Spin Chains of Haldane–Shastry Type:
A Bird’s Eye View
Federico Finkel, Artemio González-López, and Miguel A. Rodríguez
Dedicated to our friend and colleague Decio Levi in his 70th
anniversary
Abstract We present a brief report on the main properties of Haldane–Shastry type
spin chains and their relation with Calogero–Sutherland spin dynamical models.
Recent work on the thermodynamics of these chains is also discussed.
Keywords Integrable systems · Spin chains · Statistical mechanics
1 Introduction
This is a short review of the construction and main properties of integrable spin
chains of Haldane–Shastry type stemming from their relation with many-body spin
Calogero–Sutherland models, briefly outlining the exact evaluation of their partition
function and studying their thermodynamics.
The Haldane–Shastry chain was introduced independently by Haldane [16] and
Shastry [24] in the late eighties in connection with the one-dimensional Hubbard
model. The explicit computation of its spectrum and its remarkable mathematical
properties attracted very quickly the attention of many researchers in condensed
matter physics. Its extension to other similar chains was carried out through
their relation with spin Calogero–Sutherland models, and many of its integrability
properties were also related to those of the latter models. The classification of
Calogero–Sutherland type models by Olshanetsky and Perelomov [21], and their
relation with the theory of homogeneous spaces and simple Lie algebras and their
root systems, was the clue to understanding their properties in a broader context.
The freezing trick, as introduced by Polychronakos in [23], makes it possible to
F. Finkel · A. González-López · M. A. Rodríguez ()
Depto. de Física Teórica, Universidad Complutense de Madrid, Madrid, Spain
e-mail: ffinkel@ucm.es; artemio@ucm.es; rodrigue@ucm.es
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_1
3
A Bird’s Eye View
Federico Finkel, Artemio González-López, and Miguel A. Rodríguez
Dedicated to our friend and colleague Decio Levi in his 70th
anniversary
Abstract We present a brief report on the main properties of Haldane–Shastry type
spin chains and their relation with Calogero–Sutherland spin dynamical models.
Recent work on the thermodynamics of these chains is also discussed.
Keywords Integrable systems · Spin chains · Statistical mechanics
1 Introduction
This is a short review of the construction and main properties of integrable spin
chains of Haldane–Shastry type stemming from their relation with many-body spin
Calogero–Sutherland models, briefly outlining the exact evaluation of their partition
function and studying their thermodynamics.
The Haldane–Shastry chain was introduced independently by Haldane [16] and
Shastry [24] in the late eighties in connection with the one-dimensional Hubbard
model. The explicit computation of its spectrum and its remarkable mathematical
properties attracted very quickly the attention of many researchers in condensed
matter physics. Its extension to other similar chains was carried out through
their relation with spin Calogero–Sutherland models, and many of its integrability
properties were also related to those of the latter models. The classification of
Calogero–Sutherland type models by Olshanetsky and Perelomov [21], and their
relation with the theory of homogeneous spaces and simple Lie algebras and their
root systems, was the clue to understanding their properties in a broader context.
The freezing trick, as introduced by Polychronakos in [23], makes it possible to
F. Finkel · A. González-López · M. A. Rodríguez ()
Depto. de Física Teórica, Universidad Complutense de Madrid, Madrid, Spain
e-mail: ffinkel@ucm.es; artemio@ucm.es; rodrigue@ucm.es
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_1
3