SciPost Phys. 11, 067 (2021) ·
published 22 September 2021
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· pdf
The spin-1/2 Heisenberg XXZ chain is a paradigmatic quantum integrable model.
Although it can be solved exactly via Bethe ansatz techniques, there are still
open issues regarding the spectrum at root of unity values of the anisotropy.
We construct Baxter's Q operator at arbitrary anisotropy from a two-parameter
transfer matrix associated to a complex-spin auxiliary space. A decomposition
of this transfer matrix provides a simple proof of the transfer matrix fusion
and Wronskian relations. At root of unity a truncation allows us to construct
the Q operator explicitly in terms of finite-dimensional matrices. From its
decomposition we derive truncated fusion and Wronskian relations as well as an
interpolation-type formula that has been conjectured previously. We elucidate
the Fabricius-McCoy (FM) strings and exponential degeneracies in the spectrum
of the six-vertex transfer matrix at root of unity. Using a semicyclic
auxiliary representation we give a conjecture for creation and annihilation
operators of FM strings for all roots of unity. We connect our findings with
the 'string-charge duality' in the thermodynamic limit, leading to a conjecture
for the imaginary part of the FM string centres with potential applications to
out-of-equilibrium physics.
