Landau Fermi-Liquid Theory

A Landau Fermi liquid is an interacting normal state whose sufficiently low-energy excitations can still be labeled by occupations of long-lived fermionic quasiparticles near an interacting Fermi surface. The particles need not interact weakly. What must remain smooth is the low-energy energy functional: changing one quasiparticle occupation shifts every other quasiparticle energy through a residual interaction function. Its first two functional derivatives determine the dispersion, thermodynamics, static response, and stability of the liquid.

This page develops the isotropic, SU(2)-invariant, spin-$1/2$ liquid in three spatial dimensions at $T=0$. The simple charge formulas refer to a neutral system or to the irreducible short-range response after long-range Coulomb effects have been separated. The state is assumed to remain normal and to have a regular spherical Fermi surface.

Required background. Fermi-surface kinematics supplies the state count and density-of-states convention. Quasiparticle pole criteria supply the independent requirement that the pole width vanish faster than its excitation energy.

Quasiparticle occupations define the low-energy energy

Let $n^0_{\mathbf p\sigma}=\Theta(p_F-p)$ be the ground-state occupation, let $\delta n_{\mathbf p\sigma}=n_{\mathbf p\sigma}-n^0_{\mathbf p\sigma}$, and write

$$\int_{\mathbf p}\equiv \int\frac{\mathrm d^3p}{(2\pi)^3}.$$

The occupation is dimensionless; the integral counts states per unit volume. If $\mathcal E[n]$ is the energy density, the quasiparticle energy and Landau interaction are its first two functional derivatives,

$$\epsilon_{\mathbf p\sigma}[n] =\frac{\delta\mathcal E}{\delta n_{\mathbf p\sigma}}, \qquad f_{\sigma\sigma'}(\mathbf p,\mathbf p') =\frac{\delta\epsilon_{\mathbf p\sigma}} {\delta n_{\mathbf p'\sigma'}}.$$

Expanding the grand-energy density $\mathcal K=\mathcal E-\mu n$ around its ground state gives

$$\begin{aligned} \delta\mathcal K ={}&\sum_\sigma\int_{\mathbf p} \xi^*_{\mathbf p}\,\delta n_{\mathbf p\sigma}\\ &+\frac12\sum_{\sigma\sigma'} \int_{\mathbf p}\int_{\mathbf p'} f_{\sigma\sigma'}(\mathbf p,\mathbf p') \delta n_{\mathbf p\sigma}\delta n_{\mathbf p'\sigma'} +O(\delta n^3), \end{aligned}$$

where $\xi^*_{\mathbf p}=\epsilon^*_{\mathbf p}-\mu$ and $\xi^*_{p_F}=0$. Gradient terms are omitted because the functional above describes uniform or asymptotically long-wavelength deformations. The linear term is the cost of moving quasiparticles through the interacting dispersion; the quadratic term says that the cost of one occupation depends on all the others.

For a displaced zero-temperature surface, $\delta n_{\mathbf p\sigma}$ is not pointwise small: it changes by one in a thin momentum shell. The remainder is an expansion in the shell displacement and excitation density, not in the height of the occupation step.

Landau’s adiabatic correspondence is a statement about low-energy labels and quantum numbers. It does not say that the exact state is a bare Slater determinant, that the pole residue is one, or that arbitrary interactions must produce this phase. The original functional construction and its thermodynamic consequences appear in Landau 1957, §§ 1–4, pp. 921–924, Open PDF.

Spin channels and Landau-parameter convention

Use collinear spin labels $\sigma=+1$ for $\uparrow$ and $\sigma=-1$ for $\downarrow$. SU(2) invariance lets the forward interaction be decomposed as

$$f_{\sigma\sigma'}(\cos\theta) =f^s(\cos\theta)+\sigma\sigma' f^a(\cos\theta),$$

so that

$$f^s=\frac{f_{\uparrow\uparrow}+f_{\uparrow\downarrow}}2, \qquad f^a=\frac{f_{\uparrow\uparrow}-f_{\uparrow\downarrow}}2.$$

The superscripts $s$ and $a$ mean symmetric and antisymmetric under a spin-density deformation. They are often called the density and spin channels. Define the total two-spin quasiparticle density of states by

$$N(0) =\sum_\sigma\int_{\mathbf p}\delta(\xi^*_{\mathbf p}) =\frac{p_F^2}{\pi^2v_F^*} =\frac{m^*p_F}{\pi^2}, \qquad m^*\equiv\frac{p_F}{v_F^*}.$$

Each spin species therefore contributes $N(0)/2$. This page defines the dimensionless Landau parameters through the three-dimensional Legendre expansion

$$f^{s,a}(\cos\theta) =\frac1{N(0)} \sum_{\ell=0}^{\infty} F_\ell^{s,a}P_\ell(\cos\theta).$$

These declarations fix every factor of two and every factor of $2\ell+1$ below. A source using a per-spin density of states, or expanding with $(2\ell+1)P_\ell$, prints different numerical parameters. Translate the definitions before comparing formulas. Measured compressibility, susceptibility, and current are invariant; the symbol $F_\ell$ is not.

The functional derivative $f$ is also not a bare two-body potential or the on-shell collision amplitude. Repeated particle–hole scattering converts it into a dimensionless forward amplitude

$$A_\ell^{s,a} =\frac{F_\ell^{s,a}} {1+F_\ell^{s,a}/(2\ell+1)}$$

in the same harmonic convention. Collision rates use the antisymmetrized amplitude and phase space, whereas thermodynamic stiffnesses contain $F_\ell$. Baym and Pethick 1991, ch. 1 develops this distinction and the associated kinetic theory.

The chapter’s Fermi-liquid relationship map shows where this functional sits between pole data and response. Its claim-comparison table keeps Landau thermodynamics, zero sound, Luttinger counting, and instability diagnostics logically separate.

Uniform deformations give compressibility and spin response

The most useful first calculation is a uniform expansion of the two Fermi surfaces. Let

$$\delta n=\delta n_\uparrow+\delta n_\downarrow, \qquad \delta s=\delta n_\uparrow-\delta n_\downarrow$$

be changes in total particle density and spin-number density. Moving one spin surface changes its kinetic energy by $(\delta n_\sigma)^2/[2N_\sigma(0)]$, where $N_\sigma(0)=N(0)/2$. Adding the $\ell=0$ interaction and sources gives

$$\begin{aligned} \delta\mathcal K_0 ={}&\frac{1+F_0^s}{2N(0)}(\delta n)^2 +\frac{1+F_0^a}{2N(0)}(\delta s)^2\\ &-\delta\mu\,\delta n-h\,\delta s. \end{aligned}$$

This short expression is the factor-of-two check for the convention. Minimizing it yields

$$\delta n=\frac{N(0)}{1+F_0^s}\,\delta\mu, \qquad \delta s=\frac{N(0)}{1+F_0^a}\,h.$$

At zero temperature, with

$$\kappa_T\equiv \frac1{n^2}\left(\frac{\partial n}{\partial\mu}\right)_{T=0},$$

the compressibility is therefore

$$\kappa_T=\frac{N(0)}{n^2(1+F_0^s)}.$$

For a Zeeman energy $-\sigma\mu_{\rm mag}B$, take $h=\mu_{\rm mag}B$ and $M=\mu_{\rm mag}\delta s$. The magnetization susceptibility is

$$\chi_M=\frac{\mu_{\rm mag}^2N(0)}{1+F_0^a}.$$

The symbol $\mu_{\rm mag}$ is the quasiparticle magnetic moment before the self-consistent molecular field is included. A material $g$ factor, orbital response, pseudospin probe, or SI-unit convention requires separate matching.

For a free spherical reference gas at the same density and with the same magnetic moment, $N(0)/N_0(0)=m^*/m$. Thus

$$\frac{\kappa_T}{\kappa_{T,0}} =\frac{m^*/m}{1+F_0^s}, \qquad \frac{\chi_M}{\chi_{M,0}} =\frac{m^*/m}{1+F_0^a}.$$

These ratios show why an enhanced effective mass does not by itself imply an enhanced compressibility or spin response: the corresponding $F_0$ can oppose or amplify the density-of-states change.

Contact repulsion separates the two uniform channels

As a concrete matching exercise, consider a weak, renormalized low-energy contact interaction between opposite spins,

$$f_{\uparrow\uparrow}=0, \qquad f_{\uparrow\downarrow}=g>0.$$

The half-sum and half-difference definitions give

$$F_0^s=\frac{N(0)g}{2}, \qquad F_0^a=-\frac{N(0)g}{2}, \qquad F_{\ell\ge1}^{s,a}=0$$

at this order. Consequently,

$$\kappa_T =\frac{N(0)} {n^2[1+N(0)g/2]}, \qquad \chi_M =\frac{\mu_{\rm mag}^2N(0)} {1-N(0)g/2}.$$

The same repulsion stiffens a density change but favors spin polarization, because opposite spins avoid an interaction energy when their populations separate. The apparent zero of the spin denominator is the Hartree or Stoner warning. Weak-coupling matching is no longer controlled there, so this simple calculation diagnoses a tendency rather than furnishing a precise transition point. For a microscopic zero-range interaction, $g$ must be a renormalized low-energy coupling rather than a cutoff-dependent bare constant.

Heat capacity, effective mass, and backflow

The low-temperature entropy counts quasiparticle levels near the surface. Its leading coefficient is

$$\frac{C_V}{V} =\frac{\pi^2}{3}N(0)T+o(T).$$

There is no explicit multiplicative pole residue $Z$ in this leading term. That does not mean $Z$ is unphysical: it is the coherent weight of a named microscopic field. The thermodynamic effective mass instead comes from the pole dispersion through $v_F^*=p_F/m^*$. In general,

$$Z^{-1} =1-\partial_\omega\operatorname{Re}\Sigma^R, \qquad v_F^* =Z\left[ \frac{p_F}{m} +\partial_p\operatorname{Re}\Sigma^R \right]_{p_F,0}.$$

Only a momentum-independent self-energy reduces this to $m^*/m=Z^{-1}$. The microscopic self-energy treatment develops this matching and the required Ward identities.

A stronger relation follows when the continuum is Galilean invariant. The particle-number current carried by a quasiparticle contains its direct group velocity and the backflow induced in the filled sea,

$$\mathbf j_{\mathbf p\sigma} =\mathbf v_{\mathbf p}^* -\sum_{\sigma'}\int_{\mathbf p'} f_{\sigma\sigma'}(\mathbf p,\mathbf p') \boldsymbol\nabla_{\mathbf p'}n^0_{\mathbf p'\sigma'}.$$

At the Fermi surface, only the spin-symmetric $\ell=1$ harmonic survives the angular integral:

$$\mathbf j_{\mathbf p_F\sigma} =v_F^*\left(1+\frac{F_1^s}{3}\right)\hat{\mathbf p} =\frac{\mathbf p_F}{m}.$$

Hence

$$\frac{m^*}{m}=1+\frac{F_1^s}{3}.$$

for this Galilean continuum. This is a boost Ward identity, not a generic property of a crystal. On a lattice, momentum is crystal momentum; band velocity, mass tensors, and current vertices must remain separate. Landau’s original derivation and its explicit crystal caveat are in Landau 1957, § 2, pp. 922–923, Open PDF.

Shape deformations and Pomeranchuk stability

Uniform density and spin changes are the $\ell=0$ modes. Higher harmonics change the shape of the two Fermi surfaces. For one axisymmetric channel $x=s$ or $a$, write

$$\delta n_{\mathbf p\sigma} =\delta(\xi^*_{\mathbf p}) \eta_\sigma^x u_\ell^xP_\ell(\cos\vartheta), \qquad \eta_\sigma^s=1, \quad \eta_\sigma^a=\sigma.$$

The quantity $u_\ell^x$ has units of energy and is the local shift of the quasiparticle surface. Orthogonality of the Legendre polynomials diagonalizes the quadratic functional:

$$\delta\mathcal K_\ell^x =\frac{N(0)|u_\ell^x|^2}{2(2\ell+1)} \left( 1+\frac{F_\ell^x}{2\ell+1} \right).$$

Every magnetic quantum number has the same bracket. Local stability of the assumed isotropic normal state requires

$$1+\frac{F_\ell^{s,a}}{2\ell+1}>0, \qquad\text{or equivalently}\qquad F_\ell^{s,a}>-(2\ell+1).$$

The $\ell=0$ symmetric and antisymmetric inequalities are positivity of compressibility and spin susceptibility. An $\ell=2$ symmetric mode is a quadrupolar, or nematic, deformation. The $\ell=1$ sector is subtler: when an order parameter coincides exactly with a conserved charge or spin current, Ward identities can cancel an apparent susceptibility divergence, while a generic $\ell=1$ form factor need not be protected. Pomeranchuk 1959, pp. 361–362, Open PDF derives the quadratic criterion; Chubukov, Klein, and Maslov 2018, §§ II.1–II.3, Open arXiv version explains the modern conserved-current qualification.

At equality a quadratic mode becomes soft. Beyond it, the reference state is not a local minimum. This alone does not identify the final phase or the transition order: higher powers of the deformation matter, and pairing or finite-wavevector order can intervene first. The dedicated Pomeranchuk and density-wave page separates a $q=0$ shape instability from finite-$q$ order.

Where the Fermi-liquid description applies

The first requirement is a sharp fermionic pole over the scales being probed. Near the surface, its damping must obey $\Gamma/\lvert\omega\rvert\to0$ as $\omega\to0$; finite residue alone is insufficient. Pauli blocking gives the conventional three-dimensional liquid a decay rate of order $\omega^2+\pi^2T^2$, but singular interactions or critical modes can change that hierarchy. Abrikosov and Khalatnikov 1958, §§ 1 and 7, pp. 69–70 and 76 give the classic sharp-excitation criterion.

The state must also remain normal and locally stable throughout the window. Pairing, magnetism, density waves, or other order can cut off the liquid before the formal low-energy limit is reached. A regular surface and finite forward interaction are assumed; van Hove points, flat directions, perfect nesting, or coupling to a gapless gauge or order-parameter field require separate analysis.

Dimension and environment matter. A generic interacting one-dimensional metal is a Luttinger liquid rather than this quasiparticle phase. Two-dimensional Fermi liquids can exist, but their harmonic normalization and nonanalytic corrections differ from the formulas derived here. Long-range Coulomb forces distinguish thermodynamic or irreducible compressibility from the fully screened $q\to0$ charge response and turn the neutral density mode into a plasmon. On a lattice, replace spherical harmonics by point-group form factors and do not import the Galilean mass identity.

Finally, Landau theory is a conditional low-energy description, not a nonperturbative existence theorem for arbitrary Hamiltonians. Microscopic perturbative derivations establish the phenomenology under normal-state and regularity assumptions Nozières and Luttinger 1962, Part I, § I, p. 1423 and Luttinger and Nozières 1962, Part II, pp. 1431–1440. The separate Luttinger-volume page explains why a density count is not a proof of quasiparticle longevity.

Common pitfalls

Leaving the convention implicit. A total DOS, a per-spin DOS, and a $(2\ell+1)$-normalized harmonic produce different printed $F_\ell$. Reconstruct $f_{\sigma\sigma'}$ or an observable before comparing sources.

Equating the Landau function with a collision amplitude. The energy-functional interaction and the on-shell scattering amplitude are different vertex limits. Thermodynamics uses $F_\ell$; lifetimes use the antisymmetrized amplitude and collision phase space.

Treating $Z$, $m^*$, and current renormalization as one number. They measure pole weight, dispersion, and vertex/backflow response. They coincide only under additional assumptions.

Using a static formula in a dynamic limit. Compressibility and equilibrium spin susceptibility are static. Zero sound takes $q,\omega\to0$ at fixed $\omega/(qv_F)$ with $\omega\tau\gg1$.

Naming the ordered phase from one negative stiffness. A failed quadratic inequality proves only that the assumed normal state is locally unstable in that channel. Nonlinear terms and competing finite-$q$ or pairing channels decide what replaces it.

Exercises

Translate a density-of-states convention

Let $\nu(0)=N(0)/2$ be the per-spin DOS, and suppose a source defines $\widetilde F_\ell=\nu(0)f_\ell$ while keeping the same dimensional $f_\ell$ used on this page. Express $F_\ell$ and the compressibility in terms of $\widetilde F_0^s$.

Solution

Because $F_\ell=N(0)f_\ell$ and $N(0)=2\nu(0)$,

$$F_\ell=2\widetilde F_\ell.$$

Therefore

$$\kappa_T =\frac{2\nu(0)} {n^2[1+2\widetilde F_0^s]}.$$

The factor of two is fixed by the declared definitions. A source that also redefines the spin-symmetric function can have a different translation, which is why copying the symbol alone is unsafe.

Diagnose a quadrupolar instability

Find the stability bound for a symmetric $\ell=2$ deformation and identify its geometric meaning.

Solution

The stiffness is proportional to

$$1+\frac{F_2^s}{5},$$

so stability requires $F_2^s>-5$. The deformation stretches and compresses the Fermi surface with quadrupolar angular dependence. At equality its $q=0$ shape susceptibility becomes soft; the quadratic calculation does not determine the nonlinear shape or transition order.

Reconstruct three observable ratios

A Galilean liquid at fixed density has $m^*/m=2$, $F_0^s=1$, $F_0^a=-1/2$, and the same magnetic moment as its free reference gas. Find $\gamma/\gamma_0$, $\kappa_T/\kappa_{T,0}$, $\chi_M/\chi_{M,0}$, and $F_1^s$.

Solution

The heat-capacity coefficient scales with the quasiparticle DOS, so

$$\frac{\gamma}{\gamma_0}=\frac{m^*}{m}=2.$$

The two static ratios are

$$\frac{\kappa_T}{\kappa_{T,0}} =\frac2{1+1}=1, \qquad \frac{\chi_M}{\chi_{M,0}} =\frac2{1-1/2}=4.$$

Galilean invariance gives

$$F_1^s=3\left(\frac{m^*}{m}-1\right)=3.$$

Thus the heat capacity and spin response are enhanced even though the compressibility is unchanged.

Test whether the framework applies

A one-dimensional interacting wire has a threshold spectral function with no isolated fermion pole and power-law correlations. Can its low-temperature response be parameterized by the $F_\ell^{s,a}$ used here?

Solution

No. The missing long-lived single-particle pole invalidates the starting quasiparticle occupation functional, and a one-dimensional “Fermi surface” has only two points rather than the angular harmonics used above. The correct low-energy variables are collective charge and spin modes of a Luttinger liquid. Similar-looking response coefficients may exist, but they are not obtained by importing these three-dimensional Landau parameters.

Continue

Microscopic quasiparticles and self-energy connect $Z$, $v_F^*$, lifetime scaling, and the forward vertex. Fermi-liquid response and zero sound derives the collisionless kinetic equation and its order of limits. Pomeranchuk and density-wave instabilities compare $q=0$ shape modes with finite-$q$ order, while Luttinger’s theorem treats the logically separate volume count.

References

• Abrikosov, A. A., and I. M. Khalatnikov. “Theory of the Fermi Fluid (The Properties of Liquid He³ at Low Temperatures).” Soviet Physics Uspekhi 1 (1958): 68–90. doi:10.1070/PU1958v001n01ABEH003086.
• Baym, Gordon, and Christopher Pethick. Landau Fermi-Liquid Theory: Concepts and Applications. New York: Wiley, 1991, ch. 1. doi:10.1002/9783527617159.
• Chubukov, Andrey V., Avraham Klein, and Dmitrii L. Maslov. “Fermi-Liquid Theory and Pomeranchuk Instabilities: Fundamentals and New Developments.” Journal of Experimental and Theoretical Physics 127 (2018): 826–843. doi:10.1134/S1063776118110122. Open arXiv version.
• Landau, L. D. “The Theory of a Fermi Liquid.” Soviet Physics JETP 3 (1957): 920–925; English translation of Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki 30 (1956): 1058–1064. Official PDF.
• Luttinger, J. M., and P. Nozières. “Derivation of the Landau Theory of Fermi Liquids. II. Equilibrium Properties and Transport Equation.” Physical Review 127 (1962): 1431–1440. doi:10.1103/PhysRev.127.1431.
• Nozières, P., and J. M. Luttinger. “Derivation of the Landau Theory of Fermi Liquids. I. Formal Preliminaries.” Physical Review 127 (1962): 1423–1431. doi:10.1103/PhysRev.127.1423.
• Pomeranchuk, I. Ya. “On the Stability of a Fermi Liquid.” Soviet Physics JETP 8 (1959): 361–362; English translation of Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki 35 (1958): 524–525. Official PDF.

Further reading

• Shankar, Ramamurti. “Renormalization-Group Approach to Interacting Fermions.” Reviews of Modern Physics 66 (1994): 129–192. doi:10.1103/RevModPhys.66.129. Open arXiv version.