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.
Fermi liquids and Landau parameters
In a Fermi liquid, renormalized single electronpropagators are where capital momentum letters denote four vectors and the Fermi surface has zero energy. The poles of determine the quasiparticle energy-momentum dispersion relation. One can define the four-point vertex function as the diagram with two incoming electrons of momentum and two outgoing electrons of momentum and amputated external lines:
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The 2-particle-irreducible is the sum of diagrams contributing to that cannot be disconnected after cutting two electron propagators. When is very small, the T-channel becomes dominant over the S and U channels, so the Dyson equation gives Then, matrix manipulations show that is non-singular and satisfies the matrix equation, where The normalized Landau parameter is defined as, where is Fermi surface density of states. Energy is approximated by the functional where for momenta near the Fermi momentum.
Pomeranchuk stability criterion
In a 3D isotropic Fermi liquid, consider small density fluctuations where and the infinitesimal function parametrises the fluctuation. Plugging this into the energy functional, and assuming is much smaller than, gives where and is the -th Legendre polynomial. Having a positive definite energy functional requires the Pomeranchuk stability criterion, ; otherwise the Fermi surface distortion will grow unbounded until the model breaks down in what is called the Pomeranchuk instability. In 2D, a similar analysis, with circular wave fluctuations instead of spherical harmonics and Chebyshev polynomials instead of Legendre polynomials, shows the Pomeranchuk constraint to be. In non-isotropic materials, the same qualitative result is true—for sufficiently negative Landau parameters, the Fermi surface is spontaneously destroyed with unstable fluctuations. The point at which is of much theoretical interest as it indicates a quantum phase transition from a Fermi liquid to a different state of matter, and above zero temperature a quantum critical state exists.
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: Effective mass: Speed of first sound:
describes how the localized fluctuations of the momentum density function 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 near small. Physically, this describes the propagation of an electron hole pair, which is responsible for the fluctuations in. From the relation and ignoring the contributions of for, the zero sound spectrum is given by the four-vectors satisfying, or where,. When, for each real there is a real solution for, corresponding to a real zero sound dispersion relation of oscillatory waves. When, for each real there is a pure imaginary solution for, corresponding to an exponential change in zero sound amplitude over time. For, at all real, so amplitude is damped. But for, for sufficiently small, implying exponential explosion of any low-momentum zero sound fluctuation. This is a manifestation of the Pomeranchuk instability.
Pomeranchuk instabilities at are shown to not exist in non-relativistic systems by. However, instabilities at have interesting solid state applications. From the form of spherical harmonics , the Fermi surface is distorted into an ellipsoid.Specifically, in 2d, the quadrupole momentorder parameter has nonzero vacuum expectation value in the Pomeranchuk instability. The Fermi surface has eccentricity and spontaneous major axis orientation. Gradual spatial variation in 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.
