The exactly solvable spin Sutherland model of type and its related spin chain

Basu-Mallick, B.; Finkel, Federico; González‐López, A.

doi:10.1016/j.nuclphysb.2012.09.008

Cited by 11 publications

(19 citation statements)

References 52 publications

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“…A well-known property shared by all spin chains of HS type is the fact that their level density becomes asymptotically Gaussian when the number of spins is large enough. This property has been rigorously established for all HS chains of type A N−1 [38], and there is strong numerical evidence that it also holds for other root systems and in the supersymmetric case [43][44][45][46][47][48][49]. Since the Haldane-Shastry chain is the k → 0 limit of Inozemtsev's chain, it is natural to investigate whether the level density of the latter chain is also approximately Gaussian when N 1.…”

Section: Level Densitymentioning

confidence: 99%

A new perspective on the integrability of Inozemtsev’s elliptic spin chain

Finkel

González‐López

2014

Annals of Physics

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h i g h l i g h t s• Construction of Inozemtsev's elliptic spin chain using Polychronakos's freezing trick.• Numerical evidence of the Gaussian character of the level density.• Exact computation and asymptotics of the mean and standard deviation of the spectrum. • Evidence of the chain's integrability from key statistical properties of its spectrum. • Exact evaluation of finite sums of powers of Weierstrass's elliptic function. a b s t r a c tThe aim of this paper is studying from an alternative point of view the integrability of the spin chain with long-range elliptic interactions introduced by Inozemtsev. Our analysis relies on some wellestablished conjectures characterizing the chaotic vs. integrable behavior of a quantum system, formulated in terms of statistical properties of its spectrum. More precisely, we study the distribution of consecutive levels of the (unfolded) spectrum, the power spectrum of the spectral fluctuations, the average degeneracy, and the equivalence to a classical vertex model. Our results are consistent with the general consensus that this model is integrable, and that it is closer in this respect to the Heisenberg chain than to its trigonometric limit (the Haldane-Shastry chain). On the other hand, we present some numerical and analytical evidence showing that the level density of Inozemtsev's chain is asymptotically Gaussian as the number of spins tends to infinity, as is the case with the Haldane-Shastry chain. We are also able to compute analytically the mean and the standard deviation of the spectrum, showing that their asymptotic behavior coincides with that of the Haldane-Shastry chain.

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Section: Level Densitymentioning

confidence: 99%

A new perspective on the integrability of Inozemtsev’s elliptic spin chain

Finkel

González‐López

2014

Annals of Physics

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“…A characteristic property of all spin chains of Haldane-Shastry type is the fact that their level density approaches a Gaussian distribution as the number of spins tends to infinity. This property has been rigorously proved for the chains of A N −1 type and their related one-dimensional vertex models [51,52], and has been numerically checked for the B N , BC N and D N type chains with standard spin reversal operators [37][38][39]41]. More recently, it has been established that the level density of the BC N -type PF chain with PSRO shows a similar behavior [1].…”

Section: Statistical Properties Of the Spectrummentioning

confidence: 90%

“…In the spin chains studied in Refs. [35][36][37][38][39][40][41], the operators S i are represented by spin reversal operators P i (acting on the Hilbert space of the i-th particle), but this is by no means the only possible choice. As a matter of fact, in the novel version of the spin Calogero model of BC N type and its corresponding (PF) chain introduced in Ref.…”

Section: Introductionmentioning

confidence: 99%

Rational quantum integrable systems ofDNtype with polarized spin reversal operators

et al. 2015

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We study the spin Calogero model of D N type with polarized spin reversal operators, as well as its associated spin chain of Haldane-Shastry type, both in the antiferromagnetic and ferromagnetic cases. We compute the spectrum and the partition function of the former model in closed form, from which we derive an exact formula for the chain's partition function in terms of products of partition functions of Polychronakos-Frahm spin chains of type A. Using a recursion relation for the latter partition functions that we derive in the paper, we are able to numerically evaluate the partition function, and thus the spectrum, of the D N -type spin chain for relatively high values of the number of spins N . We analyze several global properties of the chain's spectrum, such as the asymptotic level density, the distribution of consecutive spacings of the unfolded spectrum, and the average degeneracy. In particular, our results suggest that this chain is invariant under a suitable Yangian group, and that its spectrum coincides with that of a Yangian-invariant vertex model with linear energy function and dispersion relation.paper, we shall extend the above results to spin Calogero models and their corresponding spin chains of Haldane-Shastry type based on the D N root system.In order to present our work in an appropriate context, let us briefly recall the origin and significance of the latter models. The Haldane-Shastry (HS) spin chain, introduced independently by these authors in the late eighties [4,5], is perhaps the best known example of an exactly solvable one-dimensional lattice model with long-range interactions. More precisely, this model describes a circular array of equispaced spins with two-body interactions inversely proportional to the square of the (chord) distance between the spins. The motivation for introducing this chain was the construction of a simple model with an exact ground state given by the U → ∞ limit of Gutzwiller's variational wavefunction for the ground state of the one-dimensional Hubbard model [6][7][8]. Over the years, the HS chain has appeared in many areas of interest both in Physics and Mathematics, such as fractional statistics and one-dimensional anyons [9-12], quantum entanglement [13], characterization of integrability vs. quantum chaos [14][15][16][17], quantum integrability via the asymptotic Bethe ansatz [18][19][20], Yangian quantum groups [12,[21][22][23], and conformal field theory [18,[24][25][26].One of the key properties of the HS chain -already noted by Haldane and Shastry in their original papers-is its intimate connection with the scalar (trigonometric) Sutherland model [27,28]. This connection was subsequently elucidated by Polychronakos in Ref. [2], who showed how to derive the HS chain from the spin Sutherland model [29][30][31] by a technique that he called the "freezing trick". The main idea behind this technique is to note that when the coupling constant in the spin Sutherland model goes to infinity the particles tend to concentrate on the coordinates of the (essential...

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“…Nor is it clearly associated with the A N root system like, e.g., the original HS chain and its rational [35,53], hyperbolic [54] and elliptic [47] counterparts, due to the presence of the operators S i in the Hamiltonian. Thus the chain Hamiltonian (2.1) is not directly associated with an extended root system, unlike most chains of HS type considered so far in the literature (see, e.g., [40,42,43,55] for other instances of HS-like chains of BC N , B N and D N types). For each k = 1, .…”

Section: The Modelsmentioning

confidence: 96%

“…The main idea in this respect is to replace the A N −1 root system associated with the interaction potential of the spin Sutherland model, which yields the original HS chain, by other (extended) root systems like, e.g., BC N , B N and D N . The spin Sutherland models associated with all of these root systems have in fact been studied in the literature, and used to construct spin chains of HS type by taking their strong coupling limits [40][41][42][43]. In particular, the exact partition function of each of these chains has been computed in closed form applying the freezing trick to their parent spin dynamical models.…”

Section: Introductionmentioning

confidence: 99%

A novel class of translationally invariant spin chains with long-range interactions

Basu-Mallick

Finkel

González‐López

2020

J. High Energ. Phys.

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We introduce a new class of open, translationally invariant spin chains with long-range interactions depending on both spin permutation and (polarized) spin reversal operators, which includes the Haldane-Shastry chain as a particular degenerate case. The new class is characterized by the fact that the Hamiltonian is invariant under "twisted" translations, combining an ordinary translation with a spin flip at one end of the chain. It includes a remarkable model with elliptic spin-spin interactions, smoothly interpolating between the XXX Heisenberg model with anti-periodic boundary conditions and a new open chain with sites uniformly spaced on a half-circle and interactions inversely proportional to the square of the distance between the spins. We are able to compute in closed form the partition function of the latter chain, thereby obtaining a complete description of its spectrum in terms of a pair of independent su(1|1) and su(m/2) motifs when the number m of internal degrees of freedom is even. This implies that the even m model is invariant under the direct sum of the Yangians Y (gl(1|1)) and Y (gl(0|m/2)). We also analyze several statistical properties of the new chain's spectrum. In particular, we show that it is highly degenerate, which strongly suggests the existence of an underlying (twisted) Yangian symmetry also for odd m.

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The exactly solvable spin Sutherland model of type and its related spin chain

Cited by 11 publications

References 52 publications

A new perspective on the integrability of Inozemtsev’s elliptic spin chain

A new perspective on the integrability of Inozemtsev’s elliptic spin chain

Rational quantum integrable systems ofDNtype with polarized spin reversal operators

A novel class of translationally invariant spin chains with long-range interactions

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