Quantum quenches in a spin-1 chain with tunable symmetry

Abstract: In recent years, the dynamics of interacting quantum systems far from equilibrium have attracted significant research interest. Driven by rapid progress in quantum simulators, various non-equilibrium phenomena have now been realized experimentally. In this work, we use the time-evolving block decimation (TEBD) method to investigate the dynamics of an anisotropic spin-1 Heisenberg chain for a wide range of experimentally accessible initial states. By adjusting the parameter $J_q$ that controls the quadrupolar interaction strength, we can tune the system from a non-integrable SU(2) Heisenberg model to an integrable SU(3) Heisenberg model. We examine the local magnetization, entanglement entropy, and spin correlations, and characterize their dependence on $J_q$. We identify a new conserved quantity at the SU(3) symmetric point and provide a theoretical framework to explain our numerical observations in terms of the number of accessible states permitted by this conservation law. Our results provide a route to realize a rich array of non-equilibrium behavior in spin-1 lattice models, which can be engineered in several experimental platforms such as ultracold atoms in optical lattices.

• The paper reveals that tuning quadrupolar interactions enables a crossover between SU(2) and SU(3) symmetry, critically impacting non-equilibrium dynamics.
• The study employs TEBD to track local observables and entanglement entropy, uncovering dynamic freezing and transport phenomena across different initial states.
• The findings provide combinatorial insights into Hilbert space fragmentation and offer experimental guidance for engineering non-ergodic quantum phases in ultracold atom systems.

Quantum Quenches in Spin-1 Chains with Tunable Symmetry: A Technical Analysis

Model Definition and Symmetry Structure

The paper investigates a one-dimensional, anisotropic spin-1 chain governed by the bilinear-biquadratic Hamiltonian, with explicit inclusion of quadrupolar couplings. The central Hamiltonian is

$$H = J_z\sum_{i}S^{z}_i S^{z}_{i+1} + J_{xy}\sum_{i}\left(S^{x}_i S^{x}_{i+1} + S^{y}_i S^{y}_{i+1}\right) + J_q\sum_{i}\mathbf{Q}_i\cdot \mathbf{Q}_{i+1},$$

where $J_q$ tunes the quadrupolar interaction, interpolating the model between the SU(2) symmetric Heisenberg point ($J_q = 0$) and the SU(3) symmetric, integrable Heisenberg point ($J_q = J_{xy} = J_z$).

Figure 1: Schematic depiction of the model and the continuous tuning from SU(2) (non-integrable) to SU(3) (integrable) symmetry via the quadrupolar interaction strength $J_q/J$.

The quadrupolar operators $\mathbf{Q}_i$ account for higher-order spin-1 degrees of freedom and enable fundamentally distinct correlations and constraints compared to the spin-1/2 case, affecting both dipolar and quadrupolar order.

Initial States and Symmetry Sectors

The dynamical response post-quantum quench is characterized for a range of product initial states, naturally grouped by their $z$-magnetization and quadrupolar content. The domain-wall (DW), antiferromagnetic (AFM), nematic (NM), and phantom helix (PH) states are explicitly constructed to probe non-equilibrium behavior across various symmetry sectors.

Figure 2: Visualization of initial states; red/blue/white balls correspond to $\sigma_i = +1,-1,0$, respectively, revealing the real-space structure of DW, AFM, NM, and related states.

In addition, the implementation of generalized spin-1 phantom helix states with tunable winding angle provides access to highly nontrivial, spatially modulated far-from-equilibrium configurations.

Figure 3: Representation of phantom helix states in the $xy$-plane for several winding parameters $Q_p$, showing non-trivial spatial winding relevant for quench dynamics.

Non-equilibrium Dynamics: Observables and Numerical Methods

The study utilizes time-evolving block decimation (TEBD) to propagate the spin chain, with matrix product state representations ensuring tractability up to chain lengths $J_q$0. Diagnostic observables include:

• Local magnetization $J_q$1
• In-plane correlations $J_q$2
• Quadrupolar correlations $J_q$3
• Bipartite entanglement entropy $J_q$4
• State fidelity $J_q$5

The system presents two globally conserved quantities across all parameter settings: total magnetization $J_q$6, and, at the SU(3) point, quadratic magnetization $J_q$7. The emergence of $J_q$8 conservation at $J_q$9 is analytically demonstrated and numerically validated, leading to block-diagonalization of the Hamiltonian into vastly reduced symmetry sectors.

Dynamics Across Symmetry Regimes

Two-Site Physics

Dynamical analysis of two-site initial configurations clarifies the fate of distinct local alignments (FM, AFM, FQ, AFQ, AFQ-dipolar). Quadrupolar conservation at the SU(3) point induces strict freezing of several observables, and the decay/oscillation amplitudes and frequencies of local observables are sharply controlled by $J_q = 0$0.

Figure 4: Graphical depiction of two-site spin and quadrupolar alignments minimizing bond energy, delineating FM, AFM, FQ, and AFQ correlations.

Large-$J_q = 0$1 Quench Dynamics

The full many-body dynamics expose rich transport and entanglement behavior:

• Magnetization front propagation from DW and NPI states demonstrates $J_q = 0$2-dependent light-cone velocities; NM states show no transport due to their eigenstate character at SU(3).
• Entanglement entropy $J_q = 0$3 typically features linear growth with time, saturating at system-dependent values. However, for symmetry-protected initial states (e.g., NM, NPI at SU(3)), $J_q = 0$4 is sharply bounded—the accessible Hilbert space is dictated by $J_q = 0$5.
• State fidelity exhibits monotonic decay except for cases with sharply reduced dynamical connectivity (e.g., revivals and freezing for NM, NPI at $J_q = 0$6).

Figure 5: Time evolution from the two-site problem, comparison of magnetization and correlation function dynamics for representative alignments and $J_q = 0$7 values.

Figure 6: Local magnetization dynamics at the chain center for varying $J_q = 0$8 and initial states, illustrating state- and parameter-dependent transport and freezing.

Figure 7: Logarithm of the number of accessible states $J_q = 0$9 as a function of $J_q = J_{xy} = J_z$0 (color denotes different $J_q = J_{xy} = J_z$1), highlighting the spectrum of dynamical constraints at the SU(3) point.

These results highlight the crucial role played by symmetry-breaking/conserving structures and show that non-integrable to integrable regime crossover can produce non-monotonic and even counterintuitive dynamical phenomena.

Combinatorial Structure and State Freezing

The analysis rigorously links dynamical freezing and constrained transport/thermalization to the number of accessible eigenstates in the symmetry sector determined by $J_q = J_{xy} = J_z$2. For $J_q = J_{xy} = J_z$3, the magnetization $J_q = J_{xy} = J_z$4 controls block structure; at $J_q = J_{xy} = J_z$5, the quadratic magnetization $J_q = J_{xy} = J_z$6 sharply restricts the dynamical manifold, which is particularly evident for nematic and impurity-in-nematic states.

Figure 8: Histograms showing the density of available state-space in central bond alignment categories for different symmetry sectors, comparing SU(3) and SU(2) limits.

Phantom Helix States

Phantom helix states illustrate that although they are eigenstates in the SU(2) (Heisenberg) limit with periodic boundary conditions, breaking SU(2) via quadrupolar coupling ($J_q = J_{xy} = J_z$7) rapidly generates entanglement and destroys the initial order, with the thermalization rate increasing with $J_q = J_{xy} = J_z$8. This demonstrates that integrability at the SU(3) point does not prevent fast entangling dynamics or fast decay of observables unless further dynamical constraints apply (e.g., as in the special nematic subspaces).

Figure 9: Time evolution of key observables and correlations for phantom helix states at various winding parameters, showing the dependence of thermalization and order melting on $J_q = J_{xy} = J_z$9.

Implications and Future Perspectives

The combinatorial control of dynamical regimes by symmetry (namely, the joint conservation of $J_q/J$0 and $J_q/J$1 at SU(3)) provides powerful handles for quantum simulation and the engineering of non-ergodic or slowly thermalizing manifolds. The results directly inform ultracold atom experiments with alkaline-earth(-like) platforms, as the models are readily quantum simulatable in these systems. The findings shed light on Hilbert space fragmentation, nonergodic subspaces, and the structure of thermalization breakdown beyond the extensively studied SU(2) case.

The formalism and combinatorial methods developed in the paper can be adapted to SU($J_q/J$2) generalizations, opening pathways for future work on non-equilibrium quantum dynamics in high-symmetry lattice systems, exploration of quantum scars, and time crystal phenomena in systems with multiple competing conservation laws.

Conclusion

This work provides a comprehensive and technically robust characterization of quantum quenches in spin-1 chains with continuously tunable symmetry between SU(2) and SU(3). The combination of dynamical simulations, analytic symmetry analysis, and combinatorial Hilbert space enumeration enables detailed understanding of ergodicity breaking, dynamical freezing, and parameter-dependent quantum transport. The results have direct implications for experimental realization and theoretical classification of non-equilibrium quantum phases, and suggest promising routes for extending such analysis to higher-$J_q/J$3 SU($J_q/J$4) magnets and related designer quantum matter platforms.