Werner Weber received his PhD from the TU Munich in 1972. He worked first at a variety of research institutions, at the MPI for Solid State Research in Stuttgart, at the Research Center in Karlsruhe (now K.I.T.), at Bell Laboratories, Murray Hill. He then became a faculty member at the TU Dortmund, where he retired in 2010. His research area is theoretical solid state physics, with main emphasis on materials science theory. He assumed many duties in university self-administration, even presently. In the spirit of Martin Gutzwiller, he recently changed his field of interest to activities in climate research, including applications.
Martin Gutzwiller and his wave function
Dionys Baeriswyl, Département de physique, Université de Fribourg, 1700 Fribourg
Werner Weber, Fakultät Physik, Universität Dortmund, DE-44221 Dortmund
Gutzwiller's work on correlated electrons is mostly concentrated in three papers, written in the time span 1962 to 1964 [1, 2, 3]. A short fourth paper was published a few years later [4]. In essence, Gutzwiller introduced a variational ansatz, where charge fluctuations are reduced as compared to Hartree-Fock theory, thus quantifying Van Vleck's qualitative idea of minimum polarity [5].
Historically, electronic correlations were first studied for the homogeneous electron gas, much less for electrons in narrow bands such as d-electrons in transition metals. A noticeable exception was Anderson's paper on the kinetic origin of antiferromagnetism in transition metal compounds, where a localized basis of Wannier functions was used [6]. In the same spirit, Gutzwiller wrote down the Hamiltonian
where the first term describes electron hopping between the neighboring sites of a lattice (
and 
are, respectively, creation and annihilation operators for electrons at site i with spin σ) and the second term is the interaction, which acts only if two electrons meet on the same site (
). Quantum chemists had previously used a similar model for π-electrons in conjugated polymers, but they had included the long-range part of the Coulomb interaction. Curiously, shortly after Gutzwiller's first paper on the subject, two publications appeared where the same Hamiltonian (1) is treated, but without reference to Gutzwiller's work, one by Hubbard [7], the other by Kanamori [8]. One has to conclude that the three papers [1, 7, 8] were written completely independently and that the Hamiltonien (1), now universally referred to as Hubbard model, was in the air, especially for investigating the problem of correlated electrons in transition metals.
In contrast to Gutzwiller, who did not care too much about the justification of the model, Hubbard estimated the different Coulomb matrix elements between localized d wave functions, and he also explained how in transition metals with partly filled 3d shells and a partly filled 4s shell the s-electrons can effectively screen the Coulomb interactions between d-electrons. The fact, pointed out by Gutzwiller [3], that the three authors, himself, Hubbard and Kanamori, obtained qualitatively different results, shows that, despite of its formal simplicity, the model was – and still is – very challenging.
Gutzwiller's main contributions to the field of correlated electrons are his ansatz for the ground state of the Hubbard model and his ingenious way of handling this wave function. He starts from the ground state 
of the hopping term, the filled Fermi sea. This would just yield the Hartree-Fock approximation, which treats neutral and "polar" configurations on the same footing. Thus he adds a projector term, now called correlator, that reduces charge fluctuations. His ansatz reads
or, written in a different way,
where 
is the number of doubly occupied sites and g is related to Gutzwiller’s parameter η by η = e-g.
The problem of evaluating the ground state energy
for this trial state still represents a formidable task. Exact results were only obtained in one dimension [9, 10]. For other dimensions, Variational Monte Carlo (VMC), pioneered for the Gutzwiller ansatz by Horsch and Kaplan [11], has been widely used in recent years [12].
Gutzwiller himself proposed an approximate way of evaluating Eq. (4) [3]. His procedure, known as "Gutzwiller approximation", involves two steps [13]. In a first step, the expectation value is factorized with respect to spin. In a second step, the remaining expectation values are assumed to be configuration-independent. This leads to a purely combinatorial problem. In the limit of infinite dimensions, the Gutzwiller approximation represents the exact solution for the Gutzwiller ansatz, as shown by Metzner and Vollhardt [14, 10]. This interesting result marked the beginning of a new era in the theory of correlated electrons, that of the Dynamical Mean-Field Theory [15].
The result of the Gutzwiller approximation can be represented in terms of a renormalized hopping, t → γt. For U → ∞, γ depends on the electron density n as γ = (1 - n)/(1 - n/2). Therefore, when approaching half filling (n → 1), the electron motion is completely suppressed, and the system is a Mott insulator. Brinkman and Rice noticed that within the Gutzwiller approximation the jamming of electrons (for n = 1) occurs at a large but finite value of U and is signaled by the vanishing of double occupancy [16]. They associated the critical point with the Mott metal-insulator transition. However, a closer scrutiny shows that this conclusion is an artifact of the Gutzwiller approximation. Indeed, for an exact treatment of the Gutzwiller ansatz (and finite lattice dimensions) double occupancy remains finite for all finite values of U. Moreover, the Gutzwiller ansatz itself is of limited validity for large values of U, as seen clearly by comparing it with the exact solution in one dimension.
Nevertheless, a Mott transition does occur for the Hubbard model, but in the sense of a topological transition from a phase with finite Drude weight for small values of U to one with vanishing Drude weight at large U, in agreement with Kohn’s distinction between metals and insulators [17]. To show this in a variational framework 1, we have used a pair of trial ground states [18], the Gutzwiller wave function together with the "inverted" ansatz
where 
is the hopping operator, 
is the ground state for U → ∞ and h is a variational parameter. One readily shows that 
has a finite Drude weight and lower energy for small U, while the Drude weight vanishes for 
, which is preferred for large U. A metal-insulator transition occurs for a value of U of the order of the band width, in good agreement with Quantum Monte Carlo results.
So far, we have assumed the Gutzwiller ansatz to be "adiabatically" linked to the filled Fermi sea , which is the main reason for the metallic character of . However, if we allow for a broken symmetry within , we may find a competing ground state with qualitatively different properties. For instance, allowing for different magnetic moments on the two sublattices of a bi-partite lattice, one can obtain an antiferromagnetic insulator already below the Mott transition, i.e., before electrons are essentially localized. This is indeed found for the square lattice (n = 1), where the Mott transition is replaced by a smooth crossover from a band (or "Slater" [19]) insulator with small alternating magnetic moments at small U to a (Heisenberg) antiferromagnetic insulator with fully developed local moments at large U. Interestingly, this is not the case for the honeycomb lattice, where antiferromagnetism sets in essentially together with the Mott transition [20], although the detailed behavior close to the transition appears to be more complicated – and quite intriguing [21].
Figure 1: Dimerization in the Hubbard-Peierls model. Circles: VMC, full lines: analytical small U expansion, broken lines: Unrestricted Hartree-Fock.
Figure 2: Superconducting order parameter as a function of doping for the Hubbard model on the square lattice.
As a second example of a broken symmetry we mention bond alternation in conjugated polymers, or, more precisely, the fate of the Peierls instability in the presence of Coulomb interaction. Eric Jeckelmann, during his Ph.D. thesis, studied the one-dimensional Peierls-Hubbard model where the bond length dependence of the hopping amplitude t provides a coupling between the electrons and the lattice [22]. He used the Gutzwiller ansatz but added both the electronic gap and the lattice dimerization as variational parameters. The result for the dimerization Δ, as a function of U and for fixed electron-lattice couplings λ, is shown in Fig. 1. In contrast to Unrestricted Hartree-Fock, where the Peierls insulator is rapidly replaced by a spin-density wave (a Slater insulator), the dimerization is found to remain finite for all values of U. It even increases initially, as discovered long before this work [23], and exhibits a maximum for U ≈ 4t, where a crossover to spin-Peierls behavior occurs. These variational results are in good agreement with subsequent calculations using the Density Matrix Renormalization Group.
As a third example we discuss results of the Ph. D. thesis of David Eichenberger [24], who studied the Hubbard model on a square lattice, using the modified Gutzwiller ansatz
The additional factor 
leads to a substantial improvement of the ground state energy and provides a kinetic exchange. We were particularly interested in the possibility of a superconducting ground state with d-wave symmetry, taken into account in the reference state . Fig. 2 shows the VMC result for the superconducting order parameter for the Hubbard model on an 8×8 square lattice with both nearest (t) and next-nearest neighbor hoppings (t') and a realistic Hubbard parameter U = 8t. Our results agree very well with other studies using completely different methods.
We turn now to the problem of itinerant ferromagnetism, which has been the main motivation for Gutzwiller (and for Hubbard and Kanamori as well) to study the Hamiltonian (1). The most simple trial state is the ground state of an effective single-particle model where the bands for up and down spins are shifted relative to each other. The "exchange splitting" is then determined by minimizing the total energy. This leads to the Stoner criterion, according to which ferromagnetism occurs if Uρ(εF) > 1, where ρ(εF) is the density of states per spin at the Fermi energy. Already in 1953 Van Vleck argued that the Stoner theory could not be the whole story, but that electronic correlations had to be taken into account. Gutzwiller’s scheme is well suited for doing that. The results obtained in this way still leave space for ferromagnetism, but the stability region in parameter space is strongly reduced as compared to that of Stoner’s theory [25]. In fact, the necessary U values are so large that one has to conclude that the single-orbital Hubbard model is not adequate for describing the ferromagnetism of transition metals.
There is another more fundamental reason why the single-band Hubbard model cannot be taken too seriously for describing transition metals. These materials are characterized by narrow partly filled 3d-bands located within a broad s-band and overlapping with even broader p bands, and therefore it is far from obvious how a one-band model should be able to describe their magnetic properties. This problem must have been clear to Gutzwiller, who used the smart title "Correlation of Electrons in a Narrow s Band" for one of his papers [3]. Notwithstanding this loophole, a realistic model should add uncorrelated electrons representing the s-band to the correlated electrons of the d-band. The Periodic Anderson Model is a first step in this direction, it admits two orbitals at each site, one of which is localized and correlated through an on-site interaction, the other is delocalized and uncorrelated. The two bands are hybridized. Using a generalized Gutzwiller ansatz together with a corresponding Gutzwiller approximation, one finds not only the usual renormalization of the correlated band by a factor &gama;, but also a renormalization of the hybridization by √γ [26, 27].
The next step is to treat two or more correlated orbitals at a site. Here, Jörg Bünemann in his Ph.D. thesis has contributed a great deal to generalize the Gutzwiller formalism [28]. The generalization leads to an enormous expansion of the Gutzwiller wave function, as many additional correlators have to be introduced. The relevant local multi-electron configurations can be represented by the eigenstates of an atomic Hamiltonian, which reproduces the atomic multiplet spectrum of the partly filled 3d shell. This extension also leads to a rapid increase of the number of variational parameters in the Gutzwiller wave function. If the number of different orbitals is N (N can be as large as 5 for an open d shell), the number of independent variational parameters can reach 22N - 2N - 1, which may be of the order of 1000 [28]. The variational parameters represent the occupancies of all possible multiplet states. At the first instance, the atomic multiplet spectrum is governed by three Slater-Condon or Racah integrals, when spherical symmetry is assumed for the atoms. Yet, the site symmetry in a crystal is lower than spherical. Incorporation of the correct site symmetry results in many further modifications and extensions of the method.
The multi-band Gutzwiller method allows the investigation of 3d transition metals and compounds on a quantitative basis. An ab initio single-particle Hamiltonian can be constructed using Density-Functional Theory (DFT). The simplest way to incorporate DFT results is to extract a tight-binding model by fitting the hopping amplitudes to the DFT bands, but more elaborate methods are available, such as down-folding the DFT bands to a reduced Wannier basis [29]. We have carried out various studies on magnetic 3d elements and on compounds of 3d elements. One paper dealt with the Fermi surface of ferromagnetic Ni. DFT predicts a hole ellipsoid around the X point of the Brillouin zone, which is missing in the data. The multi-band Gutzwiller method was based on a one-particle Hamiltonian derived from paramagnetic DFT bands for Ni including wide 4s and 4p bands.
Using typical interaction parameters for Ni, our calculations reproduced the observed Fermi surface topology [30]. Another paper dealt with the magnetic anisotropy in ferromagnetic Ni [31]. Here again, pure DFT results did not yield the correct answers, while the Gutzwiller method gave very good agreement with experiment. In all cases, the renormalization parameters γ have been found to be of the order of 0.7, indicating moderately strong correlation effects.
Finally we mention the issue of metallic anti-ferromagnetism in iron pnictides, a new class of high-temperature superconductors. Our calculations were based on down-folded DFT bands. The results indicate also in this case moderately strong correlations. The atomic magnetic moments were found to agree well with experiment, in contrast to the DFT results and also to model calculations [32].
The examples mentioned above demonstrate that Gutzwiller's simple ansatz evolved into a powerful tool for dealing with correlated electron systems. The method has recently also been applied successfully to cold bosonic atoms in an optical lattice. At the age of 50, Gutzwiller’s wave function in its extensions remains competitive for describing correlated states of matter.
1 From this point on, we will concentrate mostly on our own work, with apologies to other authors.
References
[1] M. C. Gutzwiller, Phys. Rev. Lett. 10, 169 (1963).
[2] M. C. Gutzwiller, Phys. Rev. 134, A923 (1964).
[3] M. C. Gutzwiller, Phys. Rev. 137, A1726 (1965).
[4] K. A. Chao and M. C. Gutzwiller, J. Appl. Phys. 42, 1420 (1971).
[5] J. H. van Vleck, Rev. Mod. Phys. 25, 220 (1953).
[6] P. W. Anderson, Phys. Rev. 115, 2 (1959).
[7] J. Hubbard, Proc. Roy. Soc. A276, 238 (1963).
[8] J. Kanamori, Prog. Theor. Phys. 30, 275 (1963).
[9] W. Metzner and D. Vollhardt, Phys. Rev. Lett. 59, 121 (1987); F. Gebhard and D. Vollhardt, Phys. Rev. Lett. 59, 1472 (1987).
[10] For a review see F. Gebhard, The Mott Metal-Insulator Transition: Models and Methods, Springer 1997.
[11] P. Horsch and T. A. Kaplan, J. Phys. C 16, L1203 (1983).
[12] For a recent review of the method for the fully projected Gutzwiller wave function (g → ∞) see B. Edegger, V. N. Muthukumar and C. Gros, Adv. Phys. 56, 927 (2007).
[13] For a clear presentation see P. Fulde, Electron Correlations in Molecules and Solids, Springer Series in Solid-State Sciences 100 (1990).
[14] W. Metzner and D. Vollhardt, Phys. Rev. Lett. 62, 324 (1989).
[15] For an early review see A. Georges, G. Kotliar, W. Krauth and M. J. Rozenberg, Rev. Mod. Phys. 68, 13 (1996).
[16] W. Brinkman and T. M. Rice, Phys. Rev. B 2, 4302 (1970).
[17] W. Kohn, Phys. Rev. 133, A171 (1964).
[18] L. M. Martelo, M. Dzierzawa and D. Baeriswyl, Z. Phys. B 103, 335 (1997).
[19] J. C. Slater, Phys. Rev. 82, 538 (1951).
[20] M. Dzierzawa, D. Baeriswyl and L. M. Martelo, Helv. Phys. Acta 70, 124 (1997); for a review see D. Baeriswyl, Found. Phys. 30, 2033 (2000).
[21] Z. Y. Meng, T. C. Lang, S. Wessel, F. F. Assaad and A. Muramatsu, Nature 464, 847 (2010).
[22] E. Jeckelmann, Ph D. thesis, University of Fribourg (1995); E. Jeckelmann and D. Baeriswyl, Synth. Met. 65, 211 (1994).
[23] P. Horsch, Phys. Rev. B 24, 7351 (1981); D. Baeriswyl and K. Maki, Phys. Rev. B 31, 6633 (1985).
[24] D. Eichenberger, Ph. D. thesis, University of Fribourg (2008); D. Baeriswyl, D. Eichenberger and M. Menteshashvili, New J. Phys. 11, 075010 (2009).
[25] P. Fazekas, Electron Correlation and Magnetism, World Scientific 1999.
[26] T. M. Rice and K. Ueda, Phys. Rev. Lett. 55, 995 (1985).
[27] C. M. Varma, W. Weber and L. J. Randall, Phys. Rev. B 33, 1015 (1986).
[28] J. Bünemann, Ph. D. thesis, University of Dortmund (1998); J. Bünemann, W. Weber and F. Gebhard, Phys. Rev. B 57, 6896 (1998).
[29] For a recent review see O. K. Andersen and L. Boeri, Ann. Phys. (Berlin) 523, 8 (2011).
[30] J. Bünemann et al., Europhys. Lett. 61, 667 (2003).[31] J. Bünemann, F. Gebhard, T. Ohm, S. Weiser and W. Weber, Phys. Rev. Lett. 101, 236404 (2008).
[32] T. Schickling et al., Phys. Rev. Lett. 108, 036406 (2012).
[Released: May 2012]