More is different.
God is in the details.
- Kenji Harada
- Using a tensor renormalization group method with oblique projectors for an anisotropic tensor network, we confirm that the rescaled spectrum of transfer matrices at nonequilibrium critical points in the (1+1)-dimensional directed percolation, a canonical model of nonequilibrium critical phenomena, is scale-invariant and its structure is universal.
- 6 pages, 7 figures
- Critical exponents in coupled phase-oscillator models on small-world networks
- Ryosuke Yoneda, Kenji Harada, Yoshiyuki Y. Yamaguchi
- A coupled phase-oscillator model consists of phase-oscillators, each of which has the natural frequency obeying a probability distribution and couples with other oscillators through a given periodic coupling function. This type of models is widely studied since it describes the synchronization transition, which emerges between the non-synchronized state and partially synchronized states, and which is characterized by the critical exponents. Among them, we focus on the critical exponent defined by coupling strength dependence of the order parameter. The synchronization transition is not limited in the all-to-all interaction, whose number of links is of O(N2) with N oscillators, and occurs in small-world networks whose links are of O(N). In the all-to-all interaction, values of the critical exponent depend on the natural frequency distribution and the coupling function, classified into an infinite number of universality classes. A natural question is in small-world networks, whether the dependency remains irrespective of the order of links. To answer this question we numerically compute the critical exponent on small-world networks by using the finite-size scaling method with coupling functions up to the second harmonics and with unimodal and symmetric natural frequency distributions. Our numerical results suggest that, for the continuous transition, the considered models share the critical exponent 1/2, and that they are collapsed into one universality class.
- 7 pages, 7 figures
- Finite-m scaling analysis of Berezinskii-Kosterlitz-Thouless phase transitions and entanglement spectrum for the six-state clock model
- Physical Review E 101, 062111 (2020)
- Hiroshi Ueda, Kouichi Okunishi, Kenji Harada, Roman Krčmár, Andrej Gendiar, Seiji Yunoki, and Tomotoshi Nishino
- We investigate the Berezinskii-Kosterlitz-Thouless transitions for the square-lattice six-state clock model with the corner-transfer matrix renormalization group (CTMRG). Scaling analyzes for effective correlation length, magnetization, and entanglement entropy with respect to the cutoff dimension m at the fixed point of CTMRG provide transition temperatures consistent with a variety of recent numerical studies. We also reveal that the fixed point spectrum of the corner transfer matrix in the critical intermediate phase of the six-state clock model is characterized by the scaling dimension consistent with the c=1 boundary conformal field theory associated with the effective Z_6 dual sine-Gordon model.
- 7 pages, 7 figures
- Conference: Tensor Network States: Algorithms and Applications (TNSAA) 2019-2020
- Invited talk: "New numerical approaches for directed percolation"
- Date: Dec. 4, 2019
- Conference dates: Dec. 4-6, 2019
- Venue: NCCU, Taipei, TAIWAN
- Entropy Governed by the Absorbing State of Directed Percolation
- Physical Review Letters 123, 090601 (2019)
- Kenji Harada and Naoki Kawashima
- We investigate the informational aspect of (1+1)-dimensional directed percolation, a canonical model of a nonequilibrium continuous transition to a phase dominated by a single special state called the “absorbing” state. Using a tensor network scheme, we numerically calculate the time evolution of state probability distribution of directed percolation. We find a universal relaxation of Rényi entropy at the absorbing phase transition point as well as a new singularity in the active phase, slightly but distinctly away from the absorbing transition point. At the new singular point, the second-order Rényi entropy has a clear cusp. There we also detect a singular behavior of “entanglement entropy,” defined by regarding the probability distribution as a wave function. The entanglement entropy vanishes below the singular point and stays finite above. We confirm that the absorbing state, though its occurrence is exponentially rare in the active phase, is responsible for these phenomena. This interpretation provides us with a unified understanding of time evolution of the Rényi entropy at the critical point as well as in the active phase.
Monte Carlo simulations
This demonstration shows a Monte Carlo simulation of the two-dimensional Ising model by three algorithms: Metropolis, Swendsen-Wang, and Wolff algorithms.To demo
Assistant Professor, Graduate School of Informatics, Kyoto University, Japan.
Room 203, Research Bldg. No.8, Yoshida Campus, Kyoto Univ., Kyoto, 606-8501, Japan. Map (No.59)