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arXiv:cond-mat/9707341

Zero Modes and Thermodynamics of Disordered Spin-1/2 Ladders

Alexander O. Gogolin, A. A. Nersesyan, Alexei M. Tsvelik, Lu Yu

Published Jul 31, 1997• 3 citations
arXivHTMLPDF

Abstract

The influence of nonmagnetic doping on the thermodynamic properties of two-leg S = 1/2 spin ladders is studied in this paper. It is shown that, for a weak interchain coupling, the problem can be mapped onto a model of random mass Dirac (Majorana) fermions. We investigate in detail the structure of the fermionic states localized at an individual mass kink (zero modes) in the framework of a generalized Dirac model. The low-temperature thermodynamic properties are dominated by these zero modes. We use the single-fermion density of states, known to exhibit the Dyson singularity in the zero-energy limit, to construct the thermodynamics of the spin ladder. In particular, we find that the magnetic susceptibility $χ$ diverges at $T\to 0$ as $1/T\ln^2(1/T)$, and the specific heat behaves as $C\propto 1/\ln^3(1/T)$. The predictions on magnetic susceptibility are consistent with the most recent results of quantum Monte Carlo simulations on doped ladders with randomly distributed impurities. We also calculate the average staggered magnetic susceptibility induced in the system by such defects.

DOI
10.48550/arxiv.cond-mat/9707341

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The influence of nonmagnetic doping on the thermodynamic properties of two-leg S = 1/2 spin ladders is studied in this paper. It is shown that, for a weak interchain coupling, the problem can be mapped onto a model of random mass Dirac (Majorana) fermions. We investigate in detail the structure of the fermionic states localized at an individual mass kink (zero modes) in the framework of a generalized Dirac model. The low-temperature thermodynamic properties are dominated by these zero modes. We use the single-fermion density of states, known to exhibit the Dyson singularity in the zero-energy limit, to construct the thermodynamics of the spin ladder. In particular, we find that the magnetic susceptibility χχχ diverges at T→0T\to 0T→0 as 1/Tln⁡2(1/T)1/T\ln^2(1/T)1/Tln2(1/T), and the specific heat behaves as C∝1/ln⁡3(1/T)C\propto 1/\ln^3(1/T)C∝1/ln3(1/T). The predictions on magnetic susceptibility are consistent with the most recent results of quantum Monte Carlo simulations on doped ladders with randomly distributed impurities. We also calculate the average staggered magnetic susceptibility induced in the system by such defects.
Quasi-one-dimensional spin-Peierls and spin-ladder systems are characterized by a gap in the spin-excitation spectrum, which can be modeled at low energies by that of Dirac fermions with a mass. In the presence of disorder, these systems can still be described by a Dirac fermion model, but with a random mass. Some peculiar properties, like the Dyson singularity in the density of states, are well known and attributed to creation of low-energy states due to the disorder. We take one step further and study single-particle correlations by means of Berezinskii's diagram technique. We find that, at low energy $\ensuremath{\epsilon}$, the single-particle Green function decays in real space like $G(x,\ensuremath{\epsilon})\ensuremath{\propto}{(1/x)}^{3/2}$. It follows that at these energies the correlations in the disordered system are strong --- even stronger than in the pure system without the gap. At distances $x\ensuremath{\sim}{L}_{\ensuremath{\epsilon}}\ensuremath{\propto}{\mathrm{ln}}^{2}(1/\ensuremath{\epsilon})$, the Green function crosses over to a standard exponential decay. The scale ${L}_{\ensuremath{\epsilon}}$ is different from the Thouless localization length ${\ensuremath{\lambda}}_{\ensuremath{\epsilon}}\ensuremath{\propto}\mathrm{ln}(1/\ensuremath{\epsilon})$. Additionally, we study the effects of boundaries on the local density of states. We find that the latter is logarithmically (in the energy) enhanced close to the boundary. This enhancement decays into the bulk as 1/x1/\sqrt{x}1/x​ saturating at $x\ensuremath{\sim}{L}_{\ensuremath{\epsilon}}.$ We also discuss some implications of these results for the spin systems and their relation to the investigations based on the real-space renormalization group approach.
Effects of nonmagnetic impurity doping on an AF spin-1/2 Heisenberg ladder system are studied by the QMC method. A single nonmagnetic impurity induces a localized spin-1/2 moment accompanied by "static" and enhanced AF correlations around it. Small and finite concentration of impurities induces a remarkable change of magnetic and thermodynamic properties with gapless excitations. It also shows rather sharp but continuous crossover around the concentration of about 4%. Above the crossover concentration, all the spins are strongly coupled participating in the enhanced and rather uniform power-law decay of the antiferromagnetic correlation. Below the crossover, each impurity forms an antiferromagnetic cluster only weakly coupled each other. For random distribution of impurities, large Curie-like susceptibility accompanied with small residual entropy is obtained at low temperatures in agreement with recent experimental observation in Zn-doped SrCu2O3SrCu_{2}O_{3}SrCu2​O3​. Temperature dependence of AF susceptibility shows power-law-like but weaker divergence than the single chain AFH in the temperature range studied.
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.
Abstract In this paper, we study the thermodynamic properties of spin-1/2 antiferromagnetic Heisenberg ladders by means of the stochastic series expansion quantum Monte Carlo technique. This includes the thermal properties of the specific heat, uniform and staggered susceptibilities, spin gap, and structure factor. Our numerical simulations are probed over a large ensemble of random realizations in a wide range of disorder strengths r, from the clean ( r=0r=0r=0 r = 0 ) case up to the diluted ( r→1r \rightarrow 1r→1 r → 1 ) limit, and for selected choices of number of legs LyL_yLy​ L y per site. Our results show some interesting phenomena, like the presence of crossing points in the temperature plane for both the specific heat and uniform susceptibility curves which appear to be universal in r, as well as a variable dependence of the spin gap in the amount of disorder upon increasing LyL_yLy​ L y . Graphical Abstract
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.
A two-leg quenched random bond disordered antiferromagnetic spin-1/2 Heisenberg ladder system is investigated by means of stochastic series expansion quantum Monte Carlo (QMC) method. Thermal properties of the uniform and staggered susceptibilities, the structure factor, the specific heat, and the spin gap are calculated over a large number of random realizations in a wide range of disorder strength. According to our QMC simulation results, the considered system has a special temperature point at which the specific heat takes the same value regardless of the strength of the disorder. Moreover, the uniform susceptibility is shown to display the same character except for a small difference in the location of the special point. Finally, the spin gap values are found to decrease with increasing disorder parameter and the smallest gap value found in this study is well above the weak coupling limit of the clean case.