SciPost Phys. 13, 070 (2022) ·
published 28 September 2022
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The Onsager algebra is one of the cornerstones of exactly solvable models in
statistical mechanics. Starting from the generalised Clifford algebra, we
demonstrate its relations to the graph Temperley-Lieb algebra, and a
generalisation of the Onsager algebra. We present a series of quantum lattice
models as representations of the generalised Clifford algebra, possessing the
structure of a special type of the generalised Onsager algebra. The
integrability of those models is presented, analogous to the free fermionic
eight-vertex model. We also mention further extensions of the models and
physical properties related to the generalised Onsager algebras, hinting at a
general framework that includes families of quantum lattice models possessing
the structure of the generalised Onsager algebras.
SciPost Phys. 11, 067 (2021) ·
published 22 September 2021
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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.
SciPost Phys. 11, 066 (2021) ·
published 22 September 2021
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We conjecture the existence of hidden Onsager algebra symmetries in two
interacting quantum integrable lattice models, i.e. spin-1/2 XXZ model and
spin-1 Zamolodchikov-Fateev model at arbitrary root of unity values of the
anisotropy. The conjectures relate the Onsager generators to the conserved
charges obtained from semi-cyclic transfer matrices. The conjectures are
motivated by two examples which are spin-1/2 XX model and spin-1 U(1)-invariant
clock model. A novel construction of the semi-cyclic transfer matrices of
spin-1 Zamolodchikov-Fateev model at arbitrary root of unity value of the
anisotropy is carried out via transfer matrix fusion procedure.
SciPost Phys. 10, 086 (2021) ·
published 22 April 2021
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Using the algebro-geometric approach, we study the structure of
semi-classical eigenstates in a weakly-anisotropic quantum Heisenberg spin
chain. We outline how classical nonlinear spin waves governed by the
anisotropic Landau-Lifshitz equation arise as coherent macroscopic low-energy
fluctuations of the ferromagnetic ground state. Special emphasis is devoted to
the simplest types of solutions, describing precessional motion and elliptic
magnetisation waves. The internal magnon structure of classical spin waves is
resolved by performing the semi-classical quantisation using the
Riemann-Hilbert problem approach. We present an expression for the overlap of
two semi-classical eigenstates and discuss how correlation functions at the
semi-classical level arise from classical phase-space averaging.
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