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 $$G(K)=\frac{Z}{k_0-{ϵ}_{\overrightarrow{k}}+i\eta \mathrm{sgn}(k_0)}\text{,}$$ where capital momentum letters denote four-vectors $K=(k_0,\overrightarrow{k})$ and the Fermi surface has zero energy; poles of this function determine the quasiparticle energy-momentum dispersion relation. 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:$$\begin{array}{rl}{\mathrm{\Gamma }}_{(K_3,K_4;K_1,K_2)}& =\int \prod _{i=1}^{2}d{X}_i{e}^{i{K}_i{X}_i}\prod _{i=3}^{4}d{X}_i{e}^{-i{K}_i{X}_i}⟨T{\psi }^{†}(X_3){\psi }^{†}(X_4)\psi (X_1)\psi (X_2)⟩\\ & =(2\pi {)}^8\delta (K_1-K_3)\delta (K_2-K_4)G(K_1)G(K_2)-\\ & \phantom{=}(2\pi {)}^8\delta (K_1-K_4)\delta (K_2-K_3)G(K_1)G(K_2)+\\ & \phantom{=}(2\pi {)}^4\delta (K_1+K_2-K_3-K_4)G(K_1)G(K_2)G(K_3)G(K_4)i{\mathrm{\Gamma }}_{(K_3,K_4;K_1,K_2)}\text{.}\end{array}$$ Call the momentum transfer$$K^{\prime }=(k{\text{'}}_0,\overrightarrow{k^{\prime }})=K_1-K_3\text{.}$$ When $K^{\prime }$ 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 $\tilde{\mathrm{\Gamma }}$, which corresponds to all diagrams connected after cutting two electron propagators: $${\mathrm{\Gamma }}_{K_3,K_4;K_1,K_2}={\tilde{\mathrm{\Gamma }}}_{K_3,K_4;K_1,K_2}-i\sum _Q{\tilde{\mathrm{\Gamma }}}_{K_3,Q+K^{\prime };K_1,Q}G(Q)G(Q+K^{\prime }){\mathrm{\Gamma }}_{Q,K_4;Q+K^{\prime },K_2}\text{.}$$ Solving for $\mathrm{\Gamma }$ 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$$L={\mathrm{\Gamma }}^{-1}-({\mathrm{\Gamma }}^{\omega }{)}^{-1}\text{,}$$ where$$L_{Q^{″}+K^{″},Q^{\prime }-K^{\prime };Q^{″},Q^{\prime }}=-i{\delta }_{Q^{″},Q^{\prime }}{\delta }_{K^{″},K^{\prime }}G(Q^{\prime })G(K^{\prime }+Q^{\prime })\text{.}$$ The normalized Landau parameter is defined in terms of ${\mathrm{\Gamma }}_{K_1,K_2}^{\omega }$ as $$f_{k{k}^{\prime }}=Z^{2}N{\mathrm{\Gamma }}^{\omega }(({ϵ}_{\mathrm{F}},\overrightarrow{k}),({ϵ}_{\mathrm{F}},\overrightarrow{k^{\prime }}))\text{,}$$ where $N=\frac{p_{\mathrm{F}}{m}_{\mathrm{F}}^{*}}{{\pi }^2}$ is the density of Fermi surface states. In the Legendre eigenbasis $\{P_{\ell }{\}}_{\ell }$, the parameter $f$ admits the expansion $$f_{p_{\mathrm{F}}\hat{k},p_{\mathrm{F}}\widehat{k^{\prime }}}=\sum _{\ell =0}^{\mathrm{\infty }}{P}_{\ell }(\hat{k}\cdot \widehat{k^{\prime }})f_{\ell }\text{.}$$ Pomeranchuk's analysis revealed that each $f_{\ell }$ cannot be very negative.

Stability criterion

In a 3D isotropic Fermi liquid, consider small density fluctuations $\delta n_k=\mathrm{\Theta }(|k|-p_{\mathrm{F}})-\mathrm{\Theta }(|k|-{p_{{\mathrm{F}}^{\prime }}}^{\prime }(\hat{k}))$ around the Fermi momentum $p_{\mathrm{F}}$, where the shift in Fermi surface expands in spherical harmonics as $${p_{{\mathrm{F}}^{\prime }}}^{\prime }(\hat{k})=\sum _{l=0}^{\mathrm{\infty }}{Y}_{l,m}(\hat{k})\delta {\varphi }_{lm}\text{.}$$ The energy associated with a perturbation is approximated by the functional $$E=\sum _{\overrightarrow{k}}{ϵ}_{\overrightarrow{k}}\delta n_{\overrightarrow{k}}+\sum _{\overrightarrow{k},\overrightarrow{k^{\prime }}}\frac{1}{2NV}{f}_{\overrightarrow{k}\overrightarrow{k^{\prime }}}\delta n_{\overrightarrow{k}}\delta n_{\overrightarrow{k^{\prime }}}$$ where $\overrightarrow{{ϵ}_k}=v_{\mathrm{F}}(|\overrightarrow{k}|-p_{\mathrm{F}})$. Assuming $|\delta {\varphi }_{lm}|\ll |p_{\mathrm{F}}|$, these terms are,$$\begin{array}{rl}& \sum _k{ϵ}_k\delta n_k=\frac{2}{(2\pi {)}^3}\int d^2\hat{k}\underset{p_{\mathrm{F}}}{\overset{{p_{{\mathrm{F}}^{\prime }}}^{\prime }(\hat{k})}{\int }}{v}_{\mathrm{F}}(p^{\prime }-p_{\mathrm{F}})p{\text{'}}^{2}d{p}^{\prime }=\frac{p_{\mathrm{F}}^2{v}_{\mathrm{F}}}{(2\pi {)}^3}\sum _{lm}(\delta {\varphi }_{lm}{)}^2\frac{4\pi }{2l+1}\frac{(l+m)!}{(l-m)!}\\ & \sum _{k,k^{\prime }}{f}_{k{k}^{\prime }}\delta n_k\delta n_{k^{\prime }}=\frac{2p_{\mathrm{F}}^4}{(2\pi {)}^6}\int d^2\hat{k}{d}^2\widehat{k^{\prime }}({p_{{\mathrm{F}}^{\prime }}}^{\prime }(\hat{k})-p_{\mathrm{F}})({p_{{\mathrm{F}}^{\prime }}}^{\prime }(\widehat{k^{\prime }}{)}_{\mathrm{F}})f_{p_{\mathrm{F}}\hat{k},p_{\mathrm{F}}\widehat{k^{\prime }}}\end{array}$$ and so $$E=\frac{p_{\mathrm{F}}^2{v}_{\mathrm{F}}}{2(\pi {)}^2}\sum _{lm}(\delta {\varphi }_{lm}{)}^2\frac{(l+m)!}{(2l+1)(l-m)!}(1+\frac{f_l}{2l+1})\text{.}$$ When the Pomeranchuk stability criterion $$f_l>-(2l+1)$$ 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 as 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 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: $m^{*}=\frac{p_{\mathrm{F}}}{v_{\mathrm{F}}}=m(1+f_1/3)$ Speed of first sound: $C=\sqrt{\frac{p_{\mathrm{F}}^2(1+f_0)}{m^2(3+f_1)}}$

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 $\delta n_k$ 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 $\mathrm{\Gamma }(K_3,K_4;K_1,K_2)$ 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

$K^{\prime }=(\omega (\overrightarrow{k^{\prime }}),\overrightarrow{k^{\prime }})$

satisfying

$$\frac{Z^{2}N}{f_0}=-i\sum _{Q}G(Q+K^{\prime })G(Q+K)\text{.}$$

Equivalently,

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

where

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

and

$x=\frac{|k|}{p_{\mathrm{F}}}$.

When $f_0>0$, 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 $\mathrm{\Im }(s(x))<0$, damping waves of zeroth sound. But for $-1>f_0$ and sufficiently small $x$, the imaginary part $\mathrm{\Im }(s(x))>0$, 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 $Y_{2,m}(\theta ,\varphi )$ (or $e^{2i\theta }$ in 2D), the Fermi surface is distorted into an ellipsoid (or ellipse). Specifically, in 2D, the quadrupole moment order parameter $$\tilde{Q}(q)=\sum _k{e}^{2i{\theta }_q}{\psi }_{k+q}^{†}{\psi }_k$$ has nonzero vacuum expectation value in the $l=2$ Pomeranchuk instability. The Fermi surface has eccentricity $|⟨\tilde{Q}(0)⟩|$ and spontaneous major axis orientation $\theta =\arg (⟨\tilde{Q}(0)⟩)$. Gradual spatial variation in $\theta (\overrightarrow{r})$ 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