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arXiv:cond-mat/0506120 · Physical Review B

Particle number renormalization in nearly half-filled Mott Hubbard superconductors

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

Published Oct 5, 2005• 24 citations
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Abstract

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.

DOI
10.1103/physrevb.72.134504

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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.
Gutzwiller-projected BCS wave function or the resonating-valence-bond (RVB) state in the two-dimensional (2D) extended $t\ensuremath{-}J$ model is investigated by using the variational Monte Carlo technique. We show that the results of ground-state energy and excitation spectra calculated in the grand-canonical scheme allowing particle number to fluctuate are essentially the same as previous results obtained by fixing the number of particle in the canonical scheme if the grand thermodynamic potential is used for minimization. To account for the effect of Gutzwiller projection, a fugacity factor proposed by Laughlin and Anderson a few years ago has to be inserted into the coherence factor of the BCS state. Chemical potential, particle number fluctuation, and phase fluctuation of the RVB state, difficult or even impossible to be calculated in the canonical ensemble, have been directly measured in the grand-canonical picture. We find that except for $La\ensuremath{-}214$ materials, the doping dependence of chemical potential is consistent with experimental findings on several cuprates. Similar to what has been reported by scanning tunneling spectroscopy experiments, the tunneling asymmetry becomes much stronger as doping decreases. We found a very large enhancement of phase fluctuation in the underdoped regime.
With the use of Gutzwiller-projected variational states, we study the renormalization of the current carried by the quasiparticles in high-temperature superconductors and of the quasiparticle spectral weight. The renormalization coefficients are computed by the variational Monte Carlo technique, under the assumption that quasiparticle excitations may be described by Gutzwiller-projected BCS quasiparticles. We find that the current renormalization coefficient decreases with decreasing doping and tends to zero at zero doping. The quasiparticle spectral weight Z+{Z}_{+}Z+​ for adding an electron shows an interesting structure in k\mathbf{k}k space, which corresponds to a depression of the occupation number nk{n}_{\mathbf{k}}nk​ just outside the Fermi surface. The perturbative corrections to those quantities in the Hubbard model are also discussed.
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 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 authors consider the variational Monte Carlo method of studying fermion systems in condensed matter physics. This method is a powerful tool for studying the properties of interacting electrons which are responsible for phenomena like the new high-Tc superconductors. Using a parallel algorithm for updating the determinants of large complex matrices on transputer arrays they study the energy and spin correlations for a 12*12 and a 22*22 lattice. Using the Hubbard model and the Gutzwiller projected wavefunction they have evaluated these quantities as functions of band filling and coupling constant. They find that although the Gutzwiller state is stable for densities near half filling it is not at low densities. For the stable states they find antiferromagnetic correlations. Statistically the thermodynamic limit is already attained with a 12*12 lattice.
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.