Dichotomy theorem separating complete integrability and non-integrability of isotropic spin chains

Abstract

We investigate the integrability and non-integrability of isotropic spin chains with nearest-neighbor interaction with general spin $S$ in terms of the presence or absence of local conserved quantities. We prove a dichotomy theorem that whether a single quantity is zero or not sharply separates two scenarios: (i) this system has $k$-local conserved quantities for all $k$ (completely integrable), or (ii) this system has no nontrivial local conserved quantity (non-integrable). This result excludes the possibility of an intermediate system with some but not all local conserved quantities, which solves in the affirmative the Grabowski-Mathieu conjecture. This theorem also serves as a complete classification of integrability and non-integrability for $S\le 13.5$, suggesting that all the integrable models are in the scope of the Yang-Baxter equation.

7 pages + 14 pages, 1 figure

References in corpus (26)

• Equilibration, thermalisation, and the emergence of statistical mechanics in closed quantum systems
• Quench dynamics and relaxation in isolated integrable quantum spin chains
• Experimental Observation of a Generalized Gibbs Ensemble
• Quantum Many-Body Scars and Weak Breaking of Ergodicity
• Remarks on the notion of quantum integrability
• Noncommutative Gauge Theory on Fuzzy Sphere from Matrix Model
• Yang-Baxter and the Boost: splitting the difference
• Classifying nearest-neighbour interactions and deformations of AdS
• Classifying integrable spin-1/2 chains with nearest neighbour interactions
• Thermalization without eigenstate thermalization hypothesis after a quantum quench
• Proof of the absence of local conserved quantities in the XYZ chain with a magnetic field
• Explicit Construction of Local Conserved Quantities in the XYZ Spin-1/2 Chain
• Proof of absence of local conserved quantities in the mixed-field Ising chain
• Matrix Bases for Star Products: a Review
• Reply to "Comment on "Systematic Construction of Counterexamples to the Eigenstate Thermalization Hypothesis"" by Mondaini et.al
• Exact Thermal Eigenstates of Nonintegrable Spin Chains at Infinite Temperature
• Absence of local conserved quantity in the Heisenberg model with next-nearest-neighbor interaction
• Quantum Violation of Fluctuation-Dissipation Theorem
• Complete classification of integrability and non-integrability of S=1/2 spin chains with symmetric next-nearest-neighbor interaction
• Matrix product operator representations for the local conserved quantities of the Heisenberg chain
• Algebraic classification of Hietarinta's solutions of Yang-Baxter equations~:~invertible $4\times 4$ operators
• A fuzzy bipolar celestial sphere
• Proof of absence of local conserved quantities in two- and higher-dimensional quantum Ising models
• Absence of nontrivial local conserved quantities in the quantum compass model on the square lattice
• Absence of nontrivial local conserved quantities in the spin-1 bilinear-biquadratic chain and its anisotropic extensions
• The $S=\frac{1}{2}$ XY and XYZ models on the two or higher dimensional hypercubic lattice do not possess nontrivial local conserved quantities

Cited by in corpus (1)

• Integrability from a single conservation law in quantum spin chains