CU–TP–537 New Integrable Systems from Unitary Matrix Models ∗

We show that the one dimensional unitary matrix model with potential of the form aU + bU2 + h.c. is integrable. By reduction to the dynamics of the eigenvalues, we establish the integrability of a system of particles in one space dimension in an external potential of the form a cos(x+α) + b cos(2x+ β) and interacting through two-body potentials of the inverse sine square type. This system constitutes a generalization of the Sutherland model in the presence of external potentials. The positive-definite matrix model, obtained by analytic continuation, is also integrable, which leads to the integrability of a system of particles in hyperbolic potentials interacting through two-body potentials of the inverse hypebolic sine square type. ∗ This research was supported in part by the United States Department of Energy under contract DE-AC02-76ER02271. In one space dimension an integrable class of systems is known, involving particles coupled through two-body potentials of a particular form. The generic type of these potentials is of the inverse square form. Calogero [1] first solved the three-body problem in the quantum case for inverse square interactions and quadratic external potential. Later on, the full N -body problem was solved and shown to be integrable in both the classical and the quantum case [2] and further to be related to Lie algebras [3]. Sutherland [4] solved the problem with inverse sine square interactions (and no external potentials), which can be 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 Weierstrass function type is integrable [5,6]. Again, these potentials can be thought as the inverse square potential rendered periodic on a complex torus. For a review of these systems and a comprehensive list of references see [6]. An interesting feature of the above systems is that, at least some of them, admit a matrix formulation [6,7]. Specifically, the inverse square potential arises out of a hermitian matrix 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-N expansions [8-10] and, recently, non-perturbative two-dimensional gravity [11]. Although the 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 shown to 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 interactions remains integrable at the presence of external potentials which are a general quartic polynomial in the coordinate. In this paper, we obtain a generalization of the Sutherland model; that is, we show that the system of particles with inverse sine square interactions remains integrable at the presence of external potentials of the form a cos(x + α) + b cos(2x + β). An appropriate scaling limit of this system, then, is shown to reproduce the previous quartic system. In addition, the system with all trigonometric functions in the potentials replaced by their hyperbolic counterparts (i.e., the inverse sinh square system with hyperbolic external potentials) is also integrable. 2 The fact that integrability seems to work only for the above type of external potentials is somewhat puzzling. It is known, for instance, through a collective field description of the hermitian matrix model, that the large-N inverse square system is integrable for any external potential [16]. (Although there is no corresponding result for the unitary model we have little doubt that the same is true there.) This could be, though, just another special property of the large-N system. At any rate, the method of this paper seems to work only for the above-mentioned potentials. The system to be considered is a unitary matrix model in one time dimension with lagrangian L = −2tr(U −1U̇)2 − trW (U) (1) where U is a unitary N × N matrix depending on time t, and overdot denotes time derivative. The potential W (U) must be hermitian for consistency. The equations of motion from (1) read d dt (U−1U̇)−W ′U = 0 (2) Due to the invariance of (1) under time independent unitary transformations of U , there is a conserved traceless matrix, namely [U̇ , U−1] ≡ iP (3) as can explicitly be checked using (2). P is the generator of unitary transformations of U 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 as coordinates of N particles on the circle. Take, now, the particular case where all but one of the eigenvalues of P are equal, that is P = α(N |u >< u| − 1) (4) where |u > is a constant N -dimensional unit vector. This P is naturally obtained by gauging the U(N) invariance and coupling the system to fermions [17]. Then in can be shown with a method analogous to [17] that the eigenvalues of U satisfy the equations of motion ẍn = −V (xn) + ∑ m 6=n α2 cos xn−xm 2 4 sin3 xn−xm 2 (5)