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arXiv:cond-mat/0503143 · Physical Review B

Evaluation of matrix elements in partially projected wave functions

Noboru Fukushima, Bernhard Edegger, V. N. Muthukumar, Claudius Gros

Published Oct 7, 2005• 13 citations
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Abstract

We generalize the Gutzwiller approximation scheme to the calculation of nontrivial matrix elements between the ground state and excited states. In our scheme, the normalization of the Gutzwiller wave function relative to a partially projected wave function with a single nonprojected site (the reservoir site) plays a key role. For the Gutzwiller projected Fermi sea, we evaluate the relative normalization both analytically and by variational Monte Carlo (VMC). We also report VMC results for projected superconducting states that show novel oscillations in the hole density near the reservoir site.

DOI
10.1103/physrevb.72.144505

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We generalize the Gutzwiller approximation scheme to the calculation of nontrivial matrix elements between the ground state and excited states. In our scheme, the normalization of the Gutzwiller wave function relative to a partially projected wave function with a single nonprojected site (the reservoir site) plays a key role. For the Gutzwiller projected Fermi sea, we evaluate the relative normalization both analytically and by variational Monte Carlo (VMC). We also report VMC results for projected superconducting states that show novel oscillations in the hole density near the reservoir site.
We use a generalized Gutzwiller approximation elaborated to evaluate matrix elements with partially projected wave functions and formerly applied to homogeneous systems [N. Fukushima et al., Phys. Rev. B 72, 144505 (2005); B. Edegger et al., Adv. Phys. 56, 927 (2007)]. In the present paper we consider projected single-particle (hole) excitations for electronic systems with antiferromagnetic order and obtain the corresponding tunneling probabilities. The accuracy and the reliability of our analytical approximation is tested using the variational Monte Carlo method. Possible comparisons with experimental results are also discussed.
A new approach is presented for variational Monte Carlo simulations of Hubbard-Anderson models using Gutzwiller projection operators to restrict charge fluctuations in noninteracting wave functions. The Gutzwiller matrix elements are mapped onto a statistical model, which is evaluated using a stochastic algorithm that is normally applied to finite-temperature simulation. The method has been successfully tested for one-dimensional (1D) Hubbard and 1D Anderson models, and results are presented for the 2D Hubbard model. We find a region close to the half-filled limit where the Gutzwiller states of different magnetic ordering are very close in energy.
We use a variational Monte Carlo method (VMC) and a quantum Monte Carlo method to study the two-dimensional half-filled Hubbard model. We use the Gutzwiller wave function as a trial wave function in VMC and measure energy, correlation functions, and momentum distribution for lattices up to 12\ifmmode\times\else\texttimes\fi{}12 sites. We then use a Monte Carlo algorithm to study the Hubbard model with the same boundary conditions as in VMC at very low temperatures. By comparing the results from these two methods, we assess the accuracy of the Gutzwiller wave function as a representation of the ground state of the Hubbard model. Major differences are found, implying that the Gutzwiller wave function is not a useful approximation to the ground state of the Hubbard model in two dimensions.
The effects of the Gutzwiller projection on a BCS wave function with varying particle number are considered. We show that a fugacity factor has to be introduced in these wave functions when they are Gutzwiller projected, and derive an expression for this factor within the Gutzwiller approximation. We examine the effects of the projection operator on BCS wave functions by calculating the average number of particles before and after projection. We also calculate particle number fluctuations in a projected BCS state. Finally, we point out the differences between projecting BCS wave functions in the micro- and grand-canonical schemes, and discuss the relevance of our results for variational Monte Carlo studies.
The Gutzwiller wave function for the three-band Hubbard model on a two-dimensional CuO2{\mathrm{CuO}}_{2}CuO2​ plane is studied by using the variational Monte Carlo (VMC) method in the limit Ud{\mathit{U}}_{\mathit{d}}Ud​\ensuremath{\rightarrow}\ensuremath{\infty}, where Ud{\mathit{U}}_{\mathit{d}}Ud​ is the Coulomb repulsion between holes on a Cu site. The VMC results for the energy and the fraction of d holes are compared with those of the Rice-Ueda type Gutzwiller approximation (GA). The difference between the VMC and the GA results are most pronounced at half-filling, and away from half-filling the two results agree well in both the hole-doped and the electron-doped cases. The doping dependence of the momentum distribution function is also studied.
The Gutzwiller approximation (GA) for Gutzwiller-projected grand canonical wave functions with fugacity factors is investigated in detail. Our systems in general contain inhomogeneity and local magnetic moments. In deriving renormalization formulas, we also derive or estimate terms of higher powers of intersite contractions neglected in the conventional GA. We examine several different constraints, i.e., local/global spin-dependent/independent particle-number conservation. Out of the four, the local spin-dependent constraint seems the most promising at present. An improved GA derived from it agrees with the variational Monte Carlo method better than the conventional GA does. The corrections to the conventional GA can be interpreted as two-site correlation including the phase difference of configurations. Furthermore, projected quasiparticle excited states are orthogonal to each other within the GA. Using these states, spectral weights are calculated. We show that asymmetry between electron addition and removal spectra can appear by taking into account the higher powers of the intersite contractions in the case of the ddd-wave superconductors and the Fermi sea; the addition is smaller than the removal. However, the asymmetry is quite weak especially near the Fermi level. In contrast, projected sss-wave superconductors can have the opposite asymmetry (addition larger than removal) especially near the Fermi level. In addition, formulas from the other three constraints are also derived, which may be useful depending on purposes.