Quantized Normal Matrices: Some Exact Results and Collective Field Formulation

TL;DR

This paper investigates large N normal matrix quantum models with rotation-invariant potentials, solving for eigenstates and analyzing phase transitions using collective field theory, with explicit results for quadratic and quartic potentials.

Contribution

It introduces a collective field approach to large N normal matrix models, providing exact solutions for quadratic potentials and analyzing quantum phase transitions in these models.

Findings

Exact eigenstates for quadratic potential case.

Ground state eigenvalue distribution and energy for arbitrary potentials.

Analysis of disk-annulus quantum phase transition in quartic potential.

Abstract

We formulate and study a class of U(N)-invariant quantum mechanical models of large normal matrices with arbitrary rotation-invariant matrix potentials. We concentrate on the U(N) singlet sector of these models. In the particular case of quadratic matrix potential, the singlet sector can be mapped by a similarity transformation onto the two-dimensional Calogero-Marchioro-Sutherland model at specific couplings. For this quadratic case we were able to solve the $N-$body Schr\"odinger equation and obtain infinite sets of singlet eigenstates of the matrix model with given total angular momentum. Our main object in this paper is to study the singlet sector in the collective field formalism, in the large-N limit. We obtain in this framework the ground state eigenvalue distribution and ground state energy for an arbitrary potential, and outline briefly the way to compute bona-fide quantum phase transitions in this class of models. As explicit examples, we analyze the models with quadratic and quartic potentials. In the quartic case, we also touch upon the disk-annulus quantum phase transition. In order to make our presentation self-contained, we also discuss, in a manner which is somewhat complementary to standard expositions, the theory of point canonical transformations in quantum mechanics for systems whose configuration space is endowed with non-euclidean metric, which is the basis for constructing the collective field theory.