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arXiv:2206.08679 · Physical Review Letters

Wrapping Corrections for Long-Range Spin Chains

Tamás Gombor

Published Dec 30, 2022• 9 citations
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Abstract

The long-range spin chains play an important role in the gauge-string duality. The aim of this Letter is to generalize the recently introduced transfer matrices of integrable medium-range spin chains to long-range models. These transfer matrices define a large set of conserved charges for every length of the spin chain. These charges agree with the original definition of long-range spin chains for infinite length. However, our construction works for every length, providing the definition of integrable finite-size long-range spin chains whose spectrum already contains the wrapping corrections.

DOI
10.1103/physrevlett.129.270201

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The long-range spin chains play an important role in the gauge-string duality. The aim of this Letter is to generalize the recently introduced transfer matrices of integrable medium-range spin chains to long-range models. These transfer matrices define a large set of conserved charges for every length of the spin chain. These charges agree with the original definition of long-range spin chains for infinite length. However, our construction works for every length, providing the definition of integrable finite-size long-range spin chains whose spectrum already contains the wrapping corrections.
We present a recursion relation for the explicit construction of integrable spin chain Hamiltonians with long-range interactions. Based on arbitrary short-range (e.g. nearest neighbor) integrable spin chains, it allows us to construct an infinite set of conserved long-range charges. We explain the moduli space of deformation parameters by different classes of generating operators. The rapidity map and dressing phase in the long-range Bethe equations are a result of these deformations. The closed chain asymptotic Bethe equations for long-range spin chains transforming under a generic symmetry algebra are derived. Notably, our construction applies to generalizations of standard nearest neighbor chains such as alternating spin chains. We also discuss relevant properties for its application to planar and supersymmetric gauge theories. Finally, we present a map between long-range and inhomogeneous spin chains delivering more insight into the structures of these models, as well as their limitations at wrapping order.
We present a recursion relation for the explicit construction of integrable spin chain Hamiltonians with long-range interactions. Based on arbitrary short-range (e.g. nearest-neighbor) integrable spin chains, it allows to construct an infinite set of conserved long-range charges. We explain the moduli space of deformation parameters by different classes of generating operators. The rapidity map and dressing phase in the long-range Bethe equations are a result of these deformations. The closed chain asymptotic Bethe equations for long-range spin chains transforming under a generic symmetry algebra are derived. Notably, our construction applies to generalizations of standard nearest-neighbor chains such as alternating spin chains. We also discuss relevant properties for its application to planar D=4, N=4 and D=3, N=6 supersymmetric gauge theories. Finally, we present a map between long-range and inhomogeneous spin chains delivering more insight into the structures of these models as well as their limitations at wrapping order.
An interesting type of spin chain has appeared in the context of the planar AdS/CFT correspondence: it is based on an integrable nearest-neighbor spin chain, and it is perturbatively deformed by long-range interactions which apparently preserve the integrable structure. Similar models can be constructed by demanding the existence of merely one conserved local charge. Although the latter is not a sufficient integrability condition in general, the models often display convincing signs of full integrability. Here we consider a class of long-range spin chains with spins transforming in the fundamental representation of . For the most general such model with one conserved local charge we construct a conserved Yangian generator and show that it obeys the Serre relations. We thus provide a formal proof of integrability for this class of models.