Extension of DMRG by explicit construction of Density Matrices

在有限温度密度矩阵重整化群（DMRG）中，通常通过切割一个圆柱体来构造密度矩阵，它们对应于有限温度一维量子系统。在这种情况下，2D系统（=圆柱体）的拓扑结构不同于平面，密度矩阵的常规和简单构造在自由能最小的意义上并不总是最有效的。这是因为在圆柱体周围存在信息交换，而传统方法没有适当地包括该信息交换。三维经典系统的DMRG也存在类似的困难，因此我们回到DMRG的原理，从张量积表示的变分态及其改进的角度考虑密度矩阵的有效构造。结果，我们得到了一个优化张量积状态的方程。此外，我们还发现该方程可用角转移重整化群方法（CTMRG）数值求解，并将由此得到的新变分方法--张量积变分近似（TPVA）应用于三维Ising和Potts模型，得到了相变温度和潜热，我们还将CTMRG方法应用于16顶点模型，绘制了16顶点模型的相图，在参数区域，这是迄今为止没有研究。

In the finite temperature density matrix renormalization group (DMRG) they normally construct the density matrices by cutting a cylinder which correspond ; to the finite temperature one-dimensional (ID) quantum systems. In this case, the topology of the 2D system (= cylinder) is different from the plane, conventional and simple construction of the density matrix is not always most efficient in the sense of free-energy minimum. This is because there is information exchange around the cylinder, which is not properly included by the conventional method. There is similar difficulty in DMRG for 3D classical systems.We thus return back to the principle of DMRG, we consider an efficient construction of the density matrix from the view point of the variational states represented as tensor product and their improvements. As a result, we obtained an equation that optimizes the tensor product state. In addition, we find that the equation can be solved numerically by way of the corner transfer renormalization group method (CTMRG).We apply the new variational method thus obtained, the tensor product variational approxi-mation (TPVA) to 3D Ising and Potts models, and obtained the transition temperatures and latent heats, that are comparable to those obtained by Monte Carlo simulations.We also drawed the phase diagram of the 16-vertex model by applying CTMRG method to the 16-vertex model, in the parameter region that is not investigated so far.