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Journal of Physics A Mathematical and General

The lowest energy state for a polynomial Hamiltonian and its geometrical interpretation

Wolfgang Schweizer, Peter Krämer

Published Nov 11, 1984• 0 citations
Publisher

Abstract

Consider a Hamiltonian H which is an element of the enveloping algebra of a Lie algebra L. For simplicity the authors choose L as SU(2) or SU(1,1). They look for the coherent state mod sc) which solves the variational equation delta (sc mod H mod sc)=0. This condition in state space is transcribed into a condition in a classical phase space. Geometrical concepts are given for the state of lowest energy, and for a self-consistent linearised Hamiltonian. The interpretation yields solutions of the equation for the state of lowest energy by iteration procedures.

DOI
10.1088/0305-4470/17/16/018

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Consider a Hamiltonian H which is an element of the enveloping algebra of a Lie algebra L. For simplicity the authors choose L as SU(2) or SU(1,1). They look for the coherent state mod sc) which solves the variational equation delta (sc mod H mod sc)=0. This condition in state space is transcribed into a condition in a classical phase space. Geometrical concepts are given for the state of lowest energy, and for a self-consistent linearised Hamiltonian. The interpretation yields solutions of the equation for the state of lowest energy by iteration procedures.
We study characteristic aspects of the geometric phase which is associated with the generalized coherent states. This is determined by special orbits in the parameter space defining the coherent state, which is obtained as a solution of the variational equation governed by a simple model Hamiltonian called the ”resonant Hamiltonian”. Three typical coherent states are considered: SU(2), SU(1,1) and Heisenberg-Weyl. A possible experimental detection of the phases is proposed in such a way that the geometric phases can be discriminated from the dynamical phase
We study characteristic aspects of the geometric phase which is associated with the generalized coherent states. This is determined by special orbits in the parameter space defining the coherent state, which is obtained as a solution of the variational equation governed by a simple model Hamiltonian called the "resonant Hamiltonian". Three typical coherent states are considered: SU(2), SU(1,1) and Heisenberg-Weyl. A possible experimental detection of the phases is proposed in such a way that the geometric phases can be discriminated from the dynamical phase.
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A bootstrap program is presented for algebraically solving the univariate R R -matrix of a generic integrable quantum spin chain from its Hamiltonian. The Yang-Baxter equation contains an infinite number of seemingly independent constraints on the operator valued coefficients in the expansion of the R R -matrices with respect to their spectral parameters, with the lowest order one being the Reshetikhin condition. These coefficients can be solved iteratively in a self consistent way using a lemma due to Kennedy, which reconstructs the R R -matrix after an infinite number of steps. For a generic Hamiltonian, the procedure could fail at any step, making the conditions useful as an integrability test. However, at least for the most common examples, they always turn out to be satisfied whenever the lowest order condition is. It remains to be understood whether they are indeed implied by the Reshetikhin condition.