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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.

A normalized retarded amplitude $G^R(\mathbf{k},t)=-i\Theta (t)e^{-i{\Omega }_{\mathbf{k}}t-{\Gamma }_{\mathbf{k}}t}$, with constant ${\Gamma }_{\mathbf{k}}>0$, has Fourier transform $G^R(\mathbf{k},\omega )=(\omega -{\Omega }_{\mathbf{k}}+i{\Gamma }_{\mathbf{k}}{)}^{-1}$. Consequently $A=-\mathrm{Im}{G}^R/\pi$ is the displayed positive Lorentzian line shape, of unit frequency integral. Its maximum is at ${\Omega }_{\mathbf{k}}$ and its half width at half maximum is ${\Gamma }_{\mathbf{k}}$. The energy half-width is $\hslash {\Gamma }_{\mathbf{k}}$. Amplitude decays on time $1/{\Gamma }_{\mathbf{k}}$ whereas probability decays on time $1/(2{\Gamma }_{\mathbf{k}})$. A general interacting electron has quasiparticle residue $Z$ and incoherent spectral weight in addition to this normalized local pole model; frequency-dependent width or self-energy can invalidate a global Lorentzian.

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.

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 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.

Noninteracting spectral line. Treat $\mathbf{k}$ as a wavevector, so physical crystal momentum is $\hslash \mathbf{k}$. Write ${\Omega }_{\mathbf{k}}={\epsilon }_{\mathbf{k}}/\hslash$; an energy zero at the chemical potential can be chosen throughout. The temporal factor of a noninteracting state is $e^{-i{\Omega }_{\mathbf{k}}t}$. With the Fourier transform convention $\tilde{\psi }(\omega )=\int dt{e}^{i\omega t}\psi (t)$,Thus its normalized frequency spectral function isOne should not square the delta distribution as if it were an ordinary finite-time transition amplitude. More precisely, the fermionic retarded Green function is $G_0^R(\mathbf{k},t)=-i\Theta (t)e^{-i{\Omega }_{\mathbf{k}}t}$. Its Fourier transform is $(\omega -{\Omega }_{\mathbf{k}}+i{0}^{+}{)}^{-1}$, whose negative imaginary part divided by $\pi$ is the same delta peak.

Interactions and lifetime. Interactions change the dispersion through the real part of the self-energy and permit decay into other excitations through its imaginary part. In the simplest quasiparticle-pole model, exponential decay of the retarded amplitude givesPerforming the causal integral,The corresponding Lorentzian quasiparticle spectral peak isIts total frequency integral is one. The exponential amplitude lifetime is $1/{\Gamma }_{\mathbf{k}}$; the probability lifetime is $1/(2{\Gamma }_{\mathbf{k}})$. Describing ${\Gamma }_{\mathbf{k}}$ as a scattering rate therefore requires the lifetime convention to be specified. The energy half-width is $\hslash {\Gamma }_{\mathbf{k}}$.

A general interacting electron is more accurately described by $G^R=Z_{\mathbf{k}}/(\omega -{\Omega }_{\mathbf{k}}^{\ast }+i{\Gamma }_{\mathbf{k}})+G_{\mathrm{inc}}^R$ near the pole. The coherent peak then integrates to the quasiparticle residue $Z_{\mathbf{k}}$ and an incoherent continuum carries the remaining weight. In an energy-unit self-energy convention,Linearizing the real denominator gives $Z=[1-{\hslash }^{-1}{\partial }_{\omega }\mathrm{Re}{\Sigma }^R{]}^{-1}$ and $\Gamma =-Z\mathrm{Im}{\Sigma }^R/\hslash$ at the pole. The requested unit-weight Lorentzian is the normalized quasiparticle model, not the most general exact interacting spectrum.

Peak position and width. At fixed $\mathbf{k}$, put $\gamma =\gamma (\mathbf{k})>0$ and $\Omega =\hslash k^2/(2m^{\ast })$. Differentiation givesIt is positive for $\omega <\Omega$ and negative for $\omega >\Omega$, so the only maximum is at $\Omega$, with height $1/(\pi \gamma )$. Setting $A$ to half this height gives $(\omega -\Omega {)}^2={\gamma }^2$. ThereforeIn energy units the maximum is ${\hslash }^2{k}^2/(2m^{\ast })$ and the half-width is $\hslash \gamma (\mathbf{k})$. Frequency dependence of the self-energy would require reconsidering both the shape and peak position.

Fermi gas and Fermi liquid. A noninteracting Fermi gas has a delta-function spectral line at its bare dispersion for every momentum. A normal three-dimensional Fermi liquid has a shifted coherent peak plus incoherent spectral weight; close to the Fermi surface its width scales as $(\epsilon -\mu {)}^2+(\pi k_{B}T{)}^2$. At zero temperature the peak becomes arbitrarily sharp at the Fermi surface, while it broadens farther away. Spectral functions include addition and removal weight; occupied-only photoemission contains an extra Fermi factor.

Photoemission measurement. A photon of known energy $h\nu$ ejects an electron. Measure its kinetic energy $E_{\mathrm{kin}}$ and emission direction. For binding energy $E_B\ge 0$ below the Fermi level and the appropriate calibrated work function $\Phi$, conservation of energy givesConservation of momentum parallel to a periodic surface giveswhere ${\mathbf{G}}_{\parallel }$ is a surface reciprocal-lattice vector. At suitable photon energies the photon momentum can be neglected; at higher energies it must be retained. The surface breaks perpendicular translation symmetry, so the perpendicular crystal momentum generally needs a final-state model. A two-dimensional band can be reconstructed directly from the parallel components.

Within the sudden approximation and electric-dipole approximation, photoemission intensity and spectral function are related byHere $E$ is measured relative to the chemical potential. Instrumental resolution convolves the product and inelastic scattering adds background. Thus ARPES maps the occupied electron removal spectral function, weighted by a matrix element; it does not measure the full unoccupied spectrum or an unweighted spectral function automatically. Bright ridges give dispersions, and intrinsic widths give decay information after instrumental effects are removed.

Reading the supplied image. Its bright ridges form a graphene-like cone. The crossing near $k_y=0$ is the Dirac point, not in general the Fermi level. In the paper's plotted energy coordinates its energy is roughly $E_D\sim 1.4$–$1.5\mathrm{eV}$. A velocity follows from a branch slope independently of the energy zero:For example, the lower branch moves by about $0.3\mathrm{eV}$ between $|k_y|\sim 0.03$ and $0.11{\mathring{\mathrm{A}}}^{-1}$. This gives $dE/dk\sim 3.8\mathrm{eV}\mathring{\mathrm{A}}$ andReading other points changes the estimate by tens of percent. The original exam image has unequal energy increments on nearly equally spaced labelled ticks, so it does not justify a precise fitted velocity. The value is an estimate of the cone slope extrapolated to the Fermi crossing, rather than a measured derivative exactly at a visibly identified Fermi cutoff.

To determine a Fermi energy, first calibrate the Fermi edge, then subtract the chosen reference energy. Neither a Fermi-level marker nor an explicit energy-zero convention is supplied in this crop. If its upper termination near $1.9\mathrm{eV}$ is intended to represent the Fermi cutoff, the conditional graph answer is $E_F\simeq 1.9\mathrm{eV}$ on that axis, or $E_F-E_D\sim 0.4\mathrm{eV}$ relative to the Dirac crossing. Identifying the crossing itself as the Fermi energy would incorrectly assume undoped graphene.

The research article cited by the exam identifies this as linearly polarized graphene ARPES and independently states $E_F-E_D\simeq 0.4\mathrm{eV}$ for the substrate-doped sample. That resolves the relative energy when this external calibration is admitted.^[ref] The crop alone does not uniquely fix an absolute Fermi energy; the source-supported relative value and the conditional $1.9\mathrm{eV}$ axis reading should be distinguished.

Let $A$ be the real-space area and use ${\sum }_{\mathbf{p}}\to C\int d^{2}p$, with $C=A/(2\pi \hslash {)}^2$. There is one state per momentum cell, since the fermions are spinless. An annulus between $p$ and $p+dp$ contains $C2\pi pdp$ states. Near the Fermi surface, the quasiparticle dispersion relation has $dϵ/dp=v_F^{\ast }=p_F/m^{\ast }$, so the total density of states isThe density of states per unit area and the occupied-state count are, respectively,In particular, $D^{\ast }(\mu )=2N{m}^{\ast }/p_F^2$. No factor of two for spin degeneracy is present. This counts Landau quasiparticle levels, so it is not multiplied by the quasiparticle residue of the microscopic particle Green's function.

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 $m/m^{\ast }$.