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arXiv:cond-mat/9904218 · Physical Review Letters

Multichain Mean-Field Theory of Quasi-One-Dimensional Quantum Spin Systems

Anders W. Sandvik

Published Oct 11, 1999• 106 citations
arXivHTMLPDF

Abstract

A multichain mean-field theory is developed and applied to a two-dimensional system of weakly coupled $S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/2$ Heisenberg chains. The environment of a chain ${C}_{0}$ 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 $(2n+1)$-chain Hamiltonian is solved self-consistently for $n$ up to $4$. The results are compared with simulation results for the original Hamiltonian on large rectangular lattices. Both methods show that the staggered magnetization $M$ for small interchain couplings $\ensuremath{\alpha}$ behaves as $M\ensuremath{\sim}\sqrt{\ensuremath{\alpha}}$ enhanced by a multiplicative logarithmic correction.

DOI
10.1103/physrevlett.83.3069

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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\ensuremathδ{C}_{\ensuremath{\delta}}C\ensuremathδ​, \ensuremathδ=\ifmmode±\else\textpm\fi1,…,\ifmmode±\else\textpm\fin\ensuremath{\delta}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ifmmode\pm\else\textpm\fi{}1,\dots{},\ifmmode\pm\else\textpm\fi{}n\ensuremathδ=\ifmmode±\else\textpm\fi1,…,\ifmmode±\else\textpm\fin, with the edge chains C\ifmmode±\else\textpm\fin{C}_{\ifmmode\pm\else\textpm\fi{}n}C\ifmmode±\else\textpm\fin​ 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 \ensuremathα\ensuremath{\alpha}\ensuremathα behaves as M\ensuremath∼\ensuremathαM\ensuremath{\sim}\sqrt{\ensuremath{\alpha}}M\ensuremath∼\ensuremathα​ enhanced by a multiplicative logarithmic correction.
Within a mean-field theory, the disordered spin-1/2 single-chain state is shown to be unstable towards an antiferromagnetically, long-range ordered one, when any finite transverse coupling, ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$, is introduced between chains. A mean-field Hamiltonian is derived from the Heisenberg model, using an extension of the Wigner-Jordan transformation in two dimensions. The staggered magnetization and the ground-state energy are calculated for values of transverse couplings ranging from 0 to 1. The main consequence of these calculations is that the value of ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$, required to have antiferromagnetic long-range order, must only be nonzero. This result is consistent with that of Sakai and Takahashi indicating that the critical transverse coupling should be of order of the inverse of the one-dimensional staggered magnetic susceptibility. Following a qualitative argument, it can also suggest that the Luttinger-liquid state is unstable at least at nearly half-filling for the Hubbard model in the limit of large Coulomb repulsion.
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.
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 $\ensuremath{\lambda}\ensuremath{\rightarrow}0$.
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∕21∕21∕2 transverse-field Ising model.
A one-dimensional S=1 Heisenberg antiferromagnet in a magnetic field H (\ensuremath{\parallel}z axis) at T=0 is studied by numerical diagonalizations up to N=16. We give the magnetization curve at the thermodynamic limit, and derive an anomaly at Hc1{\mathit{H}}_{\mathit{c}1}Hc1​ (=\ensuremath{\Delta}), where \ensuremath{\Delta} is the Haldane gap. It is also found that the transverse spin correlation has the asymptotic form 〈S0x{\mathit{S}}_{0}^{\mathit{x}}S0x​Srx{\mathit{S}}_{\mathit{r}}^{\mathit{x}}Srx​〉\ensuremath{\sim}(-1)r{)}^{\mathit{r}})r${\mathit{r}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\eta}}}$, and the transverse staggered susceptibility ${\mathrm{\ensuremath{\chi}}}_{\mathrm{st}}^{\mathit{x}\mathit{x}}$ diverges between Hc1{\mathit{H}}_{\mathit{c}1}Hc1​ and Hc2{\mathit{H}}_{\mathit{c}2}Hc2​ (=4). The exponent \ensuremath{\eta} has a minimum (\ensuremath{\eta}\ensuremath{\simeq}0.3) at magnetization m\ensuremath{\simeq}1/3 and \ensuremath{\eta}\ensuremath{\simeq}0.5 at Hc1{\mathit{H}}_{\mathit{c}1}Hc1​ and Hc2{\mathit{H}}_{\mathit{c}2}Hc2​. If the system is quasi-one-dimensional, even small interchain couplings can create a canted N\'eel order, within a mean-field approximation for interchain interactions. This is consistent with a recent NMR measurement for Ni(C2{\mathrm{C}}_{2}C2​H8{\mathrm{H}}_{8}H8​N2{\mathrm{N}}_{2}N2​)2{)}_{2})2​NO2{\mathrm{NO}}_{2}NO2​(ClO4{\mathrm{ClO}}_{4}ClO4​) (NENP) at low temperature.