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arXiv:cond-mat/9712003 · Physical review. B, Condensed matter

Dynamics of spin ladders

R. Eder

Published May 15, 1998• 21 citations
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Abstract

We derive an approximate theory for Heisenberg spin ladders with two legs by mapping the spin dynamics onto the problem of hard-core ``bond bosons.'' The parameters of the bosonic Hamiltonian are obtained by matching anomalous Green's functions to Lanczos results and we find evidence for a strong renormalization due to quantum fluctuations. Various dynamical spin correlation functions are calculated and found to be in good agreement with Lanczos results. We then enlarge the effective Hamiltonian to describe the coupling of the bosonic spin fluctuations to a single hole injected into the system and treat the hole dynamics within the rainbow-diagram approximation of Schmitt-Rink et al. Theoretical predictions for the single-hole spectral function are obtained and found to be in good agreement with Lanczos results.

DOI
10.1103/physrevb.57.12832

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We derive an approximate theory for Heisenberg spin ladders with two legs by mapping the spin dynamics onto the problem of hard-core ``bond bosons.'' The parameters of the bosonic Hamiltonian are obtained by matching anomalous Green's functions to Lanczos results and we find evidence for a strong renormalization due to quantum fluctuations. Various dynamical spin correlation functions are calculated and found to be in good agreement with Lanczos results. We then enlarge the effective Hamiltonian to describe the coupling of the bosonic spin fluctuations to a single hole injected into the system and treat the hole dynamics within the rainbow-diagram approximation of Schmitt-Rink et al. Theoretical predictions for the single-hole spectral function are obtained and found to be in good agreement with Lanczos results.
We present an analytic theory for the energy spectrum of a two-leg spin ladder doped with two holes. Starting from a pseudo-fermion-bond-boson representation of the corresponding t1,2−J1,2t_{1,2}-J_{1,2}t1,2​−J1,2​ Hamiltonian we apply a diagrammatic approach adapted to the limit of strong rung coupling, which includes both, the coupling of holes to the spin background as well as the two-hole interactions. The two-hole spectrum is calculated and the formation of bound states is discussed. Additionally the evolution of the spin gap of the ladder upon doping is analyzed. A comparison with existing exact diagonalization data is presented and good agreement is found.
We present a unified account for the coupled single-hole and spin dynamics in the spin-gap phase of dimerized and frustrated spin chains and two-leg spin ladders. Based on the strong dimer limit of a one-dimensional t1,2,3{t}_{1,2,3}t1,2,3​-J1,2,3{J}_{1,2,3}J1,2,3​ model a diagrammatic approach is presented that employs a mapping of the spin Hamiltonian onto a pseudofermion bond-boson model. Results for the single-hole spectrum are detailed. A finite quasiparticle weight is observed and studied for a variety of system parameters. A comparison with existing exact-diagonalization data is performed and good agreement is found.
We numerically investigate a bosonic representation for hole pairs on a two-leg $t\ensuremath{-}J$ ladder where hard core bosons on a chain represent the hole pairs on the ladder. The interaction between hole pairs is obtained by fitting the density profile obtained with the effective model to the one obtained with the $t\ensuremath{-}J$ model, taking into account the inner structure of the hole pair given by the hole-hole correlation function. For these interactions we calculate the Luttinger liquid parameter, which takes the universal value ${K}_{\ensuremath{\rho}}=1$ as half filling is approached, for values of the rung exchange ${J}^{\ensuremath{'}}$ between strong coupling and the isotropic case. The long distance behavior of the hole-hole correlation function is also investigated. Starting from large ${J}^{\ensuremath{'}},$ the correlation length first increases as expected but diminishes significantly as ${J}^{\ensuremath{'}}$ is reduced and bound holes sit mainly on adjacent rungs. As the isotropic case is approached, the correlation length increases again. This effect is related to the different kind of bonds in the region between the two holes of a hole pair when they move apart.
In this thesis, we work on three problems. In the first problem, we work on a local time-dependent potential in a one-dimensional interacting bath. This work is motivated by cold atomic experiments, where one can shake the ultracold systems by using a time-dependent potential. We use linear response theory, bosonization, and renormalization group to compute the energy deposition in the one-dimensional bath by the time-dependent potential. We find that the energy deposition both for weak and strong coupling regimes shows a power-law behavior on the frequency of oscillation, and shows a different power-law exponent in strong and weak coupling limit. In the second problem, we work on the dynamics of a mobile impurity in a two-leg bosonic ladder. The theoretical and experimental studies of a mobile impurity in a one-dimensional bath in context of ultracold systems motivate this work. We confine the impurity to move in one of the legs of the ladder. We combine bosonization, renormalization group, and density matrix renormalization group to understand the dynamics of the impurity. We compute the Green's function of the impurity both theoretically and numerically. For a small interaction between the impurity and the bath, we find theoretically that the Green's function of the impurity decays as a power-law, and our numerical results show a good agreement with analytical result. Furthermore, we also give an analytical result of the Green's function for an infinite interaction, we find again that the Green's function decays as a power-law, and shows a good agreement with numerical results. Moreover, finally we give a semi-analytical expression for the power-law exponent at zero momentum as a function of interaction, again in good agreement with the numerical results. In the third problem, we are again investigating the dynamics of a mobile impurity in a two-leg bosonic ladder, but in this case, the impurity also tunnels in both longitudinal and transverse directions. We use similar methods than for the second problem. We compute the Green's function in the bonding and the anti-bonding modes of the impurity. The Green's function in the anti-bonding mode decays as power-law at small momentum and small interaction, while the Green's function in the bonding decays exponentially, and our numerical results show a good agreement with analytical ones.
Holes weakly doped into a four-leg t-JJJ ladder bind in pairs. At dopings exceeding a critical doping of ${\ensuremath{\delta}}_{c}\ensuremath{\simeq}\frac{1}{8}$ four-hole clusters are observed to form in density-matrix renormalization-group calculations. The symmetry of the ground-state wave function does not change and we are able to reproduce this behavior qualitatively with an effective bosonic model in which the four-leg ladder is represented as two coupled two-leg ladders and hole pairs are mapped on hard-core bosons moving along and between these ladders. At lower dopings, $\ensuremath{\delta}<{\ensuremath{\delta}}_{c},$ a one-dimensional bosonic representation for hole pairs works and allows us to calculate accurately the Luttinger liquid parameter ${K}_{\ensuremath{\rho}}$ that takes the universal value ${K}_{\ensuremath{\rho}}=1$ as half-filling is approached.
We introduce a set of discrete modular transformations and in order to study the relationships between the different phases of the Heisenberg ladders obtained with all possible exchange coupling constants. For the two-legged ladder we show that the resonating valence bond (RVB) phase is invariant under the transformation, while the Haldane phase is invariant under . These two phases are related by . Moreover, there is a `mixed' phase, that is invariant under , and which under becomes the RVB phase, while under becomes the Haldane phase. For odd ladders there exists only the transformation which, for strong coupling, maps the effective antiferromagnetic spin chain onto the spin chain. Our work is based on a combination of approximate methods such as bosonization, perturbation theory and the nonlinear sigma model, adapted to the different regimes of coupling constants.