Nonequilibrium dynamical mean-field calculations based on the noncrossing approximation and its generalizations

We solve the impurity problem which arises within nonequilibrium dynamical mean-field theory for the Hubbard model by means of a self-consistent perturbation expansion around the atomic limit. While the lowest order, known as the noncrossing approximation (NCA), is reliable only when the interaction $U$ is much larger than the bandwidth, low-order corrections to the NCA turn out to be sufficient to reproduce numerically exact Monte Carlo results in a wide parameter range that covers the insulating phase and the metal-insulator crossover regime at not too low temperatures. As an application of the perturbative strong-coupling impurity solver we investigate the response of the double occupancy in the Mott insulating phase of the Hubbard model to a dynamical change in the interaction or the hopping, a technique which has been used as a probe of the Mott insulating state in ultracold fermionic gases.

Figure 1

The L-shaped contour $\mathcal{C}$ for the description of transient nonequilibrium states with initial state density matrix $\propto \text{exp}\left[-\beta H\left(0\right)\right]$. The indicated time arguments are in cyclic order, $t_1\prec t_2\prec t_3$ (see text).

Figure 2

(a) A fifth-order contribution to the self-energy ${\Sigma }_{m{m}^{\prime }}^{\lambda }\left(t,t^{\prime }\right)$, consisting of Green’s functions ${\mathcal{G}}_0^{\lambda }$ (solid lines) and hybridization functions $\Lambda$ (dotted lines). (b) Building blocks of the diagrams in the hybridization expansion. The pseudoparticle line (solid line) corresponds to the interacting propagator $\mathcal{G}$ in skeleton expansions and to ${\mathcal{G}}_0$ otherwise. (c) All diagrams for ${\Sigma }^{\text{scel}}\left[\mathcal{G},\Lambda \right]$ up to third order. In the topologies indicated, the hybridization line can point in any direction, which gives 2, 4, and $4\times 8$ diagrams in first, second, and third order, respectively. (d) Factorization of third-order diagrams [(iii)–(vi) in (c)] by separating out the vertex part.

Figure 3

All diagrams for $G^{\text{scel}}\left[\mathcal{G},\Lambda \right]$ in first order [diagram (i)], second-order [diagram (ii)], and third-order [diagram (iii)–(vi)]. Internal hybridization lines can point in any direction, which gives 1, 2, and $4\times 4$ diagrams in first, second, and third order, respectively.

Figure 4

(Color online) Double occupancy $d_{eq}\left(\beta ,U\right)$ in the thermal equilibrium state. The impurity solver is either CTQMC, or the self-consistent hybridization expansion up to first (NCA), second (OCA), and third order. (a) $\beta =5$. (b) $\beta =10$. (c) $\beta$ close to the critical temperature $T_c$ of the first-order metal-insulator transition.

Figure 5

(Color online) Matsubara component defined in Eq. (5) of the local Green’s function $G_{\sigma }\left(\tau \right)$ [Eq. (3)], which is independent of $\sigma$ in the paramagnetic phase. The impurity solver is either CTQMC, or the self-consistent hybridization expansion up to first (NCA), second (OCA), and third order. (a) $U=4.5,\beta =20$ (crossover regime). (b) $U=6,\beta =20$ (insulating phase). Note that the CTQMC results in the insulating phase are only accurate for values larger than the “noise floor” of about ${10}^{-3}$. Panels (c) and (d) show the same data as (a) and (b), respectively, but plotted for small $\tau$ on a linear scale.

Figure 6

(Color online) Double occupancy $d\left(t\right)$ after an interaction quench from $U_0$ to $U$. [(a) and (b)] Initial states in the crossover regime [$U_0=3$, $\beta =5$], [(c) and (d)] Initial states in the insulating phase [$U_0=5$, $\beta =5$].

Figure 7

(Color online) (a) Double occupancy $d\left(t\right)$ during a periodic modulation of the interaction around $U_0=7$ ($\beta =10$, $\delta U=\alpha U_0=2$). The frequency is below the resonance peak $\left(\omega =1.5\right)$, close to the resonance $\left(\omega =7\right)$, and above the resonance $\left(\omega =10\right)$. Bold solid lines indicate the average $d_{av}\left(t\right)$ over one period $\left[t-\pi /\omega ,t+\pi /\omega \right]$. OCA was used as an impurity solver. (b) Modulation spectrum for $U_0=7$ and $\beta =10$, using either NCA or OCA as an impurity solver: For large amplitude $\delta U=2$, the data have been obtained from $d_{av}\left(t_0\right)$ at given time $t_0=10$ (open symbols); for small amplitudes $\delta U=0.5$, the slope $\left(d/dt\right)d_{av}\left(t\right)$ in the interval $8<t<10$ is plotted (full symbols). The curves have been scaled with a constant factor to match their peak heights. (c) Averaged double occupancy $d_{av}\left(t\right)$ at $U_0=7$ for various modulation amplitudes $\delta U=\alpha U_0$, plotted against $t\delta U^2$. The three curves labeled NCA almost fall on top of each other.

Figure 8

(Color online) (a) Modulation spectrum at $\beta =10$ and various interactions $U_0$, obtained from the increase $d_{av}\left(t_0\right)-d\left(0\right)$ at $t_0=10$ $\left(\delta U=2\right)$. (b) Location of the maximum ${\omega }_{\text{max}}$ of the spectra in (a), plotted against $U_0$.

Figure 9

(Color online) (a) The double occupancy after quenches from $U=U_0$ to $U=U_0-\delta U$, with $\delta U=0.5$, $\beta =10$, and $U_0=7,6,5.6,5,4.5$. OCA has been used as an impurity solver. (b) Fourier transform in Eq. (40) of the data in (a).