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For a resting uniform continuum gas with pair interactions depending on relative positions, shifting every particle momentum by ${\mathbf{p}}_0$ leaves interaction energy unchanged and changes the microscopic kinetic energy by ${\mathbf{p}}_0\cdot \mathbf{P}/m+N{p}_0^2/(2m)$. The resting ground state has total momentum $\mathbf{P}=0$. The resulting boost energy uses the bare mass even in an interacting Fermi liquid; matching this with the Landau quasiparticle energy functional constrains its interaction kernel.

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.

Landau quasiparticles describe the long-lived low-energy excitations of an interacting normal Fermi liquid. Starting from a free Fermi gas, imagine switching on interactions without changing the phase or destroying the one-to-one labelling of low-energy states. The resulting excitation carries the quantum numbers of a dressed particle but has a renormalized energy and interactions with other excitations. Its microscopic wavefunction need not resemble a bare electron state. The occupied momenta define a Fermi surface, and the quasiparticle energy measured from the chemical potential vanishes on that surface.

The Landau quasiparticle energy functional expresses the change near equilibrium in terms of occupation changes:Differentiation gives the energy of one quasiparticle in the distorted distribution. This is a phenomenological expansion around the true interacting ground state, not the assertion that its total energy equals that of independent particles. For an isotropic system, spin-symmetric and spin-antisymmetric components of $f$ are expanded in Legendre polynomials. Multiplication by the appropriate Fermi-surface density of states defines dimensionless Landau parameters $F_l^s,F_l^a$. The original formulation relates a change in the distribution to the energy shift.^[ref]

Pauli blocking makes the low-energy excitations sharp. The phase space for a quasiparticle to scatter while creating another particle–hole pair shrinks as the Fermi surface is approached, giving the regular three-dimensional low-energy Fermi-liquid quasiparticle lifetimeThus at zero temperature its width is small compared with its excitation energy. The single-particle spectral function has a pole with quasiparticle residue $Z$, generally less than one, and an incoherent remainder. In a conventional Fermi liquid the occupation distribution has a discontinuity of height $Z$ at the Fermi surface, rather than a unit free-particle step.

For an isotropic three-dimensional system, define $N^{\ast }(0)$ to include both spins. The quasiparticle density of states determines the low-temperature electronic heat capacity, $C_V/V=({\pi }^2/3)k_B^2{N}^{\ast }(0)T$. In the usual Landau-parameter normalization, the spin response is modified by $1/(1+F_0^a)$ and the compressibility by $1/(1+F_0^s)$. In a Galilean-invariant continuum, $m^{\ast }/m=1+F_1^s/3$; that particular relation must not be imposed unchanged on a crystal band. A denominator approaching zero signals a possible instability of the assumed normal state.

The same interaction functional and a kinetic equation describe collective zero sound when the excitation frequency exceeds the collision rate. Ordinary first sound is the collision-dominated hydrodynamic limit. The theory organizes thermodynamics, response and transport through a small set of measured effective parameters while explaining why interactions coexist with long-lived fermionic excitations. It is not applicable without qualification in phases where the Fermi-liquid quasiparticle pole disappears or a gap opens over the relevant Fermi surface.

A Landau quasiparticle is an electron-like excitation dressed by the interacting medium. Adiabatic continuity connects low-energy states of an ordinary Fermi liquid to those of a free Fermi gas without changing their particle statistics or conserved quantum numbers. The excitation has a renormalized dispersion, effective mass, magnetic response and finite lifetime. Pauli exclusion principle restricts collisions near the Fermi surface; in a regular three-dimensional Fermi liquid,Thus the width is small compared with excitation energy sufficiently near the surface at low temperature. This permits particle-like counting and transport even when microscopic interactions are strong. It does not make every interaction disappear: the Landau quasiparticle energy functional contains residual interactions,Specific heats and magnetic susceptibilities need not equal their bare-gas values. Logarithmic corrections can modify the lifetime scale in two dimensions, and a Luttinger liquid or a fractional Hall fluid is not an ordinary weakly perturbed electron gas.

An electron-like excitation in a normal metal and a neutral fermionic excitation in normal helium-3 are Landau quasiparticles. More generally, phonons, magnons and excitons are particle-like collective excitations, with their own statistics and quantum numbers; they are not all Landau quasiparticles. composite fermions are another emergent description, reorganizing correlations in a partly occupied Landau level rather than simply replacing the electron mass in an ordinary band.

Attach two statistical flux quanta ${\Phi }_0=h/e$ to each electron. Even attachment preserves fermionic exchange statistics, while the average statistical field opposes the applied field:This is an effective description of correlations, not literal flux tubes produced by small solenoids. An integer number of filled effective levels can give an incompressible electronic fractional Hall state. This is the correspondence between integer Hall quantization of composite fermions and selected fractional Hall states of electrons.^[ref]

Setting the signed effective filling to integer $p$ givesPositive $p$ gives $1/3,2/5,3/7,\dots$ below $1/2$. Negative $p=-q$, $q>0$, gives $q/(2q-1)$ above it, including $2/3,3/5,4/7,\dots$. The sign records reversed $B^{\ast }$. At $\nu =1/2$, the mean effective field vanishes, so the simplest description is a compressible composite-fermion sea rather than another member with finite integer $p$. Fractional charged excitations of gapped Hall states and the charge used in this semiclassical composite-fermion orbit description should not be conflated.

A direct probe away from the incompressible fractions is composite-fermion geometric resonance. One implementation patterned a high-mobility GaAs/AlGaAs Hall bar into periodic and random antidot arrays. electron beam lithography followed by an approximately $130\mathrm{nm}$ reactive-ion etch made arrays with periods $400$–$700\mathrm{nm}$. Four-terminal low-current measurements swept a perpendicular magnetic field at low temperature; red-light illumination varied density and the modulation. Periodic arrays showed commensurability structure near half filling and a Hall anomaly; matched random arrays did not. Unpatterned sections supplied another reference.^[ref]

The semiclassical explanation uses, for a fully spin-polarized sea,The mass cancels from the orbit radius. A selected orbit fits the lattice when $2R_c^{\ast }\simeq Ca$, with a geometry-dependent constant $C$ of order unity. Pinned orbits around antidots then alter longitudinal transport, producing features nearThe simplest single-antidot scale has $C\simeq 1$; detailed peak positions require the actual potential and orbit family. The two field directions bend orbits oppositely. The diagnostic is an orbit scale controlled by the small residual $B^{\ast }$, rather than the much larger applied $B$. At the same density the fully polarized $k_F^{\ast }$ is $\sqrt{2}$ times the zero-field spin-degenerate electron value; this is an independent counting test.

Calibrate density from independent Hall/filling measurements and compare several lattice periods. Use small excitation and temperature sweeps, and check periodic, random and unpatterned regions. Density variation changes both $B_{1/2}$ and the orbit scale, so simply plotting against uncorrected field can mislead. mean free path and density inhomogeneity limit resolution; illumination can soften or substantially change the imposed potential, so it cannot automatically be treated as density adjustment alone. These qualifications are part of the inference from geometry to emergent particles, rather than a claim that every feature near half filling is a new fractional plateau.

A normal Fermi liquid describes an interacting fermion system whose low-energy excitations remain continuously connected to adding particles to or removing particles from a Fermi gas. The elementary long-lived excitation is a Landau quasiparticle: adding a particle also changes the surrounding many-body state. Its energy, velocity and interactions are renormalized, even though its electric charge and spin remain those appropriate to the excitation.

Near an isotropic Fermi surface,Residual interactions make the quasiparticle energy depend on the occupations of other quasiparticles, expressed by the Landau quasiparticle energy functional. Its angular components give the Landau parameters controlling density and spin responses. Thus the successful particle picture does not mean that interactions have become zero.

Pauli blocking restricts the final states available for low-energy collisions. In an ordinary three-dimensional Fermi liquid with regular interactions, a representative decay law isAt $T=0$, the width vanishes faster than $|\xi |$, so an excitation becomes increasingly sharp as it approaches the Fermi surface. Its coherent spectral weight is the quasiparticle residue $Z$, and the physical electron occupation has a discontinuity of height $Z$ at zero temperature. Incoherent spectral weight accounts for the remainder. The quasiparticle occupations themselves obey a Fermi-Dirac distribution.

Several leading thermodynamic signatures retain their gas-like form but have interaction-dependent coefficients. In particular,for an isotropic spin-one-half model, with $N^{\ast }(0)$ the spin-summed quasiparticle density of states per volume. effective mass changes $N^{\ast }(0)$, while the dimensionless interaction parameters $F_0^a,F_0^s$ separately change the spin susceptibility and compressibility. Collisionless collective density motion can propagate as zero sound, distinct from the single-particle excitations.

The $T^2$ quasiparticle scattering rate often underlies a low-temperature $T^2$ contribution to metallic electrical resistivity, but a mechanism relaxing electrical electric current is needed. In a clean Galilean-invariant continuum, electron-electron collisions conserve total momentum and do not alone give a finite dc electrical resistivity. Crystal Umklapp scattering or other current-relaxing processes change that conclusion.

Normal liquid helium-3 and many metals motivate the Fermi-liquid framework. Pairing, magnetic ordering or other transitions can reconstruct the low-energy states. Generic interacting one-dimensional fermion systems instead form a different low-energy state, so the three-dimensional quasiparticle conclusions should not simply be transferred to a chain.

A normal Fermi liquid has interacting fermions whose low-energy excitations can be followed continuously from those of a Fermi gas, provided no phase transition destroys that correspondence. A Landau quasiparticle is a particle together with the surrounding many-body response. It has the appropriate charge and spin, a renormalized dispersion and a long lifetime close to the Fermi surface. Interactions remain in both the quasiparticle parameters and the interactions between quasiparticles; they have not been discarded.

Near an isotropic Fermi surface,The last relation defines the local isotropic Fermi-liquid effective mass. The Landau quasiparticle energy functional expresses how changing occupations shifts other quasiparticle energies:with normalization of the sum absorbed into $f$. Its angular components give Landau parameters describing collective density and spin response.

The Pauli exclusion principle severely restricts the available final states for low-energy collisions. In an ordinary three-dimensional Fermi liquid with regular interactions,At zero temperature the decay width decreases faster than $|\xi |$, making quasiparticles increasingly sharp near the Fermi surface. Their coherent spectral weight is the quasiparticle residue $0<Z\le 1$, and the true electron momentum distribution has a discontinuity of height $Z$ at zero temperature. The rest of the spectral weight is incoherent. The quasiparticle occupation itself still follows a Fermi-Dirac distribution.

The low-temperature electronic heat capacity is linear, $C_V/\mathcal{V}=({\pi }^2/3)k_B^2{N}^{\ast }(0)T$, where $N^{\ast }(0)$ is the spin-summed quasiparticle density of states. In an isotropic spin-one-half model, the uniform spin susceptibility and compressibility take the formsThus enhanced effective mass and residual interactions can separately change observable responses. The familiar $T^2$ electrical resistivity requires a mechanism relaxing electrical current, such as allowed Umklapp processes; collisions in a perfectly clean Galilean-invariant continuum conserve total momentum and do not alone produce a finite dc resistivity.

Zero sound is a collisionless collective density oscillation sustained by the quasiparticle interaction; ordinary first sound occurs when collisions establish local equilibrium during a cycle. Metals and normal liquid helium-3 supply important examples. The description fails in phases with no sufficiently sharp low-energy quasiparticles, such as generic interacting one-dimensional systems, and must be modified after pairing or another ordering transition.

For a uniform continuum gas with pair interactions depending on relative positions, a Galilean boost changes every particle momentum by $p_0\hat{\mathbf{x}}$ and leaves the interaction energy unchanged. Write ${\hat{P}}_x={\sum }_j{\hat{p}}_{j,x}$. The transformed many-body Hamiltonian operator isThe resting ground state has $⟨{\hat{P}}_x⟩=0$, so the Galilean boost energy of an interacting gas isThe mass here is the bare particle mass, because this is an exact transformation of the microscopic kinetic energy. Substituting the Fermi-liquid effective mass $m^{\ast }$ would omit the change in interaction energy encoded by the Landau quasiparticle energy functional. This conclusion requires the Principle of Galilean relativity; a gas in a lattice or a fixed external potential energy landscape need not obey it.

Within a stable normal Fermi liquid, the low-energy states are described by long-lived Landau quasiparticles and holes in a thin shell about the Fermi surface; the Pauli exclusion principle suppresses their decay. Expanding the energy about the equilibrium occupations gives the one-quasiparticle dispersion relation at first order and the interaction between occupation changes at second order, namely the Landau quasiparticle energy functional. For a smooth interaction, radial deviations in that interaction may be neglected at leading order, leaving its angular dependence; higher occupation terms contribute higher powers of the small shell thickness. Collective occupation deformations are included in the same description. The sufficiency claim is conditional on the normal Fermi liquid regime, and on supplementing this static energy functional with dynamics when required: it is not a description of arbitrary low-energy phases such as a superconductor or non-Fermi liquid.

Use the total density of states $D^{\ast }=A{m}^{\ast }/(2\pi {\hslash }^2)$ and $C=A/(2\pi \hslash {)}^2$ from part (b). In the paper's double-sum convention, $f(\varphi )$ has units of energy and scales as $A^{-1}$. Define its first angular moment byThis convention matters: the Landau quasiparticle energy functional written with an explicit $1/A$ instead uses the intensive kernel $\tilde{f}=Af$.

At zero temperature, the displayed functional is the change in $E-\mu N$; a true Galilean boost preserves $N$, so $\Delta \mathcal{F}=\Delta E$. For a radial displacement $\nu (\theta )$, the first term in the Landau quasiparticle energy functional becomesThe oriented radial integral also works for negative $\nu$: removing occupied levels below the chemical potential costs positive energy. Substituting $\nu =p_0\cos \theta +O(p_0^2/p_F)$ and using $N=C\pi p_F^2$ givesThe exact second-order correction to the boundary is unnecessary here because the unperturbed excitation energy vanishes on the Fermi surface; it changes this term only beyond quadratic order. It must nevertheless be understood when identifying $\Delta \mathcal{F}$ with a fixed-particle number energy change.

For the quadratic interaction term, the first-order occupation change alone suffices. The radial Dirac delta function integrals givePut $\varphi =\theta -{\theta }^{\prime }$. The remaining angular integral usesso only the first cosine component of the interaction survives. Since $D^{\ast }=2\pi C{m}^{\ast }$,Adding both contributions and matching the bare-mass result of part (a) yields the two-dimensional Landau effective-mass relation:Equivalently, using the density of states per unit area $g^{\ast }=D^{\ast }/A$,If dimensionless Landau parameters are defined through the Fourier series $D^{\ast }f(\varphi )=F_0+{\sum }_{\ell \ge 1}{F}_{\ell }\cos (\ell \varphi )$, then $D^{\ast }{\overline{f}}_1=F_1/2$ andDefining the cosine expansion with an extra factor of two would instead absorb that factor into $F_1$; the angular-integral formula above fixes the convention unambiguously. The physical content is that interaction changes accompanying a moving Fermi sea compensate the renormalized quasiparticle velocity, so the total boost energy is governed by the bare mass. For an interaction independent of angle, ${\overline{f}}_1=0$ and this consistency condition requires $m^{\ast }=m$.

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.