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arXiv:0711.4813 · Journal of Statistical Mechanics Theory and Experiment

Yangian symmetry of long-range \mathfrak {g}\mathfrak {l}(N) integrable spin chains

Niklas Beisert, Denis Erkal

Published Mar 7, 2008• 32 citations
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Abstract

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.

DOI
10.1088/1742-5468/2008/03/p03001

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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.
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.
Quantum spin chains arise naturally from perturbative large- N field theories and matrix models. The Hamiltonian of such a model is a long-range deformation of nearest-neighbour type interactions. Here, we study the most general long-range integrable spin chain with spins transforming in the fundamental representation of . We derive the Hamiltonian and the corresponding asymptotic Bethe ansatz at the leading four perturbative orders with several free parameters. Furthermore, we propose Bethe equations for all orders and identify the moduli of the integrable system. We finally apply our results to plane-wave matrix theory and show that the Hamiltonian in a closed sector is not of this form and therefore not integrable beyond the first perturbative order. This also implies that the complete model is not integrable.
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) S U ( 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.
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.