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arXiv:1907.11728 · Physical Review Research

Ground state phase diagram of the doped Hubbard model on the four-leg cylinder

Yi‐Fan Jiang, Jan Zaanen, Thomas Devereaux, Hong‐Chen Jiang

Published Jul 15, 2020• 137 citations
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Abstract

The authors determine the ground state phase diagram of the lightly-doped Hubbard model on four leg cylinders with next-nearest-neighbor hopping, t'. The paper shows that, while the system remains insulating without t', it quickly evolves into two distinct superconducting Luther-Emery liquids for small t' with different spin and single particle properties.

DOI
10.1103/physrevresearch.2.033073

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The authors determine the ground state phase diagram of the lightly-doped Hubbard model on four leg cylinders with next-nearest-neighbor hopping, t'. The paper shows that, while the system remains insulating without t', it quickly evolves into two distinct superconducting Luther-Emery liquids for small t' with different spin and single particle properties.
The authors employ the density-matrix renormalization group to investigate the doped Hubbard model on six-leg square cylinders. It uncovers a rich quantum phase diagram, intricately sensitive to next-nearest-neighbor electron hopping ($t\ensuremath{'}$). The positive-ttt\ensuremath{'} region shows a robust ddd-wave superconducting phase with intertwined superconducting and charge-density-wave orders. In contrast, the negative-ttt\ensuremath{'} side remains insulating, where doped holes form either long-range charge stripe order at small ttt\ensuremath{'} or a holon Wigner crystal with one doped hole per emergent unit cell and short-range spin correlations at larger ttt'.
A lightly doped single-band Hubbard model on a two-leg ladder exhibits a Luther-Emery phase, whereas the three-band Hubbard ladder behaves as a Luttinger liquid upon hole doping. To understand this discrepancy, we present a systematic density-matrix renormalization group study of the three-band Hubbard model on two-leg cylinders with further-neighbor particle hoppings. The inclusion of the longer-range hopping is motivated by the studies of the single-band Hubbard model in which the further-neighbor hopping terms are suggested to be crucial for the unconventional superconductivity. When the longer-range hopping parameters are small, the ground state is a Luttinger liquid having mutually commensurate superconducting, charge, and spin density wave correlations. Increasing the longer-range hopping drives a transition into a Luther-Emery phase with quasi-long-ranged superconducting and charge orders but short-ranged spin-spin correlations. By down-folding the three-band Hubbard model into an effective $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}\text{\ensuremath{-}}J\text{\ensuremath{-}}{J}^{\ensuremath{'}}$ model, we find that in the Luther-Emery phase, both the nearest- and second-neighbor kinetic energies are enhanced due to an effective increase of copper-oxygen hybridization. Amplifying intercell oxygen orbital hopping mirrors the benefits of reducing the charge transfer energy, causing doped holes to favor oxygen orbitals and strengthening superconducting pairing.
A lightly doped single-band Hubbard model on a two leg ladder exhibits a Luther-Emery phase, while the three-band Hubbard ladder behaves as a Luttinger liquid upon hole doping. In order to understand this discrepancy, we present a systematic density-matrix renormalization group study of the three-band Hubbard model on two-leg cylinders with further-neighbor particle hoppings. The inclusion of the longer-range hopping is motivated by the studies of the single-band Hubbard model in which the further-neighbor hopping terms are suggested to be crucial for the unconventional superconductivity. When the longer-range hopping parameters are small, the ground state is a Luttinger liquid having mutually commensurate superconducting, charge and spin density wave correlations. Increasing the longer-range hopping drives a transition into a Luther-Emery phase with quasi-long ranged superconducting and charge orders but short-ranged spin-spin correlations. By down-folding the three-band Hubbard model into an effective ttt-t′t't′-JJJ-J′J'J′ model, we find that in the Luther-Emery phase, both the nearest and second neighbor kinetic energies are enhanced due to an effective increase of copper-oxygen hybridization. Amplifying inter-cell oxygen orbital hopping mirrors the benefits of reducing the charge transfer energy, causing doped holes to favor oxygen orbitals and strengthening superconducting pairing.
We report a density-matrix renormalization group study of the lightly doped $t\text{\ensuremath{-}}J$ model on a four-leg cylinder with doped hole concentrations per site $\ensuremath{\delta}=5%\text{--}12.5%$. By keeping an unusually large number of states and long system sizes, we are able to accurately document the interplay between superconductivity, spin, and charge-density-wave orders. The long-distance behavior is consistent with that of a Luther-Emery liquid with a spin gap and power-law charge-density-wave and superconducting correlations. This is the widest $t\text{\ensuremath{-}}J$ or Hubbard system in which power-law superconducting correlations have been established.
We study the one-dimensional Hubbard model with nearest-neighbor and next-nearest-neighbor hopping integrals by using the density-matrix renormalization group method and Hartree-Fock approximation. Based on the calculated results for the spin gap, total-spin quantum number, and Tomonaga-Luttinger-liquid parameter, we determine the ground-state phase diagram of the model in the entire filling and wide parameter region. We show that, in contrast to the weak-coupling regime where a spin-gapped liquid phase is predicted in the region with four Fermi points, the spin gap vanishes in a substantial region in the strong-coupling regime. It is remarkable that a large variety of phases, such as the paramagnetic metallic phase, spin-gapped liquid phase, singlet and triplet superconducting phases, and fully polarized ferromagnetic phase, appear in such a simple model in the strong-coupling regime.
Inspired by the observation of a robust ddd-wave superconducting phase driven by tuning the next-nearest-neighbor (NNN) electron hopping in recent density matrix renormalization group (DMRG) studies of six- and eight-leg $t\text{\ensuremath{-}}J$ model, we systematically study the phase diagram of the two-leg $t\text{\ensuremath{-}}J$ ladder with the NNN couplings (including NNN hopping and spin interaction) in a large region of doping level, by means of the DMRG calculations. Upon doping from half filling, we identify the Luther-Emery liquid (LEL) phase, which can be distinguished as the pairing-dominant and charge density-dominant regime by comparing the Luttinger parameter ${K}_{\ensuremath{\rho}}$. With the growing NNN couplings, pairing correlations are enhanced and correspondingly ${K}_{\ensuremath{\rho}}$ increases, driving the system from the density-dominant to the pairing-dominant regime. In the Tomonaga-Luttinger liquid (TLL) phase in the larger doping region, we identify two TLL regimes with different features of charge density correlation. At the quarter filling (1/21/21/2 doping level), we find that the strong dimer orders of bond energy in the open system actually decay algebraically and thus do not indicate a spontaneous translational symmetry breaking. Our results show that in the LEL phase of the two-leg ladder, the NNN couplings seem to play the similar role as that on the wider $t\text{\ensuremath{-}}J$ cylinder, and studies on this more accessible system can be helpful toward understanding the emergence of the remarkable ddd-wave superconducting phase on the wider system.