An Impurity Solver Using the Time-Dependent Variational Matrix Product State Approach We use the time dependent variational matrix product state (tVMPS) approach to investigate the dynamical properties of the single impurity Anderson model (SIAM). Under the Jordan-Wigner transformation, the SIAM is reformulated into two spin-1/2 XY chains with local magnetic fields along the z -axis. The chains are connected by the longitudinal Ising coupling at the end points. The ground state of the model is searched variationally within the space spanned by the matrix product state (MPS). The temporal Green’s functions are calculated both by the imaginary and real time evolutions, from which the spectral information can be extracted. The possibility of using the tVMPS approach as an impurity solver for the dynamical mean field theory is also addressed. Finite temperature density operator is obtained by the ancilla approach. The results are compared to those from the Lanczos and the Hirsch-Fye quantum Monte-Carlo methods. PACS numbers: 71.27.+a, 71.10.Fd, 71.20.Be Lei Wang $^{1}$, Jia-Ning Zhuang$^{1}$, Xi Dai$^{1}$, and X. C. Xie2 , 1 $^{1}$Beijing National Lab for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China and $^{2}$Department of Physics, Oklahoma State University, Stillwater, Oklahoma 74078, USA During the past twenty years, the dynamical mean field theory (DMFT) 1,2 has been quickly developed into a powerful method to solve the strongly correlated models on the lattice (for a review see Georges et al. $^{3}$). DMFT maps the lattice models to the corresponding quantum impurity models subject to the self-consistency condi- tions. Unlike the normal static mean field approaches, DMFT keeps the full local dynamics induced by the lo- cal interactions. DMFT has been successfully applied to various correlation problems, such as the Mott tran- sition in the Hubbard model 3,4 and the heavy fermion systems $^{5,6}$. In DMFT one encounters the problem of how to effi- ciently solve the quantum impurity problems with self- consistently determined bath. The impurity solver can be regarded as the engine of DMFT, which influences the efficiency and accuracy of DMFT calculations. Since the invention of DMFT, many impurity solvers have been developed. With the development of the mod- ern computers, the essentially exact numerical methods have received much attentions. Among them the most used methods include the exact diagonalization (ED) $^{7}$, Hirsch-Fye Quantum Monte Carlo (HFQMC) 8 and the numerical renormalization group (NRG) $^{9}$. Most recently the continuous-time quantum Monte Carlo (CTQMC) solver 10,11 has been introduced. These methods can be divided into two classes. ED and NRG 12 approaches are based on the Hamiltonains, while the QMC solvers are based on the Lagrangians (action). Generally the La- grangian approach is favored since the DMFT theories it- self is derived in the Lagrangian representation, and thus in the self-consistent process there is no need to map the continuous hybridization function to the discrete Hamil- tonians. However the Hamiltonian approaches have the merit that they work well at low temperatures and for real frequency which is more relevant to the experimental quantities, since most of the novel quantum phenomena in condensed matter physics happen at very low tem- perature. By the term of "impurity solver", one means not only to compute the ground state but also the whole spectral functions, which include the lower energy quasi- particle parts as well as the higher energy Hubbard bands in general. NRG approach could resolve exponentially small scales at the expense of the accuracy at intermedi- ate and high energies. The extensions of DMFT also call for the develop- ment of impurity solvers. In recent years, LDA+DMFT has been developed very quickly and successfully ap- plied to many systems $^{13}$, see Kotliar et al. 14 and Held 15 for the reviews of the recent developments and applica- tions. Since in the real materials there are usually more than one orbitals involved, therefore one needs to solve the quantum impurity problems with multi-orbitals effi- ciently. A second direction of the development is to study not the impurities but small clusters embedded in bath to capture the spatial fluctuations $^{16}$. Therefor one needs an efficient method to calculate the spectral properties of smaller clusters. The requirement of solving complex impurity problems (multi-orbital or cluster) puts hurdles on the ED and NRG approaches. A third direction of extension of the DMFT is to deal with the non-equilibrium systems $^{17}$, which do not have the translational symmetry in the time domain. It re- quires the solver being able to work in the time domain, while to our knowledge most of the solvers work in the frequency (imaginary or real) domain. A recent attempt is made in the CTQMC approach $^{18}$. While this approach can handle arbitrary interaction strengths, it suffers from a dynamical sign problem which becomes severe at long time or for large bandwidth. Therefore it is urgent to develop an impurity solver working at zero temperature, which satisfies the following criteria. i) It can capture both the low energy quasi- arXiv:1001.2943v2 [cond-mat.str-el] 4 Feb 2010 I. INTRODUCTION