School of Mathematics and Statistics The University of ... · 2020 · 09 · 30 Jules Lamers · New...

2020 · 09 · 30 Jules Lamers · New connections · UQ/zoom 1/18

How coordinate Bethe ansatzworks for Inozemtsev model

Jules Lamers

School of Mathematics and Statistics

The University of Melbourne

joint work with Rob Klabbers (Nordita)

building on [V. I. Inozemtsev, ’90–’99]


Exactly solvablelong-range spin chains

anisotropic(elliptic)

partially isotropic(trig)

isotropic(rational)

nearest-neighbour(contact)

intermediate(elliptic)

long range(trigonometric)

Heisenberg XXXInozemtsev

Haldane–Shastry

q-deformed Haldane–Shastry

q-deformed Inozemtsev ?(some version of)

Heisenberg XXZ

(some version of)Heisenberg XYZ

anisotropic Inozemtsev ??

anisotropic Haldane–Shastry ??


Inozemtsev’s elliptic spin chainAnatomy

long-range pairwise

elliptic pair potential

[Inozemtsev ’90, ’95]

periods

[Haldane ’88][Shastry ’88][Inozemtsev ’92]

[Heisenberg ’28]

spin exchange

Weierstrass


Inozemtsev’s elliptic spin chainExactly solvable interpolation

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2π2

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5π2

6

π2

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E EHeisenberg● Exactly solvable

up to solving Bethe-ansatz equations

● Quantum integrable

Haldane–Shastry ● Exactly solvable

in closed form(with Jacks)

● Quantum integrable

● Yangian symmetry

Inozemtsev● Exactly solvable

spectrum up to solving BAE● Quantum integrable?


Inozemtsev’s elliptic spin chainSome highlights

● Spectrum for M=2 using Hermite’s solution of Lamé equation

● Spectrum in hyperbolic limitvia connection to hyperbolic Calogero–Sutherland

● Spectrum for general Mvia connection to elliptic Calogero–Sutherland

● Asymptotic Yangian symmetry

● Thermodynamic Bethe ansatz

● Proposal for higher Hamiltonians

● Guest appearance in AdS/CFT

[Inozemtsev ’90]

[Inozemtsev ’92]using [Chalykh Veselov ’90]

[Inozemtsev ’95, ’99]using [Felder Varchenko ’95]

[Inozemtsev ’96]

[Serban Staudacher ’04]

[Dittrich Inozemtsev ’97][Klabbers ’16]

[Ha Haldane ’93][De La Rosa Gomez et al ’16]


Our goal

Understand Inozemtsev’s solution and its limits

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κ

E E


StrategyGeneral considerations

● Isotropy ( -invariance): fix , focus on highest weight

● Use coordinate basis (so will have to ensure cyclicity)

● Homogeneity (translational invariance) determinesDispersion relation

Heisenberg Haldane–Shastry

reciprocal periods [Inozemtsev ’90][Klabbers JL]


Strategy Extended coordinate Bethe ansatz

For general seek wave functions of the form

Assumptions● [technical] has simple poles at equal arguments

● double quasiperiodicity

Cyclicity of requires Bethe-ansatz equations

plane wave

[Inozemtsev ’95][Klabbers JL]

depends on positions!


Connection with eCSResult of extended CBA

If for quantum elliptic Calogero–Sutherland

(for with simple poles at equal arguments)and if BAE

Then as long as we identify [→ next slide]

has


Connection with eCSComments

● rapidities

● is additive iff is so (which doesn’t seem to be the case)

● will have simple poles or double zeroes at equal arguments; latter would be better in HS limit [but see later]

eCS wave function ( )plane wavequasimomenta

[Klabbers JL]

[Klabbers JL]


Two magnonsFrom Lamé to Inozemtsev

● LaméHermite

● Quasiperiodicity fixes in terms of

● Inozemtsev

provided BAEscattering phase


Two magnonsRationalisation & completeness

● Eigenvectors with all have highest weight

● Together the BAE

give an equation elliptic in

● In elliptic coordinates the latter gives a polynomial equation

– Very efficient for numerical solutions

– Can count number of solutions: matches # highest weight

● Energy function is elliptic in too

● Full spectral problem becomes rational in elliptic coordinates


Two magnonsHeisenberg limit

● We have

so

● Energy becomes additive

● BAE

remains finite

becomes Bethe’s wave function

[Klabbers JL]

Bethe’s equation


Two magnonsRecap of Heisenberg

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1520○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

304050100

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6.125

6.25

6.5

7

89

10

1520

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0 π

8

π

4

3π

8

π

2

0

1

2

3

Re pm

Impm

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2021

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2223242526

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2021

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Lcr

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22

0.41 0.42 0.43 0.44 0.45 0.46 0.47

-0.05

0.00

0.05

‘critical length’

π

2π

3π

2

ptot

2

4

6

8

E

descendants

scattering

cf ◦ free (φ=0)

bound

unbound

exceptional

π

2π

3π

2

ptot

2

4

6

8

E

descendants

scattering

cf ◦ free (φ=0)

bound

unboundπ

2π

3π

2

ptot

2

4

6

8

E

descendants

scattering

cf ◦ free (φ=0)

bound

unbound


0.5 1 1.5 2 2.5 3 ∞0

10

20

30

40

50

60

70

410

21.9

61.3

κ

L

n=5

n=4

n=3

n=2

Two magnonsInozemtsev: critical loci

π

2π

3π

2

ptot

2

4

6

8

E

descendants

scattering

cf ◦ free (φ=0)

bound

unbound (beyond cr,r)

exceptional

bound (beyond cr,i)π

2π

3π

2

ptot

2

4

6

8

E

descendants

scattering

cf ◦ free (φ=0)

bound

unbound (beyond cr,r)

exceptional

bound (beyond cr,i)

[Klabbers JL]


Two magnonsHaldane–Shastry limit (I)

π

2π

3π

2

ptot

2

4

6

8

10

E

descendants

motifsπ

2π

3π

2

ptot

2

4

6

8

10

E

descendants

motifs


Two magnons Haldane–Shastry limit (II)

● We have

so

with ‘evaluation’

● BAE become trivial

● Energy becomes (strictly) additive

[Klabbers JL]

Scattering states:Schur expansion of

Jack (upon ev)


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5π2

6

π2

κ

E E

● Solution via relation with elliptic Calogero–Sutherland

● ‘Quasi-additive’ energy

● limits in detail, recover known results

● Spectral problem rationalises

● Open questions

– ?

– Quantum integrability (R, etc) ?

– XXZ-like (q-)analogue ?

Conclusion

School of Mathematics and Statistics The University of ... · 2020 · 09 · 30 Jules Lamers · New...

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