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Physical review. B, Condensed matter

Antiferromagnetic exchange in two-leg spin-1/2 ladders

D. C. Johnston

Published Nov 1, 1996• 74 citations
Publisher

Abstract

Extraction of antiferromagnetic (AF) exchange constants J and J\ensuremath{'} in two-leg spin-1/2 nearest-neighbor (NN) Heisenberg ladders from experimental magnetic spin susceptibility \ensuremath{\chi}(T) data is studied, where J\ensuremath{'} is the NN exchange constant in the rungs and J is that in the legs. Two low-T (T\ensuremath{\ll}J/${\mathit{k}}_{\mathit{B}}$) approximations of Troyer, Tsunetsugu, and W\"urtz [Phys. Rev. B 50, 13 515 (1994)] for \ensuremath{\chi}(T) of the isolated spin-1/2 two-leg ladder with J\ensuremath{'}/J=1 are shown to be accurate to \ensuremath{\sim}10% when extrapolated to T\ensuremath{\sim}J\ensuremath{'}/${\mathit{k}}_{\mathit{B}}$. The variations in the magnetic excitation dispersion relation parameters [which enter the low-T expression of Troyer et al. for \ensuremath{\chi}(T)] for J\ensuremath{'}/J1 are inferred from previous theoretical results. Next, the values of the maxima in \ensuremath{\chi}(T)\ensuremath{\equiv}${\mathrm{\ensuremath{\chi}}}^{\mathrm{max}}$ from many numerical calculations in the literature for various nonfrustrated spin-1/2 NN Heisenberg antiferromagnets are considered. The values of the product ${\mathrm{\ensuremath{\chi}}}^{\mathrm{max}}$${\mathit{z}}_{\mathrm{eff}}$ are found to depend systematically and almost uniqely on an effective magnetic coordination number ${\mathit{z}}_{\mathrm{eff}}$, defined in the text, but are nearly independent of the dimensionality of the spin lattice and of the magnitude of any gap in the magnetic excitation spectrum. This observation allows accurate bounds on the exchange coupling constants in arbitrary quasi-low-dimensional spin-1/2 antiferromagnets to be inferred from experimental ${\mathrm{\ensuremath{\chi}}}^{\mathrm{max}}$ values. Finally, a mean-field-type expression for \ensuremath{\chi}(T) for arbitrary quasi-low-dimensional spin-1/2 Heisenberg antiferromagnets is derived, using the above ${\mathrm{\ensuremath{\chi}}}^{\mathrm{max}}$ phenomenology as input, which allows the influence of intersystem coupling on \ensuremath{\chi}(T) to be evaluated. These results, and the \ensuremath{\chi}(T) calculations of Barnes and Riera [Phys. Rev. B 50, 6817 (1994)] for isolated two-leg spin-1/2 Heisenberg ladders, are used to analyze the experimental \ensuremath{\chi}(T) data of Azuma et al. [Phys. Rev. Lett. 73, 3463 (1994)] for the spin-1/2 two-leg ladder compound ${\mathrm{SrCu}}_{2}$${\mathrm{O}}_{3}$. The analyses together suggest that J\ensuremath{'}/J\ensuremath{\sim}0.5, contrary to the expectation that J\ensuremath{'}/J\ensuremath{\approxeq}1, and that J is very large (\ensuremath{\sim}2000 K), similar to the value of J in the linear-chain cuprate ${\mathrm{Sr}}_{2}$${\mathrm{CuO}}_{3}$. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physrevb.54.13009

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Extraction of antiferromagnetic (AF) exchange constants J and J\ensuremath{'} in two-leg spin-1/2 nearest-neighbor (NN) Heisenberg ladders from experimental magnetic spin susceptibility \ensuremath{\chi}(T) data is studied, where J\ensuremath{'} is the NN exchange constant in the rungs and J is that in the legs. Two low-T (T\ensuremath{\ll}J/kB{\mathit{k}}_{\mathit{B}}kB​) approximations of Troyer, Tsunetsugu, and W"urtz [Phys. Rev. B 50, 13 515 (1994)] for \ensuremath{\chi}(T) of the isolated spin-1/2 two-leg ladder with J\ensuremath{'}/J=1 are shown to be accurate to \ensuremath{\sim}10% when extrapolated to T\ensuremath{\sim}J\ensuremath{'}/kB{\mathit{k}}_{\mathit{B}}kB​. The variations in the magnetic excitation dispersion relation parameters [which enter the low-T expression of Troyer et al. for \ensuremath{\chi}(T)] for J\ensuremath{'}/J1 are inferred from previous theoretical results. Next, the values of the maxima in \ensuremath{\chi}(T)\ensuremath{\equiv}\ensuremathχmax{\mathrm{\ensuremath{\chi}}}^{\mathrm{max}}\ensuremathχmax from many numerical calculations in the literature for various nonfrustrated spin-1/2 NN Heisenberg antiferromagnets are considered. The values of the product {\mathrm{\ensuremath{\chi}}}^{\mathrm{max}}$${\mathit{z}}_{\mathrm{eff}} are found to depend systematically and almost uniqely on an effective magnetic coordination number zeff{\mathit{z}}_{\mathrm{eff}}zeff​, defined in the text, but are nearly independent of the dimensionality of the spin lattice and of the magnitude of any gap in the magnetic excitation spectrum. This observation allows accurate bounds on the exchange coupling constants in arbitrary quasi-low-dimensional spin-1/2 antiferromagnets to be inferred from experimental \ensuremathχmax{\mathrm{\ensuremath{\chi}}}^{\mathrm{max}}\ensuremathχmax values. Finally, a mean-field-type expression for \ensuremath{\chi}(T) for arbitrary quasi-low-dimensional spin-1/2 Heisenberg antiferromagnets is derived, using the above \ensuremathχmax{\mathrm{\ensuremath{\chi}}}^{\mathrm{max}}\ensuremathχmax phenomenology as input, which allows the influence of intersystem coupling on \ensuremath{\chi}(T) to be evaluated. These results, and the \ensuremath{\chi}(T) calculations of Barnes and Riera [Phys. Rev. B 50, 6817 (1994)] for isolated two-leg spin-1/2 Heisenberg ladders, are used to analyze the experimental \ensuremath{\chi}(T) data of Azuma et al. [Phys. Rev. Lett. 73, 3463 (1994)] for the spin-1/2 two-leg ladder compound {\mathrm{SrCu}}_{2}$${\mathrm{O}}_{3}. The analyses together suggest that J\ensuremath{'}/J\ensuremath{\sim}0.5, contrary to the expectation that J\ensuremath{'}/J\ensuremath{\approxeq}1, and that J is very large (\ensuremath{\sim}2000 K), similar to the value of J in the linear-chain cuprate {\mathrm{Sr}}_{2}$${\mathrm{CuO}}_{3}. \textcopyright{} 1996 The American Physical Society.
A comprehensive theoretical and experimental study is presented of the magnetic susceptibility versus temperature χ(T) of spin S = 1/2 two- and three-leg Heisenberg ladders and ladder oxide compounds. Extensive quantum Monte Carlo simulations of χ(T) were carried out for both isolated and coupled two-leg ladders with spatially anisotropic intraladder exchange. Accurate fits to these and related literature QMC data were obtained. We have also calculated the one- and two-magnon dispersion relations and the dynamical spin structure factor for anisotropic isolated 2 x 12 ladders. The exchange constants in the two-leg ladder compound SrCu2O3 are estimated from LDA+U calculations. We report the detailed crystal structure of SrCu2O3 and of the three-leg ladder compound Sr2Cu3O5. New experimental χ(T) data are reported for the two-leg ladder cuprates SrCu2O3 and LaCuO_{2.5}, and for the (nominally) two-leg ladder vanadates CaV2O5 and MgV2O5. The new and literature χ(T) data for these compounds and for Sr2Cu3O5 are modeled using our QMC χ(T) simulation fits, and the exchange coupling constants between the spins-1/2 are thereby estimated for each material. The surpisingly strong spatial anisotropy of the bilinear intraladder exchange constants in the cuprate compounds is discussed together with the results of other experiments sensitive to this anisotropy. Recent theoretical predictions are discussed including those which indicate that a four-spin cyclic exchange interaction within a Cu4 plaquette is important to determining the magnetic properties and which can significantly influence the exchange interactions estimated from χ(T) data assuming the presence of only bilinear exchange.
We consider asymmetric spin-12\frac{1}{2}21​ two-leg ladders with nonequal antiferromagnetic (AF) couplings ${J}_{\ensuremath{\parallel}}$ and $\ensuremath{\kappa}{J}_{\ensuremath{\parallel}}$ along legs $(\ensuremath{\kappa}\ensuremath{\le}1)$ and ferromagnetic rung coupling, ${J}_{\ensuremath{\perp}}$. This model is characterized by a gap $\ensuremath{\Delta}$ in the spectrum of spin excitations. We show that in the large ${J}_{\ensuremath{\perp}}$ limit this gap is equivalent to the Haldane gap for the AF spin-1 chain, irrespective of the asymmetry of the ladder. The behavior of the gap at small rung coupling falls in two different universality classes. The first class, which is best understood from the case of the conventional symmetric ladder at $\ensuremath{\kappa}=1$, admits a linear scaling for the spin gap $\ensuremath{\Delta}\ensuremath{\sim}{J}_{\ensuremath{\perp}}$. The second class appears for a strong asymmetry of the coupling along legs, $\ensuremath{\kappa}{J}_{\ensuremath{\parallel}}⪡{J}_{\ensuremath{\perp}}⪡{J}_{\ensuremath{\parallel}}$ and is characterized by two energy scales: the exponentially small spin gap $\ensuremath{\Delta}\ensuremath{\sim}{J}_{\ensuremath{\perp}}\text{ }\text{exp}(\ensuremath{-}{J}_{\ensuremath{\parallel}}/{J}_{\ensuremath{\perp}})$, and the bandwidth of the low-lying excitations induced by a Suhl-Nakamura indirect exchange $\ensuremath{\sim}{J}_{\ensuremath{\perp}}^{2}/{J}_{\ensuremath{\parallel}}$. We report numerical results obtained by exact diagonalization, density-matrix renormalization group and quantum Monte Carlo simulations for the spin gap and various spin correlation functions. Our data indicate that the behavior of the string order parameter, characterizing the hidden AF order in Haldane phase, is different in the limiting cases of weak and strong asymmetries. On the basis of the numerical data, we propose a low-energy theory of effective spin-1 variables, pertaining to large blocks on a decimated lattice.
We consider a two-chain, spin-12\frac{1}{2}21​ antiferromagnetic Heisenberg spin ladder in an external magnetic field HHH. The spin ladder is known to undergo second-order quantum phase transitions (QPTs) at two critical values Hc1{H}_{c1}Hc1​ and Hc2{H}_{c2}Hc2​ of the magnetic field. There are now known examples of strongly coupled (where the rung exchange interaction is much stronger than the nearest-neighbor intrachain exchange interaction) organic ladder compounds in which QPTs have been experimentally observed. In this paper, we investigate whether well-known bipartite entanglement meaures such as one-site von Neumann entropy, two-site von Neumann entropy, and concurrence develop special features close to the quantum critical points. As suggested by an earlier theorem, the first derivatives of the measures with respect to magnetic field are expected to diverge as $H\ensuremath{\rightarrow}{H}_{c1}$ and $H\ensuremath{\rightarrow}{H}_{c2}$. Based on numerical diagonalization data and a mapping of the strongly coupled ladder Hamiltonian onto the XXZXXZXXZ chain Hamiltonian, for which several analytical results are known, we find that the derivatives of the entanglement measures diverge as $H\ensuremath{\rightarrow}{H}_{c2}$ but remain finite as $H\ensuremath{\rightarrow}{H}_{c1}$. The reason for this discrepancy is analyzed. We further calculate two recently proposed quantum information theoretic measures, the reduced fidelity and reduced fidelity susceptibility, and show that these measures provide appropriate signatures of the QPTs occuring at the critical points H=Hc1H={H}_{c1}H=Hc1​ and H=Hc2H={H}_{c2}H=Hc2​.
The antiferromagnetic Heisenberg model on the three-leg ladder is studied using the generalized Jordan-Wigner transformation in dimensions higher than 1, and the bond-mean-field theory. The magnetic susceptibility and other thermodynamic quantities are analyzed as a function of the rung-to-leg coupling ratio $\ensuremath{\alpha}$ and temperature TTT. We fit the experimental susceptibility data of the three-leg material Sr2Cu3O5{\mathrm{Sr}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{5}Sr2​Cu3​O5​ of Azuma and co-workers with good agreement. One of the main findings of this work is the proposal that close to two-thirds of the spin degrees of freedom on each of the rungs of the ladder lock at low TTT for small $\ensuremath{\alpha}$, then collectively almost 2∕32∕32∕3 of the spin degrees of freedom on all the rungs freeze completely at low TTT for $\ensuremath{\alpha}$ greater than a threshold value. The approach developed here can be used to study the three-leg ladder for all values of $\ensuremath{\alpha}$, and is thus suitable for the description of the crossover regime between the weak- and strong-coupling regimes.
Various forms of antiferromagnetic exchange interaction among quantized spins in one-dimensional and quasi-one-dimensional lattices are examined. Primary results are reported in two real compounds, the sodium vanadate NaV2O5 and the copper germanate CuGeO3, and in a class of compounds of real materials modeled as dimerized antiferromagnetically coupled spin-1/2 chains susceptible to spin-1 impurity doping and that include such examples as the strontium cuprate Sr14Cu24O41 and vanadium pyrophosphate (VO)2P2O7. Sodium vanadate, NaV2O5, a quarter-filled two-leg ladder compound that was originally thought to be composed of magnetic legs and nonmagnetic legs was subsequently shown to be a charge ordered system below room temperature. Initial models of the material as isolated antiferromagnetically coupled spin-1/2 chains can be mapped to the subsequently resolved magnetic system of the charge ordered state with the spins on molecular orbitals of V-O-V rungs antiferromagentically coupled between neighboring rungs either on the same ladder or on neighboring ladders. Comparing the model's dynamic structure factor to inelastic neutron scattering data could distinguish between the proposed spin coupling schemes and consequently reveal the details of the interaction between the lattice and the "zig-zag" charge density as well as the magnetic stabilization of the low temperature phase. Copper germanate, the first and only known inorganic spin-Peierls compound, exhibits a strong temperature dependent exchange coupling pattern among the spin-1/2 Cu2+ chains. Above the spin-Peierls transition temperature an unmodulated antiferromagnetic nearest neighbor exchange interaction J1 competes with an antiferromagnetic next nearest neighbor interaction J2. The next nearest neighbor exchange represents an effective coupling equivalent to the net effect of all longer range exchange interactions. This frustration to the nearest neighbor spin exchange produces a distinctive magnetic susceptibility χ (T) much different from the Bonner-Fisher susceptibility of the spin-1/2 Heisenberg chain with nearest neighbor antiferromagnetic exchange only. The ratio of the second to first nearest neighbor couplings α = J2/J1 ≈ 0.36 is sufficient to open a spontaneous gap in the spin-wave excitation spectrum at low temperatures. Below TSP = 14K, the dynamical structure factor is used to fit the dimerization δ and the exchange interactions J1 and J2 to the inelastic neutron scattering data of CuGeO3 at T = 10K. It is found that both δ and α increase significantly at lower temperatures, relative to the values obtained in the high temperature phase and at the onset of dimeration at 14 K. Static structure factor calculations how Scattering inconsistent with the δ−J1−J2 model and can be attributed primarily to the phonon degrees of freedom but possibly also to the couplings between chains in the b and a directions of the crystal lattice. Structurally dimerized compounds of antiferromagnetic spin-1/2 chains possess dynamic structure factor and magnetic susceptibility features that are very sensitive to doping with magnetic and nonmagnetic impurities. It is shown that the effects of spin-1 impurities are very similar to those of nonmagnetic (S = 0) impurities if the coupling between impurity and native spin is about the same magnitude as or larger than between native S = 1/2 spins. The microscopic origins of the similarity can be appreciated with the consideration that neighboring spins to the impurity find it more energetically favorable to couple to the impurity than the other spins of the lattice. Thus the spin-1 impurity and its neighboring S = 1/2 spins decouple from the lattice, creating a nonmagnetic break in the chain. The spins that now observe a nonmagnetic cluster on one side can couple to the S = 1/2 spins on the other side with a higher AF correlation than in the pure compound since they do not have to have a fluctuating dimer resonating between two neighbors. Such enhancements to the local correlations can stabilize the global AF order in one dimensional compounds as shown in theoretical and experimental studies of CuGeO3. ED calculations of the dynamic structure factor show that S(q, ω) developes states with ω = 0 in the gap at q = π. Magnetic susceptibilites in the presense of spin-1 impurities were obtained for 80 site chains by QMC simulations and for 16 site chains by ED calculations. An impurity concentration-dependent second peak in the low temperature region reveals an increasing second maximum with T as observed in doped copper germanates and suggests that the results of the study are applicable even to phonon mediated dimerization. This is not immediately apparent because spin-phonon coupling could distort the dimerization patterns. However, experimental and theoretical results show that elastic coupling between chains can stabilize the dimerization pattern due to the impurities and the results will still apply.
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