5 amount of raw data. Generally there are two approaches to avoid the overboadening: the linear prediction (LP) and the fitting and extrapolate (FE) procedure $^{39,40}$. We leave the LP approach for further investigation and adopt the FE approach which we believe is more under control. We fit parts of the raw data with A cos( ω ( t − t$_{0}$ )) /t $^{α}$+ Be − $^{βt}$, where A, B, t$_{0}$, α, β are fitting parameters. Then the function is extrapolate to very large time. By this method we have taken use of most of the raw data. A comparison of DOS from directly FT and after "fit and extrapolate" is shown in Fig.6. Even from directly FT data one could recognize zero component Kondo peak and the development of Hubbard band as U increases. However, the Fridel sum rule which states the pinning of the A (0) = 2 πD is violated. Actually, the conservation of zero frequency peak relies on the conservation of the areas enclosed by the G $^{r}$( t ) curve. This means the res- olution of the Kondo peak relies on very accurate long time behavior of the Green’s function. By the FE proce- dure before FT, this conservation could be fulfilled, Fig.6. However, the fitting ansatz inevitably introduces prior knowledge about the spectral and gives much sharper peaks in present case. Note that causality is violated (negative dos) in U = 1 case. We find it is more stable to extract the spectral function from the G ( τ ) by maximum entropy method (MEM). Since the data is free from statistical error, the procedure is more stable and less involved than the con- tinuation of Monte Carlo results. As stated in early work of Naef et al $^{41}$, although the continuation approach may be still insufficient for resolving specific line shapes, it is appropriate for identifying gaps and peak positions, which is more relevant in DMFT studies. So in present paper MEM is still adopted to get DMFT converged spec- tral functions. After getting the temporal Green’s function, one has essentially solved the impurity problem. One should plug it into the DMFT self-consistent loop to produce the SIAM chain Hamiltonian for the next iteration step. For demonstration purpose we consider the case of the Bethe lattice where the self-consistent procedure is greatly sim- plified. By multiplying D 2 4 with G ( τ ) one gets the hy- bridization functions Δ( τ ) , where D is the half-bandwith of the semi-ellipse DOS. For the general lattice, the self- consistent loop should be 1) Using Fourier transform or MEM to get the Green’s function in the frequency do- main, and do the self-consistent following the conven- tional routine. This kind of self-consistent procedure was adopted in PQMC solver $^{38}$. or 2) Just do all of the com- putation in the time domain. In that case, one needs to do inverse of the Green’s function matrix in time domain and convolution of non-interacting DOS with each matrix element. It is a formidable task but still could be done, as one encountered in the non-equilibrium DMFT. 17 The zero temperature spectral information can be ex- tracted from the imaginary time Green’s function G ( τ ) , by the maximum entropy continuation similar to the pro- cedures in the PQMC approach to the SIAM $^{38}$. Or one could get the spectral information from the Fourier trans- form of real time Green’s function G $^{r}$( t ) . $^{39,40}$. In the Fourier transformation (FT) approach, to avoid the negative DOS, generally one multiplies the time do- main data with a window function ( W ( t ) = $_{e}$− 4( t T $_{max}$) 2 is used here) which is a decayed function from zero time to the large time cutoff, but this method inevitably broaden the peak in the spectral function and also drops a large Figure 5: (Color online) The zero temperature retarded Green’s function G $^{r}$( t ) for a 20 -site SIAM chain. Where ε$_{n}$ = 0 , γ$_{n}$ = 0 . 5 , the interacting strength U = 0 , 1 , 2 , 3 , 4 , 5 from the bottom to top. The diamond shaped dots indicate the analytical result for U = 0 . Finite size effect of the chain does not show up for this time scale, so the Fourier transform of the real time Green’s function gives spectral functions. Figure 6: (Color online) DOS of a SIAM for U = 1 , 2 , 3 , 4 from directly FT (full line) of real time data (Fig.5) and fit the long time behavior and then perform FT (dash line). IV. EXTRACT SPECTRAL FUNCTION 0 4 8 12 16 20 –1.0 –0.5 0.0 0.5 Im Gr(t) t –4 –3 –2 –1 0 1 2 3 4 0.0 2.0 4.0 6.0 ω DOS U = 1 U = 2 U = 3 U = 4 V. DMFT SELF-CONSISTENT