2 particle physics and the high energy Hubbard bands. ii) It is easy to be generalized to multi-orbital or cluster cases. iii) It works with real frequency and gives the real time dynamical properties directly. In this paper, we develop an impurity solver satisfy- ing all these criteria based on the time-dependent vari- ational matrix product states (tVMPS) approach. The variational approach directly attacks the strongly corre- lated problems by an educated guess of the many body wavefuncitions. By introducing more variational pa- rameters one could enlarge the dimensions of the vari- ational space but it makes the variation process harder in general. For example, in the case of Gutzwiller varia- tional wave function, the evaluation of the expectation value is often done approximately by introducing the Gutzwiller approximation $^{19,20}$. Matrix product states (MPS) are states where the coefficients of the wave func- tion are a product of matrices depending on the local lattice site states. They are generated naturally from the NRG and DMRG calculations. Actually the latter ap- proach can be reformulated into the variational approach within the space spanned by the MPS $^{21}$. Time evolution algorithm 22 based on MPS was first proposed from the quantum information perspective, and then been trans- lated into the language more access to the many-body theorists $^{23,24}$, see Verstraete et al. $^{25}$, García-Ripoll 26 and Schollwoeck and White 27 for reviews of the time evolution algorithm of the MPS. There are some previous attempts using the VMPS ap- proach to study the quantum impurity problems. Weich- selbaum et al. 28 studied the spectral property of SIAM in the presence of a magnetic field using the correc- tion vector approach, which is first proposed in the con- text of dynamical density matrix renormalization group (DDMRG) $^{29}$. Saberi et al. 30 performed detailed compar- ison of the VMPS and the NRG approach on the SIAM. Holzner et al. 31 used the VMPS formalism to study the two-lead, multi-level Anderson impurity model. Along the line of developing fast impurity solvers, several of the authors have developed the Gutzwiller based impurity solver $^{32}$, which is suited for combining with the LDA and studying the low temperature properties of multi- orbital models. There were previous attempt of using the DMRG as the impurity solver of the DMFT. 33,34 The organization of this paper is as follows. In Sec. II we give an overview of the computation of the ground state of the SIAM in the VMPS formalism, where an unfold 30 technique greatly reduces the computational ef- fort. In Sec. III we describe the algorithm for time evolu- tion in both real and imaginary times, by which one could calculate the Green’s functions in the time domain. In Sec.IV we describe the fit and extrapolate scheme to ex- tract the spectral function from the real time data. In Sec. V we address the DMFT self-consistent loop where a fitting procedure from the continuous hybridization func- tion to the SIAM chain Hamiltonian is needed. Finally, we conclude the paper by making remarks on the pros and cons of the present solver and make an outlook for It is a zero dimensional problem, and there only exists the fluctuation in the temporal axis, which is captured by the time dependent hybridization function Δ( τ, τ ′ ) . One could de-integral the action by introducing the non- interacting bath degree of freedoms. The electron hops into the bath, travels in it for a while and then hops back to the impurity site thus brings in the temporal nonlocal correlations. The de-integral process is not unique, and in the DMFT context, both the star and the chain ge- ometry of the bath degree of freedom have been studied. In the present study we reformulate the SIAM into a one dimensional chain with the nearest neighbor hopping, see Figure 1. In the framework of DMFT the Hamiltonian parameters of the chain are determined self-consistently, and generally has no translational invariance. We deal with the chain with typical chain length of 10 to 20 sites in the present study. By introducing the Jordan-Wigner transformation (JWT) for the spin up and down fermions separately, $_{s}$− i ( $_{τ}$− i ) = ( ∏ k<i p$_{kσ}$ $^{)}$c$_{iσ}$ where p$_{kσ}$ = ( − 1) n$_{kσ}$ is the future developments. In the appendix we generalize the solver to finite temperature, where ancillary sites which play the role of the heat bath are introduced. The action of the single impurity Anderson model (SIAM) is S$_{SIAM}$ = ∫ β 0 dτ ∫ β 0 dτ ′ c † $_{σ}$( τ )[( ∂$_{τ}$ − µ ) δ$_{τ,τ}$ ′ + Δ( τ, τ ′ )] c$_{σ}$ ( τ ′ ) +$^{∫}$ β 0 $^{dτ U n}$↑$^{n}$↓ (1) Figure 1: (Color online) Unfold the SIAM chain. The SIAM chain is separated into two parts with different spins, they are connected at the leftmost end where the bold bond denotes the Hubbard interaction U . II. VMPS APPROACH TO THE GROUND STATE OF THE SIAM U γ$_{1}$ γ$_{2}$ U ε$_{1}$ ε$_{2}$ ↑ ↓ ε$_{N}$ γ$_{N}$ UNFOLD