Physical review. B, Condensed matter

Monte Carlo study of the one-dimensional quantum Heisenberg ferromagnet near T=0

J. W. Lyklema

Published Mar 1, 1983• 49 citations

Abstract

An exact Monte Carlo method to study quantum spin systems is applied to the one-dimensional spin-\textonehalf{} Heisenberg Hamiltonian. We show that it is possible to perform accurate calculations at a very low temperature. We find a power-law divergence for the susceptibility, with a critical exponent of 1.75.

DOI: 10.1103/physrevb.27.3108

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Physical review. B, Condensed matter

Monte Carlo study of the one-dimensional quantum Heisenberg ferromagnet near T=0

J. W. Lyklema

Published Mar 1, 1983• 49 citations

Publisher

Abstract

DOI: 10.1103/physrevb.27.3108

Related papers

Quantum Monte Carlo study of the spin-1/2 Heisenberg model

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Thermodynamic properties of the spin-1/2 Heisenberg ferromagnet are calculated by using Handscomb's Monte Carlo method. Several methods of analyses are used to determine critical properties of the model. For a square lattice we find that the susceptibility diverges exponentially at zero temperature. That is, \ensuremath{\chi}\ensuremath{\sim}exp[b(J/ ${\mathit{k}}_{\mathit{B}}$ T)] with the constant b=4.5\ifmmode\pm\else\textpm\fi{}0.5, which is lower than the prediction (b=2\ensuremath{\pi}) of a modified spin-wave theory. For the simple-cubic lattice, we find that the critical temperature ${\mathit{k}}_{\mathit{B}}$ ${\mathit{T}}_{\mathit{c}}$ /J=1.68\ifmmode\pm\else\textpm\fi{}0.01, and the ratio of the exponents \ensuremath{\gamma}/\ensuremath{\nu}=2.0\ifmmode\pm\else\textpm\fi{}0.05, which are in good agreement with the estimates of the high-temperature series-expansion method.Thermodynamic properties of the spin-1/2 Heisenberg ferromagnet are calculated by using Handscomb's Monte Carlo method. Several methods of analyses are used to determine critical properties of the model. For a square lattice we find that the susceptibility diverges exponentially at zero temperature. That is, \ensuremath{\chi}\ensuremath{\sim}exp[b(J/${\mathit{k}}_{\mathit{B}}$T)] with the constant b=4.5\ifmmode\pm\else\textpm\fi{}0.5, which is lower than the prediction (b=2\ensuremath{\pi}) of a modified spin-wave theory. For the simple-cubic lattice, we find that the critical temperature ${\mathit{k}}_{\mathit{B}}$${\mathit{T}}_{\mathit{c}}$/J=1.68\ifmmode\pm\else\textpm\fi{}0.01, and the ratio of the exponents \ensuremath{\gamma}/\ensuremath{\nu}=2.0\ifmmode\pm\else\textpm\fi{}0.05, which are in good agreement with the estimates of the high-temperature series-expansion method.

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Shinji Kadowaki, Akira Ueda

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{\mathit{k}}_{\mathit{B}}

T)] with the constant b=4.5\ifmmode\pm\else\textpm\fi{}0.5, which is lower than the prediction (b=2\ensuremath{\pi}) of a modified spin-wave theory. For the simple-cubic lattice, we find that the critical temperature

{\mathit{k}}_{\mathit{B}}

{\mathit{T}}_{\mathit{c}}

/J=1.68\ifmmode\pm\else\textpm\fi{}0.01, and the ratio of the exponents \ensuremath{\gamma}/\ensuremath{\nu}=2.0\ifmmode\pm\else\textpm\fi{}0.05, which are in good agreement with the estimates of the high-temperature series-expansion method.

The Suzuki-Trotter transformation has been used to transform

N

-spin

S=\frac{1}{2}

Heisenberg and

\mathrm{XY}

chains into two-dimensional ($N\ifmmode\times\else\texttimes\fi{}2m$) classical Ising models with complicated interactions. A Monte Carlo method using multispin flipping techniques is then used to study these models for $\frac{\mathrm{kT}}{|J|}\ensuremath{\ge}0.0125$ and $m\ensuremath{\le}10$. Finite-size scaling is used to analyze the low-temperature susceptibility of the ferromagnetic Heisenberg chain, and we find a power-law divergence $\ensuremath{\chi}={T}^{\ensuremath{-}\ensuremath{\gamma}}$ with $\ensuremath{\gamma}=1.32\ifmmode\pm\else\textpm\fi{}0.13$. Difficulties in the method, particularly due to singular bonding, are discussed.

{\mathit{k}}_{\mathit{B}}

=1). For S=3/2 we obtain the exponent \ensuremath{\gamma}=1.45\ifmmode\pm\else\textpm\fi{}0.05 for the divergence of the staggered susceptibility.

{100}^{3}

spins. This enables us to compute independent estimates of \ensuremath{\nu} and \ensuremath{\gamma} from power-law fits of their critical divergencies.