Topology and Edge Modes in Quantum Critical Chains

We show that topology can protect exponentially localized, zero energy edge modes at critical points between one-dimensional symmetry-protected topological phases. This is possible even without gapped degrees of freedom in the bulk—in contrast to recent work on edge modes in gapless chains. We present an intuitive picture for the existence of these edge modes in the case of noninteracting spinless fermions with time-reversal symmetry (BDI class of the tenfold way). The stability of this phenomenon relies on a topological invariant defined in terms of a complex function, counting its zeros and poles inside the unit circle. This invariant can prevent two models described by the same conformal field theory (CFT) from being smoothly connected. A full classification of critical phases in the noninteracting BDI class is obtained: Each phase is labeled by the central charge of the CFT, $c\in \frac{1}{2}\mathbb{N}$, and the topological invariant, $\omega \in \mathbb{Z}$. Moreover, $c$ is determined by the difference in the number of edge modes between the phases neighboring the transition. Numerical simulations show that the topological edge modes of critical chains can be stable in the presence of interactions and disorder.

Figure 1

Representation of the critical Hamiltonian $H_1+H_2$ with its edge mode [each fermionic site is decomposed into Majorana modes: $\gamma$ (blue) and $\tilde{\gamma }$ (red); a bond signifies a term in the Hamiltonian]. Also shown are the gapped Hamiltonians $H_{\alpha }$ ($\alpha =0$ is trivial, and $\alpha =1$ is the Kitaev chain). $H_0+H_1$ is the standard critical Majorana chain. The associated complex function $f(z)$ [Eq. (3)] and zeros in the complex plane are shown.

Figure 2

The middle figure shows the zeros of $f(z)$. The zero $z_0$ within the disk (blue) corresponds to an edge mode (per edge) with localization length $\xi =-1/\ln |z_0|$. Each zero on the unit circle (red) implies a massless Majorana field in the low-energy limit ($c=\frac{1}{2}$).

Figure 3

Phase diagram illustrating how critical points with the same CFT description but different topological invariants cannot be connected: Interpolating $H_0+H_1$ and $H_1+H_2$ induces a point where the dynamical critical exponent $z_{\mathrm{dyn}}$ changes discontinuously.

Figure 4

Phase transition at strong disorder: (a) Entanglement scaling (averaging $10^5$ states) suggests an infinite randomness fixed point with $c_{\mathrm{eff}}=\ln \sqrt{2}$ (black lines guide the eye; gray is the clean case), and (b) the distribution of edge mode localization length over disorder realizations (inset: edge mode for one realization).

Figure 5

Finite-size scaling for the interacting Hamiltonian (4) with open boundaries and $U=0.3$: (a) The bulk is described by the $c=\frac{1}{2}$ Majorana CFT (black line guides the eye), and (b) energy splitting between fermionic parity sectors is exponentially small, $\mathrm{\Delta }E\sim e^{-L/\xi }$ with $\xi \approx 2.42$. The two ground states are related by an edge mode.