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arXiv:cond-mat/9705196 · Physical review. B, Condensed matter

Quasi-one-dimensional spin- 12 Heisenberg magnets in their ordered phase: Correlation functions

Fabian H. L. Eßler, Alexei M. Tsvelik, Gesualdo Delfino A

Published Nov 1, 1997• 87 citations
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Abstract

We study weakly coupled antiferromagnetic spin chains in their ordered phase by combining an exact solution of the single-chain problem with a random-phase-approximation analysis of the interchain interaction. A single chain is described by a quantum Sine-Gordon model and dynamical staggered susceptibilities are determined by employing the form-factor approach to quantum correlation functions. We consider both antiferromagnetic order encountered in quasi-one-dimensional materials such as ${\mathrm{KCuF}}_{3}$ and spin-Peierls order as found in ${\mathrm{CuGeO}}_{3}.$

DOI
10.1103/physrevb.56.11001

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We study the dynamical response of frustrated, quasi-one-dimensional spin-1/2 Heisenberg antiferromagnets at finite temperatures. We allow for the presence of a Dzyaloshinskii-Moriya interaction. We concentrate on a model of weakly coupled planes of anisotropic triangular lattices. Combining exact results for the dynamical response of one-dimensional Heisenberg chains with a random-phase approximation in the frustrated interchain couplings, we calculate the dynamical susceptibility in the disordered phase. We investigate the instability of the disordered phase to the formation of collective modes. We find a very weak instability to the formation of incommensurate magnetic order and determine the ordering temperature and wave vector. We also determine the effects of uniform magnetic fields on the ordering transition.
The ground-state properties of the quasi-one-dimensional spin-1/21/21/2 antiferromagnetic Heisenberg model is investigated by using a variational method. Spins on chains along the xxx direction are antiferromagnetically coupled with exchange J>0J>0J>0, while spins between chains in the yyy direction are coupled either ferromagnetically (${J}^{\ensuremath{'}}<0$) or antiferromagnetically (${J}^{\ensuremath{'}}>0$). The staggered and the colinear antiferromagnetic magnetizations are computed and their dependence on the anisotropy parameter $\ensuremath{\lambda}=|{J}^{\ensuremath{'}}|/J$ is analyzed. It is found that an infinitesimal interchain coupling parameter is sufficient to stabilize a long-range order with either a staggered magnetization ms{m}_{s}ms​ (${J}^{\ensuremath{'}}>0$) or a colinear antiferromagnetic magnetization mcaf{m}_{caf}mcaf​ (${J}^{\ensuremath{'}}<0$), both behaving as $\ensuremath{\simeq}{\ensuremath{\lambda}}^{1/2}$ for $\ensuremath{\lambda}\ensuremath{\rightarrow}0$.
The ordering of weakly coupled random antiferromagnetic S=1/2S=1/2S=1/2 chains, as relevant for the recent experimentally investigated spin chain materials, is considered theoretically. The one-dimensional isotropic Heisenberg model with random exchange interactions is treated numerically on finite chains with the density-matrix renormalization-group approach as well as with the standard renormalization analysis, both within the mean-field approximation for interchain coupling ${J}_{\ensuremath{\perp}}$. Results for the ordering temperature TN{T}_{N}TN​ and for the ordered moment m0{m}_{0}m0​ are presented and are both reduced with increasing disorder, agreeing with the experimental observations. The most pronounced effect of the random singlet concept appears to be a very large span of local ordered moments, becoming wider with decreasing ${J}_{\ensuremath{\perp}}$, consistent with the $\ensuremath{\mu}$-SR experimental findings.
We present a systematic study of the anisotropic spin-1/2 Heisenberg model in staggered magnetic fields in two dimensions. To mimic real materials, we consider a system of coupled, antiferromagnetic chains, whose interchain interaction can be either ferro- or antiferromagnetic. When the staggered field is commensurate with the magnetic interactions, an energy gap opens immediately and follows a power law as a function of the applied field, similar to the situation in one dimension. When the field competes with the interactions, a quantum phase transition (QPT) occurs from a gapless, magnetically ordered phase at low fields to a gapped, disordered regime. We use a continuous-time Monte Carlo method to compute the staggered moment of the ordered phases and the spin gap of the disordered phases. We deduce the phase diagrams as functions of the anisotropy ratio and the applied field, and calculate the scaling behavior of the models in both quantities. We show that in the competitive case, the staggered field acts to maintain a regime of quasi-1D behavior around the QPT, and we discuss as a consequence the nature of the crossover from one-dimensional (1D) to 2D physics.