Emergent topology and dynamical quantum phase transitions in two-dimensional closed quantum systems

Dynamical quantum phase transitions (DQPTs) manifested in the nonanalyticities in the temporal evolution of a closed quantum system generated by the time-independent final Hamiltonian, following a quench (or ramping) of a parameter of the Hamiltonian, is an emerging frontier of nonequilibrium quantum dynamics. We, here, introduce the notion of a dynamical topological order parameter (DTOP) that characterizes these DQPTs occurring in quenched (or ramped) two-dimensional closed quantum systems; this is quite a nontrivial generalization of the notion of DTOP introduced in Budich and Heyl [Phys. Rev. B 93, 085416 (2016)] for one-dimensional situations. This DTOP is obtained from the “gauge-invariant” Pancharatnam phase extracted from the Loschmidt overlap, i.e., the modulus of the overlap between the initially prepared state and its time-evolved counterpart reached following a temporal evolution generated by the time-independent final Hamiltonian. This generic proposal is illustrated considering DQPTs occurring in the subsequent temporal evolution following a sudden quench of the staggered mass of the topological Haldane model on a hexagonal lattice where it stays fixed to zero or unity and makes a discontinuous jump between these two values at critical times at which DQPTs occur. What is remarkable is that while the topology of the equilibrium model is characterized by the Chern number, the emergent topology associated with the DQPTs is characterized by a generalized winding number.

Figure 1

The rhomboidal Brillouin zone (BZ) that we have chosen in this paper is illustrated for a $10\times 10$ system. The Dirac points are shown in black (biggest dots) while other small dots represent the points of the BZ. The four corner points and the highest symmetry point at the center [with coordinates $(2\pi /3,0)]$ marked by squares of medium size are time-reversal invariant momentum (TRIM) points where $M_H\left(\mathbf{k}\right)=0$; while the left corner point $(0,0)$ (in green) and the central point (in cyan) are included in the BZ, other TRIM points (in red) are not. The EBZ along $k_x$ used in Eq. (6) is actually along the $k_y=0$ line from $k_x=0$ to $\frac{2\pi }{3}$ whereas the BZ along $k_y$ is the circle closed between $-\sqrt{3}{k}_x$ to $\sqrt{3}{k}_x$ (see also [70]).

Figure 2

(a) Fisher zeros for sectors $n=0$ and 1, in the complex $z$ plane following a sudden quench of the staggered mass of the Haldane model from the nontopological phase to the topological phase. The FZs form areas which cut the imaginary (real-time) axis [46, 47]. (b) At those instants of real time when the boundary points of these areas lie on the real-time axis, the derivative of the return probability $I^{\prime }\left(t\right)$ also exhibits nonanalyticities (cusp singularities) clearly marking the presence of a DQPT. The temporal evolution of the DTOP which is superimposed on the same curve makes a discontinuous jump precisely at $t_n^{*}\mathrm{s}$, thereby perfectly characterizing associated DQPTs. Note that a factor of 0.5 has been added to $I^{\prime }\left(t\right)$ to show an exact correspondence with $\nu \left(t\right)$.