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relations and related quantum glass models

B. Sriram Shastry

Published Jan 1, 2011• 0 citations
OpenAlex

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 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.

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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 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.
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.
This part of the book is fully devoted to the physics of (1 + 1)-dimensional quantum or twodimensional classical models. As usual, I shall distinguish between classical and quantum systems only if it is necessary. Those models where the difference between quantum and classical is not relevant will be called ‘two-dimensional’; those where for some reason or other I want to emphasize the quantum aspect will be called ‘one-dimensional’. The reason I spend so much time discussing two-dimensional physics is that here one finds a kind of paradise for strong interactions and nonperturbative effects. Those effects which are tricky or even impossible to achieve in higher dimensions appear in almost every two-dimensional model. To our greatest satisfaction there are theoretical tools at our disposal which allow us to solve many of the corresponding problems and describe these phenomena. The availability of these strong mathematical tools is a unique feature of two-dimensional physics. Even if it turns out to be impossible to generalize these tools for higher dimensions, I hope that the reader will be rewarded for the time spent by the pleasure obtained from contemplating their beauty. The world we are about to enter has certain distinct features which are worth mentioning in the introduction. The first one concerns the second-order phase transitions. In (1 + 1)-dimensional systems such transitions may occur only at zero temperature.
Degeneracy of the eigenvalues of hermitian matrices is analyzed in terms of algebraic relations between the matrix elements. Only finite-dimensional matrices are considered, representing Hamiltonians of spin systems in particular.The analysis is in terms of linear relations in the space of 'diagonal' operators, i.e. a complete set of powers of the Hamiltonian. A relation is found with the theorem formulated by Von Neumann and Wigner that only the variation of three parameters of the Hamiltonian may result in a 'crossing' or degeneracy of two eigenvalues.