  We present an alternative to the conventional matrix product state
representation, which allows us to avoid the explicit local Hilbert space
truncation many numerical methods employ. Utilising chain mappings
corresponding to linear and logarithmic discretizations of the spin-boson model
onto a semi-infinite chain, we apply the new method to the sub-ohmic SBM. We
are able to reproduce many well-established features of the quantum phase
transition, such as the critical exponent 1/2 predicted by mean-field theory.
Via extrapolation of finite-chain results, we are able to determine the
infinite-chain critical couplings at which the transition occurs and, in
general, study the behaviour of the system well into the localised phase.
