Kohei Fukai, Raphael Kleinemühl, Balázs Pozsgay, Eric Vernier
SciPost Phys. 16, 003 (2024) ·
published 5 January 2024

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We deal with quantum spin chains whose Hamiltonian arises from a representation of the TemperleyLieb algebra, and we consider the mean values of those local operators which are generated by the TemperleyLieb algebra. We present two key conjectures which relate these mean values to existing literature about factorized correlation functions in the XXZ spin chain. The first conjecture states that the finite volume mean values of the current and generalized current operators are given by the same simple formulas as in the case of the XXZ chain. The second conjecture states that the mean values of products of TemperleyLieb generators can be factorized: they can expressed as sums of products of current mean values, such that the coefficients in the factorization depend neither on the eigenstate in question, nor on the selected representation of the algebra. The coefficients can be extracted from existing work on factorized correlation functions in the XXZ model. The conjectures should hold for all eigenstates that are nondegenerate with respect to the local charges of the models. We consider concrete representations, where we check the conjectures: the socalled golden chain, the $Q$state Potts model, and the trace representation. We also explain how to derive the generalized current operators from concrete expressions for the local charges.
SciPost Phys. Core 6, 069 (2023) ·
published 19 October 2023

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We present the explicit expressions for the matrix product operator (MPO) representation for the local conserved quantities of the Heisenberg chain. The bond dimension of the MPO grows linearly with the locality of the charges. The MPO has more simple form than the local charges themselves, and their Catalan tree patterns naturally emerge from the matrix products. The MPO representation of local conserved quantities is generalized to the integrable SU(N) invariant spin chain.