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arXiv:0801.2280 · Physical Review B

Grand canonical Gutzwiller approximation for magnetic inhomogeneous systems

Noboru Fukushima

Published Sep 4, 2008• 45 citations

Abstract

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 $d$-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 $s$-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.

DOI: 10.1103/physrevb.78.115105

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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

d

-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

s

-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.

The generalization of the Gutzwiller approximation to inhomogeneous systems is considered, with extra spin-and-site-dependent fugacity factors included. It is found that the inclusion of fugacity factors reconciles the seemingly contradictory choices of Gutzwiller factors used in the literature. Moreover, from the derivation of the Gutzwiller factors, it is shown that the Gutzwiller approximation breaks the rotational symmetry of the trial wave functions and that different components of the spin-spin interaction need to be renormalized differently under the approximation. Various schemes to restore the rotational symmetry are discussed and are compared with results from variational Monte Carlo calculations for the two-dimensional square-lattice antiferromagnet. Results along different paths within the full parameter space, which correspond to different choices of fugacity factors in the literature, are also compared.

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 description of electronic inhomogeneities in transition metal compounds requires a technique which is able to deal with the excitations of an inhomogeneous, strongly correlated system. In this talk I will outline the formalism of the timedependent Gutzwiller approximation [1] for the Hubbard model which allows for the computation of random phase approximation (RPA) like fluctuations on top of the Gutzwiller approximation (GA). The performance of the method is demonstrated by comparing static quantities like ground state energy and double occupancies as well as dynamical correlation functions in the charge, spin and pair channels with HartreeFock+RPA and exact results. I will further show that the method gives a good account of several spectroscopic quantities related to stripes in highTc superconductors. A major part of the talk will address the recent generalization of the method to the case of multiple bands [2,3]. Contrary to the standard RPA, the incorporation of the local multipletstructure within our approach leads to a much larger sensitivity of ferromagnetism on the the ratio between Hund coupling and intraorbital repulsion J/U. Besides, we find that the Brinkman Rice localization transition, which in the paramagnetic GA appears at integer doping is always masked by an antiferromagnetic or an orbitally ordered ground state. In addition, a quantum Lifshitz point is found at larger values of J/U which corresponds to the common boundary of ferro, para and incommensurate magnetic phases. [1] G. Seibold and J. Lorenzana, Phys. Rev. Lett. 86, 2605 (2001) [2] E. von Oelsen, G. Seibold, and J. Bunemann, New J. Phys. 13, 113031 (2011) [3] E. v. Oelsen, G. Seibold, and J. Bunemann, Phys. Rev. Lett. 107, 076402 (2011)

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.

A systematic diagrammatic expansion for Gutzwiller wavefunctions (DE-GWFs) proposed very recently is used for the description of the superconducting (SC) ground state in the two-dimensional square-lattice t – J model with the hopping electron amplitudes t (and ) between nearest (and next-nearest) neighbors. For the example of the SC state analysis we provide a detailed comparison of the methodʼs results with those of other approaches. Namely, (i) the truncated DE-GWF method reproduces the variational Monte Carlo (VMC) results and (ii) in the lowest (zeroth) order of the expansion the method can reproduce the analytical results of the standard Gutzwiller approximation (GA), as well as of the recently proposed 'grand-canonical Gutzwiller approximation' (called either GCGA or SGA). We obtain important features of the SC state. First, the SC gap at the Fermi surface resembles a wave only for optimally and overdoped systems, being diminished in the antinodal regions for the underdoped case in a qualitative agreement with experiment. Corrections to the gap structure are shown to arise from the longer range of the real-space pairing. Second, the nodal Fermi velocity is almost constant as a function of doping and agrees semi-quantitatively with experimental results. Third, we compare the doping dependence of the gap magnitude with experimental data. Fourth, we analyze the k -space properties of the model: Fermi surface topology and effective dispersion. The DE-GWF method opens up new perspectives for studying strongly correlated systems, as it (i) works in the thermodynamic limit, (ii) is comparable in accuracy to VMC, and (iii) has numerical complexity comparable to that of the GA (i.e., it provides the results much faster than the VMC approach).

We analyze the electronic Green's functions in the superconducting ground state of the $t\text{\ensuremath{-}}J$ model using Gutzwiller-projected wave functions, and compare them to the conventional BCS form. Some of the properties of the BCS state are preserved by the projection: the total spectral weight is continuous around the quasiparticle node and approximately constant along the Fermi surface. On the other hand, the overall spectral weight is reduced by the projection with a momentum-dependent renormalization, and the projection produces electron-hole asymmetry in renormalization of the electron and hole spectral weights. The latter asymmetry leads to the bending of the effective Fermi surface, which we define as the locus of equal electron and hole spectral weight.