# 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

In a Fermi liquid, renormalized single electron propagators (ignoring spin) are $G(K)={\frac {Z}{k_{0}-\epsilon _{\vec {k}}+i\eta \operatorname {sgn}(k_{0})}}{\text{,}}$ where capital momentum letters denote four-vectors K=(k$_0$, k$^→$){\textstyle K=(k_{0}, {\vec {k}})} and the Fermi surface has zero energy; poles of this function determine the quasiparticle energy-momentum dispersion relation. The four-point vertex function Γ$_{(K3, K4;K1, K2)}${\textstyle \Gamma _{(K_{3}, K_{4};K_{1}, K_{2})}} describes the diagram with two incoming electrons of momentum K$_1${\textstyle K_{1}} and K$_2${\textstyle K_{2}}; two outgoing electrons of momentum K$_3${\textstyle K_{3}} and K$_4${\textstyle K_{4}}; and amputated external lines:${\begin{aligned}\Gamma _{(K_{3},K_{4};K_{1},K_{2})}&=\int {\prod _{i=1}^{2}{dX_{i}\,e^{iK_{i}X_{i}}}\prod _{i=3}^{4}{dX_{i}\,e^{-iK_{i}X_{i}}}\langle T\psi ^{\dagger }(X_{3})\psi ^{\dagger }(X_{4})\psi (X_{1})\psi (X_{2})\rangle }\\&=(2\pi )^{8}\delta (K_{1}-K_{3})\delta (K_{2}-K_{4})G(K_{1})G(K_{2})-\\&{\phantom {=}}(2\pi )^{8}\delta (K_{1}-K_{4})\delta (K_{2}-K_{3})G(K_{1})G(K_{2})+\\&{\phantom {=}}(2\pi )^{4}\delta ({K_{1}+K_{2}-K_{3}-K_{4}})G(K_{1})G(K_{2})G(K_{3})G(K_{4})i\Gamma _{(K_{3},K_{4};K_{1},K_{2})}{\text{.}}\end{aligned}}$ Call the momentum transfer$K'=(k'_{0},{\vec {k'}})=K_{1}-K_{3}{\text{.}}$ When K$^′${\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 Γ$^~${\textstyle {\tilde {\Gamma }}}, which corresponds to all diagrams connected after cutting two electron propagators: $\Gamma _{K_{3},K_{4};K_{1},K_{2}}={\tilde {\Gamma }}_{K_{3},K_{4};K_{1},K_{2}}-i\sum _{Q}{\tilde {\Gamma }}_{K_{3},Q+K';K_{1},Q}G(Q)G(Q+K')\Gamma _{Q,K_{4};Q+K',K_{2}}{\text{.}}$ Solving for $\Gamma$ shows that, in the similar-momentum, similar-wavelength limit k$^′$≪ω$^′$≪1{\textstyle k'\ll \omega '\ll 1}, the former tends towards an operator Γ$_{K1, K2}$$^ω${\textstyle \Gamma _{K_{1}, K_{2}}^{\omega }} satisfying$L=\Gamma ^{-1}-(\Gamma ^{\omega })^{-1}{\text{,}}$ where$L_{Q''+K'',Q'-K';Q'',Q'}=-i\delta _{Q'',Q'}\delta _{K'',K'}G(Q')G(K'+Q'){\text{.}}$ The normalized Landau parameter is defined in terms of Γ$_{K1, K2}$$^ω${\textstyle \Gamma _{K_{1}, K_{2}}^{\omega }} as $f_{kk'}=Z^{2}N\Gamma ^{\omega }((\epsilon _{\rm {F}},{\vec {k}}),(\epsilon _{\rm {F}},{\vec {k'}})){\text{,}}$ where N=p$_F$m$_F$$^∗$/π$^2$ {\textstyle N={\frac {p_{\mathrm {F} }m_{\mathrm {F} }^{*}}{\pi ^{2}}}} is the density of Fermi surface states. In the Legendre eigenbasis {P$_ℓ$}$_ℓ${\textstyle \{P_{\ell }\}_{\ell }}, the parameter f{\textstyle f} admits the expansion $f_{p_{\rm {F}}{\hat {k}},p_{\rm {F}}{\hat {k'}}}=\sum _{\ell =0}^{\infty }{P_{\ell }({\hat {k}}\cdot {\hat {k'}})f_{\ell }}{\text{.}}$ Pomeranchuk's analysis revealed that each f$_ℓ${\textstyle f_{\ell }} cannot be very negative.

## Stability criterion

In a 3D isotropic Fermi liquid, consider small density fluctuations δn$_k$=Θ(|k|−p$_F$)−Θ(|k|−p$_F$$^′$(k$^^$)){\textstyle \delta n_{k}=\Theta (|k|-p_{\mathrm {F} })-\Theta (|k|-p_{\mathrm {F} }'({\hat {k}}))} around the Fermi momentum p$_F${\textstyle p_{\mathrm {F} }}, where the shift in Fermi surface expands in spherical harmonics as $p_{\rm {F}}'({\hat {k}})=\sum _{l=0}^{\infty }Y_{l,m}({\hat {k}})\delta \phi _{lm}{\text{.}}$ The energy associated with a perturbation is approximated by the functional $E=\sum _{\vec {k}}\epsilon _{\vec {k}}\delta n_{\vec {k}}+\sum _{{\vec {k}},{\vec {k'}}}{{\frac {1}{2NV}}f_{{\vec {k}}{\vec {k'}}}\delta n_{\vec {k}}\delta n_{\vec {k'}}}$ where ϵ$_k$$^→$=v$_F$(|k$^→$|−p$_F$){\textstyle {\vec {\epsilon _{k}}}=v_{\mathrm {F} }(|{\vec {k}}|-p_{\mathrm {F} })}. Assuming |δϕ$_{lm}$|≪|p$_F$|{\textstyle |\delta \phi _{lm}|\ll |p_{\rm {F}}|}, these terms are,${\begin{aligned}&\sum _{k}\epsilon _{k}\delta n_{k}={\frac {2}{(2\pi )^{3}}}\int d^{2}{\hat {k}}\int _{p_{\rm {F}}}^{p_{\rm {F}}'({\hat {k}})}v_{\rm {F}}(p'-p_{\rm {F}})p'^{2}dp'={\frac {p_{\rm {F}}^{2}v_{\rm {F}}}{(2\pi )^{3}}}\sum _{lm}(\delta \phi _{lm})^{2}{\frac {4\pi }{2l+1}}{\frac {(l+m)!}{(l-m)!}}\\&\sum _{k,k'}f_{kk'}\delta n_{k}\delta n_{k'}={\frac {2p_{\rm {F}}^{4}}{(2\pi )^{6}}}\int d^{2}{\hat {k}}d^{2}{\hat {k'}}(p_{\rm {F}}'({\hat {k}})-p_{\rm {F}})(p_{\rm {F}}'({\hat {k'}})_{\rm {F}})f_{p_{\rm {F}}{\hat {k}},p_{\rm {F}}{\hat {k'}}}\end{aligned}}$ and so $E={\frac {p_{\rm {F}}^{2}v_{\rm {F}}}{2(\pi )^{2}}}\sum _{lm}(\delta \phi _{lm})^{2}{\frac {(l+m)!}{(2l+1)(l-m)!}}\left(1+{\frac {f_{l}}{2l+1}}\right){\text{.}}$
When the Pomeranchuk stability criterion $f_{l}>-(2l+1)$ is satisfied, this value is positive, and the Fermi surface distortion δϕ$_{lm}${\textstyle \delta \phi _{lm}} requires energy to form. Otherwise, δϕ$_{lm}${\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 ∝e$^{ilθ}${\textstyle \propto e^{il\theta }} instead of spherical harmonics and Chebyshev polynomials instead of Legendre polynomials, shows the Pomeranchuk constraint to be f$_l$>−1{\textstyle f_{l}>-1}. 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 $F_{l}=-(2l+1)$ 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.

### 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: $\kappa =-{\frac {1}{V}}{\frac {\partial V}{\partial P}}={\frac {N/n^{2}}{1+f_{0}}}$
Effective mass: $m^{*}={\frac {p_{\rm {F}}}{v_{\rm {F}}}}=m(1+f_{1}/3)$
Speed of first sound: $C={\sqrt {\frac {p_{\rm {F}}^{2}(1+f_{0})}{m^{2}(3+f_{1})}}}$

## 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 δn$_k${\textstyle \delta n_{k}} 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 Γ(K$_3$, K$_4$;K$_1$, K$_2$){\textstyle \Gamma (K_{3}, K_{4};K_{1}, K_{2})} near small K$_1$−K$_3${\textstyle K_{1}-K_{3}}. Physically, this describes the propagation of an electron hole pair, which is responsible for the fluctuations in δn$_k${\textstyle \delta n_{k}}.
From the relation Γ=((Γ$^ω$)$^{−1}$−L)$^{−1}${\textstyle \Gamma =((\Gamma ^{\omega })^{-1}-L)^{-1}} and ignoring the contributions of f$_ℓ${\textstyle f_{\ell }} for ℓ>0{\textstyle \ell >0}, the zero sound spectrum is given by the four-vectors $K'=(\omega ({\vec {k'}}),{\vec {k'}})$ satisfying ${\frac {Z^{2}N}{f_{0}}}=-i\sum _{Q}G(Q+K')G(Q+K){\text{.}}$ Equivalently,
| ${\frac {-1}{f_{0}}}=\Phi (s,x)={\frac {(s-x/2)^{2}-1}{4x}}\ln {\!\left({\frac {(s-x/2)+1}{(s-x/2)-1}}\right)}-{\frac {(s+x/2)^{2}-1}{4x}}\ln {\!\left({\frac {(s+x/2)+1}{(s+x/2)-1}}\right)}+{\frac {1}{2}}$ | (1) |
where s=ω(k$^→$)/|k$^→$|p$_F$ {\textstyle s={\frac {\omega ({\vec {k}})}{|{\vec {k}}|p_{\rm {F}}}}} and x=|k|/p$_F$ {\textstyle x={\frac {|k|}{p_{\rm {F}}}}}.
When $f_{0}>0$, the equation (1) can be implicitly solved for a real solution $s(x)$, corresponding to a real dispersion relation of oscillatory waves.
When $f_{0}<0$, the solution $s(x)$ is pure imaginary, corresponding to an exponential change in amplitude over time. For $-1<f_{0}<0$, the imaginary part $\Im (s(x))<0$, damping waves of zeroth sound. But for $-1>f_{0}$ and sufficiently small $x$, the imaginary part $\Im (s(x))>0$, implying exponential growth of any low-momentum zero sound perturbation.

## Nematic phase transition

Pomeranchuk instabilities in non-relativistic systems at $l=1$ cannot exist. However, instabilities at $l=2$ have interesting solid state applications. From the form of spherical harmonics $Y_{2,m}(\theta ,\phi )$ (or $e^{2i\theta }$ in 2D), the Fermi surface is distorted into an ellipsoid (or ellipse). Specifically, in 2D, the quadrupole moment order parameter ${\tilde {Q}}(q)=\sum _{k}e^{2i\theta _{q}}\psi _{k+q}^{\dagger }\psi _{k}$ has nonzero vacuum expectation value in the $l=2$ Pomeranchuk instability. The Fermi surface has eccentricity $|\langle {\tilde {Q}}(0)\rangle |$ and spontaneous major axis orientation $\theta =\arg(\langle {\tilde {Q}}(0)\rangle )$. Gradual spatial variation in $\theta ({\vec {r}})$ 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.