap BCS state, the zero-temperature electron occupation is $n(\xi )=(1-\xi /\sqrt{{\xi }^2+{\Delta }^2})/2$. The number above the normal-state Fermi surface is approximately $D(\mu ){\int }_0^{\infty }n(\xi )d\xi$. The integral equals $\Delta /2$, while a three-dimensional parabolic band including both spins has $D(\mu )=3N/(2\mu )$. This gives the displayed leading fraction.

The result measures redistribution relative to a normal Fermi sea, not bosonic single-mode condensate depletion or quasiparticle occupation. A different band dimension or anisotropic gap requires its actual density-of-states or momentum integral. Material estimates need specified gap and chemical-potential scales.

In a conventional isotropic spin-singlet superconductor, the nuclear spin-lattice relaxation rate can rise just below the transition before falling at lower temperature. The BCS coherence factors and enhanced near-gap density of states produce the coherence enhancement. Nuclear relaxation and ultrasonic attenuation have different coherence factors, so their contrasting temperature dependences test pairing structure rather than only the existence of a gap.^[ref]

The BCS anomalous average is the pair expectation $⟨b_k⟩=⟨c_{-k\downarrow }{c}_{k\uparrow }⟩$. In the BCS ground state it is $u_k{v}_k={\Delta }_k/(2E_k)$ for real gap at zero temperature. At temperature $T$ it becomes ${\Delta }_k\tanh (E_k/(2k_{B}T))/(2E_k)$. Substitution into the pairing-field definition gives the BCS gap equation.

The reduced BCS pairing Hamiltonian retains scattering of opposite-momentum, opposite-spin Cooper pairs:A mean-field approximation introduces the BCS anomalous average and pairing field ${\Delta }_k=-{\sum }_{k^{\prime }}{V}_{k,k^{\prime }}⟨b_{k^{\prime }}⟩$. The subtraction $-\sum V_{k,k^{\prime }}⟨b_k^{†}⟩⟨b_{k^{\prime }}⟩$ corrects the double counting of the mean-field interaction energy.

The Cooper instability is the logarithmic growth of an attractive BCS coupling at low energy. If ${\nu }_F$ is the density of states and $V<0$, perturbation theory breaks down at a scale proportional to $\Lambda e^{-1/({\nu }_F|V|)}$.

The BCS gap equation makes the superconducting gap function self-consistent with the BCS anomalous average that creates it. For a translation-invariant spin-singlet mean-field approximation, define ${\xi }_k={ϵ}_k-\mu$ and $E_k=\sqrt{{\xi }_k^2+|{\Delta }_k{|}^2}$. ThenThe interaction $V_{k{k}^{\prime }}$ and its normalization/cutoff are part of the model. A nonzero solution describes pairing; ${\Delta }_k=0$ is always a solution, so finding any solution alone does not establish which state minimizes the free energy.

For constant attraction $-V$ in a symmetric shell $|\xi |<\Lambda$ with constant single-spin density of states ${\nu }_s$, the zero-temperature equation reduces toAt weak coupling, this gives $\Delta \simeq 2\Lambda e^{-1/({\nu }_{s}V)}$. The exponential explains how a small attraction produces an extremely small pairing scale, matching the Cooper instability. The finite-temperature equation can be linearized as the gap tends to zero to determine the mean-field transition. ^[ref]

Differentiate the subtracted grand potential of the reduced BCS model with respect to the squared pairing amplitude. The displayed fermionic Matsubara sum converts the derivative of the trace logarithm into ${\sum }_k\tanh (\beta E_k/2)/(2E_k)$. Setting it equal to $V/\lambda$ gives the uniform nonzero-gap saddle. The limit E→0 is finite at positive temperature, $\beta /4$. The sum relies on antiperiodic temporal boundary conditions from the fermionic coherent-state trace.

For a fixed-density balanced two-spin Fermi gas with masses $(m,m,\gamma m)$, the Fermi energy scales as ${\gamma }^{-1/3}$ while the single-spin Fermi-level density of states scales as ${\gamma }^{1/3}$. A constant attractive interaction $g$ within a narrow symmetric energy window ${\epsilon }_c\ll {\epsilon }_F$ therefore gives $1={\lambda }_0{\gamma }^{1/3}\mathrm{arsinh}({\epsilon }_c/\Delta )$. Solving gives $\Delta ={\epsilon }_c/\sinh [1/({\lambda }_0{\gamma }^{1/3})]$, whose weak-coupling limit is displayed. Extrapolating to arbitrarily large anisotropy violates the assumptions that the Fermi energy dominates the window and that the pairing coupling is small.

An attractive interaction $V_{k,k^{\prime }}=-V/\mathcal{V}$ within $|{\xi }_k|,|{\xi }_{k^{\prime }}|<{\Omega }_D$ gives a constant gap inside that energy shell and zero outside. With approximately constant single-spin density of states ${\nu }_s$ per volume,The weak coupling limit is $\Delta \simeq 2{\Omega }_D{e}^{-1/({\nu }_{s}V)}$. A density counting both spin species is twice ${\nu }_s$ and must be divided by two. The shell cutoff is a relative-energy condition, even when every Cooper pair has zero total momentum.

Use $N_0={\nu }_s$ for the normal-state single-spin density of states per unit energy and volume at the Fermi surface. Write the attraction as $-V/\mathcal{V}$, with $V>0$, and define $\lambda =N_{0}V$.^[ref]

Since $V/\mathcal{V}$ has units of energy, $V$ has units of energy times volume, making $\lambda$ dimensionless. Set ${\Omega }_D=\hslash {\omega }_D$, an energy cutoff using the Debye frequency ${\omega }_D$, rather than a frequency itself. These conventions also fix the meaning of the earlier constant-shell BCS gap equation.

Here is the reduction of the BCS gap equation in these conventions. Substituting the zero-temperature BCS anomalous average givesThe $k$ sum counts one spin species. For $\Delta >0$, cancelling $\Delta$ and replacing this sum by an integral with constant $N_0$ givesThe factor two from the symmetric integral cancels the $1/2$ in the BCS anomalous average; it is not an extra spin degeneracy. Substituting $\xi =\Delta \sinh t$ evaluates the integral as $\mathrm{arsinh}({\Omega }_D/\Delta )$. Solving givesThe first equality is exact within the constant-density of states, sharp-cutoff mean-field approximation; the last is the weak coupling asymptote. Extending the first equality to large $\lambda$ does not make this simplified model an accurate description of a strongly coupled material.

For an original numerical example, choose $\hslash {\omega }_D=20\mathrm{meV}$ and $\lambda =0.25$. The exact constant-shell BCS gap equation gives $\Delta (0)=0.732871\mathrm{meV}$; the weak coupling formula gives $0.732626\mathrm{meV}$. Its fractional underestimate, measured against the exact result, is exactly $e^{-2/\lambda }=e^{-8}\simeq 0.03355\%$. At $\lambda =1$ that error is $e^{-2}\simeq 13.53\%$, so the two formulas are no longer interchangeable. No material parameters have been fitted here.

At nonzero temperature, substitute the finite-temperature BCS anomalous average instead. The same cancellation yieldsAt the superconducting transition temperature take the continuous $\Delta \to 0$ limit. The integrand at $\xi =0$ then has the finite limiting value $1/(2k_B{T}_c)$, so no divergent endpoint needs to be inserted into a numerical quadrature. For $k_B{T}_c\ll {\Omega }_D$, the resulting integral gives $k_B{T}_c\simeq 1.134{\Omega }_D{e}^{-1/\lambda }$ and $2\Delta (0)/(k_B{T}_c)\simeq 3.528$. These ratios concern the isotropic weak coupling model.^[ref]

The BCS quasiparticle spectrum is the positive excitation energy obtained by diagonalizing the paired particle-hole block. Here the weak coupling shell model has $\Delta \ll {\Omega }_D$: the minimum lies inside the pairing shell, and states outside it cost more. To derive it for a uniform isotropic real $\Delta >0$, measure the normal-state energy from the chemical potential, ${\xi }_k={ϵ}_k-\mu$, and use the same sign convention as the BCS coherence factors:Its eigenvalues are ${\epsilon }_{\pm }=\pm E_k$, where $E_k=\sqrt{{\xi }_k^2+{\Delta }^2}$. The positive $E_k$ is the cost of creating a quasiparticle above the BCS ground state. The lower branch $-E_k$ is its particle-hole partner in this doubled matrix description, not an excitation that lowers the ground state energy without bound.

The location and value of the minimum follow directly:The unique minimum as a function of $\xi$ is $E(0)=\Delta$, at the normal-state Fermi surface. On an isotropic three-dimensional Fermi surface it occurs for every direction of $k$, rather than at a unique point of momentum space. Near that minimum,Farther away, $E(\xi )\simeq |\xi |$ when $\Delta \ll |\xi |\ll {\Omega }_D$. For the illustrative $\lambda =0.25$ example, a state at $\xi =\Delta$ costs $\sqrt{2}\Delta \simeq 1.03644\mathrm{meV}$ rather than the minimum $0.732871\mathrm{meV}$.

An elementary excitation of the paired mean-field state whose electron operator has mixed creation and annihilation components. Its positive excitation energy is $E_k=\sqrt{{\xi }_k^2+|{\Delta }_k{|}^2}$. At zero temperature the BCS ground state has zero occupation of these positive-energy modes, even though physical electrons have a smooth nontrivial momentum occupation.

For a particle-hole-symmetric normal density of states $g_F$ and isotropic superconducting energy gap $\Delta >0$, the electron spectral density is zero for $|E|<\Delta$ and has the displayed form outside the gap. The BCS coherence factors weight the two roots $\xi =\pm \sqrt{E^2-{\Delta }^2}$, where $\xi$ is normal-state energy relative to the Fermi energy. At positive $E$, their electron weights $(1+\xi /E)/2$ sum to one. Multiplying their sum by the Jacobian $|d\xi /dE|=E/\sqrt{E^2-{\Delta }^2}$ proves the formula. This electron spectral density differs from the unweighted count of all positive-energy quasiparticles, which receives both roots. Its gap-edge divergence is integrable and approaches the normal electronic density at large $|E|$.

In this fully gapped isotropic BCS quasiparticle spectrum, $\Delta$ is the minimum single-quasiparticle excitation energy. The interval between $-\Delta$ and $+\Delta$ has width $2\Delta$. A process that creates two quasiparticles from the BCS ground state has an energetic threshold $2\Delta$, subject also to its transition-selection rules.^[ref]

For example, two independently created excitations with ${\xi }_1={\xi }_2=0$ cost $E_1+E_2=2\Delta$. With the chosen cutoff example this is $1.46574\mathrm{meV}$, while adding one excitation costs at least $0.732871\mathrm{meV}$. This factor of two matters when comparing photon absorption with the Giaever superconducting tunnelling experiment. Quantum tunnelling from a normal metal can add one excitation to a sample exhibiting superconductivity; with two gapped electrodes, their excitation costs add. A minimum excitation cost is an energetic condition; the measured onset can additionally depend on temperature and on the available transitions.

The clean isotropic weak-coupling BCS coherence length is conventionally ${\xi }_0=\hslash v_F/(\pi \Delta (0))$, with Fermi velocity $v_F$. It characterizes the spatial extent of pairing correlations.^[ref]

The scale follows from the same spectral calculation. Linearizing the normal energy near the Fermi surface gives ${\xi }_k\simeq \hslash v_F(k-k_F)$. The region $|{\xi }_k|\lesssim \Delta$ therefore has radial width $\delta k\sim \Delta /(\hslash v_F)$. Fourier analysis associates this width with a distance $\ell \sim 1/\delta k\sim \hslash v_F/\Delta$. This argument establishes the scale, not the conventional factor $1/\pi$. A small superconducting energy gap consequently permits pairing correlations over many interparticle spacings.

The superconducting isotope effect is the dependence of the superconducting transition temperature on isotopic mass $M$ at fixed chemical composition. Its discovery in 1950 was a crucial clue that phonons participate in conventional superconductivity.^[ref]

The model prediction can be derived from two assumptions. For harmonic motion with a fixed restoring-force constant $K$, a phonon frequency scales as $\omega \propto \sqrt{K/M}$. If the BCS dimensionless coupling is independent of $M$, the weak coupling result $k_B{T}_c\simeq 1.134\hslash {\omega }_D{e}^{-1/\lambda }$ then gives $T_c\propto M^{-1/2}$ and $\alpha =1/2$. Changes in the effective interaction can change this exponent; it is not a universal law for every material exhibiting superconductivity. For a stipulated mass ratio $M_2/M_1=1.04$, the ideal model predicts $T_{c,2}/T_{c,1}=1/\sqrt{1.04}\simeq 0.98058$. This is an original scaling example, not a pair of measured transition temperatures.

A Landau quasiparticle is a fermionic excitation connected continuously to a Fermi gas particle when interactions preserve the phase. Near an isotropic Fermi surface, its energy relative to the chemical potential is ${\xi }_p={ϵ}_p-\mu \simeq v_F(p-p_F)$, with Fermi momentum $p_F$ and Fermi velocity $v_F$.^[ref]

The Pauli exclusion principle restricts decay. For a three-dimensional normal Fermi liquid with regular short-range interactions, a representative low-energy lifetime model isHere $E_{\ast }$ is an energy scale and $A$ is dimensionless; sharpness requires $\hslash /{\tau }_p\ll |{\xi }_p|$ at zero temperature.^[ref]

For an original example, stipulate $A=1$ and $T=0$. At $|{\xi }_p|/E_{\ast }=0.02$, the decay energy is $0.0004E_{\ast }$, just $2\%$ of the excitation energy; halving $|{\xi }_p|$ reduces that ratio to $1\%$. Approaching the Fermi surface improves the particle description even though the microscopic interaction has not weakened. This quadratic illustration is three-dimensional; low-dimensional corrections require separate analysis.

In a regular three-dimensional Fermi liquid, the Pauli exclusion principle leaves only a small set of final states available to a low-energy Landau quasiparticle. Writing $\xi =ϵ-\mu$, the leading decay rate has the scaleThe coefficient depends on the interaction and band structure.

At zero temperature, an excitation of energy $\xi >0$ creates an additional particle–hole pair. Conservation of energy leaves two small independent energy intervals, producing the quadratic phase-space factor. Its spectral width $\Gamma \sim \hslash /\tau$ then satisfies $\Gamma /|\xi |\to 0$ as $\xi \to 0$. This zero-temperature limit is the sense in which excitations become sharp at the Fermi surface. ^[ref]

At fixed nonzero temperature, the rate instead approaches a finite value proportional to $T^2$ as $\xi \to 0$. One must not infer a vanishing ratio $\Gamma /|\xi |$ in that different limit. A joint low-temperature, low-energy limit compares the width with the relevant excitation scale.

A particle loss rate can include the factor $1-f(\xi )$, whereas a spectral decay width also contains the complementary inverse collision process. Their sum has the regular quadratic low-energy scale ${\xi }^2+(\pi k_{B}T{)}^2$. Particle and hole decay must not be confused with an occupied-state loss probability.

Changing one Landau quasiparticle occupation changes the energies of others. To second order around equilibrium,Here $\mathcal{V}$ is volume, occupations are per momentum and spin state, and $f$ has units of energy times volume.^[ref]

For a toy two-state change at fixed volume, take $\delta n_a=\delta n_b=\eta$, $f_{aa}=f_{bb}=0$, $f_{ab}=f_{ba}=g\mathcal{V}$, and unperturbed energies ${\xi }_a={\xi }_b=e$. The expression gives $2e\eta +g{\eta }^2$. The factor $1/2$ counts the interaction of the same pair once; it does not halve each excitation's energy. This finite-state illustration only explains the bookkeeping of the Landau quasiparticle energy functional.

For a symmetric interaction kernel, $\delta E/\delta n_a={ϵ}_a+V^{-1}{\sum }_b{f}_{ab}\delta n_b$. Smooth Fermi-surface kernels and fixed-density chemical-potential shifts give an $O(T^2)$ thermal correction by the Sommerfeld expansion. The reference dispersion already includes zero-temperature interactions.

For a three-dimensional isotropic unpolarized spin-one-half Fermi liquid, define $f^s=(f_{\uparrow \uparrow }+f_{\uparrow \downarrow })/2$ and $f^a=(f_{\uparrow \uparrow }-f_{\uparrow \downarrow })/2$. Dimensionless Landau parameters use the spin-summed density of states per volume:$P_{\ell }$ is a Legendre polynomial; $s,a$ label spin symmetry. No extra $(2\ell +1)$ multiplies this expansion.^[ref] The superscript $s$ on $F_0^s$ is unrelated to the dimensionless wave speed $s$ below.

In a three-dimensional uniform isotropic continuum obeying the Principle of Galilean relativity, with one bare particle mass $m$, the Fermi-liquid effective mass satisfiesA band effective mass need not satisfy this relation.^[ref]

For the stipulated $m^{\ast }=2m$ in the occupation figure, this relation requires $F_1^s=3$ if those assumptions apply. At fixed $p_F$, $v_F$ is half its free-particle value, so moving by $0.04p_F$ from the Fermi surface costs approximately $0.02p_F^2/m$ instead of $0.04p_F^2/m$. The factor of three belongs to the three-dimensional angular average, not to a universal definition of effective mass.

For an isotropic spinless continuum Fermi liquid with the Principle of Galilean relativity, let $D^{\ast }=A{m}^{\ast }/(2\pi {\hslash }^2)$ be the total density of states and use a Landau quasiparticle energy functional whose double sum has no explicit $1/A$. ThenIndeed, a small boost has radial displacement $p_0\cos \theta$. The one-quasiparticle term costs $N{p}_0^2/(2m^{\ast })$, and the quadratic interaction multiplies that by $D^{\ast }\int f\cos \varphi d\varphi /(2\pi )$, since $\int \cos (\theta +\varphi )\cos \theta d\theta =\pi \cos \varphi$. Matching the Galilean boost energy of an interacting gas proves the formula. With Fourier series $D^{\ast }f=F_0+{\sum }_{\ell \ge 1}{F}_{\ell }\cos (\ell \varphi )$, the angular average is $F_1/2$. If the functional instead contains $1/A$, use its intensive kernel $Af$ and the density of states per area.

The quasiparticle residue is the coherent pole weight of the single-particle Green's function. Its coherent spectral term $Z\delta (E-{\xi }_p)$ contributes $Z\Theta (-{\xi }_p)$ to the zero-temperature momentum occupation. A smooth remainder leaves a jump $Z$ at the Fermi surface; Landau quasiparticle occupations have a unit step.^[ref]

For the local illustrative distribution in the figure, put $x=(p-p_F)/p_F$ and$\Theta$ is the Heaviside step function. Its one-sided limits are $0.825$ and $0.175$, giving $Z=0.65$. The smooth term is stipulated; this local curve does not specify a normalized distribution over all momenta. In particular, $Z$ is neither the fraction of atoms above the Fermi surface nor $m/m^{\ast }$.

The particle-hole continuum is the range of energy and momentum transfers that can excite an occupied fermion state into an unoccupied state, leaving a hole behind. For a free gas at zero temperature, choose $|\mathbf{k}|<k_F$ and $|\mathbf{k}+\mathbf{q}|>k_F$; the excitation energy is ${ϵ}_{\mathbf{k}+\mathbf{q}}-{ϵ}_{\mathbf{k}}$.

Varying the initial momentum produces a continuum at fixed transferred wavevector. For small $q$ in an isotropic gas, its positive frequencies extend approximately up to $v_{F}q$. A collective mode overlapping these excitations can decay by creating particle–hole pairs; a zero sound branch outside that region can avoid this particular damping channel. Interactions change the response and effective parameters, so the free-gas boundary is a useful starting model rather than a universal exact spectrum.

Zero sound is a collisionless collective Fermi surface deformation sustained by Landau quasiparticle interactions.^[ref]

A useful analogy is a crowd whose members respond to their neighbors' density without repeatedly bumping into them: the restoring interaction can remain effective when collisions cannot relax the distribution within one cycle. Zero sound is collective; it is not a single Landau quasiparticle outrunning every other excitation.

Consider a neutral, isotropic, unpolarized three-dimensional spin-one-half Fermi liquid, with only $F_0^s>0$ nonzero. A longitudinal density wave with angular frequency $\omega$ and wavevector magnitude $q$ has $s=\omega /(q{v}_F)=c_0/v_F$. The collisionless angular equation for the energy displacement $\nu (u)$ of the Fermi surface isAveraging givesThe bracket decreases from infinity to zero on $1<s<\infty$: each $F_0^s>0$ gives one undamped root above the small-$q$ particle-hole continuum.^[ref]

For an original example, $F_0^s=2$ gives $s=1.16474145$. Since $F_1^s=0$ here, the Principle of Galilean relativity gives $m^{\ast }=m$ and taking the hydrodynamic limit before $T\to 0$ gives $c_1/v_F=\sqrt{(1+F_0^s)/3}=1$. Stipulating $v_F=50\mathrm{m}{\mathrm{s}}^{-1}$ gives $c_0\simeq 58.24\mathrm{m}{\mathrm{s}}^{-1}$ and $c_1=50\mathrm{m}{\mathrm{s}}^{-1}$. These are model choices, not helium-3 measurements. At large $F_0^s$, expanding the logarithm gives $s\sim \sqrt{F_0^s/3}$; for small positive $F_0^s$, the root approaches $1$ exponentially closely, so numerical resolution matters. A charged electron liquid needs its long-range interaction included; this neutral model is not its long-wavelength density mode.