We analyze the thermodynamics and the critical behavior of the supersymmetric su($m$) $t$-$J$ model with long-range interactions. Using the transfer matrix formalism, we obtain a closed-form expression for the free energy per site both for a finite n
umber of sites and in the thermodynamic limit. Our approach, which is different from the usual ones based on the asymptotic Bethe ansatz and generalized exclusion statistics, can in fact be applied to a large class of models whose spectrum is described in terms of supersymmetric Young tableaux and their associated Haldane motifs. In the simplest and most interesting su(2) case, we identify the five ground state phases of the model and derive the complete low-temperature asymptotic series of the free energy per site, the magnetization and charge densities, and their susceptibilities. We verify the models characteristic spin-charge separation at low temperatures, and show that it holds to all orders in the asymptotic expansion. Using the low-temperature asymptotic expansions of the free energy, we also analyze the critical behavior of the model in each of its ground state phases. While the standard su(1|2) phase is described by two independent CFTs with central charge $c=1$ in correspondence with the spin and charge sectors, we find that the low-energy behavior of the su(2) and su(1|1) phases is that of a single $c=1$ CFT. We show that the model exhibits an even richer behavior on the boundary between zero-temperature phases, where it can be non-critical but gapless, critical in the spin sector but not in the charge one, or critical with central charge $c=3/2$.
We study the spectrum of the long-range supersymmetric su$(m)$ $t$-$J$ model of Kuramoto and Yokoyama in the presence of an external magnetic field and a charge chemical potential. To this end, we first establish the precise equivalence of a large cl
ass of models of this type to a family of su$(1|m)$ spin chains with long-range exchange interactions and a suitable chemical potential term. We exploit this equivalence to compute in closed form the partition function of the long-range $t$-$J$ model, which we then relate to that of an inhomogeneous vertex model with simple interactions. From the structure of this partition function we are able to deduce an exact formula for the restricted partition function of the long-range $t$-$J$ model in subspaces with well-defined magnon content in terms of its analogue for the equivalent vertex model. This yields a complete analytical description of the spectrum in the latter subspaces, including the precise degeneracy of each level, by means of the supersymmetric version of Haldanes motifs and their related skew Young tableaux. As an application, we determine the structure of the motifs associated with the ground state of the spin $1/2$ model in the thermodynamic limit in terms of the magnetic field strength and the charge chemical potential. This leads to a complete characterization of the distinct ground state phases, determined by their spin content, in terms of the magnetic field strength and the charge chemical potential.
We apply a recently developed dispersive formalism to calculate the Regge trajectories of the $f_2(1270)$ and $f_2(1525)$ mesons. Trajectories are calculated, not fitted to a family of resonances. Assuming that these spin-2 resonances can be treated
in the elastic approximation the only input are the pole position and residue of the resonances. In both cases, the predicted Regge trajectories are almost real and linear, with slopes in agreement with the universal value of order 1 GeV$^{-2}$.
