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JSIAM Letters

3-dimensional solvable XX spin lattice Hamiltonian derived from 3-variable Krawtchouk polynomials

Hiroshi Miki, Kengo Miura

Published Jan 1, 2016• 4 citations
Publisher

Abstract

A new solvable 3-dimensional spin lattice Hamiltonian with inhomogeneous couplings, which can be diagonalized by 3-variable Krawtchouk polynomials, is proposed. The model is defined on the lattice of rectangular pyramid and describes near-neighbor interactions instead of nearest-neighbor ones. Using the properties of 3-variable Krawtchouk polynomials, quantum state transfer in the model is analyzed. In particular, for some value of the parameters, perfect state transfer is shown to occur from the apex to the diagonal plane.

DOI
10.14495/jsiaml.8.41

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A new solvable 3-dimensional spin lattice Hamiltonian with inhomogeneous couplings, which can be diagonalized by 3-variable Krawtchouk polynomials, is proposed. The model is defined on the lattice of rectangular pyramid and describes near-neighbor interactions instead of nearest-neighbor ones. Using the properties of 3-variable Krawtchouk polynomials, quantum state transfer in the model is analyzed. In particular, for some value of the parameters, perfect state transfer is shown to occur from the apex to the diagonal plane.
Quantum-state transfer in a triangular domain of a two-dimensional, equally spaced, spin lattice with nonhomogeneous nearest-neighbor couplings is analyzed. An exact solution of the one-excitation dynamics is provided in terms of two-variable Krawtchouk orthogonal polynomials that have been recently defined. The probability amplitude for an excitation to transit from one site to another is given. For some values of the parameters, perfect transfer is shown to take place from the apex of the lattice to the boundary hypotenuse.
New exactly solvable one-dimensional XX spin chain models that exhibit perfect state transfer are defined. These models have inhomogeneous couplings and magnetic fields determined from the three-term recurrence relations satisfied by the q-Racah and para q-Racah polynomials. Due to this connection with orthogonal polynomials, the one-excitation sector can be solved analytically. This allows us to provide explicit sets of conditions on the polynomial parameters that guarantee the occurrence of perfect state transfer across these spin chains.
Abstract New exactly solvable one-dimensional XX spin chain models that exhibit perfect state transfer (PST) are defined. These models have inhomogeneous couplings and magnetic fields determined from the three-term recurrence relations satisfied by the q -Racah and para q -Racah polynomials. Due to this connection with orthogonal polynomials, the one-excitation sector can be solved analytically. This allows us to provide explicit sets of conditions on the polynomial parameters that guarantee the occurrence of PST across these spin chains.
We study the time evolution of a single spin excitation state in certain linear spin chains, as a model for quantum communication. Some years ago it was discovered that when the spin chain data (the nearest neighbour interaction strengths and the magnetic field strengths) are related to the Jacobi matrix entries of Krawtchouk polynomials or dual Hahn polynomials, so-called perfect state transfer takes place. The extension of these ideas to other types of discrete orthogonal polynomials did not lead to new models with perfect state transfer, but did allow more insight in the general computation of the correlation function. In the present paper, we extend the study to discrete orthogonal polynomials of q-hypergeometric type. A remarkable result is a new analytic model where perfect state transfer is achieved: this is when the spin chain data are related to the Jacobi matrix of q-Krawtchouk polynomials. The other cases studied here (affine q-Krawtchouk polynomials, quantum q-Krawtchouk polynomials, dual q-Krawtchouk polynomials, q-Hahn polynomials, dual q-Hahn polynomials and q-Racah polynomials) do not give rise to models with perfect state transfer. However, the computation of the correlation function itself is quite interesting, leading to advanced q-series manipulations.