The Application Of The Density Matrix Renormalization Group To The Spin-boson Model

The spin-boson model,which consists of a two-level system coupling to a phonon bath with continuous spectrum,is an ideal model for investigating quantum dissipation. For the critical behavior of the spin-boson model with a sub-Ohmic bath,the results obtained by analytical approximations,such as the variational approach,are inconsistent to the numerical renormalization group(NRG),we therefore study this problem with a different numerical technique-density matrix renormalization group(DMRG),in order to clarify the discrepancies;meanwhile,we generalize the DMRG algorithm to tackle the model with a bosonic bath with continuous spectrum,e.g.,the spin-boson model.In our DMRG calculations,we apply the optimized phonon technique to a phonon block rather than a phonon mode,separate the warm up procedure of the system block and environment block,and use only one center site in the sweeping processes.With these improvements,we can calculate hundreds of phonon modes,or thousands if we are needed.However,our DMRG algorithm has limitations that we must divide the phonon spectrum linearly.In this case,the critical behavior,i.e.,the quantum delocalizedlocalized phase transition,cannot be obtained.To overcome this difficulty,we mimic NRG to devise a so-called DMRG flow to extract the quantum critical couplings and extrapolate them to thermodynamic limit.The quantum phase transition boundary obtained by our DMRG algorithm is greatly in agreement to the NRG result in both Ohmic and sub-Ohmic cases.