Nonequilibrium dynamical mean-field theory and its applications

The study of nonequilibrium phenomena in correlated lattice systems has developed into one of the most active and exciting branches of condensed matter physics. This research field provides rich new insights that could not be obtained from the study of equilibrium situations, and the theoretical understanding of the physics often requires the development of new concepts and methods. On the experimental side, ultrafast pump-probe spectroscopies enable studies of excitation and relaxation phenomena in correlated electron systems, while ultracold atoms in optical lattices provide a new way to control and measure the time evolution of interacting lattice systems with a vastly different characteristic time scale compared to electron systems. A theoretical description of these phenomena is challenging because, first, the quantum-mechanical time evolution of many-body systems out of equilibrium must be computed and second, strong-correlation effects which can be of a nonperturbative nature must be addressed. This review discusses the nonequilibrium extension of the dynamical mean field theory (DMFT), which treats quantum fluctuations in the time domain and works directly in the thermodynamic limit. The method reduces the complexity of the calculation via a mapping to a self-consistent impurity problem, which becomes exact in infinite dimensions. Particular emphasis is placed on a detailed derivation of the formalism, and on a discussion of numerical techniques, which enable solutions of the effective nonequilibrium DMFT impurity problem. Insights gained into the properties of the infinite-dimensional Hubbard model under strong nonequilibrium conditions are summarized. These examples illustrate the current ability of the theoretical framework to reproduce and understand fundamental nonequilibrium phenomena, such as the dielectric breakdown of Mott insulators, photodoping, and collapse-and-revival oscillations in quenched systems. Furthermore, remarkable novel phenomena have been predicted by the nonequilibrium DMFT simulations of correlated lattice systems, including dynamical phase transitions and field-induced repulsion-to-attraction conversions.

Figure 1

A schematic picture of the nonequilibrium DMFT formalism. $\mathrm{\Delta }(t,t^{\prime })$ is the hybridization function, while $\mathrm{\Sigma }(t,t^{\prime })$ is the self-energy.

Figure 2

Left panel: Schematic time evolution of the system in a pump-probe experiment with various physical processes (see text). Right panel: Comparison of a short-time approximation ([233]; [80]), which approaches a prethermalization plateau, and nonequilibrium DMFT ([331]), which also describes the crossover toward a thermal state, for a sudden switching on of the Hubbard interaction to $U=0.375W$.

Figure 3

(a) Temporal evolution of the density of photoinduced carriers in a correlated electron system (a Ni complex) for two values of the excitation density. From [159]. The inset schematically shows a pump-probe experiment. (b) Temporal evolution of the doublon occupation in a cold-atom system on an optical lattice (as schematically depicted in the inset). From [315].

Figure 4

The L-shaped contour $={}_1\cup {}_2\cup {}_3$ in the Kadanoff-Baym formalism. The arrows indicate the contour ordering. For example, $t$ lies ahead of $t^{\prime }$ in the ordering ($t\succ t^{\prime }$).

Figure 5

Causal structure of the Kadanoff-Baym equation (41), when $G$ is computed on one time slice of an equally spaced grid (shaded area): The computation of derivatives for the propagation from the previous time slice (bold arrows) addresses $G$ only at earlier times, marked by a diagonal pattern for ${\partial }_t{G}^R$, diamond pattern for ${\partial }_t{G}^{\neg }$, and checkerboard pattern for ${\partial }_t{G}^{<}$. Hermitian symmetries (19) are used to relate $G^A$ with $G^R$, $G^{<}(t,t^{\prime })$ with $G^{<}(t^{\prime },t)$ and $G^{⌐}$ with $G^{\neg }$.

Figure 6

The Keldysh contour ${}_K={}_1\cup {}_2$ with the two branches ranging from $-\infty$ to $\infty$.

Figure 7

Schematic representation of a free-fermion bath model. Each site of the system is connected to an electrode with a coupling $\mathrm{\Gamma }$ at temperature $T$. The arrows indicate the motion of particles.

Figure 8

Illustration of the CTQMC method. Top panel: Example of a sixth-order diagram appearing in the weak-coupling expansion. Interaction vertices are linked by bath Green’s functions ${}_0$. Bottom panel: Example of a fourth-order diagram appearing in the strong coupling expansion: full (empty) dots represent impurity creation (annihilation) operators. Pairs of creation and annihilation operators are linked by hybridization functions $\mathrm{\Delta }$ (dashed lines). In both methods, all diagrams obtained by linking a given collection of operators by Green’s functions or hybridization lines are summed up into a determinant.

Figure 9

The self-energy diagrams up to third order. The thin lines represent ${}_{0\sigma }(t,t^{\prime })$, the bold line $G_{\sigma }(t,t^{\prime })$, and the dashed lines the interaction vertices.

Figure 10

Nonequilibrium DMFT results for the kinetic, potential, and total energies of the Hubbard model after an interaction quench from $U=0$ to $U=2$ (left panel) and $U=5$ (right panel). Symbols show the weak-coupling CTQMC result, dashed lines the result from bare second-order perturbation theory, and the solid lines the result from self-consistent second-order perturbation theory. Adapted from [80].

Figure 11

Elements of the strong-coupling expansion: (a) vertices [Eq. (158)] and lines [Eq. (157)]. (b) Some diagrams for $Z$, in the open-contour representation (from $0^{+}$ on the upper branch of $$, to $-i\beta$). (c) Diagram for $⟨(t)⟩$ (star) in the open-contour representation. The shaded part can be viewed as an initial-state correction, or lesser self-energy. (d) All skeleton diagrams $\mathfrak{S}[,\mathrm{\Delta }]$ up to third order. The first two diagrams define the NCA and OCA, respectively. (e) All skeleton diagrams $G[,\mathrm{\Delta }]$ up to third order.

Figure 12

Nonequilibrium DMFT results for the time evolution of the double occupancy in the Hubbard model after a quench from $U(t=0)=U_0$ to $U$. The symbols show weak-coupling CTQMC results, and the lines results from strong-coupling perturbation theory. From [82].

Figure 13

Diagrammatic representation of the Floquet Green’s function ${\mathit{G}}_{mn}(\omega )$. The wavy lines denote photon propagators of the pump light, and the arrows indicate the energy flow.

Figure 14

Illustration of a system consisting of $N$ correlated layers (full dots) between noninteracting leads (open dots). The intralayer hopping $t$, interlayer hopping $t_p$, interaction $U$, and chemical potential $\mu$ can be layer and time dependent.

Figure 15

Time evolution of the magnetization in a semi-infinite chain with interaction $U=1$ on the first (impurity) site. The hopping between the noninteracting bath sites is $t=1$ and the initial state is the impurity, occupied with a spin-up electron, decoupled from the bath. The time evolution is triggered by the sudden switch-on of the hopping between the impurity and the bath. The curves show the time evolution of the spin polarization in the impurity, computed in a finite chain with five bath sites, the CPT result for a reference cluster $C^{\prime }$ that consists of only the impurity and one bath site (“zeroth order”), and for the isolated cluster $C^{\prime }$ (“reference system”). From [167].

Figure 16

Time evolution of the impurity occupation in a two-site system (one impurity with interaction $U=2$, one noninteracting bath site, hopping $v=0.5$ switched on at $t=0$). The exact solution is compared to various approximations within the dual-fermion method (see text). From [167].

Figure 17

Various regimes of electric field-induced phenomena plotted against the field strength $F$ and frequency $\mathrm{\Omega }$.

Figure 18

Damped Bloch oscillations in the Falicov-Kimball model with a constant electric field $F=1$, and indicated values of the interaction $U$. From [105].

Figure 19

Many-body density of states of the Falicov-Kimball model with $U=0.5$ and a constant electric field (a) $F=0.5$ and (b) $F=1$. From [101].

Figure 20

dc field-induced current in the Hubbard model with coupling to a thermostat. Current for $U=6$ and $F=4.7$ and indicated values of the heat-bath coupling $\mathrm{\Lambda }$. The field is suddenly switched on at time $t=0$. The DMFT impurity problem is solved with second-order perturbation theory (IPT). (Inset) Time evolution of the effective temperature for $U=6$ and $F=1.9$. Adapted from [6].

Figure 21

(a) Illustration of the dielectric breakdown of a Mott insulator in a strong electric field due to many-body quantum tunneling. The many-body energy (solid curve) is plotted against the separation of doublons and holes $l_{\text{dh}}$. For the dielectric breakdown the state has to tunnel (dashed line) through an energy barrier ${\mathrm{\Delta }}_{\text{Mott}}\sim U$ related to the creation of charge excitations (doublon-hole pairs). Since the electric field reduces the energy of a pair by $F{l}_{\text{dh}}$, the quantum tunneling among many-body states takes place when ${\mathrm{\Delta }}_{\text{Mott}}-F{l}_{\text{dh}}\sim 0$, i.e., when $l_{\text{dh}}={\ell }_U$. The tunneling probability depends on the overlap between the ground state and excited-state wave functions. Adapted from [254]. (b) The threshold field $F_{\mathrm{th}}(U)$ from DMFT calculations for the hypercubic lattice with density of states $\rho (E)\propto \text{exp}(-E^2)$ [OCA results from 81].

Figure 22

Time-resolved photoemission spectrum of the half-filled, spinless Falicov-Kimball model, after excitation by a monocycle pulse. (a) Pulse shape. (b) Metallic system with $U=0.5t^{*}$. (c) Insulating system with $U=2t^{*}$. From [238].

Figure 23

(a) Double occupancy in the Hubbard model after photodoping (hypercubic lattice, field along the body diagonal of the lattice; see Sec. 2b4). The system is excited with a Gaussian field pulse $F(t)=F_0\cos (\mathrm{\Omega }t)e^{-t^2/\mathrm{\Delta }{t}^2}$ (frequency $\mathrm{\Omega }=U$). The initial temperature is $T=0.2$ (above the critical end point of the metal-insulator line), and the field amplitude is chosen such that the final effective temperature is $T_{\text{eff}}=0.5$. (b) Thermalization time, obtained from exponential fits to the data in (a). Solid lines correspond to $\tau \propto \exp [\alpha (U/v)\log (U/v)]$. The strong-coupling expansion (see Sec. 2c5) was used to solve the DMFT impurity model, and the results for first- (NCA), second- (OCA), and third-order (3rd o.) converge. Adapted from [84].

Figure 24

Dynamical modification of the hopping amplitude observed for Bose-Einstein condensed cold atoms in a periodically shaken optical lattice. $A$ is the ratio of the amplitude to the frequency of the driving ac field and $v_{\text{eff}}$ ($v$) is the effective (bare) hopping amplitude of particles. From [201].

Figure 25

Nonequilibrium DMFT result for the time evolution of the double occupancy for the ac-field-driven Hubbard model (symbols with error bars) with $U=1$, $\mathrm{\Omega }=2\pi$, and various values of $A\equiv E/\mathrm{\Omega }$. Solid curves are the corresponding results for the interaction quench $U\to U_{\text{eff}}=U/{}_0(A)$ with time rescaled as $t/|{}_0(A)|$. The inset shows the Bessel function ${}_0(A)$. From [330].

Figure 26

(a) The Floquet spectrum $A(\mathit{k},\omega )$ for the ac-field-driven Falicov-Kimball model with $U=3$, $E=0.8$, and $\mathrm{\Omega }=1.8$. (b) A schematic band structure in the ac field, where the upper (lower) Hubbard band are represented, while their Floquet sidebands with a spacing $\mathrm{\Omega }$ are shown by dashed lines. From [327].

Figure 27

Optical conductivity for the ac-field-driven Falicov-Kimball model coupled to a fermionic heat bath with $U=3$, $\mathrm{\Gamma }=0.05$, $T=0.05$, and (a) $\mathrm{\Omega }=1.8$, (b) 2.7, and (c) 3.3. The dashed curves illustrate (for specific values of $E$) the results without vertex correction. The arrows indicate the frequency $\mathrm{\Omega }$ of the ac field. Inset: diagrams of the bubble and vertex correction. From [328].

Figure 28

Double occupation $d(t)$ for quenches to (a) $U_{+}=1$, (b) $U_{+}=3$, and (c) $U_{+}=8$, starting from an initial metallic ($U_{-}<2$) or insulating state ($U_{-}>2$). The half bandwidth is 2. For (a) and (b) the energy is the same after both quenches, but the long-time limit (left-pointing arrows) is different for both and also different from the expected thermal value (thick right-pointing arrows). The inset in (a) shows a magnification. From [76].

Figure 29

Momentum distribution $n(ϵ,t)$ after an interaction quench in the Hubbard model from the noninteracting ground state ($U=0$) to interaction (a) $U=2$, (b) $U=3.3$, and (c) $U=5$; the half bandwidth is 2. The solid line at $t=3.5$ in (b) is the equilibrium expectation value for the momentum distribution at the same total energy (temperature $T=0.84$). Adapted from [80].

Figure 30

Fermi surface discontinuity $\mathrm{\Delta }n$ and double occupation $d(t)$ after quenches to $U\le 3$ (left panels) and $U\ge 3.5$ (right panels). Horizontal dotted lines in panel (c) are the prethermalization plateaus predicted by [233]. Horizontal arrows indicate thermal values of the double occupation. Adapted from [79].

Figure 31

Top panel: Antiferromagnetic phase diagram for the half-filled Hubbard model (semielliptic DOS, bandwidth 4). The QMC data are from [185], the NCA and OCA phase-boundaries from [352], and the third-order weak-coupling perturbation results from [326]. The time evolution of the order parameter (staggered magnetization) is shown for quenches $U=2\to 1.9,1.8,\dots ,1.0$ [bottom left, from 326], and for quenches $U=4\to 6$, 7, 8 [bottom right, from 352]. The arrows indicate the corresponding thermal values of the order parameter reached in the long-time limit.

Figure 32

The results of the sample program for the time evolution of the double occupancy $d(t)$ after interaction quenches from $U_i=0$ to $U_f$ with $\beta =16$.