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Strongly Correlated Systems pp 203–236Cite as

Dynamical Mean-Field Theory

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Part of the book series: Springer Series in Solid-State Sciences ((SSSOL,volume 171))

Abstract

The dynamical mean-field theory (DMFT) is a widely applicable approximation scheme for the investigation of correlated quantum many-particle systems on a lattice, e.g., electrons in solids and cold atoms in optical lattices. In particular, the combination of the DMFT with conventional methods for the calculation of electronic band structures has led to a powerful numerical approach which allows one to explore the properties of correlated materials. In this introductory article we discuss the foundations of the DMFT, derive the underlying self-consistency equations, and present several applications which have provided important insights into the properties of correlated matter.

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Notes

  1. 1.

    The coordination number Z is determined by the dimension d and the lattice structure. Already in d = 3 the coordination number can be quite large, e.g., Z = 6 for a simple cubic lattice, Z = 8 for a bcc-lattice and Z = 12 for an fcc-lattice, making its inverse, 1 ∕ Z, rather small. It is then natural to consider the limit Z → ∞ to simplify the problem. For a hypercubic lattice, obtained by generalizing the simple cubic lattice in d = 3 to arbitrary dimensions, one has Z = 2d. The limit d → ∞ is then equivalent to Z → ∞. Several standard approximation schemes which are commonly used to explain experimental results in dimension d = 3 are exact only in d, Z = ∞ [9].

  2. 2.

    In the following we set the Planck constant \(\hslash \), the Boltzmann constant k B, and the lattice constant a equal to unity.

  3. 3.

    Here a “path” is any sequence of lines in a diagram; they are “separate” when they have no lines in common.

  4. 4.

    In d = ∞ limit the notion of a Fermi surface of a lattice system is complicated by the fact that the dispersion ε k is not a simple smooth function.

  5. 5.

    The CPA is the best single-site approximation for disordered, non-interacting lattice electrons [27,  28,  29]; it becomes exact in the limit d, Z → ∞ [30,  8].

  6. 6.

    We note that the sign of the hopping amplitude t ij used here (see the definition in 7.1b) is opposite to that in [33].

  7. 7.

    In principle, any one of the local functions \({\mathcal{G}}_{\sigma }(i{\omega }_{n})\), \({\Sigma }_{\sigma }(i{\omega }_{n})\), or \({\Delta }_{\sigma }(i{\omega }_{n})\) can be viewed as a “dynamical mean field” acting on particles on a site, since they all appear in the bilinear term of the local action (7.32).

  8. 8.

    In the following we only consider the paramagnetic phase, whereas magnetic order is assumed to be suppressed (“frustrated”).

  9. 9.

    Here we assume for simplicity that the metal remains a Fermi liquid and the insulator stays paramagnetic down to the lowest temperatures.

  10. 10.

    We note that \(\hat{{H}}_{\mathrm{LDA}}^{0}\) may include additional non-interacting orbitals.

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Acknowledgments

We thank Jim Allen, Vladimir Anisimov, Nils Blümer, Ralf Bulla, Liviu Chioncel, Theo Costi, Vlad Dobrosavljević, Peter van Dongen, Martin Eckstein, Volker Eyert, Florian Gebhard, Karsten Held, Walter Hofstetter, Vaclav Janiš, Anna Kauch, Stefan Kehrein, Georg Keller, Gabi Kotliar, Jan Kuneš, Ivan Leonov, Walter Metzner, Michael Moeckel, Igor Nekrasov, Thomas Pruschke, Xinguo Ren, Shigemasa Suga, Götz Uhrig, Martin Ulmke, Ruud Vlaming, Philipp Werner, and Unjong Yu for valuable collaborations. Support by the Deutsche Forschungsgemeinschaft through TRR 80 and FOR 1346 is gratefully acknowledged. KB was also supported by the grant N N202 103138 of the Polish Ministry of Science and Education.

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  1. Theoretical Physics III, Center for Electronic Correlations and Magnetism, Institute of Physics, University of Augsburg, 86135, Augsburg, Germany

    Dieter Vollhardt & Marcus Kollar

  2. Faculty of Physics, Institute of Theoretical Physics, University of Warsaw, ul. Hoża 69, 00-681, Warsaw, Poland

    Krzysztof Byczuk

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  1. Dieter Vollhardt
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Correspondence to Dieter Vollhardt .

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  1. Dipartimento di Fisica "E.R. Caianiello", Università degli Studi di Salerno, Via Ponte don Melillo, Fisciano (SA), 84084, Italy

    Adolfo Avella

  2. Dipartimento di Fisica "E.R. Caianiello", Università degli Studi di Salerno, Via Ponte don Melillo, Fisciano (SA), 84084, Italy

    Ferdinando Mancini

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Vollhardt, D., Byczuk, K., Kollar, M. (2012). Dynamical Mean-Field Theory. In: Avella, A., Mancini, F. (eds) Strongly Correlated Systems. Springer Series in Solid-State Sciences, vol 171. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21831-6_7

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