GENERATINGFUNCTIONSFOREFFECTIVEHAMILTONIANS....

ЯДЕРНАЯФИЗИКА, 2011, том 74, № 7, с. 1–9

ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ

GENERATING FUNCTIONS FOR EFFECTIVE HAMILTONIANS.SYMMETRY, TOPOLOGY, COMBINATORICS

c© 2011 B. I. Zhilinskii*

Universite du Littorale Cote d’Opale, Dunkerque, FranceReceived July 19, 2010

Recent developments associated with old technique of generating functions and invariant theory which Ihave started to apply to molecular problems due to my collaboration with Yu.F. Smirnov about 25 yearsago are discussed. Three aspects are presented: the construction of diagonal in polyad quantum numbereffective resonant vibrational Hamiltonians using the symmetrized Hadamard product; the decompositionof the number of state generating function into regular and oscillatory contributions and its relationwith Todd polynomials and topological characterization of energy bands; qualitative aspects of resonantoscillators and fractional monodromy as one of new generalizations of Hamiltonian monodromy.

1. HISTORICAL INTRODUCTION

At the end of 1980s I began my collaborationwith Yurii Fedorovich Smirnov. We both worked atthat time in Moscow State University. He workedat Physics Department (Nuclear Physics ResearchInstitute), myself –– at Chemistry Department. Thereason of our collaboration was our mutual interestin applications of invariant theory, in particular ofthe Molien generating function technique, to con-struction of effective Hamiltonians for nuclear models(main field interest of Yu.F. Smirnov) and to molec-ular systems (my proper field of research) [1, 2].Yu.F. Smirnov guided me through mathematicalaspects of classical invariant theory and throughcompletely unfamiliar to me at that time generatingfunction technique. I tried to formulate mathematicalideas of invariant theory in a way most directly relatedto natural questions arising in my study of qualita-tive theory of highly-excited finite particle quantumsystems. We also started to work on new possibleapplications of another modern technique, quantumalgebra, to molecular and nuclear systems [3], butat the beginning of 1990s I left Moscow and settledin France, whereas Yu.F. Smirnov moved to Mex-ico. We continued to meet and to discuss scientificproblems but had no occasion to publish somethingtogether. Nevertheless, the subject that we havestarted together with Yu.F. Smirnov becomes one ofthe powerfull tools which I used during last 20 yearsin relation to different problems of qualitative theoryof highly-excited quantum molecular systems [4–9].Some of these problems I present in the current paperwhich I devote to the memory of Yurii FedorovichSmirnov.

*E-mail: [email protected]

2. HADAMARD PRODUCT OF GENERATINGFUNCTIONS

I start with simple molecular example whichnevertheless has many common points with gen-eral effective vibrational Hamiltonians in molecularor nuclear physics. Let us consider vibrations ofmethane molecule CH4, which has equilibrium con-figuration of Td symmetry. There are one nondegen-erate νA1

1 , one doubly degenerate νE2 , and two triply

degenerate νF2i , i = 3, 4, vibrations, which moreover

satisfy special resonance condition ν1 : ν3 : ν2 : ν4 == 2 : 2 : 1 : 1. Assuming slight anharmonicity, theexcited vibrational states arrange into so-called vi-brational polyads formed by groups of nearly degen-erate vibrational states (polyads) characterized by thesame effective number of quanta N = 2n1 + n2 ++ 2n3 + n4. In order to construct effective vibrationalHamiltonian describing inter-polyad structure, weneed to take into account all properly symmetrizedoperators formed from harmonic oscillator creationand annihilation operators{ [(

a+1

)s1 ×(a+

2

)s2 ×(a+

3

)s3 ×(a+

4

)s4]Γ ×

×[(a1)t1 × (a2)t2 × (a3)t3 × (a4)t4

]Γ

}A1

satisfying additional restriction 2s1 + s2 + 2s3 + s4 == 2t1 + t2 + 2t3 + t4 which follows directly from theresonance conditions. Along with Td geometricalinvariance imposed on the Hamiltonian by phys-ical requirements, the invariance with respect totime reversal should typically be imposed as well.

1

2 ZHILINSKII

The difference between spatial symmetry and time-reversal invariance is technically important becauseelementary creation (annihilation) operators in mostcases could be chosen as transforming according toirreducible representations of the symmetry group,whereas time reversal operation transforms creationoperators into annihilation and vice versa.

The crucial initial point is the Molien theorem [4,10–14] which in its simplest form enables one toconstruct the generating function for the numberof irreducible representations of a given type in thedecomposition of the N th symmetric power of oneirreducible representation. More exactly, for the finitegroup G, for the initial representation Γi and for thefinal representation Γf the generating function

MG(Γf ← Γi;λ) = |G|−1 × (1)

×∑g∈G

χΓf (g)det (In − λΓi(g))−1 =

∞∑N=0

CNλN

gives the numbersCN indicating how many times thefinal representation Γf appears in the decompositionof the N th symmetric power of the initial representa-tion Γi. In (1) |G| is the order of the finite group G,χΓ is the character of the irreducible representationΓ, tilde means complex conjugation. λ is a dummyvariable. This generating function typically can bewritten in the form of a rational function of somespecial form ∑

k ckλk

(1− λd1)(1 − λd2) · · · (1− λds),

which has symbolic meaning specifying existence ofa certain number of functionally independent denom-inator invariants and a number of linearly independentthough algebraically dependent numerator invariants.This symbolic interpretation gives the description ofthe integrity basis [4] (or in other words, the homo-geneous system of parameters [15]). An alternativeinterpretation of generating function can be donein terms of a set of generators (which are typicallyfunctionally dependent), a set of relations (syzygies)between them, a set of relations between relations,etc. [16].

Calculation of the generating functions for thenumber of invariants or covariants constructed fromelementary tensors transforming according to a givenirreducible or reducible representation of finite groupsis just a relatively simple task as soon as the charactertable is known. A less trivial problem is the con-struction of generating function for operators whichare diagonal within polyads. This can be achieved byextending the geometrical symmetry group with thedynamical symmetry responsible for the resonance orby constructing the needed generating function from

two generating functions, one for creation operators,another for annihilation operators. The nontrivial partconsists in introducing the important restriction onthe numbers of creation and annihilation operatorswhich leads to operators diagonal within polyads.Such construction was realized, for example, in [17].More recently a slightly different approach was pro-posed which is based on the construction knownas Hadamard product of formal power series [15,18–20]. The advantage of this approach is due tothe possibility to take into account in a simple wayan additional symmetry requirements imposed si-multaneously on creation and annihilation operators,like invariance with respect to time reversal, and torepresent the answer again in the rational functionform.

Let us remind here that the Hadamard productof two formal power series f(z) =

∑n≥0 fnz

n andg(z) =

∑n≥0 gnz

n is defined as their term-by-termproduct [15, 18–20]

f(z) � g(z) =∑n≥0

fngnzn. (2)

The following fact is known: the Hadamard product oftwo rational functions is a rational function [18]. Thisstatement allows to convert the formal power seriesinto a rational function form.

We start with an extremely simple example oftwo A modes in the 1 : 1 resonance with trivialsymmetry group. The generating function for creation(or equivalently for annihilation) operators takes theform depending on two dummy variables (λ, k) whichcount independently the degree of creation operatorsand the associated modifications of the polyad quan-tum numbers:

g(A← 2A,λ, k) =1

(1− λk)2 = (3)

=∞∑

n=0

(n+ 1)λnkn.

Now in order to construct the generating functionfor diagonal operators we need to form the powerseries in k with coefficients being squares of coeffi-cients in (3). This operation is exactly the Hadamardproduct (square) of formal power series [15, 18–20].We denote the Hadamard product by �k and itsapplication to generating function (3) gives for theHadamard square

1(1− λk)2 � k

1(1− λk)2 = (4)

=∞∑

n=0

(n+ 1)2λ2nkn =1 + λ2k

(1− λ2k)3.

ЯДЕРНАЯ ФИЗИКА том 74 № 7 2011

GENERATING FUNCTIONS 3

The degree of auxiliary variable λ counts now the totaldegree of creation and annihilation operators. At thesame time the variable k counts ∆n associated withonly creation (or only annihilation) operators formingthe diagonal operator.

Formula (4) follows directly from the identities(where t replaces the λ2k)

d

dttd

dt

(1 + t+ t2 + t3 + . . .

)=

∞∑n=0

(n+ 1)2tn, (5)

d

dttd

dt

(1

1− t

)=

1 + t(1− t)3 . (6)

In order to take into account only time-reversalinvariant operators we need to use instead of simpleHadamard square the symmetrized Hadamard square[(g(λ) � kg(λ) + g(λ2)]/2 which is an analog of asymmetrized square of an irreducible representation.As soon as normal Hadamard square is known, the

calculation of the symmetrized square meets no diffi-culties. Thus, for the generating function (3) its sym-metrized Hadamard square, counting the number ofdiagonal time-reversal invariant operators, becomes

12

[(g � kg) + g(λ2)

]=

=12

(1 + λ2k

(1 − λ2k)3+

1(1− λ2k)2

)=

1(1− λ2k)3

.

The nontrivial character of k dependence appearsin the case of any k1 : k2 resonances different form1 : 1. We give as an example the 2 : 1 : 1 resonancebetween nondegenerate A1 and doubly degenerate Emodes of tetrahedral molecules.

The total generating function describing all diag-onal time reversal and Td-invariant operators con-structed through arbitrary intermediate representa-tions has the following form

g(A1,TR,∆N = 0,← A1 ⊕ E (2 : 1);λ, k) = (7)

=1 + λ7k4 + λ8k5 − λ12k7 − λ13k8 − λ20k12

(1− λ2k) (1− λ2k2) (1− λ3k2) (1− λ4k2) (1− λ6k3) (1− λ9k6).

The generating function (7) is given in its simplestmost reduced form which has six terms in the denom-inator and both positive and negative contributionsin the numerator. Naturally, there should be only fivefunctionally independent invariants for this problemand consequently the six terms in the denominatorcorrespond to a system of invariants related by syzy-gies. We do not want to discuss here the explicitconstruction of a system of syzygies. This example isgiven in order to demonstrate that even in the case ofrelatively simple generating functions the structure ofthe associated module of invariant functions can berather complicated.

3. DENSITY OF STATE FUNCTION ANDTOPOLOGY OF ENERGY BANDS

The generating function for the number of stateswithin vibrational polyads formed by resonant oscil-lators is an extremely interesting object from com-pletely different points of view.

First of all, the number of states in polyads, inthe case of nontrivial resonance relation betweenvibrational frequencies, is a quasipolynomial, i.e. thisfunction can be split into a smooth polynomial partand an additional oscillatory contribution, whose

period is related to the resonance condition [4–6, 21].The generating function technique allows us to ex-tract directly from the total generating function bothregular polynomial and oscillatory contributions tothe density of states.

Namely, the generating function for the number ofstates in polyads formed by d1 : d2 : . . . : dk resonantoscillators can be written as

gd1:d2:...:dK(t) = (8)

=1

(1− td1)(1 − td2) · · · (1− tdK ).

The coefficient CN in the formal series

gd1:d2:...:dK(t) =

∑N

CN tN , (9)

can in its turn be represented in the form

CN = aK−1NK−1 + aK−1N

K−1 + . . . + (10)

+ a0 + oscillatory part,

with aj being the polynomials of di exposants [4]

aK−1−i(d1, d2, . . . , dK)∼ (11)

∼ 1(K − 1− i)!Toddi([d1, d2, . . . , dK ]).


4 ZHILINSKII

1.0

0.5

0

–0.5

–1.0

C

N

20 40 60 80 100

N

Fig. 1. Oscillatory part of the density of states for 3 : 5 : 7 resonance oscillators.

The oscillatory part of the number of state functionhas complicated form related to the problem of count-ing the lattice points of rational polyhedra [22, 23], butagain it can be represented with a number of the sameTodd polynomials each of which now reproduces onlypart of values of oscillatory function for certain integervalues of arguments.

Figure 1 shows an example of oscillatory con-tribution to the number of state function for threeoscillators in the 3 : 5 : 7 resonance. It is interestingto see that a series of shifted parabolic dependenciesrecover all oscillatory contributions. Moreover, thequadratic polynomial turns out to be the same as thequadratic polynomial describing the regular polyno-mial contribution.

The appearence of Todd polynomials forces us tothink about a topological interpretation and this isreally the case. To see that we can use the deforma-tion quantization approach which gives the followingexpression for the number of quantum states [24, 25](the so-called general index formula):

Tr1 =∫

Ch(V )Ch(h)Todd(M). (12)

Here, Tr1 gives the number of states for the vectorbundle V over manifold M characterized by its rankand Chern numbers; Todd(M) is the Todd polynomialfor the manifold M , i.e. the Todd polynomial for thetangent fibre bundle over M ; Ch(V ) is the Cherncharacter of the vector fibre bundle V ; Ch(h) is theChern character for the quantization of the trivial linebundle overM .

The relation between the topology of the bands andthe number of states within the band, which followsdirectly from the index formula (12), is supported bythe description of the rearrangement of bands withinthe semi-quantum model [7, 26, 27].

Another interesting aspect of generating functionsfor the number of states within vibrational polyads canbe illustrated by examinating two simple and quitesimilar examples of vibrational resonances, namelythe three-degree-of-freedom oscillators with reso-nance conditions 1 : 1 : 2 and 1 : 2 : 2. In both cases,the reduction of the classical problem is based onthe harmonic aproximation and takes into accountthe integral of motion corresponding to the totalaction of harmonic problem. From the point of viewof topology, the reduced classical phase spaces are,in both cases, the weighted projective spaces P{1,1,2}and P{1,2,2}, respectively. The singularity structure ofthese reduced spaces is naturally quite different. It isinteresting to note that there is certain equivalencebetween P{1,2,2} space and a standard complex pro-jective space CP2, which is a reduced space for triplydegenerate harmonic oscillator. At the same time, theP{1,1,2} space is essentially different [28, 29].

Working with generating functions for the num-ber of states, it is easy to formulate an equivalentproposition. Let us start with the 1 : 2 : 2 resonanceexample. We can rewrite the generating function forthe number of states in this case as follows

g1:2:2 =1

(1− t)(1− t2)2 = 1 + t+ 3t2 + (13)

+ 3t3 + 6t4 + 6t5 + . . . =1 + t

(1− t2)3 =

=1

(1− u)3 + t1

(1− u)3 .

A formal Taylor series expansion of the generatingfunction (13) shows clearly that looking separatelyon even and odd polyads we find exatly the samenumbers as in the case of triply degenerate oscillator.



Fig. 2. Typical images of the energy–momentum map for completely integrable Hamiltonian systems with two degrees offreedom in the case of (from left to right) integer monodromy, fractional monodromy, nonlocal monodromy, and bidromy. Valuesin light shaded area lift to single 2-tori; values in dark shaded area lift to two 2-tori.

Fig. 3. Two-dimensional singular fibers in the case of integrable Hamiltonian systems with two degrees of freedom (left toright): singular torus, bitorus, pinched and curled tori.

a b c

Value of the first action

Ene

rgy

Fig. 4. Quantum joint spectra for typical regions of the image of energy–momentum map for 2D-integrable problems. Fromleft to right: regular lattice, lattice with an isolated defect, lattice with two-component region.

The same effect can be seen after rearrangement ofthe initial generating function into a sum of two termswith denominator coinciding with the denominator ofgenerating function for the 1 : 1 : 1 oscillator (afterreplacing t2 = u).

The same kind of transformation applied to thegenerating function for the 1 : 1 : 2 oscillator givesmore complicated contributions

g1:1:2 =1

(1− t)2(1− t2) = (14)

=1 + 2t+ t2

(1− t2)3 =2t

(1− u)3 +1 + u

(1− u)3 .

We have again (after replacing t2 = u) different terms

with the same denominator (1− u)3, which is similarto the 1 : 1 : 1 oscillator problem. At the same time,one of two terms has nontrivial numerator, (1 + u).The nontrivial numerator is present for bands overcomplex projective space in the case of quantizationof topologically nontrivial vector bundles (accordingto general index formula). The relation between thegenerating function for the number of states (overCP 2) written in the form

A+Bu+ Cu2

(1− u)3

and the topology of the fiber bundle over CP 2 has the


6 ZHILINSKII

Fig. 5. Construction of the 1:(−1) lattice defect startingfrom the regular Z2 lattice. Dark grey quadrangles showthe evolution of an elementary lattice cell along a closedpath around the defect point.

following form (see [7, 26])

r = A+B + C,c1 = −B − 2C,

c2 = −B/2− 2C +B2/2 + 2BC + 2C2,

where r is the rank and c1, c2 are the first and secondChern classes of the fiber bundle.

Returning now back to the interpretation of thedecomposition of generating function (14) for the1 : 1 : 2 resonance, we can state that the internalstructure of polyads formed by oscillators in the1 : 1 : 2 resonance is equivalent to superposition oftwo trivial bands over CP2 space for odd polyadsand to superposition of two bands (one with trivialtopology and another with nontrivial first Chern class,c1 = −1) for even polyads. Namely, on the basis ofthis concrete example it is possible to formulate ageneral tentative interpretation of the internal struc-ture of polyads.

The internal structure of polyads for general res-onance (over weigted projective space) can be de-scribed as formed by several polyads over standardprojective space but with nontrivial topological prop-erties (Chern classes).

4. FRACTIONAL MONODROMYAND DEFECTS OF LATTICES

In this section I want to demonstrate how thegenerating functions formalism and, in particular,the splitting of the number of state functions intoregular and oscillatory contributions turns out tobe the stimulating argument for introducing a newnotion of fractional monodromy. For this I need first toremind several facts about standard (integer) Hamil-tonian monodromy for completely integrable classicalHamiltonian systems, and about its quantum analogand its relation to defects of regular lattices.

For classical integrable Hamiltonian systems withtwo degrees of freedom the image of the correspond-ing energy–momentum map contains regular and

singular values (see Fig. 2). In order to stress thatdifferent images of energy momentum map are quitecharacteristic for concrete physical problems we pro-vide here physical examples for each of four images:(i) a particle in a “mexican hat” potential [30]; (ii) twononlinear oscillators in the 1 : (−2) resonance [31];(iii) quadratic spherical pendulum [32], or LiCNmolecule [33]; (iv) three-dimensional oscillator with1 : 1 : 2 resonance or vibrational stretching modes ofCO2 like molecule [34].

Whereas inverse images of the regular values arealways regular tori, the inverse images for singularvalues are topologically different objects (shown inFig. 3).

The presence of isolated singular values (like inFig. 2, left) leads to an obstruction to the existenceof global action–angle variables in spite of the factthat the problem is completely integrable one. Atypical generic example of an isolated critical valueof the energy–momentum map for classical dynamicsystems with two degrees of freedom is a so-calledpinched torus (see Fig. 3). Due to its presence, thetoric fibration over any closed contour surroundingcritical value should be nontrivial. This nontrivialcharacter of classical integrable (almost) toric fi-bration is known under the name of Hamiltonianmonodromy [35, 36]. In order to see, in the simplestway, the manifestation of Hamiltonian monodromyin corresponding quantum systems we can analyzethe joint spectrum of mutually commuting quan-tum observables. The joint spectrum is a quantumequivalent of the classical energy momentum map.For two-degree-of-freedom systems, any simply con-nected region of regular values of classical energy–momentum map is represented in quantum mechan-ics as a locally regular lattice of commun eigenvaluesof two commuting integrals of motion.

Critical value manifests itself in quantum jointspectrum as a special defect of regular lattice [37]. Tosee the effect of a defect we can check the evolutionof the elementary cell of the lattice while it makesa closed loop surrounding the critical value [38].Figure 4 demonstrates the Hamiltonian monodromyby comparing the initial cell with the final cell. Thematrix transforming initial cell into final one is thequantum monodromy matrix.

The construction of the “monodromy defect” ofa regular lattice (as it is shown in Fig. 5) is basedessentially on the “cutting and glue” procedure widelyused in solid state physics in order to explain thestructure of dislocations and disclinations [39]. Themain feature of the construction consists of removinga slice from the regular lattice in such a way that thenumber of removed points with a given value of one ofthe integrals of motion varies linearly with the value



Fig. 6.Construction of 1 : 2 rational lattice defect. Left: Elementary cell does not pass (the result is ambigouos). Right: Doublecell passes unambigously.

–1.0

–1.0 0

Ene

rgy

J

z

/(2

|

S

|

+

|

N

|

)0.5–0.5

–1.51.0

λ

= 0.5

–0.50

0.51.01.5

–0.7–0.5

0

0.5

Ene

rgy

λ

= 0.5

–0.6 –0.5 –0.4 –0.3

J

z

/(2

|

S

|

+

|

N

|

)

Fig. 7. Quantum fractional monodromy for two coupled angular momenta.

of that integral. The mathematical origin of such con-struction is the Duistermaat–Heckman theorem [40]which formulates, in particular, that the volume ofreduced phase space for integrable system with twodegrees of freedom is a piece-wise linear functionof the integral value. In quantum mechanics thevolume of the reduced phase space should be replacedby the number of quantum states for the reducedproblem. It is interesting to note, that so constructed“elementary monodromy defect” is quite differentfrom standard dislocations, disclinations, and otherdefects suggested and studied in solid state physics.The natural question, which appears together withan interpretation of Hamiltonian monodromy as adefect of regular lattice, is: how can we generalizethe “elementary monodromy defect” and the notionof Hamiltonian monodromy itself just by trying togeneralize the concept of lattice defects and at thesame time keeping in mind the realization of latticeas a joint spectrum of several commuting operators?Here it is just the time to remind the typical formolecular problem oscillatory behavior of the densityof state function. Instead of removing a wedge fromthe regular lattice with the number of removed states

being the linear function of the integral of motion,we can remove more complicated wedge with thenumber of removed states being the quasi-polynomialfunction (having ocsillatory contribution) of the in-tegral of motion. The simplest situation of such kindcorresponds to oscillatory contribution with period 2.Naturally, in this case the gluing of the cut afterremoving the wedge should lead to linear rather thanpoint defect (in the case of two-dimensional lattices).

The construction of the rational 1 : 2 defect isshown in Fig. 6. It is clear, that in the presenceof such fractional defect the elementary cell cannotunambigously cross the defect line. In the case of1 : 2 fractional defect the elementary cell can take twodifferent forms after crossing the defect line dependingon the place of crossing point. At the same time,looking at the evolution of the double cell duringcrossing the defect line, one can see that the resultis completely independent of the crossing point.

A formal interpretation of the monodromy matrixin terms of elementary cell transformation leads insuch a case to monodromy matrix with fractionalentries. Nevertheless, one should not forget thatthe elementary cell itself is not passable through


8 ZHILINSKII

the fractional cut. Only multiple cells are passableand writing monodromy matrix in terms of multiplecells (or, in more mathematical terms, in terms of asubgroup of the homology group) leads naturally to amonodromy matrix with integer entries. This allowsus to give another description of the fractional defect,namely, the fractional defect (which has codimensionone) can be removed by going to the sublattice withbigger “elementary” cells. These “bigger” cells are,in fact, multiple cells of the original lattice, and thesemultiple cells are passable from the point of view offractional defect under study.

From the classical dynamics point of view thetopological structure of critical fibers associated withfractional defect line is the curled torus, shown inFig. 3. The appearence of corresponding singularity inintegrable fibration is due to nonlinear resonance be-tween two modes. The simplest mathematical modelof fractional defect is related to two nonlinear oscilla-tors with 1 : (−2) resonance [31]. Similar singularitycan equally appear in models of coupling of twoangular momenta (see Fig. 7). Detailed analysis ofsuch model problem with interaction of two angularmomenta leading to fractional monodromy defect wasdone in [41] by using the effective Hamiltonian

H =1− λ|S| Sz + (15)

+ λ(

1|S||N|SzNz +

12|S||N|2

(N2

−S+ +N2+S−

)),

written in terms of two angular momenta N == (Nx, Ny, Nz) and S = (Sx, Sy, Sz). As a more re-alistic example, we can cite such simple and in thesame time fundamental physical system as hydrogenatom in presence of static electric and magneticfields which was recently shown to possess manydifferent qualitative effects [42] including fractionalmonodromy and a new, recently introduced notion ofbidromy [43].

5. CONCLUSION

I would like to note in the conclusion that the gen-erating function technique, being rather old math-ematical tool, remains unfortunately until now arelatively exotic method which has a few applicationsin molecular and atomic physics. At the same timethe generating functions are at the intersection ofa number of quite different mathematical fields. Wehave mentioned combinatorics, topology, symmetry,dynamical systems. We hope that this multidisci-plinary aspect of generating function will stimulatevarious application in molecular physics and in nat-ural science in general.

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