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

从张量积变分的观点出发，我们发展了密度矩阵重整化群(DMRG)方法的新变体。利用角点转移矩阵重整化群(CTMRG)方法，解析地给出了加速DMRG中无限系统算法收敛的乘积波函数重整化群(PWFRG)方法的计算背景。应用一种特殊的转移矩阵，扩展了矩阵乘积形式的变化态的宽度，该转移矩阵是通过将重整化矩阵应用于恒等式算符而产生的。实时DMRG跟踪量子态的时间演化，存在误差堆积问题。为了改进这个问题，我们考虑了一种作用，它是由每个时间片上张量的乘积计算出来的。作用的最小化得到了系统的时间演化，从而将实时DMRG的数值算法转化为一个最小化问题，可以通过并行计算非常快速地处理。作为CTMRG方法的一个实际应用，我们观察了一个10点模型的相图。结果，我们发现了一个新的临界点，该临界点服从伊辛普适类。作为与Andre j Gendiar博士的联合研究，我们用DMRG方法观察了二维ANNNI模型的磁化壁能。观察到了从反相到无序相的直接相变，排除了浮相的存在。

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.