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Hole doped three-band \emph{d-p} model: Superconducting properties in view of minimal-size real-space $d$-wave pairing operators by means of Variational Monte-Carlo method

Andrzej Biborski, M. Zegrodnik, Józef Spałek

Published Mar 6, 2020• 0 citations
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Abstract

The three-band \emph{d-p} model is investigated by means of Variational Monte-Carlo method with the BCS-like wave-function supplemented with the Gutzwiller and Jastrow correlators. Optimization by means of Variational Monte-Carlo approach has lead to reproduction of the $d$-$wave$ superconducting state with the characteristic dome-like shape of the order parameter for hole doping $\delta \lesssim 0.4$ with a good agreement with both the diagrammatic expansion of Gutzwiller wave-function (DE-GWF) method and the experimental observations. The nature of the $d$-$wave$ pairing is also investigated by means of recently proposed \emph{Minimal-size real-space $d$-wave pairing operators} [A. Moreo and E. Dagotto, Phys. Rev. B \textbf{100}, 214502 (2019)]. Simultaneous emergence of the long-range superconducting ordering observed by means of the correlation functions is reported. Additionally, the trial wave-function is used to investigate the magnetic properties of the system. The analysis of spin-spin correlation functions is carried out, resulting in the evidence of antiferromagnetic $\mathbf{q}=(\pi,\pi)$, short-range order, as expected. For the sake of completeness, estimation of charge gap resulted in $\Delta_{CG}\approx1.2\text{eV}$, which is close to values reported experimentally for cuprates.

Related papers

Superconducting properties of the hole-doped three-band d−p model studied with minimal-size real-space d -wave pairing operators

Andrzej Biborski, M. Zegrodnik, Józef Spałek

The three-band d-p model is investigated by means of a variational Monte Carlo (VMC) method with the BCS-like wave-function supplemented by the Gutzwiller and Jastrow correlators. The VMC optimization leads to a $d\text{\ensuremath{-}}\mathit{wave}$ superconducting state with a characteristic domelike shape of the order parameter for hole doping $\ensuremath{\delta}\ensuremath{\lesssim}0.4$, in good agreement with the experimental observations. Also, the off-diagonal pair-pair correlation functions, calculated within VMC, vindicates the results obtained very recently within the diagrammatic expansion of the Gutzwiller wave function method [cf. Zegrodnik, Biborski, Fidrysiak, and Spa\l{}ek, Phys. Rev. B 99, 104511 (2019)]. Subsequently, the nature of the $d\text{\ensuremath{-}}\mathit{wave}$ pairing is investigated by means of recently proposed minimal-size real-space d-wave pairing operators [Moreo and Dagotto, Phys. Rev. B 100, 214502 (2019)]. An emergence of the long-range superconducting ordering for both $d$ and $p$ orbitals is reported by analyzing the corresponding off-diagonal pair-pair correlation functions. The dominant character of d-wave pairing on $d$ orbitals is confirmed. Additionally, the trial wave function is used to investigate the magnetic properties of the system. The analysis of spin-spin correlation functions is carried out and shows antiferromagnetic $\mathbf{q}=(\ensuremath{\pi},\ensuremath{\pi})$, short-range order, as expected. For the sake of completeness, the charge gap has been estimated, which for the parent compound takes the value ${\mathrm{\ensuremath{\Delta}}}_{\text{CG}}\ensuremath{\approx}1.78\ifmmode\pm\else\textpm\fi{}0.51\phantom{\rule{4.pt}{0ex}}\text{eV}$, and agrees with values reported experimentally for the cuprates.

arXiv:2003.03117•Physical review. B./Physical review. B•Jun 4, 2020•12 citations

Correlated-Electron Systems and High-Temperature Superconductivity

Takashi Yanagisawa, Mitake Miyazaki, Yamaji K.

We present recent theoretical results on superconductivity in correlated-electron systems, especially in the two-dimensional Hubbard model and the three-band d-p model. The mechanism of superconductivity in high-temperature superconductors has been extensively studied on the basis of various electronic models and also electron-phonon models. In this study, we investigate the properties of superconductivity in correlated-electron systems by using numerical methods such as the variational Monte Carlo method and the quantum Monte Carlomethod. The Hubbard model is one of basic models for strongly correlated electron systems, and is regarded as the model of cuprate high temperature superconductors. The d-p model is more realistic model for cuprates. The superconducting condensation energy obtained by adopting the Gutzwiller ansatz is in reasonable agreement with the condensation energy estimated for YBa2Cu3O7. We show the phase diagram of the ground state using this method. We have further investigated the stability of striped and checkerboard states in the under-doped region. Holes doped in a half-filled square lattice lead to an incommensurate spin and charge density wave. The relationship of the hole density x and incommensurability δ, δ~x, is satisfied in the lower doping region, as indicated by the variationalMonte Carlocalculations for the two-dimensional Hubbard model. A checkerboard-like charge-density modulation with a roughly period has also been observed by scanning tunneling microscopy experiments in Bi2212 and Na-CCOC compounds. We have performed a variational Monte Carlo simulation on a two-dimensional t-t′-t″- U Hubbard model with a Bi-2212 type band structure and found that the period checkerboard spin modulation, that is characterized by multi Q vectors, is indeed stabilized. We have further performed an investigation by using a quantumMonte Carlomethod, which is a numerical method that can be used to simulate the behavior of correlated electron systems. We present a new algorithm of the quantum Monte Carlo diagonalization that is a method for the evaluation of expectation value without the negative sign problem. We compute pair correlation functions and show that pair correlation is indeed enhanced with hole doping.

arXiv:1307.2456•Journal of Modern Physics•Jan 1, 2013•23 citations

Antiferromagnetism, superconductivity and phase diagram in the three-band model of high-temperature cuprates

Takashi Yanagisawa

Abstract We investigate the ground state of the two-dimensional three-band d-p model for for high-temperature superconductors on the basis of a variational Monte Carlo method. We employ an optimization variational Monte Carlo method that we have developed recently. The many-body wave function is improved beyond the Gutzwiller ansatz by adopting the wave function in the form ψ = exp(−S)ψG where ψG represents the Gutzwiller function and S is a kinetic operator. The strong magnetic correlation and also superconductivity (SC) are induced by the on-site Coulomb repulsive interaction. It is important to clarify the phase diagram that includes superconductive phase and antiferromagnetic phase. We show the phase diagram to show the antiferromagnetic region in the parameter space. High-temperature superconductivity may occur in the strongly correlated region near the antiferromagnetic boundary.

Journal of Physics Conference Series•Sep 1, 2019•1 citation

Superconductivity in high- Tc and related strongly correlated systems from variational perspective: Beyond mean field theory

Józef Spałek, Maciej Fidrysiak, M. Zegrodnik +1 more

The principal purpose of this topical review is to single out some of the universal features of high-temperature (high-Tc) and related strongly-correlated systems, which can be compared with experiment in a quantitative manner. The description starts with the concept of exchange-interaction-mediated (real-space) pairing, combined with strong correlations among narrow band electrons, for which the reference state is that of the Mott–Hubbard insulator. The physical discussion of concrete properties relies on variational approach, starting from generalized renormalized mean-field theory (RMFT) in the form of statistically-consistent Gutzwiller approximation (SGA), and its subsequent generalization, i.e., a systematic Diagrammatic Expansion of the Variational (Gutzwiller-type) Wave Function (DE-GWF). The solution leads to the two energy scales, one involving unconventional quasiparticles close to the Fermi energy, and the other reflecting the fully correlated state, involving electrons deeper below the Fermi surface. Those two regimes are separated by a kink in the dispersion relation, which is observed in photoemission. As a result, one obtains both the doping dependent properties in the correlated state, and effective renormalized quasiparticles. The reviewed ground-state characteristics for high-Tc systems encompass high-Tc superconductivity, nematicity, charge- (and pair-) density-wave effects, as well as non-BCS kinetic energy gain in the paired state. The calculated dynamic properties are: the universal Fermi velocity, the Fermi wave-vector, the effective mass enhancement, the pseudogap; and the d-wave gap magnitude. We discuss that, within the variational approach, the minimal realistic model is represented by the so-called t-J-U Hamiltonian. Inadequacy of the t-J and Hubbard models, particularly of their RMFT versions, is discussed explicitly. For heavy fermion systems, modeled by the Anderson lattice model and with the DE-GWF approach, we discuss the phase diagram, encompassing superconducting, Kondo insulating, ferro- and anti-ferromagnetic states. The superconducting state is then a two-d-wave gap system. If the orbital degeneracy of f-electrons is included, the coexistent ferromagnetic- (spin–triplet) superconducting phases appear and match those observed for UGe2 in a semiquantitative manner. Finally, in the second part, we generalize our approach to the collective spin and charge fluctuations in high-Tc systems, starting from variational approach, defining the saddle-point state, and combining it with 1/N expansion. The present scheme differs essentially from that starting from the saddle-point Hartree–Fock approximation, and incorporating the fluctuations in the random phase approximation (RPA). The spectrum of collective spin and charge excitations is determined for the Hubbard and t-J-U models, and subsequently compared quantitatively with recent experiments. The Appendices provide formal details to make this review self-contained.

arXiv:2202.05113•Physics Reports•Mar 17, 2022•46 citations

Phase diagram of the three-band d-p model based on the optimization variational Monte Carlo method

Takashi Yanagisawa, Mitake Miyazaki, Yamaji K.

The phase diagram of cuprate high-temperature superconductors is investigated on the basis of the three-band d-p model. We use the optimization variational Monte Carlo method, where improved many-body wave functions have been proposed to make the ground-state wave function more precise. We investigate the stability of antiferromagnetic state by changing the band parameters such as the hole number, level difference $Δ_{dp}$ between $d$ and $p$ electrons and transfer integrals. We show that the antiferromagnetic correlation weakens when $Δ_{dp}$ decreases and the pure $d$-wave superconducting phase may exist in this region. We present phase diagrams including antiferromagnetic and superconducting regions by varying the band parameters. The phase diagram obtained by changing the doping rate $x$ contains antiferromagnetic, superconducting and also phase-separated phases. We propose that high-temperature superconductivity will occur near the antiferromagnetic boundary in the space of band parameters.

arXiv:2105.00464•May 2, 2021•0 citations

Universal properties of high-temperature superconductors from real-space pairing: Role of correlated hopping and intersite Coulomb interaction within the t−J−U model

M. Zegrodnik, Józef Spałek

We study the effect of the correlated hopping term and the intersite Coulomb interaction term on principal features of the $d$-wave superconducting (SC) state, in both the electron- and hole-doped regimes within the $t\text{\ensuremath{-}}J\text{\ensuremath{-}}U$ model. In our analysis, we use the approach based on the diagrammatic expansion of the Gutzwiller wave function (DE-GWF), which allows us to go beyond the renormalized mean-field theory (RMFT). We show that the correlated hopping term enhances the pairing at the electron-doped side of the phase diagram. Moreover, the so-called non-BCS regime (which manifests itself by the negative kinetic energy gain at the transition to the SC phase) is narrowed down with the increasing magnitude of the correlated hopping $\ensuremath{\sim}K$. Also, the doping dependencies of the nodal Fermi velocity and Fermi momentum, as well as the average number of double occupancies, are analyzed with reference to the experimental data for selected values of the parameter $K$. For the sake of completeness, the influence of the intersite Coulomb repulsion on the obtained results is provided. Additionally, selected results concerning the Hubbard-model case are also presented. A complete model with all two-site interactions is briefly discussed in Appendix for reference.

arXiv:1705.06038•Physical review. B./Physical review. B•Aug 14, 2017•29 citations
Hole doped three-band \emph{d-p} model: Superconducting properties in view of minimal-size real-space ddd-wave pairing operators by means of Variational Monte-Carlo method
The three-band \emph{d-p} model is investigated by means of Variational Monte-Carlo method with the BCS-like wave-function supplemented with the Gutzwiller and Jastrow correlators. Optimization by means of Variational Monte-Carlo approach has lead to reproduction of the ddd-wavewavewave superconducting state with the characteristic dome-like shape of the order parameter for hole doping δ≲0.4\delta \lesssim 0.4δ≲0.4 with a good agreement with both the diagrammatic expansion of Gutzwiller wave-function (DE-GWF) method and the experimental observations. The nature of the ddd-wavewavewave pairing is also investigated by means of recently proposed \emph{Minimal-size real-space ddd-wave pairing operators} [A. Moreo and E. Dagotto, Phys. Rev. B \textbf{100}, 214502 (2019)]. Simultaneous emergence of the long-range superconducting ordering observed by means of the correlation functions is reported. Additionally, the trial wave-function is used to investigate the magnetic properties of the system. The analysis of spin-spin correlation functions is carried out, resulting in the evidence of antiferromagnetic q=(π,π)\mathbf{q}=(\pi,\pi)q=(π,π), short-range order, as expected. For the sake of completeness, estimation of charge gap resulted in ΔCG≈1.2eV\Delta_{CG}\approx1.2\text{eV}ΔCG​≈1.2eV, which is close to values reported experimentally for cuprates.
The three-band d-p model is investigated by means of a variational Monte Carlo (VMC) method with the BCS-like wave-function supplemented by the Gutzwiller and Jastrow correlators. The VMC optimization leads to a $d\text{\ensuremath{-}}\mathit{wave}$ superconducting state with a characteristic domelike shape of the order parameter for hole doping $\ensuremath{\delta}\ensuremath{\lesssim}0.4$, in good agreement with the experimental observations. Also, the off-diagonal pair-pair correlation functions, calculated within VMC, vindicates the results obtained very recently within the diagrammatic expansion of the Gutzwiller wave function method [cf. Zegrodnik, Biborski, Fidrysiak, and Spa\l{}ek, Phys. Rev. B 99, 104511 (2019)]. Subsequently, the nature of the $d\text{\ensuremath{-}}\mathit{wave}$ pairing is investigated by means of recently proposed minimal-size real-space d-wave pairing operators [Moreo and Dagotto, Phys. Rev. B 100, 214502 (2019)]. An emergence of the long-range superconducting ordering for both ddd and ppp orbitals is reported by analyzing the corresponding off-diagonal pair-pair correlation functions. The dominant character of d-wave pairing on ddd orbitals is confirmed. Additionally, the trial wave function is used to investigate the magnetic properties of the system. The analysis of spin-spin correlation functions is carried out and shows antiferromagnetic $\mathbf{q}=(\ensuremath{\pi},\ensuremath{\pi})$, short-range order, as expected. For the sake of completeness, the charge gap has been estimated, which for the parent compound takes the value ${\mathrm{\ensuremath{\Delta}}}_{\text{CG}}\ensuremath{\approx}1.78\ifmmode\pm\else\textpm\fi{}0.51\phantom{\rule{4.pt}{0ex}}\text{eV}$, and agrees with values reported experimentally for the cuprates.
We present recent theoretical results on superconductivity in correlated-electron systems, especially in the two-dimensional Hubbard model and the three-band d-p model. The mechanism of superconductivity in high-temperature superconductors has been extensively studied on the basis of various electronic models and also electron-phonon models. In this study, we investigate the properties of superconductivity in correlated-electron systems by using numerical methods such as the variational Monte Carlo method and the quantum Monte Carlomethod. The Hubbard model is one of basic models for strongly correlated electron systems, and is regarded as the model of cuprate high temperature superconductors. The d-p model is more realistic model for cuprates. The superconducting condensation energy obtained by adopting the Gutzwiller ansatz is in reasonable agreement with the condensation energy estimated for YBa2Cu3O7. We show the phase diagram of the ground state using this method. We have further investigated the stability of striped and checkerboard states in the under-doped region. Holes doped in a half-filled square lattice lead to an incommensurate spin and charge density wave. The relationship of the hole density x and incommensurability δ, δ~x, is satisfied in the lower doping region, as indicated by the variationalMonte Carlocalculations for the two-dimensional Hubbard model. A checkerboard-like charge-density modulation with a roughly period has also been observed by scanning tunneling microscopy experiments in Bi2212 and Na-CCOC compounds. We have performed a variational Monte Carlo simulation on a two-dimensional t-t′-t″- U Hubbard model with a Bi-2212 type band structure and found that the period checkerboard spin modulation, that is characterized by multi Q vectors, is indeed stabilized. We have further performed an investigation by using a quantumMonte Carlomethod, which is a numerical method that can be used to simulate the behavior of correlated electron systems. We present a new algorithm of the quantum Monte Carlo diagonalization that is a method for the evaluation of expectation value without the negative sign problem. We compute pair correlation functions and show that pair correlation is indeed enhanced with hole doping.
Abstract We investigate the ground state of the two-dimensional three-band d-p model for for high-temperature superconductors on the basis of a variational Monte Carlo method. We employ an optimization variational Monte Carlo method that we have developed recently. The many-body wave function is improved beyond the Gutzwiller ansatz by adopting the wave function in the form ψ = exp(−S)ψG where ψG represents the Gutzwiller function and S is a kinetic operator. The strong magnetic correlation and also superconductivity (SC) are induced by the on-site Coulomb repulsive interaction. It is important to clarify the phase diagram that includes superconductive phase and antiferromagnetic phase. We show the phase diagram to show the antiferromagnetic region in the parameter space. High-temperature superconductivity may occur in the strongly correlated region near the antiferromagnetic boundary.
The principal purpose of this topical review is to single out some of the universal features of high-temperature (high-Tc) and related strongly-correlated systems, which can be compared with experiment in a quantitative manner. The description starts with the concept of exchange-interaction-mediated (real-space) pairing, combined with strong correlations among narrow band electrons, for which the reference state is that of the Mott–Hubbard insulator. The physical discussion of concrete properties relies on variational approach, starting from generalized renormalized mean-field theory (RMFT) in the form of statistically-consistent Gutzwiller approximation (SGA), and its subsequent generalization, i.e., a systematic Diagrammatic Expansion of the Variational (Gutzwiller-type) Wave Function (DE-GWF). The solution leads to the two energy scales, one involving unconventional quasiparticles close to the Fermi energy, and the other reflecting the fully correlated state, involving electrons deeper below the Fermi surface. Those two regimes are separated by a kink in the dispersion relation, which is observed in photoemission. As a result, one obtains both the doping dependent properties in the correlated state, and effective renormalized quasiparticles. The reviewed ground-state characteristics for high-Tc systems encompass high-Tc superconductivity, nematicity, charge- (and pair-) density-wave effects, as well as non-BCS kinetic energy gain in the paired state. The calculated dynamic properties are: the universal Fermi velocity, the Fermi wave-vector, the effective mass enhancement, the pseudogap; and the d-wave gap magnitude. We discuss that, within the variational approach, the minimal realistic model is represented by the so-called t-J-U Hamiltonian. Inadequacy of the t-J and Hubbard models, particularly of their RMFT versions, is discussed explicitly. For heavy fermion systems, modeled by the Anderson lattice model and with the DE-GWF approach, we discuss the phase diagram, encompassing superconducting, Kondo insulating, ferro- and anti-ferromagnetic states. The superconducting state is then a two-d-wave gap system. If the orbital degeneracy of f-electrons is included, the coexistent ferromagnetic- (spin–triplet) superconducting phases appear and match those observed for UGe2 in a semiquantitative manner. Finally, in the second part, we generalize our approach to the collective spin and charge fluctuations in high-Tc systems, starting from variational approach, defining the saddle-point state, and combining it with 1/N expansion. The present scheme differs essentially from that starting from the saddle-point Hartree–Fock approximation, and incorporating the fluctuations in the random phase approximation (RPA). The spectrum of collective spin and charge excitations is determined for the Hubbard and t-J-U models, and subsequently compared quantitatively with recent experiments. The Appendices provide formal details to make this review self-contained.
The phase diagram of cuprate high-temperature superconductors is investigated on the basis of the three-band d-p model. We use the optimization variational Monte Carlo method, where improved many-body wave functions have been proposed to make the ground-state wave function more precise. We investigate the stability of antiferromagnetic state by changing the band parameters such as the hole number, level difference ΔdpΔ_{dp}Δdp​ between ddd and ppp electrons and transfer integrals. We show that the antiferromagnetic correlation weakens when ΔdpΔ_{dp}Δdp​ decreases and the pure ddd-wave superconducting phase may exist in this region. We present phase diagrams including antiferromagnetic and superconducting regions by varying the band parameters. The phase diagram obtained by changing the doping rate xxx contains antiferromagnetic, superconducting and also phase-separated phases. We propose that high-temperature superconductivity will occur near the antiferromagnetic boundary in the space of band parameters.
We study the effect of the correlated hopping term and the intersite Coulomb interaction term on principal features of the ddd-wave superconducting (SC) state, in both the electron- and hole-doped regimes within the $t\text{\ensuremath{-}}J\text{\ensuremath{-}}U$ model. In our analysis, we use the approach based on the diagrammatic expansion of the Gutzwiller wave function (DE-GWF), which allows us to go beyond the renormalized mean-field theory (RMFT). We show that the correlated hopping term enhances the pairing at the electron-doped side of the phase diagram. Moreover, the so-called non-BCS regime (which manifests itself by the negative kinetic energy gain at the transition to the SC phase) is narrowed down with the increasing magnitude of the correlated hopping $\ensuremath{\sim}K$. Also, the doping dependencies of the nodal Fermi velocity and Fermi momentum, as well as the average number of double occupancies, are analyzed with reference to the experimental data for selected values of the parameter KKK. For the sake of completeness, the influence of the intersite Coulomb repulsion on the obtained results is provided. Additionally, selected results concerning the Hubbard-model case are also presented. A complete model with all two-site interactions is briefly discussed in Appendix for reference.