First-Order Dynamical Phase Transitions

Recently, dynamical phase transitions have been identified based on the nonanalytic behavior of the Loschmidt echo in the thermodynamic limit [Heyl et al., Phys. Rev. Lett. 110, 135704 (2013)]. By introducing conditional probability amplitudes, we show how dynamical phase transitions can be further classified, both mathematically, and potentially in experiment. This leads to the definition of first-order dynamical phase transitions. Furthermore, we develop a generalized Keldysh formalism which allows us to use nonequilibrium dynamical mean-field theory to study the Loschmidt echo and dynamical phase transitions in high-dimensional, nonintegrable models. We find dynamical phase transitions of first order in the Falicov-Kimball model and in the Hubbard model.

Figure 1

Generalized contour-dependent Hamiltonian on the Keldysh contour $\mathcal{C}$. The upper, lower, and imaginary branches of the contour are denoted by ${\mathcal{C}}_1$, ${\mathcal{C}}_2$, and ${\mathcal{C}}_3$, respectively. The arrows indicate the contour ordering, in this case $t_1$ comes earlier than $t_2$, i.e., $t_1\prec t_2$. For a Green function $G(t_1,t_2)$, if $t_1$ lies on ${\mathcal{C}}_1$ and $t_2$ lies on ${\mathcal{C}}_2$, the latter cannot be shifted to the upper contour.

Figure 2

Time-dependent generalized expectation value of the double occupancy $d_{FK}(t^{\prime })\equiv ⟨d(t^{\prime }){⟩}_{H_{\mathcal{C}}}$ in the FKM for $U=3.0$ and increasing values of $t$ from $t=0.2$ to $t=2.0$ ($t$ is evident from the length of the contour, $0\le t^{\prime }\le t$). Upper panel: real part, lower panel: imaginary part. Data obtained with $\beta =50$, a real-time discretization step $dt=0.02$ and a mesh of $N_{\tau }=200$ points on the imaginary axis (see the Supplemental Material [39] for technical details).

Figure 3

Dynamical phase diagram of the FKM. Top: real and imaginary part of the integrated double occupation ${\mathrm{\Delta }}_{\mathrm{FKM}}$ obtained by increasing (decreasing) $U$ in steps of $\mathrm{\Delta }U=0.1$ from $U=0.1$ ($U=6.0$), and using the solution at $U$ as a seed for the iterative solution of DMFT at $U+\mathrm{\Delta }U$ ($U-\mathrm{\Delta }U$). Bottom: Blue squares show the coexistence region around the first transition branch, obtained at each $t$ as described in the upper panel. For the other transition branches, we provide only lower-bound estimates for the coexistence region: In the region between red dots at the same $t$, two coexisting solutions are found by different choices in the update of the Green function at each DMFT iteration (see Supplemental Material [39]).

Figure 4

DPT in the Hubbard model. Panel (a): Real part of the integrated double occupation for a quench to $U=10$ at different $t$. To confirm the convergence of the results with the fictitious temperature, we show data for $\beta =20$ (red circles) and $\beta =50$ (blue stars). Panel (b): coexisting solutions for ${\mathrm{\Delta }}_H(U)$ at $t=0.8$ at different values of $U$. Red circles and blue stars are obtained using as an initial guess for the hybridization function in DMFT the noninteracting Green function on the Bethe lattice and the converged solution at $U=11$, respectively.