Multiconfiguration time-dependent Hartree impurity solver for nonequilibrium dynamical mean-field theory

Nonequilibrium dynamical mean-field theory (DMFT) solves correlated lattice models by obtaining their local correlation functions from an effective model consisting of a single impurity in a self-consistently determined bath. The recently developed mapping of this impurity problem from the Keldysh time contour onto a time-dependent single-impurity Anderson model (SIAM) [C. Gramsch et al., Phys. Rev. B 88, 235106 (2013)] allows one to use wave-function-based methods in the context of nonequilibrium DMFT. Within this mapping, long times in the DMFT simulation become accessible by an increasing number of bath orbitals, which requires efficient representations of the time-dependent SIAM wave function. These can be achieved by the multiconfiguration time-dependent Hartree (MCTDH) method and its multilayer extensions. We find that MCTDH outperforms exact diagonalization for large baths in which the latter approach is still within reach and allows for the calculation of SIAMs beyond the system size accessible by exact diagonalization. Moreover, we illustrate the computation of the self-consistent two-time impurity Green's function within the MCTDH second quantization representation.

Figure 1

(a) DMFT impurity problem according to the action of Eq. (3). Arrows illustrate the effect of the two-time hybridization function ${\Lambda }_{\sigma }(t,t^{\prime })$, which describes all possible processes where a particle of spin $\sigma$ jumps at time $t^{\prime }$ from the interacting impurity site (red) to a lattice site $i$ (green), propagates forward or backward in time from site $i$ to site $j$, and at a time $t$ jumps back to the impurity. (b) Representation of the DMFT bath by a single-impurity Anderson model (SIAM) with six bath orbitals and hopping matrix elements $V_{0l}^{\sigma }$, $l\ge 1$. The variable $Q_1$ illustrates the combination of physical degrees of freedom into combined modes according to the MCTDH ansatz of Eq. (19).

Figure 2

From top to bottom, tensor networks representing the coefficients of the MCTDH ansätze of Eqs. (9), (18), and (20), respectively. Each box represents a tensor with as many indices as outgoing lines. A sum is performed over all indices at a line connecting two tensors.

Figure 3

Time evolution of the complex hopping matrix elements $V_{0l}\left(t\right)=V_{0l}^{\uparrow }\left(t\right)=V_{0l}^{\downarrow }\left(t\right)$ for a SIAM with $L=4$ bath orbitals and a reference bath which is governed by the equilibrium Green's function of Eq. (28) with inverse temperature $\beta =1$.

Figure 4

Hilbert-space dimension of the SIAM as a function of the number of bath sites $L$ (dashed black line) and dimensionality of the corresponding $A$ vectors in MCTDH-SQR (colored lines) for a setup where four bath spin orbitals are treated in a combined mode $(N_{\mathrm{CM}}^{\mathrm{b}}=4)$ and $N_{\mathrm{SPF}}^{\mathrm{b}}$ single-particle functions are involved [cf. Eq. (30)].

Figure 5

Time-dependent double occupancy $\langle d\rangle \left(t\right)$ of the impurity site for a SIAM with $L=4$ bath orbitals at (a) $U=2$ and (b) $U=6$, calculated by exact diagonalization (ED) and MCTDH-SQR with various numbers of SPFs $\left(N_{\mathrm{SPF}}^{\mathrm{b}}\right)$.

Figure 6

Comparison of MCTDH-SQR with four combined modes ($N_{\mathrm{CM}}=4$) and $N_{\mathrm{SPF}}$ single-particle functions to exact diagonalization (ED) at an on-site interaction of $U=10$. Shown is the time-dependent impurity double occupancy $\langle d\rangle \left(t\right)$ for a SIAM with (a) $L=10$, (b) $L=12$, and (c) $L=16$ bath orbitals. The solid black line, showing the ED result for $L=14$, acts as a reference to determine the maximum time $t_{\max }$ in Fig. 7.

Figure 7

Numerical complexity of the time evolution to a given time $t_{\max }$ in the SIAM, measured in terms of the number $D$ of parameters in the wave-function ansatz [i.e., the required Hilbert-space dimension $D_{\mathrm{H}}\left(L\right)$ for exact diagonalization and the size of the tensor $D_A(L;N_{\mathrm{SPF}})$ in MCTDH]. To determine the maximum time $t_{\max }$ which can be accurately simulated for a given configuration $(L,N_{\mathrm{SPF}})$, and hence with a given $D$, we require the error in the double occupation to be lower than a given bound $\epsilon$ [see Figs. (a)–6(c)]: yellow and red symbols and lines correspond to exact diagonalization and an $\epsilon$ value of 1% and 10%, respectively. Black circles correspond to MCTDH and $\epsilon =5\%$.

Figure 8

(a) Imaginary part of the Green's function $g_{\sigma }^{<}(t,t^{\prime })$ of Eq. (28), which is used to compute the initial guess for the hybridization function in the first DMFT iteration. (b)–(d) Self-consistent results for the local impurity Green's function $G_{0\sigma }(t,t^{\prime })$ as obtained from an MCTDH-SQR calculation with $L=8$ bath orbitals in the single-impurity Anderson model; the on-site Coulomb repulsion is $U=4$ (note that $G_{0\sigma }^{>}=-G_{0\sigma }^{<}$ because of particle-hole symmetry). Black arrows indicate the early time domain, where the transient dynamics due to the switch-on of hopping is most pronounced. The gray (red) arrow in (c) points to the formation of small artifacts in the final “steady” state which are due to the representation of the DMFT bath with finitely many bath orbitals. Furthermore, in (d), the solid black line shows the time evolution of the double occupation $\langle d\rangle$ in the system, and the dotted (red) line indicates the switch-on of hopping.