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 $G(K)=\frac\text$
where capital momentum letters denote four-vectors $K=(k_0,\vec)$ and the Fermi surface has zero energy; poles of this function determine the quasiparticle energy-momentum dispersion relation. The four-point vertex function $\Gamma_$ describes the diagram with two incoming electrons of momentum $K_1$ and two outgoing electrons of momentum $K_3$ and and amputated external lines:$\begin \Gamma_&=\int \\ &=(2\pi)^8 \delta(K_1-K_3)\delta(K_2-K_4) G(K_1) G(K_2) - \\ &\phantom(2\pi)^8 \delta(K_1-K_4)\delta(K_2-K_3) G(K_1) G(K_2) + \\ &\phantom(2\pi)^4 \delta() G(K_1)G(K_2)G(K_3)G(K_4) i\Gamma_\text \end$ Call the momentum transfer$K'=(k'_0,\vec)=K_1-K_3\text$ When $K'$ 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 which corresponds to all diagrams connected after cutting two electron propagators: $\Gamma_ = \tilde\Gamma_ - i \sum_Q \tilde\Gamma _ G(Q)G(Q+K') \Gamma_\text$ Solving for $\Gamma$ shows that, in the similar-momentum, similar-wavelength limit the former tends towards an operator $\Gamma_^$ satisfying$L=\Gamma^-(\Gamma^\omega)^\text$ where$L_ = -i\delta_\delta_G(Q')G(K'+Q')\text$ The normalized Landau parameter is defined in terms of $\Gamma_^$ as $f_ = Z^2 N \Gamma^\omega ( (\epsilon_, \vec) , (\epsilon_, \vec))\text$ where $N=\frac$ is the density of Fermi surface states. In the Legendre eigenbasis the parameter $f$ admits the expansion $f_ = \sum_^\text$ Pomeranchuk's analysis revealed that each $f_\ell$ cannot be very negative.

Stability criterion

In a 3D isotropic Fermi liquid, consider small density fluctuations $\delta n_k=\Theta(, k, -p_)-\Theta(, k, -p_'(\hat))$ around the Fermi momentum where the shift in Fermi surface expands in spherical harmonics as $p_'(\hat) = \sum_^\infty Y_(\hat) \delta \phi_\text$ The energy associated with a perturbation is approximated by the functional $E = \sum_ \epsilon_ \delta n_ + \sum_$ where Assuming , these terms are,$\begin &\sum_ \epsilon_k \delta n_k = \frac\int d^2 \hat \int_^ v_ (p'-p_) p'^2 d p' = \frac \sum_ (\delta \phi_)^2 \frac \frac \\ &\sum_ f_ \delta n_k \delta n_ = \frac \int d^2 \hat d^2 \hat (p_'(\hat)-p_)(p_'(\hat)_)f_ \end$ and so $E = \frac \sum_ (\delta \phi_)^2 \frac\left( 1+ \frac\right)\text$

When the Pomeranchuk stability criterion $f_l >-(2l+1)$ is satisfied, this value is positive, and the Fermi surface distortion $\delta\phi_$ requires energy to form. Otherwise, $\delta\phi_$ 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 $\propto e^$ instead of spherical harmonics and Chebyshev polynomials instead of Legendre polynomials, shows the Pomeranchuk constraint to be 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: $\kappa = -\frac \frac =\frac$

Effective mass: $m^* = \frac = m(1+f_1/3)$

Speed of first sound: $C = \sqrt$

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 $\Gamma(K_3, K_4; K_1, K_2)$ near small Physically, this describes the propagation of an electron hole pair, which is responsible for the fluctuations in

From the relation $\Gamma= ((\Gamma^\omega)^ - L)^$ and ignoring the contributions of $f_\ell$ for the zero sound spectrum is given by the four-vectors $K' = (\omega(\vec), \vec)$ satisfying $\frac =-i \sum_Q G(Q+K')G(Q+K)\text$ Equivalently, where $s = \frac$ and

When the equation () can be implicitly solved for a real solution $s(x)$, corresponding to a real dispersion relation of oscillatory waves.

When the solution $s(x)$ is pure imaginary, corresponding to an exponential change in amplitude over time. For the imaginary part damping waves of zeroth sound. But for $-1 >f_0$ and sufficiently small the imaginary part 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 $Y_ (\theta, \phi)$ (or $e^$ in 2D), the Fermi surface is distorted into an ellipsoid (or ellipse). Specifically, in 2D, the quadrupole moment order parameter $\tilde(q) = \sum_k e^ \psi^_ \psi_k$ has nonzero vacuum expectation value in the $l=2$ Pomeranchuk instability. The Fermi surface has eccentricity $, \langle \tilde(0) \rangle,$ and spontaneous major axis orientation $\theta =\arg(\langle \tilde(0) \rangle)$. Gradual spatial variation in $\theta(\vec)$ 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

References

Lifshitz, E.M. and Pitaevskii, L.P., Statistical Physics, Part 2 (Pergamon, 1980)

Pomeranchuk, I. Ya., Sov.Phys.JETP,8,361 (1958)
Baym, G., and Pethick, Ch., Landau Fermi-Liquid Theory (Wiley-VCH, Weinheim, 2004, 2nd. Edition).





Reidy, K. E. Fermi liquids near Pomeranchuk instabilities. Diss. Kent State University, 2014.
{{cite journal , last=Nilsson , first=Johan , last2=Castro Neto , first2=A. H. , title=Heat bath approach to Landau damping and Pomeranchuk quantum critical points , journal=Physical Review B , publisher=American Physical Society (APS) , volume=72 , issue=19 , date=2005-11-14 , issn=1098-0121 , doi=10.1103/physrevb.72.195104 , page=195104, arxiv=cond-mat/0506146

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