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.
In a Fermi liquid, renormalized single electron propagators (ignoring spin) are
where capital momentum letters denote four-vectors
and the Fermi surface has zero energy; poles of this function determine the quasiparticle energy-momentum dispersion relation.
describes the diagram with two incoming electrons of momentum
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:
The normalized Landau parameter is defined in terms of
In a 3D isotropic Fermi liquid, consider small density fluctuations
, where the shift in Fermi surface expands in spherical harmonics as
is satisfied, this value is positive, and the Fermi surface distortion
releases energy, and will grow without bound until the model breaks down.
[4] In anisotropic materials, the same qualitative result is true—for sufficiently negative Landau parameters, unstable fluctuations spontaneously destroy the Fermi surface.
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.
[5] 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.
The Pomeranchuk instability manifests in the dispersion relation for the zeroth sound, which describes how the localized fluctuations of the momentum density function
[1] 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
Physically, this describes the propagation of an electron hole pair, which is responsible for the fluctuations in
, corresponding to a real dispersion relation of oscillatory waves.
is pure imaginary, corresponding to an exponential change in amplitude over time.
, implying exponential growth of any low-momentum zero sound perturbation.
in 2D), the Fermi surface is distorted into an ellipsoid (or ellipse).
Specifically, in 2D, the quadrupole moment order parameter
Oganesyan et al.'s analysis [8] 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[9] 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.