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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.

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.

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.

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]

A normal Fermi liquid is an interacting fermion phase continuously connected to a Fermi gas without broken symmetry. Its Landau quasiparticles are dressed particle/hole excitations near the Fermi surface, with the same conserved quantum numbers, finite quasiparticle residue and a decay width small compared with their excitation energy. Their reference dispersion already includes zero-temperature interaction renormalization; “bare quasiparticle” here does not mean a free atom.^[ref]

The rate assumes $k_{B}T,|\xi |\ll E_F$, $\xi =ϵ-\mu$, regular scattering amplitudes and density of states, equilibrium Fermi-Dirac distributions, and two-body collisions conserving energy and momentum. Pauli exclusion principle restricts the available phase space. It describes a regular three-dimensional normal liquid, not a one-dimensional interacting electron liquid. On the surface, ${\tau }^{-1}\propto T^2$; at $T=0$ and $\xi >0$, it is proportional to ${\xi }^2$. Hence excitations become increasingly sharp. The quoted asymmetric occupation factor is for particle out-scattering: a hole decay or full spectral width also needs the complementary in-scattering process; it does not imply exponentially stable deep holes.

Symmetry $f_{ab}=f_{ba}$ cancels the quadratic functional's factor one-half, giving the thermal Landau quasiparticle energy shiftHere $a=(\mathbf{k},\lambda )$, and the derivative is with respect to an occupation, not particle density. Replacing the homogeneous momentum sum by an energy integral defines a smooth angularly averaged kernel $H_a(ϵ)$. With $\delta n=n_F(ϵ;\mu ,T)-\Theta ({\mu }_0-ϵ)$,Fixed density gives $\delta \mu =-({\pi }^2/6)(k_{B}T{)}^2{D}^{\prime }({\mu }_0)/D({\mu }_0)+O(T^4)$; fixed chemical potential instead has $\delta \mu =0$. Thus ${\widetilde{ϵ}}_a={ϵ}_a+O(T^2)$ in either case, provided the kernels are smooth and no instability intervenes. This is the Sommerfeld expansion; singular spectra require separate treatment.^[ref]

The leading entropy is therefore that of quasiparticles with their zero-temperature effective mass:$D^{\ast }$ here is the total, spin-summed density of states. The plotted low-temperature slopes require the enhanced masses $5.4m$ at molar volume $26.2\mathrm{c}{\mathrm{m}}^3\mathrm{mo}{\mathrm{l}}^{-1}$ and $2.7m$ at $36.8\mathrm{c}{\mathrm{m}}^3\mathrm{mo}{\mathrm{l}}^{-1}$, rather than the bare-mass predictions. At fixed molar particle number, ${\gamma }_{\mathrm{mol}}\propto m^{\ast }{V}_{\mathrm{mol}}^{2/3}$. Agreement is restricted to the low-temperature limit: the measured curves depart markedly from a fixed-mass ideal-gas extrapolation at higher temperature. This supports long-lived quasiparticles with smooth low-energy parameters, but heat capacity alone does not measure their lifetime or prove adiabatic continuity.