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arXiv:cond-mat/0412186 · Physical Review B

Gap function ϕ(k,ω) for a two-leg t−J ladder and the pairing interaction

Didier Poilblanc, Douglas J. Scalapino

Published May 5, 2005• 7 citations
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Abstract

The gap function $\ensuremath{\phi}(k,\ensuremath{\omega})$, determined from a Lanczos calculation for a doped 2-leg $t\text{\ensuremath{-}}J$ ladder, is used to provide insight into the spatial and temporal structure of the pairing interaction. It implies that this interaction is a local near-neighbor coupling which is retarded. The onset frequency of the interaction is set by the energy of an $S=1$ magnon-hole pair and it is spread out over a frequency region of the order of the bandwidth.

DOI
10.1103/physrevb.71.174403

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The gap function \ensuremathϕ(k,\ensuremathω)\ensuremath{\phi}(k,\ensuremath{\omega})\ensuremathϕ(k,\ensuremathω), determined from a Lanczos calculation for a doped 2-leg t\ensuremath-Jt\text{\ensuremath{-}}Jt\ensuremath-J ladder, is used to provide insight into the spatial and temporal structure of the pairing interaction. It implies that this interaction is a local near-neighbor coupling which is retarded. The onset frequency of the interaction is set by the energy of an S=1S=1S=1 magnon-hole pair and it is spread out over a frequency region of the order of the bandwidth.
The evolution of the spin gap of a 2-leg ladder upon doping depends upon the nature of the lowest triplet excitations in a ladder with two holes. Here we study this evolution using various numerical techniques for a $t\ensuremath{-}{t}^{\ensuremath{'}}\ensuremath{-}J$ ladder as the next-near-neighbor hopping ${t}^{\ensuremath{'}}$ is varied. We find that depending on the value of ${t}^{\ensuremath{'}},$ the spin gap can evolve continuously or discontinuously and the lowest triplet state can correspond to a magnon, a bound magnon-hole-pair, or two separate quasiparticles. Previous experimental results on the superconducting two-leg ladder ${\mathrm{Sr}}_{14\ensuremath{-}x}{\mathrm{Ca}}_{x}{\mathrm{Cu}}_{24}{\mathrm{O}}_{41}$ are discussed.
The truncated Lanczos method using a variational scheme based on Hilbert space reduction as well as a local basis change [Dagotto et al., Phys. Rev. B 58,58,58, 12 063 (1998)] is reexamined. The energy is extrapolated as a power-law function of the Hamiltonian variance. This systematic extrapolation procedure is tested quantitatively on the two-leg t-JJJ ladder doped with two holes. For this purpose, we have carried out calculations of the spin gap and of the pair dispersion up to size $2\ifmmode\times\else\texttimes\fi{}15.$
The two-leg $t\ensuremath{-}J$ ladder forms a spin liquid at half filling that evolves to a Luther-Emery liquid upon doping. Our aim is to obtain a complete phase diagram for isotropic coupling (i.e., rungs and legs equal) as a function of electron density n and the ratio J/t.J/t.J/t. Two known limiting cases are n<12,n<\frac{1}{2},n<21​, which is a single band Luttinger liquid, and small hole doping $\ensuremath{\delta}\ensuremath{\ll}1$ for $J/\stackrel{\ensuremath{\rightarrow}}{t}0,$ which is a Nagaoka ferromagnet. Using Lanczos techniques we examine the region between the Nagaoka and Luther-Emery phases for 1>n>12.1>n>\frac{1}{2}.1>n>21​. We find evidences for gapless behavior in both spin and charge channels for J/t<0.3,J/t<0.3,J/t<0.3, consistent with Luttinger liquids in both bonding and antibonding bands (i.e., C2S2).C2S2).C2S2). This proposal is based on the behavior of spin and charge correlation functions. For example, the hole-hole correlation function that displays hole pairing at larger J/tJ/tJ/t shows hole-hole repulsion in this region. As a further test, we examined the dependence of the energy on a relative phase shift between bonding and antibonding bands. For J/t<0.3J/t<0.3J/t<0.3 this is very weak, indicating a lack of pairing between these channels.
Two-leg t-J ladders are investigated in the framework of a combination of the phass string formulation and bond-operator representation. We develope a mean-field theory in the strong rung interaction regime, i.e. J┴ ＞＞ J, t, which provides a unified description of the undoped insulating phase and the low doping spin gap phase. The ground state of the doped phase is intrinsically a superconductor with a d-wave symmetry gap driven by the t- term. The ground-state energy is in good agreement with numerical results. Phase separation is shown to occur beyond some critical value of J/t for given doping concentration. The local structure of hole pairs as well as the spectra of various spin and charge modes are also analyzed.