Manifestation of a topological gapless phase in a two-dimensional chiral symmetric system through Loschmidt echo

Unlike the edge state of a topological insulator where its energy level lives in the bulk energy gap, the edge state of a topological semimetal hides in the bulk spectrum and is difficult to be identified by the energy. We investigate the sensitivity of bulk and edge states of the gapless phase for a topological semimetal to the disordered perturbation via a concrete two-dimensional chiral symmetric lattice model. The topological gapless phase is characterized by two opposite vortices in the momentum space and nonzero winding numbers, which correspond to the edge flatband when the open boundary condition is applied. For this system, numerical results reveal that a distinguishing feature is that the robustness of the edge states against weak disorder and the flatband edge modes remain locked at zero energy in the presence of weak chiral-symmetry-preserving disorder. We employ the Loschmidt echo (LE) for both bulk and edge states to study the dynamic effect of disordered perturbation. We show that, for an initial bulk state, the LE decays exponentially, whereas it converges to a constant for an initial edge state in the presence of weak disorder. Furthermore, the convergent LE can be utilized to identify the positions of vortices as well as the phase diagram. We discuss the realization of such dynamic investigations in a topological photonic system.

Figure 1

(a) Schematic of the 2D tight-binding model with Hamiltonian Eq. (4), which is essentially a bipartite $M\times N$ honeycomb lattice. The solid lines and dashed lines represent hopping terms with strengths $\mu$ and 1, respectively. (b) Phase diagram of the lattice system with parameter $\mu$. The orange lines indicate the phase boundary, which separate the topologically trivial gapped phases (gray) and topological gapless phases (blue). The system with the parameter at the boundary (orange lines) is a topologically trivial gapless phase. Points (a)–(f) represent the systems in each phase, which are extensively investigated in Fig. 3.

Figure 2

(a) Schematic of the geometry of the system with the cylindrical boundary condition. The locations of the initial local states at the edge (blue dot) and bulk (orange dot) of the 2D lattice system are indicated. (b) Schematic of the modified Su-Schrieffer-Heeger (SSH) chain $H_k$ represented in Eq. (18). The arrows and dashed lines represent complex and real hopping terms $(\mu +e^{ik})$ and 1, respectively. The edge modes of a set of modified SSH chains form the flatband edge modes as bound states located at two edges of the cylinder when the system is in the blue region of the phase diagram in Fig. 1.

Figure 3

Kaleidoscope of quantum phases. (a1)–(f1) Plots of energy spectra from Eq. (10) at six typical points (a–f) marked in the phase diagram in Fig. 1. There, the band structure exhibits a bulk gap in (a1); a single degeneracy point with parabolic dispersion in (b1) and (f1); two degeneracy points with linear dispersion in (c1) and (e1); and two degeneracy lines in (d1). (a2)–(f2) Plots of the Bloch vector field defined in Eq. (9) in the momentum space for six cases corresponding to (a1)–(f1). There are two vortices in (c2) and (e2) with opposite winding numbers $\pm 1$. As $\mu$ increases or decreases, two vortices get close and merge into a single point in (b2) or (f2), and disappear in (a2). (a3)–(f3) Plots of the spectra of a set of modified SSH chains [Eq. (17)] in open boundary condition with $N=40$ for six cases corresponding to (a1)–(f1). It indicates that the existence of pair of vortices links to a flatband of the square lattice. (a4)–(f4) Plots of the LE obtained by numerical simulations from Eq. (1) at $t=3000J^{-1}$ and analytical expressions from Eq. (28) in which the initial state is taken as the site state at the edge. Here, $J$ is the scale of the Hamiltonian, and we take $J=1.R$ is the disorder strength, and $m$ denotes the measurement index. The average value of the LEs are plotted on the right of each panel. The size of the system is $M\times N=80\times 80$.

Figure 4

Linear and logarithmic scale plots of eigenenergy around zero for $H,H_D$, and $H^{\prime }$ with the cylindrical boundary condition. $n$ denotes the sorting index. The parameters are $\mu =1$ for (a) and (b); $\mu =1.8$ for (c) and (d); $R=0.1$ for (a) and (c); $R=0.3$ for (b) and (d). It indicates that the number of zero modes is dependent on $\mu$, and the zero modes remain unchanged in the presence of the chiral-symmetry-preserving disordered perturbation whereas they do not survive under the chiral-symmetry-breaking disordered perturbation. The results are obtained by numerical diagonalization for the system with $M\times N=60\times 60$.

Figure 5

Numerical results of inverse participation ratio (IPR) for the gapless phase with $\mu =1$ corresponding to Fig. 4. (a) System without disorder. (b) System with chiral-symmetry-preserving disorder $R=0.1$. The size of the system is $M\times N=60\times 60$.

Figure 6

(a) Profiles of the initial states of numerical simulations for the LEs. The edge state (blue) is taken as the eigenstate of the system in the cylindrical boundary condition without disorder, and the bulk state (orange) is taken as the 2D Gaussian wave packet. (b) Plots of numerical simulations for the LEs as the functions of time. The initial states are taken as the edge state and bulk state shown in (a). It can be seen that the LEs have diametrically opposite behaviors for the initial bulk and edge states. The time $t$ is in units of $J^{-1}$, where $J$ is the scale of the Hamiltonian, and we take $J=1$. The size of the system is $M\times N=60\times 60$, and the disorder strength is $R=0.1$.

Figure 7

Comparison of the average convergent LEs and analytical expression from Eq. (28) as the functions of $\mu$. The initial state is a edge site state, and the final time is $t=1000J^{-1}$, where $J$ is the scale of the Hamiltonian, and we take $J=1$. The size of the system is $M\times N=60\times 60$, and the disorder strength is $R=0.1$. It is found that the two results agree with each other well. This means that the measurement of the LE can identify the phase diagram.