Abstract
Fourth-order strong-coupling degenerate perturbation theory is used to derive an effective low-energy Hamiltonian for the Kondo-lattice model with a depleted system of localized spins. In the strong-J limit, completely local Kondo singlets are formed at the spinful sites which bind a fraction of conduction electrons. The low-energy theory describes the scattering of the excess conduction electrons at the Kondo singlets as well as their effective interactions generated by virtual excitations of the singlets. Besides the Hubbard term, already discussed by Nozières, we find a ferromagnetic Heisenberg interaction, an antiferromagnetic isospin interaction, a correlated hopping and, in more than one dimensions, three- and four-site interactions. The interaction term can be cast into highly symmetric and formally simple spin-only form using the spin of the bonding orbital symmetrically centered around the Kondo singlet. This spin is non-local. We show that, depending on the geometry of the depleted lattice, spatial overlap of the non-local spins around different Kondo singlets may cause ferromagnetic order. This is sustained by a rigorous argument, applicable to the half-filled model, by a variational analysis of the stability of the fully polarized Fermi sea of excess conduction electrons as well as by exact diagonalization of the effective model. A similar fourth-order perturbative analysis is performed for the depleted Anderson lattice in the limit of strong hybridization. Even in a parameter regime where the Schrieffer-Wolff transformation does not apply, this yields the same effective theory albeit with a different coupling constant.
Similar content being viewed by others
References
S. Doniach, Physica B 91, 321 (1977)
C. Lacroix, M. Cyrot, Phys. Rev. B 20, 1969 (1979)
H. Tsunetsugu, M. Sigrist, K. Ueda, Rev. Mod. Phys. 69, 809 (1997)
Y. Kuramoto, Y. Kitaoka, Dynamics of Heavy Electrons (Oxford University Press, New York, 2000)
P. Coleman, in Handbook of Magnetism and Advanced Magnetic Materials (Wiley, 2007), Vol. 1, p. 95
M.A. Ruderman, C. Kittel, Phys. Rev. 96, 99 (1954)
T. Kasuya, Prog. Theor. Phys. 16, 45 (1956)
K. Yosida, Phys. Rev. 106, 893 (1957)
K.G. Wilson, Rev. Mod. Phys. 47, 773 (1975)
A.C. Hewson, The Kondo Problem to Heavy Fermions (Cambridge University Press, Cambridge, 1993)
M. Troyer, D. Würtz, Phys. Rev. B 47, 2886 (1993)
I.P. McCulloch, A. Juozapavicius, A. Rosengren, M. Gulácsi, Phys. Rev. B 65, 052410 (2002)
M. Gulácsi, Adv. Phys. 53, 769 (2004)
R. Peters, N. Kawakami, Phys. Rev. B 86, 165107 (2012)
J. Otsuki, H. Kusunose, Y. Kuramoto, J. Phys. Soc. Jpn 78, 034719 (2009)
R. Peters, N. Kawakami, T. Pruschke, Phys. Rev. Lett. 108, 086402 (2012)
C. Lacroix, Solid State Commun. 54, 991 (1985)
Y. Nagaoka, Phys. Rev. 147, 392 (1966)
J. Kondo, Prog. Theor. Phys. 32, 37 (1964)
A. Schwabe, I. Titvinidze, M. Potthoff, Phys. Rev. B 88, 121107(R) (2013)
F.F. Assaad, Phys. Rev. B 65, 115104 (2002)
P. Nozières, J. Low Temp. Phys. 17, 31 (1974)
P. Nozières, J. Phys. C 37, C1-271 (1976)
P. Nozières, A. Blandin, J. Phys. 41, 193 (1980)
S.R. White, Phys. Rev. Lett. 69, 2863 (1992)
U. Schollwöck, Ann. Phys. 326, 96 (2011)
J.R. Schrieffer, P.A. Wolff, Phys. Rev. 149, 491 (1966)
P. Sinjukow, W. Nolting, Phys. Rev. B 65, 212303 (2002)
M. Sigrist, H. Tsunetsugu, K. Ueda, T.M. Rice, Phys. Rev. B 46, 13838 (1992)
I. Titvinidze, A. Schwabe, M. Potthoff, Phys. Rev. B 90, 045112 (2014)
A. Schwabe, M. Hänsel, M. Potthoff, Phys. Rev. A 90, 033615 (2014)
R. Peters, Y. Tada, N. Kawakami, Phys. Rev. B 88, 155134 (2013)
L. Zhou, J. Wiebe, S. Lounis, E. Vedmedenko, F. Meier, S. Blügel, P. Dederichs, R. Wiesendanger, Nat. Phys. 6, 187 (2010)
A.A. Khajetoorians, J. Wiebe, B. Chilian, R. Wiesendanger, Science 332, 1062 (2011)
A.A. Khajetoorians, J. Wiebe, B. Chilian, S. Lounis, S. Blügel, R. Wiesendanger, Nat. Phys. 8, 497 (2012)
R.K. Kaul, M. Vojta, Phys. Rev. B 75, 132407 (2007)
S. Burdin, C. Lacroix, Phys. Rev. Lett. 110, 226403 (2013)
F.H.L. Essler, H. Frahm, F. Göhmann, A. Klümper, V. Korepin, The One-Dimensional Hubbard Model (Cambridge University Press, Cambridge, 2005)
E. Lange, Mod. Phys. Lett. B 12, 915 (1998)
I. Titvinidze, A. Schwabe, N. Rother, M. Potthoff, Phys. Rev. B 86, 075141 (2012)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Titvinidze, I., Schwabe, A. & Potthoff, M. Strong-coupling limit of depleted Kondo- and Anderson-lattice models. Eur. Phys. J. B 88, 9 (2015). https://doi.org/10.1140/epjb/e2014-50772-1
Received:
Revised:
Published:
DOI: https://doi.org/10.1140/epjb/e2014-50772-1