Correlation functions from a unified variational principle: Trial Lie groups

Abstract

Time-dependent expectation values and correlation functions for many-body quantum systems are evaluated by means of a unified variational principle. It optimizes a generating functional depending on sources associated with the observables of interest. It is built by imposing through Lagrange multipliers constraints that account for the initial state (at equilibrium or off equilibrium) and for the backward Heisenberg evolution of the observables. The trial objects are respectively akin to a density operator and to an operator involving the observables of interest and the sources. We work out here the case where trial spaces constitute Lie groups. This choice reduces the original degrees of freedom to those of the underlying Lie algebra, consisting of simple observables; the resulting objects are labeled by the indices of a basis of this algebra. Explicit results are obtained by expanding in powers of the sources. Zeroth and first orders provide thermodynamic quantities and expectation values in the form of mean-field approximations, with dynamical equations having a classical Lie-Poisson structure. At second order, the variational expression for two-time correlation functions separates-as does its exact counterpart-the approximate dynamics of the observables from the approximate correlations in the initial state. Two building blocks are involved: (i) a commutation matrix which stems from the structure constants of the Lie algebra; and (ii) the second-derivative matrix of a free-energy function. The diagonalization of both matrices, required for practical calculations, is worked out, in a way analogous to the standard RPA. The ensuing structure of the variational formulae is the same as for a system of non-interacting bosons (or of harmonic oscillators) plus, at non-zero temperature, classical gaussian variables. This property is explained by mapping the original Lie algebra onto a simpler Lie algebra. The results, valid for any trial Lie group, fulfill consistency properties and encompass several special cases: linear responses, static and time-dependent fluctuations, zero-and high-temperature limits, static and dynamic stability of small deviations.

Quantum and thermal fluctuations in two-component Bose gases

“…where ψj (r) = ψj (r) − Φ j (r) is the noncondensed part of the field operator with Φ j (r) = ψj (r) being the condensate wave-function. Equations ( 2) and (3) are obtained using the Balian-Vénéroni variational principle [63] that optimizes a generating functional related to the observables of interest. The single component BEC version of Eqs.…”

Effects of quantum fluctuations on the dynamics of dipolar Bose polarons

“…The TDHFB theory governs selfconsistently the dynamics and the static of the condensate, impurity, thermal cloud and the anomalous density. It is based on the time-dependent Balian-Vénéroni variational principle which optimizes the evolution of the state according to the relevant observable in a given variational space [57]. Recently, the TDHFB theory has been revisited with renormalized coupling constant and more accurate description of dynamical and thermodynamic properties of BEC has been achieved [47,[51][52][53][54][55].…”

Quantum correlations in dipolar droplets: Time-dependent Hartree-Fock-Bogoliubov theory

“…Previous models neglected these, difficult to compute, terms. The TD-HFB equations are based on the Balian-Vénéroni variational principle [27]. This latter requires that both the state of the system and the observable of interest vary in their own variational space.…”

“…The main difference between the TDHFB approach and the other variational treatments is that, in our variational theory, we do not * a.boudjemaa@univ-chlef.dz minimize only the expectation values of a single operator such as the free energy in the variational HF and HFB approximations but we minimize an action in addition to a Gaussian variational ansatz. In spite of the Gaussian character of the variational ansatz, the approximations obtained in this way go beyond the usual mean-field theory and include correlations between particles [24,27,28]. These latter permit us to extract in a useful way the pair correlation function.…”