We express the free energy of a system in its critical region as a line integral along a renormalization group trajectory. The kernal associated with the trajectory integral is just the spin independent part of the Hamiltonian generated by each renormalization group iteration. If the trajectory integral is parametrized in terms of nonlinear scaling fields, a closed expression for the crossover scaling function is obtained. Crossover scaling functions associated with free energy are calculated explicitly for recursion relation models treated by Riedel and Wegner, and the relevance to ε‐expansions near four dimensions is discussed. A breakdown of hyperscaling above the borderline dimensions (d=4 for critical points, and d=3 for tricritical points) arises naturally in the formalism.
Functional renormalization group (FRG) has become a diverse and powerful tool to derive effective low-energy scattering vertices of interacting many-body systems. Starting from a free expansion point of the action, the flow of the RG parameter $\ensuremath{\Lambda}$ allows us to trace the evolution of the effective one- and two-particle vertices towards low energies by taking into account the vertex corrections between all parquet channels in an unbiased fashion. In this work, we generalize the expansion point at which the diagrammatic resummation procedure is initiated from a free UV limit to a cluster product state. We formulate a cluster FRG scheme where the noninteracting building blocks (i.e., decoupled spin clusters) are treated exactly, and the intercluster couplings are addressed via RG. As a benchmark study, we apply our cluster FRG scheme to the spin-
21 bilayer Heisenberg model (BHM) on a square lattice where the neighboring sites in the two layers form the individual two-site clusters. Comparing with existing numerical evidence for the BHM, we obtain reasonable findings for the spin susceptibility, the spin-triplet excitation energy, and quasiparticle weight even in coupling regimes close to antiferromagnetic order. The concept of cluster FRG promises applications to a large class of interacting electron systems.
It is shown that some recently proposed iterative approaches to the ground state of quantum systems can be obtained as zero-temperature limits of free-energy-preserving renormalization transformations, if the perturbative expansion is handled in the appropriate way. Besides giving new insight into the ground-state approaches and their mutual relationships, these results allow us to perform consistent approximate calculations of the free energy at all temperatures and to get global descriptions of the critical properties. Free-energy and specific-heat calculations are reported for
XY and Heisenberg spin-\textonehalf{} chains and for the triangular
XY model. It is also demonstrated how this extension into finite temperatures allows us to compute in a consistent way the
z exponent, and to obtain substantial improvement in the numerical values for the ground-state energy density of the transverse Ising model.