Critical behaviors of Ising model in a fractal lattice

Abstract

Fractal lattices are a special kind of lattice: they have non-integer Hausdorff dimensions and break the translation invariance. Studying these lattices can help us understand the influence of non-integer dimensions and lacking of translational symmetry on critical behaviors. We study the Ising model in a fractal lattice with a non-integer dimension of <inline-formula><tex-math id="M1">\begin{document}$\log_4(12)\approx 1.7925$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20222432_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20222432_M1.png"/></alternatives></inline-formula>by using the higher order tensor network renormalization group (HOTRG) algorithm. The partition function is represented in terms by a tensor network, and is finally calculated by a coarse graining process based on higher order singular value decomposition. When the truncation length and the time of coarse graining increase, the results are found convergent. Magnetic moment, internal energy and correlation properties are calculated by inserting impurity tensors into the tensor network at different temperatures and in different external magnetic fields. The magnetic susceptibility is obtained by differentiating the magnetic moment with respect to the magnetic field, and the capacity is calculated by differentiating the internal energy with respect to the temperature. Our numerical results show that there is a continuous order-disorder phase transition in this system, and the critical temperature is found to be <inline-formula><tex-math id="M2">\begin{document}$T_{\rm{c}}/J = 1.317188$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20222432_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20222432_M2.png"/></alternatives></inline-formula>. Physical quantities show singular behaviours around the critical point, and the correlation length is found to be divergent at the critical point, which is consistent with the result of the renormalization group theory. The corresponding critical exponent is obtained by fitting the numerical data around the critical point. We also calculate the critical exponents at different positions by inserting impurity tensors into different places of the lattice. Owing to the lack of translational symmetry, it is found that the critical exponents <i>α</i>, <i>β</i>, <i>δ</i> fitted at different positions vary, but the critical exponent <i>γ</i> remains almost the same. From the scaling hypothesis, it can be deduced that the critical exponents satisfy the hyperscaling relations which contain the dimension of the lattice. Our numerical results show that all of the hyperscaling relations are satisfied when the fractional dimension and the critical exponents we have obtained are substituted into them on some sites of the fractal lattice, but only two of the four hyperscaling relations are satisfied on other sites.