Dynamically Induced Exceptional Phases in Quenched Interacting Semimetals

We report on the dynamical formation of exceptional degeneracies in basic correlation functions of nonintegrable one- and two-dimensional systems quenched to the vicinity of a critical point. Remarkably, fine-tuned semimetallic points in the phase diagram of the considered systems are thereby promoted to topologically robust non-Hermitian (NH) nodal phases emerging in the coherent time evolution of a dynamically equilibrating system. Using nonequilibrium Green’s function methods within the conserving second Born approximation, we predict observable signatures of these NH nodal phases both in equilibrated spectral functions and in the nonequilibrium dynamics of momentum distribution functions.

Figure 1

Illustration of the quench protocol and the complex energy bands of the effective NH Hamiltonian [cf. Eq. (1)] around the Fermi energy. As inital state, we consider the ground state of a band insulator at half filling (left). The Hamiltonian system then undergoes a parameter quench in which the band gap is closed thus leading to a critical nonequilibrium state (middle), and thereafter an interaction between the particles is switched on so as to make the Hamiltonian nonintegrable. During the subsequent coherent postquench dynamics, the excitations created during the quench dynamically equilibrate. In this process, a nodal NH phase featuring exceptional degeneracies (hallmarked by a degeneracy of both the real and the imaginary part) emerges (right).

Figure 2

(a) Complex band structure of the effective Hamiltonian $H_{\mathrm{NH}}(k,\omega =0)$ [cf. Eq. (1)] at the Fermi energy in the long-time limit. (b) Visualization of the phase $\varphi =\arg (\sqrt{{\mathbf{d}}_R^2-{\mathbf{d}}_I^2+2i{\mathbf{d}}_R·{\mathbf{d}}_I})$ of the complex energy gap function $\mathrm{\Delta }E$. Contours at which conditions of Eqs. (4) and (5) are satisfied are indicated in violet and black, respectively. EPs (red dots) occur at the intersections of those contours. All EPs are connected pairwise by Fermi arcs along which Eq. (5) is satisfied and where ${\mathbf{d}}_R^2-{\mathbf{d}}_I^2<0$. When crossing a Fermi arc, the phase $\varphi$ jumps by $\pi$. Parameters in all plots are $J(t_i)=-2.0,U_a=0.0\to J(t_f)=1.0,U_a=1.5$ over the quench; all other parameters constant in time at $U_b=0.0$, $d=0.5$, $\tau =1.0$. System size $L=2\times 250$ (i.e., 250 particles and unit cells). Inset of (a): stability of the NH exceptional phase against opening a small gap in $H_f^0(k)$ by detuning $J(t_f)$ from its critical value 1.0, as determined by the length of the Fermi arc ${\mathrm{\Delta }}_{\mathrm{EP}}$ (system size $L=2\times 80$).

Figure 3

(a) Spectral function $A(k,\omega )=-(1/\pi )\mathrm{Im}\{\mathrm{Tr}[G^R(k,\omega )]\}$ in 1D. Besides energy bands with moderate intensity, the Fermi arc characteristic of the exceptional NH phase is visible as a strip with high intensity around momentum $k=\pi$ which slightly broadens close to the EPs. (b) Decay times $\tau (k)$ for the decay of the nonequilibrium occupations $n(k,t)$. The decay times (black crosses) exhibit fluctuations, but follow a characteristic local mean (solid line): Within the Fermi arc (violet part, violet dots represent the EPs) the decay times are striking higher than outside of it (orange part). Insets: real-time evolution of the decay of $n(k,t)$, distinguishing the Fermi arc region from the outer region. Parameters in all plots identical to Fig. 2).

Figure 4

(a) Phase of the complex energy gap function analogous to Fig. 2 for $t_q=t_I=0.5$ and $U_a=2$. The violet (black) contours represent the condition Eq. (4) [Eq. (5)], while the red dots indicate the EPs. (b) Position of the EPs as function of the quench time $t_q$. For $U_a=1$ we used the GKBA for an $N_k=96\times 96$ sampling of the Brilloin zone, while for $U_a=2$ the full KBEs were solved on a $N_k=40\times 40$ grid. $U_b=0$ is fixed in all calculations. Inset: the spectral function $A(\mathbf{k},\omega )$ with $\mathbf{k}=(k_x,0)$.