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arXiv:0711.1973 · Regular and Chaotic Dynamics

The commutativity of integrals of motion for quantum spin chains and elliptic functions identities

J. Dittrich, V. Inozemtsev

Published Feb 1, 2008• 3 citations
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Abstract

We prove the commutativity of the first two nontrivial integrals of motion for quantum spin chains with elliptic form of the exchange interaction. We also show thair linear independence for the numbers of spins larger than 4. As a byproduct, we obtained several identities between elliptic Weierstrass functions of three and four arguments.

DOI
10.1007/s11819-008-1003-3

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We prove the commutativity of the first two nontrivial integrals of motion for quantum spin chains with elliptic form of the exchange interaction. We also show thair linear independence for the numbers of spins larger than 4. As a byproduct, we obtained several identities between elliptic Weierstrass functions of three and four arguments.
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Infinitely many commuting operators for the elliptic quantum group Uq,p(slN^)U_{q,p}(\hat{sl_N})Uq,p​(slN​^​)
We construct two classes of infinitely many commuting operators associated with the elliptic quantum group Uq,p(slN^)U_{q,p}(\hat{sl_N})Uq,p​(slN​^​). We call one of them the integral of motion Gm{\cal G}_mGm​, (m∈N)(m \in {\mathbb N})(m∈N) and the other the boundary transfer matrix TB(z)T_B(z)TB​(z), (z∈C)(z \in {\mathbb C})(z∈C). The integral of motion Gm{\cal G}_mGm​ is related to elliptic deformation of the NNN-th KdV theory. The boundary transfer matrix TB(z)T_B(z)TB​(z) is related to the boundary Uq,p(slN^)U_{q,p}(\hat{sl_N})Uq,p​(slN​^​) face model. We diagonalize the boundary transfer matrix TB(z)T_B(z)TB​(z) by using the free field realization of the elliptic quantum group, however diagonalization of the integral of motion Gm{\cal G}_mGm​ is open problem even for the simplest case Uq,p(sl2^)U_{q,p}(\hat{sl_2})Uq,p​(sl2​^​).