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arXiv:cond-mat/0206344 · Physical review. B, Condensed matter

Inhomogeneously doped two-leg ladder systems

Stefan Weßel, Martin Indergand, Andreas Martin Läuchli, Urs Ledermann, Manfred Sigrist

Published May 22, 2003• 6 citations
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Abstract

A chemical potential difference between the legs of a two-leg ladder is found to be harmful for the Cooper pairing. The instability of superconductivity in such systems is analyzed by compairing results of various analytical and numerical methods. Within a strong-coupling approach for the $t\ensuremath{-}J$ model, supplemented by exact numerical diagonalization, hole binding is found unstable beyond a finite, critical chemical potential difference. The spinon-holon mean field theory for the $t\ensuremath{-}J$ model shows a clear reduction of the BCS gaps upon increasing the chemical potential difference leading to a breakdown of superconductivity. Based on a renormalization group approach and an Abelian bosonization, the doping dependent phase diagram for the weakly interacting Hubbard model with different chemical potentials was determined.

DOI
10.1103/physrevb.67.184517

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A chemical potential difference between the legs of a two-leg ladder is found to be harmful for the Cooper pairing. The instability of superconductivity in such systems is analyzed by compairing results of various analytical and numerical methods. Within a strong-coupling approach for the t\ensuremath−Jt\ensuremath{-}Jt\ensuremath−J model, supplemented by exact numerical diagonalization, hole binding is found unstable beyond a finite, critical chemical potential difference. The spinon-holon mean field theory for the t\ensuremath−Jt\ensuremath{-}Jt\ensuremath−J model shows a clear reduction of the BCS gaps upon increasing the chemical potential difference leading to a breakdown of superconductivity. Based on a renormalization group approach and an Abelian bosonization, the doping dependent phase diagram for the weakly interacting Hubbard model with different chemical potentials was determined.
Accurate numerical simulations of a doped $t\ensuremath{-}J$ model on a two-leg ladder are presented for the particle number, chemical potential, magnetic susceptibility, and entropy in the limit of large exchange coupling on the rung using a finite-temperature density matrix renormalization group method. This required an improved algorithm to achieve numerical stability down to low temperatures. The thermal dissociation of hole pairs and of the rung singlets are separately observed, and the evolution of the hole pair binding energy and magnon spin gap with hole doping is determined.
We study the effect of doping away from half-filling in weakly (but finitely) interacting N-leg Hubbard ladders using renormalization-group and bosonization techniques. For a small on-site repulsion U, the N-leg Hubbard ladders are equivalent to an N-band model, where at half-filling the Fermi velocities are ${v}_{1}{=v}_{N}<{v}_{2}{=v}_{N\ensuremath{-}1}<\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot}.$ We then obtain a hierarchy of energy scales, where the band pairs $(j,N+1\ensuremath{-}j)$ are successively frozen out. The low-energy Hamiltonian is then the sum of N/2N/2N/2 [or $(N\ensuremath{-}1)/2$ for N odd] two-leg ladder Hamiltonians without gapless excitations (plus a single chain for N odd with one gapless spin mode)---similar to the N-leg Heisenberg spin ladders. The energy scales lead to a hierarchy of gaps. Upon doping away from half-filling, the holes first enter the band(s) with the smallest gap: For odd N, the holes enter first the nonbonding band (N+1)/2(N+1)/2(N+1)/2 and the phase is a Luttinger liquid, while for even N, the holes enter first the band pair (N/2,N/2+1)(N/2,N/2+1)(N/2,N/2+1) and the phase is a Luther-Emery liquid, similar to numerical treatments of the $t\ensuremath{-}J$ model, i.e., at and close to half-filling, the phases of the Hubbard ladders for small and large U are the same. For increasing doping, hole pairs subsequently enter at critical dopings the other band pairs $(j,N+1\ensuremath{-}j)$ (accompanied by a diverging compressibility): The Fermi surface is successively opened by doping, starting near the wave vector $(\ensuremath{\pi}/2,\ensuremath{\pi}/2).$ Explicit calculations are given for the cases N=3N=3N=3 and 4.
Models of interacting Majorana modes may be realized in vortex lattices in superconducting films in contact with topological insulators and may be tuned to the strong interaction regime by adjusting the chemical potential. Extending the results on one- and two-dimensional Majorana-Hubbard models, here we study two- and four-leg ladders using both field theory and the {density-matrix renormalization group (DMRG)} methods, finding a phase diagram largely consistent with that proposed for the two-dimensional model on the square lattice.
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.
We revisit the ground state of the Hubbard model on two-legged ladders in this paper. We perform density matrix renormalization group (DMRG) calculations on large system sizes with large numbers of kept states and perform extrapolation of DMRG results with truncation errors in the converged region. We find that the superconducting correlation exponent Ksc{K}_{sc}Ksc​ extracted from the pair-pair correlation is very sensitive to the position of the reference bond, reflecting a huge boundary effect on it. By systematically removing the effects from boundary conditions, finite sizes, and truncation errors in DMRG, we obtain more accurate values of Ksc{K}_{sc}Ksc​ and ${K}_{\ensuremath{\rho}}$ with DMRG. With these exponents, we confirm that the two-legged Hubbard model is in the Luther-Emery liquid phase with ${K}_{sc}\ifmmode\cdot\else\textperiodcentered\fi{}{K}_{\ensuremath{\rho}}=1$ from tiny doping near half filling to 1/81/81/8 hole doping. When the doping is increased to $\ensuremath{\delta}⪆1/6$, the behaviors of charge, pairing, and spin correlations do not change qualitatively, but the relationship ${K}_{sc}\ifmmode\cdot\else\textperiodcentered\fi{}{K}_{\ensuremath{\rho}}=1$ is likely to be violated. With the further increase of the doping to $\ensuremath{\delta}=1/3$, the quasi-long-ranged charge correlation turns into a true long-ranged charge order, and the spin gap is closed, while the pair-pair correlation still decays algebraically. Our work provides a standard way to analyze the correlation functions when studying systems with open boundary conditions.