New Trend in DMRG - from the Optimization of the Tensor Product Decomposition

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From the view point of the tensor product variational method, we have developed new variants of the density matrix renormalization group (DMRG) method.1. The computational background of the product wave function renormalization group (PWFRG) method, which accelerate the numerical convergence of the infinite system algorithm in DMRG, is analytically given by way of the corner transfer matrix renormalization group (CTMRG) method. The width of the variational state in the form of matrix product is extended applying a special transfer matrix, which is created by applying renormalization matrices to the identity operator.2. The real time DMRG, which trace the time evolution of quantum states, has a problem of error stacking. In order to improve the problem, we have considered a kind of action, which is calculated from product of tensors on each time slice. Minimization of the action draws the time evolution of the system, and thus the numerical algorithm of the real time DMRG is reformulated as a minimization problem, which can be treated very rapidly by parallel computation.3. As a practical application of the CTMRG method, we observe the phase diagram of a 10-vertex model. As a result, we find a new critical point, which subject to the Ising universality class.4. As a joint research with Dr. Andre j Gendiar, we observe the domain wall energy of the two-dimensional ANNNI model by the DMRG method. Phase transition directly from the anti-phase to the disordered phase is observed, and the presence of the floating phase is excluded.