  We have carried out analytical and numerical studies of the spin-boson model
in the sub-ohmic regime with the influence of both the diagonal and
off-diagonal coupling accounted for via the Davydov D1 variational ansatz.
While a second-order phase transition is known to be exhibited by this model in
the presence of diagonal coupling only, we demonstrate the emergence of a
discontinuous first order phase transition upon incorporation of the
off-diagonal coupling. A plot of the ground state energy versus magnetization
highlights the discontinuous nature of the transition between the isotropic
(zero magnetization) state and nematic (finite magnetization) phases. We have
also calculated the entanglement entropy and a discontinuity found at a
critical coupling strength further supports the discontinuous crossover in the
spin-boson model in the presence of off-diagonal coupling. It is further
revealed via a canonical transformation approach that for the special case of
identical exponents for the spectral densities of the diagonal and the
off-diagonal coupling, there exists a continuous crossover from a single
localized phase to doubly degenerate localized phase with differing
magnetizations.
