Thermodynamic Bethe ansatz

The thermodynamic Bethe ansatz (TBA) was developed more than twenty years ago by Yang and Yang [119 ], initially as a technique which allows to compute thermodynamic quantities for a system of bosons interacting dynamically via factorizable scattering. This method was generalized thirteen years ago by Al.B. Zamolodchikov [124 ] to a system of particles which interact dynamically in a relativistic manner through a scattering matrix which belongs to an IQFT. This latter approach, nowadays usually referred to as thermodynamic Bethe ansatz, has triggered numerous further investigations. The reason for this activity is at least twofold, on one hand the TBA serves as an interface between conformally invariant theories and massive integrable deformations of them and on the other hand it serves as a complementary approach to other methods and techniques The TBA-approach allows to extract various types of informations from a massive integrable quantum field theory once its scattering matrix is known. Most easily one obtains the central charge of the Virasoro algebra of the underlying ultraviolet conformal field theory, the conformal dimension and the factor of proportionality of the perturbing operator, the vacuum expectation values of the energy-momentum tensor and other interesting physical and mathematical quantities. Thus, the TBA provides a test laboratory in which certain conjectured scattering matrices and solutions to the above mentioned system of equations may be probed for consistency. In addition, the TBA is useful since it provides quantities which may be employed in other contexts, such as the computation of correlation functions. For instance the constant of proportionality, the dimension of the perturbing field and the vacuum expectation value of the trace of the energy-momentum tensor may be used in a perturbative approach around the operator product expansion of a two-point function within a conformal field theory. The vacuum expectation value of the trace of the energy-momentum tensor serves further as an initial value for the recursive equations between different n-particle form factors, as described in the previous section. In general the statistical interaction between the particles in the multi-particle system is assumed to be of fermionic type. Motivated by considerations in the context of conformal field theories we have generalized this to systems whose statistical interaction is of Haldane type [ 11 ]. Technically the TBA-approach consists essentially of three main steps, first solving the TBA-equations with a given scattering matrix for the pseudo-energies, using thereafter the solution to compute the so-called scaling function and extract finally the quantities of interest from it (some interesting quantities can already be obtained from the knowledge of the pseudo-energies). Due to the fact that the TBA-equations are a system of coupled nonlinear integral equations there exist no closed analytical solutions for them. Nonetheless, numerically this is in principle a well controllable problem (it becomes, however, quite complex for an increasing number of particle types) and the equations may be solved iteratively. The convergence of this procedure to a solution and its very existence may be investigated most naturally by means of the Banach fixed point theorem, which we have carried out for some theories [ 72 ]. Approximated analytical expressions in terms of easy accessible Lie algebraic quantities for the scaling function in the ultraviolet regime of ATFT in which the underlying Lie algebra is simply laced were presented in [72 ,71 ]. For theories which contain also unstable particles the scaling function becomes most interesting as it develops a complicated staircase pattern. As the TBA is a probing of the theory at different energy scales these steps can be associated to the possible formation of certain unstable particles. In [34 ,16 ] we carried out the TBA for various HSG models and identified the unstable particles inside the spectrum whose mass scale characterize the flow between different coset models. Whereas the temperature of the onsets can be predicted by means of the Breit-Wigner formula [10 ], the corresponding values for the central charges require a more sophisticated analysis. In [19 ,16 ] we formulated a new Lie algebraic decoupling rule which predicts the height of these flows. To accommodate new physical situations the TBA has to be adapted sometimes, for instance in [34 ] we provided a new formulation which includes parity breaking and in [ 26 ] we newly included impurities which allow simultaneously for reflection and transmission . Besides these aforementioned extensions, the status of the TBA is quite settled and it is often only a matter of carrying out the task for the concrete model. One of the main challenges which persists is to improve the analytic picture trying to overcome the dependence on the numerics. In [24 ] we proposed a method which extrapolates away from the ultraviolet fixed points, where a fairly good analytical understanding exists, and might allow to handle solutions in terms of q-deformed Weyl characters.