Dynamical bulk-edge correspondence for degeneracy lines in parameter space

A degeneracy line in parameter space, at which the energy gap closes up, can be the boundary separating either two topological quantum phases or two conventional phases, depending on the surface containing the line. We study the topological feature of degeneracy line in parameter space via a dimerized Kitaev spin chain with staggered transverse field, which can be mapped onto the system of dimerized spinless fermions with $p$-wave superconductivity. The quantum phase boundaries are straight crossing degeneracy lines in three-dimensional (3D) parameter space. We show that the degeneracy line acts as a vortex filament associated with a vector field, which is generated from the Zak phase of the Bogoliubov–de Gennes band. We also investigate the topological invariant in Majorana fermion representation for open chains. The edge modes are not the zero mode but are still protected by the energy gap. The exact midgap states of the Majorana lattice allow us to obtain the corresponding Majorana probability distribution for the whole 3D parameter space, which can establish a scalar field to identify the topological feature of the degeneracy lines. Furthermore, when we switch on a weak tunneling between two ends of the Majorana lattice, the topological invariants of the degeneracy lines can be obtained by the pumped Majorana probability from an adiabatic passage encircling the lines. Numerical simulations of quasiadiabatic time evolution are performed in a small system. We compute the current to monitor the transport of the Majorana fermion, which exhibits an evident character of topological pumping. Our results link the bulk topological invariant to the dynamical behavior of the edge mode, extending the concept of bulk-edge correspondence.

Figure 1

Energy spectra from Eq. (9) at typical points (a) $g_e=0.5$, $g_o=0.5$, $\lambda =0.8$; (b) $g_e=0.5$, $g_o=0.5$, $\lambda =1$; (c) $g_e=0$, $g_o=0$, $\lambda =0.8$; and (d) $g_e=0$, $g_o=0$, $\lambda =1$. We see that (a) in general, there are four energy bands with symmetry about zero energy, (b) the two positive (negative) bands touch at a single point when $g_e=\pm g_o$, (c) when $g_e=g_o=0,$ two energy bands near the zero point become a flat band, and (d) the top and bottom bands touch at a single point at the flat band.

Figure 2

Schematic illustration of the degeneracy lines and the corresponding topological invariant in three-dimensional parameter space. There are four lines, labeled a, b, c, and d, which cross at the $\lambda$ axis. The topological feature of the degeneracy lines is characterized by the path integral along the loops, such as the topological invariant, which is denoted for each loop. The obtained results indicate that the topological feature of the degeneracy lines is equivalent to that of the magnetic fields produced by the current-carrying wires (a)–(d), with the directions denoted by arrows.

Figure 3

Lattice geometries for the Majorana models described in Eqs. (61) and (65), with (a) an open boundary and (b) weak tunneling $\kappa$ between two ends. Solid (open) circles indicates (anti-) Majorana modes, while solid and dashed lines indicate intrachain and end-to-end weak-hopping terms, respectively. Both (a) and (b) represent a SSH chain with staggered side couplings, forming a comblike lattice.

Figure 4

(a) Majorana dimensionless center-of-mass distribution on the parameter space, the $(g_o-g_e)\text{−}\lambda$ plane for the midgap state in Eq. (64). There are four regions filled with different colors. If we calculate the change in $\eta$ along a loop enclosing the degeneracy point $(0,\pm 1)$ in the space, then $\eta$ will be changed by $\pm 1$. On the other hand, if the loop does not contain the degeneracy point, then the change in $\eta$ is zero. It is important to note that the degeneracy point mentioned above is part of the degeneracy line, and the topological feature of the degeneracy point is consistent with the degeneracy line. (b) and (c) Plots of the spectra of the Majorana lattice with parameters along the cycle in (a) with open and weak-link boundary conditions, respectively. The spectrum in (c) indicates that the change in $\eta$ along a loop can be obtained by an adiabatic process for the midgap state marked in red.

Figure 5

(a) Schematics of four cycles around several typical points in the parameter space, the $(g_o-g_e)\text{−}\lambda$ plane. The radius $R=0.95$; point $b$ at $(0,-1)$ and point $d$ at (0,1) are two degeneracy points with opposite topological invariants, while point $c$ at $(0,0)$ and point $e$ at (0,4) are two trivial points. (b)–(e) Plots of Majorana current and the corresponding total probability transfer for the quasiadiabatic process. The results are obtained numerically for a system with $N=8$ and $\kappa =0.02$. The speed of time evolution is $\omega =5\times {10}^{-7}$. It indicates that the topological invariant can be obtained with the dynamical process.