Pomeranchuk instability

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

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

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

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

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

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

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

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

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

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

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

${\displaystyle L_{Q''+K'',Q'-K';Q'',Q'}:=-i\delta _{Q'',Q'}\delta _{K'',K'}G(Q')G(K'+Q')}$.[2]

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

${\displaystyle E=\sum _{\vec {k}}\epsilon _{\vec {k}}\delta n_{\vec {k}}+\sum _{{\vec {k}},{\vec {k'}}}{{\frac {1}{2NV}}f_{{\vec {k}}{\vec {k'}}}\delta n_{\vec {k}}\delta n_{\vec {k'}}}}$

where ${\displaystyle \epsilon _{\vec {k}}=v_{\rm {F}}(|{\vec {k}}|-p_{\rm {F}})}$ for momenta ${\displaystyle {\vec {k}}}$ near the Fermi momentum ${\displaystyle p_{\rm {F}}}$.

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

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

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

gives

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

where ${\displaystyle f_{p_{\rm {F}}{\hat {k}},p_{\rm {F}}{\hat {k'}}}=\sum _{l=0}^{\infty }P_{l}({\hat {k}}\cdot {\hat {k'}})f_{l}}$ and ${\displaystyle P_{l}}$ is the ${\displaystyle l}$-th Legendre polynomial.[3] Having a positive definite energy functional requires the Pomeranchuk stability criterion, ${\displaystyle f_{l}>-(2l+1)}$; otherwise the Fermi surface distortion ${\displaystyle \delta \phi _{lm}}$ will grow unbounded until the model breaks down in what is called the Pomeranchuk instability.

In 2D, a similar analysis, with circular wave fluctuations ${\displaystyle \propto e^{il\theta }}$ instead of spherical harmonics and Chebyshev polynomials instead of Legendre polynomials, shows the Pomeranchuk constraint to be ${\displaystyle f_{l}>-1}$.[4] 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 ${\displaystyle F_{l}=-(2l+1)}$ is of much theoretical interest as it indicates a quantum phase transition from a Fermi liquid to a different state of matter, and above zero temperature a quantum critical state exists.[5]

Zero sound describes how the localized fluctuations of the momentum density function ${\displaystyle \delta n_{k}}$ 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 ${\displaystyle \Gamma (K_{3},K_{4};K_{1},K_{2})}$ near small ${\displaystyle K_{1}-K_{3}}$. Physically, this describes the propagation of an electron hole pair, which is responsible for the fluctuations in ${\displaystyle \delta n_{k}}$. From the relation ${\displaystyle \Gamma =((\Gamma ^{\omega })^{-1}-L)^{-1}}$ and ignoring the contributions of ${\displaystyle f_{\ell }}$ for ${\displaystyle \ell >0}$, the zero sound spectrum is given by the four-vectors ${\displaystyle K'=(\omega ({\vec {k'}}),{\vec {k'}})}$ satisfying ${\displaystyle {\frac {Z^{2}N}{f_{0}}}=-i\sum _{Q}G(Q+K')G(Q+K)}$, or

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

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

When ${\displaystyle f_{0}>0}$, for each real ${\displaystyle x}$ there is a real solution for ${\displaystyle s(x)}$, corresponding to a real zero sound dispersion relation of oscillatory waves. When ${\displaystyle f_{0}<0}$, for each real ${\displaystyle x}$ there is a pure imaginary solution for ${\displaystyle s(x)}$, corresponding to an exponential change in zero sound amplitude over time. For ${\displaystyle -1<f_{0}<0}$, ${\displaystyle \Im (s(x))<0}$ at all real ${\displaystyle x}$, so amplitude is damped. But for ${\displaystyle -1>f_{0}}$, ${\displaystyle \Im (s(x))>0}$ for sufficiently small ${\displaystyle x}$, implying exponential explosion of any low-momentum zero sound fluctuation. This is a manifestation of the Pomeranchuk instability.[2]

Pomeranchuk instabilities at ${\displaystyle l=1}$ are shown to not exist in non-relativistic systems by [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

${\displaystyle {\tilde {Q}}(q)=\sum _{k}e^{2i\theta _{q}}\psi _{k+q}^{\dagger }\psi _{k}}$

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]

• Kohn anomaly
• Pomeranchuk's theorem
• Lindhard theory