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physica status solidi (b)

Renormalization Group and Critical Exponents for a Two‐Dimensional Ising Ferromagnet with Long‐Range Interaction

S. Fakioǧlu

Published Jan 1, 1982• 3 citations
Publisher
DOI
10.1002/pssb.2221090163

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The one-dimensional Ising model with ferromagnetic interactions which decay as 1/${\mathit{r}}^{\mathrm{\ensuremath{\alpha}}}$ is considered. Using a real-space renormalization group scheme (RG) we calculate the critical temperature and the correlation-length critical exponent as a function of \ensuremath{\alpha}. General asymptotic properties are obtained for arbitrary values of the rescaling length b of the RG transformation. Several rigorous results are recovered exactly in the limit b\ensuremath{\rightarrow}\ensuremath{\infty}. We obtain a b=\ensuremath{\infty} extrapolation of the critical temperature for arbitrary values of \ensuremath{\alpha}>1, which we conjecture aproximates with high precision the exact one. In particular, we obtain the value Tc{\mathit{T}}_{\mathit{c}}Tc​/J=${\mathrm{\ensuremath{\pi}}}^{2}$/12 for the 1/r2{\mathit{r}}^{2}r2 model.
Critical behavior of the one- and two-dimensional general continuous Ising models with long-range ferromagnetic interactions decaying as ${1/r}^{d+\ensuremath{\sigma}}$ is studied using a histogram Monte Carlo technique. A continuous Ising model means that a spin can take any value between $\ensuremath{-}1$ and 1. It is found that the system exhibits a second-order phase transition with nonstandard critical exponents which depend on $\ensuremath{\sigma}.$ Results for various values of $\ensuremath{\sigma}$ will be shown and compared to predictions from renormalization-group theory. Though there is an agreement with the overall tendency predicted, there are several fundamental differences. Discussion is given.
We study the criticality of a long-range quantum ferromagnetic Ising chain with algebraically decaying interactions $1/{r}^{\ensuremath{\alpha}}$ via the fidelity susceptibility based on the exact diagonalization and the density-matrix renormalization-group techniques. We find that critical exponents change monotonically from the mean-field universality class to the short-range Ising universality class for intermediate $\ensuremath{\alpha}$, which are consistent with recent results obtained from renormalization-group techniques. In addition, we determine the critical values for $1.8\ensuremath{\le}\ensuremath{\alpha}\ensuremath{\le}3$ from the finite-size scaling of the fidelity susceptibility. Our work provides very nice numerical data from the fidelity susceptibility for the quantum long-range ferromagnetic Ising chain.