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On the Q operator and the spectrum of the XXZ model at root of unity

by Yuan Miao, Jules Lamers, Vincent Pasquier

Submission summary

As Contributors: Yuan Miao
Arxiv Link: (pdf)
Date submitted: 2021-03-25 10:36
Submitted by: Miao, Yuan
Submitted to: SciPost Physics
Academic field: Physics
  • Mathematical Physics
  • Quantum Algebra
  • Condensed Matter Physics - Theory
Approach: Theoretical


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.

Current status:
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Submission 2012.10224v2 on 25 March 2021

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