Physics: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 [math]\displaystyle{ G(K)=\frac{Z}{k_0 -\epsilon_{\vec{k}} + i\eta \sgn(k_0)}\text{,} }[/math] where capital momentum letters denote four-vectors [math]\displaystyle{ K=(k_0,\vec{k}) }[/math] and the Fermi surface has zero energy; poles of this function determine the quasiparticle energy-momentum dispersion relation. The four-point vertex function [math]\displaystyle{ \Gamma_{(K_3,K_4;K_1,K_2)} }[/math] describes the diagram with two incoming electrons of momentum [math]\displaystyle{ K_1 }[/math] and [math]\displaystyle{ K_2 }[/math]; two outgoing electrons of momentum [math]\displaystyle{ K_3 }[/math] and [math]\displaystyle{ K_4 }[/math]; and amputated external lines:[math]\displaystyle{ \begin{align} \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{align} }[/math] Call the momentum transfer[math]\displaystyle{ K'=(k'_0,\vec{k'})=K_1-K_3\text{.} }[/math] When [math]\displaystyle{ K' }[/math] 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 [math]\displaystyle{ \tilde{\Gamma} }[/math], which corresponds to all diagrams connected after cutting two electron propagators: [math]\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{.} }[/math] Solving for [math]\displaystyle{ \Gamma }[/math] shows that, in the similar-momentum, similar-wavelength limit [math]\displaystyle{ k'\ll\omega'\ll1 }[/math], the former tends towards an operator [math]\displaystyle{ \Gamma_{K_1,K_2}^{\omega} }[/math] satisfying[math]\displaystyle{ L=\Gamma^{-1}-(\Gamma^\omega)^{-1}\text{,} }[/math] where[math]\displaystyle{ L_{Q''+K'', Q'-K'; Q'', Q'} = -i\delta_{Q'',Q'}\delta_{K'',K'}G(Q')G(K'+Q')\text{.} }[/math] The normalized Landau parameter is defined in terms of [math]\displaystyle{ \Gamma_{K_1,K_2}^{\omega} }[/math] as [math]\displaystyle{ f_{kk'} = Z^2 N \Gamma^\omega ( (\epsilon_{\rm F}, \vec{k}) , (\epsilon_{\rm F}, \vec{k'}))\text{,} }[/math] where [math]\displaystyle{ N=\frac{p_{\mathrm{F}}m_{\mathrm{F}}^*}{\pi^2} }[/math] is the density of Fermi surface states. In the Legendre eigenbasis [math]\displaystyle{ \{P_\ell\}_\ell }[/math], the parameter [math]\displaystyle{ f }[/math] admits the expansion [math]\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{.} }[/math] Pomeranchuk's analysis revealed that each [math]\displaystyle{ f_\ell }[/math] cannot be very negative.

Stability criterion

In a 3D isotropic Fermi liquid, consider small density fluctuations [math]\displaystyle{ \delta n_k=\Theta(|k|-p_{\mathrm{F}})-\Theta(|k|-p_{\mathrm{F}}'(\hat{k})) }[/math] around the Fermi momentum [math]\displaystyle{ p_\mathrm{F} }[/math], where the shift in Fermi surface expands in spherical harmonics as [math]\displaystyle{ p_{\rm F}'(\hat{k}) = \sum_{l=0}^\infty Y_{l,m}(\hat{k}) \delta \phi_{lm}\text{.} }[/math] The energy associated with a perturbation is approximated by the functional [math]\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'} } }[/math] where [math]\displaystyle{ \vec{\epsilon_k}=v_\mathrm{F}(|\vec{k}|-p_\mathrm{F}) }[/math]. Assuming [math]\displaystyle{ |\delta\phi_{lm}|\ll|p_{\rm F}| }[/math], these terms are,[math]\displaystyle{ \begin{align} &\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 d p' = \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_{k k'} \delta n_k \delta n_{k'} = \frac{2 p_{\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{align} }[/math] and so [math]\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{.} }[/math]

When the Pomeranchuk stability criterion [math]\displaystyle{ f_l \gt -(2l+1) }[/math] is satisfied, this value is positive, and the Fermi surface distortion [math]\displaystyle{ \delta\phi_{lm} }[/math] requires energy to form. Otherwise, [math]\displaystyle{ \delta\phi_{lm} }[/math] 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 [math]\displaystyle{ \propto e^{i l \theta} }[/math] instead of spherical harmonics and Chebyshev polynomials instead of Legendre polynomials, shows the Pomeranchuk constraint to be [math]\displaystyle{ f_l \gt -1 }[/math]. 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 [math]\displaystyle{ F_l = - (2l+1) }[/math] 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: [math]\displaystyle{ \kappa = -\frac{1}{V} \frac{\partial V}{\partial P} =\frac{N/n^2}{1+f_0} }[/math]

Effective mass: [math]\displaystyle{ m^* = \frac{p_{\rm F}}{v_{\rm F}} = m(1+f_1/3) }[/math]

Speed of first sound: [math]\displaystyle{ C = \sqrt{\frac{p_{\rm F}^2 (1+ f_0)}{m^2( 3+f_1)}} }[/math]

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 [math]\displaystyle{ \delta n_k }[/math] 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 [math]\displaystyle{ \Gamma(K_3, K_4; K_1, K_2) }[/math] near small [math]\displaystyle{ K_1-K_3 }[/math]. Physically, this describes the propagation of an electron hole pair, which is responsible for the fluctuations in [math]\displaystyle{ \delta n_k }[/math].

From the relation [math]\displaystyle{ \Gamma= ((\Gamma^\omega)^{-1} - L)^{-1} }[/math] and ignoring the contributions of [math]\displaystyle{ f_\ell }[/math] for [math]\displaystyle{ \ell \gt 0 }[/math], the zero sound spectrum is given by the four-vectors [math]\displaystyle{ K' = (\omega(\vec{k'}), \vec{k'}) }[/math] satisfying [math]\displaystyle{ \frac{Z^2 N}{f_0} =-i \sum_Q G(Q+K')G(Q+K)\text{.} }[/math] Equivalently,

[math]\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} }[/math]

(1)

where [math]\displaystyle{ s = \frac{\omega(\vec{k})}{|\vec{k}|p_{\rm F}} }[/math] and [math]\displaystyle{ x = \frac{|k|}{p_{\rm F}} }[/math].

When [math]\displaystyle{ f_0\gt 0 }[/math], the equation (1) can be implicitly solved for a real solution [math]\displaystyle{ s(x) }[/math], corresponding to a real dispersion relation of oscillatory waves.

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

Nematic phase transition

Pomeranchuk instabilities in non-relativistic systems at [math]\displaystyle{ l=1 }[/math] cannot exist. However, instabilities at [math]\displaystyle{ l=2 }[/math] have interesting solid state applications. From the form of spherical harmonics [math]\displaystyle{ Y_{2,m} (\theta, \phi) }[/math] (or [math]\displaystyle{ e^{2i\theta} }[/math] in 2D), the Fermi surface is distorted into an ellipsoid (or ellipse). Specifically, in 2D, the quadrupole moment order parameter [math]\displaystyle{ \tilde{Q}(q) = \sum_k e^{2i \theta_q} \psi^{\dagger}_{k+q} \psi_k }[/math] has nonzero vacuum expectation value in the [math]\displaystyle{ l=2 }[/math] Pomeranchuk instability. The Fermi surface has eccentricity [math]\displaystyle{ |\langle \tilde{Q}(0) \rangle| }[/math] and spontaneous major axis orientation [math]\displaystyle{ \theta =\arg(\langle \tilde{Q}(0) \rangle) }[/math]. Gradual spatial variation in [math]\displaystyle{ \theta(\vec{r}) }[/math] 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