Fluctuations around Periodic BPS-Density Waves in the Calogero Model

TL;DR

This paper analyzes the stability and fluctuation spectrum of periodic BPS density waves in the Calogero model, revealing that fluctuations are independent of wave amplitude and do not alter the energy density compared to the ground state.

Contribution

It provides a complete analysis of quadratic fluctuations around BPS density waves, deriving the fluctuation spectrum and showing its independence from wave amplitude.

Findings

Fluctuation spectrum is independent of wave amplitude.

Zero-point energy contributions are identical to the ground state.

Quadratic fluctuations do not shift the energy density.

Abstract

The collective field formulation of the Calogero model supports periodic density waves. An important set of such density waves is a two-parameter family of BPS solutions of the equations of motion of the collective field theory. One of these parameters is essentially the average particle density, which determines the period, while the other parameter determines the amplitude. These BPS solutions are sometimes referred to as "small amplitude waves" since they undulate around their mean density, but never vanish. We present complete analysis of quadratic fluctuations around these BPS solutions. The corresponding fluctuation hamiltonian (i.e., the stability operator) is diagonalized in terms of bosonic creation and annihilation operators which correspond to the complete orthogonal set of Bloch-Floquet eigenstates of a related periodic Schr\"odinger hamiltonian, which we derive explicitly. Remarkably, the fluctuation spectrum is independent of the parameter which determines the density wave's amplitude. As a consequence, the sum over zero-point energies of the field-theoretic fluctuation hamiltonian, and its ensuing normal-ordering and regularization, are the same as in the case of fluctuations around constant density background, namely, the ground state. Thus, quadratic fluctuations do not shift the energy density tied with the BPS-density waves studied here, compared to its ground state value. Finally, we also make some brief remarks concerning fluctuations around non-BPS density waves