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arXiv:1307.6613 · Physical Review B

Magnetic responses of randomly depleted spin ladders

Arthur Lavarélo, Guillaume Roux, Nicolas Laflorencie

Published Oct 21, 2013• 14 citations
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Abstract

The magnetic responses of a spin-$1/2$ ladder doped with nonmagnetic impurities are studied combining both analytical and numerical methods. The regime where frustration induces incommensurability is taken into account. Several improvements are made on the results of the seminal work by Sigrist and Furusaki [M. Sigrist and A. Furusaki, J. Phys. Soc. Jpn. 65, 2385 (1996)] and deviations from the Brillouin magnetization curve due to interactions are also analyzed. We first discuss the magnetic profile around a single impurity and the effective interactions between impurities within the bond-operator mean-field theory. The results are compared to density-matrix renormalization group calculations. In particular, these quantities are shown to be sensitive to the transition to the incommensurate regime. We then focus on the behavior of the zero-field susceptibility through an effective Curie constant. At zero temperature, we give doping-dependent corrections to the results of Sigrist and Furusaki on general bipartite lattices and compute exactly the distribution of ladder clusters due to chain breaking effects. Solving the effective model with exact diagonalization and quantum Monte Carlo simulations gives the temperature dependence of the Curie constant. Its high-temperature limit is understood within a random dimer model, while the low-temperature tail is compatible with a real-space renormalization group scenario. Interestingly, solving the full microscopic model does not show a plateau corresponding to the saturation of the impurities in isotropic ladders. The second magnetic response that is analyzed is the magnetic curve. Below fields of the order of the spin gap, the magnetization process is controlled by the physics of interacting impurity spins. The random dimer model is shown to capture the bulk of the curve, accounting for the deviation from a Brillouin behavior due to interactions. The effective model calculations agree rather well with density-matrix renormalization group calculations at zero temperature and with quantum Monte Carlo simulations at finite temperature. In all, the effect of incommensurability does not display a strong qualitative effect on both the magnetic susceptibility and the magnetic curve. Consequences for experiments on the BiCu${}_{2}$PO${}_{6}$ compound and other spin-gapped materials are briefly mentioned.

DOI
10.1103/physrevb.88.134420

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The magnetic responses of a spin-1/21/21/2 ladder doped with nonmagnetic impurities are studied combining both analytical and numerical methods. The regime where frustration induces incommensurability is taken into account. Several improvements are made on the results of the seminal work by Sigrist and Furusaki [M. Sigrist and A. Furusaki, J. Phys. Soc. Jpn. 65, 2385 (1996)] and deviations from the Brillouin magnetization curve due to interactions are also analyzed. We first discuss the magnetic profile around a single impurity and the effective interactions between impurities within the bond-operator mean-field theory. The results are compared to density-matrix renormalization group calculations. In particular, these quantities are shown to be sensitive to the transition to the incommensurate regime. We then focus on the behavior of the zero-field susceptibility through an effective Curie constant. At zero temperature, we give doping-dependent corrections to the results of Sigrist and Furusaki on general bipartite lattices and compute exactly the distribution of ladder clusters due to chain breaking effects. Solving the effective model with exact diagonalization and quantum Monte Carlo simulations gives the temperature dependence of the Curie constant. Its high-temperature limit is understood within a random dimer model, while the low-temperature tail is compatible with a real-space renormalization group scenario. Interestingly, solving the full microscopic model does not show a plateau corresponding to the saturation of the impurities in isotropic ladders. The second magnetic response that is analyzed is the magnetic curve. Below fields of the order of the spin gap, the magnetization process is controlled by the physics of interacting impurity spins. The random dimer model is shown to capture the bulk of the curve, accounting for the deviation from a Brillouin behavior due to interactions. The effective model calculations agree rather well with density-matrix renormalization group calculations at zero temperature and with quantum Monte Carlo simulations at finite temperature. In all, the effect of incommensurability does not display a strong qualitative effect on both the magnetic susceptibility and the magnetic curve. Consequences for experiments on the BiCu2{}_{2}2​PO6{}_{6}6​ compound and other spin-gapped materials are briefly mentioned.
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We present a 31P^{31}\text{P}31P NMR study of the coupled spin 12\frac{1}{2}21​ ladder compound BiCu2PO6{\text{BiCu}}_{2}{\text{PO}}_{6}BiCu2​PO6​. In the pure material, intrinsic susceptibility, and dynamics show a spin gap of about $\ensuremath{\Delta}\ensuremath{\approx}35--40\text{ }\text{K}$. Substitution of nonmagnetic Zn or magnetic Ni impurity at Cu site induces a staggered magnetization which results in a broadening of the 31P^{31}\text{P}31P NMR line while susceptibility far from the defects is unaffected. The effect of Ni on the NMR line broadening is twice that of Zn, which is consistent with quantum Monte Carlo calculations assuming that Ni couples ferromagnetically to its adjacent Cu. The induced moment follows a 1/T1/T1/T temperature dependence due to the Curie-type development of the moment amplitude while its extension saturates and does not depend on impurity content or nature. This allow us to verify the generically expected scenario for impurity doping and to extend it to magnetic impurity case: in an antiferromagnetically correlated low-dimensional spin system with antiferromagnetic correlations, any type of impurity induces a staggered moment at low temperature, whose extension is not linked to the impurity nature but to the intrinsic physics at play in the undoped pure system, from one-dimensional to two-dimensional systems.
The phase diagram of an antiferromagnetic ladder with frustrating next-nearest-neighbor couplings along the legs is determined using numerical methods (exact diagonalization and density-matrix renormalization group) supplemented by strong-coupling and mean-field analysis. Interestingly, this model displays remarkable features, bridging the physics of the J1{J}_{1}J1​-J2{J}_{2}J2​ chain and of the unfrustrated ladder. The phase diagram as a function of the transverse coupling ${J}_{\ensuremath{\perp}}$ and the frustration J2{J}_{2}J2​ exhibits an Ising transition between a columnar phase of dimers and the usual rung-singlet phase of two-leg ladders. The transition is driven by resonating valence bond fluctuations in the singlet sector while the triplet spin gap remains finite across the transition. In addition, frustration brings incommensurability in the real-space spin-correlation functions, the onset of which evolves smoothly from the J1{J}_{1}J1​-J2{J}_{2}J2​ chain value to zero in the large-${J}_{\ensuremath{\perp}}$ limit. The onset of incommensurability in the spin structure factor and in the dispersion relation is also analyzed. The physics of the frustrated rung-singlet phase is well understood using perturbative expansions and mean-field theories in the large-${J}_{\ensuremath{\perp}}$ limit. Last, we discuss the effect of the nontrivial magnon dispersion relation on the thermodynamical properties of the system. The relation of this model and its physics to experimental observations on compounds that are currently investigated, such as BiCu2{}_{2}2​PO6{}_{6}6​, is eventually addressed.
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We derive and investigate the microscopic model of the quantum magnet BiCu2PO6{\text{BiCu}}_{2}{\text{PO}}_{6}BiCu2​PO6​ using band-structure calculations, magnetic susceptibility and high-field magnetization measurements, as well as exact diagonalization (ED) and density-matrix renormalization group (DMRG) techniques. The resulting quasi-one-dimensional spin model is a two-leg antiferromagnetic ladder with frustrating next-nearest-neighbor couplings along the legs. The individual couplings are estimated from band-structure calculations and by fitting the magnetic susceptibility with theoretical predictions, obtained using full diagonalizations. The nearest-neighbor leg coupling J1{J}_{1}J1​, the rung coupling J4{J}_{4}J4​, and one of the next-nearest-neighbor couplings J2{J}_{2}J2​ amount to 120--150 K while the second next-nearest-neighbor coupling is ${J}_{2}^{\ensuremath{'}}\ensuremath{\simeq}{J}_{2}/2$. The spin ladders do not match the structural chains, and although the next-nearest-neighbor interactions J2{J}_{2}J2​ and ${J}_{2}^{\ensuremath{'}}$ have very similar superexchange pathways, they differ substantially in magnitude due to a tiny difference in the O-O distances and in the arrangement of nonmagnetic PO4{\text{PO}}_{4}PO4​ tetrahedra. An extensive ED study of the proposed model provides the low-energy excitation spectrum and shows that the system is in the strong rung coupling regime. The strong frustration by the next-nearest-neighbor couplings leads to a triplon branch with an incommensurate minimum. This is further corroborated by a strong-coupling expansion up to second order in the inter-rung coupling. Based on high-field magnetization measurements, we estimate the spin gap of $\ensuremath{\Delta}\ensuremath{\simeq}32\text{ }\text{K}$ and suggest the likely presence of antisymmetric Dzyaloshinskii-Moriya anisotropy and interladder coupling J3{J}_{3}J3​. We also provide a tentative description of the physics of BiCu2PO6{\text{BiCu}}_{2}{\text{PO}}_{6}BiCu2​PO6​ in magnetic field, in the light of the low-energy excitation spectra and numerical calculations based on ED and DMRG. In particular, we raise the possibility for a rich interplay between one- and two-component Luttinger liquid phases and a magnetization plateau at 1/2 of the saturation value.
Low-temperature thermodynamic properties of the spin-1/2 Heisenberg ladder system with non-magnetic impurities are discussed using an effective Hamiltonian for the impurity-induced spins in the background of a spin liquid with a gap. It is shown that the uniform susceptibility shows Curie-like behavior in high-, intermediate-, and low-temperature regimes, with three different Curie constants. The ratio of specific heat to temperature, C/T, is divergent in the low-temperature limit, reflecting a singular distribution of the effective couplings among the impurity spins.