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Scaling limit of the ground state Bethe roots for the inhomogeneous XXZ spin-1/2 chain

https://doi.org/10.1016/j.nuclphysb.2024.116624

Gehrmann, Sascha; Kotousov, Gleb A; Lukyanov, Sergei L (July 2024, Nuclear physics B)

It is known that for the Heisenberg XXZ spin - chain in the critical regime, the scaling limit of the vacuum Bethe roots yields an infinite set of numbers that coincide with the energy spectrum of the quantum mechanical 3D anharmonic oscillator. The discovery of this curious relation, among others, gave rise to the approach referred to as the ODE/IQFT (or ODE/IM) correspondence. Here we consider a multiparametric generalization of the Heisenberg spin chain, which is associated with the inhomogeneous six-vertex model. When quasi-periodic boundary conditions are imposed the lattice system may be explored within the Bethe Ansatz technique. We argue that for the critical spin chain, with a properly formulated scaling limit, the scaled Bethe roots for the ground state are described by second order differential equations, which are multi-parametric generalizations of the Schrödinger equation for the anharmonic oscillator.

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A<sc>bstract</sc> Using the analytic Bethe ansatz, we initiate a study of the scaling limit of the quasi-periodic$$ {D}_3^{(2)} $$

D_{3}^{(2)}

spin chain. Supported by a detailed symmetry analysis, we determine the effective scaling dimensions of a large class of states in the parameter regimeγ∈ (0,$$ \frac{\pi }{4} $$

\frac{π}{4}

). Besides two compact degrees of freedom, we identify two independent continuous components in the finite-size spectrum. The influence of large twist angles on the latter reveals also the presence of discrete states. This allows for a conjecture on the central charge of the conformal field theory describing the scaling limit of the lattice model.

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