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arXiv:cond-mat/0606790 · Physical Review B

Universal reduction of effective coordination number in the quasi-one-dimensional Ising model

Synge Todo

Published Sep 22, 2006• 6 citations
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Abstract

Critical temperature of quasi-one-dimensional general-spin Ising ferromagnets is investigated by means of the cluster Monte Carlo method performed on infinite-length strips, $L\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$ or $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$. We find that in the weak interchain coupling regime the critical temperature as a function of the interchain coupling is well-described by a chain mean-field formula with a reduced effective coordination number, as the quantum Heisenberg antiferromagnets recently reported by Yasuda et al. [Phys. Rev. Lett. 94, 217201 (2005)]. It is also confirmed that the effective coordination number is independent of the spin size. We show that in the weak interchain coupling limit the effective coordination number is, irrespective of the spin size, rigorously given by the quantum critical point of a spin-$1∕2$ transverse-field Ising model.

DOI
10.1103/physrevb.74.104415

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The large-NNN expansion of the quasi-two-dimensional quantum nonlinear $\ensuremath{\sigma}$ model is used in order to establish experimentally applicable universal scaling relations for the quasi-two-dimensional Heisenberg antiferromagnet. We show that, at $N=\ensuremath{\infty}$, the renormalized coordination number introduced by Yasuda et al. [Phys. Rev. Lett. 94, 217201 (2005)] is a universal number in the limit of ${J}^{\ensuremath{'}}∕J\ensuremath{\rightarrow}0$. Moreover, similar scaling relations proposed by Hastings and Mudry [Phys. Rev. Lett. 96, 027215 (2006)] are derived at $N=\ensuremath{\infty}$ for the three-dimensional static spin susceptibility at the wave vector $(\ensuremath{\pi},\ensuremath{\pi},0)$, as well as for the instantaneous structure factor at the same wave vector. We then use 1∕N1∕N1∕N corrections to study the relation between interplane coupling, correlation length, and critical temperature, and show that the universal scaling relations lead to logarithmic corrections to previous mean-field results.
A multichain mean-field theory is developed and applied to a two-dimensional system of weakly coupled S=1/2S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/2S=1/2 Heisenberg chains. The environment of a chain C0{C}_{0}C0​ is modeled by a number of neighboring chains ${C}_{\ensuremath{\delta}}$, $\ensuremath{\delta}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ifmmode\pm\else\textpm\fi{}1,\dots{},\ifmmode\pm\else\textpm\fi{}n$, with the edge chains ${C}_{\ifmmode\pm\else\textpm\fi{}n}$ coupled to a staggered field. Using a quantum Monte Carlo method, the effective (2n+1)(2n+1)(2n+1)-chain Hamiltonian is solved self-consistently for nnn up to 444. The results are compared with simulation results for the original Hamiltonian on large rectangular lattices. Both methods show that the staggered magnetization MMM for small interchain couplings $\ensuremath{\alpha}$ behaves as $M\ensuremath{\sim}\sqrt{\ensuremath{\alpha}}$ enhanced by a multiplicative logarithmic correction.
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We study the quenching of the Haldane gap in quasi-one-dimensional systems of weakly coupled spin-1 antiferromagnetic Heisenberg chains. The critical interchain coupling Jc required to stabilize long range magnetic order can be accurately determined from large scale quantum Monte Carlo calculations. Several different geometries of coupled chains are studied, illustrating the dependence of Jc on the coordination of chains. For bipartite geometries, ferromagnetically coupled chains yield similar magnitudes for Jc.
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