Fluctuations around periodic BPS-density waves in the Calogero model

Abstract

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 Schrödinger 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.

Finiteness of entanglement entropy in collective field theory

“…While the singlet sector of single matrix quantum mechanics becomes a theory of free fermions, non-singlet sectors lead to models of interacting fermions, notably the Spin-Calogero models, particularly in the study of the long string sector [44,45]. Collective field theory for Calogero models have been developed in [46][47][48][49]. In these cases, the entanglement…”

“…The problem of target space entanglement in the many-body quantum mechanics of these particles can be formulated exactly as above. In fact there is a well known collective field theory formulation of the Calogero model using its bosonized current-algebra representation, so that this can be reformulated in terms of collective fields [46][47][48][49]…”

Supersymmetric many-particle quantum systems with inverse-square interactions

“…The Calogero-Marchioro model [100] is one such example which describes a (D > 1)-dimensional quantum system of particles interacting with each other through two-body and three-body inverse-square interaction terms. The importance of the (D = 2)-dimensional Calogero-Marchioro model lies in its relevance in the study of a host of different subjects, like the normal matrix model [103][104][105]42], two-dimensional Bose system [104], quantum Hall effect [106], quantum dot [107] and extended superconformal symmetry [64]. It is known that the two-dimensional Calogero-Marchioro model at some specific value of the coupling constant describes the dynamics of a Gaussian ensemble of normal matrices in the large N limit [103,51,105].…”

“…The study of Calogero-Moser-Sutherland systems have produced many interesting results which are relevant in the context of a diverse branche of physics, including exclusion statistics [18,19], quantum Hall effect [20], Tomonaga-Luttinger liquid [21], quantum chaos [22], electric transport in mesoscopic systems [23], novel correlations [24], spin-chains [25][26][27][28][29][30][31][32][33], etc. These developments are also important in the context of mathematical physics, for example, algebraic and integrable structure [34][35][36], mapping of rational model to Calogero model with Coulomb-like potential [37], self-adjoint extensions [38,39], equivalence to a system of free oscillators [40,41], collective field formulation of many-particle systems [42], etc.…”

On the bound states and correlation functions of a class of Calogero-type quantum many-body problems with balanced loss and gain

“…The CMS models and various generalizations of them have been extensively studied in the literature [19][20][21]. These models have many implications in distinct areas of physics and mathematics such as in exclusion statistics [22], quantum chaos [23], spin chains [24], algebraic and integrable structure [25], selfadjoint extensions [26], collective field formulation of many-particle systems [27], quantum Hall effect [28], Tomonaga-Luttinger liquid [29] etc. Therefore, an extension of CMS model in the context of system having balanced loss and gain and the study of their integrability and/ or exact solvability is an obvious curiosity.…”