Slow Modes Reveal Conserved Quantities in Dissipative Quantum Systems.

Research reveals integrals of motion in dissipative many-body systems evolve as slow modes, exhibiting slower decay rates than other operators. Numerical analysis and perturbation theory confirm a strong correlation between these slowest-decaying modes and the system’s integrals of motion, offering a novel method for their identification.

The behaviour of complex quantum systems subject to environmental influences remains a central challenge in physics. Understanding how conserved quantities – known as integrals of motion – persist within systems experiencing dissipation is crucial for modelling realistic physical processes. Researchers at Princeton University, Tian-Hua Yang and Dmitry A. Abanin, alongside colleagues, have investigated this phenomenon in many-body systems. Their work, titled ‘Integrals of motion as slow modes in dissipative many-body operator dynamics’, demonstrates a direct link between these conserved quantities and the slowest decaying modes within the system’s dynamics, offering a novel approach to identifying integrals of motion and gaining insight into how these systems evolve when subject to energy loss.

Identifying Conserved Quantities via Dissipative Dynamics

Researchers have established a connection between integrals of motion (IOMs) – quantities remaining constant within a quantum system – and the slow modes arising in systems subject to dissipation. This alters approaches to characterising complex quantum systems by linking static conservation laws to dynamic processes. The study investigates how weak, local dissipation affects systems possessing one or more IOMs, revealing these conserved quantities manifest as slow modes within Lindbladian dynamics – a mathematical framework describing open quantum systems that evolve due to interaction with an environment. This provides a novel method for identifying IOMs by analysing the decay rates of operators, offering a powerful tool for understanding many-body systems.

The core finding centres on the observation that IOMs with limited spatial extent exhibit slower decay rates within the dissipative dynamics compared to other operators, providing a clear signature for their identification. Consequently, the eigenoperators associated with the slowest decay rates demonstrate a significant overlap with the system’s IOMs, strengthening the validity of the observed connection through both numerical simulations and perturbative calculations.

Specifically, the methodology was applied to several many-body models, including the transverse field Ising model and the Heisenberg model, showcasing its versatility. Within the anisotropic Heisenberg model (XXZ), researchers identified a domain-wall swap operator as a key IOM, manifesting as a slow mode under dissipation, suggesting a connection between emergent conserved quantities and the physics of domain walls within the system. Numerical results for the isotropic Heisenberg model (XXX) show a strong correspondence between up to 25 eigenoperators and known IOMs, further validating the approach and demonstrating its accuracy in identifying conserved quantities.

This research establishes a clear link between IOMs and slow modes in Lindbladian operator dynamics subject to weak local dissipation, providing a new lens through which to view the interplay between conservation laws and dynamic processes. IOMs with limited spatial extent function as slow modes, exhibiting slower decay rates in the Frobenius norm – a measure of the ‘size’ of a matrix – compared to generic operators, offering a quantifiable measure of their influence on the system’s evolution. Consequently, the slowest decaying eigenoperators of the Lindbladian demonstrate a substantial overlap with the IOMs of the underlying Hamiltonian, providing a direct connection between static conservation laws and dynamic dissipative processes.

Researchers identify and characterise integrals of motion (IOMs) within the transverse field Ising model (TFIM) and the Heisenberg spin chain, utilising the eigenoperator expansion as a primary numerical technique. The study demonstrates the existence of a complete set of Majorana-bilinear IOMs for the TFIM, notable for their translational invariance and, crucially, their freedom from the sign problem that often complicates numerical simulations involving Majorana operators – operators that are their own inverse. These findings advance understanding of the TFIM’s dynamics and its potential to exhibit many-body localisation, offering insights into the emergence of complex behaviour in quantum systems.

Researchers confirm known conserved quantities, such as total spin, and identify a family of quasi-local IOMs for the Heisenberg model, expanding our knowledge of the system’s symmetries. The identified methodology is free from the sign problem often encountered in simulations involving Majorana operators.

By leveraging the dynamics of dissipative systems, scientists can gain insights into the underlying symmetries and conserved quantities that govern their evolution, paving the way for new discoveries in quantum physics and materials science. Future research will focus on extending this methodology to more complex systems and exploring its potential applications in areas such as quantum information processing and materials design.