diagrammatic representation of the spherical part of calogero …lechtenf/sis11/sistalks/... ·...

Diagrammatic representation of the spherical part of Calogeromodel

Tigran Hakobyan

Yerevan State University & Yerevan Physics Institute, Armenia

The talk is based on the results obtained in collaboration withO. Lechtenfeld, A. Nersessian

International Workshop ”Supersymmetry in Integrable Systems”,Hannover, August 1-4, 2011

Tigran Hakobyan Yerevan State University & Yerevan Physics Institute, Armenia

Diagrammatic representation of the spherical part of Calogero model

The goal of the talk

The main goal of this talk is the investigation of the angular part of the Calogeromodel. Considered as a separate Hamiltonian, it describes a mechanical systemon the sphere, which we call briefly a ”spherical mechanics”.

In particular, we will

Construct the constants of motion for spherical mechanics related to theCalogero model using the matrix model approach.

Introduce a diagramatic representation for constructed invariants.

Study the relation with valence-bonds singlet states for usual quantumspin states.

Study the free-particle limit.

Our previous related publications: (1) Hakobyan, Lechtenfeld, Nersessian, Saghatelyan,

Invariants of the spherical sector in conformal mechanics, J. Phys.A (2011); (2) Hakobyan,

Krivonos, Lechtenfeld, Nersessian, Hidden symmetries of integrable conformal mechanical systems,

Phys. Lett. A (2010); (3) Hakobyan, Nersessian, Yeghikyan, Cuboctahedric Higgs oscillator from

the Calogero model, J. Phys. A (2009).



Motivation

The spherical system of the Calogero model is superintegrable, theintegrals of motion are more elaborated and not investigated much.

To compare approach used recently [with Krivonos, Saghatelyan, Yeghikyan]for the construction of invariants for the spherical system obtained fromgeneral conformal mechanics.

The study of the spherical mechanics has its own interest. It describes aparticle motion on the sphere interacting with some potential fields, whichcan be considered as some multi-center high dimensional generalization ofthe spherical Higgs oscillator.

The spherical mechanics of the Calogero model was used for theexplanation of the non-equivalence of different quantizations of theCalogero model [Feher, Tsutsui, Fulop, 2005], as well as for the constructionits superconformal generalizations [Bellucci, Krivonos, Sutulin, 2008].

The shift operators mapping the spherical part to the Laplace-Beltramioperators have been constructed [?, ?].



Calogero model

The system [Calogero, 69,71]

H =1

2

N∑

i=1

p2i +

∑

i<j

g2

(qi − qj)2,

can be obtained form the free Hermitian matrix model

Hmat =1

2Tr P

2 := Tr P2, {Pij , Qj′i′} = δii′δjj′

by the SU(N) reduction [Kazhdan, Kostant, Sternberg, 78, Olshanetskii, Perelomov,

81]P → UPU

+, Q → UQU+.

The conserved current isJ = i[Q, P].

The gauge fixation

Qjk → qjδjk , Jiji 6=j= −g .



Calogero model

Provides the reduction Hmat → H:

J → g(−1 + |u〉〈u|), ui = 1 =⇒ [P, Q] = const.

Pjk → Ljk = δjkpk + (1−δjk)ig

qj − qk= Lax matrix

Chose an orthogonal basis T0 ∼ 1 ∈ u(1) and Ta ∈ su(N):

Tr TaTb = δab , [Ta, Tb] =∑

c

ifabcTc ,

Q =

N2−1∑

a=0

QaTa, P =

N2−1∑

a=0

PaTa, J =

N2−1∑

b=1

JbTb

{Pa, Qb} = δab, {Ja, Jb} = i∑

c

fabcJc , Jc = −i

2

∑

a,b

fabcMab ,

where the angular momentum tensor is

M = P ∧ Q =∑

a,b

Mab Ta ⊗ Tb , Mab = PaQb − PbQa.



Reduction of the center of mass

The U(1) symmetry of Hmat = 12Tr P2 under

Q → Q + ǫ1, P → P.

The related conserved current is

Tr P ∼ P0SU(N) reduction−−−−−−−−−→

N∑

i=1

piU(1) reduction−−−−−−−−→ 0

{P0, Jb} = 0

The U(1) reduction eliminates of the center-of-mass momenta and coordinatesof the Calogero system. The reduces Hamiltonian is

Hr.c.m. =

1

2

N−1∑

i=1

P2i +

∑

1≤i<j≤N−1

g2

(αVij Q)2

,

αVij = Eii − Ejj = root vectors of su(N).



Conformal invariance and Casimir element

The action preserves with respect to conformal so(2, 1) ≡ sl(2, R)

{H, D} = 2H, {K , D} = −2K , {H, K} = D ,

D = Tr PQ, K =1

2Tr Q

2, H =1

2Tr P

2.

The invariant basis with signature (1,−1,−1):

S1,3 = H ± K , S2 = D , {Sα, Sβ} = −2ǫαβγSγ .

Although the conformal symmetry is not a symmetry of the Hamiltonian, itresults in additional constant of motion given by the Casimir element:

I =∑

α

SαSα = 4KH − D

2 = Tr P2Tr Q

2 − (TrPQ)2.



The spherical part I: spherical mechanics

The Casimir element I is the angular part of the Calogero Hamiltonian. Inany spherical coordinate system, we have:

D = rpr , K =r2

2, H =

p2r

2+

I(pφα, φα)

2r2.

The spherical part I depending on angular coordinates. It is quadratic onangular momenta pφα

.

It can be expressed aslo in terms of the angular momentum

I =1

2(Tr ⊗ Tr)M2 =

∑

a<b

M2ab .

The spherical mechanics given by the Hamiltonian I(pφα, φα) describes a particle

on (N − 1)-dimensional sphere moving in the presence of the external potential.



The spherical mechanics of the four particle Calogero model

Two-dimensional spherical system ob-tained from the N = 4 Calogeromodel with reduced center of mass[Hakobyan, Nersessian, Yeghikyan, Cuboc-

tahedric Higgs oscillator from the Calogero

model, 2009]

I =p2

θ

2+

p2ϕ

2 sin2 θ+

9g(8 − tan2 θ)2

2(3 tan2 θ − 8 + tan3 θ cos 3ϕ)2+

9g

4 sin2 θ(1 + cos 6ϕ)

+12g

3 tan2 θ − 8 + tan3 θ cos 3ϕ



Relation of I with the conventional integrals

Calogero model has N Liouville integrals [Moser, 1975]

Ik = Tr Pk =

∑

a1,...,ak

da1...ak Pa1 . . . Pak ,

I1, I2reduction−−−−−−→

∑

pi , H.

Here da1...ak = Tr (Ta1 . . . Tak ) is the invariant tensor.

In particular, da = δa0, dab = δab .

and N − 1 additional integrals (the system is max. superintegrable)[Wojciechowski, 1983]

The angular part can be expressed in terms of these integralsI = I(Ik , Gk′).

It maps the Liuoville integrals to the additional integrals:

Gk = {I, Ik}, k 6= 2



Constants of motion of spherical system I

The integrals of motion of the spherical mechanics are expressed in termsof the angular coordinates and momenta

Ik = Ik (pθα, θα).

D ,K depend only on radial pr , r .

{Ik , H} = 0, since {Ik , I} = 0 and H = p2r /2 + I/r2.

Therefore, the integrals Ik are involution with the whole conformal algebra{Ik , sl(2, R)} = 0 and are conformal singlets.

From the other side, the integrals must be also a SU(N) scalar in order toassure the valid reduction map from the matrix model.

So,

The algebra of integrals of the spherical mechanics is formed by SU(N)×SL(2,R)singlets.



Construction of invariants of the spherical system

Any observable from the system I(pφα, φα) can be expressed in terms of

(overcompleted) set of Mab.

The conformal algebra is in involution with the angular momentum tensor{Mab , Sα} = 0. So, Mab are conformal singlets.

In remains to form from Mab a SU(N) invariant.

The indexes a, b belong to the adjoint representation of SU(N).

So, SU(N) invariant can be constructed by contraction of Ma1b1 . . . Makbkwith

some number of invariant tensors dc1...cl .

This will be a polynomial Ik(Mab) of k-th order on Mab .

In general, there are different Ik of the same k , which are independentfrom each other and lower-order invariants Ik′<k .

There is a simple diagramatic representation of variety of such integrals, aswell as their description in terms of representations of sl(2, R) algebra.



Diagrammatic representations of the spherical invariants Ik

The graphical representations of Mab and da1...an :

Mab =a b

dabcde = b

ab

bb

cb

db

e

The bond-crossing relations reduces significantly the number of functionallyindependent integrals.

Ma′bMab′ = MabMa′b′ − Maa′Mbb′

Diagrammatic representation of the crossing relations:

a

b

a′

b′

a

b

a′

b′

a

b

a′

b′

= +



Examples of spherical invariants

The analogue of the Liouville integralsIk of H:

Ik = Tr M2k = (Tr ⊗ Tr)M2k

=∑

ai ,bi

da1...a2k db1...b2kMa1b1 . . . Ma2k b2k

b

b

b

b

b

b

b

b

b

b

b

b

a1

b1

a2

b2

a3

b3

a4

b4

a5

b5

a6

b6

In particular, I1 = I, and since dab = δab, the adjoint indexes coincide (on theleft):

I =

b b

b b

=

b

b b

b

b b

b

b b

b

b

a1 a2 a3 a4 a5 a6

b1 b2 b3

The right diagram defined by R = (Tr ⊗ 1)M2 [Wojciechowski, 84]

Tr Rk =

∑

ai ,bi ,b′

i

db1b′1...bkb′

kMa1b1Ma1b′

1. . . Mak bk

Mak b′k.



Commutators

Commutators:

{In, Im} = 4nmIn,m , {Ir.c.m.n ,Ir.c.m.

m } = 4nm

(

Ir.c.m.n,m −

1

NI

′r.c.m.n,m

)

,

where In,m, I′n,m are combined diagrams:

I4,6 =b b b b b b b b b b

b b b b b b b b

I′

4,6 =b b b b b b b b b b

b b b b b b b b

The consequence of cyclic symmetry of da1,...ak , and U(N), SU(N) completenessrelations

N−1∑

a=0

da1...an−1adaan ...an+m−1 = da1...an+m−1 ,

N−1∑

b=1

db1...bn−1bdbbn ...bn+m−1= db1b2...bn+m−1

+1

Ndb1...bn−1

dbn ...bn+m−1.



Free-particle limit g → 0

In g → 0 limit only Cartan subalgebra gererators remain

P →

N−1∑

i=0

piTi , Q →

N−1∑

i=0

qiTi , M →

N−1∑

i,j=0

Mij Ti ⊗ Tj

dk1k2...kn →

{

1 if k1 = · · · = kn

0 otherwise

b

b

b

b

b

b

b

b

b

b

b

b

a1

b1

a2

b2

a3

b3

a4

b4

a5

b5

a6

b6

b

b

=⇒

i

j

I = I1 →N−1∑

i,j=0

M2ij , Ik →

∑

i,j

M2kij , Ik →

∑

i

pki .



Relation with so(N) invariant rigid body

Higher order Casimir invariants of so(N) are not independent:

TrMn =∑

a1...an

Ma1a2Ma2a3 . . . Mana1 = In/2, n = 2k

since due to the crossing relation Mn =1

2

(

TrM2)

Mn−2 = IMn−2.

b

b

b

b

b

b

=

b

b

b

b

b

b

The quadratic Casimirs

k−1∑

i<j=0

Mij for embedded

so(2) ⊂ so(3) ⊂ · · · ⊂ so(N) are not SU(N) invariant



The independent four-order invariants

J1

J2J3



The independent six-order invariants

J1 J2 J3

J4

J5 J6 J7 J8

J9 J10

J11 J12

J13

J14 J15

J16



Dual representation of diagrams

The Liouville integral of Calogero model is the highest weight state of so(2, 1)representation of conformal spin s = n

2formed by

Ijk = TrsymP

k−jQ

j = |+ · · ·+︸︷︷︸

k−j

− · · ·−︸︷︷︸

j

〉, 0 ≤ j ≤ k , I0k = Ik .

where |σ1 . . . σn〉 = TrsymAσ1 . . . Aσn , σi = ±, A

+ = P, A− = Q.

The angular momentum components can be expressed

Mab =∑

σ1,σ2=±

εσ1σ2Aσ1a A

σ2b , εσ1σ2 = −εσ2σ1

The dual graphical representations:

εσσ′ =σ σ

′

|σ1σ2σ3σ4σ5〉 = b

σ1b

σ2b

σ3b

σ4b

σ5



Valence-bond states

The crossing relation

1

2

3

4

1

2

3

4

1

2

3

4

= +

is a relation among dimer pairs [Temperley, Lieb, 1971]

[32][14] = [12][34] − [13][24]

where [ij] = | +i −j〉 − | −i +j〉 is a spin-1/2 singlet.

Temperley-Lieb basis for spin lattice: Put the spins on a line (cycle), thentake all possible dimer coverings without bond intersections.

To any dimer covering of a spin lattice an integral of spherical mechanicscorresponds, which respects the bond crossing relation.

The linearly independent integrals correspond to the Temperley-Lieb basis.

Only 2N-3 integrals are functionally independent.



Conclusion

Summary

The integrals of motion for the spherical system associated with theangular part of Calogero model is constructed.

The diagrammatic representation of the integrals is given in terms ofdimer valence-bond states.

The problems

To extract Liouville integrals of motion.

To compute Poisson algebra of invariants.

Possible relation with WN algebras.



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