Abstract:
Vertical density matrix algorithm (VDMA), a tensor product state formulation of the ``higher-dimensional'' density matrix renormalization group, is applied to the spin 1/2 antiferromagnetic XXZ model on the checkerboard lattice. The VDMA was, in the preceding study, applied to the transverse field Ising model on the square lattice and the three-dimensional classical Ising model. In the present paper, its implementation procedure is modified in order to apply the VDMA to the XXZ model. Numerical accuracy of the VDMA is investigated for the XXZ model on the square lattice, which shows that the method gives reliable results for the ground state energy. In the frustrated region, VDMA results are compared with a simple calculation based on a magnetically disordered state. It is found that the weakly frustrated region is in the N\'{e}el ordered phase, while in the strongly frustrated region the realized phase cannot be identified clearly by the obtained results.

Abstract:
We study operators acting on a tensor product Hilbert space and investigate their product numerical range, product numerical radius and separable numerical range. Concrete bounds for the product numerical range for Hermitian operators are derived. Product numerical range of a non-Hermitian operator forms a subset of the standard numerical range containing the barycenter of the spectrum. While the latter set is convex, the product range needs not to be convex nor simply connected. The product numerical range of a tensor product is equal to the Minkowski product of numerical ranges of individual factors.

Abstract:
The uniform two-dimensional variational tensor product state is applied to the transverse-field Ising, XY, and Heisenberg models on a regular hyperbolic lattice surface. The lattice is constructed by tessellation of the congruent pentagons with the fixed coordination number being four. As a benchmark, the three models are studied on the flat square lattice simultaneously. The mean-field-like universality of the Ising phase transition is observed in full agreement with its classical counterpart on the hyperbolic lattice. The tensor product ground state in the thermodynamic limit has an exceptional three-parameter solution. The variational ground-state energies of the spin models are calculated.

Abstract:
The ground state phase diagram of spin-1/2 J1-J2 antiferromagnetic Heisenberg model on square lattice around the maximally frustrated regime (J2~0.5J1) has been debated for decades. Here we study this model using a recently proposed novel numerical method - the cluster update algorithm for tensor product states (TPSs). The ground state energies at finite sizes and in the thermodynamic limit (with finite size scaling) are in good agreement with exact diagonalization study. At the largest bond dimension available D = 9 and through finite size scaling of the magnetization order near the transition point, we accurately determine the critical point Jc=0.5321(2)J1 and the critical exponent beta = 0.499(3). In the range of 0.5321

Abstract:
Using the Hermitian tensor product description of the extremal even unimodular lattice of dimension 72 found by Nebe in 2010 we show its extremality with the methods from Coulangeons article in Acta Arith. 2000.

Abstract:
We present a new algorithm to calculate the thermodynamic quantities of three-dimensional (3D) classical statistical systems, based on the ideas of the tensor product state and the density matrix renormalization group. We represent the maximum-eigenvalue eigenstate of the transfer matrix as the product of local tensors which are iteratively optimized by the use of the ``vertical density matrix'' formed by cutting the system along the transfer direction. This algorithm, which we call vertical density matrix algorithm (VDMA), is successfully applied to the 3D Ising model. Using the Suzuki-Trotter transformation, we can also apply the VDMA to two-dimensional (2D) quantum systems, which we demonstrate for the 2D transverse field Ising model.

Abstract:
Using quantum Monte Carlo (QMC) simulations and a mean field (MF) theory, we investigate the spin-1/2 XXZ model with nearest neighbor interactions on a periodic depleted square lattice. In particular, we present results for 1/4 depleted lattice in an applied magnetic field and investigate the effect of depletion on the ground state. The ground state phase diagram is found to include an antiferromagnetic (AF) phase of magnetization $m_{z}=\pm 1/6$ and an in-plane ferromagnetic (FM) phase with finite spin stiffness. The agreement between the QMC simulations and the mean field theory based on resonating trimers suggests the AF phase and in-plane FM phase can be interpreted as a Mott insulator and superfluid of trimer states respectively. While the thermal transitions of the in-plane FM phase are well described by the Kosterlitz-Thouless transition, the quantum phase transition from the AF phase to in-plane FM phase undergo a direct second order insulator-superfluid transition upon increasing magnetic field.

Abstract:
We study a two dimensional XXZ-Ising on square-hexagon (4-6) lattice with spin-1/2. The phase diagram of the ground state energy is discussed, shown two different ferrimagnetic states and two type of antiferromagnetic states, beside of a ferromagnetic state. To solve this model, it could be mapped into the eight-vertex model with union jack interaction term. Imposing exact solution condition we find the region where the XXZ-Ising model on 4-6 lattice have exact solutions with one free parameter, for symmetric eight-vertex model condition. In this sense we explore the properties of the system and analyze the competition of the interaction parameters providing the region where it has an exact solution. However the present model does not satisfy the \textit{free fermion} condition, unless for a trivial situation. Even so we are able to discuss their critical points region, when the exactly solvable condition is ignored.