Phase diagrams and critical behavior of the quantum spin-1/2 XXZ model on diamond-type hierarchical lattices

TL;DR

This study investigates the phase diagrams and critical behavior of the quantum spin-1/2 XXZ model on diamond-type hierarchical lattices using real-space renormalization, revealing phase transitions, fixed points, and critical exponents across different fractal dimensions.

Contribution

It provides the first detailed analysis of the XXZ model's phase behavior on fractal lattices with varying dimensions, including critical exponents and fluctuation effects.

Findings

Finite temperature phase transitions exist for the isotropic Heisenberg limit.

Critical exponents match classical models for specific fractal dimensions.

Quantum fluctuations are stronger in XY-like models at low temperatures.

Abstract

In this paper, the phase diagrams and the critical behavior of the spin-1/2 anisotropic XXZ ferromagnetic model (the anisotropic parameter {\Delta}\in(-\infty,1]) on two kinds of diamond-type hierarchical (DH) lattices with fractal dimensions d_{f}=2.58 and 3, respectively, are studied via the real-space renormalization group method. It is found that in the isotropic Heisenberg limit ({\Delta}=0), there exist finite temperature phase transitions for the two kinds of DH lattices above. The systems are also investigated in the range of -\infty<{\Delta}<0 and it is found that they exhibit XY-like fixed points. Meanwhile, the critical exponents of the above two systems are also calculated. The results show that for the lattice with d_{f}=2.58, the value of the Ising critical exponent {\nu}_{I} is the same as that of classical Ising model and the isotropic Heisenberg critical exponent {\nu}_{H} is a finite value, and for the lattice with d_{f}=3, the values of {\nu}_{I} and {\nu}_{H} agree well with those obtained on the simple cubic lattice. We also discuss the quantum fluctuation at all temperatures and find the fluctuation of XY-like model is stronger than the anistropic Heisenberg model at the low-temperature region. By analyzing the fluctuation, we conclude that there will be remarkable effect of neglecting terms on the final results of the XY-like model. However, we can obtain approximate result at bigger temperatures and give qualitatively correct picture of the phase diagram.