We study a recently proposed quantum integrable model defined on a lattice with N sites, with fermions or bosons populating each site, as a close relative of the well known spin- Gaudin model. This model has 2 N arbitrary parameters, a linear dependence on an interaction type parameter x , and can be solved exactly. It has N known constants of motion that are linear in x . We display further constants of motion with higher fermion content that are linearly independent of the known conservation laws. Our main result is that despite the existence of the higher conservation laws, the model has only N functionally independent conservation laws . Therefore we propose that N can be viewed as the number of degrees of freedom, in parallel to the classical definition of integrability.
We investigate the connection between energy level crossings in integrable systems and their integrability, i.e. the existence of a set of non-trivial integrals of motion. In particular, we consider a general quantum Hamiltonian linear in the coupling u , H ( u ) = T + uV , and require that it has the maximum possible number of nontrivial commuting partners also linear in u . We demonstrate how this commutation requirement alone leads to: (1) an exact solution for the energy spectrum and (2) level crossings, which are always present in these Hamiltonians in violation of the Wigner–von Neumann non-crossing rule. Moreover, we construct these Hamiltonians explicitly by resolving the above commutation requirement and show their equivalence to a sector of Gaudin magnets (central spin Hamiltonians). In contrast, fewer than the maximum number of conservation laws does not guarantee level crossings.
We examine the nature, number, and interrelation of conservation laws in the one-dimensional Hubbard model. In previous work by Shastry [Phys. Rev. Lett. 56, 1529 (1986); 56, 2334 (1986); 56, 2453 (1986); J. Stat. Phys. 50, 57 (1988)], who studied the model on a large but finite number of lattice sites
(Na), only
Na+1 conservation laws, corresponding to
Na+1 operators that commute with themselves and the Hamiltonian, were explicitly identified, rather than the $\ensuremath{\sim}{2N}_{a}$ conservation laws expected from the solvability and integrability of the model. Using a pseudoparticle approach related to the thermodynamic Bethe ansatz, we discover an additional
Na+1 independent conservation laws corresponding to nonlocal, mututally commuting operators, which we call transfer-matrix currents. Further, for the model defined in the whole Hilbert space, we find there are two other independent commuting operators (the squares of the \ensuremath{\eta}-spin and spin operators) so that the total number of local plus nonlocal commuting conservation laws for the one-dimensional Hubbard model is
2Na+4. Finally, we introduce an alternative set of
2Na+4 conservation laws which assume particularly simple forms in terms of the pseudoparticle and Yang-particle operators. This set of mutually commuting operators lends itself more readily to calculations of physically relevant correlation functions at finite energy or frequency than the previous set.
We study general quantum integrable Hamiltonians linear in a coupling constant and represented by finite N × N real symmetric matrices. The restriction on the coupling dependence leads to a natural notion of nontrivial integrals of motion and classification of integrable families into types according to the number of such integrals. A type M family in our definition is formed by N – M nontrivial mutually commuting operators linear in the coupling. Working from this definition alone, we parameterize type M operators, i.e. resolve the commutation relations, and obtain an exact solution for their eigenvalues and eigenvectors. We show that our parameterization covers all type 1, 2 and 3 integrable models and discuss the extent to which it is complete for other types. We also present robust numerical observation on the number of energy-level crossings in type M integrable systems and analyze the taxonomy of types in the 1D Hubbard model.
We prove that, for translationally invariant quantum spin chains with finite-range interactions, the existence of a specific conservation law implies the presence of infinitely many local conserved quantities, i.e., integrability. This shows that the standard hierarchy of local conserved quantities arising for nearest-neighbor Hamiltonians obtained via the canonical Yang-Baxter construction is already encoded in the lowest nontrivial conservation law, known as the Reshetikhin condition. Combined with recent rigorous results on nonintegrability, our theorem strongly restricts the possibility of partially integrable systems that admit only a finite but large number of local conserved quantities. Our work establishes a rigorous foundation for the systematic identification of new integrable models and deepens the algebraic understanding of conservation-law structures in quantum spin chains.
Abstract In this work we demonstrate how one can, in a generic approach, derive a set of N simple quadratic Bethe equations for integrable Richardson–Gaudin (RG) models built out of N spins-1/2. These equations depend only on the N eigenvalues of the various conserved charges so that any solution of these equations defines, indirectly through the corresponding set of eigenvalues, one particular eigenstate. The proposed construction covers the full class of integrable RG models of the XYZ (including the subclasses of XXZ and XXX models) type realised in terms of spins-1/2, coupled with one another through , , terms, including, as well, magnetic field-like terms linear in the Pauli matrices. The approach exclusively requires integrability, defined here only by the requirement that N conserved charges R i (with ) such that exist. The result is therefore valid, and equally simple, for models with or without symmetry, with or without a properly defined pseudo-vacuum as well as for models with non-skew symmetric couplings.
We study general quantum integrable Hamiltonians linear in a coupling constant and represented by finite NxN real symmetric matrices. The restriction on the coupling dependence leads to a natural notion of nontrivial integrals of motion and classification of integrable families into Types according to the number of such integrals. A Type M family in our definition is formed by N-M nontrivial mutually commuting operators linear in the coupling. Working from this definition alone, we parameterize Type M operators, i.e. resolve the commutation relations, and obtain an exact solution for their eigenvalues and eigenvectors. We show that our parameterization covers all Type 1, 2, and 3 integrable models and discuss the extent to which it is complete for other types. We also present robust numerical observation on the number of energy level crossings in Type M integrable systems and analyze the taxonomy of types in the 1d Hubbard model.