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arXiv:1010.3445 · Journal of Physics A Mathematical and Theoretical

Parameter-dependent commuting matrices, Plücker relations and related quantum glass models

B. Sriram Shastry

Published Jan 10, 2011• 11 citations
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Abstract

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.

DOI
10.1088/1751-8113/44/5/052001

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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.
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 Pl¨ ucker 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.
Starting from a generalization of Weyl's relations in finite dimension NNN, we show that the Heisenberg commutation relations can be satisfied in a specific N−1N-1N−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 NNN 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.
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 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.
The 1D Hubbard model is shown to be an exactly integrable system. A "covering" model of 2D statistical mechanics which I proposed recently was shown to provide a one-parameter family of transfer matrices, commuting with the Hamiltonian of the Hubbard model. I show in this work that any two transfer matrices of a family commute mutually. At the root of the commutation relation is the ubiquitous Yang-Baxter factorization condition. The form of the RRR operator is displayed explicitly.
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.