View source for Gutzwiller wave function

==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 [http://en.wikipedia.org/wiki/Hydrogen hydrogen] [http://en.wikipedia.org/wiki/Molecule molecule], <math>H_2\ ,</math>
where the two sites represent the protons. In a toy model description, the electrons can occupy only the <math>1s</math> orbital
of either proton with spin <math>\sigma=\uparrow,\downarrow\ .</math>
The four possible configurations are shown in  <figref>Gutzwiller_Wave_Function_GWF-hydrogen.jpg</figref>.
Naturally, the basis set is far too small for a quantitative description of the hydrogen molecule.
[[Image:Gutzwiller Wave Function GWF-hydrogen.jpg|thumb|300px|right|fig:two-siteHM|Four possible electronic configurations 
for the two-site hydrogen molecule.]]
The electrons can tunnel between the protons with amplitude <math>-t</math> (<math>t>0</math>), and the ground
state <math>|\Psi\rangle</math>
is a linear combination of the four configurations.
In the Hund-Mulliken [http://en.wikipedia.org/wiki/Molecular_orbital molecular-orbital] (MO) description, 
all four configurations have the same probability.
The MO ground state <math>|\Psi_0\rangle</math>
is incorrect because the configurations <math>|{\rm iii}\rangle</math>
and <math>|{\rm iv}\rangle</math> correspond to a negatively charged <math>H^-</math> atom
next to a bare proton (<math>H^+</math>). Such configurations where the <math>1s</math>-orbital is 
doubly occupied are energetically unfavorable because the We added
electrons on the same site repel each other due to their mutual 
[http://en.wikipedia.org/wiki/Electrostatic_force Coulomb interaction] (generally called the Hubbard <math>U\ ,</math> <math>U>0</math>). 
Therefore, the weight of doubly occupied orbitals is reduced in the 
ground state <math>|\Psi\rangle\ .</math> 

If one starts with some simple single-particle product wave function <math>|\Psi\rangle_0</math> like
the MO function, a better wave function <math>|\Psi\rangle</math>
can be obtained by reducing the weight of the configurations containing double occupancies, such
as <math>|{\rm iii}\rangle</math> and <math>|{\rm iv}\rangle</math> in  <figref>Gutzwiller_Wave_Function_GWF-hydrogen.jpg</figref>.
The [http://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanics operator] that
counts the number of doubly occupied sites is <math>\hat{D}</math> and
the ground state can be written in the form of a GWF [[#gutzwiller1963|(M.C. Gutzwiller, 1963)]],
:<math>\label{eq:def-Psi-hydro}
|\Psi\rangle = g^{\hat{D}}|\Psi_0\rangle \; ,
</math>

where <math>g=g(U)</math> is a function of the electrons' intra-orbital
Coulomb interaction. 
The toy model Hamiltonian is the two-site Hubbard model,
:<math>\label{eq:two-siteH}
\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) \; .
</math>

The MO ground state is <math>|\Psi_0\rangle=(|{\rm i}\rangle -|{\rm ii}\rangle +|{\rm iii}\rangle
+|{\rm iv}\rangle)/2</math> and the GWF reads
:<math>\label{eq:GWF-two-site}
|\Psi\rangle = \frac{1}{2} \left[ |{\rm i}\rangle -|{\rm ii}\rangle +g\left(|{\rm iii}\rangle +|{\rm iv}\rangle\right)\right] \;.
</math>

For <math>U\to\infty\ ,</math> <math>g\to 0\ ,</math> and 
<math>|\Psi\rangle</math> reduces to the Heitler-London [http://en.wikipedia.org/wiki/Valence_bond_theory valence-bond] (VB) state, 
which is a linear combination of the configurations
<math>|{\rm i}\rangle</math> and <math>|{\rm ii}\rangle</math> in  <figref>Gutzwiller_Wave_Function_GWF-hydrogen.jpg</figref>.
The GWF \eqref{eq:def-Psi-hydro} smoothly interpolates between
the Hund-Mulliken and the Heitler-London wave functions.

For the two-site Hubbard model, the variational energy is given by
:<math>\label{eq:2site-Evar}
E_{\rm var}(g) = -\frac{4tg}{1+g^2} + U \frac{g^2}{1+g^2} \; .
</math>

The variational minimum of the energy is obtained from the condition
<math>\partial E_{\rm var}(g)/\partial g=0\ ,</math> which is fulfilled for 
:<math>\label{eq:gopt-twosite}
g_{\rm opt}(U) = \frac{1}{4t}\left[ -U + \sqrt{U^2+(4t)^2\,}  \right] \; .
</math>

For the two-site Hubbard model, the variational upper bound becomes the exact ground-state energy,
:<math>\label{eq:groundstateenergy-twosite}
E_{\rm var}(g_{\rm opt}(U))=E_0(U)=\frac{1}{2} \left(U - \sqrt{U^2+(4t)^2\,}\right)\; .
</math>

For large interaction strengths, <math>U\gg 4t\ ,</math> the Gutzwiller parameter and the ground-state energy become small,
<math>g\to 2t/U\ ,</math> <math> E_0\to -4t^2/U\ ,</math> but they remain finite, of order <math>1/U\ .</math>
===Gutzwiller correlator===
Crystal lattices contain a large number <math>L</math> 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 <math>|\Psi_0\rangle</math>
is the lattice generalization of the Hund-Mulliken molecular-orbital (MO) wave function
(also known as [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock] wave function). 
For example, <math>|\Psi_0\rangle=|\hbox{FS}\rangle</math>
denotes the [http://en.wikipedia.org/wiki/Fermi_energy Fermi sea] of noninteracting electrons. 

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

The Gutzwiller correlator (GC) reduces the weight
of those configurations in <math>|\Psi_0\rangle</math>
which are energetically unfavorable, i.e., which have
a large atomic energy.
The general GWF is thus defined by [[#buenemannetal1998|(J. B&uuml;nemann, F. Gebhard, and W. Weber, 1998)]]  
:<math>\label{eq:def}
|\Psi_{\rm G}\rangle = \hat{P}_{\rm G}|\Psi_0\rangle \; .
</math>

The symbol <math>\hat{P}_{\rm G}</math> is an operator that modifies the single-particle
product wave function <math>|\Psi_0\rangle\ .</math>
The operator <math>\hat{m}_{\Gamma,\vec{R}}=
|\Gamma\rangle_{\vec{R}} {}_{\vec{R}}\langle \Gamma|</math> 
picks out a particular atom on site <math>\vec{R}</math> and then 
projects the atom onto the atomic 
configuration <math>\Gamma\ .</math>
The GC assigns
a weight factor <math>\lambda_{\Gamma,\vec{R}}</math> to each
atomic configuration,
:<math>\label{eq:def-PG1}
\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 \eqref{eq:def-Psi-hydro} for the one-band model is recovered for <math>\lambda_{1}=\lambda_{2}=\lambda_{3}=1</math> and 
<math>\lambda_{4}=g</math> for the four hydrogen atomic states <math>\bigl\{ H^+,H_{\uparrow},H_{\downarrow}, H^-\bigr\}\ .</math>

The Gutzwiller correlator suppresses
charge fluctuations which are too large in <math>|\Psi_0\rangle\ .</math>
For example, a [http://en.wikipedia.org/wiki/Fermi_energy Fermi-gas] description of the <math>3d</math>-electrons 
of [http://en.wikipedia.org/wiki/Nickel nickel] employs the Fermi-sea ground state.
It predicts a certain probability to detect
five-fold ionized nickel ions <math>\hbox{Ni}^{5+}</math>
in a nickel crystal. The energy <math>E_{{\rm Ni}^{5+}}</math> 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 <math>\hbox{Ni}^{5+}</math> 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 <math>3d</math>-shell and the <math>4s</math>-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 [http://en.wikipedia.org/wiki/Superfluid superfluid] [http://en.wikipedia.org/wiki/Helium-4 Helium-4].
===Expectation values and variational ground-state energy===
Physical quantities such as the magnetization <math>M_z</math>
are determined by [http://en.wikipedia.org/wiki/Expectation_value_(quantum_mechanics) expectation values] of corresponding
quantum-mechanical operators <math>\hat{M_z}\ .</math>
Quite generally, the Gutzwiller variational expressions for expectation values of observables are 
:<math>\label{eq:expectationvalues}
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 <math>|\Psi_0\rangle</math> and <math>\hat{P}_{\rm G}\ ,</math>
the variational energy <math>E_{\rm var}=\langle\hat{H}\rangle</math>
for a given model [http://en.wikipedia.org/wiki/Hamiltonian_operator Hamiltonian], e.g., 
for the Hubbard model
(read a [[Gutzwiller wave function/historical note|historical note]] on its invention), must be calculated.
Then, <math>E_{\rm var}</math> must be minimized with respect
to all variational parameters to find 
the optimal variational energy, <math>E_{\rm var}^{\rm opt}\ .</math>