Tensor renormalization group and the solution of classical lattice models

We consider the sign problem for classical spin models at complex $\ensuremath{\beta}=1/{g}_{0}^{2}$ on $L\ifmmode\times\else\texttimes\fi{}L$ lattices. We show that the tensor renormalization group method allows reliable calculations for larger Im$\ensuremath{\beta}$ than the reweighting Monte Carlo method. For the Ising model with complex $\ensuremath{\beta}$ we compare our results with the exact Onsager-Kaufman solution at finite volume. The Fisher zeros can be determined precisely with the tensor renormalization group method. We check the convergence of the tensor renormalization group method for the $O(2)$ model on $L\ifmmode\times\else\texttimes\fi{}L$ lattices when the number of states ${D}_{s}$ increases. We show that the finite size scaling of the calculated Fisher zeros agrees very well with the Kosterlitz-Thouless transition assumption and predict the locations for larger volume. The location of these zeros agree with Monte Carlo reweighting calculation for small volume. The application of the method for the $O(2)$ model with a chemical potential is briefly discussed.We consider the sign problem for classical spin models at complex $\ensuremath{\beta}=1/{g}_{0}^{2}$ on $L\ifmmode\times\else\texttimes\fi{}L$ lattices. We show that the tensor renormalization group method allows reliable calculations for larger Im$\ensuremath{\beta}$ than the reweighting Monte Carlo method. For the Ising model with complex $\ensuremath{\beta}$ we compare our results with the exact Onsager-Kaufman solution at finite volume. The Fisher zeros can be determined precisely with the tensor renormalization group method. We check the convergence of the tensor renormalization group method for the $O(2)$ model on $L\ifmmode\times\else\texttimes\fi{}L$ lattices when the number of states ${D}_{s}$ increases. We show that the finite size scaling of the calculated Fisher zeros agrees very well with the Kosterlitz-Thouless transition assumption and predict the locations for larger volume. The location of these zeros agree with Monte Carlo reweighting calculation for small volume. The application of the method for the $O(2)$ model with a chemical potential is briefly discussed.

arXiv:1309.6623Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fieldsJan 9, 201460 citations

Renormalization Group on the Lattice

Andreas Wipf

Lecture notes in physicsOct 26, 20120 citations

Tensor renormalization group centered about a core tensor

Wangwei Lan, Glen Evenbly

We propose a modified form of tensor renormalization-group algorithm for evaluating partition functions of classical statistical mechanical models on two-dimensional lattices. This algorithm coarse-grains only the rows and columns of the lattice adjacent to a single core tensor at each step, such that the lattice size shrinks linearly with the number of coarse-graining steps as opposed to shrinking exponentially as in the usual tensor renormalization group (TRG). However, the cost of this approach only scales as $O({\ensuremath{\chi}}^{4})$ in terms of the bond dimension $\ensuremath{\chi}$, significantly cheaper than the $O({\ensuremath{\chi}}^{6})$ cost scaling of TRG, whereas numerical benchmarking indicates that both approaches have comparable accuracy for the same bond dimension $\ensuremath{\chi}$. It follows that the algorithm can allow for more accurate investigation of classical models on modestly sized lattices than is possible with standard TRG.We propose a modified form of tensor renormalization-group algorithm for evaluating partition functions of classical statistical mechanical models on two-dimensional lattices. This algorithm coarse-grains only the rows and columns of the lattice adjacent to a single core tensor at each step, such that the lattice size shrinks linearly with the number of coarse-graining steps as opposed to shrinking exponentially as in the usual tensor renormalization group (TRG). However, the cost of this approach only scales as $O({\ensuremath{\chi}}^{4})$ in terms of the bond dimension $\ensuremath{\chi}$, significantly cheaper than the $O({\ensuremath{\chi}}^{6})$ cost scaling of TRG, whereas numerical benchmarking indicates that both approaches have comparable accuracy for the same bond dimension $\ensuremath{\chi}$. It follows that the algorithm can allow for more accurate investigation of classical models on modestly sized lattices than is possible with standard TRG.

arXiv:1906.09283Physical review. B./Physical review. BDec 11, 20198 citations

Non-perturbative renormalization-group approach to lattice models

N. Dupuis, K. Sengupta

arXiv:0806.4257The European Physical Journal BNov 1, 200817 citations

Tensor network algorithm by coarse-graining tensor renormalization on finite periodic lattices

Hui‐Hai Zhao, Z. Y. Xie, Tao Xiang +1 more

We develop coarse-graining tensor renormalization group algorithms to compute physical properties of two-dimensional lattice models on finite periodic lattices. Two different coarse-graining strategies, one based on the tensor renormalization group and the other based on the higher-order tensor renormalization group, are introduced. In order to optimize the tensor network model globally, a sweeping scheme is proposed to account for the renormalization effect from the environment tensors under the framework of second renormalization group. We demonstrate the algorithms by the classical Ising model on the square lattice and the Kitaev model on the honeycomb lattice, and show that the finite-size algorithms achieve substantially more accurate results than the corresponding infinite-size ones.We develop coarse-graining tensor renormalization group algorithms to compute physical properties of two-dimensional lattice models on finite periodic lattices. Two different coarse-graining strategies, one based on the tensor renormalization group and the other based on the higher-order tensor renormalization group, are introduced. In order to optimize the tensor network model globally, a sweeping scheme is proposed to account for the renormalization effect from the environment tensors under the framework of second renormalization group. We demonstrate the algorithms by the classical Ising model on the square lattice and the Kitaev model on the honeycomb lattice, and show that the finite-size algorithms achieve substantially more accurate results than the corresponding infinite-size ones.

arXiv:1510.03333Physical review. B./Physical review. BMar 9, 201640 citations

O(2)

model on $L\ifmmode\times\else\texttimes\fi{}L$ lattices when the number of states

{D}_{s}

increases. We show that the finite size scaling of the calculated Fisher zeros agrees very well with the Kosterlitz-Thouless transition assumption and predict the locations for larger volume. The location of these zeros agree with Monte Carlo reweighting calculation for small volume. The application of the method for the

O(2)

model with a chemical potential is briefly discussed.