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arXiv:1906.06060 · Physical review. D/Physical review. D.

Phase transition of four-dimensional Ising model with higher-order tensor renormalization group

Shinichiro Akiyama, Y. Kuramashi, Takumi Yamashita, Yusuke Yoshimura

Published Sep 23, 2019• 54 citations
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Abstract

We apply the higher-order tensor renormalization group to the four-dimensional ferromagnetic Ising model, which has been attracting interest in the context of the triviality of the scalar ${\ensuremath{\phi}}_{d=4}^{4}$ theory. We investigate the phase transition of this model with the higher-order tensor renormalization group enlarging the lattice size up to $102{4}^{4}$ with parallel computation. The results for the internal energy and the magnetization are consistent with the weak first-order phase transition.

DOI
10.1103/physrevd.100.054510

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We apply the higher-order tensor renormalization group to the four-dimensional ferromagnetic Ising model, which has been attracting interest in the context of the triviality of the scalar \ensuremathϕd=44{\ensuremath{\phi}}_{d=4}^{4}\ensuremathϕd=44​ theory. We investigate the phase transition of this model with the higher-order tensor renormalization group enlarging the lattice size up to 10244102{4}^{4}10244 with parallel computation. The results for the internal energy and the magnetization are consistent with the weak first-order phase transition.
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We investigate the phase transition of the four-dimensional single-component ${\ensuremath{\phi}}^{4}$ theory on the lattice using the tensor renormalization group method. We have examined the hopping parameter dependence of the bond energy and the vacuum condensation of the scalar field $⟨\ensuremath{\phi}⟩$ at a finite quartic coupling $\ensuremath{\lambda}$ on large volumes up to V=10244V=102{4}^{4}V=10244 in order to detect the spontaneous breaking of the Z2{\mathbb{Z}}_{2}Z2​ symmetry. Our results show that the system undergoes the weak first-order phase transition at a certain critical value of the hopping parameter. We also make a comparative study of the three-dimensional ${\ensuremath{\phi}}^{4}$ theory and find that the properties of the phase transition are consistent with the universality class of the three-dimensional Ising model.
Phase transitions of the J_{1}-J_{2} Ising model on a square lattice are studied using the higher-order tensor renormalization group (HOTRG) method. This system involves a competition between the ferromagnetic interaction J_{1} and antiferromagnetic interaction J_{2}, and in previous studies, weak first-order and second-order transitions were observed near the ratio g=J_{2}/|J_{1}|=1/2. It has also been suggested that the universality class of the second-order phase transition connected to the first-order transition line for g>1/2 belongs to the Ashkin-Teller class, which is characterized by a continuously varying critical exponent with g, as predicted by field-theoretical and other studies. Our results, based on the HOTRG calculations for significantly larger sizes, indicate that the region of the first-order transition is marginally narrower than that in previous studies. Furthermore, it is suggested that the region where the critical exponent changes does not necessarily coincide with the Ashkin-Teller region.
The phase transition of the Ising model is investigated on a planar lattice that has a fractal structure. On the lattice, the number of bonds that cross the border of a finite area is doubled when the linear size of the area is extended by a factor of 4. The free energy and the spontaneous magnetization of the system are obtained by means of the higher-order tensor renormalization group method. The system exhibits the order-disorder phase transition, where the critical indices are different from those of the square-lattice Ising model. An exponential decay is observed in the density-matrix spectrum even at the critical point. It is possible to interpret that the system is less entangled because of the fractal geometry.
We use the higher-order tensor renormalization group method to study the two-dimensional generalized XY model that admits integer and half-integer vortices. This model is the deformation of the classical XY model and has a rich phase structure consisting of nematic, ferromagnetic, and disordered phases and three transition lines belonging to the Berezinskii-Kosterlitz-Thouless and Ising class. We explore the model for a wide range of temperatures, T , and the deformation parameter, Δ , and compute specific heat along with integer and half-integer magnetic susceptibility, finding both Berezinskii-Kosterlitz-Thouless-like and Ising-like transitions and the region where they meet. Published by the American Physical Society 2024
The critical behavior of the classical Ising model on a three-dimensional fractal lattice with Hausdorff dimension dH=ln⁡32/ln⁡4=2.5d_H = \ln32 / \ln4 = 2.5dH​=ln32/ln4=2.5 is investigated using the higher-order tensor renormalization group (HOTRG) method. We determine the critical temperature Tc≈2.65231T_c \approx 2.65231Tc​≈2.65231 and the critical exponents for magnetization β≈0.059β\approx 0.059β≈0.059 and field response δ≈35δ\approx 35δ≈35. Unlike a previously studied 2D fractal with dH≈1.792d_H \approx 1.792dH​≈1.792, the specific heat for this 3D fractal exhibits a divergent singularity at TcT_cTc​. The results are compared with those for regular lattices and other fractal structures to elucidate the role of dimensionality in critical phenomena.