7 decomposition and by increasing the number of the fit- ting parameters, i.e. the length of the chain. However, the first error is more severe and prohibit long time evo- lution. It is known that the algorithm will breakdown at runaway time, which approximately shows the loga- rithmic dependence on χ . At the runaway time the en- tanglement of the states increases to a value which could not be sufficiently represented by the given finite trunca- tion dimension. The computational effort of the tVMPS solver is mainly determined by the truncation dimension χ and is insensitive to the Hamiltonian parameters of the problem. We then summarize the limitations due to present im- plementation of the algorithm, and point out possible solutions. First, the algorithm does not have a good res- olution on the Kondo resonant peak, i.e. it does not pin down at the noninteracting value as U increases. This could be cured by using the logarithmic discretization in the bath, and 44,45 has attained some success in using this technique to resolve Kondo effect in the transport properties. Second, to reduce computational effort we limit chain length only slightly larger than that acces- sible to exact diagonalization, but by using conserved quantity such as total particle number, magnetization or even non-abelian SU (2) symmetry of SIAM, much larger system size is accessible within the same computational time. Third, accurate determination of low temperature Green’s function needs long time evolution without hit- ting the runaway time. Whether the tVMPS could pro- duce result that competes to the Monte Carlo method should be answered by future investigation. To conclude, we have shown that the tVMPS approach could work at zero temperature as well as at finite tem- peratures. By the post-selfconsistent real time evolution, one could also get the real time dynamics, from which the real frequency can be extracted. These merits make it a promising solver for DMFT to investigate the low tem- perature properties of the strongly correlated systems. off of the matrix size in the MPS representation, there are errors in the ground state and the evolution proce- dure. Also there are Trotter decomposition errors from the discretization of the evolution operators. Finally, er- rors also come from the mapping from the hybridization function to the chain Hamiltonian. The latter two errors can be easily reduced by adopting higher order Trotter Figure 9: (Color online) Physical quantities at T = 0 from the tVMPS solver compared with the Lanczos solver. Up- per panel: The average kinetic energy per site, Lower panel: The average double occupancy per site versus the interacting strength U/D . Figure 10: (Color online) The DOS extracted from the con- verged G ( τ ) by MEM. Figure 8: (Color online) Main figure: The DMFT con- verged zero temperature Green’s functions by the tVMPS solver with 18 sites. Where the interaction strength U = 0 , 0 . 8 D, 1 . 6 D, 2 . 4 D, 3 . 2 D, 4 . 0 D from bottom to the top. The long time asymptotic behavior of G ( τ ) differen tiates the metallic and the Mott insulator states. In set: The density of states on the Fermi surface A (0) , which are compared with the results from a Lanczos solver with 8 -site. Slightly dropping of A (0) in the metallic side is due to finite β , whic h prohibits to resolve very small frequency. –4 –2 0 2 4 0.0 0.2 0.4 0.6 ω /D ρ ( ω )D U/D = 1 U/D = 2 U/D = 4 0 10 20 30 –0.5 –0.4 –0.3 –0.2 –0.1 0.0 τ G( τ ) 0 1 2 3 4 0.0 0.2 0.4 0.6 U/D A(0) D Lanczos tVMPS –0.4 –0.3 –0.2 –0.1 Ekin/D Lanczos tVMPS 0 1 2 3 4 0.0 0.1 0.2 0.3 Double Occupancy U/D