Tensor Product Variational Formulation for Quantum Systems

TL;DR

The paper addresses the challenge of applying DMRG to infinite 2D quantum systems and proposes a tensor-product variational approach (TPVA) that uses a uniform 2D product of local weights to create a tractable trial state. The variational state is built from a uniform isotropic IRF with 3 adjustable parameters, and its energy is evaluable via CTMRG-based calculations of IRF partition functions for both the norm and the bond structure, applied to the square-lattice $S=1/2$ XXZ model with $ ilde{H}$ chosen for convenience. For the Heisenberg ($α=1$) and XY ($α=0$) cases, the optimal parameters yield a disordered state with no long-range AF order, and the TPVA energies exceed recent Monte Carlo estimates by about 2.3% and 1.2%, respectively; the authors discuss limitations relative to earlier variational work and suggest including auxiliary variables in the local weight to improve accuracy. This work demonstrates the feasibility of TPVA for infinite 2D quantum systems and highlights a clear path to enhancement by expanding the local variational space.

Abstract

We consider a variational problem for the two-dimensional (2D) Heisenberg and XY models, using a trial state which is constructed as a 2D product of local weights. Variational energy is calculated by use of the the corner transfer matrix renormalization group (CTMRG) method, and its upper bound is surveyed. The variational approach is a way of applying the density matrix renormalization group method (DMRG) to infinite size 2D quantum systems.