Majorana string simulation of nonequilibrium dynamics in two-dimensional lattice fermion systems

TL;DR

The paper tackles the challenge of simulating real-time dynamics of interacting fermions in two dimensions, where classical methods struggle due to entanglement growth and sign problems. It introduces a Heisenberg-picture Majorana-string propagation (MP) algorithm that evolves observables expanded in a Majorana string basis, with Trotter-consistent truncations to cap growth and maintain accuracy. The method is exact for quadratic (Gaussian) dynamics and extends to interacting regimes through controlled truncations, with benchmarks on 1D and 2D Fermi-Hubbard quenches showing competitive short-to-intermediate-time accuracy against MPS, fPEPS, and experimental data. Importantly, MP can incorporate variational initial states via MPO representations, enabling flexible benchmarking against state-of-the-art results, and providing a scalable classical tool for 2D fermionic dynamics relevant to cuprate-like systems and quantum simulators.

Abstract

The study of real-time dynamics of fermions remains one of the last frontiers beyond the reach of classical simulations and is key to our understanding of quantum behavior in chemistry and materials, with implications for quantum technology. Here we introduce a Heisenberg-picture algorithm that propagates observables expressed in a Majorana-string basis using a truncation scheme that preserves Trotter accuracy and aims at maintaining computational efficiency. The framework is exact for quadratic Hamiltonians--remaining restricted to a fixed low-weight sector determined by the physical observable--admits variational initial states, and can be extended to interacting regimes via systematically controlled truncations. We benchmark our approach on one- and two-dimensional Fermi-Hubbard quenches, comparing against tensor network methods (MPS and fPEPS) and recent experimental data. The method achieves high accuracy on timescales comparable to state-of-the-art variational techniques and experiments, demonstrating that controlled Majorana-string truncation is a practical tool for simulating two-dimensional fermionic dynamics.

Figures (10)

• Figure 1: Scattering of two spinless fermions on a $12\times 12$ lattice at $U/t=0$. a-c) Plot of the local densities before ($\tau/t=1.50$), during ($\tau/t=3.80$) and after ($\tau/t=4.80$) the scattering process, using a time-step $\delta \tau/t = 0.01$. MP reproduces the dynamics exactly. d) Excitation error $\sum_{i \in \mathcal{I}_e} \Delta n_i / n_{\text{tot }}$ introduced in Ref. dai2025fermionic with $\eta=0.6$ (see \ref{['eq:exc_sites']} for the definition of the excited sites $\mathcal{I}_e$ and $\eta$), compared with the data reported in Ref. dai2025fermionic for $\mathrm{W}_{\mathrm{II}}$zaletel2015time and TDVP haegeman2016unifying using MPS, and 2D isoTNS. The excitation error for Majorana propagation is mainly determined by the Trotter error, and can be made arbitrarily small.
• Figure 2: a) Expectation value of the density $n_{50,\uparrow}$ at a central site for a $1$D lattice with $100$ (spinful) sites, subject to second-order Trotter dynamics with $\delta\tau=0.12$. We compare the results from Majorana propagation at various $S$ truncations (solid colored lines) with MPS of different bond dimensions (dashed grey lines). The coefficient truncation for all Majorana propagation simulations is $\varepsilon=10^{-5}$. b) Growth of the number of Majorana strings with time at various cutoffs $S$. The right axis also shows the maximum bond dimension obtained with the MPS simulations.
• Figure 3: Space-time map of the OTOC $C^2_k(\tau):= ||[n_{50,\uparrow}(\tau), L_k]||$ of $n_{50,\uparrow}(\tau)$ with the local operator $L_k$ on a 100 site 1D chain. Panels a) and b) show the results for different coefficient truncations $\varepsilon = 10^{-9}$ (a) and $\varepsilon = 10^{-5}$ (b). In both simulations $S=4$. c) Plot of the OTOC differences $e_\varepsilon$ for $\varepsilon=10^{-5}$ against the most accurate result $\varepsilon=10^{-9}$. The full 1D lattice consists of 100 sites.
• Figure 4: Expectation values of the doublon occupation probability $n_{40,\uparrow\downarrow}$ for a ground state $\ket\psi$ prepared with $U_0/t=30$ and the different shifts in chemical potential $\mu=50, 0.1$, that is then quenched with the Fermi-Hubbard Hamiltonian with $U/t=1$. The staggered state $\ket{\psi_{st}}:=\ket{\uparrow\downarrow\cdots\uparrow\downarrow}$ corresponds to the ground state of $U_0/t=\infty, \mu> 0$. For the overlap of ground states $\ket\psi$ at finite $U_0,\mu$ with $\ket{\psi_{st}}$ we use the notation $o_\psi = |\braket{\psi}{\psi_{st}}|^2$. The MP truncations are $S=10, \varepsilon=5\cdot 10^{-6}$.
• Figure 5: a) Comparison of $p_\mathrm{hole}(\tau)$ obtained with Majorana propagation on a $7\times 7$ lattice compared to the experimental results reported in Ref. Ji2021PRX. As in the experiment, we set $U/t=8.72$. The three columns report the results for the truncations $S=2,4,6$ respectively. For each choice of $S$ we truncate at different values of $\varepsilon$. For $S=4,6$ we observe good qualitative agreement between the experimental data and the simulated ones. b) The number of strings in the observables for different choices of $S,\varepsilon$.

Theorems & Definitions (1)

• proof