Note on the thermodynamic Bethe Ansatz approach to the quantum phase diagram of the strong coupling ladder compounds

We investigate the low-temperature phase diagram of the exactly solved su(4) two-leg spin ladder as a function of the rung coupling $J_{\perp}$ and magnetic field $H$ by means of the thermodynamic Bethe Ansatz (TBA). In the absence of a magnetic field the model exhibits three quantum phases, while in the presence of a strong magnetic field there is no singlet ground state for ferromagnetic rung coupling. For antiferromagnetic rung coupling, there is a gapped phase in the regime H < H_{c1}, a fully polarized gapped phase for H > H_{c2} and a Luttinger liquid magnetic phase in the regime H_{c1} < H < H_{c2}. The critical behaviour derived using the TBA is consistent with the existing experimental, numerical and perturbative results for the strong coupling ladder compounds. This includes the spin excitation gap and the critical fields H_{c1} and H_{c2}, which are in excellent agreement with the experimental values for the known strong coupling ladder compounds (5IAP)_2CuBr_4 2H_2 O, Cu_2(C_5 H_{12} N_2)_2 Cl_4 and (C_5 H_{12} N)_2 CuBr_4. In addition we predict the spin gap $\Delta \approx J_{\perp}-{1/2}J_{\parallel}$ for the weak coupling compounds with $J_{\perp} \sim J_{\parallel}$, such as (VO)_2 P_2 O_7, and also show that the gap opens for arbitrary $J_{\perp}/ J_{\parallel}$.