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.
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.
Starting from a generalization of Weyl's relations in finite dimension
N, we show that the Heisenberg commutation relations can be satisfied in a specific
N−1 dimensional subspace, and display a linear map for projecting operators to this subspace. This setup is used to construct a hierarchy of parameter-dependent commuting matrices in
N dimensions. This family of commuting matrices is then related to Type-1 matrices representing quantum integrable models. The commuting matrices find an interesting application in quantum computation, specifically in Grover's database search problem. Each member of the hierarchy serves as a candidate Hamiltonian for quantum adiabatic evolution and, in some cases, achieves higher fidelity than standard choices -- thus offering improved performance.
Studies of integrable quantum many-body systems have a long history with an impressive record of success. However, quite surprisingly, an unambiguous definition of quantum integrability remains a matter of an ongoing debate. We contribute to this debate by presenting several observation motivated by a particular quantum model of
N spins
1/2. Remarkably, although this system is nonintegrable, it possesses a large number of mutually commuting local binary integrals of motion. We address the question of the number of independent integrals of motion. Strikingly, we find that, according to a common definition of independence of quantum integrals of motion (QIMs), this number equals to the number of spins,
N. A common wisdom would then suggest that the system is completely integrable, which is not the case. To resolve this conundrum, we dwell upon the notion of independence of QIMs. We argue that the common definition should be amended in order to match the intuitive notion of independence. We suggest a new definition, which, in particular, implies that some of the
N QIMs of the considered model are dependent. Still, even according to this strict definition,
N/2 integrals of motion remain independent. This challenges the intuition on the integrable quantum many-body systems as well as some of the most elaborated definitions of quantum integrability.
We construct ensembles of random integrable matrices with any prescribed number of nontrivial integrals and formulate integrable matrix theory (IMT)-a counterpart of random matrix theory (RMT) for quantum integrable models. A type-M family of integrable matrices consists of exactly N-M independent commuting N×N matrices linear in a real parameter. We first develop a rotationally invariant parametrization of such matrices, previously only constructed in a preferred basis. For example, an arbitrary choice of a vector and two commuting Hermitian matrices defines a type-1 family and vice versa. Higher types similarly involve a random vector and two matrices. The basis-independent formulation allows us to derive the joint probability density for integrable matrices, similar to the construction of Gaussian ensembles in the RMT.
Type-I matrices were introduced recently as finite-dimensional prototypes of quantum integrable systems. These matrices are linearly dependent on an 'interaction' type parameter, and possess interesting properties such as commuting partner matrices and generically violate the von Neumann Wigner noncrossing rule. The important role of Plücker relations in this construction is noted. Type-I matrices are given a transparent formulation in terms of Fermi or Bose-type particle operators; they represent a quantum glass model with either Fermi or Bose statistics, with several free parameters that may be chosen at will.