Nonequilibrium dynamical mean-field theory based on weak-coupling perturbation expansions: Application to dynamical symmetry breaking in the Hubbard model

We discuss the general formalism and validity of weak-coupling perturbation theory as an impurity solver for nonequilibrium dynamical mean-field theory. The method is implemented and tested in the Hubbard model, using expansions up to fourth order for the paramagnetic phase at half filling and third order for the antiferromagnetic and paramagnetic phase away from half filling. We explore various types of weak-coupling expansions and examine the accuracy and applicability of the methods for equilibrium and nonequilibrium problems. We find that in most cases an expansion of local self-energy diagrams including all the tadpole diagrams with respect to the Weiss Green's function (bare-diagram expansion) gives more accurate results than other schemes such as self-consistent perturbation theory using the fully interacting Green's function (bold-diagram expansion). In the paramagnetic phase at half filling, the fourth-order bare expansion improves the result of the second-order expansion in the weak-coupling regime, while both expansions suddenly fail at some intermediate interaction strength. The higher-order bare perturbation is especially advantageous in the antiferromagnetic phase near half filling. We use the third-order bare perturbation expansion within the nonequilibrium dynamical mean-field theory to study dynamical symmetry breaking from the paramagnetic to the antiferromagnetic phase induced by an interaction ramp in the Hubbard model. The results show that the order parameter, after an initial exponential growth, exhibits an amplitude oscillation around a nonthermal value followed by a slow drift toward the thermal value. The transient dynamics seems to be governed by a nonthermal critical point, associated with a nonthermal universality class, which is distinct from the conventional Ginzburg-Landau theory.

Figure 1

The Kadanoff-Baym contour $\mathcal{C}$.

Figure 2

The self-energy diagrams up to third order (except for the Hartree diagrams). The solid lines represent the fermion propagator, while the dashed lines are interaction vertices.

Figure 3

The Hartree diagram. The bold line represents the interacting Green's function $G_{\sigma }$.

Figure 4

An explicit expansion of the Hartree term in the bare Green's function ${\mathcal{G}}_{0,\sigma }$ up to third order.

Figure 5

The self-energy diagrams at fourth order for the PM phase at half filling.

Figure 6

The double occupancy for the Hubbard model in equilibrium at half filling with $\beta =16$ calculated in DMFT with various impurity solvers.

Figure 7

The density for the Hubbard model in equilibrium with $\beta =16$ calculated by DMFT with QMC, Hartree approximation, and the second-order perturbation theories. The labels (I)–(V) correspond to the classification in Table .

Figure 8

The density for the Hubbard model in equilibrium with $\beta =16$ calculated by DMFT with the third-order perturbation theories. The labels (I)–(V) correspond to the classification in Table .

Figure 9

The AFM transition temperature for the Hubbard model at half filling derived from DMFT with various impurity solvers. The region below (above) the curves represents the AFM (PM) phase. The labels (I)–(III) correspond to the classification in Table . The QMC data are taken from Ref. 67.

Figure 10

The AFM phase diagram of the Hubbard model covering the weak-coupling and strong-coupling limits. The QMC data are taken from Ref. 67.

Figure 11

The staggered magnetization for the Hubbard model at half filling evaluated by DMFT with QMC and the second-order perturbation theory of type (I) in the weak-coupling (top panel) and strong-coupling (bottom panel) regimes. The QMC data for $U=2,10$ are taken from Ref. 67.

Figure 12

The spectral function of the majority (minority) spin component [blue (red) solid curve] for the AFM phase of the Hubbard model at half filling calculated by DMFT with the second-order perturbation theory of type (I). The dashed lines show the spectral function calculated by DMFT with the noncrossing approximation. The parameters are $U=3,\beta =20$ (top panel) and $U=6,\beta =16$ (bottom panel).

Figure 13

The AFM transition temperature for the Hubbard model at half filling derived from DMFT with various third-order perturbation impurity solvers. The region below (above) the curves represents the AFM (PM) phase. The labels (I)–(IV) correspond to the classifications in Table .

Figure 14

The staggered magnetization of the Hubbard model at half filling evaluated by DMFT with QMC and the third-order perturbation theory of type (I).

Figure 15

Time evolution of the double occupancy (top panel) and the jump of the momentum distribution function (bottom panel) after the interaction quenches $U=0\to 0.5,1,\cdots ,3$ (from top to bottom) in the PM phase of the Hubbard model at half filling calculated by the nonequilibrium DMFT with QMC (circles, taken from Ref. 23), the bare second-order (dashed curves), and bare fourth-order (solid curves) perturbation theories.

Figure 16

Time evolution of the double occupancy (top panel) and the jump of the momentum distribution function (bottom) after the interaction quenches $U=0\to 4,5,6,8$ (from bottom to top in the first minima of $d$, and from top to bottom in the initial decrease of $\Delta n$) in the PM phase of the Hubbard model at half filling calculated by the nonequilibrium DMFT with QMC (circles, taken from Ref. 23), the bare second-order (dashed curves), and bare fourth-order (solid curves) perturbation theories.

Figure 17

The drift of the total energy $E\left(t\right)-E\left(0^{+}\right)$ (measured at $t=5,10$) in the simulation of the interaction quench $U=0\to U_f$ using the bare second-order and bare fourth-order perturbation solvers.

Figure 18

Time evolution of the double occupancy (top), the jump of the momentum distribution function (middle), and the total energy (bottom) after the interaction quenches $U=0\to 16,24,32$ (from long- to short-time data) in the PM phase of the Hubbard model at half filling calculated by nonequilibrium DMFT with the bare second-order perturbation theory.

Figure 19

Time evolution of the double occupancy (top panel) and the jump of the momentum distribution function (bottom) after the interaction quenches $U=0\to 0.5,1,\cdots ,3$ (from top to bottom) in the PM phase of the Hubbard model at half filling calculated by the nonequilibrium DMFT with QMC (circles, taken from Ref. 23), the bold second-order (dashed curves), and the bold fourth-order (solid curves) self-consistent perturbation theories.

Figure 20

Time evolution of the double occupancy (top panel) and the jump of the momentum distribution function (bottom) after the interaction quenches $U=4,5,6,8$ (from bottom to top in the long-time limit of $d$, and from top to bottom in $\Delta n$) in the PM phase of the Hubbard model at half filling calculated by the nonequilibrium DMFT with QMC (circles, taken from Ref. 23), the bold second-order (dashed curves), and the bold fourth-order (solid curves) self-consistent perturbation theories.

Figure 21

Time evolution of the double occupancy (a), the jump of the momentum distribution function (b), the density (c), and the total-energy drift (d) after interaction quenches in the PM phase of the Hubbard model at quarter filling calculated by the nonequilibrium DMFT with the type (I) second-order (dashed curves, $U=0\to 0.5,1,\cdots ,3$) and type (I) third-order (solid curves, $U=0\to 0.5,1,1.5,2$) perturbation theories.

Figure 22

Time evolution of the double occupancy (a), the jump of the momentum distribution function (b), the density (c), and the total-energy drift (d) after interaction quenches in the PM phase of the Hubbard model at quarter filling calculated by the nonequilibrium DMFT with the type (IV) second-order (dashed curves, $U=0\to 0.5,1,\cdots ,3$) and type (IV) third-order (solid curves, $U=0\to 0.5,1,\cdots ,2.5$) perturbation theories.

Figure 23

Time evolution of the staggered magnetization after the interaction ramps $U=1.75\to 1.8,1.9,\cdots ,2.6$ (from bottom to top) in the Hubbard model. $\beta =11$, $h={10}^{-4}$, and $t_{\mathrm{ramp}}=10$. The arrows indicate the corresponding thermal values reached in the long-time limit.

Figure 24

The initial (solid circle, $U_i=1.75$) and final (open circles, $U_f=1.9,2,\cdots ,2.6$ from left to right) thermal states in the simulation of the dynamical symmetry breaking in Fig. 23. The labels “PM” and “AFM” indicate the PM and AFM phases, respectively.

Figure 25

Time evolution of the total energy for the interaction ramps that correspond to Fig. 23.

Figure 26

Various quantities that characterize the critical behavior of the dynamical symmetry breaking with $U_i=1.75$: $m_{\max }$ ($m_{\min }$) is the maximum (minimum) of the first cycle of the oscillation in $m\left(t\right)$, $m_{\mathrm{th}}$ is the thermal value taken in the long-time limit, and ${\tau }_i$ is the rate of the initial exponential growth. The dashed line is an extrapolation of the middle points of $m_{\max }$ and $m_{\min }$.

Figure 27

Comparison between the DMFT result for $m\left(t\right)$ for the quench $U=1.25\to 2$, $\beta =22$ with the phenomenological Ginzburg-Landau theory. The arrow indicates the thermal value of $m$ reached in the long-time limit.

Figure 28

The time evolution of the double occupancy for the quench $U=1.25\to 2$, $\beta =22$. The arrow on the left indicates the thermal value of $m$ for the PM phase, while the one on the right shows the value in the AFM phase.

Figure 29

The time-resolved spectral function $A(\omega ,t)$ of the minority spin component for the quench $U=1.25\to 2$, $\beta =22$.

Figure 30

The time evolution of $m$ for the interaction ramp $U=0\to 1.25$ in the Hubbard model at half filling with $\beta =40$, $h={10}^{-4}$, and $t_{\mathrm{ramp}}=10$ obtained from the Hartree approximation.

Figure 31

Time evolution of $m$ for interaction ramps $U=1.0\to 1.05,1.1,\cdots ,1.5$ in the Hubbard model at half filling with $\beta =40$, $h={10}^{-4}$, and $t_{\mathrm{ramp}}=10$.

Figure 32

The initial exponential growth rate ${\tau }_i$ and the maximum (minimum) value of $m$ at the first peak (dip) of the oscillations for the interaction ramps in Fig. 31.