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New Integrable Systems from Unitary Matrix

arXiv:hep-th/9110064v1 22 Oct 1991CU–TP–537New Integrable Systems from Unitary MatrixModels∗Alexios P. PolychronakosPupin Physics Laboratories, Columbia University,New York, NY 10027Abstract: We show that the one dimensional unitary matrix model with poten-tial of the form aU + bU2 + h.c. is integrable. By reduction to the dynamics ofthe eigenvalues, we establish the integrability of a system of particles in one spacedimension in an external potential of the form a cos(x + α) + b cos(2x + β) and in-teracting through two-body potentials of the inverse sine square type.

This systemconstitutes a generalization of the Sutherland model in the presence of externalpotentials. The positive-definite matrix model, obtained by analytic continuation,is also integrable, which leads to the integrability of a system of particles in hyper-bolic potentials interacting through two-body potentials of the inverse hypebolicsine square type.∗This research was supported in part by the United States Department of Energy undercontract DE-AC02-76ER02271.

In one space dimension an integrable class of systems is known, involving particlescoupled through two-body potentials of a particular form.The generic type of thesepotentials is of the inverse square form. Calogero [1] first solved the three-body problem inthe quantum case for inverse square interactions and quadratic external potential.

Lateron, the full N-body problem was solved and shown to be integrable in both the classicaland the quantum case [2] and further to be related to Lie algebras [3]. Sutherland [4] solvedthe problem with inverse sine square interactions (and no external potentials), which canbe thought as the inverse square potential rendered periodic on the circle.

Eventually,it was realized that the system of particles with two-body potentials of the Weierstrassfunction type is integrable [5,6]. Again, these potentials can be thought as the inversesquare potential rendered periodic on a complex torus.

For a review of these systems anda comprehensive list of references see [6].An interesting feature of the above systems is that, at least some of them, admit amatrix formulation [6,7]. Specifically, the inverse square potential arises out of a hermitianmatrix model, and the inverse sine square potential arises from a unitary matrix model.The matrix formulation is in many respects a better framework to study these systems.Such hermitian matrix models have been studied in physics in the context of large-Nexpansions [8-10] and, recently, non-perturbative two-dimensional gravity [11].

Althoughthe one dimensional (c = 1) unitary model has not been studied in this context, discrete(c < 1) models have been considered [12]. Further, the inverse square potential was shownto be of relevance to fractional statistics [13] and anyon physics [14].

It becomes, therefore,of interest to look for integrable systems with more general potentials.In a previous paper [15] we achieved such a generalization for the Calogero system;specifically, it was shown that the system of particles with inverse square potential in-teractions remains integrable at the presence of external potentials which are a gen-eral quartic polynomial in the coordinate. In this paper, we obtain a generalization ofthe Sutherland model; that is, we show that the system of particles with inverse sinesquare interactions remains integrable at the presence of external potentials of the forma cos(x + α) + b cos(2x + β).

An appropriate scaling limit of this system, then, is shownto reproduce the previous quartic system. In addition, the system with all trigonometricfunctions in the potentials replaced by their hyperbolic counterparts (i.e., the inverse sinhsquare system with hyperbolic external potentials) is also integrable.2

The fact that integrability seems to work only for the above type of external potentialsis somewhat puzzling. It is known, for instance, through a collective field description ofthe hermitian matrix model, that the large-N inverse square system is integrable for anyexternal potential [16].

(Although there is no corresponding result for the unitary modelwe have little doubt that the same is true there.) This could be, though, just anotherspecial property of the large-N system.

At any rate, the method of this paper seems towork only for the above-mentioned potentials.The system to be considered is a unitary matrix model in one time dimension withlagrangianL = −12tr(U−1 ˙U)2 −trW(U)(1)where U is a unitary N × N matrix depending on time t, and overdot denotes timederivative.The potential W(U) must be hermitian for consistency.The equations ofmotion from (1) readddt(U−1 ˙U) −W ′U = 0(2)Due to the invariance of (1) under time independent unitary transformations of U, thereis a conserved traceless matrix, namely[ ˙U, U−1] ≡iP(3)as can explicitly be checked using (2). P is the generator of unitary transformations ofU and constitutes a kind of conserved “angular momentum” in the (curved) space U(N).The eigenvalues of U, on the other hand, written as eixn, n = 1, .

. .N, can be thought ascoordinates of N particles on the circle.

Take, now, the particular case where all but oneof the eigenvalues of P are equal, that isP = α(N|u >< u| −1)(4)where |u > is a constant N-dimensional unit vector.This P is naturally obtained bygauging the U(N) invariance and coupling the system to fermions [17]. Then in can beshown with a method analogous to [17] that the eigenvalues of U satisfy the equations ofmotion¨xn = −V ′(xn) +Xm̸=nα2 cos xn−xm24 sin3 xn−xm2(5)3

where the potential of the particles is definedV (x) = W(eix)(6)and prime denotes x-derivative. These are exactly the equations of motion of particlesof unit mass moving in the external potential V and interacting through the two-bodypotentialV2(x) =α22 sin x22(7)In the abscence of external potentials, the above system is the Sutherland system which isknown to be integrable.

It is the purpose of this paper to show that the system remainsintegrable in the presence of external potentials V (x) of the formV (x) = a1 cos x + a2 sin x + b1 cos 2x + b2 sin 2x(8)The above potential can be thought as a generalization on the circle of the quartic potentialon the line. Indeed, the quartic potential is in fact a special case of the above potential andcan be obtained from it in the limit of infinite radius of the circle.

To see this, introduceexplicitly the radius R by rescalingx →xR ,t →tR2(9)In terms of the new variables the potential becomesV (x) →1R2V ( xR)(10)Then, by choosing the coefficients of the potential to scale asa1 = −83R4a −8R6c ,a2 = 2R5bb1 = 2R6c + 16R4a ,b2 = −R5b(11)we see that upon taking the limit R →∞we recover the potentialV (x) = ax2 + bx3 + cx4(12)It should be clear that this is essentially the only scaling that leads to a finite potentialat the large R limit. Therefore, the integrability of the present model also contains theintegrability of the quartic system as a limiting special case.4

We shall then consider the unitary matrix problem with potentialW(U) = A2 U + A∗2 U−1 + B2 U2 + B∗2 U−2 ,A ≡a1 −ia2 , B ≡b1 −ib2(13)Through a rotation of the circle, that is, through a redefinition of U of the form U →eiφU,we can always redefine the phase of B. So, for simplicity we will choose B real.

We shalldefine the “left-hamiltonian matrix” H asH = −12(U−1 ˙U)2 + W(U)(14)whose trace gives the energy of the system. Similarly, we can define the “right-hamiltonianmatrix”˜H = −12( ˙UU−1)2 + W(U) = UHU−1(15)Although the trace of the above matrices is a conserved quantity, neither of them is con-served as a matrix, nor are their eigenvalues.

Consider, now, the matricesL = H + κ2[U−1 ˙U, U + U−1] ,M = −12U−1 ˙U + κ2(U −U−1)(16)Then, upon using the equations of motion it is possible to show that˙L + [L, M] = 0(17)provided that we choose κ asκ2 = B(18)Therefore, the matrices L and M constitute a Lax pair [18] and the eigenvalues of L areconserved. Equivalently, the tracesIn = trLn(19)are constants of the motion.

Restoring, now, an arbitrary phase to B, through a phaseredefinition of U, we get that the Lax matrices in the most general case are2L = −12(U−1 ˙U)2 + AU + BU2 + κ[U−1 ˙U, U] + h.c.(20)M = −12U−1 ˙U + κU −h.c. (21)where h.c. denotes hermitian conjugate.

Notice that L is hermitian while M is antihermi-tian. Relations (17) and (18) remain unchanged.5

Alternatively, we could write the “right-Lax pair” matrices ˜L,˜M, by substituting˙UU−1 for U−1 ˙U in (20) and (21) and flipping the sign of the first term in M. The twochoices are really equivalent, connected through a unitary transformation generated by Uitself.The above are true without any assumptions for the constant commutator P. Assum-ing, now, that P is of the form (4), we can express the conserved quantities In in terms ofparticle coordinates xn and momenta pn = ˙xn. To see this, perform a (time-dependent)unitary transformation on U which brings it to a diagonal form while it doesn’t change In.Then, using (4), we see that the matrices of the system take the formUij = δijeixj ,Pij = α(δij −1)(U−1 ˙U)ij = i ˙xiδij + iα(1 −δij)ei(xi−xj) −1(22)The In, therefore, can be expressed using (22) in terms of particle quantities and constituteN independent integrals of motion for the system of particles in the potential (8) interactingthrough inverse sine square two-body potentials of strength α2.

I1 is the total energy ofthe system. I2 has the formI2 =Xi12p2i +Xj̸=iα22s2ij+ V (xi)2+Xi̸=j̸=k̸=iα22sij sjk2+Xi̸=jα(pi + pj)2sij2−|B|α22Xi̸=jcos(xi + xj + β)(23)where we definedsij = 2 sin xi −xj2,B = |B|eiβ(24)and used the identityX{i,j,k,l distinct}1sij sjk skl sli= 0(25)We shall not give here the explicit form of the rest of In in terms of xi, pi, which is quitecomplicated.

Their independence is obvious from the fact that, just as in the hermitiancase, In is a polynomial in pi of order 2n with highest order term of the form Pi p2ni , andsuch terms cannot be obtained from lower order terms for n ≤N.6

The above Lax pair matrices go over to the Lax matrices of the hermitian problem inthe limit of infinite radius, under the simultaneous scaling (9) and (11) for the coordinatesand the potential, and the rescalingL →1R2L ,M →1R2M(26)In the case c = 0, though, there is no scaling limit, expressing the fact that the purelycubic potential does not admit a Lax pair formulation although it is still integrable.It should also be noted that, just as in the hermitian case, there are two distinct Laxpairs, corresponding to the two possible choices of sign for κ in (18). Again, this has noimpact on the conserved quantities In since the two choices give essentially the same setof conserved quantities, modulo traces of P n, which are trivial nondynamical constantsin the space of eigenvalues.

Note also that, in the hermitian case, when the coefficient ofthe quartic term became negative the Lax matrix L became non-hermitian, indicating therunaway nature of the system. The integrals of motion, nevertheless, remained real.

Inthe present case no such thing happens, since there can be no runaway behavior on thecircle, and the Lax matrix (20) remains hermitian for all values of the coefficients of thepotential.The special case B = 0 is interesting: it corresponds to the quadratic hermitian model(i.e., the Calogero system) which is obtained as the large R scaling limit in this case. Inparticular, L = H just as in the hermitian case.

Contrary to the Calogero system, however,where H is a constant matrix, in this case there is still some nontrivial unitary rotationwith time, due to the nonzero curvature of the space. Therefore, the gauge matrix of theLax pair M = −12U−1 ˙U, generating this rotation, does not vanish.

At the scaling limit,of course, M goes to zero.It remains to show the fact that the above quantities are in involution, that is, thattheir Poisson brackets vanish. Again, as in the hermitian case, trying to use the explicitexpressions of In in terms of particle phase space variables and using{xi, pj} = δij(27)is completely hopeless.

Instead, we shall work with the canonical structure of the originalmatrix problem and make use of the fact that the projection from the full matrix phase7

space of U to the phase space of its eigenvalues is a hamiltonian reduction [6,7]. To seethis, define from (1) a canonical momentum PUPU = δLδ ˙U= −U−1 ˙UU−1(28)Then the symplectic two-form ω isω = tr (dPU dU)(29)giving rise to canonical Poisson brackets for U and PU.

For convenience, we shall workwith the antihermitian matrixΠ ≡−PUU = U−1 ˙U(30)and the symplectic one-form A, in terms of which we haveA = −tr (Π U−1dU) ,ω = dA(31)Decompose, now, U and Π in terms of diagonal and angular degrees of freedom, namelyU = V −1ΛV ,Π = V −1N + Λ−1[A, Λ]V(32)with Λ and N diagonal unitary and antihermitian matrices, respectively:Λij = exiδij ,Nij = ipi δij(33)and A an off-diagonal antihermitian matrix which will be identified with the off-diagonalpart of ˙V V −1 by the equations of motion. Then, in this parametrization (31) becomesA = tr−NΛ−1dΛ + (ΛAΛ−1 + Λ−1AΛ −2A) dV V −1=Xipi dxi + triPV −1dV(34)So A decomposes into two non-mixing parts, the first one being the particle phase spaceone-form leading to (27) and the second part identifying P as the variable canonicallyconjugate to the angular degrees of freedom.

Therefore, the Poisson brackets of quantitiesexpressible in terms of xi and pi can equally well be evaluated using the full matrix phasespace Poisson structure.To show the involution of In it is much more convenient to use the geometric formu-lation of symplectic manifolds, rather than work explicitly with Poisson brackets. In this8

approach, we map to each function f on the phase space a vector field vf on the phasespace through the relation< vf, ω >= df(35)where < , > denotes internal product (contraction). The matrix differential operators δΠand δUU form a basis for the vector fields, satisfying< (δΠ)ij, dΠkl >=< (δUU)ij, (U−1dU)kl >= δilδjk ,else < .

, . >= 0(36)In this basis, we can express the vector field vf asvf = tr(v1δΠ + v2δUU)(37)Then the Poisson bracket of any two functions f, g can be calculated as{f, g} = vf(g) = trv1 δgδΠ + v2 δgδU U(38)Therefore, defining vn = vIn, we have{In, Im} = vn(Im) = m trLm−1vn(L)(39)On the other hand, vn satisfies< vn, ω >= dIn = tr(Ln−1dL)(40)By expressing L in (20) in terms of Π and U and expanding dL asdL =XiRi dΠ Si +Xi˜Ri U−1dU ˜Si(41)and using the expression for ω derived from (31), that isω = tr−dΠU−1dU + Π(U−1dU)2(42)we get from (40)vn = −nXitr˜SiLn−1 ˜RiδΠ + [SiLn−1Ri, Π]δΠ −SiLn−1RiδUU(43)Finally, substituting (43) in (39) we obtain{In, Im} = −nmXi,jtrLm−1Ri ˜SjLn−1 ˜RjSi + Lm−1RiSjLn−1RjΠSi−(m ⇀↽n) (44)9

The rest is a tedious algebraic exercise, calculating the explicit expressions for Ri, Si, ˜Rjand ˜Sj and evaluating (44). We shall omit all detail and simply give the result{Im, In} = mn(κ2 −B) trLm−1ULn−1[Π, U]+ h.c. −(m ⇀↽n)(45)Choosing then κ as in (18) the Poisson brackets (45) vanish and the In are in involution.We have therefore proved the full integrability of the system.The expression for L can be brought to a particularly suggestive form in a somewhatspecial case.

Define the matricesA = Π + Y (U) ,A† = −Π + Y (U) ,with Y (U) = κU + κ∗U−1 + λ ,λ = real(46)Then L = 12A†A is exactly the previous Lax matrix (20), with B and κ properly connected.The coefficients of the potential, however, are constrained to obey the relationBA2 = real(47)The situation is again analogous to the hermitian case, where such an expression for Lalso existed for a similarly constrained potential. Note that, in this caseIn(κ, λ) = tr(12A†A)n = tr(12AA†) = In(−κ, −λ)(48)which shows that the two possible choices of sign for κ in (18) give exactly the sameconserved quantities.

The proof of involution of the In is also simplified in this case.The above results can readily be extended to the case of positive definite matrices X.Such matrices can be expressed asX = eM ,M hermitian. (49)They differ therefore from unitary matrices by a mere analytic continuation, and canbe thought as unitary models with imaginary radius R. We can explicitly perform thiscontinuation in the expressions for the Lax pair matrices, by putting R = i in (9), (10),(11)and (26), which amounts to the redefinitionsU →X ,L →−La1 ± ia2 →−(a1 ± a2) ,b1 ± ib2 →−(b1 ± b2)(50)10

and now κ and κ∗are distinct, satisfyingκ2 = −b1 −b2 ,κ∗2 = −b1 + b2(51)The new potential of the system becomesV (x) = a1 cosh x + a2 sinh x + b1 cosh 2x + b2 sinh 2x(52)Notice that, for (52) to be bounded from below, we must haveb1 ± b2 > 0(53)and for such values κ and κ∗both become imaginary. The Lax matrix becomes, explicitlyL = 12Π2 + W(X) + iκ1[Π, X] + iκ2[Π, X−1](54)where we put κ = 2iκ1, κ∗= 2iκ2.

The momentum Π = X−1 ˙X now is not antihermitianany more, but rather satisfiesΠ† = XΠX−1(55)Therefore, although L is not hermitian, it satisfiesL† = XLX−1(56)which ensures that the conserved quantities In are real. Alternatively, we can directlymake the analytic continuationxi →ixi ,pi →ipi(57)into the expressions for In and obtain the new expressions.

These will be conserved quan-tities for particles on the line, in an external potential of the type (52) interacting throughtwo-body potentials of the typeV2(x) =α22 sinh x22(58)Systems of particles with the above two-body potential and no external potentials wereknown to be integrable. As we proved, they remain integrable in the presence of potentialsof the form (52).11

In conclusion, we see that the above unitary system (and its positive definite ana-lytic continuation) is integrable and closely parallels the hermitian case. It is interestingthat integrability again stops at potentials with only their two lowest nontrivial Fouriercoefficients nonvanishing and does not extend to arbitrary periodic potentials.

Obviousattempts to generalize the result to higher potentials, e.g., through a construction similarto (46) with arbitrary hermitian Y (U), fail. This may be just a shortcoming of the methodused.

If, on the other hand, it turns out that integrability indeed stops at this level, itwould be interesting to understand the deeper reason for that and the special significanceof these potentials.We should stress that, just as in the hermitian case, integrability does not necessarilyimply complete solvability.The significance of the results is that the problem in thepresence of the interparticle interaction remains as solvable as in the case of decoupledparticles. The solution of these systems could conceivably be achieved in terms of thesolutions of the one-body problem (which in general involves elliptic functions).

The proofof integrability in the quantum domain is also of interest, especially in view of the conjectureput forth in [13] that the inverse square two-body potential on the line and the inverse sinesquare potential on the circle simply endow the particles with fractional statistics. These,as well as possible generalizations for other two-body potentials (e.g., of the Weierstrassform) remain interesting topics for further work.REFERENCES[1] F. Calogero, J.

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