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Register a new userCalculation of fermionic Green functions with Grassmann higher-order tensor renormalization group
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We develop calculational method for fermionic Green functions in the framework of Grassmann higher-order tensor renormalization group. The validity of the method is tested by applying it to three-dimensional free Wilson fermion system. We compare the numerical results for chiral condensate and two-point correlation functions with the exact ones obtained by analytical methods.
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A calculation method for higher-order moments of physical quantities, including magnetization and energy, based on the higher-order tensor renormalization group is proposed. The physical observables are represented by impurity tensors. A systematic summation scheme provides coarse-grained tensors including multiple impurities. Our method is compared with the Monte Carlo method on the two-dimensional Potts model. While the nature of the transition of the $q$-state Potts model has been known for a long time owing to the analytical arguments, a clear numerical confirmation has been difficult due to extremely long correlation length in the weakly first-order transitions, e.g., for $q=5$. A jump of the Binder ratio precisely determines the transition temperature. The finite-size scaling analysis provides critical exponents and distinguishes the weakly first-order and the continuous transitions.
We present our progress on a study of the $O(3)$ model in two-dimensions using the Tensor Renormalization Group method. We first construct the theory in terms of tensors, and show how to construct $n$-point correlation functions. We then give results for thermodynamic quantities at finite and infinite volume, as well as 2-point correlation function data. We discuss some of the advantages and challenges of tensor renormalization and future directions in which to work.
We calculate thermodynamic potentials and their derivatives for the three-dimensional $O(2)$ model using tensor-network methods to investigate the well-known second-order phase transition. We also consider the model at non-zero chemical potential to study the Silver Blaze phenomenon, which is related to the particle number density at zero temperature. Furthermore, the temperature dependence of the number density is explored using asymmetric lattices. Our results for both zero and non-zero magnetic field, temperature, and chemical potential are consistent with those obtained using other methods.
We study the $SU(2)$ gauge-Higgs model in two Euclidean dimensions using the tensor renormalization group (TRG) approach. We derive a tensor formulation for this model in the unitary gauge and compare the expectation values of different observables between TRG and Monte Carlo simulations finding excellent agreement between the two methods. In practice we find the TRG method to be far superior to Monte Carlo simulation for calculations of the Polyakov loop correlation function which is used to extract the static quark potential.
An algorithm of the tensor renormalization group is proposed based on a randomized algorithm for singular value decomposition. Our algorithm is applicable to a broad range of two-dimensional classical models. In the case of a square lattice, its computational complexity and memory usage are proportional to the fifth and the third power of the bond dimension, respectively, whereas those of the conventional implementation are of the sixth and the fourth power. The oversampling parameter larger than the bond dimension is sufficient to reproduce the same result as full singular value decomposition even at the critical point of the two-dimensional Ising model.
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