Kosterlitz-Thouless phase transitions on discretized random surfaces

Abstract

The large N limit of a one-dimensional infinite chain of random matrices is investigated.It is found that in addition to the expected Kosterlitz-Thouless phase transition this model exhibits an infinite series of phase transitions at special values of the lattice spacing ǫ pq = sin(πp/2q). An unusual property of these transitions is that they are totally invisible in the double scaling limit. A method which allows us to explore the transition regions analytically and to determine certain critical exponents is developed. It is argued that phase transitions of this kind can be induced by the interaction of two-dimensional vortices with curvature defects of a fluctuating random lattice.

Eigenvalue dynamics and the matrix chain

“…In addition to this standard behaviour it was argued in [16] that there exist a countable infinity of additional points in this phase where the eigenvalue distribution ρ 0 exhibits subleading non-analyticities. In fact we can reproduce these results if we assume, incorrectly, that the origin z = 0 is a relevant stationary point in the evaluation of ρ 0 .…”

“…It is well known that at degenerate stationary points there is a change in the analytic structure of a saddle-point approximated integral so it is natural to assume that when (5.19) is satisfied, shown to occur [13,15,16]. The only overt sign of the transition here is that the integral kernel K again changes analytic structure to…”

“…This form of the solution of the problem is well-suited to detailed local analysis as demonstrated in [16,17]. However, for a global analysis which can accommodate the existence of multiple stationary points, we will take the (well-studied) integral form (3.4) over the functional equation (3.8).…”

“…In this form it is clear that the solution will depend only on the constant b which sets a scale [16], otherwise it is a universal quantity. Unfortunately the asymptotic (ǫ → 0) solutions of this equation are not known and so remains the eigenvalue distribution ρ KT of the matrix chain at the Kosterlitz-Thouless transition.…”

“…In addition to a = 0 and a = 1, it was argued in [16] and [17] that at a discrete infinity of lattice spacings non-analytic behaviour arises in the distribution of eigenvalues of the matrix variables φ x . Considering iterative approximation of a functional solution to the equations of motion, it was argued these critical points arise when a = sin (pπ/2q) where p ≤ q are each positive integers.…”