Overview
This study investigates the local and quasilocal conservation laws of specific three-state interaction-round-a-face (IRF) cellular automata (CA) and their corresponding quantum deformations. The research utilizes patch-matrix-product methods to analyze two reversible three-state IRF CA rules: the species-preserving rule and the species-flipping rule, both previously introduced by Klobas and Prosen in 2022. Additionally, the analysis extends to a coherent quantum deformation that interpolates between these two classical rules.
Research Context
The cellular automata under examination are three-state IRF models, specifically the reversible species-preserving and species-flipping rules. These rules were initially presented by Klobas and Prosen in J. Phys. A 55, 094003 (2022). The present work builds upon this foundation by exploring the conservation laws inherent to these systems and their quantum counterparts.
Approach
The methodology employed in this study involves patch-matrix-product methods. This technique facilitates the investigation of conservation laws within an explicit translationally invariant ansatz. For the quantum deformation, the approach involves studying the algebraic structures to identify invariant states, quasimodes, and diagonal quasi-local charges.
Findings
Classical Rules: Species-Preserving and Species-Flipping
- The two classical rules, the species-preserving and species-flipping rules, share a one-parameter family of quasi-local conservation laws.
- This family of conservation laws has an auxiliary-dimension-three realization.
- Every regular member of this family is also identified as the first centered logarithmic derivative of an analytic auxiliary-dimension-two family.
- This auxiliary-dimension-two family passes through the identity observable.
- Within the closed span of this family and the three elementary local charges, the only non-zero Euler velocities are $\pm\sqrt{3/23}$.
- Consequently, $\sqrt{3/23}$ constitutes a rigorous lower bound on the maximal Euler speed in any larger conserved sector.
Species-Flipping Rule Specifics
- For the species-flipping rule, the value of $\sqrt{3/23}$ agrees with the reported extrapolated value of $0.361$ by Klobas and Prosen.
- This agreement is interpreted as strongly supporting the completeness of the known sound-active charges for this rule.
Species-Preserving Rule Specifics
- The species-preserving rule additionally admits a staggered generating family.
- The logarithmic derivatives of this staggered generating family form an infinite tower of strictly local charges.
Quantum Deformation
In the coherent quantum deformation, the same algebraic structures observed in the classical rules yield several phenomena:
- Exact low-bond-dimension invariant states are produced, which are identified as scar candidates.
- Exponentially long-lived quasi-local quasimodes are observed.
- Diagonal quasi-local charges are generated. After projection away from the three elementary local charges, these diagonal quasi-local charges produce a non-zero Mazur bound.
Why This Matters
The identification of specific conservation laws, Euler velocities, and their bounds provides insight into the fundamental dynamics of these three-state IRF cellular automata. The agreement between calculated and reported values for the species-flipping rule strengthens the understanding of its sound-active charges. Furthermore, the manifestations of these algebraic structures in the quantum deformation, such as scar candidates and long-lived quasimodes, are significant for characterizing the behavior of quantum systems derived from these classical models.