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arXiv:1110.6399 · Physical Review B

Grand canonical variational approach for the t−J model

Chung-Pin Chou, Fan Yang, Ting-Kuo Lee

Published Feb 23, 2012• 6 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.85.054510

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Gutzwiller-projected BCS wave function or the resonating-valence-bond (RVB) state in the two-dimensional (2D) extended t\ensuremath−Jt\ensuremath{-}Jt\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−214La\ensuremath{-}214La\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.
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.
We investigate analytically a connection between the t-J model and the strongly correlated Bardeen-Cooper-Schrieffer (BCS) Hamiltonian, with the effect of strong electron correlations accounted by the Gutzwiller projection. We show that in the immediate vicinity of half filling the projected 2D BCS Hamiltonian with strong pairing develops an antiferromagnetically (AF) ordered ground state. This result explicitly demonstrates that antiferromagnetism in this model appears as a natural consequence of the strong Coulomb repulsion in a low doped regime. At moderate doping the ground state of the Gutzwiller-projected BCS Hamiltonian becomes qualitatively similar to Anderson's resonating valence bond state which is known to fit nicely the properties of the t-J model in this regime. These two properties taken together indicate that the projected BCS Hamiltonian captures the essential low-energy physics of the t-J model in the whole underdoped region.
We investigate a connection between the $t\text{\ensuremath{-}}J$ model and the strongly correlated Bardeen-Cooper-Schrieffer (BCS) Hamiltonian, with the effect of strong electron correlations accounted by the Gutzwiller projection. We show that in the immediate vicinity of half filling the projected two-dimensional BCS Hamiltonian with strong pairing develops an antiferromagnetically ordered ground state. This result explicitly demonstrates that antiferromagnetism in this model appears as a natural consequence of the strong Coulomb repulsion in a low doped regime. At moderate doping the ground state of the Gutzwiller-projected BCS Hamiltonian is, in finite-system studies, known to become qualitatively similar to Anderson's resonating valence bond state which, in turn, fits nicely with the properties of the $t\text{\ensuremath{-}}J$ model in that regime. Combined together, these two properties indicate that the projected BCS Hamiltonian at least qualitatively captures the essential low-energy physics of the $t\text{\ensuremath{-}}J$ model in the underdoped region.
Motivated by the observation of inhomogeneous patterns in some high-Tc{T}_{c}Tc​ cuprate compounds, several variational Gutzwiller-projected wave functions with built-in charge and bond-order parameters are proposed for the extended $t\text{\ensuremath{-}}J\text{\ensuremath{-}}V$ model on the square lattice at low doping. First, following a recent Gutzwiller-projected mean-field approach by one of us [D. Poilblanc, Phys. Rev. B 72, 060508(R) (2005)], we investigate, as a function of doping and Coulomb repulsion the relative stability of a wide variety of modulated structures with square unit cells of size $2\ifmmode\times\else\texttimes\fi{}2$, $\sqrt{8}\ifmmode\times\else\texttimes\fi{}\sqrt{8}$, $4\ifmmode\times\else\texttimes\fi{}4$, and $\sqrt{32}\ifmmode\times\else\texttimes\fi{}\sqrt{32}$. It is found that the $4\ifmmode\times\else\texttimes\fi{}4$ bond-order wave function with staggered-flux pattern (and small charge and spin current density wave) is a remarkable competitive candidate for hole doping around 1∕81∕81∕8 in agreement with scanning tunneling microscopy observations in the underdoped regime of some cuprates. This wave function is then optimized accurately and its properties studied extensively using a variational Monte Carlo scheme. Moreover, we find that under increasing the Coulomb repulsion, the ddd-wave superconducting RVB wave function is rapidly destabilized with respect to the $4\ifmmode\times\else\texttimes\fi{}4$ bond-order wave function. The stability of the bond-modulated wave function is connected to a gain of Coulomb and exchange energies. We suggest that such ordering patterns could be dynamical or could spontaneously appear in the vicinity of an impurity or a vortex in the mixed phase of the cuprates. Finally, we consider also a commensurate flux phase, but this wave function turns out not to be competitive because of its rather poor kinetic energy. However, we find it has very competitive exchange and Coulomb energies.
Superconductivity is usually described in the framework of the Bardeen-Cooper-Schrieffer (BCS) wavefunction, which even includes the resonating-valence-bond (RVB) wavefunction proposed for the high-temperature superconductivity in the cuprate. A natural question is ififif any fundamental physics could be possibly missed by applying such a scheme to strongly correlated systems. Here we study the pairing wavefunction of two holes injected into a Mott insulator/antiferromagnet in a two-leg ladder using variational Monte Carlo approach. By comparing with density-matrix renormalization group (DMRG) calculation, we show that a conventional BCS or RVB pairing of the doped holes makes qualitatively wrong predictions and is incompatible with the fundamental pairing force in the $t\text{\ensuremath{-}}J$ model, which is kinetic-energy driven by nature. By contrast, a non-BCS-like wavefunction incorporating such novel effect will result in a substantially enhanced pairing strength and improved ground state energy as compared to the DMRG results. We argue that the non-BCS form of such a new ground state wavefunction is essential to describe a doped Mott antiferromagnet at finite doping.
We carry out the variational Monte Carlo calculation to examine spatially inhomogeneous states in hole- and electron-doped cuprates. By using Gutzwiller approximation, we consider the excitations, arising from charge density, spin density, and pair field, of the mean-field ground state of the ttt-JJJ model. It leads to the stripe patterns we have found numerically in a generalized ttt-JJJ-type model including mass renormalization from the electron-phonon coupling. In the hole-doped side, a robust ddd-wave superconducting order results in the formation of the half-doped antiferromagnetic resonating-valence-bond (AF-RVB) stripes shown by the well-known Yamada plot. On the other hand, due to a long-range AF order in electron-doped materials, a stripe structure with the ``in-phase'' magnetic domain (IPMD) is obtained in the underdoped regime instead of the AF-RVB stripe. The IPMD stripe with the largest period permitted by lattice size is stabilized near the underdoped region and it excludes the Yamada plot from electron-doped cases. Based on finite lattice size to which we can reach, the existence of IPMD stripes may imply an electronic phase separation into an electron-rich and an insulating half-filled AF long-range ordered domains in electron-doped compounds.