Gutzwiller wave function

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Florian Gebhard and Martin Gutzwiller (2009), Scholarpedia, 4(4):7288.

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Curator: Dr. Florian Gebhard, Physics, University of Marburg, Germany
Curator: Dr. Martin Gutzwiller, Yale University, New Haven, CT

Figure 1: Grey-scale plot of the negative second derivative of the ARPES intensity for nickel with respect to the energy on a logarithmic scale (insets: linear scale) for two directions of the Brillouin zone. Theoretical curves are the predictions from the Gutzwiller theory.

The Gutzwiller wave function (GWF) is a variational many-particle wave function which describes interacting particles on a rigid lattice. It provides an illustrative example for Landau's Fermi-liquid theory. Based on the GWF, the dispersion of quasi-particle excitations in the Fermi liquid state of transition metals can be calculated and compared to angle-resolved photo-emission (ARPES) experiments.

1 Definition
2 Evaluation
3 Applications
4 Appendix: historical note
5 References
6 Further reading
7 See also

Definition

Toy model description of the hydrogen molecule

In order to illustrate the basic idea behind the GWF, consider the simplest interacting electron model, the hydrogen molecule, $H_2$ , where the two sites represent the protons. In a toy model description, the electrons can occupy only the $1s$ orbital of either proton with spin $\sigma=\uparrow,\downarrow$ . The four possible configurations are shown in Fig. 2. Naturally, the basis set is far too small for a quantitative description of the hydrogen molecule.

Figure 2: Four possible electronic configurations for the two-site hydrogen molecule.

The electrons can tunnel between the protons with amplitude $-t$ ( $t>0$ ), and the ground state $|\Psi\rangle$ is a linear combination of the four configurations. In the Hund-Mulliken molecular-orbital (MO) description, all four configurations have the same probability. The MO ground state $|\Psi_0\rangle$ is incorrect because the configurations $|{\rm iii}\rangle$ and $|{\rm iv}\rangle$ correspond to a negatively charged $H^-$ atom next to a bare proton ( $H^+$ ). Such configurations where the $1s$ -orbital is doubly occupied are energetically unfavorable because the We added electrons on the same site repel each other due to their mutual Coulomb interaction (generally called the Hubbard $U$ , $U>0$ ). Therefore, the weight of doubly occupied orbitals is reduced in the ground state $|\Psi\rangle$ .

If one starts with some simple single-particle product wave function $|\Psi\rangle_0$ like the MO function, a better wave function $|\Psi\rangle$ can be obtained by reducing the weight of the configurations containing double occupancies, such as $|{\rm iii}\rangle$ and $|{\rm iv}\rangle$ in Fig. 2. The operator that counts the number of doubly occupied sites is $\hat{D}$ and the ground state can be written in the form of a GWF (M.C. Gutzwiller, 1963),

(1)

$|\Psi\rangle = g^{\hat{D}}|\Psi_0\rangle \; ,$

where $g=g(U)$ is a function of the electrons' intra-orbital Coulomb interaction. The toy model Hamiltonian is the two-site Hubbard model,

(2)

$\hat{H} = \hat{T}+U\hat{D} = (-t)\sum_{\sigma=\uparrow,\downarrow} \left( \hat{c}_{1,\sigma}^+\hat{c}_{2,\sigma} + \hat{c}_{2,\sigma}^+\hat{c}_{1,\sigma} \right) + U \left( \hat{c}_{1,\uparrow}^+\hat{c}_{1,\uparrow}\hat{c}_{1,\downarrow}^+\hat{c}_{1,\downarrow}+ \hat{c}_{2,\uparrow}^+\hat{c}_{2,\uparrow}\hat{c}_{2,\downarrow}^+\hat{c}_{2,\downarrow} \right) \; .$

The MO ground state is $|\Psi_0\rangle=(|{\rm i}\rangle -|{\rm ii}\rangle +|{\rm iii}\rangle +|{\rm iv}\rangle)/2$ and the GWF reads

(3)

$|\Psi\rangle = \frac{1}{2} \left[ |{\rm i}\rangle -|{\rm ii}\rangle +g\left(|{\rm iii}\rangle +|{\rm iv}\rangle\right)\right] \;.$

For $U\to\infty$ , $g\to 0$ , and $|\Psi\rangle$ reduces to the Heitler-London valence-bond (VB) state, which is a linear combination of the configurations $|{\rm i}\rangle$ and $|{\rm ii}\rangle$ in Fig. 2. The GWF (1) smoothly interpolates between the Hund-Mulliken and the Heitler-London wave functions.

For the two-site Hubbard model, the variational energy is given by

(4)

$E_{\rm var}(g) = -\frac{4tg}{1+g^2} + U \frac{g^2}{1+g^2} \; .$

The variational minimum of the energy is obtained from the condition $\partial E_{\rm var}(g)/\partial g=0$ , which is fulfilled for

(5)

$g_{\rm opt}(U) = \frac{1}{4t}\left[ -U + \sqrt{U^2+(4t)^2\,} \right] \; .$

For the two-site Hubbard model, the variational upper bound becomes the exact ground-state energy,

(6)

$E_{\rm var}(g_{\rm opt}(U))=E_0(U)=\frac{1}{2} \left(U - \sqrt{U^2+(4t)^2\,}\right)\; .$

For large interaction strengths, $U\gg 4t$ , the Gutzwiller parameter and the ground-state energy become small, $g\to 2t/U$ , $E_0\to -4t^2/U$ , but they remain finite, of order $1/U$ .

Gutzwiller correlator

Crystal lattices contain a large number $L$ of atoms and a similarly large number of electrons. To a first approximation, each electron can be treated as moving in an average potential. The resulting single-electron product state $|\Psi_0\rangle$ is the lattice generalization of the Hund-Mulliken molecular-orbital (MO) wave function (also known as Hartree-Fock wave function). For example, $|\Psi_0\rangle=|\hbox{FS}\rangle$ denotes the Fermi sea of noninteracting electrons.

The spectrum and eigenstates of the individual atoms on the lattice sites are $E_{\Gamma}$ and $|\Gamma\rangle$ . In the single-band example, there are four atomic states, $|1\rangle=|H^+\rangle$ , $|2\rangle=|H_{\uparrow}\rangle$ , $|3\rangle=|H_{\downarrow}\rangle$ , and $|4\rangle=|H^-\rangle$ , with the atomic energies $E_1=E_2=E_3=0$ and $E_4=U$ .

The Gutzwiller correlator (GC) reduces the weight of those configurations in $|\Psi_0\rangle$ which are energetically unfavorable, i.e., which have a large atomic energy. The general GWF is thus defined by (J. Bünemann, F. Gebhard, and W. Weber, 1998)

(7)

$|\Psi_{\rm G}\rangle = \hat{P}_{\rm G}|\Psi_0\rangle \; .$

The symbol $\hat{P}_{\rm G}$ is an operator that modifies the single-particle product wave function $|\Psi_0\rangle$ . The operator $\hat{m}_{\Gamma,\vec{R}}= |\Gamma\rangle_{\vec{R}} {}_{\vec{R}}\langle \Gamma|$ picks out a particular atom on site $\vec{R}$ and then projects the atom onto the atomic configuration $\Gamma$ . The GC assigns a weight factor $\lambda_{\Gamma,\vec{R}}$ to each atomic configuration,

(8)

$\hat{P}_{\rm G} = \prod_{\vec{R}} \hat{P}_{{\rm G},\vec{R}} \quad ,\quad \hat{P}_{{\rm G},\vec{R}} = \prod_{\Gamma} \lambda_{\Gamma,\vec{R}}^{\hat{m}_{\Gamma,\vec{R}}} = \sum_{\Gamma} \lambda_{\Gamma,\vec{R}} \hat{m}_{\Gamma,\vec{R}} \; .$

The GWF (1) for the one-band model is recovered for $\lambda_{1}=\lambda_{2}=\lambda_{3}=1$ and $\lambda_{4}=g$ for the four hydrogen atomic states $\bigl\{ H^+,H_{\uparrow},H_{\downarrow}, H^-\bigr\}$ .

The Gutzwiller correlator suppresses charge fluctuations which are too large in $|\Psi_0\rangle$ . For example, a Fermi-gas description of the $3d$ -electrons of nickel employs the Fermi-sea ground state. It predicts a certain probability to detect five-fold ionized nickel ions $\hbox{Ni}^{5+}$ in a nickel crystal. The energy $E_{{\rm Ni}^{5+}}$ of the corresponding atomic configuration is more than one hundred eV. Since the ground state has the lowest possible total energy, the probability to find five-fold ionized nickel ions $\hbox{Ni}^{5+}$ in the ground state of a nickel crystal must be exponentially small. This is ensured by the Gutzwiller correlator. In the GWF, only atomic configurations with approximately 10 electrons in the $3d$ -shell and the $4s$ -shell have non-negligible probability.

The Gutzwiller correlator belongs to the class of Jastrow-Feenberg correlators which are used, e.g., for the investigation of superfluid Helium-4.

Expectation values and variational ground-state energy

Physical quantities such as the magnetization $M_z$ are determined by expectation values of corresponding quantum-mechanical operators $\hat{M_z}$ . Quite generally, the Gutzwiller variational expressions for expectation values of observables are

(9)

$A_{\rm var}=\langle \hat{A}\rangle_{\rm G} = \frac{\langle \Psi_0 | \hat{P}_{\rm G}^+ \hat{A} \hat{P}_{\rm G}|\Psi_0 \rangle} {\langle \Psi_0 | \hat{P}_{\rm G}^+ \hat{P}_{\rm G}|\Psi_0 \rangle} \; .$

The variational parameters in the GWF are obtained as follows: Starting from $|\Psi_0\rangle$ and $\hat{P}_{\rm G}$ , the variational energy $E_{\rm var}=\langle\hat{H}\rangle$ for a given model Hamiltonian, e.g., for the Hubbard model (read a /historical note on its invention), must be calculated. Then, $E_{\rm var}$ must be minimized with respect to all variational parameters to find the optimal variational energy, $E_{\rm var}^{\rm opt}$ .

Evaluation

The evaluation of expectation values within the GWF poses a many-particle problem which is unsolvable in general. The numerical evaluation of expectation values is possible on finite lattice with the help of the Variational Monte Carlo (VMC) method as introduced by C. Gros et al., 1987, and by H. Yokoyama and H. Shiba, 1987; for a review, see Edegger et al., 2007.

As in standard Feynman-Dyson perturbation theory, the analytical evaluation of expectation values starts from a series expansion around the noninteracting limit, $\lambda_{\vec{R}}=1$ , and the individual orders are expressed in terms of diagrams. In contrast to the standard calculation of Green functions for interacting electron systems, the GWF permits various choices for the bare vertex. This flexibility of the GWF allows its exact evaluation in one spatial dimension and in the limit of infinite dimensions. In his three articles, Gutzwiller provided the first realistic picture of the ferromagnetism of nickel and its alloys with copper; further developments have been due to other authors.

Exact results in one spatial dimension

For electrons on a ring, expectation values for the Gutzwiller-correlated Fermi sea can be evaluated for all electron densities $n=n_{\uparrow}+n_{\downarrow}$ , magnetizations $m=(n_{\uparrow}-n_{\downarrow})$ , and interaction parameters $g$ without making further approximations. When expectation values are expanded in terms of $(g^2-1)$ and particle-hole symmetry is used, all coefficients of the series expansion can be determined (W. Metzner and D. Vollhardt, 1988) (F. Gebhard and D. Vollhardt, 1988).

Average double occupancy

Figure 3: Average double occupancy as a function of the interaction parameter for the paramagnetic Gutzwiller-correlated Fermi sea for various particle densities in one dimension (full lines, eq. (10)) and in infinite dimensions (dashed lines, eq. (18)).

The average number of doubly occupied sites is defined by $\overline{d} = \langle \hat{D}/L\rangle_{\rm G} = (1/L) \sum_{\ell=1}^L \langle \hat{n}_{\ell,\uparrow}\hat{n}_{\ell,\downarrow} \rangle_{\rm G}$ . The result for the Gutzwiller-correlated Fermi sea in one dimension for all interaction parameters $g\geq 0$ , electron densities $0\leq n\leq 1$ , and magnetizations $0\leq m\leq n$ is (M. Kollar and D. Vollhardt, 2002)

(10)

$\overline{d}(g,n,m)= \frac{g^2}{2(1-g^2)^2}\left[ -(1-g^2)(n-m) +\ln\left(\frac{1-(1-g^2)m}{1-(1-g^2)n}\right)\right] \; .$

For $g=1$ (noninteracting limit), the result reproduces the Hartree-Fock value, $\overline{d}(g=1,n,m) = (n^2-m^2)/4$ . For $g\to 0$ (strong-coupling limit), the double occupancy vanishes, $\overline{d}(g=0,n,m) = 0$ .

Momentum distribution

Figure 4: Momentum distribution as a function of crystal momentum for the paramagnetic Gutzwiller-correlated Fermi sea at half band filling for various interaction parameters in one dimension (full lines, eq. (11)). For comparison, the result based on the limit of infinite dimensions (dashed lines, eq. (18)) is also shown.

The momentum distribution $n_{k,\sigma}= \langle \hat{n}_{k,\sigma}\rangle_{\rm G}$ is $2\pi$ -periodic and inversion symmetric for a symmetric dispersion relation $\varepsilon(-k)= \varepsilon(k)$ of the underlying Hubbard model.

The exact formulae for the Gutzwiller-correlated Fermi sea are rather involved. For the paramagnetic half-filled Fermi sea, $n=1$ and $m=0$ , and with $0\leq \tilde{k}=2|k|/\pi\leq 2$ , $G=1-g^2$ , they can be written as

(11)

$n_{|k|\leq \pi/2,\sigma}(g)=\frac{g^2+4g+1}{2(1+g)^2} + \frac{g^2}{(1+g)^2} \frac{4}{\pi\sqrt{(2-G)^2-(\tilde{k}G)^2}} K\left[\frac{G\sqrt{1-\tilde{k}^2}}{\sqrt{(2-G)^2-(\tilde{k}G)^2}}\right] \; ,$

where

(12)

$K(k)=\int\limits_{0}^{\pi}{\rm d}\phi [1-k^2\sin^2(\phi)]^{-1/2}$

is the complete elliptic integral of the first kind. Moreover, $n_{\pi/2<|k|\leq \pi,\sigma}(g) = 1-n_{\pi-|k|,\sigma}(g)$ .

In general, the momentum distribution is discontinuous at the Fermi wave number $k_{\rm F}=\pi/2$ , as is characteristic for a metal. For the half-filled paramagnetic Fermi sea ( $n=1$ , $m=0$ ), the jump is given by

(13)

$q(g)=n_{k=\pi/2^-,\sigma}(g)- n_{k=\pi/2^+,\sigma}(g)= \frac{4g}{(1+g)^2} \; .$

The discontinuity vanishes for strong coupling, $g=0$ , when all electrons are localized (Brinkman-Rice insulator).

From the exact solution in one dimension via Bethe Ansatz it is known that the Hubbard model away from half band-filling provides an example of a Luttinger Liquid whose momentum distribution is continuous with a divergent slope at the Fermi wave number. The GWF does not reproduce this generic behavior of one-dimensional metals.

Variational energy for the single-band Hubbard model

Figure 5: Ground-state energy for the Hubbard model with nearest-neighbor electron transfer and dispersion $\varepsilon(k)=-2\cos(k)$ at half band filling as a function of the Hubbard interaction (full line). The Gutzwiller variational upper bound is shown for comparison (dashed line).

The variational energy provides an exact upper bound for the ground-state energy of the Hubbard model, which is exactly known for the Hubbard model from the Bethe Ansatz for the dispersion relation $\varepsilon(k)=-2t\cos(k)$ (electron transfer between nearest neighbors only). For the paramagnetic case, $m=0$ , the comparison shows that the ground-state energy for half band filling, $n=1$ , deviates substantially from the exact result for large interactions, $U/t\to \infty$ (W. Metzner and D. Vollhardt, 1988).

In this limit, double occupancies and empty sites are constrained to be adjacent to each other in the exact ground state, but this correlation is absent in the GWF (F. Gebhard and D. Vollhardt, 1988). To cure this problem, various extensions of the GWF have been proposed, e.g., the Baeriswyl-GWF wave function (M. Dzierzawa, D. Baeriswyl, and M. Di Stasio, 1995) and the Local-Ansatz wave functions (P. Fulde, 1995), which introduce correlations between neighboring sites. Typically, these wave functions can be evaluated analytically only in limiting cases. Otherwise, they must be treated numerically (Variational Monte Carlo) (B. Edegger, V.N. Muthukumar et al., 2007).

Spin correlations and Haldane-Shastry model

The $z$ -component of the spin-spin correlation function is defined by $C^{\rm SS}(r) = (1/L)\sum_{\ell=1}^L \langle \hat{S}_{r+\ell}^{z}\hat{S}_{\ell}^{z} \rangle_{\rm G}$ , where $\hat{S}_{\ell}^{z} =(\hat{n}_{\ell,\uparrow}- \hat{n}_{\ell,\downarrow})/2$ is the operator for the z-component of the electron spin on site $\ell$ .

While the correlations between double occupancies and holes are poorly described by the GWF in the strong-coupling limit, the spin-spin correlations at half band filling correctly show the characteristic behavior of Heisenberg-type spin models in one dimension. The spin-spin correlations of the Gutzwiller-projected paramagnetic Fermi sea ( $g=0$ , $n=1$ , $m=0$ ) are given by (F. Gebhard and D. Vollhardt, 1988)

(14)

$C^{\rm SS}(r>0;g=0) = (-1)^r \frac{{\rm Si}(\pi r)}{4\pi r} \; ,$

where $\hbox{Si}(x)=\int_0^x{\rm d}u \frac{\sin u}{u}$ is the sine integral. For large distances, $r\gg 1$ , the spin-spin correlation function decays to zero proportionally to $(-1)^r/(8r)$ . The absence of long-range order is characteristic of a RVB (resonating valence-bond) state (P. Fazekas and P.W. Anderson, 1974). Nevertheless, the Fourier-transformed spin-spin correlation function diverges logarithmically at $q=\pi$ .

The Gutzwiller-projected half-filled Fermi sea is the exact ground state of the spin-1/2 Heisenberg model with $1/r^2$ -exchange (Haldane-Shastry model) (F.D.M. Haldane, 1988) (B.S. Shastry, 1988)

(15)

$\hat{H}_{\rm HS}= \sum_{r=1}^{L-1} \left(\frac{\pi}{L\sin(\pi r/L)}\right)^2 \sum_{\ell=1}^L \hat{\vec{S}}_{\ell+r} \cdot\hat{\vec{S}}_{\ell} \quad \hbox{with} \quad \hat{\vec{S}}_{\ell}: \hbox{spin-1/2 vector operator.}$

At finite hole density, $n\leq 1$ , the Gutzwiller-projected paramagnetic Fermi sea is the exact ground state of the supersymmetric $t$ - $J$ model with $1/r^2$ -exchange (Y. Kuramoto and H. Yokoyama, 1991).

Exact results in the limit of infinite spatial dimensions

In the limit of infinite spatial dimensions, the number of nearest neighbors $Z$ of a given lattice site (coordination number) tends to infinity, $Z\to \infty$ . For example, $Z=2d$ in a simple cubic lattice in $d$ dimensions tends to infinity in the limit of infinite spatial dimensions.

Nickel crystallizes in a fcc structure, which has coordination number $Z=12$ . Therefore, one may view the limit of infinite spatial dimensions as a starting point of a $1/Z$ -expansion, and corrections can be expected to be small, of the order of $1/Z$ .

Simplifications

In the limit $1/Z\to 0$ , expectation values $\langle \hat{A}\rangle_{\rm G}$ for the Gutzwiller-correlated wave functions as defined in eq. (7) can be evaluated without further approximations. It is possible to set up a diagrammatic series expansion around the uncorrelated limit, $\lambda_{\Gamma,\vec{R}}=1$ . In this case, not a single diagram must be calculated in the $Z\to \infty$ limit. The theory remains nontrivial because the single-particle density matrix $P_{\vec{R},\sigma;\vec{R}',\sigma'}=\langle \hat{c}_{\vec{R},\sigma}^{+} \hat{c}_{\vec{R}',\sigma'}^{}\rangle_{\rm G}$ and the average atomic occupancy $m_{\Gamma,\vec{R}} = \langle \hat{m}_{\Gamma,\vec{R}}\rangle_{\rm G}$ are renormalized in the procedure (J. Bünemann, F. Gebhard, and W. Weber, 1998).

Variational ground-state energy

In the limit of infinite coordination number, the variational parameters $\lambda_{\Gamma,\vec{R}}$ may be replaced by the physical expectation values $m_{\Gamma,\vec{R}}$ for the occupation of an atomic configuration $\Gamma$ on lattice site $\vec{R}$ . Moreover, the local density matrix for noninteracting electrons $C^0_{\vec{R};\sigma,\sigma'}=P^0_{\vec{R},\sigma;\vec{R},\sigma'}$ must obey certain constraints which can be included with the help of Lagrange parameters $\eta_{\vec{R};\sigma,\sigma'}$ .

For a translationally invariant symmetric multi-band Hubbard model, the variational ground-state energy functional for a normalized single-particle product state $|\Psi_0\rangle$ with fixed average particle density reads

(16)

$E_{\rm var}\left(m_{\Gamma},\eta_{\sigma,\sigma'}, C_{\sigma,\sigma'},\left\{ |\Psi_0\rangle \right\}\right) =\langle \Psi_0 | \hat{T}^{\rm eff} | \Psi_0 \rangle + L \sum_{\Gamma} E_{\Gamma} m_{\Gamma} -L \sum_{\sigma,\sigma'}\eta_{\sigma,\sigma'} C^0_{\sigma,\sigma'}\quad , \quad \hat{T}_{\rm eff} =\sum_{\vec{k};\sigma,\sigma'} \varepsilon_{\sigma,\sigma'}^{\rm eff}(\vec{k}) \hat{c}_{\vec{k},\sigma}^+ \hat{c}_{\vec{k},\sigma'}^{} \; ,$

where $\sigma=1,2,\ldots 2N$ labels the atomic orbitals. In particular, $N=1,3,5$ for atomic $s$ , $p$ , $d$ shells. The minimization of the functional with respect to $|\Psi_0\rangle$ shows that it is the ground state of the effective kinetic energy $\hat{T}^{\rm eff}$ with the effective dispersion relation

(17)

$\varepsilon_{\sigma,\sigma'}^{\rm eff}(\vec{k}) = \sum_{\gamma,\gamma'} Q_{\gamma,\gamma'}^{\sigma,\sigma'} \varepsilon_{\gamma,\gamma'}^0(\vec{k}) +\eta_{\sigma,\sigma'} \; .$

The matrix $Q_{\gamma,\gamma'}^{\sigma,\sigma'}$ is a known but, in general, complicated function of the variational parameters. The matrices $Q_{\gamma,\gamma'}^{\sigma,\sigma'}$ and $\eta_{\sigma,\sigma'}$ express the fact that the electron-electron interaction reduces the bandwidth of the bare bands with dispersion $\varepsilon_{\gamma,\gamma'}^0(\vec{k})$ and changes their hybridization as well as their relative positions.

The minimization of the energy functional with respect to all variational parameters is a numerically demanding task for real materials because it requires the minimization of a functional with several thousands of parameters.

Landau-Gutzwiller quasi-particles

The Gutzwiller variational theory provides an explicit example of the Landau Fermi-liquid theory. The Gutzwiller correlator $\hat{P}_{\rm G}$ continuously transforms the single-particle ground state $|\Psi_0\rangle$ to the (variational) ground state for interacting particles $|\Psi\rangle_{\rm G}$ . For example, the momentum distribution $n_{\vec{k},\sigma}$ displays a jump discontinuity at the Fermi energy both in $|\Psi_0\rangle$ and in $|\Psi\rangle_{\rm G}$ , see Fig. 4 and eq. (13). This jump represents the Fermi surface in the reciprocal (or Fourier) space where the individual single-electron states depend on the wave vector $\vec{k}$ . The discontinuity is smaller than unity and depends on the strength of the coupling. Gutzwiller (1965) provided a relatively simple derivation of this generic result within the GWF. In contrast, Hubbard's original theory led to a band splitting into an upper and a lower Hubbard band, so that a system at half band filling would describe an insulator, irrespective of the strength of the Coulomb interaction. The Gutzwiller theory for the ground state of an interacting many-particle system provides the dominant Landau parameters $F_{0,1}^{s,a}$ , which determine the thermodynamics as well as the dispersion relation of the dominant hydrodynamic modes (zero and first sound). The excitation energies and the temperature must be small compared to the Fermi temperature.

In the spirit of the Landau Fermi-liquid theory, the Landau-Gutzwiller theory describes quasi-hole (quasi-particle) excitations as Gutzwiller-correlated holes (particles) in $|\Psi_0\rangle$ (J. Bünemann, F. Gebhard, and R. Thul, 2003). Their dispersion relation is the same as that obtained for the effective kinetic energy $\hat{T}^{\rm eff}$ , eq. (17). Therefore, $\varepsilon_{\sigma,\sigma'}(\vec{k})$ defines the quasi-particle band structure, which can be compared to experimental data from angle-resolved photo-emission spectroscopy (ARPES).

A time-dependent version of the Gutzwiller wave function (G. Seibold et al., 2004) permits the direct calculation of the dynamic magnetic and charge correlation functions.

Relation to other methods

Gutzwiller Approximation (GA) and Brinkman-Rice transition

The general formulae in infinite dimensions considerably simplify for the Gutzwiller-correlated paramagnetic Fermi sea which is a variational ground state for the single-band Hubbard model. In a particle-hole symmetric system at half band filling, the average double occupancy $\overline{d}$ , the bandwidth reduction factor $q_{\sigma}$ , the dispersion of the quasi-particles $\varepsilon_{\sigma}^{\rm eff}(\vec{k})$ , and the momentum distribution $n_{\vec{k},\sigma}$ are given by

(18)

$\overline{d}(g)= \frac{g}{2(1+g)} \quad , \quad q_{\sigma}(\overline{d}) = 8\overline{d}\left(1-2\overline{d}\right)=\frac{4g}{(1+g)^2}\quad , \quad \varepsilon_{\sigma}^{\rm eff}(\vec{k})= q_{\sigma}(\overline{d}) \varepsilon_{\sigma}^0(\vec{k})\quad ,\quad \hbox{and} \quad n_{\vec{k},\sigma}(\overline{d})= \left\{ \begin{array}{lcr} (1+q_{\sigma}(\overline{d}))/2 & \hbox{for} & \varepsilon(\vec{k}) \leq 0\\ (1-q_{\sigma}(\overline{d}))/2 & \hbox{for} & \varepsilon(\vec{k}) > 0 \end{array} \right. .$

These approximation-free results in infinite dimensions are identical to those obtained from the Gutzwiller Approximation (GA), which was based on a semi-classical counting of configurations (M.C. Gutzwiller, 1965) (J. Bünemann, 1998).

For the single-band Hubbard model, the GA describes a metal-to-insulator transition in which all electrons are localized above a finite critical strength of the Hubbard interaction (Brinkman-Rice transition) (W.F. Brinkman and T.M. Rice, 1970). For $U\geq U_{\rm BR}$ , the minimization of the variational ground-state energy functional leads to $\overline{d}^{\rm opt}(U>U_{\rm BR})=q_{\sigma}^{\rm opt}(U>U_{\rm BR})= E_{\rm var}^{\rm opt}(U>U_{\rm BR})=0$ , where $U_{\rm BR}$ is of the order of the bandwidth of the noninteracting electrons.

The Brinkman-Rice transition is an artifact of the limit of infinite dimensions, i.e. the variational result $g_{\rm opt}>0$ holds for the Hubbard model in finite dimensions for all finite interaction strengths $U>0$ . The Brinkman-Rice transition cannot be removed by any finite-order expansion in $1/Z$ (F. Gebhard, 1990). Nevertheless, the Brinkman-Rice transition provides an illustrative example of the breakdown of the Fermi liquid state at the Mott metal-to-insulator transition.

Kotliar-Ruckenstein slave-boson mean-field theory

In the Kotliar-Ruckenstein slave-boson approach, each atomic configuration is represented by a boson, so that the Hubbard interaction becomes simple. In contrast, the kinetic energy becomes much more complicated in terms of the bosons because the motion of an electron from one site to another changes the atomic configuration and thus the boson number on both sites (G. Kotliar and A.E. Ruckenstein, 1986).

After an ingenious transformation of the boson transfer operators between sites, the replacement of the bosonic operators by average values (saddle-point approximation) leads to an effective Hamiltonian with the same dispersion relation as eq. (17). Therefore, the Kotliar-Ruckenstein slave-boson mean-field theory is identical to the result of the Gutzwiller theory in the limit of infinite coordination number (J. Bünemann and F. Gebhard, 2007).

Applications

The application of the GWF to real materials (Gutzwiller theory) involves a three stage approximation. The starting point is the parameterization of the (multi-band) Hubbard model for which the bare dispersion relation $\varepsilon_{\sigma,\sigma'}^0(\vec{k})$ and the energies of the atomic levels must be specified. The second approximation is the GWF itself, which is a variational ground state. Thirdly, the GWF is evaluated in infinite dimensions, but the corresponding expressions are applied to three-dimensional systems.

Liquid Helium-3

Like an electron, a Helium-3 atom is a spin-1/2 fermion. At low temperatures, Helium-3 is a normal liquid and also a Landau Fermi liquid. Its properties are well described by the Gutzwiller theory for the single-band Hubbard model because the Helium atoms repel each other strongly. The assumption that the fermions move on a lattice is an additional approximation in the liquid phase.

A reasonable agreement between the theoretical prediction and experimental data is obtained for the pressure dependence of the magnetic susceptibility $\chi_{\rm s}$ and the compressibility $\kappa$ . They are related to the Landau Fermi-liquid parameters $F_0^{\rm s,a}$ (D. Vollhardt, 1984) via $\chi_{\rm s}/\chi_{\rm s}^0=(m^*/m) (1+F_0^{\rm a})^{-1}$ and $\kappa/\kappa^0=(m^*/m) (1+F_0^{\rm s})^{-1}$ . Here the upper index `0' refers to the noninteracting system. The two Landau parameters are predicted to be functions of the experimentally accessible mass enhancement $m^*(p)/m=1/q_{\sigma}(p)=1/(1-(U(p)/U_{\rm BR})^2)=1+F_1^{\rm s}/3$ , which increases with pressure $p$ . The theoretical predictions $F_0^{\rm s}(p)=-1+1/(1-I(p))^2$ and $F_0^{\rm a}=-1+1/(1+I(p))^2$ , with $I(p)=\sqrt{1-1/q_{\sigma}(p)}$ , reproduce the experimental values to an accuracy of 50% and 10%, respectively.

Band structure of nickel

Figure 6: Cross section of the Fermi surface in two planes in the Brillouin zone. Open symbols and filled dots are experimental data, lines are the predictions from the Gutzwiller theory.

For nickel, the local spin-density approximation (LSDA) to density-functional theory (DFT) does not provide a good description of the quasi-particle bands as measured in ARPES experiments. Essentially all of the discrepancies are resolved by the Gutzwiller theory.

The Gutzwiller theory employs a multi-band Hubbard model with bare band structure $\epsilon_{\sigma,\sigma'}^0(\vec{k})$ , which is obtained from a paramagnetic LDA calculation. Only the bands close to the Fermi energy, i.e., the $3d$ , $4s$ , and $4p$ bands, are taken into account in the Gutzwiller theory. In spherical approximation, the atomic spectrum of the $3d$ -shell depends on three Racah parameters A, B, and C, whereby the parameters $B$ and $C$ are close to their values for free Ni $^{++}$ -ions. The Racah- $A$ corresponds to the Hubbard $U$ and is a free parameter of the Gutzwiller theory. When only the correlations in the $3d$ bands are considered, $A=8\, {\rm eV}{\rm -}10\, {\rm eV}$ leads to a good agreement between the quasi-particle bands from the Gutzwiller theory and experimental ARPES data. This is shown in Fig. 1 (J. Bünemann, F. Gebhard et al., 2003).

The quality of the theoretical predictions improves when the spin-orbit coupling is taken into account. The theory explains the observed <111>-direction for the magnetic moment, the magnitude of the magnetic anisotropy energy, and the change in the Fermi-surface topology around the X-point in the presence of a strong external magnetic field along the <001>-direction (Gersdorf effect). A Fermi-surface cross section in the presence of the spin-orbit coupling is shown in Fig. 6 (J. Bünemann, F. Gebhard et al., 2008).

Further applications

Correlated superconductors

The Gutzwiller theory is not only applicable to ferromagnetism but to many other types of ground states with broken symmetry, e.g., to antiferromagnetism or to lattice disorder in which the ground state lacks translational symmetry. In the same way, Gutzwiller-correlated BCS (Bardeen-Cooper-Schrieffer) wave functions are candidates for superconductors with strong electronic correlations such as the high-temperature superconductors (P.W. Anderson, 1987). This idea was worked out in more detail by F.C. Zhang et al., 1988; for a review, see (B. Edegger, V.N. Muthukumar et al., 2007). The concept of gossamer superconductivity put forward by Bernevig, Laughlin, and Santiago, 2003, is also based on Gutzwiller-correlated BCS wave functions.

Atoms in optical lattices

The GWF can equally be applied to the Bose-Hubbard model which is suitable for ultracold bosonic atoms in optical lattices. The Gutzwiller theory reproduces the mean-field result for the phase boundary between the superfluid phase and the Mott phase at zero temperature (D. Jaksch, C. Bruder et al., 1998). The mean-field approach becomes exact in the limit of infinite dimensions and provides a reasonable approximation to bosons in two-dimensional and three-dimensional confined geometries.

Appendix: historical note

A historical note on the invention of the Hubbard model

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Suggested by:	Dr. Jian K. Liu, Laboratory of Neurophysics and Physiology, Universite Paris Descartes
Invited by:	Dr. Eugene M. Izhikevich, Editor-in-Chief of Scholarpedia, the peer-reviewed open-access encyclopedia
Action editor:	Dr. Riccardo Guida, Institut de Physique Théorique; CEA, IPhT; CNRS; Gif-sur-Yvette, France