Tree tensor networks for many-body localization in two dimensions

TL;DR

The paper tackles the challenge of simulating many-body localization in a disordered two-dimensional spin-1/2 Heisenberg model by employing tree tensor networks with a physics-informed, adaptive tree layout. It demonstrates that TTNs capture two-dimensional entanglement more effectively than MPS and are easier to contract than PEPS, enabling simulations of larger lattices and longer times. Using a TDVP-based evolution on TTNs together with a structural optimization procedure, the authors achieve higher accuracy per parameter across disorder strengths and system sizes, including near the ergodic-to-localization crossover. The results provide evidence for a finite-size crossover to slow dynamics in 2D, aligning with the view that true 2D MBL may be unstable in the thermodynamic limit, and establish TTN-based TDVP as a scalable tool for exploring disordered 2D quantum systems.

Abstract

We investigate the disordered spin-$\frac12$Heisenberg model in two dimensions and employ tree tensor networks (TTNs) with a physics-informed structural optimization of the tree layout, to simulate dynamics in the many-body localization problem. We find that TTNs are able to capture two-dimensional entanglement patterns more effectively than matrix product states (MPS) while being more efficient to contract than projected entangled pair states (PEPS) to probe larger systems and longer times. Structural optimization of the trees based on time evolution of the entanglement in the system allows to keep the necessary bond dimensions low and to maximally exploit the increased expressiveness of TTNs over MPS. In this way, we achieve more accurate results in all considered parameter regimes both below and above the ergodicity-to-localization crossover at a comparable compute-time cost.

Figures (11)

• Figure 1: MPS mappings for an $L=8$ square lattice. (a) snaking preserves row locality but separates columns by $L$, (b) Hilbert curve maintains $\mathcal{O}(\log L)$ proximity for neighbors.
• Figure 2: Projected entangled pair state for a square lattice.
• Figure 3: (a) Tree tensor network with one physical index per tensor. (b) The norm of a TTN. (c) Extension of the TTN ansatz towards matrix product operators (TTO). (d) The expectation value of an observable.
• Figure 4: Step–by–step orthogonalization of the tree tensor network.
• Figure 5: Example of a TTN with a tensor at each physical lattice site.