Pomeranchuk instability explained

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 $$G(K)=\backslash frac\backslash text$$where capital momentum letters denote four-vectors $K=(k\_0,\backslash vec)$ and the Fermi surface has zero energy; poles of this function determine the quasiparticle energy-momentum dispersion relation. The four-point vertex function $\backslash Gamma\_$ describes the diagram with two incoming electrons of momentum $K\_1$ and two outgoing electrons of momentum $K\_3$ and and amputated external lines: Call the momentum transfer$$K\text{'}=(k\text{'}\_0,\backslash vec)=K\_1-K\_3\backslash text$$ When $K\text{'}$ 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: $$\backslash Gamma\_\; =\; \backslash tilde\backslash Gamma\_\; -\; i\; \backslash sum\_Q\; \backslash tilde\backslash Gamma\; \_\; G(Q)G(Q+K\text{'})\; \backslash Gamma\_\backslash text$$ Solving for

\Gamma shows that, in the similar-momentum, similar-wavelength limit the former tends towards an operator $\backslash Gamma\_^$ satisfying$$L=\backslash Gamma^-(\backslash Gamma^\backslash omega)^\backslash text$$ where$$L\_\; =\; -i\backslash delta\_\backslash delta\_G(Q\text{'})G(K\text{'}+Q\text{'})\backslash text$$ The normalized Landau parameter is defined in terms of $\backslash Gamma\_^$ as $$f\_\; =\; Z^2\; N\; \backslash Gamma^\backslash omega\; ((\backslash epsilon\_,\; \backslash vec),\; (\backslash epsilon\_,\; \backslash vec))\backslash text$$ where $N=\backslash frac$ is the density of Fermi surface states. In the Legendre eigenbasis the parameter $f$ admits the expansion $$f\_\; =\; \backslash sum\_^\backslash text$$ Pomeranchuk's analysis revealed that each $f\_\backslash ell$ cannot be very negative.

Stability criterion

In a 3D isotropic Fermi liquid, consider small density fluctuations $\backslash delta\; n\_k=\backslash Theta(|k|-p\_)-\backslash Theta(|k|-p\_\text{'}(\backslash hat))$ around the Fermi momentum where the shift in Fermi surface expands in spherical harmonics as $$p\_\text{'}(\backslash hat)\; =\; \backslash sum\_^\backslash infty\; Y\_(\backslash hat)\; \backslash delta\; \backslash phi\_\backslash text$$ The energy associated with a perturbation is approximated by the functional $$E\; =\; \backslash sum\_\; \backslash epsilon\_\; \backslash delta\; n\_\; +\; \backslash sum\_$$ where Assuming, these terms are, and so $$E\; =\; \backslash frac\; \backslash sum\_\; (\backslash delta\; \backslash phi\_)^2\; \backslash frac\backslash left(1+\; \backslash frac\backslash right)\backslash text$$

When the Pomeranchuk stability criterion $$f\_l\; >-(2l+1)$$ is satisfied, this value is positive, and the Fermi surface distortion $\backslash delta\backslash phi\_$ requires energy to form. Otherwise, $\backslash delta\backslash phi\_$ 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 $\backslash propto\; e^$ 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} = - ( 2 l +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 =- (1)/(V) (\partial V)/(= \partial P) (N/n ^{2})/(1+ f _{0})

Effective mass

m ^{*} = (p _{\rm})/(v _{\rm}) = m( 1+ f _{1/3)}

Speed of first sound:

C = \sqrt{ (p ^{2} _{\rm F} ( 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 $\backslash delta\; n\_k$ 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 $\backslash Gamma(K\_3,\; K\_4;\; K\_1,\; K\_2)$ near small Physically, this describes the propagation of an electron hole pair, which is responsible for the fluctuations in

From the relation $\backslash Gamma=\; ((\backslash Gamma^\backslash omega)^\; -\; L)^$ and ignoring the contributions of $f\_\backslash ell$ for the zero sound spectrum is given by the four-vectors

K' = (\omega(\vec{k'}), \vec{k'}) satisfying $$\backslash frac\; =-i\; \backslash sum\_Q\; G(Q+K\text{'})G(Q+K)\backslash text$$ Equivalently, where $s\; =\; \backslash frac$

p_

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 _{2,m} (\theta, \phi) (or e ^{2i\theta} in 2D), the Fermi surface is distorted into an ellipsoid (or ellipse). Specifically, in 2D, the quadrupole moment order parameter $$\backslash tilde(q)\; =\; \backslash sum\_k\; e^\; \backslash psi^\_\; \backslash psi\_k$$ has nonzero vacuum expectation value in the l =2 Pomeranchuk instability. The Fermi surface has eccentricity |\langle \tilde{Q}( 0 ) \rangle| and spontaneous major axis orientation \theta = \arg(\langle \tilde{Q}( 0 ) \rangle) . Gradual spatial variation in \theta(\vec{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

References

^{[1] [2] [3] [4] [5] [6] [7] [8] [9] [10]}

Notes and References

1. Lifshitz, E.M. and Pitaevskii, L.P., Statistical Physics, Part 2 (Pergamon, 1980)
2. Kolomeitsev . E. E. . Voskresensky . D. N. . Scalar quanta in Fermi liquids: Zero sounds, instabilities, Bose condensation, and a metastable state in dilute nuclear matter . The European Physical Journal A . Springer Nature . 52 . 12 . 2016 . 1434-6001 . 10.1140/epja/i2016-16362-0 . 362. 1610.09748 .
3. Pomeranchuk, I. Ya., Sov.Phys.JETP,8,361 (1958)
4. Baym, G., and Pethick, Ch., Landau Fermi-Liquid Theory (Wiley-VCH, Weinheim, 2004, 2nd. Edition).
5. Kiselev . Egor I. . Scheurer . Mathias S. . Wölfle . Peter . Schmalian . Jörg . Limits on dynamically generated spin-orbit coupling: Absence ofl=1Pomeranchuk instabilities in metals . Physical Review B . American Physical Society (APS) . 95 . 12 . 2017-03-20 . 2469-9950 . 10.1103/physrevb.95.125122 . 125122. 1611.01442 .
6. Oganesyan . Vadim . Kivelson . Steven A. . Fradkin . Eduardo . Quantum theory of a nematic Fermi fluid . Physical Review B . American Physical Society (APS) . 64 . 19 . 2001-10-17 . 0163-1829 . 10.1103/physrevb.64.195109 . 195109. cond-mat/0102093 .
7. Halboth . Christoph J. . Metzner . Walter . d-Wave Superconductivity and Pomeranchuk Instability in the Two-Dimensional Hubbard Model . Physical Review Letters . American Physical Society (APS) . 85 . 24 . 2000-12-11 . 0031-9007 . 10.1103/physrevlett.85.5162 . 5162–5165. cond-mat/0003349 .
8. Kitatani . Motoharu . Tsuji . Naoto . Aoki . Hideo . Interplay of Pomeranchuk instability and superconductivity in the two-dimensional repulsive Hubbard model . Physical Review B . American Physical Society (APS) . 95 . 7 . 2017-02-03 . 2469-9950 . 10.1103/physrevb.95.075109 . 075109. free . 1609.05759 .
9. Reidy, K. E. Fermi liquids near Pomeranchuk instabilities. Diss. Kent State University, 2014.
10. Nilsson . Johan . Castro Neto . A. H. . Heat bath approach to Landau damping and Pomeranchuk quantum critical points . Physical Review B . American Physical Society (APS) . 72 . 19 . 2005-11-14 . 1098-0121 . 10.1103/physrevb.72.195104 . 195104. cond-mat/0506146 .