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Physics Procedia

Stability of Staggered Flux State in d-p Model Studied Using Variational Monte Carlo Method

Shun Tamura, Hisatoshi Yokoyama

Published Jan 1, 2015• 0 citations
Publisher

Abstract

The stability of a staggered flux (SF) or d-density wave state is studied in a d-p model using a variational Monte Carlo (VMC) method. This state possesses a local circular current and possibly causes the pseudogap phase in high-Tc cuprate superconductors. We introduce into the trial function a configuration dependent phase factor, which was recently shown to be indispensable to stabilize current-carrying states. We pay attention to the energy gain as a function of adjacent O-O hopping and Cu-site coulomb repulsion U. It is found the SF state becomes stable for hole doping rates of and , owing to the reduction in kinetic energy, but seems fragile because the energy gain rapidly decays on doping holes for any . The loop current tends to increase as increases.

DOI
10.1016/j.phpro.2015.05.092

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The stability of a staggered flux (SF) or d-density wave state is studied in a d-p model using a variational Monte Carlo (VMC) method. This state possesses a local circular current and possibly causes the pseudogap phase in high-Tc cuprate superconductors. We introduce into the trial function a configuration dependent phase factor, which was recently shown to be indispensable to stabilize current-carrying states. We pay attention to the energy gain as a function of adjacent O-O hopping and Cu-site coulomb repulsion U. It is found the SF state becomes stable for hole doping rates of and , owing to the reduction in kinetic energy, but seems fragile because the energy gain rapidly decays on doping holes for any . The loop current tends to increase as increases.
With cuprate superconductors in mind, we check whether the d -wave superconducting ( d SC) state coexists with or excludes the staggered flux (SF) state, which was revealed to be a strong candidate for the pseudogap state. To this end, we use a variational Monte Carlo method for the square-lattice t - J model with diagonal transfer t '. We construct a trial wave function which can simultaneously represent d SC and SF orders for a continuous description of their interplay. Furthermore, a band renormalization effect is introduced into both orders independently. It is found that the SF state does not coexist with d SC and is unstable for a wide area in the parameter space. However, a strange state having both SF and antiferromagnetic orders is found to be stable for very low doping rate.
With cuprate superconductors in mind, we check whether the d -wave superconducting ( d SC) state coexists with or excludes the staggered flux (SF) state, which was revealed to be the strong candidate for the pseudogap state. To this end, we use a variational Monte Carlo method for the square Hubbard model with diagonal transfer t '. In the trial wave function, d SC and SF orders coexist, which allows a continuous description of their interplay; essential factors for Mott physics and band renormalization are also included. It is found that the SF state does not coexist with d SC and is unstable for U/t = 12 regardless of the value of t'/t within the present work.
Abstract In connection with the symmetry-breaking phenomena found in cuprate superconductors, the instability toward x−y and (x + y)−(x − y) anisotropy (Pomeranchuk instability) is studied for a strongly correlated Hubbard model (U/t = 12) on a square lattice with the next-nearest-neighbor transfer t′ using a variational Monte Carlo method. As a variational function, a staggered flux state ΨSF is considered, which is a candidate for the pseudogap state in cuprates. ΨSF is stabilized in a underdoped regime and reduced to the normal state (projected Fermi sea) for larger doping area. By analyzing ΨSF for t′/t = 0, ±0.3, we argue that the appearance of Pomeranchuk instability is limited to the normal state with t′/t = 0 and 0.08 ≤ δ ≤ 0.20 (σ: doping rate) in the x−y anisotropy. Band-renormalization effects (Fermi-surface modification) owing to strong correlations play a crucial role for the argument.
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.
Kinetic and magnetic energies of the staggered flux phase, with a fictitious flux of equal magnitude but opposite direction in adjacent square plaquettes, are calculated for the two-dimensional t-J model using the variational Monte Carlo method. They are compared to the energies of the resonating-valence-bond state, the flux phase with half a quantum per plaquette, and the projected Fermi-liquid state. For about 10% hole concentration the staggered flux phase has the lowest energy of the nonsuperconducting states, but its energy is still higher than that of the superconducting d-wave state.