Post on 21-Mar-2021
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).
Tigran Hakobyan Yerevan State University & Yerevan Physics Institute, Armenia
Diagrammatic representation of the spherical part of Calogero model
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 [?, ?].
Tigran Hakobyan Yerevan State University & Yerevan Physics Institute, Armenia
Diagrammatic representation of the spherical part of Calogero model
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 .
Tigran Hakobyan Yerevan State University & Yerevan Physics Institute, Armenia
Diagrammatic representation of the spherical part of Calogero model
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.
Tigran Hakobyan Yerevan State University & Yerevan Physics Institute, Armenia
Diagrammatic representation of the spherical part of Calogero model
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).
Tigran Hakobyan Yerevan State University & Yerevan Physics Institute, Armenia
Diagrammatic representation of the spherical part of Calogero model
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.
Tigran Hakobyan Yerevan State University & Yerevan Physics Institute, Armenia
Diagrammatic representation of the spherical part of Calogero model
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.
Tigran Hakobyan Yerevan State University & Yerevan Physics Institute, Armenia
Diagrammatic representation of the spherical part of Calogero model
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
Tigran Hakobyan Yerevan State University & Yerevan Physics Institute, Armenia
Diagrammatic representation of the spherical part of Calogero model
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
Tigran Hakobyan Yerevan State University & Yerevan Physics Institute, Armenia
Diagrammatic representation of the spherical part of Calogero model
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.
Tigran Hakobyan Yerevan State University & Yerevan Physics Institute, Armenia
Diagrammatic representation of the spherical part of Calogero model
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.
Tigran Hakobyan Yerevan State University & Yerevan Physics Institute, Armenia
Diagrammatic representation of the spherical part of Calogero model
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′
= +
Tigran Hakobyan Yerevan State University & Yerevan Physics Institute, Armenia
Diagrammatic representation of the spherical part of Calogero model
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.
Tigran Hakobyan Yerevan State University & Yerevan Physics Institute, Armenia
Diagrammatic representation of the spherical part of Calogero model
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.
Tigran Hakobyan Yerevan State University & Yerevan Physics Institute, Armenia
Diagrammatic representation of the spherical part of Calogero model
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 .
Tigran Hakobyan Yerevan State University & Yerevan Physics Institute, Armenia
Diagrammatic representation of the spherical part of Calogero model
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
Tigran Hakobyan Yerevan State University & Yerevan Physics Institute, Armenia
Diagrammatic representation of the spherical part of Calogero model
The independent four-order invariants
J1
J2J3
Tigran Hakobyan Yerevan State University & Yerevan Physics Institute, Armenia
Diagrammatic representation of the spherical part of Calogero model
The independent six-order invariants
J1 J2 J3
J4
J5 J6 J7 J8
J9 J10
J11 J12
J13
J14 J15
J16
Tigran Hakobyan Yerevan State University & Yerevan Physics Institute, Armenia
Diagrammatic representation of the spherical part of Calogero model
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
Tigran Hakobyan Yerevan State University & Yerevan Physics Institute, Armenia
Diagrammatic representation of the spherical part of Calogero model
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.
Tigran Hakobyan Yerevan State University & Yerevan Physics Institute, Armenia
Diagrammatic representation of the spherical part of Calogero model
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.
Tigran Hakobyan Yerevan State University & Yerevan Physics Institute, Armenia
Diagrammatic representation of the spherical part of Calogero model