Hamiltonian-based impurity solver for nonequilibrium dynamical mean-field theory

We derive an exact mapping from the action of nonequilibrium dynamical mean-field theory (DMFT) to a single-impurity Anderson model (SIAM) with time-dependent parameters, which can be solved numerically by exact diagonalization. The representability of the nonequilibrium DMFT action by a SIAM is established as a rather general property of nonequilibrium Green functions. We also obtain the nonequilibrium DMFT equations using the cavity method alone. We show how to numerically obtain the SIAM parameters using Cholesky or eigenvector matrix decompositions. As an application, we use a Krylov-based time propagation method to investigate the Hubbard model in which the hopping is switched on, starting from the atomic limit. Possible future developments are discussed.

Figure 1

The L-shaped time contour $C$ consists of three parts. The integration runs along $C_1$ from $t_{\min }$ to $t_{\max }$ on the real axis. It then goes back along $C_2$, and follows $C_3$ into the complex plane to $t_{\min }-i\beta$, where $\beta$ is the inverse temperature. In the sense of the contour, the time $t\in C_2$ is therefore larger than $t^{\prime }\in C_1$ in this example, i.e., $t{>}_C{t}^{\prime }$. For simplicity, we set $t_{\min }=0$ throughout this work.

Figure 2

Possible time-evolution schemes. (a) DMFT iteration involving all times $t,t^{\prime }\le t_{\max }$ simultaneously, (b) progressive time-propagation scheme for which the DMFT self-consistency needs to be established only on the current timestep $N$ at a time. In (b), the notation $G(N;t,t^{\prime })$ indicates that $t$ and $t^{\prime }$ belong to the time slice $N$, i.e., one time (either $t$ or $t^{\prime }$) is identical to $N\times \Delta t$ and the other is smaller or equal to $N\times \Delta t$.

Figure 3

Comparison of low-rank Cholesky [${(-i{\Lambda }_{+}^{<})}^{\mathrm{ch}}$] and eigenvector [${(-i{\Lambda }_{+}^{<})}^{\mathrm{ev}}$] approximation. The top panel on the left shows the real part of an typical input Weiss field $-i{\Lambda }_{+}^{<}$ (obtained from a calculation with six bath sites, i.e., $L_{\mathrm{bath}}=2L=6$ according to the setup described in Sec. 6 with $U=2$). The Weiss field in the top panel of the second (third) column displays the an approximate hybridization $V_{np}^{\mathrm{ch}}$ with rank $L=2$ ($L=3$) that was calculated with the low-rank Cholesky approach. In the lower panel of the second (third) column, an approximate Weiss field obtained from a rank $L=2$ ($L=3$) eigenvector decomposition is shown. The bottom left panel compares the stepwise error of both approximations as defined in (67).

Figure 4

Time evolution of the hybridization $V_{0p}^{+}\left(t\right)$. Both panels show a rank $L=3$ approximation that corresponds to the input Weiss field $-i{\Lambda }_{+}^{<}$ displayed in Fig. 3. To obtain $V_{0p}^{\mathrm{ch}}\left(t\right)$, in the top panel, we used the low-rank Cholesky approach. In the lower panel, we plot the hybridization $V_{0p}^{\mathrm{ev}}\left(t\right)$, which was calculated using the low-rank eigenvector approximation. The inset shows the decay of the eigenvalues $a_p$ of $-i{\Lambda }_{+}^{<}$.

Figure 5

Atomic limit: SIAM representation of the DMFT initial state for the paramagnetic phase of the Bethe lattice and $L_{\mathrm{bath}}=6$. While the full dot denotes the impurity site with on-site Coulomb repulsion $U$, the open dots mark the bath with $V_{0p}\left(0\right)=0$ in the initial state ($1\le p\le L_{\mathrm{bath}}$).

Figure 6

DMFT iteration scheme for a fixed maximal time, i.e., $t,t^{\prime }\le t_{\max }$. The initial input Weiss field ${\Lambda }_1(t,t^{\prime })$ is given by (71). $d_n\left(t\right)$ denotes the double occupation after the $n$th iteration.

Figure 7

From every iteration we obtain a time evolution for the double occupation $d_n\left(t\right)$, represented by a single line in this plot. We used $t_{\max }=4$, $L_{\mathrm{bath}}=4$, $U=5$, and $v\left(t\right)$ as defined in Eq. (69) with $t_{\mathrm{q}}=0.25$ (vertical dotted line), cf. Fig. 10 for its profile. Seven iterations were performed. (Top) Results obtained using the eigenvector approximation. (Bottom) Results obtained using the Cholesky approximation.

Figure 8

Converged DMFT results obtained using eigenvector and Cholesky approximation. Calculations were performed for $U=5$ and $v\left(t\right)$ as defined in Eq. (69), with $t_{\mathrm{q}}=0.25$ (vertical dotted line). Each line displays the final result for the double occupation $d\left(t\right)$, which depends on the maximal time $t_{\max }$ and on the number of bath sites $L_{\mathrm{bath}}$. (Top) Eigenvector approximation. (Bottom) Cholesky approximation. The size of the time step was fixed to $\delta t=0.04$.

Figure 9

Test of energy conservation. Numerical results for the time-dependent energies $\langle E_{\mathrm{kin}}\rangle$ (green), $\langle E_{\mathrm{int}}\rangle$ (blue) and $\langle E_{\mathrm{tot}}\rangle$ (red) for $U=2$ (solid lines) and $U=4$ (dashed lines). The thick curves correspond to the SIAM representation with $L_{\mathrm{bath}}=8$, the thinner curves to $L_{\mathrm{bath}}=6$ and 4, respectively. All data were obtained with the Cholesky decomposition. The vertical dotted line indicates the time $t_{\mathrm{q}}$ at the end of the ramp, after which the Hamiltonian is time-independent.

Figure 10

Time dependence of the double occupation $d\left(t\right)$ in the Bethe lattice for the approximate SIAM representation with $L_{\mathrm{bath}}=4$, 6, and 8 (using the Cholesky decomposition) and for different $U$ ranging from zero and small to large Coulomb interaction. The ramp of the hopping parameter (orange curve) is as in Fig. 9.

Figure 11

Comparison of the SIAM-based results for $L_{\mathrm{bath}}=8$ (violet curves) to the noncrossing approximation (NCA), the one-crossing approximation (OCA) and the corresponding third-order approximation of the hybridization expansion. While (a) covers small to moderate interactions and (b) addresses moderate to strong couplings. The ramp $v\left(t\right)$ (orange curve) is as in Figs. 9 and 10.