Liquid state theories and critical phenomena

What can be learnt from the nonperturbative renormalization group?

“…Computing it requires to keep track all along the RG flow of both relevant and irrelevant couplings since they all contribute to the non-universal behavior of the model. While this is in general a formidable task in perturbation theory, that anyway is useless if these couplings are large, it just consists in the NPRG approach in integrating the flow equations for the potential with the bare potential of equation (27) as initial condition at scale Λ [42,44,45]. …”

A microscopic approach to phase transitions in quantum systems

“…(6) in three dimensions. As already pointed out in the classical case [6], the HRT approach is able to correctly implement Maxwell construction at first order phase transitions and in fact the free energy density a Q (m) at the end of the integration, i.e. in the Q → 0 limit, becomes rigorously flat in a finite region of the magnetization axis for T < T c .…”

“…As an example, we plot in Fig. 1 the RG flux of g Q obtained by the integration of the full QHRT equation (6). We clearly see the effect of the unstable zero temperature weak coupling fixed point while, for the nearest neighbor Heisenberg model, the other fixed point (g c ), governing the quantum critical regime, has no effect on the RG trajectories.…”

“…By contrast, in classical models, numerical simulations are quite efficient even in the neighborhood of critical points [5] and, from the analytical side, microscopic approaches especially devised for the quantitative description of the phase diagram of classical fluids and magnets are available. For instance, the hierarchical reference theory of fluids (HRT) [6] has proven quite accurate in locating the phase transition lines both in lattice and in continuous models.In this Letter we sketch the derivation of the quantum hierarchical reference theory of fluids (QHRT) which we then apply to the Heisenberg antiferromagnet. We will demonstrate that the known renormalization group equations near four and near two dimensions are naturally recovered within our approach, which therefore unifies two complimentary techniques.…”

“…By contrast, in classical models, numerical simulations are quite efficient even in the neighborhood of critical points [5] and, from the analytical side, microscopic approaches especially devised for the quantitative description of the phase diagram of classical fluids and magnets are available. For instance, the hierarchical reference theory of fluids (HRT) [6] has proven quite accurate in locating the phase transition lines both in lattice and in continuous models.…”