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arXiv:cond-mat/0312572 · Physical Review B

Crossover effects in the random-exchange spin- 12 antiferromagnetic chain

Nicolas Laflorencie, Heiko Rieger, Anders W. Sandvik, Patrik Henelius

Published Aug 31, 2004• 36 citations
arXivHTMLPDF

Abstract

The random antiferromagnetic spin-$1∕2$ XX and XXZ chain is studied numerically for varying strength of the disorder, using exact diagonalization and stochastic series expansion methods. The spin-spin correlation function as well as the stiffness display a clear crossover from the pure behavior (no disorder) to the infinite randomness fixed point or random singlet behavior predicted by the real space renormalization group. The crossover length scale is shown to diverge as $\ensuremath{\xi}\ensuremath{\sim}{\mathcal{D}}^{\ensuremath{-}\ensuremath{\gamma}}$, where $\mathcal{D}$ is the variance of the random bonds. Our estimates for the exponent $\ensuremath{\gamma}$ agrees well within the error bars with the one for the localization length exponent emerging within an analytical bosonization calculation. Exact diagonalization and stochastic series expansion results for the string correlation function are also presented.

DOI
10.1103/physrevb.70.054430

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Combining quantum Monte Carlo simulations with the maximum entropy method, we study the dynamical structure factor S(k,ω)S(k,ω)S(k,ω) of spin-1/2 antiferromagnetic Heisenberg chains with various random bond distributions. We emphasize the crossover behavior in the dynamical properties from pure chain to disorder-dominated random singlet phase due to bond randomness. For the distribution corresponding to the infinite randomness fixed point, S(k,ω)S(k,ω)S(k,ω) develops broad non-spinon excitations as well as the random-singlet peak near (k,ω)=(π,0)(k, ω) = (π, 0)(k,ω)=(π,0), consistent with the known results obtained by the real-space renormalization group method. For weak disorder, however, we find clear signature of spinon excitations, reminiscent of pure spin chains, blurred by disorder. We discuss the implication for experiments on random-bond antiferromagnetic spin chains, realizable, e.g., in BaCu2_22​(Six_xx​Ge1−x_{1-x}1−x​)2_22​O7_77​.
The decay of (disorder-averaged) static spin correlation functions at T = 0 for the one-dimensional spin-1/2 XXZ antiferromagnet with uniform longitudinal coupling J ∆ and random transverse coupling Jλi is investigated by numerical calculations for ensembles of finite chains. At ∆ = 0 (XX model) the calculation is based on the Jordan-Wigner mapping to free lattice fermions for chains with up to N = 100 sites. At ∆ ̸ = 0 Lanczos diagonalizations are carried out for chains with up to N = 22 sites. The longitudinal correlation function 〈Sz 0Sz r 〉 is found to exhibit a power-law decay with an exponent that varies with ∆ and, for nonzero ∆, also with the width of the λi-distribution. The results for the transverse correlation function 〈S x 0 Sx r 〉 show a crossover from power-law decay to exponential decay as the exchange disorder is turned on. 1 Typeset using REVTEXThe combination of randomness and quantum fluctuations is well-known to be a fertile ground for interesting physical phenomena including Anderson localization. In onedimensional (1D) tight-binding systems the rule is that disorder always leads to localization. However, if the randomness is purely off-diagonal, the localization length diverges at the band center, 1 which is bound to affect the decay law of correlation functions. One particular off-diagonally disordered fermion model, the 1D half-filled tight-binding model with random hopping, is equivalent to the special XX case ( ∆ = 0) of the 1D s = 1/2 XXZ model with random transverse exchange coupling, described by the Hamiltonian H = J ∑
Stochastic series expansion quantum Monte Carlo is used to study the ground state of the antiferromagnetic spin-1 Heisenberg chain with bond disorder. Typical spin- and string-correlation functions behave in accordance with real-space renormalization group predictions for the random-singlet phase. The average string-correlation function decays algebraically with an exponent of $\ensuremath{-}0.378(6),$ in very good agreement with the prediction of $\ensuremath{-}(3\ensuremath{-}\sqrt{5})/2\ensuremath{\simeq}\ensuremath{-}0.382,$ while the average spin-correlation function is found to decay with an exponent of about $\ensuremath{-}1,$ quite different from the expected value of $\ensuremath{-}2.$ By implementing the concept of directed loops for the spin-1 chain we show that autocorrelation times can be reduced by up to two orders of magnitude.
We reconsider the random bond antiferromagnetic spin-1/2 chain for weak disorder and demonstrate the existence of crossover length scale x_W that diverges with decreasing strength of the disorder. Recent DMRG calculations [Phys. Rev. Lett. 89, 127202 (2002); cond-mat/0111027] claimed to have found evidence for a non-universal behavior in this model and found no indications of a universal infinite randomness fixed point (IRFP) scenario. We show that these data are not in the asymptotic regime since the system sizes that have been considered are of the same order of magnitude or much smaller than the crossover length x_W. We give a scaling form for the xx-spin-correlation function that takes this crossover length into account and that is compatible with the IRFP scenario and a random singlet phase.
The decay of (disorder-averaged) static spin correlation functions at T=0 for the one-dimensional spin-1/2 XXZ antiferromagnet with uniform longitudinal coupling JΔ and random transverse coupling Jλi is investigated by numerical calculations for ensembles of finite chains. At Δ=0 (XX model) the calculation is based on the Jordan-Wigner mapping to free lattice fermions for chains with up to N=100 sites. At Δ≠0 Lanczos diagonalizations are carried out for chains with up to N=22 sites. The longitudinal correlation function 〈Sz0Szr〉 is found to exhibit a power-law decay with an exponent that varies with Δ and, for nonzero Δ, also with the width of the λi-distribution. The results for the transverse correlation function 〈Sx0Sxr〉 show a crossover from power-law decay to exponential decay as the exchange disorder is turned on.
The effects of randomness and dimerization on the spin-1/2 XX\mathrm{XX}XX chain are studied by a mapping to free fermions. Results are presented for the transverse and longitudinal components of the average and typical spin- and string-correlation functions. Contrary to previous numerical evidence, the decay exponents of all the above correlation functions are found to be in good agreement with the theoretical renormalization-group predictions for the random singlet phase. We also present crossover functions for the above correlation functions in the random dimer phase. Disorder averages have been taken for system sizes up to N=1024,N=1024,N=1024, about ten times larger than in previous studies.