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Physical review. B, Condensed matter

Role of next-nearest-neighbor hopping in thet-t’-Jmodel

Takami Tohyama, Sadamichi Maekawa

Published Feb 1, 1994• 167 citations
Publisher

Abstract

A next-nearest-neighbor hopping t' in the t-t'-J model has a different sign between the hole- and electron-doped systems of high-${\mathit{T}}_{\mathit{c}}$ cuprates: t'0 for the hole-doped system and t'>0 for the electron-doped system. We show that the sign is responsible for the remarkable difference of antiferromagnetic (AF) phases between the two systems. To reveal this and clarify the role of t', we examine magnetic excitations of the t-t'-J model by employing exact diagonalization techniques for 4\ifmmode\times\else\texttimes\fi{}4 and \ensuremath{\surd}18 \ifmmode\times\else\texttimes\fi{} \ensuremath{\surd}18 lattices. In the low-doping region (two-hole case of the \ensuremath{\surd}18 \ifmmode\times\else\texttimes\fi{} \ensuremath{\surd}18 lattice), AF spin correlations are stabilized for the case of t'>0, i.e., electron-doping case, but not for the case of t'0. In the high-doping region (four-hole case of the 4\ifmmode\times\else\texttimes\fi{}4 lattice and more) magnetic excitations are mainly controlled by the geometry of the Fermi surface of the noninteracting system.

DOI
10.1103/physrevb.49.3596

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A next-nearest-neighbor hopping t' in the t-t'-J model has a different sign between the hole- and electron-doped systems of high-Tc{\mathit{T}}_{\mathit{c}}Tc​ cuprates: t'0 for the hole-doped system and t'>0 for the electron-doped system. We show that the sign is responsible for the remarkable difference of antiferromagnetic (AF) phases between the two systems. To reveal this and clarify the role of t', we examine magnetic excitations of the t-t'-J model by employing exact diagonalization techniques for 4\ifmmode\times\else\texttimes\fi{}4 and \ensuremath{\surd}18 \ifmmode\times\else\texttimes\fi{} \ensuremath{\surd}18 lattices. In the low-doping region (two-hole case of the \ensuremath{\surd}18 \ifmmode\times\else\texttimes\fi{} \ensuremath{\surd}18 lattice), AF spin correlations are stabilized for the case of t'>0, i.e., electron-doping case, but not for the case of t'0. In the high-doping region (four-hole case of the 4\ifmmode\times\else\texttimes\fi{}4 lattice and more) magnetic excitations are mainly controlled by the geometry of the Fermi surface of the noninteracting system.
Using exact diagonalization techniques, we study one dynamical hole in the two-dimensional t-J and Hubbard models on a square lattice including a next-nearest-neighbor hopping t'. We present the phase diagram in the parameter space (J/t,t'/t), discussing the ground-state properties of the hole. At J=0, a crossing of levels exists at some value of t' separating a ferromagnetic from an antiferromagnetic ground state. For nonzero J, at least four different regions appear where the system behaves like an antiferromagnet or a (not fully saturated) ferromagnet. We study the quasiparticle behavior of the hole, showing that for small values of \ensuremath{\Vert}t'\ensuremath{\Vert} the previously presented string picture is still valid. We also find that, for a realistic set of parameters derived from the Cu-O Hamiltonian, the hole has momentum (\ensuremath{\pi}/2,\ensuremath{\pi}/2), suggesting an enhancement of the p-wave superconducting mode due to the second-neighbor interactions in the spin-bag picture. Results for the t-t'-U model are also discussed with conclusions similar to those of the t-t'-J model. In general we found that t'=0 is not a singular point of these models.
We systematically examine the asymmetry of the electronic states in the hole- and electron-doped cuprates by using a $t\text{\ensuremath{-}}J$ model with the second-neighbor hopping ${t}^{\ensuremath{'}}$ and third one ${t}^{\ensuremath{''}}$ (the $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}\text{\ensuremath{-}}{t}^{\ensuremath{''}}\text{\ensuremath{-}}J$ model). Numerically exact diagonalization method is employed for a 20-site square lattice. We impose twisted boundary conditions (BC) instead of standard periodic BC. For static and dynamical correlation functions, averaging procedure over the twisted BC is used to reduce the finite-size effect. We find that antiferromagnetic spin correlation remains strong in electron doping in contrast to the case of hole doping, being similar to the case of the periodic BC. This leads to a remarkable electron-hole asymmetry in the dynamical spin structure factor and two-magnon Raman scattering. By changing the twist, the single-particle spectral function is obtained for all momenta in the Brillouin zone. Examining the spectral function in detail, we find a gap opening at around the $\mathbf{k}=(\ensuremath{\pi},0)$ region for 10% doping of holes (the carrier concentration x=0.1x=0.1x=0.1), leading to a Fermi arc that is consistent with experiments. In electron doping, however, a gap opens at around $\mathbf{k}=(\ensuremath{\pi}∕2,\ensuremath{\pi}∕2)$ and persists up to x=0.2x=0.2x=0.2, being correlated with the strength of the antiferromagnetic correlation. We find that the magnitude of the gaps is sensitive to ${t}^{\ensuremath{'}}$ and ${t}^{\ensuremath{''}}$. A pseudogap is also seen in the optical conductivity for electron doping, and its magnitude is found to be the same as that in the spectral function. We compare calculated quantities with corresponding experimental data, and discuss similarities and differences between them as well as their implications.
The Hubbard model is an iconic model in quantum many-body physics and has been intensely studied, especially since the discovery of high-temperature cuprate superconductors. Combining the complementary capabilities of two computational methods, we found superconductivity in both the electron- and hole-doped regimes of the two-dimensional Hubbard model with next-nearest-neighbor hopping. In the electron-doped regime, superconductivity was weaker and was accompanied by antiferromagnetic Néel correlations at low doping. The strong superconductivity on the hole-doped side coexisted with stripe order, which persisted into the overdoped region with weaker hole-density modulation. These stripe orders varied in fillings between 0.6 and 0.8. Our results suggest the applicability of the Hubbard model with next-nearest hopping for describing cuprate high–transition temperature ( T c ) superconductivity.
We investigate the electronic states in the antiferromagnetic (AF) phase of electron-doped cuprates by using numerically exact diagonalization technique for a $t\ensuremath{-}{t}^{\ensuremath{'}}\ensuremath{-}{t}^{\ensuremath{''}}\ensuremath{-}J$ model. When AF correlation develops with decreasing temperature, a gaplike behavior emerges in the optical conductivity. Simultaneously, the coherent motion of carriers due to the same sublattice hoppings is enhanced. We propose that the phase is characterized as an AF state with small Fermi surface around the momentum $\mathbf{k}=(\ensuremath{\pi},0)$ and $(0,\ensuremath{\pi}).$ This is a remarkable contrast to the behavior of hole-doped cuprates.
We study the phase diagram of the two-dimensional t-J-V model of holes in a two-dimensional antiferromagnet in the J\ensuremath{\gg}t limit. Here t is the hole hopping parameter, J is the exchange coupling between nearest-neighbor spins, and V is the nearest-neighbor electron-electron repulsion. We also include the possibility of Heisenberg-Ising anisotropy, ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$\ensuremath{\ne}Jz{\mathit{J}}_{\mathit{z}}Jz​. We obtain a complete description for ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$\ensuremath{\ll}Jz{\mathit{J}}_{\mathit{z}}Jz​ and approximate results for ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$=Jz{\mathit{J}}_{\mathit{z}}Jz​. There is no region of the zero-temperature phase diagram that supports a dilute gas of holes in an antiferromagnet. Rather, the system always phase separates into hole-rich and no-hole phases, a result we believe to be more general than the model we have considered. We find various hole-crystal phases and, at high hole density, we find a ``Fermi-liquid'' phase and at least two superfluid phases, one with a charge 2e and one with a charge 4e order parameter.
The Hubbard model has attracted considerable interest due to its prototypical role in describing strongly interacting electronic systems, such as high-critical-temperature superconductors as well as many novel quantum materials. By introducing next-nearest-neighbor (NNN) hoppings to the Hubbard model, the phase diagram becomes richer, and fascinating phenomena arise in both, one-dimensional chains and square lattices, such as: antiferromagnetism, ferromagnetism, superconductivity (SC), as well as charge orders, among others. Moreover, NNN hoppings play a fundamental role in understanding effects of doping on magnetism and pairing orders in strongly interacting regimes. In this article, we review the recent progress in understanding the different competing phases of this model in one and two dimensions from a computational perspective. We comment on the pressing technical challenges, illustrate the controversial results concerning the emergence of the SC phase, and conclude with our perspectives on future explorations.