Two-hole and four-hole bound states in a t -J ladder at half-filling

The spin dynamics of a doped 2-leg spin ladder is investigated by numerical techniques. We show that a hole pair-magnon boundstate evolves at finite hole doping into a sharp magnetic excitation below the two-particle continuum. This is supported by a field theory argument based on a SO(6)-symmetric ladder. Similarities and differences with the resonant mode of the high-Tc cuprates are discussed.PACS numbers: PACS numbers: 75.10.Jm, 75.40.Mg Spin ladders materials are built from weakly coupled ladder units consisting of chain sub-units of spins S=1/2 (so-called "legs") connected via some "rung" couplings and exhibit numerous fascinating and intriguing properties [1]. Among them, the role of the parity of the number of legs is remarkable: only ladders with an even number of legs exhibit a spin gap (ie a finite energy scale for triplet excitations) as seen from different characteristic behaviors of the spin susceptibility observed in SrCu 2 O 3 and SrCu 3 O 5 [2], typical 2-leg and 3-leg compounds respectively.Besides exotic experimental properties, spin ladders are of special interest for theorists as well, especially the simple 2-leg ladder: indeed, it is believed that its ground state (GS) is a close realization of the Resonating Valence Bond (RVB) state proposed by Anderson [3] in the context of high-T C superconductivity. The striking difference between "even" and "odd" ladders mentioned above can naturally be explained in this simple picture since pairing nearest neighbor (NN) spins into spin singlets is more easily realized on rungs with an even number of sites. While the generic two dimensional (2D) Mott insulator (for one electron per site) is antiferromagnetic (AF) and may evolve into an RVB state only at finite doping, the 2-leg spin ladder exhibits a finite spin correlation length. It is therefore an ideal system to investigate doping into the RVB-like spin liquid [4] leading to d x 2 −y 2 -like pairing [5]. As a matter of fact, pressure was shown to induced superconductivity in the intrinsically doped Sr 2 Ca 12 Cu 24 O 41+δ ladder material [6] hence bridging the apparent gap between 2-leg ladder and layerbased cuprates.In this Letter, we investigate the spin dynamics of a 2-leg spin ladder doped with mobile holes. In 2D superconducting cuprates, a resonant mode at an energy around 40 meV was observed by Inelastic Neutron Scattering (INS) [7,8]. This mode was shown to be quite sharp both in energy and in momentum space (centered around the AF wavevector) and its observation seems to be directly linked to the appearance of superconductiv- ity. Anomalous spectral lineshape in photoemission experiments was also interpreted as the effect of a coupling of the quasiparticles to a collective mode [9] related to the pairing interaction. On the other hand, in ladders, the spin dynamics remains largely to be explored. We believe that a closer inspection, both experimentally and theoretically, of the low energy spin excitations in such a basic system will provide insights into the mechanism of pairing mediat...

Resonant magnetic mode in superconducting two-leg ladders

Poilblanc

Orignac²,

White³

et al. 2004

Gap functionϕ(k,ω)for a two-legt−Jladder and the pairing interaction

Poilblanc

Scalapino

2005

The gap function φ(k, ω), determined from a Lanczos calculation for a doped 2-leg t-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 order the bandwith. provides information on the spatial and dynamic structure of the pairing interaction. Traditionally, electron tunneling has been used to explore the frequency dependence of the gap [1]. Presently, angular resolved photoemission spectroscopy [2] (ARPES) and scanning tunneling microscopy [3,4] (STM) open the possibility of obtaining both k and ω information about the superconducting gap. However, approaches to extract this information on φ(k, ω) from such experiments are still being explored. As has been previously discussed, a k and ω dependent gap function for finite t-J lattices can be obtained using Lanczos exact diagonalization [5,6]. Here, using results for φ(k, ω) obtained for a doped 2-leg t-J ladder, we explore what can be learned once data for φ(k, ω) becomes available.The Hamiltonian for a 2-leg t-J ladder can be written aswhere c iα are projected hole operators (spin indices are omitted) and α = (1, 2) labels the two legs of the ladder. We will consider the isotropic case in which J rung = J leg = J and t rung = t leg = t and our calculations will be carried out for a periodic 2 × 12 ladder at 1/8 and 1/6 hole doping.As previously discussed, a gap function can be extracted by combining Lanczos results for the usual one-electron Green's function G(k, ω) with the Fourier transform of Gorkov's off-diagonal Green's functionHere, for a finite system, this expectation value is taken between the ground states for N and N − 2 particles and we choose the phase of F to be zero. We will take N = 22, N − 2 = 20 corresponding to an average filling n ≃ 21 24 = 0.875. The Dyson equations [7] relating G and2 with Z and X the usual Nambu self energies and φ(k, ω) the gap function. Then, using theAlternatively, Z(k, ω) can be eliminated to obtain an expression for the superconductingFrom a numerical calculation of G(k, ω) and F (k, ω), ∆(k, ω) has been obtained for a 32-site, t-J cluster [5]. This extended previous work by Ohta et. al [8] who fit the spectral weight Im F (k, ω) to a d x 2 −y 2 BCS-Bogoliubov quasiparticle form in which the frequency dependence of the gap was neglected.In contrast to the long range order of the superconducting ground state of a 2D lattice, a 2-leg ladder exhibits power law pair field correlations [10, 11] which decay as x −1/κρ . Here, κ ρ is the Luttinger liquid parameter associated with the massless charge mode. This implies that for a ladder of length L, the off-diagonal Green's function F (k, ω) decays [12,13,14] as (ξ/L) 1 2κρ. Here, the coherence length ξ is proportional to the inverse of the gap. For our doped ladder L = 12 is of order this coherenc...

t−Jzladder: Density-matrix renormalization group and series expansion calculations of the phase diagram

Weiße

Bursill²,

Hamer³

et al. 2006

Self Cite

The phase diagram of the two-leg t-Jz ladder is explored, using the density matrix renormalization group method. Results are obtained for energy gaps, electron density profiles and correlation functions for the half-filled and quarter-filled cases. The effective Lagrangian velocity parameter vρ is shown to vanish at half-filling. The behaviour of the one-hole gap in the Nagaoka limit is investigated, and found to disagree with theoretical predictions. A tentative phase diagram is presented, which is quite similar to the full t-J ladder, but scaled up by a factor of about two in coupling. Near half-filling a Luther-Emery phase is found, which may be expected to show superconducting correlations, while near quarter-filling the system appears to be in a Tomonaga-Luttinger phase.