Gutzwiller wave function

Post-publication activity

Figure 1: Grey-scale plot of the negative second derivative of the ARPES intensity for nickel with respect to 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.

Definition

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. The electrons can occupy only the \(1s\) orbital of either proton with spin \(s=\uparrow,\downarrow\). The four possible configurations are shown in Fig.<ref>fig:two-siteHM</ref>.

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

The electrons can tunnel between the protons, and the ground state \(|\Psi\rangle\) is a linear combination of the four configurations. In the Hund-Mullikan 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 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 wave function \(|\Psi\rangle_0\) like the Hund-Mullikan function, a better wave function \(|\Psi\rangle\) can be obtained by reducing the weight of the configurations with double occupancies, such as \(|{\rm iii}\rangle\) and \(|{\rm iv}\rangle\) in Fig.<ref>fig:two-siteHM</ref>. The operator which 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)

|\Psi\rangle = g^{\hat{D}}|\Psi_0\rangle , </math> where \(g=g(U)\) is a function of the electrons' intra-orbital Coulomb interaction. For \(U\to\infty\), \(g\to 0\), and \(|\Psi\rangle\) reduces to the Heitler-London state, which is the sum of the configurations \(|{\rm i}\rangle\) and \(|{\rm ii}\rangle\) in Fig.<ref>fig:two-siteHM</ref>. The GWF (<ref>eq:def-Psi-hydro</ref>) smoothly interpolates between the Hund-Mullikan and the Heitler-London wave functions.

Gutzwiller correlator

Crystal lattices contain a large number \(L\) of atoms, with a similrly large number of electrons. At first, each electron is treated as moving in an average potential. The resulting single-electron product state \(|\Psi_0\rangle\) is the lattice-generalization of the Hund-Mullikan 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 non-interacting 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)

|\Psi_{\rm G}\rangle = \hat{P}_{\rm G}|\Psi_0\rangle </math> 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,

\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}} </math> The GWF (<ref>eq:def-Psi-hydro</ref>) 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 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 guaranteed by the Gutzwiller correlator. In the GWF, only atomic configurations with approximately \(n_{3d}\approx 9\) electrons in the \(3d\)-shell have a non-negligible probability.

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

Expectation values and variational ground-state energy

Physical quantities like the magnetization \(M_z\) are derived from the expectation values of their corresponding quantum-mechanical operators \(\hat{M_z}\). Quite generally, the Gutzwiller variational expressions are

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} </math> 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 must be calculated, e.g., for the Hubbard model (read a /historical note on the invention of the Hubbard model). 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 with the GWF poses a many-particle problem which is unsolvable, in general. As in standard Feynman-Dyson perturbation theory, expectation values can be calculated in a series expansion around the non-interacting 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 Gutzwiller approach permits various choices for the bare vertex. This flexibility permits an exact evaluation of the GWF in one spatial dimension and in the limit of infinite dimensions.

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) and in infinite dimensions (dashed lines).

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) \[ \overline{d}(g,n,m)= \frac{g^2}{2(1-g^2)^2}\left[ -(1-g^2)(n-m) +\ln\left(\frac{1-(1-g^2)m}{1-(1-g^2)n}\right)\right] \]. For \(g=1\) (non-interacting 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). For comparison, we also show the result based on the limit of infinite dimensions (dashed lines).

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. (M. Kollar and D. Vollhardt, 2002) 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

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] </math> where

K(k)=\int\limits_{0}^{\pi}{\rm d}\phi [1-k^2\sin^2(\phi)]^{-1/2} </math> 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 \(k_{\rm F}=\pi/2\) which is characteristic for a metal. The jump for the half-filled paramagnetic Fermi-sea (\(n=1\), \(m=0\)) is given by

q(g)=n_{k=\pi/2^-,\sigma}(g)- n_{k=\pi/2^+,\sigma}(g)= \frac{4g}{(1+g)^2}. </math> The discontinuity vanishes for strong coupling, \(g=0\), when all electrons are localized (Brinkman-Rice insulator).

Variational energy for the single-band Hubbard model

Figure 5: Ground-state energy for the Hubbard model with nearest-neighbor electron transfer (amplitude t) and cosine dispersion 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 has been solved exactly by Bethe Ansatz in one dimension for the dispersion relation \(\varepsilon(k)=-2t\cos(k)\) (electron transfer between nearest neighbors only). The comparison for the paramagnetic case, \(m=0\), 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 next 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 and the Local-Ansatz wave functions 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).

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 electron spin on site \(\ell\) in \(z\)-direction.

Albeit 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 are characteristic for Heisenberg-type 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)

C^{\rm SS}(r>0;g=0) = (-1)^r \frac{{\rm Si}(\pi r)}{4\pi r} \qquad \hbox{(Si: sine integral).} </math> For large distances, \(r\gg 1\), the spin-spin correlation function decays to zero proportional to \((-1)^r/(8r)\). The absence of long-range order is characteristic for an RVB (resonating valence-bond) state. 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)

\hat{H}_{\rm HS}= \sum_{r=1}^{L} \left(\frac{\pi}{L\sin(\pi r/L)}\right)^2 \sum_{\ell=1}^L \hat{\vec{S}}_{\ell+r} \cdot\hat{\vec{S}}_{\ell} \qquad \hat{\vec{S}}_{\ell}: \hbox{spin-1/2 vector operator.} </math> 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 \(Z\) to 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 the 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. (<ref>eq:def</ref>) can be evaluated without further approximations. It is possible to set up a diagrammatic series expansion around the uncorrelated limit, \(\lambda_{\Gamma,\vec{R}}=1\), where not a single diagram must be calculated in the limit \(Z\to \infty\). The theory remains non-trivial 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 non-interacting local density matrix \(C^0_{\vec{R};\sigma,\sigma'}=P^0_{\vec{R},\sigma;\vec{R},\sigma'}\) must obey certain constraints which are included with the help of Lagrange parameters \(\eta_{\vec{R};\sigma,\sigma'}\).

For a translational symmetric multi-band Hubbard model, the variational ground-state energy functional for a normalized single-particle product state \(|\Psi_0\rangle\) with fixed avarage particle density reads

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'}^{} , </math> where \(\sigma=1,2,\ldots 2N\) labels the atomic orbitals; \(N=1,3,5\) 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

\varepsilon_{\sigma,\sigma'}^{\rm eff}(\vec{k}) = \sum_{\gamma,\gamma'} Q_{\gamma,\gamma'}^{\sigma,\sigma'} \varepsilon_{\gamma,\gamma'}^0(\vec{k}) +\eta_{\sigma,\sigma'} . </math> The matrix \(Q_{\gamma,\gamma'}^{\sigma,\sigma'}\) is a known but, in general, complicated function of the variational parameters. (J. Bünemann, F. Gebhard, and W. Weber, 1998) The matrices \(Q_{\gamma,\gamma'}^{\sigma,\sigma'}\) and \(\eta_{\sigma,\sigma'}\) express the fact that the electron-electron reduces the bandwidth of the bare bands with dispersion \(\varepsilon_{\gamma,\gamma'}^0(\vec{k})\), and changes their hybridization and relative positions, too.

The minimization of the energy functional with respect to all variational parameters is a demanding numerical 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 for 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 \(|\Psi\rangle_{\rm G}\). Within this framework, the Gutzwiller variational theory for the ground-state of an interacting many-particle system provides the dominant Landau parameters which determine the thermodynamics and the dispersion relation of the leading hydrodynamic modes. 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 obtained for the effective kinetic energy \(\hat{T}^{\rm eff}\), Eq. (<ref>eq:epseffective</ref>). 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).

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

For the single-band Hubbard model, the GA describes a metal-to-insulator transition where all electrons are localized above a 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 non-interacting electrons.

The Brinkman-Rice transition is not contained in the GWF for any finite dimensions, i.e., the transition is an artifact of the limit of infinite dimensions. It cannot be removed by any finite-order expansion in \(1/Z\). (F. Gebhard, 1990) Nevertheless, the Brinkman-Rice transition provides an illustrative example for the breakdown of the Fermi liquid state at the 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 transfers between sites, the replacement of the bosonic operators by numbers (saddle-point approximation) leads to an effective Hamiltonian with the same dispersion relation as in Eq. (<ref>eq:epseffective</ref>). Therefore, the Kotliar-Ruckenstein slave-boson mean-field theory is identical to the results from 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 three approximation steps. 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 only. 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 can well be described by the Gutzwiller theory for a 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 for the liquid phase.

A reasonable agreement between the theoretical prediction and experimental data is obtained for the pressure-dependence of the magnetic susceptibility and the compressibility which are closely related to the Landau Fermi-liquid parameters. (D. Vollhardt, 1984) They are predicted to be functions of the experimentally accessible mass enhancement \(m^*(p)/m=1/q_{\sigma}(p)=1/(1-(U(p)/U_{\rm BR})^2)\) 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 experiments with an accuracy of 50% and 10%, respectively.

Band structure of nickel

Figure 6: Cuts of the Fermi surface with 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 are taken into account in the Gutzwiller theory, i.e., the \(3d\), \(4s\), and \(4p\) bands. In spherical approximation, the atomic spectrum of the \(3d\)-shell depends on three Racah parameters 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. <ref>fig:ARPES</ref>. (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 reproduces the observed band anti-crossings. A Fermi-surface cut in the presence of the spin-orbit coupling is shown in Fig. <ref>fig:Fermisurface1</ref>. (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 a broken symmetry, e.g., to antiferromagnetism or to lattice disorder where 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. (B. Edegger, V.N. Muthukumar et al., 2007)

Atoms in optical lattices

The GWF can equally be applied to the Bose-Hubbard model which is suitable for ultracold bosonic atoms in optical traps. 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 confinement geometries.

References

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Bünemann, Jörg; Gebhard, Florian and Weber, Werner (1998). Multiband Gutzwiller wave functions for general on-site interactions. Physical Review B 57: 6896.
Bünemann, Jörg et al. (2003). Atomic correlations in itinerant ferromagnets: Quasi-particle bands of nickel. Europhysics Letters 61: 667.
Bünemann, Jörg; Gebhard, Florian and Thul, Rüdiger (2003). Landau-Gutzwiller quasiparticles. Physical Review B 67: 075103.
Bünemann(2007). Equivalence of Gutzwiller and slave-boson mean-field theories for multiband Hubbard models. Physical Review B 76: 193104.
Bünemann, Jörg et al. (2008). Spin-orbit coupling in ferromagnetic nickel. Physical Review Letters 102: 1.
Edegger, Bernd; Muthukumar, Vangal N and Gros, Claudius (2007). Gutzwiller-RVB theory of high temperature superconductivity: results from renormalised mean field theory and variational Monte Carlo calculations. Advances in Physics 56: 927.
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Jaksch, Dieter et al. (1998). Cold bosonic atoms in optical lattices. Physical Review Letters 81: 3108.
Kollar(2002). Exact analytic results for the Gutzwiller wave function with finite magnetization. Physical Review B 65: 155121.
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Vollhardt, Dieter (1984). Normal Helium-3: an almost localized Fermi liquid. Reviews of Modern Physics 56: 99.

Gutzwiller wave function

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