An overview on the theoretic formalism and up to date applications in quantum condensed matter physics of the effective potential and effective Hamiltonian methods is given. The main steps of their unified derivation by the so-called pure quantum self-consistent harmonic approximation (PQSCHA) are reported and explained. What makes this framework attractive is its easy implementation as well as the great simplification in obtaining results for the statistical mechanics of complicated quantum systems. Indeed, for a given quantum system the PQSCHA yields an effective system, i.e. an effective classical Hamiltonian with dependence on h(cross) and beta and classical-like expressions for the averages of observables, that has to be studied by classical methods. Anharmonic single-particle systems are analysed in order to get insight into the physical meaning of the PQSCHA, and its extension to the investigation of realistic many-body systems is pursued afterwards. The power of this approach is demonstrated through a collection of applications in different fields, such as soliton theory, rare gas crystals and magnetism. Eventually, the PQSCHA allows us also to approach quantum dynamical properties.

The path-integral method is used for determination of the quantum corrections to the free energy of nonlinear systems. All quantum effects of the harmonic part of the potential are considered and a variational principle is used to account for the quantum corrections due to the anharmonic part. Correct renormalized frequencies are obtained at any temperature and an effective potential to be inserted in the configurational integral is found. A new general expression for the partition function at any temperature in the low-coupling limit is obtained.PACS numbers: 05.30.Ch, 75.10.Jm Methods of reducing quantum statistical calculations to classical ones 1,2 are finding nowadays increasing applications to fluid 3 and solid 4 models. In one-dimensional magnetic chains, classical models are unsatisfactory 5 (see, e.g., the CsNiF 3 case) and quantum corrections have been taken into account 6,7 in the low-coupling limit, restricted to the noninteracting soliton approximation, which fails in the range of temperature where the peak of the specific heat occurs. 8 Different expansions have also been proposed. 9, 10 We use here a path-integral approach which improves upon previous variational treatments 11 by considering all the quantum effects of the harmonic part of the potential, while the variational principle, in the first cumulant approximation, is used to account for the renormalized quantum corrections, both of potential and of frequencies, due to the anharmonic part. Besides unifying the previous methods, 6,9,10 our approach gives a correct effective potential for all temperatures in the low-coupling limit and can be applied to systems where the energy scales of linear and nonlinear excitations are well separated. Relegating the mathematical details of the theory to an extended paper, we present in this Letter a brief account of the formalism and a concrete application to the sine-Gordon chain, including a numerical evaluation of the specific heat throughout the temperature range, thus clarifying some controversial issues on soliton behavior in planar magnetic chains.The path-integral form of the partition function Z = e ~& F isThe functional integral is evaluated over all the closed paths and the potential V is taken in the discrete form K=(m/2) Z x&jXjg Z U{x,).(2)We assume that Bjj = B i + nj + n contains all the harmonic interaction [£/"(0) = 0]. The anharmonic part U(x) has been taken to be local for simplicity, since the extension to a general U(x) is straightforward. We defineThe functions W, w,-, and w u must be chosen in order to minimize the right-hand side of the first-order cumulant inequalitywhere F 0 and the average are defined with use of V 0 instead of V. It is worthwhile to note that w u (y) are not taken equal to djj V(y) = mB^•+gd iJ U"(y), but are determined by the variational principle. As will be shown in the following, this is a crucial point to avoid unphysical results at lowest temperatures and just this full application of the variational principle leads to the correct frequency renormaliza...

A contraction procedure starting from SO(4)q is used to determine the quantum analog E(3)q of the three-dimensional Euclidean group and the structure of its representations. A detailed analysis of the contraction of the R-matrix is then performed and its explicit expression has been found. The classical limit of R is shown to produce an integrable dynamical system. By means of the R-matrix the pseudogroup of the noncommutative representative functions is considered. It will finally be shown that a further contraction made on E(3)q produces the two-dimensional Galilei quantum group and this, in turn, can be used to give a new realization of E(3)q and E(2,1)q.

The structure of the quantum Heisenberg group is studied in the two different frameworks of the Lie algebra deformations and of the quantum matrix pseudogroups. The R-matrix connecting the two approaches, together with its classical limit r, are explicitly calculated by using the contraction technique and the problems connected with the limiting procedure discussed. Some unusual properties of the quantum enveloping Heisenberg algebra are shown.