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arXiv:cond-mat/0203472 · Physical review. B, Condensed matter

Cyclic spin exchange in cuprate spin ladders

Tamara S. Nunner, Philipp Brune, Thilo Kopp, Marco Windt, M. Grüninger

Published Nov 27, 2002• 63 citations
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Abstract

We investigate the influence of a cyclic spin exchange ${J}_{\mathrm{cyc}}$ on the one- and two-triplet excitations of an undoped two-leg $S=1/2$ ladder, using the density-matrix renormalization group (DMRG). The dispersion of the $S=0$ two-triplet bound state is dramatically reduced by ${J}_{\mathrm{cyc}}$ due to a repulsion between triplets on neighboring rungs. In $(\mathrm{La},\mathrm{Ca}{)}_{14}{\mathrm{Cu}}_{24}{\mathrm{O}}_{41}$ a consistent description of both the spin gap and the $S$=0 bound state requires ${J}_{\mathrm{cyc}}{/J}_{\ensuremath{\perp}}\ensuremath{\approx}0.20--0.27$ and ${J/J}_{\ensuremath{\perp}}\ensuremath{\approx}1.25--1.35.$ With these coupling ratios the recently developed dynamical DMRG yields an excellent description of the entire $S=0$ excitation spectrum observed in the optical conductivity, including the continuum contribution.

DOI
10.1103/physrevb.66.180404

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We investigate the influence of a cyclic spin exchange Jcyc{J}_{\mathrm{cyc}}Jcyc​ on the one- and two-triplet excitations of an undoped two-leg S=1/2S=1/2S=1/2 ladder, using the density-matrix renormalization group (DMRG). The dispersion of the S=0S=0S=0 two-triplet bound state is dramatically reduced by Jcyc{J}_{\mathrm{cyc}}Jcyc​ due to a repulsion between triplets on neighboring rungs. In (La,Ca)14Cu24O41(\mathrm{La},\mathrm{Ca}{)}_{14}{\mathrm{Cu}}_{24}{\mathrm{O}}_{41}(La,Ca)14​Cu24​O41​ a consistent description of both the spin gap and the SSS=0 bound state requires Jcyc/J\ensuremath⊥\ensuremath≈0.20−−0.27{J}_{\mathrm{cyc}}{/J}_{\ensuremath{\perp}}\ensuremath{\approx}0.20--0.27Jcyc​/J\ensuremath⊥​\ensuremath≈0.20−−0.27 and J/J\ensuremath⊥\ensuremath≈1.25−−1.35.{J/J}_{\ensuremath{\perp}}\ensuremath{\approx}1.25--1.35.J/J\ensuremath⊥​\ensuremath≈1.25−−1.35. With these coupling ratios the recently developed dynamical DMRG yields an excellent description of the entire S=0S=0S=0 excitation spectrum observed in the optical conductivity, including the continuum contribution.
Resonant inelastic x-ray scattering (RIXS) has become an important tool for studying elementary excitations in correlated materials. Here, we present a systematic theoretical investigation of the Cu LLL-edge RIXS spectra of undoped and doped cuprate two-leg spin-ladders in both the non-spin-conserving (NSC) and spin-conserving (SC) channels. The spectra are rich and host many exotic excitations. In the NSC channel of the undoped case, we identify one-triplon and bound triplet two-triplon excitations in the strong rung coupling limit, as well as confined spinons in the weak rung coupling limit. In the doped case, we observe a quasiparticle excitation formed from a bound charge and spin-1/21/21/2 in the strong rung coupling limit. In the SC channel, we also identify several new features, including bound singlet two-triplon excitations and confined spinons in the undoped ladders in the strong and weak rung coupling limits, respectively. Conversely, in the doped case, the SC channel primarily probes both gapless and gapped charge excitations. Finally, we revisit the available data for the ladder compound Sr14Cu24O41{\mathrm{Sr}}_{14}{\mathrm{Cu}}_{24}{\mathrm{O}}_{41}Sr14​Cu24​O41​ in the context of our results.
The effect of a ring exchange on doped two-leg ladders is investigated combining exact diagonalization (ED) and density matrix renormalization group (DMRG) computations. We focus on the nature and weights of the low-energy magnetic excitations and on superconducting pairing. The stability with respect to this cyclic term of a remarkable resonant mode originating from a hole pair-magnon bound state is examined. We also find that, near the zero-doping critical point separating rung-singlet and dimerized phases, doping reopens a spin gap.
This thesis describes the neutron scattering measurements of magnetic excitations in \nspin-chains and ladders. The first part discusses an experimental investigation of the copper oxide family Srâ â Cuâ â Oâ â composed of edge-sharing chains and spin-ladders. \nThe study of Laâ Srâ â Cuâ â Oâ â comprises a slightly hole-doped chain and an undoped ladder structure where the chain can be modeled by a ferromagnetic nearest and an \nantiferromagnetic next-nearest neighbor coupling. The hole effects are apparent in gaps in the dispersion relation and can be described by a charge-density wave agreeing with the commensuration of the dispersion. Investigating the undoped ladder establishes \nthe exchange constants including a cyclic exchange manifested by the two-magnon \ncontinuum and the suppression of the S = 1 bound mode. An orbital consideration \nprovides an explanation for the exchanges including the different sizes of rung and \nleg coupling. The excitation spectrum of the doped ladder in Caâ .â Srâ â .â Cuâ â Oâ â can be \ndescribed by a direct comparison with the undoped ladder and the differences consisting \nof a higher energy mode and subgap scattering can be successfully modeled by the charge spectrum of the ladder calculated from the free electron model. The second part of the thesis investigates the alternating chain material Cu(NOâ )â · 2.5D2O and establishes the gapped one-magnon dispersion, the two-magnon continuum and for the first time the S =1 bound mode. Applying magnetic field drives the system through two critical field transitions, condensation of magnons into the ground state and saturation. The modes beyond saturation can be modeled by spin wave theory and the excitations at the first critical field follow Luttinger Liquid behavior. Additionally investigated are the temperature effects with the excitations being of a different nature but containing the signature of a strong correlated system. For an outlook the measurements including temperature and field are provided with further theoretical descriptions necessary.
In the framework of this thesis the infrared spectra of two-leg spin ladders were measured. This study presents the first determination of the magnetic contribution to the optical conductivity of spin ladders. A comparison between experiment and new theoretical calculations provided a comprehensive understanding of the magnetic spectrum of the undoped telephone-number compounds La(y)Ca(14-y)Cu(24)O(41). To be able to study the weak magnetic contribution to the spectra, it was necessary to measure not only the reflectance but in particular the transmittance of thin single crystals. When both quantities are known, one can directly calculate the optical conductivity. The low-temperature spectra were compared with results of three different theoretical approaches in close collaboration with the theory groups of Uhrig et al. from the University of Cologne and of Kopp et al. from the University of Augsburg. The excellent agreement between theory and experiment allowed us to identify the two sharp peaks in the mid-infrared range unambiguously as the signature of a true bound state of two triplets. The occurrence of bound states in spin ladders had been predicted already by several groups, but we provided the first experimental verification. Further spectral weight at higher frequencies was attributed to the multi-triplet continuum. The exchange couplings were derived, and the importance of the four-spin cyclic exchange was demonstrated. With increasing temperature especially the line shape for polarization along the legs changes considerably. The spectral weight of the continuum strongly grows, whereas the bound-state peaks lose weight. The resemblance between the measured room-temperature data of the S=1/2 ladder and theoretical results for the S=1 Haldane chain was discussed. The corresponding calculations were performed by Nunner et al. in Augsburg. Another main part of this thesis was to examine the influence of doping on the infrared spectra. We observed the same bound-state features in Sr(14)Cu(24)O(41) as in the undoped compounds, but we found additional structure in the spectra. Several scenarios were discussed in order to explain the new features, including the modulation of the exchange coupling along the legs due to the incommensurate structure and due to the charge order in the chains.
We investigate an S=1/2S=1/2S=1/2 two-leg spin ladder with a cyclic four-spin exchange interaction whose interaction constant is denoted by J4J_4J4​, by using the density matrix renormalization group method. The interchain and the intrachain interaction constant are denoted by JrungJ_{rung}Jrung​ and JlegJ_{\rm leg}Jleg​, respectively and assumed to be antiferromagnetic. It turns out that a spin gap between the singlet (Stotz=0S_{tot}^z=0Stotz​=0) and the triplet (Stotz=1S_{tot}^z=1Stotz​=1) states vanishes at J4/Jleg≃0.3J_4/J_{leg} \simeq 0.3J4​/Jleg​≃0.3 for Jrung=JlegJ_{rung}=J_{leg}Jrung​=Jleg​. This result is in contrast with the fact that the S=1/2S=1/2S=1/2 antiferromagnetic Heisenberg ladder, that is the case of Jrung≠Jleg,J4=0J_{rung} \neq J_{leg}, J_4=0Jrung​=Jleg​,J4​=0, has a spin gap for all nonzero value of interchain interaction Jrung>0J_{\rm rung}>0Jrung​>0. We find a larger value of the correlation length for the spin-pair correlation function than a linear size LLL of the system at J4/Jleg=0.3J_4/J_{\rm leg}=0.3J4​/Jleg​=0.3 and Jrung=JlegJ_{rung}=J_{leg}Jrung​=Jleg​: the correlation length ξξξ is about 204 times of the lattice constant for L=84 for these values of interactions. We also find that the string correlation function decays rather algebraically than exponentially at J4/Jleg=0.3J_4/J_{leg}=0.3J4​/Jleg​=0.3 and Jrung=JlegJ_{rung}=J_{leg}Jrung​=Jleg​. These results suggest that there is a quantum phase transition at J4/Jleg≃0.3J_4/J_{leg} \simeq 0.3J4​/Jleg​≃0.3 for Jrung=JlegJ_{rung}=J_{leg}Jrung​=Jleg​. We estimate a phase boundary where the spin gap vanishes in a J4/Jleg−Jrung/JlegJ_4/J_{leg}-J_{rung}/J_{leg}J4​/Jleg​−Jrung​/Jleg​ plane and obtain a consistent result with that by a perturbation theory for Jrung/Jleg>1J_{\rm rung}/J_{leg}>1Jrung​/Jleg​>1.
This article summarizes and extends the recent developments in the microscopic modeling of the magnetic excitations in cuprate two-leg ladder systems. The microscopic Hamiltonian comprises dominant Heisenberg exchange terms plus an additional four-spin interaction which is about five times smaller. We give an overview over the relevant energies like the one-triplon dispersion, the energies of two-triplon bound states and the positions of multi-triplon continua and over relevant spectral properties like spectral weights and spectral densities in the parameter regime appropriate for cuprate systems. It is concluded that an almost complete understanding of the magnetic excitations in undoped cuprate ladders has been obtained as measured by inelastic neutron scattering, inelastic light (Raman) scattering and infrared absorption.