Pomeranchuk instability

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

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

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:

$${\displaystyle {\begin{aligned}\Gamma _{(K_{3},K_{4};K_{1},K_{2})}&=\int {\prod _{i=1}^{2}{dX_{i}\,e^{iK_{i}X_{i}}}\prod _{i=3}^{4}{dX_{i}\,e^{-iK_{i}X_{i}}}\langle T\psi ^{\dagger }(X_{3})\psi ^{\dagger }(X_{4})\psi (X_{1})\psi (X_{2})\rangle }\\&=(2\pi )^{8}\delta (K_{1}-K_{3})\delta (K_{2}-K_{4})G(K_{1})G(K_{2})-{}\\&{\phantom {{}={}}}(2\pi )^{8}\delta (K_{1}-K_{4})\delta (K_{2}-K_{3})G(K_{1})G(K_{2})+{}\\&{\phantom {{}={}}}(2\pi )^{4}\delta ({K_{1}+K_{2}-K_{3}-K_{4}})G(K_{1})G(K_{2})G(K_{3})G(K_{4})i\Gamma _{(K_{3},K_{4};K_{1},K_{2})}{\text{.}}\end{aligned}}}$$

Call the momentum transfer

$${\displaystyle K'=(k'_{0},{\vec {k'}})=K_{1}-K_{3}{\text{.}}}$$

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:

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

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

$${\displaystyle L=\Gamma ^{-1}-(\Gamma ^{\omega })^{-1}{\text{,}}}$$

where[2]

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

The normalized Landau parameter is defined in terms of ${\textstyle \Gamma _{K_{1},K_{2}}^{\omega }}$ as

$${\displaystyle f_{kk'}=Z^{2}N\Gamma ^{\omega }((\epsilon _{\rm {F}},{\vec {k}}),(\epsilon _{\rm {F}},{\vec {k'}})){\text{,}}}$$

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

$${\displaystyle f_{p_{\rm {F}}{\hat {k}},p_{\rm {F}}{\hat {k'}}}=\sum _{\ell =0}^{\infty }{P_{\ell }({\hat {k}}\cdot {\hat {k'}})f_{\ell }}{\text{.}}}$$

Pomeranchuk's analysis revealed that each ${\textstyle f_{\ell }}$ cannot be very negative.

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

$${\displaystyle p_{\rm {F}}'({\hat {k}})=\sum _{l=0}^{\infty }Y_{l,m}({\hat {k}})\delta \phi _{lm}{\text{.}}}$$

The energy associated with a perturbation is approximated by the functional

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

where ${\textstyle {\vec {\epsilon _{k}}}=v_{\mathrm {F} }(|{\vec {k}}|-p_{\mathrm {F} })}$. Assuming ${\textstyle |\delta \phi _{lm}|\ll |p_{\rm {F}}|}$, these terms are,[3]

$${\displaystyle {\begin{aligned}&\sum _{k}\epsilon _{k}\delta n_{k}={\frac {2}{(2\pi )^{3}}}\int d^{2}{\hat {k}}\int _{p_{\rm {F}}}^{p_{\rm {F}}'({\hat {k}})}v_{\rm {F}}(p'-p_{\rm {F}})p'^{2}dp'={\frac {p_{\rm {F}}^{2}v_{\rm {F}}}{(2\pi )^{3}}}\sum _{lm}(\delta \phi _{lm})^{2}{\frac {4\pi }{2l+1}}{\frac {(l+m)!}{(l-m)!}}\\&\sum _{k,k'}f_{kk'}\delta n_{k}\delta n_{k'}={\frac {2p_{\rm {F}}^{4}}{(2\pi )^{6}}}\int d^{2}{\hat {k}}d^{2}{\hat {k'}}(p_{\rm {F}}'({\hat {k}})-p_{\rm {F}})(p_{\rm {F}}'({\hat {k'}})_{\rm {F}})f_{p_{\rm {F}}{\hat {k}},p_{\rm {F}}{\hat {k'}}}\end{aligned}}}$$

and so

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

When the Pomeranchuk stability criterion

$${\displaystyle f_{l}>-(2l+1)}$$

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]

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

$${\displaystyle {\frac {Z^{2}N}{f_{0}}}=-i\sum _{Q}G(Q+K')G(Q+K){\text{.}}}$$

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]

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

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

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