On the Optimal Layout of Two-Dimensional Lattices for Density Matrix Renormalization Group

TL;DR

The paper addresses the problem of finding optimal 2D lattice layouts for DMRG by mapping to 1D chains and proposes using Hamiltonian-path layouts that minimize a geometric cost $LA_{1/2}$. It defines $LA_q(\varphi,G)=\sum_{uv\in E}\lambda(u,v)^q$ and argues that $q=1/2$ is particularly well-correlated with DMRG performance, due to entanglement considerations. The authors implement simulated annealing to minimize $C[\varphi]=LA_{1/2}(\varphi)-(N-1)$ and apply it to square-lattice AFM and related lattices, finding substantial improvements over snake or fractal-like layouts; in several cases, the same accuracy is achieved with roughly half the bond dimension $\chi$, yielding large speedups because cost scales as $\chi^3$. They conclude that $LA_{1/2}$-driven Hamiltonian-path layouts are a practical route to enhance DMRG on 2D lattices and advocate extending the approach to other lattices and non-geometric cost terms for inhomogeneous Hamiltonians.

Abstract

For quantum spin models defined on a two-dimensional lattice, we look for the best numbering of the lattice sites (a layout) that, at fixed bond dimension and other parameters of the density matrix renormalization group (DMRG) algorithm, gives the lowest value of the variational energy, maximum entropy and truncation error. We consider the conjecture that the optimal layout is a Hamiltonian path, and that it optimizes a simply computable geometric cost function. Finding the minimum of such a function, which is a variant of the minimum linear arrangement problem, provides the DMRG with an efficient layout of the lattice and improves both accuracy and convergence time. We present applications to the antiferromagnetic and spin glass spin-1/2 models on the square and triangular lattices.

Figures (10)

• Figure 1: A square lattice with a simple layout. The layout is the numbering of the vertices, which induces the graph on the bottom.
• Figure 2: Square lattice for the definition of the antiferromagnetic model in Eq.s(\ref{['eq:SqLAFM']}) and (\ref{['eq:SqL_SG']}), and triangular lattice for the model in Eq.(\ref{['eq:TlAFM']}). The little arrows represent the spin degrees of freedom.
• Figure 3: The number of paths obtained with exact enumeration with fixed extrema on adjacent sides $L=4,...,7$. The exponential fit $\ln(N)/L^2=0.36-0.95/L$. The leading term agrees within error with that for the total number of paths $0.352$ in jacobsen2007exact.
• Figure 4: Square lattice antiferromagnet: Correlation between the geometric lowest cost in Eq.(\ref{['eq:CLA12']}) and its best performance of the DMRG algorithm for $L=6$. All the paths in $L=6$ are found but only 9 of them are selected. DRMG uses TenPy with Maximum bond dimension $\chi=100$, maximum number of sweeps 100.
• Figure 5: Cost function (\ref{['eq:CLA12']}) of the generalized Hilbert curves (blue) vs the optimal path (yellow) found by simulated annealing up to $L=128$. The best fit of the data taking the last 4 points ($L=16,32,64,128$) gives $C_{\mathrm{H}}/L^2=1.376\ln L-0.694$ and $C_{\mathrm{O}}/L^2=1.36\ln L-0.90$. The optimal layout conjectured in mitchison1986optimal would give $C/L^2=1.422\ln L$. Although the scaling form is almost certainly correct, it seems that both the Hilbert curve and the optimal curve have a smaller cost than hypothesized in mitchison1986optimal.