The q-deformed Haldane–Shastry modelfrom the affine Hecke algebra, via 'freezing', to exact eigenvectors
Jules Lamers[1801.05728]

ongoing work withV. Pasquier and D. Serban

building on D. Uglov[hep-th/950814]
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Outline●

Recap of the Haldane–Shastry model●

q-deformed Haldane–Shastry model– Hamiltonian– Key properties– Exact highest-weight vectors– Overview of underlying algebraic structure●Hamiltonian (Macdonald operator) of spin-Ruijsenaars

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Haldane–Shastry modelspin-1/2 chain,Hilbert space:sites

long-range
pairwise interactions

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[Haldane ’88]
[Shastry ’88]

exchange interactions

pair potential

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’s key properties●●

Infinite-dim symmetry algebra already for
isotropy
Yangian invariance
Exact highest-weight wave functions:

Vandermonde 2●:Jack (zonal)

Free gas of anyons

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Underlying structure
algebradegenerateafine Hecke[Bernard et al ’93]
[Talstra Haldane ’95]

generalised modelspin-Sutherland●
Yangian invariance‘freezing’

spin chain

Haldane–Shastry●
Yangian invariance●exact h.-w. vectors
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quantum many-body system(trig) Sutherland●
exact eigenfunctions:
Jack
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q-deformed underlying structure[Bernard et al ’93]
[Uglov ’95]
[JL Pasquier Serban]

generalised model

algebraspin-Ruijsenaars●
quantum-afine invarianceafine Hecke‘freezing’
spin chain

q-Haldane–Shastry●
quantum-afine invariance●exact h.-w. vectors
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quantum many-body system(trig) Ruijsenaars●
exact eigenfunctions:
Macdonald

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q-deformed (chiral) Hamiltonian
spin-1/2 chain,
Hilbert space:long-range
pairwise form

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[Uglov ’95]
[JL ’18]

sites

q-deformed
exchange interactions

point-split potential

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q-deformed long-range interactions
The long-range exchangetransport

is deformed to2019 · 09 · 18

nearest-neighbour exchange

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Key properties●

Quantum-afine symmetry at finite size

●Multi-spin interactions●●[Bernard et al ’93]
[Uglov ’95]

Spectrum is real iffor ‘full’ Hamiltonian also if[JL Pasquier Serban]

Exact highest-weight wave functions[JL Pasquier Serban]

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Exact highest-weight wave functions[JL Pasquier Serban]

Hecke

symmetric square ofq-Vandermonde2019 · 09 · 18

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Macdonald(q-zonal)
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Spin-Ruijsenaars model
Physical space:[Bernard et al ’93]
[JL Pasquier Serban]

Spin-Macdonald:

Monodromy matrix:[Bernard et al ’93]
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Conclusion

Summary●Haldane–Shastry model can be q-deformed– guiding property:

quantum-afine symmetry already at finite length– pairwise form, accounting for multi-spin interactions– from explicit spin-Macdonald–Ruijsenaars operators●

Exact spectrum at finite size
– highest-weight vectors feature Macdonalds

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Conclusion

Future directions●More detail– explicit (level c = 0) quantum-afine action– (conformal?) field theory describing low energy– q root of unity●Generalizations– higher rank, super case, type BC, etc– q-deformed Inozemtsev spin chain?
– XYZ-like version??

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Bonus material

Spectrum at finite length
pass to ‘motifs’

q-translation

addition isallowed if

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q-momentum

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Bonus material

Exact highest-weight vectors revisited
In the M-particle sector: via XXZ Bethe vectors[JL Pasquier Serban]

as for HS
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Macdonald(q-zonal)
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