Introduction to modelling radical pair quantum spin dynamics with tensor networks

TL;DR

This work presents a scalable, fully quantum framework for simulating radical-pair spin dynamics with large nuclear baths using tensor-network representations. By combining $MPS$, $vMPDO$, and $LPMPS$ with TDVP propagation and Lindblad-type relaxation, the authors overcome the exponential growth of Hilbert and Liouville spaces to open-system dynamics, benchmarking up to 18–60 nuclei and applying the method to biologically relevant cryptochrome-inspired systems. Key results include quantitative convergence benchmarks showing how bond dimensions and field strength affect accuracy, and demonstration of anisotropic magnetosensitivity in a two-radical-pair model, linking nuclear environment and field orientation to spin-selective yields. The framework provides a robust computational tool for spin chemistry, quantum biology, and spintronics, offering insights into avian magnetoreception and enabling predictive investigations with ab initio hyperfine and dipolar couplings in complex molecular networks. Code and tooling are available publicly, broadening accessibility and reproducibility for the community.

Abstract

Radical pairs (also known as spin qubit pairs, electron-hole pairs) are transient reaction intermediates that are found and utilised in all areas of science. Radical pair spin dynamics simulations including all nuclear spins have been a computational barrier due to exponential scaling memory requirements. We address this issue with a tensor network method for accurately simulating the full open quantum dynamics of radical pair systems, explicitly accounting for hyperfine interactions with up to 30 nuclear spins with additional benchmarking including 60 nuclei. By employing the matrix product state (MPS) and matrix product density operator (MPDO) representations, we mitigate the exponential scaling of Hilbert and Liouville spaces typically encountered in full quantum non-Markovian treatments. We demonstrate the power of these methods with biologically relevant flavin-tryptophan radical pair systems, where we investigate electron hopping processes between multiple radical pairs using Lindblad jump operators. These simulations precisely capture anisotropic spin dynamics, clearly identifying orientational dependence of the magnetic field, which enhances or diminishes the spin-selective product yield. These directional sensitivities highlight the critical dependence of the nuclear environment and underscore the necessity of fully quantum treatments in spin biophysics, offering critical insights into avian magnetoreception mechanisms. This work provides a robust computational framework applicable to a broad range of scientific realms, which include spin chemistry, quantum biology, and spintronics.

Figures (10)

• Figure 1: Radical-pair model and workflow. Schematic of a typical radical-pair system. Spin evolution is computed with tensor-network simulators (MPS, vMPDO, LPMPS), and spin-selective reaction yields are obtained from a kinetic scheme driven by cumulative (time-integrated) state populations produced by the spin dynamics.
• Figure 2: Tensor-network representations and time evolution. (a) Schematic tensor-network layout for a radical-pair system with five nuclear spins. Purple nodes denote nuclear spins on molecule $i=1$, light-blue nodes denote the two electronic-spin sites, and orange nodes denote nuclear spins on molecule $i=2$. (b) Illustration of time evolution on the MPS/MPDO manifold via tangent-space projection under the time-dependent variational principle.
• Figure 3: Convergence and classical-quantum comparison for an 18-nuclear-spin radical pair. (a) Population dynamics from classical-vector approaches (Schulten-Wolynes, SW; semiclassical, SC) compared against tensor-network MPS and the full wavefunction reference. (b) Convergence with bond dimension for $m\in\{16, 64, 128\}$ (stochastic MPS with $K=4096$ trajectories) and $\chi, r \in \{256, 1024, 1536\}$ (vMPDO/LPMPS). Each trace shows the trajectory-averaged diagonal of the reduced density matrix. (c) Dependence of convergence on magnetic-field strength.
• Figure 4: Electron hopping geometry and anisotropic singlet yields. (a) Sequential electron transfer from $\text{Trp}_{\text{A}}$ to $\text{Trp}_{\text{D}}$ and relative orientations of aromatic backbones in the crystal. Spin bases are defined in the FAD frame, with the $z$-axis normal to the isoalloxazine plane and right-handed hyperfine tensors. (b) Cumulative singlet yield $\Phi^W_{\mathrm{ref}}(t)$ for different hyperfine-cutoff thresholds. Included nuclear counts are $\left(N_{\mathrm{FAD}}, N_{\mathrm{C}}, N_\mathrm{D}\right) = (1, 3, 3)$ for $\mathrm{cutoff} > 0.5 \;\text{mT}\xspace$, $\left(N_{\mathrm{FAD}}, N_{\mathrm{C}}, N_\mathrm{D}\right) = (5, 6, 5)$ for $\mathrm{cutoff} > 0.3 \;\text{mT}\xspace$, and $\left(N_{\mathrm{FAD}}, N_{\mathrm{C}}, N_\mathrm{D}\right) = (12, 9, 9)$ for $\mathrm{cutoff} > 0.1 \;\text{mT}\xspace$. (c-e) Transient singlet-yield ratios vs. magnetic-field azimuth $\theta$, referenced to the azimuthal average. Panel (c) uses cutoff > $0.3\;\text{mT}\xspace$. Panel (d) uses cutoff > $0.1\;\text{mT}\xspace$. Panel (e) uses cutoff > $0.1\;\text{mT}\xspace$ and strong field $|B| = 5.0 \;\text{mT}\xspace$.
• Figure 5: Tensor network diagram of the operation $\hat{\mathcal{P}}_L^{[1:2]}\otimes\hat{\mathbb{1}}_3\otimes\hat{\mathcal{P}}_R^{[4:6]}\hat{O}$ and $\hat{\mathcal{P}}_L^{[1:3]}\otimes\hat{\mathcal{P}}_R^{[4:6]}\hat{O}$ on MPS $\ket{X}$ to obtain $\ket{X^\prime}$.