The Pomeranchuk instability is an instability in the shape of the Fermi surface of a material with interacting fermions, causing Landau’s Fermi liquid theory to break down. It occurs when a Landau parameter in Fermi liquid theory has a sufficiently negative value, causing deformations of the Fermi surface to be energetically favourable. It is named after the Soviet physicist Isaak Pomeranchuk.
Introduction: Landau parameter for a Fermi liquid.
In a Fermi liquid, renormalized single electron propagators (ignoring spin) are formula_1
where capital momentum letters denote four-vectors formula_2 and the Fermi surface has zero energy; poles of this function determine the quasiparticle energy-momentum dispersion relation. The four-point vertex function formula_3 describes the diagram with two incoming electrons of momentum formula_4 and two outgoing electrons of momentum formula_5 and and amputated external lines:formula_6 Call the momentum transferformula_7 When formula_8 is very small (the regime of interest here), the "T"-channel dominates the "S"- and "U"-channels. The Dyson equation then offers a simpler description of the four-point vertex function in terms of the 2-particle irreducible which corresponds to all diagrams connected after cutting two electron propagators: formula_9 Solving for formula_10 shows that, in the similar-momentum, similar-wavelength limit the former tends towards an operator formula_11 satisfyingformula_12 whereformula_13 The normalized Landau parameter is defined in terms of formula_11 as formula_15 where formula_16 is the density of Fermi surface states. In the Legendre eigenbasis the parameter formula_17 admits the expansion formula_18 Pomeranchuk's analysis revealed that each formula_19 cannot be very negative.
Stability criterion.
In a 3D isotropic Fermi liquid, consider small density fluctuations formula_20 around the Fermi momentum where the shift in Fermi surface expands in spherical harmonics as formula_21 The energy associated with a perturbation is approximated by the functional formula_22 where \vec{k} Assuming \delta\phi_{lm}, these terms are,formula_23 and so formula_24
When the Pomeranchuk stability criterion formula_25 is satisfied, this value is positive, and the Fermi surface distortion formula_26 requires energy to form. Otherwise, formula_26 releases energy, and will grow without bound until the model breaks down. That process is known as Pomeranchuk instability.
In 2D, a similar analysis, with circular wave fluctuations formula_28 instead of spherical harmonics and Chebyshev polynomials instead of Legendre polynomials, shows the Pomeranchuk constraint to be In anisotropic materials, the same qualitative result is true—for sufficiently negative Landau parameters, unstable fluctuations spontaneously destroy the Fermi surface.
The point at which formula_29 is of much theoretical interest as it indicates a quantum phase transition from a Fermi liquid to a different state of matter Above zero temperature a quantum critical state exists.
Physical quantities with manifest Pomeranchuk criterion.
Many physical quantities in Fermi liquid theory are simple expressions of components of Landau parameters. A few standard ones are listed here; they diverge or become unphysical beyond the quantum critical point.
Isothermal compressibility: formula_30
Effective mass: formula_31
Speed of first sound: formula_32
Unstable zero sound modes.
The Pomeranchuk instability manifests in the dispersion relation for the zeroth sound, which describes how the localized fluctuations of the momentum density function formula_33 propagate through space and time.
Just as the quasiparticle dispersion is given by the pole of the one-particle propagator, the zero sound dispersion relation is given by the pole of the "T"-channel of the vertex function formula_34 near small Physically, this describes the propagation of an electron hole pair, which is responsible for the fluctuations in 
When formula_40, the equation (1) can be implicitly solved for a real solution formula_41, corresponding to a real dispersion relation of oscillatory waves.
When formula_42, the solution formula_41 is pure imaginary, corresponding to an exponential change in amplitude over time. For formula_44, the imaginary part formula_45, damping waves of zeroth sound. But for formula_46 and sufficiently small formula_47, the imaginary part formula_48, implying exponential growth of any low-momentum zero sound perturbation.
Nematic phase transition.
Pomeranchuk instabilities in non-relativistic systems at formula_49 cannot exist. However, instabilities at formula_50 have interesting solid state applications. From the form of spherical harmonics formula_51 (or formula_52 in 2D), the Fermi surface is distorted into an ellipsoid (or ellipse). Specifically, in 2D, the quadrupole moment order parameter formula_53 has nonzero vacuum expectation value in the formula_50 Pomeranchuk instability. The Fermi surface has eccentricity formula_55 and spontaneous major axis orientation formula_56. Gradual spatial variation in formula_57 forms gapless Goldstone modes, forming a nematic liquid statistically analogous to a liquid crystal. Oganesyan et al.'s analysis of a model interaction between quadrupole moments predicts damped zero sound fluctuations of the quadrupole moment condensate for waves oblique to the ellipse axes.
The 2d square tight-binding Hubbard Hamiltonian with next-to-nearest neighbour interaction has been found by Halboth and Metzner to display instability in susceptibility of "d"-wave fluctuations under renormalization group flow. Thus, the Pomeranchuk instability is suspected to explain the experimentally measured anisotropy in cuprate superconductors such as LSCO and YBCO.