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数理解析研究所講究録

Domain Wall Renormalization Group Approach to the 2d Ising Model with External Magnetic Field (Applications of Renormalization Group Methods in Mathematical Sciences)

健一 青木, 康弘 藤井, 玉青 小林, 大輔 佐藤, 洋 富田

Published Jul 1, 2014• 0 citations
OpenAlex

Abstract

We represent the spin configuration of 2-dimensional Ising model by the do- main wall configuration defined on the dual link and formulate the domain wall renormalization group (DWRG) according to the tensor network renormalization technique.In this report, DWRG is extended to include the external magnetic field by introducing the oriented domain wall variables.The renormalization group transformation is obtained in 6 dimensional parameter space completely analyti- cally and its eigenvalues around the non-trivial fixed point is calculated to give the magnetic susceptibility exponent.

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We represent the spin configuration of 2-dimensional Ising model by the do- main wall configuration defined on the dual link and formulate the domain wall renormalization group (DWRG) according to the tensor network renormalization technique.In this report, DWRG is extended to include the external magnetic field by introducing the oriented domain wall variables.The renormalization group transformation is obtained in 6 dimensional parameter space completely analyti- cally and its eigenvalues around the non-trivial fixed point is calculated to give the magnetic susceptibility exponent.
Using a recently proposed new renormalization group method (tensor renormalization group), we analyze the Ising model on the two-dimensional square lattice. For the lowest-order approximation with two-domain wall states, it realizes the idea of coarse graining of domain walls. We write down explicit analytic renormalization transformation and prove that the picture of the coarse graining of the physical domain walls does hold for all physical renormalization group flows. We solve it to get the fixed point structure and obtain the critical exponents and the critical temperature. These results are very near to the exact values. We also briefly report the improvement using four-domain wall states.
The mixed spin-7/2 and spin-1/2 Ising model with a crystal field in the absence and presence of an external longitudinal magnetic field is investigated within the framework both of the renormalization group (RG) method and Monte Carlo simulation (MCS). First, we have determined the ground-state phase diagram. Then, using RG technique, the finite temperature phase diagrams of this model are plotted for the two- and three-dimensional cases by following the flow in the parameters space of the Hamiltonian and evaluating the free energy and its isotherm derivative at low temperatures. Our results show that the model exhibits first- and second-order phase transitions as well as critical end-points. In addition, by linearizing the RG transformation at the vicinity of the fixed points of the second-order transition, we have determined the critical exponents. The MCS is also employed to validate the results obtained previously by renormalization, and to examine the influence of an external magnetic field on the magnetic properties and hysteresis loops of the system.
The three-dimensional random Ising model with a Gaussian distribution of nearest-neighbor interactions is studied for the pure spin-glass case where the average interaction vanishes. The distribution of domain-wall energies at zero temperature is calculated using a Monte Carlo-quench algorithm to find the ground-state energy for finite lattices. A renormalization-group transformation is set up which preserves the domain-wall energy distribution when the lattice parameter is changed. In the strong-coupling regime (zero temperature) the model iterates toward strong coupling and therefore exhibits a spin-glass phase transition at nonzero temperature. The thermal exponent is $\ensuremath{\nu}=3.0\ifmmode\pm\else\textpm\fi{}1.0$ and the heat capacity exponent is $\ensuremath{\alpha}=\ensuremath{-}7\ifmmode\pm\else\textpm\fi{}3$.
We apply a recently proposed real-space renormalization-group method to the two-dimensional square Ising model in a transverse magnetic field at zero temperature. We do the calculation to first and second order in the intercell coupling; in both cases we find a nontrivial fixed point. We compare the obtained critical exponents with accepted values for the three-dimensional Ising model from high-temperature series expansions. In first order the agreement is poor, in second order we obtain good agreement with the expected values.