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.
Origin : Files produced by the author(s)
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