Topological blocking in quantum quench dynamics

We study the nonequilibrium dynamics of quenching through a quantum critical point in topological systems, focusing on one of their defining features: ground-state degeneracies and associated topological sectors. We present the notion of “topological blocking,” experienced by the dynamics due to a mismatch in degeneracies between two phases, and we argue that the dynamic evolution of the quench depends strongly on the topological sector being probed. We demonstrate this interplay between quench and topology in models stemming from two extensively studied systems, the transverse Ising chain and the Kitaev honeycomb model. Through nonlocal maps of each of these systems, we effectively study spinless fermionic $p$-wave paired topological superconductors. Confining the systems to ring and toroidal geometries, respectively, enables us to cleanly address degeneracies, subtle issues of fermion occupation and parity, and mismatches between topological sectors. We show that various features of the quench, which are related to Kibble-Zurek physics, are sensitive to the topological sector being probed, in particular, the overlap between the time-evolved initial ground state and an appropriate low-energy state of the final Hamiltonian. While most of our study is confined to translationally invariant systems, where momentum is a convenient quantum number, we briefly consider the effect of disorder and illustrate how this can influence the quench in a qualitatively different way depending on the topological sector considered.

Figure 1

An example of topological blocking in which the quench goes from a doubly degenerate gapped phase to a nondegenerate gapped phase, as happens, for instance, in the one-dimensional spinless $p$-wave superconductor. Upon closing the gap at the critical point, one of the degenerate states in the initial phase is lifted into the continuum of excited states in the final phase. This time-evolved initial state has no overlap with the final ground state.

Figure 2

Typical quench data for the one-dimensional spinless $p$-wave superconductor on an $N=60$ site ring show the overlap of the time-evolved ground state of the initial Hamiltonian in one topological sector with the lowest-energy state of the final Hamiltonian within that sector. (This is not always the global ground state of the final Hamiltonian.) As a result of the topological blocking, the odd-fermion sector with periodic boundary conditions has a higher final overlap than the even-fermion sector. See Sec. 2 for details.

Figure 3

Schematic of the spectrum of the transverse Ising ring as a function of $h-J$. In the ferromagnetic phase, the ground state is doubly degenerate in the thermodynamic limit, and the excitation spectrum consists of bands of states with both even and odd fermion numbers. These states are created from the two ground states using pairs of ${\gamma }^{†}$ operators. In particular, there are no energy levels with an odd number of ${\gamma }^{†}$ excitations over one of the ground states. We explicitly indicate these levels as the parity-blocked regions. In the paramagnetic phase there is a unique ground state with an even number of fermions. The lowest excited band consists of odd-fermion-number states which are, however, not created by single ${\gamma }^{†}$ from the ground state. Further bands are created from ground state and the lowest band using pairs of ${\gamma }^{†}$ operators. The purple dashed line indicates that, in the adiabatic limit, the odd sectoral ground state of the ferromagnetic phase flows to the lowest-energy state in the paramagnetic phase.

Figure 4

Overlap vs time for four two-level systems corresponding to $k\sqrt{JT}=0.2$, $0.4$, $0.6$. and $0.8$, with $h\left(t\right)=2Jt/T$ and $JT=100$. We have set $J=1$.

Figure 5

(a) Typical shift of probability amplitude in a Landau-Zener transition for states associated with generic $(k,-k)$ pairs. (b) The level crossing for the occupied $k=0$ state in the odd-fermion sector is at the heart of topological blocking; lack of coupling with the unoccupied mode and parity constraints force the $k=0$ state to go into the postquench excited state.

Figure 6

The honeycomb spin model. The $K^{\alpha }$ are directional spin exchange terms which appear in the Hamiltonian (see Section 3b). On a torus, we identify opposite sides of the diamond shape. Symmetries in the model are made by making closed product loops of overlapping $K$-terms. The plaquette operators $W$ are the simplest symmetries and exist on the surface of the torus. The homologically non-trivial symmetries (of which we only indicate $L_x$) are made with overlapping products of $K^{\alpha }$ that loop around the torus.

Figure 7

Typical overlap quench data for the Kitaev spin model. In this example we examined a torus of $N_x=N_y=100$ with a fixed $J_x=J_y=1$, $\kappa =0.25$ and time-dependent $J_z=3-(2t/T)$ with $T=80$. The blocked sector ({++}) has a generically higher end overlap. Small differences between the other sectors are due to finite system size.

Figure 8

Schematic of the Kitaev model phase diagram. The $A$ regions correspond to toric code phases, while the $B$ region corresponds to the non-Abelian Ising phase. The dashed line indicates the quench parametrization. Note that this diagram uses Kitaev's normalization $J_x+J_y+J_z=1$. In our quench protocol, we use $J_x=J_y=1$, with $J_z$ ranging between 1 and 3, but on renormalizing, this gives a path similar to the dashed line shown in the diagram.

Figure 9

Schematic of the spectrum of the vortex-free sector of the honeycomb model as a function of $J_x+J_y-J_z$. In the toric code phase ($J_z>J_x+J_y$), the ground state is fourfold degenerate in the thermodynamic limit with all states being constructed from an even number of fermion excitations. The excitation spectrum consists of bands of states created from the four ground states using pairs of ${\gamma }^{†}$ operators. As in the transverse Ising case, there are no energy levels with an odd number of ${\gamma }^{†}$ excitations over one of the ground states. In the non-Abelian Ising phase ($J_z<J_x+J_y$), there is a threefold degenerate ground state. The parity-blocking mechanism means that the fully periodic sector is gapped away from the other three states and that the lowest-lying states in this sector form part of a band. Further bands are created from the ground states and this lowest band using pairs of ${\gamma }^{†}$ operators. The purple dashed line indicates that, in the adiabatic limit, the ({++}) sectoral ground state in the toric code phase flows into this lowest band but does not necessarily flow to the lowest-energy state in the ({++}) sector of the Ising phase (the lowest state in the band usually occurs at a different momentum from the initial state, and our quench conserves momentum).

Figure 10

The numerical scaling of the final overlap as a function of $N^2/T$ against the predicted behavior in Eq. (56). The system size used here is $N_x=N_y=100$ with $J_x=J_y=1$, $\kappa =0.25$, and $J_z=3-(2t/T)$.

Figure 11

Instantaneous ground-state overlap for a number of disorder configurations which are constant in time. Disorder has a negligible effect on the overlap profile of the unblocked (blue) sector. On the other hand, we see that disorder allows the blocked sector (red) to disperse within the band, inducing a clear instability in the overlap profile. The zero-disorder values are indicated here by the dashed gray line for comparison. Here, we examined a $N=60$ site chain with $J=1$ over a time $T=50$. The disorder configurations used have standard deviations $\sigma =0.1$.