ThePomeranchuk instability is an instability in the shape of theFermi surface of a material with interactingfermions, causingLandau'sFermi 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 theSoviet physicistIsaak Pomeranchuk.

In aFermi liquid,renormalized singleelectronpropagators (ignoringspin) are$${\displaystyle G(K)=\frac{Z}{k_0-{ϵ}_{\overrightarrow{k}}+i\eta \mathrm{sgn}(k_0)}\text{,}}$$where capital momentum letters denotefour-vectors$K=(k_0,\overrightarrow{k})$ and theFermi surface has zero energy; poles of this function determine thequasiparticle energy-momentumdispersion relation.^[1] The four-point vertex function${\mathrm{\Gamma }}_{(K_3,K_4;K_1,K_2)}$ describes the diagram with two incoming electrons of momentum$K_1$ and$K_2$; two outgoing electrons of momentum$K_3$ and$K_4$; and amputated external lines: Call the momentum transfer$${\displaystyle K^{\prime }=(k_0^{\prime },\overrightarrow{k^{\prime }})=K_1-K_3\text{.}}$$ When$K^{\prime }$ is very small (the regime of interest here), theT-channel dominates theS- andU-channels. TheDyson equation then offers a simpler description of the four-point vertex function in terms of the 2-particle irreducible$\tilde{\mathrm{\Gamma }}$, which corresponds to all diagrams connected after cutting two electron propagators:$${\displaystyle {\mathrm{\Gamma }}_{K_3,K_4;K_1,K_2}={\tilde{\mathrm{\Gamma }}}_{K_3,K_4;K_1,K_2}-i\sum _Q{\tilde{\mathrm{\Gamma }}}_{K_3,Q+K^{\prime };K_1,Q}G(Q)G(Q+K^{\prime }){\mathrm{\Gamma }}_{Q,K_4;Q+K^{\prime },K_2}\text{.}}$$ Solving for${\displaystyle \mathrm{\Gamma }}$ shows that, in the similar-momentum, similar-wavelength limit$k^{\prime }\ll {\omega }^{\prime }\ll 1$, the former tends towards an operator${\mathrm{\Gamma }}_{K_1,K_2}^{\omega }$ satisfying$${\displaystyle L={\mathrm{\Gamma }}^{-1}-({\mathrm{\Gamma }}^{\omega }{)}^{-1}\text{,}}$$ where^[2]$${\displaystyle L_{Q^{″}+K^{″},Q^{\prime }-K^{\prime };Q^{″},Q^{\prime }}=-i{\delta }_{Q^{″},Q^{\prime }}{\delta }_{K^{″},K^{\prime }}G(Q^{\prime })G(K^{\prime }+Q^{\prime })\text{.}}$$ The normalized Landau parameter is defined in terms of${\mathrm{\Gamma }}_{K_1,K_2}^{\omega }$ as$${\displaystyle f_{k{k}^{\prime }}=Z^{2}N{\mathrm{\Gamma }}^{\omega }(({ϵ}_{\mathrm{F}},\overrightarrow{k}),({ϵ}_{\mathrm{F}},\overrightarrow{k^{\prime }}))\text{,}}$$ where$N=\frac{p_{\mathrm{F}}{m}_{\mathrm{F}}^{\ast }}{{\pi }^2}$ is the density of Fermi surface states. In theLegendre eigenbasis$\{P_{\ell }{\}}_{\ell }$, the parameter$f$ admits the expansion$${\displaystyle f_{p_{\mathrm{F}}\hat{k},p_{\mathrm{F}}\widehat{k^{\prime }}}=\sum _{\ell =0}^{\mathrm{\infty }}{P}_{\ell }(\hat{k}\cdot \widehat{k^{\prime }})f_{\ell }\text{.}}$$ Pomeranchuk's analysis revealed that each$f_{\ell }$ cannot be very negative.

In a 3D isotropic Fermi liquid, consider small density fluctuations$\delta n_k=\mathrm{\Theta }(|k|-p_{\mathrm{F}})-\mathrm{\Theta }(|k|-p_{\mathrm{F}}^{\prime }(\hat{k}))$ around theFermi momentum$p_{\mathrm{F}}$, where the shift in Fermi surface expands inspherical harmonics as$${\displaystyle p_{\mathrm{F}}^{\prime }(\hat{k})=\sum _{l=0}^{\mathrm{\infty }}{Y}_{l,m}(\hat{k})\delta {\varphi }_{lm}\text{.}}$$ The energy associated with a perturbation is approximated by the functional$${\displaystyle E=\sum _{\overrightarrow{k}}{ϵ}_{\overrightarrow{k}}\delta n_{\overrightarrow{k}}+\sum _{\overrightarrow{k},\overrightarrow{k^{\prime }}}\frac{1}{2NV}{f}_{\overrightarrow{k}\overrightarrow{k^{\prime }}}\delta n_{\overrightarrow{k}}\delta n_{\overrightarrow{k^{\prime }}}}$$ where$\overrightarrow{{ϵ}_k}=v_{\mathrm{F}}(|\overrightarrow{k}|-p_{\mathrm{F}})$. Assuming$|\delta {\varphi }_{lm}|\ll |p_{\mathrm{F}}|$, these terms are,^[3] and so$${\displaystyle E=\frac{p_{\mathrm{F}}^2{v}_{\mathrm{F}}}{2(\pi {)}^2}\sum _{lm}(\delta {\varphi }_{lm}{)}^2\frac{(l+m)!}{(2l+1)(l-m)!}\left(1+\frac{f_l}{2l+1}\right)\text{.}}$$

When thePomeranchuk stability criterion$${\displaystyle f_l>-(2l+1)}$$ is satisfied, this value is positive, and the Fermi surface distortion$\delta {\varphi }_{lm}$ requires energy to form. Otherwise,$\delta {\varphi }_{lm}$ releases energy, and will grow without bound until the model breaks down. That process is known asPomeranchuk instability.

In 2D, a similar analysis, with circular wave fluctuations$\propto e^{il\theta }$ instead of spherical harmonics andChebyshev polynomials instead of Legendre polynomials, shows the Pomeranchuk constraint to be$f_l>-1$.^[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 aquantum 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.^[6]

Isothermalcompressibility:${\displaystyle \kappa =-\frac{1}{V}\frac{\mathrm{\partial }V}{\mathrm{\partial }P}=\frac{N/n^2}{1+f_0}}$

Effective mass:${\displaystyle m^{\ast }=\frac{p_{\mathrm{F}}}{v_{\mathrm{F}}}=m(1+f_1/3)}$

Speed of first sound:${\displaystyle C=\sqrt{\frac{p_{\mathrm{F}}^2(1+f_0)}{m^2(3+f_1)}}}$

The Pomeranchuk instability manifests in the dispersion relation for thezeroth sound, which describes how the localized fluctuations of the momentum density function$\delta n_k$ 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 theT-channel of the vertex function$\mathrm{\Gamma }(K_3,K_4;K_1,K_2)$ near small$K_1-K_3$. Physically, this describes the propagation of an electron hole pair, which is responsible for the fluctuations in$\delta n_k$.

From the relation$\mathrm{\Gamma }=(({\mathrm{\Gamma }}^{\omega }{)}^{-1}-L{)}^{-1}$ and ignoring the contributions of$f_{\ell }$ for$\ell >0$, the zero sound spectrum is given by the four-vectors${\displaystyle K^{\prime }=(\omega (\overrightarrow{k^{\prime }}),\overrightarrow{k^{\prime }})}$ satisfying$${\displaystyle \frac{Z^{2}N}{f_0}=-i\sum _{Q}G(Q+K^{\prime })G(Q+K)\text{.}}$$ Equivalently,

$${\displaystyle \frac{-1}{f_0}=\mathrm{\Phi }(s,x)=\frac{(s-x/2{)}^2-1}{4x}\ln \left(\frac{(s-x/2)+1}{(s-x/2)-1}\right)-\frac{(s+x/2{)}^2-1}{4x}\ln \left(\frac{(s+x/2)+1}{(s+x/2)-1}\right)+\frac{1}{2}}$$

1

where$s=\frac{\omega (\overrightarrow{k})}{|\overrightarrow{k}|p_{\mathrm{F}}}$ and$x=\frac{|k|}{p_{\mathrm{F}}}$.

When${\displaystyle f_0>0}$, the equation (1) can beimplicitly solved for a real solution${\displaystyle s(x)}$, corresponding to a real dispersion relation of oscillatory waves.

When, the solution${\displaystyle s(x)}$ is pureimaginary, corresponding to an exponential change in amplitude over time. For, the imaginary part, damping waves of zeroth sound. But for${\displaystyle -1>f_0}$ and sufficiently small${\displaystyle x}$, the imaginary part${\displaystyle \mathrm{\Im }(s(x))>0}$, implying exponential growth of any low-momentum zero sound perturbation.^[2]

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${\displaystyle Y_{2,m}(\theta ,\varphi )}$ (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$${\displaystyle \tilde{Q}(q)=\sum _k{e}^{2i{\theta }_q}{\psi }_{k+q}^{†}{\psi }_k}$$ has nonzerovacuum expectation value in the${\displaystyle l=2}$ Pomeranchuk instability. The Fermi surface has eccentricity${\displaystyle |⟨\tilde{Q}(0)⟩|}$ and spontaneous major axis orientation${\displaystyle \theta =\arg (⟨\tilde{Q}(0)⟩)}$. Gradual spatial variation in${\displaystyle \theta (\overrightarrow{r})}$ forms gaplessGoldstone 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 ofd-wave fluctuations underrenormalization group flow. Thus, the Pomeranchuk instability is suspected to explain the experimentally measured anisotropy incuprate superconductors such as LSCO andYBCO.^[10]

• Kohn anomaly
• Pomeranchuk's theorem
• Lindhard theory

• Fermions