Quantum Integrability of Super-Spin Chains From Discrete Hirota Dynamics

Vladimir Kazakov

Nikos Theodorakopoulos

V. I. Inozemtsev

Stefan Weigert, Gerhard H. Muller

Laurent Delisle, Amine Jaouadi

We present a classification of supersymmetric bilinear KdV-type equations admitting unconstrained three-super-soliton solutions. Extending Hirota's classical three-soliton criterion to the supersymmetric setting, we derive the complete bosonic and fermionic compatibility conditions governing the existence of three-super-soliton solutions. We prove that every supersymmetric bilinear KdV-type equation possesses unconstrained one- and two-super-soliton solutions, whereas three-super-soliton solutions exist only when eight integrability conditions are satisfied. These conditions provide a supersymmetric analogue of Hirota's classical integrability criterion and naturally recover the fermionic relations previously introduced by Carstea, revealing their structural origin. As a consequence, we classify the supersymmetric extensions of Hirota bilinear KdV-type equations and show that only a subset of Hietarinta's classical classification remains valid in the unrestricted supersymmetric framework.

Stanislav Zdislavovich Pakuliak, С. М. Сергеев

We investigate an N‐state spin model called quantum relativistic Toda chain and based on the unitary finite‐dimensional representations of the Weyl algebra with q being Nth primitive root of unity. Parameters of the finite‐dimensional representation of the local Weyl algebra form the classical discrete integrable system. Nontrivial dynamics of the classical counterpart corresponds to isospectral transformations of the spin system. Similarity operators are constructed with the help of modified Baxter′s Q‐operators. The classical counterpart of the modified Q‐operator for the initial homogeneous spin chain is a Bäcklund transformation. This transformation creates an extra Hirota‐type soliton in a parameterization of the chain structure. Special choice of values of solitonic amplitudes yields a degeneration of spin eigenstates, leading to the quantum separation of variables, or the functional Bethe ansatz. A projector to the separated eigenstates is constructed explicitly as a product of modified Q‐operators.