6 parameters of the SIAM chain by minimizing The key problem then is how to determine the Hamil- tonian parameters of the SIAM chain from the contin- uous hybridization function Δ( τ ) . Similar problem is encountered in other Hamiltonian based solvers such as ED, NRG and DDMRG approaches. In the ED ap- proach the step is determined by the conjugated gradi- ent minimization 3 or continued fraction expansion 42 of the Green’s function in the frequency domain, but due to the finite size of the effective bath that can be dealt with, one has to truncate and leads to errors in this step. In the NRG approach the logarithmic discretization pro- cedure is adopted $^{9,43}$, while in the DDMRG approach, no logarithmic energy separation is assumed, so a direct tri-diagonalization scheme can be used, i.e. first fit the hybridization function with a star shaped Hamiltonian, then using Lanczos method to tri-diagonalize the Hamil- tonian, and the diagonal/off diagonal matrix elements are the onsite/hopping parameters for a chain Hamiltonian. In the present case, we fit Δ( τ ) with the Hamiltonian Here we discuss various sources of error and computa- tional effort of the tVMPS solver. First due to the cut- It is also possible to do the DMFT self-consistent in the real time domain, but in the DMFT self-consistent loop we adopt the imaginary time Green’s function which is more stable. And since all of the evolution operators are real, this choice would reduce the numerical effort. Once convergence is achieved, a real time evolution could be redone to get the retarded Green’s function G $^{r}$( t ) and the spectral function could be extracted from it with ap- propriate FT method. After the DMFT loop converges, one could get various information from the converged Green’s function G ( τ ) and the MPS. e . g. the kinetic energy and the double occupancy rate. See Fig.9 for the comparison of the re- sults from the tVMPS and the Lanczos solver at zero temperature. The long time asymptotic behavior for the interaction strength U larger than the critical U$_{c}$ shows the presence of the charge excitation gap. One could also recognize the metal-insulator transition clearly from the DOS on the Fermi surface A (0) = − $^{lim$_{β}$}$→∞ β $_{2}$G ( β $_{2}$) and the suppression of the double occupancy. The real fre- quency information can be extracted from the analytical continuation approaches, such as the Pade approximation or the MEM. The MEM inverted the integral transfor- mation − G ( τ ) = ∫ ∞ −∞ K ( ω, τ ) A ( ω ) , where symmetrical kernel K ( ω, τ ) = 1 2 e − $^{ωτ}$+ e ( τ − β ) ω e − $^{βω}$+1 is used to persevere the particle hole symmetry A ( ω ) = A ( − ω ) , β is a fictitious temperature β = 128 in our calculation. The results are shown in Fig.10. ˜ Δ( τ ) is the hybridization function for a noninteracting SIAM chain with onsite energy ε$_{i}$ and hopping amplitude γ$_{i}$ , i = 1 , 2 , ...N$_{bath}$ . ˜ Δ( τ ) = ∑ l γ 2 1 | U$_{1}$$_{l}$ | $^{2}$e − E$_{l}$$^{τ}$[ θ ( τ )( n$_{l}$ − 1) + θ ( − τ ) n$_{l}$ ] , where U and E are eigenstate and eigen- value of the noninteracting Hamiltonian H$_{bath}$ , satisfying H$_{bath}$U = UE . n$_{l}$ is the occupation number on level l , and n$_{l}$ = θ ( − E$_{l}$ ) for zero temperature. We use conjugate gradient method to search within the parameter space spanned by ε$_{i}$ and γ$_{i}$ which minimize χ 2 $_{Δ}$. Generally there are 2 N$_{bath}$ parameters, while in the particle-hole symmetric case all of the ε$_{n}$ is kept as zero and there- fore one has N$_{bath}$ fitting parameters. Since we are deal- ing with relatively larger number of bath sites comparing with ED approach and the function to be fitted is more well behaviored (different from the fitting G ( iω$_{n}$ ) in fre- quency domain, one has Δ( τ ) which is more smooth and featureless), the fitting is more reliable. In the present study, the average fitting error can be reduced to 10 − 6 per data point. Figure 7: Flow diagram of DMFT self-consistent calculations. tVMPS solves Green’s function for a SIAM chain Hamilto- nian in temporal domain. Self-consistent process gives new hybridization function Δ( τ ) , from which the Hamiltonian pa- rameters { ε$_{i}$, γ$_{i}$ } are fitted, thus closes the loop. (a). For general lattice, one performs Fourier transform or analytical continuation to translate G ( τ ) to frequency domain, and then determines new hybridization function, information of non- interaction DOS enters in this step. (b). The self-consistent loop simplifies for infinite dimension Bethe lattice, where Mat- subara Green’s function G ( τ ) directly determines the new hy- bridization function. Spectral function A ( ω ) could be gotten by MEM continuation of the converged Matsubara Green’s function G ( τ ) or FT of retarded Green’s function G $^{r}$( t ) , in the latter case, real time evolution using converged Hamilto- nian parameters is firstly performed. χ 2 Δ = N$_{τ}$ ∑ i =1 | Δ( τ$_{i}$ ) − ˜ Δ( τ$_{i}$ ) | 2 (5) Δ( iω$_{n}$ ) Δ( τ ) { ϵ$_{n}$, γ$_{n}$ } G ( iω$_{n}$ ) tVMPS FT or  MEM iFT Fit G ( τ ) Δ( τ ) { ϵ$_{n}$, γ$_{n}$ } tVMPS Bethe Lattice Fit G ( τ ) G $^{r}$( t ) A ( ω ) General   Lattice (a) (b) MEM FT  VI. SUMMARY AND OUTLOOKS