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arXiv:cond-mat/0410037 · Physical Review B

Weak-coupling phase diagrams of bond-aligned and diagonal, doped Hubbard ladders

Shirit Baruch, Dror Orgad

Published May 10, 2005• 2 citations
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Abstract

We study, using a perturbative renormalization group technique, the phase diagrams of bond-aligned and diagonal Hubbard ladders defined as sections of a square lattice with nearest-neighbor and next-nearest-neighbor hopping. We find that for not too large hole doping and small next-nearest-neighbor hopping, the bond-aligned systems exhibit a fully spin-gapped phase while the diagonal systems remain gapless. Increasing the next-nearest-neighbor hopping typically leads to a decrease of the gap in the bond-aligned ladders, and to a transition into a gapped phase in the diagonal ladders. Embedding the ladders in an antiferromagnetic environment can lead to a reduction in the extent of the gapped phases. These findings may suggest a relation between the orientation of hole-rich stripes and superconductivity as observed in ${\mathrm{La}}_{2\ensuremath{-}x}{\mathrm{Sr}}_{x}\mathrm{Cu}{\mathrm{O}}_{4}$.

DOI
10.1103/physrevb.71.184503

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We study, using a perturbative renormalization group technique, the phase diagrams of bond-aligned and diagonal Hubbard ladders defined as sections of a square lattice with nearest-neighbor and next-nearest-neighbor hopping. We find that for not too large hole doping and small next-nearest-neighbor hopping, the bond-aligned systems exhibit a fully spin-gapped phase while the diagonal systems remain gapless. Increasing the next-nearest-neighbor hopping typically leads to a decrease of the gap in the bond-aligned ladders, and to a transition into a gapped phase in the diagonal ladders. Embedding the ladders in an antiferromagnetic environment can lead to a reduction in the extent of the gapped phases. These findings may suggest a relation between the orientation of hole-rich stripes and superconductivity as observed in La2\ensuremath−xSrxCuO4{\mathrm{La}}_{2\ensuremath{-}x}{\mathrm{Sr}}_{x}\mathrm{Cu}{\mathrm{O}}_{4}La2\ensuremath−x​Srx​CuO4​.
We investigate the ground-state phase diagram of the half-filled one-dimensional Hubbard model with next-nearest-neighbor hopping using the density-matrix renormalization-group technique as well as an unrestricted Hartree-Fock approximation. We find commensurate and incommensurate disordered magnetic insulating phases and a spin-gapped metallic phase in addition to the one-dimensional Heisenberg phase. At large on-site Coulomb repulsion U, we make contact with the phase diagram of the frustrated Heisenberg chain, which has spin-gapped phases for sufficiently large frustration. For weak U, sufficiently large next-nearest-neighbor hopping t2{t}_{2}t2​ leads to a band structure with four Fermi points rather than two, producing a spin-gapped metallic phase. As U is increased in this regime, the system undergoes a Mott-Hubbard transition to a frustrated antiferromagnetic insulator.
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.
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'.
The effect of next-nearest-neighbor hopping t2{t}_{2}t2​ on the ground-state phase diagram of the one-dimensional Kondo lattice is studied with density-matrix renormalization-group techniques and by comparing with the phase diagram of the classical-spin variant of the same model. For a finite t2{t}_{2}t2​, i.e., for a zigzag-ladder geometry, indirect antiferromagnetic interactions between the localized spins are geometrically frustrated. We demonstrate that t2{t}_{2}t2​ at the same time triggers several magnetic phases which are absent in the model with nearest-neighbor hopping only. For strong JJJ, we find a transition from antiferromagnetic to incommensurate magnetic short-range order, which can be understood entirely in the classical-spin picture. For weaker JJJ, a spin-dimerized phase emerges, which spontaneously breaks the discrete translation symmetry. The phase is not accessible to perturbative means but is explained, on a qualitative level, by the classical-spin model as well. Spin dimerization alleviates magnetic frustration and is interpreted as a key to understand the emergence of quasi-long-range spiral magnetic order, which is found at weaker couplings. The phase diagram at weak JJJ, with gapless quasi-long-range order on top of the twofold degenerate spin-dimerized ground state, competing with a nondegenerate phase with gapped spin (and charge) excitations, is unconventional and eludes an effective low-energy spin-only theory.
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 study the phase diagram of the one-dimensional Hubbard model with next-nearest-neighbor hopping using exact diagonalization, the density-matrix renormalization group, the Edwards variational ansatz, and an adaptation of weak-coupling calculations on the two-chain Hubbard model. We find that a substantial region of the strong-coupling phase diagram is ferromagnetic, and that three physically different limiting cases are connected in one ferromagnetic phase. At a point in the phase diagram at which there are two Fermi points at weak coupling, we study carefully the phase transition from the paramagnetic state to the fully polarized one as a function of the on-site Coulomb repulsion. We present evidence that the transition is second order and determine the critical exponents numerically. In this parameter regime, the system can be described as a Luttinger liquid at weak coupling. We extract the Luttinger-liquid parameters and show how their behavior differs from that of the nearest-neighbor Hubbard model. The general weak-coupling phase diagram can be mapped onto that of the two-chain Hubbard model. We exhibit explicitly the adapted phase diagram and determine its validity by numerically calculating spin and charge gaps using the density-matrix renormalization group.