High-accuracy tensor renormalization group algorithms and their applications (高精度テンソル繰り込み群法とその応用) 氏名足立大樹We propose novel real-space renormalization group algorithms for classical and quantum lattice models.Conventionally, the tensor renormalization group (TRG) algorithm proposed by Levin and Nave in 2007 and the higher-order tensor renormalization group (HOTRG) algorithm proposed by Xie et al. in 2012 have been widely used as efficient numerical renormalization methods to study large-scale classical and quantum spin systems.The accuracy of the tensor network algorithms is generally controlled by the bond dimension χ, which is the cutoff of tensor indices.With a fixed bond dimension χ, TRG can renormalize the two-dimensional squarelattice tensor network within O ( χ 5 ) computational time.However, it is difficult to apply the TRG algorithm to models in three or higher dimensions.On the other hand, HOTRG can be applied, at least in principle, to higher-dimensional models, and it has a higher accuracy than TRG when comparing with the same bond dimension, while the computational cost of O ( χ 4d-1 ) for d-dimensional models is higher than TRG even in two dimensions and too high to execute simulations for higher-dimensional models in practice.New algorithms that can perform the real-space renormalization in general dimensions with a reasonable cost are strongly demanded.High-accuracy tensor renormalization group algorithms and their applications (高精度テンソル繰り込み群法とその応用) 氏名足立大樹We propose novel real-space renormalization group algorithms for classical and quantum lattice models.Conventionally, the tensor renormalization group (TRG) algorithm proposed by Levin and Nave in 2007 and the higher-order tensor renormalization group (HOTRG) algorithm proposed by Xie et al. in 2012 have been widely used as efficient numerical renormalization methods to study large-scale classical and quantum spin systems.The accuracy of the tensor network algorithms is generally controlled by the bond dimension χ, which is the cutoff of tensor indices.With a fixed bond dimension χ, TRG can renormalize the two-dimensional squarelattice tensor network within O ( χ 5 ) computational time.However, it is difficult to apply the TRG algorithm to models in three or higher dimensions.On the other hand, HOTRG can be applied, at least in principle, to higher-dimensional models, and it has a higher accuracy than TRG when comparing with the same bond dimension, while the computational cost of O ( χ 4d-1 ) for d-dimensional models is higher than TRG even in two dimensions and too high to execute simulations for higher-dimensional models in practice.New algorithms that can perform the real-space renormalization in general dimensions with a reasonable cost are strongly demanded.

Mar 23, 20200 citations

The Real Space Dynamic Renormalization Group

Gene F. Mazenko, Oriol T. Valls

Topics in current physicsJan 1, 19823 citations

A real-space renormalization group for site and bond percolation

PETER J. REYNOLDS, Eugene Stanley, W. Klein

A real-space renormalization group is developed which renormalizes probabilities directly in the percolation problem. An exact transformation is given in one dimension, and a cluster approach is presented for other lattices. The results are in excellent agreement with series calculations for the critical percolation concentration p c (both site and bond), and in good agreement for the correlation length exponent nu p . Additionally, in one dimension, a field-like variable is included and the remaining exponents are calculated.A real-space renormalization group is developed which renormalizes probabilities directly in the percolation problem. An exact transformation is given in one dimension, and a cluster approach is presented for other lattices. The results are in excellent agreement with series calculations for the critical percolation concentration p c (both site and bond), and in good agreement for the correlation length exponent nu p . Additionally, in one dimension, a field-like variable is included and the remaining exponents are calculated.

Journal of Physics C Solid State PhysicsApr 28, 1977270 citations

Real Space Renormalization Group

A. J. Berlinsky, Amanda Harris

Graduate texts in physicsJan 1, 20194 citations

Dynamics of the one-dimensional Potts model

P. O. Weir, J. Michael Kosterlitz

We consider the q-state Potts model in one dimension for the cases q ${=2}^{m}$ , m=1,2,3,. . . . The dynamical critical exponent is calculated with use of the real-space renormalization-group method. We find z=2 in the 4- and 8-state models, which agrees with the Ising case, and speculate that this generalizes to any m.We consider the q-state Potts model in one dimension for the cases q${=2}^{m}$, m=1,2,3,. . . . The dynamical critical exponent is calculated with use of the real-space renormalization-group method. We find z=2 in the 4- and 8-state models, which agrees with the Ising case, and speculate that this generalizes to any m.

Physical review. B, Condensed matterJan 1, 198612 citations

We consider the q-state Potts model in one dimension for the cases q

{=2}^{m}

, m=1,2,3,. . . . The dynamical critical exponent is calculated with use of the real-space renormalization-group method. We find z=2 in the 4- and 8-state models, which agrees with the Ising case, and speculate that this generalizes to any m.