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Efficient Simulation of the 2D Hubbard Model via Hilbert Space-Filling Curve Mapping

Ashkan Abedi, Vittorio Giovannetti, Dario De Santis

TL;DR

This work tackles the computational challenge of simulating the 2D Hubbard model by mapping the lattice to a 1D chain via the Hilbert space-filling curve, preserving locality and enabling efficient MPS-based ground-state searches. By implementing a fermionic MPS/MPO framework and comparing against the conventional snake mapping, the authors demonstrate lower variational energies at fixed bond dimension, with gains that grow with system size and are most pronounced near the physically relevant intermediate coupling $U/t$. They report reliable results up to $32\times32$ lattices under both open and periodic boundaries and explore doped regimes, achieving close agreement with state-of-the-art benchmarks while reducing computational cost. To push beyond the MPS limit, the paper also proposes Hilbert-inspired tree tensor networks (TTN) that further mitigate long-range terms at modest overhead, outlining a path toward scalable simulations of larger 2D fermionic systems.

Abstract

We investigate tensor network simulations of the two-dimensional Hubbard model by mapping the lattice onto a one-dimensional chain using space-filling curves. In particular, we focus on the Hilbert curve, whose locality-preserving structure minimizes the range of effective interactions in the mapped model. This enables a more compact matrix product state (MPS) representation compared to conventional snake mapping. Through systematic benchmarks, we show that the Hilbert curve consistently yields lower ground-state energies at fixed bond dimension, with the advantage increasing for larger system sizes and in physically relevant interaction regimes. Our implementation reaches clusters up to $32\times32$ sites with open and periodic boundary conditions, delivering reliable ground-state energies and correlation functions in agreement with established results, but at significantly reduced computational cost. These findings establish space-filling curve mappings, particularly the Hilbert curve, as a powerful tool for extending tensor-network studies of strongly correlated two-dimensional quantum systems beyond the limits accessible with standard approaches.

Efficient Simulation of the 2D Hubbard Model via Hilbert Space-Filling Curve Mapping

TL;DR

This work tackles the computational challenge of simulating the 2D Hubbard model by mapping the lattice to a 1D chain via the Hilbert space-filling curve, preserving locality and enabling efficient MPS-based ground-state searches. By implementing a fermionic MPS/MPO framework and comparing against the conventional snake mapping, the authors demonstrate lower variational energies at fixed bond dimension, with gains that grow with system size and are most pronounced near the physically relevant intermediate coupling . They report reliable results up to lattices under both open and periodic boundaries and explore doped regimes, achieving close agreement with state-of-the-art benchmarks while reducing computational cost. To push beyond the MPS limit, the paper also proposes Hilbert-inspired tree tensor networks (TTN) that further mitigate long-range terms at modest overhead, outlining a path toward scalable simulations of larger 2D fermionic systems.

Abstract

We investigate tensor network simulations of the two-dimensional Hubbard model by mapping the lattice onto a one-dimensional chain using space-filling curves. In particular, we focus on the Hilbert curve, whose locality-preserving structure minimizes the range of effective interactions in the mapped model. This enables a more compact matrix product state (MPS) representation compared to conventional snake mapping. Through systematic benchmarks, we show that the Hilbert curve consistently yields lower ground-state energies at fixed bond dimension, with the advantage increasing for larger system sizes and in physically relevant interaction regimes. Our implementation reaches clusters up to sites with open and periodic boundary conditions, delivering reliable ground-state energies and correlation functions in agreement with established results, but at significantly reduced computational cost. These findings establish space-filling curve mappings, particularly the Hilbert curve, as a powerful tool for extending tensor-network studies of strongly correlated two-dimensional quantum systems beyond the limits accessible with standard approaches.

Paper Structure

This paper contains 14 sections, 10 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Space-Filling Curves
  4. Snake Mapping
  5. Hilbert Curve Mapping
  6. Recursive Construction of the Hilbert Curve
  7. Tensor Network Implementation
  8. Optimization Algorithm
  9. Results and Discussion
  10. Open Boundary Conditions (OBC) at Half-Filling ($\langle n_{\text{occ}}\rangle=1$)
  11. Periodic Boundary Conditions (PBC) at Half-Filling ($\langle n_{\text{occ}}\rangle=1$)
  12. Doped System
  13. Tree Tensor Networks (TTN) from Hilbert-Inspired Topologies
  14. Conclusions

Figures (10)

  • Figure 1: Construction of the Hilbert curve. (a) First-order curve $\mathcal{H}_1$ connecting 4 sites. (b) Second-order curve $\mathcal{H}_2$ for a $4 \times 4$ lattice, showing quadrant divisions (TL: top-left, TR: top-right, BL: bottom-left, BR: bottom-right) and rotation directions.
  • Figure 2: Comparison of (a) Hilbert curve and (b) snake curve mappings for an $8 \times 8$ lattice. Numbers indicate the site ordering in the 1D chain.
  • Figure 3: Relative energy difference $\Delta E/|E_{\mathrm{Hilbert}}|$ between snake and Hilbert mappings for the half-filled Hubbard model under OBC at bond dimension $m=1000$. Results are shown for $4\times4$, $8\times8$, and $16\times16$ lattices as a function of $U$, where $t=1$.
  • Figure 4: (a) Energy per site for Hilbert (blue) and snake (yellow) as a function of system size for the half-filled Hubbard model with $U = 6$ and $t = 1$. (b) The increasing trend of the gap between the two demonstrates the growing advantage of the Hilbert mapping for larger systems. At fixed bond dimension $m=1000$, the energy-per-site difference grows from $0$ for $4\times4$ to $0.085$ for $32\times32$.
  • Figure 5: Ground state energy per site for the half-filled Hubbard model (OBC) with $U = 6$ and $t = 1$ as a function of bond dimension for different lattice sizes: (a) $4 \times 4$, (b) $8 \times 8$, and (c) $16 \times 16$. Results are shown for both Hilbert (blue) and snake (yellow) mappings.
  • ...and 5 more figures