Pomeranchuk instability

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[edit]

In a Fermi liquid, renormalized single electron propagators (ignoring spin) are

where capital momentum letters denote four-vectors ${\textstyle K=(k_{0},{\vec {k}})}$ and the Fermi surface has zero energy; poles of this function determine the quasiparticle energy-momentum dispersion relation.^[1] The four-point vertex function ${\textstyle \Gamma _{(K_{3},K_{4};K_{1},K_{2})}}$ describes the diagram with two incoming electrons of momentum ${\textstyle K_{1}}$ and ${\textstyle K_{2}}$; two outgoing electrons of momentum ${\textstyle K_{3}}$ and ${\textstyle K_{4}}$; and amputated external lines: Call the momentum transfer When ${\textstyle K'}$ 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 ${\textstyle {\tilde {\Gamma }}}$, which corresponds to all diagrams connected after cutting two electron propagators: Solving for ${\displaystyle \Gamma }$ shows that, in the similar-momentum, similar-wavelength limit ${\textstyle k'\ll \omega '\ll 1}$, the former tends towards an operator ${\textstyle \Gamma _{K_{1},K_{2}}^{\omega }}$ satisfying where^[2] The normalized Landau parameter is defined in terms of ${\textstyle \Gamma _{K_{1},K_{2}}^{\omega }}$ as where ${\textstyle N={\frac {p_{\mathrm {F} }m_{\mathrm {F} }^{*}}{\pi ^{2}}}}$ is the density of Fermi surface states. In the Legendre eigenbasis ${\textstyle \{P_{\ell }\}_{\ell }}$, the parameter ${\textstyle f}$ admits the expansion Pomeranchuk's analysis revealed that each ${\textstyle f_{\ell }}$ cannot be very negative.

Stability criterion[edit]

In a 3D isotropic Fermi liquid, consider small density fluctuations ${\textstyle \delta n_{k}=\Theta (|k|-p_{\mathrm {F} })-\Theta (|k|-p_{\mathrm {F} }'({\hat {k}}))}$ around the Fermi momentum ${\textstyle p_{\mathrm {F} }}$, where the shift in Fermi surface expands in spherical harmonics as

The energy associated with a perturbation is approximated by the functional where ${\textstyle {\vec {\epsilon _{k}}}=v_{\mathrm {F} }(|{\vec {k}}|-p_{\mathrm {F} })}$. Assuming ${\textstyle |\delta \phi _{lm}|\ll |p_{\rm {F}}|}$, these terms are,^[3] and so

When the Pomeranchuk stability criterion

is satisfied, this value is positive, and the Fermi surface distortion ${\textstyle \delta \phi _{lm}}$ requires energy to form. Otherwise, ${\textstyle \delta \phi _{lm}}$ 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 ${\textstyle \propto e^{il\theta }}$ instead of spherical harmonics and Chebyshev polynomials instead of Legendre polynomials, shows the Pomeranchuk constraint to be ${\textstyle f_{l}>-1}$.^[4] 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 ${\displaystyle F_{l}=-(2l+1)}$ 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]

Physical quantities with manifest Pomeranchuk criterion[edit]

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.^[6]

Isothermal compressibility: ${\displaystyle \kappa =-{\frac {1}{V}}{\frac {\partial V}{\partial P}}={\frac {N/n^{2}}{1+f_{0}}}}$

Effective mass: ${\displaystyle m^{*}={\frac {p_{\rm {F}}}{v_{\rm {F}}}}=m(1+f_{1}/3)}$

Speed of first sound: ${\displaystyle C={\sqrt {\frac {p_{\rm {F}}^{2}(1+f_{0})}{m^{2}(3+f_{1})}}}}$

Unstable zero sound modes[edit]

The Pomeranchuk instability manifests in the dispersion relation for the zeroth sound, which describes how the localized fluctuations of the momentum density function ${\textstyle \delta n_{k}}$ propagate through space and time.^[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 ${\textstyle \Gamma (K_{3},K_{4};K_{1},K_{2})}$ near small ${\textstyle K_{1}-K_{3}}$. Physically, this describes the propagation of an electron hole pair, which is responsible for the fluctuations in ${\textstyle \delta n_{k}}$.

From the relation ${\textstyle \Gamma =((\Gamma ^{\omega })^{-1}-L)^{-1}}$ and ignoring the contributions of ${\textstyle f_{\ell }}$ for ${\textstyle \ell >0}$, the zero sound spectrum is given by the four-vectors ${\displaystyle K'=(\omega ({\vec {k'}}),{\vec {k'}})}$ satisfying

Equivalently,

$${\displaystyle {\frac {-1}{f_{0}}}=\Phi (s,x)={\frac {(s-x/2)^{2}-1}{4x}}\ln {\!\left({\frac {(s-x/2)+1}{(s-x/2)-1}}\right)}-{\frac {(s+x/2)^{2}-1}{4x}}\ln {\!\left({\frac {(s+x/2)+1}{(s+x/2)-1}}\right)}+{\frac {1}{2}}}$$

(1)

where ${\textstyle s={\frac {\omega ({\vec {k}})}{|{\vec {k}}|p_{\rm {F}}}}}$ and ${\textstyle x={\frac {|k|}{p_{\rm {F}}}}}$.

When ${\displaystyle f_{0}>0}$, the equation (1) can be implicitly solved for a real solution ${\displaystyle s(x)}$, corresponding to a real dispersion relation of oscillatory waves.

When ${\displaystyle f_{0}<0}$, the solution ${\displaystyle s(x)}$ is pure imaginary, corresponding to an exponential change in amplitude over time. For ${\displaystyle -1<f_{0}<0}$, the imaginary part ${\displaystyle \Im (s(x))<0}$, damping waves of zeroth sound. But for ${\displaystyle -1>f_{0}}$ and sufficiently small ${\displaystyle x}$, the imaginary part ${\displaystyle \Im (s(x))>0}$, implying exponential growth of any low-momentum zero sound perturbation.^[2]

Nematic phase transition[edit]

Pomeranchuk instabilities in non-relativistic systems at ${\displaystyle l=1}$ cannot exist.^[7] However, instabilities at ${\displaystyle l=2}$ have interesting solid state applications. From the form of spherical harmonics ${\displaystyle Y_{2,m}(\theta ,\phi )}$ (or ${\displaystyle e^{2i\theta }}$ in 2D), the Fermi surface is distorted into an ellipsoid (or ellipse). Specifically, in 2D, the quadrupole moment order parameter

has nonzero vacuum expectation value in the ${\displaystyle l=2}$ Pomeranchuk instability. The Fermi surface has eccentricity ${\displaystyle |\langle {\tilde {Q}}(0)\rangle |}$ and spontaneous major axis orientation ${\displaystyle \theta =\arg(\langle {\tilde {Q}}(0)\rangle )}$. Gradual spatial variation in ${\displaystyle \theta ({\vec {r}})}$ forms gapless Goldstone modes, forming a nematic liquid statistically analogous to a liquid crystal. 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.^[10]

See also[edit]

• Kohn anomaly
• Pomeranchuk's theorem
• Lindhard theory

References[edit]