Long-range deformations for integrable spin chains

Abstract

We present a recursion relation for the explicit construction of integrable spin chain Hamiltonians with long-range interactions. Based on arbitrary shortrange (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 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.

Dynamic Lattice Supersymmetry in $$\mathfrak {gl}\left( {n}|{m}\right) $$ gl n | m Spin Chains

“…Finally, gauge theory suggests that we should also generalize to spin chains that are longranged and for which the Hamiltonian itself is length-changing. For this purpose the works of [5] and [35,36] might provide some clue. Of course, if might also be possible to find a nonperturbative way to freeze the length-changing via some complicated basis transformation generalizing [15,16,19].…”

The spin Sutherland model of type and its associated spin chain

“…In the context of high-energy physics, the dynamics of particles or fields in the near-horizon region of black holes has been described through such integrable systems [14][15][16]. More recently, quantum integrable spin chains with long-range interaction have played a key role in calculating higher loop effects in the spectra of trace operators of planar N = 4 super Yang-Mills theory [17][18][19]. Furthermore, this type of quantum integrable systems are found to be connected with different areas of mathematics like random matrix theory [20], multivariate orthogonal polynomials [21][22][23][24], Dunkl operators [25,26], and Yangian quantum groups [27][28][29][30].…”

Review of AdS/CFT Integrability, Chapter I.3: Long-Range Spin Chains

“…A method for constructing integrable closed long-range spin chains with generic Lie (super)algebras and spin representations has been introduced in [12,13] inspired by the findings of [33]. Interestingly, it is a bottom-up approach.…”

“…The freedom encountered in the previous sub-section while determining the generic form of the higher-loop corrections corresponds to freedom in choosing the X(λ) operator. It has been advocated in [12,13] that there are three different admissible classes of such operators: boost charges, bi-local charges and local charges. The first two act inhomogeneously on the spin chain and are parametrised by α r (λ) and β r,s (λ) respectively.…”

“…These findings were subsequently generalised to arbitrary Lie (super)algebra in [12]. Moreover, a novel recursion relation has been proposed, which allows to lift an integrable nearest-neighbour spin chain to its long-range counterpart, see also [13]. This has laid solid foundations for the theory of perturbative long-range systems.…”