Abstract
We consider the finite volume mean values of current operators in integrable spin chains with local interactions, and provide an alternative derivation of the exact result found recently by the author and two collaborators. We use a certain type of long range deformation of the local spin chains, which was discovered and explored earlier in the context of the AdS/CFT correspondence. This method is immediately applicable also to higher rank models: as a concrete example we derive the current mean values in the SU(3)SU(3)-symmetric fundamental model, solvable by the nested Bethe Ansatz. The exact results take the same form as in the Heisenberg spin chains: they involve the one-particle eigenvalues of the conserved charges and the inverse of the Gaudin matrix.
Highlights
There has been some interest in the computation of current mean values in one dimensional integrable models
The motivation mainly came from the recent theory of Generalized Hydrodynamics [1, 2], which aims to describe the non-equilibrium dynamics of integrable models on the Euler scale
The connection between the deformed chains and the current mean values is quite simple, and surprisingly it has not been noticed before: For each current operator there is a long-range deformation such that the given current operator itself is the leading perturbing operator. This provides a direct link towards the current mean values, which we explore in the present paper
Summary
There has been some interest in the computation of current mean values in one dimensional integrable models. One of the central elements of the theory is the set of continuity relations for the conserved charges of the models, because they lead to the generalized Euler-equations describing the ballistic flow of the quasi-particles For this purpose it is essential to know the exact mean values of the currents associated to the conserved charges, assuming local equilibration on some intermediate length and time scales. In [6] the author and two collaborators derived an exact finite volume result for the current mean values, valid in a large class of Bethe Ansatz solvable quantum models. This derivation only uses a specific form factor expansion for the finite volume mean values, and the continuity equations that define the current operators.
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