# Conductivity of Weakly Disordered Metals Close to a “Ferromagnetic” Quantum Critical Point

## Abstract

We calculate analytically the conductivity of weakly disordered metals close to a “ferromagnetic” quantum critical point in the low-temperature regime. Ferromagnetic in the sense that the effective carrier potential \(V(q,\omega )\), due to critical fluctuations, is peaked at zero momentum \(q=0\). Vertex corrections, due to both critical fluctuations and impurity scattering, are explicitly considered. We find that only the vertex corrections due to impurity scattering, combined with the self-energy, generate appreciable effects as a function of the temperature *T* and the control parameter *a*, which measures the proximity to the critical point. Our results are consistent with resistivity experiments in several materials displaying typical Fermi liquid behaviour, but with a diverging prefactor of the \(T^2\) term for small *a*.

## Keywords

Conductivity calculation Vertex corrections Quantum critical point Fermi liquid Weak disorder## References

- 1.J. Paglione, M.A. Tanatar, D.G. Hawthorn, F. Ronning, R.W. Hill, M. Sutherland, L. Taillefer, C. Petrovic, Phys. Rev. Lett.
**97**, 106606 (2006)ADSCrossRefGoogle Scholar - 2.A. Bianchi, R. Movshovich, I. Vekhter, P.G. Pagliuso, J.L. Sarrao, Phys. Rev. Lett.
**91**, 257001 (2003)ADSCrossRefGoogle Scholar - 3.S.A. Grigera, R.S. Perry, A.J. Schofield, M. Chiao, S.R. Julian, G.G. Lonzarich, S.I. Ikeda, Y. Maeno, A.J. Millis, A.P. Mackenzie, Science
**294**, 329 (2001)ADSCrossRefGoogle Scholar - 4.P. Gegenwart, J. Custers, C. Geibel, K. Neumaier, T. Tayama, K. Tenya, O. Trovarelli, F. Steglich, Phys. Rev. Lett.
**89**, 056402 (2002)ADSCrossRefGoogle Scholar - 5.P. Gegenwart, J. Custers, Y. Tokiwa, C. Geibel, F. Steglich, Phys. Rev. Lett.
**94**, 076402 (2005)ADSCrossRefGoogle Scholar - 6.N.P. Butch, K. Jin, K. Kirshenbaum, R.L. Greene, J. Paglione, PNAS
**109**, 8440 (2012)ADSCrossRefGoogle Scholar - 7.T. Shibauchi, L. Krusin-Elbaum, M. Hasegawa, Y. Kasahara, R. Okazaki, Y. Matsuda, PNAS
**105**, 7120 (2008)ADSCrossRefGoogle Scholar - 8.L. Balicas, S. Nakatsuji, H. Lee, P. Schlottmann, T.P. Murphy, Z. Fisk, Phys. Rev. B
**72**, 064422 (2005)ADSCrossRefGoogle Scholar - 9.S. Nakatsuji, K. Kuga, Y. Machida, T. Tayama, T. Sakakibara, Y. Karaki, H. Ishimoto, S. Yonezawa, Y. Maeno, E. Pearson, G.G. Lonzarich, L. Balicas, H. Lee, Z. Fisk, Nat. Phys.
**4**, 603 (2008)CrossRefGoogle Scholar - 10.J.G. Analytis, H.-H. Kuo, R.D. McDonald, M. Wartenbe, P.M.C. Rourke, N.E. Hussey, I.R. Fisher, Nat. Phys.
**10**, 194 (2014)CrossRefGoogle Scholar - 11.H.V. Löhneysen, A. Rosch, M. Vojta, P. Wölfle, Rev. Mod. Phys.
**79**, 1015 (2007)ADSCrossRefGoogle Scholar - 12.G. Kastrinakis, Europhys. Lett.
**112**, 67001 (2015)ADSCrossRefGoogle Scholar - 13.A.A. Abrikosov, L.P. Gorkov, I.E. Dzyaloshinski,
*Methods of Quantum Field Theory in Statistical Physics*(Prentice-Hall, Cliffwoods, NY, 1964)zbMATHGoogle Scholar - 14.P.A. Lee, T.V. Ramakrishnan, Rev. Mod. Phys.
**57**, 287 (1985)ADSCrossRefGoogle Scholar - 15.J. Hertz, Phys. Rev. B
**14**, 1165 (1976)ADSCrossRefGoogle Scholar - 16.A.J. Millis, Phys. Rev. B
**48**, 7183 (1993)ADSCrossRefGoogle Scholar - 17.G. Kastrinakis, Phys. Rev. B
**72**, 075137 (2005)ADSCrossRefGoogle Scholar - 18.G.D. Mahan,
*Many-Particle Physics*, 2nd edn. (Plenum Press, New York, 1990). (the relevant material is mostly in section 3 of chapter 7)CrossRefGoogle Scholar - 19.L. Dell’ Anna, W. Metzner, Phys. Rev. Lett.
**98**, 136402 (2007). (erratum Phys. Rev. Lett.**103**, 159904 (2009))ADSCrossRefGoogle Scholar - 20.A.V. Chubukov, D.L. Maslov, Phys. Rev. Lett.
**103**, 216401 (2009)ADSCrossRefGoogle Scholar - 21.S.S. Lee, Phys. Rev. B
**80**, 165102 (2009)ADSCrossRefGoogle Scholar