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.

Fermi liquids and Landau parameters

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

${\frac {Z}{k_{0}-\epsilon _{\vec {k}}+i\eta \operatorname {sgn}(k_{0})}}=G_{}(K)$,

where capital momentum letters denote four vectors $K=(k_{0},{\vec {k}})$ and the Fermi surface has zero energy. The poles of $G(K)$ determine the quasiparticle energy-momentum dispersion relation. One can define the four-point vertex function $\Gamma _{(K_{3},K_{4};K_{1},K_{2})}$ as the diagram with two incoming electrons of momentum $K_{1},K_{2}$ and two outgoing electrons of momentum $K_{3},K_{4}$ and amputated external lines:

$\int \prod _{i=1}^{2}dX_{i}e^{iK_{i}X_{i}}\prod _{i=3}^{4}dX_{i}e^{-iK_{i}X_{i}}\langle T\psi _{}^{\dagger }(X_{3})\psi _{}^{\dagger }(X_{4})\psi _{}(X_{1})\psi _{}(X_{2})\rangle =$

$=(2\pi )^{8}\delta (K_{1}-K_{3})\delta (K_{2}-K_{4})G(K_{1})G(K_{2})-(2\pi )^{8}\delta (K_{1}-K_{4})\delta (K_{2}-K_{3})G(K_{1})G(K_{2})+$

$+(2\pi )^{4}\delta ({K_{1}+K_{2}-K_{3}-K_{4}})G(K_{1})G(K_{2})G(K_{3})G(K_{4})i\Gamma _{(K_{3},K_{4};K_{1},K_{2})}$.

The 2-particle-irreducible ${\tilde {\Gamma }}$ is the sum of diagrams contributing to $\Gamma$ that cannot be disconnected after cutting two electron propagators. When $K':=K_{1}-K_{3}:=(k'_{0},{\vec {k'}})$ is very small (the regime of interest here), the T-channel becomes dominant over the S and U channels, so the Dyson equation gives

$\Gamma _{K_{3},K_{4};K_{1},K_{2}}={\tilde {\Gamma }}_{K_{3},K_{4};K_{1},K_{2}}-i\sum _{Q}{\tilde {\Gamma }}_{K_{3},Q+K';K_{1},Q}G(Q)G(Q+K')\Gamma _{Q,K_{4};Q+K',K_{2}}$

Then, matrix manipulations (treating these quantities like infinite matrices with indices labelled by pairs $(K_{1},K_{3})$ and $(K_{2},K_{4})$) show that

$\Gamma _{K_{1},K_{2}}^{\omega }:=\lim _{k'\to \ 0}\lim _{k'/\omega '\to 0}\Gamma _{K_{3},K_{4};K_{1},K_{2}}$

is non-singular and satisfies the matrix equation $\Gamma =((\Gamma ^{\omega })^{-1}-L)^{-1}$, where

$L_{Q''+K'',Q'-K';Q'',Q'}:=-i\delta _{Q'',Q'}\delta _{K'',K'}G(Q')G(K'+Q')$.

The normalized Landau parameter is defined as $f_{kk'}=Z^{2}N\Gamma ^{\omega }((\epsilon _{\rm {F}},{\vec {k}}),(\epsilon _{\rm {F}},{\vec {k'}}))$, where $N={\frac {p_{\rm {F}}m_{\rm {F}}^{*}}{\pi ^{2}}}$ is Fermi surface density of states. Energy 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 $\epsilon _{\vec {k}}=v_{\rm {F}}(|{\vec {k}}|-p_{\rm {F}})$ for momenta ${\vec {k}}$ near the Fermi momentum $p_{\rm {F}}$.

Pomeranchuk stability criterion

In a 3D isotropic Fermi liquid, consider small density fluctuations $\delta n_{k}=\Theta (|k|-p_{\rm {F}})-\Theta (|k|-p_{\rm {F}}'({\hat {k}}))$ where $p_{\rm {F}}'({\hat {k}})=\sum _{l=0}^{\infty }Y_{l,m}({\hat {k}})\delta \phi _{lm}({\hat {k}})$ and the infinitesimal function $\delta \phi _{lm}({\hat {k}})$ parametrises the fluctuation ($Y_{l,m}$ denote the spherical harmonics). Plugging this into the energy functional, and assuming $\delta \phi _{lm}$ is much smaller than $p_{\rm {F}}$,

$\sum _{k}\epsilon _{k}\delta n_{k}={\frac {2}{(2\pi )^{3}}}\int d^{2}{\hat {k}}\int _{p_{\rm {F}}}^{p_{\rm {F}}'({\hat {k}})}v_{\rm {F}}(p'-p_{\rm {F}})p'^{2}dp'={\frac {p_{\rm {F}}^{2}v_{\rm {F}}}{(2\pi )^{3}}}\sum _{lm}(\delta \phi _{lm})^{2}{\frac {4\pi }{2l+1}}{\frac {(l+m)!}{(l-m)!}}$<

$\sum _{k,k'}f_{kk'}\delta n_{k}\delta n_{k'}={\frac {2p_{\rm {F}}^{4}}{(2\pi )^{6}}}\int d^{2}{\hat {k}}d^{2}{\hat {k'}}(p_{\rm {F}}'({\hat {k}})-p_{\rm {F}})(p_{\rm {F}}'({\hat {k'}})_{\rm {F}})f_{p_{\rm {F}}{\hat {k}},p_{\rm {F}}{\hat {k'}}}$

gives

$E={\frac {p_{\rm {F}}^{2}v_{\rm {F}}}{2(\pi )^{2}}}\sum _{lm}(\delta \phi _{lm})^{2}{\frac {(l+m)!}{(2l+1)(l-m)!}}\left(1+{\frac {f_{l}}{2l+1}}\right)$,

where $f_{p_{\rm {F}}{\hat {k}},p_{\rm {F}}{\hat {k'}}}=\sum _{l=0}^{\infty }P_{l}({\hat {k}}\cdot {\hat {k'}})f_{l}$ and $P_{l}$ is the $l$-th Legendre polynomial. Having a positive definite energy functional requires the Pomeranchuk stability criterion, $f_{l}>-(2l+1)$; otherwise the Fermi surface distortion $\delta \phi _{lm}$ will grow unbounded until the model breaks down in what is called the Pomeranchuk instability.

In 2D, a similar analysis, with circular wave fluctuations $\propto e^{il\theta }$ instead of spherical harmonics and Chebyshev polynomials instead of Legendre polynomials, shows the Pomeranchuk constraint to be $f_{l}>-1$. 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 $F_{l}=-(2l+1)$ is of much theoretical interest as it indicates a quantum phase transition from a Fermi liquid to a different state of matter, and 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

Zero sound describes how the localized fluctuations of the momentum density function $\delta n_{k}$ 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 $\Gamma (K_{3},K_{4};K_{1},K_{2})$ 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 $\Gamma =((\Gamma ^{\omega })^{-1}-L)^{-1}$ and ignoring the contributions of $f_{\ell }$ for $\ell >0$, the zero sound spectrum is given by the four-vectors $K'=(\omega ({\vec {k'}}),{\vec {k'}})$ satisfying ${\frac {Z^{2}N}{f_{0}}}=-i\sum _{Q}G(Q+K')G(Q+K)$, or

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

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

When $f_{0}>0$, for each real $x$ there is a real solution for $s(x)$, corresponding to a real zero sound dispersion relation of oscillatory waves. When $f_{0}<0$, for each real $x$ there is a pure imaginary solution for $s(x)$, corresponding to an exponential change in zero sound amplitude over time. For $-1<f_{0}<0$, $\Im (s(x))<0$ at all real $x$, so amplitude is damped. But for $-1>f_{0}$, $\Im (s(x))>0$ for sufficiently small $x$, implying exponential explosion of any low-momentum zero sound fluctuation. This is a manifestation of the Pomeranchuk instability.

Nematic phase transition

Pomeranchuk instabilities at $l=1$ are shown to not exist in non-relativistic systems by . 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.

See also

• Kohn anomaly
• Pomeranchuk's theorem
• Lindhard theory