We develop a theoretical model aimed to explain the results of a recent experiment on the spin-1/2 ladder compound  (BPCB) [i] which measured a dramatic dependence of the thermal conductivity on a magnetic field H (Magnetothermal conductivity). The results show a double-minimum feature in the thermal conductivity isotherms, symmetric with respect to the  field H₀=(H_c1+H_c2)/2 which corresponds to the center of the spinon-band in these systems [ii]. Our theory for the thermal transport accounts for the  H-dependent coupling of spinons to lattice phonon modes. To this end, we employ a mapping of the ladder Hamiltonian in a strong magnetic field onto an XXZ spin-chain in a weak effective field (H_eff=H-H₀ ) and consequently to a Luttinger liquid of Jordan-Wigner Fermions at a chemical potential H_eff . The resulting Bosonized representation provides a low-energy theory for the spinon excitations and their coupling to the phonons. The latter gives rise to hybridization of spinons and phonons and the formation of new eigenmodes. Similarly to an earlier work on spin-chains[iii], we show that the interplay of umklapp and disorder scattering dominates the relaxation of heat current, yielding magnetothermal effects consistent with the experimental observation.

[i] A. V. Sologubenko et al., Phys. rev. B 80, 220411(R) (2009).

[ii] R. Chitra and T. Giamarchi, Phys. rev. B 55, 5816 (1997).

[iii] E. Shimshoni, D. Rasch, P. Jung, A. V. Sologubenko and A. Rosch, Phys. rev B 79, 064406 (2009)