Entanglement spectrum of gapless topological phases: a case study with topological superconductors

TL;DR

This paper addresses whether entanglement-based diagnostics extend to gapless topological phases, focusing on 2D nodal topological superconductors as free-fermion models. It leverages the correlation spectrum (CS), the single-particle ES for free fermions, to detect signatures of both edge modes and bulk nodal states on a cylinder, showing a $1/2$ CS mode corresponding to Majorana edge modes in the dx^2−y^2 case and a cut-dependent, but edge/bulk–aware, CS in the dxy case. A key finding is that the spectral-flattening argument that underpins ES–edge-mode correspondence in gapped systems breaks down in gapless bulk, with CS degeneracy exhibiting power-law scaling at criticality; nonetheless, trace-index jumps reveal half-quantized signatures associated with Majorana edges. Overall, the work generalizes the Li–Haldane entanglement spectrum framework to gapless topological phases, providing a practical CS-based diagnostic for edge-bulk interplay and motivating future work on interactions.

Abstract

Using bulk gapless topological superconductors in both 1d and 2d as free fermion model examples, we demonstrate the power of subsystem correlation spectrum (the spectrum of correlation matrix), or equivalently the entanglement spectrum for the case of free fermions, in characterizing the topology of the non-trivial ground state. For the systems considered, we show that signatures of the lowenergy spectrum, including both the edge modes and the bulk modes, appear in the correlation spectrum, albeit with different behaviors. This work generalizes the 2d Li-Haldane entanglement spectrum characterization of topological edge states to 2d topological systems with gapless bulk.

Figures (5)

• Figure 1: (a) Energy spectrum under open boundary condition (OBC) and (b) correlation spectrum for gapped $\alpha$-chain at $g=0.5$, with winding number $\omega_1=1$; (c) energy spectrum under OBC and (d) correlation spectrum for gapped $\alpha$-chain at $g=1.5$, with winding number $\omega_2=2$. The length of the chain is taken to be $100$ unit cells and correlation spectra are calculated for a single virtual cut on the open chain dividing it into two subsystems.
• Figure 2: (a) Energy spectrum and (b) half-chain correlation spectrum under OBC for the critical point corresponding to $g=1$.
• Figure 3: Scaling of degeneracy splitting $\delta$ between the $1/2$ modes of the correlation spectrum with respect to subsystem size $L_{A}$ for two cuts on an open chain for (a) the gapped $\alpha$-chain with $g=0.5$ and (b) the critical $\alpha$-chain with $g=1.0$.
• Figure 4: (a) Energy spectrum, (b) CS and (c) trace index of the nodal $d_{x^2-y^2}$-wave TSC. (d)(e)(f) show the case with gapped nodes respectively. $L_x=160$, $L_y=300$, PBC along $y$-direction and OBC along $x$-direction with one virtual cut on the cylinder for correlation spectrum and trace index between unit cells 76 and 77. Parameters: $t=1$, $\mu=-4t$, $\lambda=0.5t$, $h=2t$, $\Delta=t$ for bulk gapless case. To gap out bulk gapless nodes, we use a $C_4$-breaking term $d(\tau_y\otimes\sigma_y)$, with $d=1.5$ and $\tau_y$ being the Pauli matrix in particle-hole space, reducing it to class $D$ of the ten-fold way which has a Chern number classification.
• Figure 5: (a) Energy spectrum, (b) CS and (c) trace index of the nodal $d_{xy}$-wave TSC. (d)(d)(f) show the case with gapped nodes respectively. $L_x=160$, $L_y=300$, PBC along $y$-direction and OBC along $x$-direction with one virtual cut on the cylinder for correlation spectrum and trace index between unit cells 76 and 77. Parameters: $t=1$, $\mu=-4t$, $\lambda=0.5t$, $h=2t$, $\Delta=t$ for bulk gapless case. To gap out bulk gapless nodes, we use a $C_4$-breaking term $d(\tau_y\otimes\sigma_y)$, with $d=1$ and $\tau_y$ being the Pauli matrix in particle-hole space, reducing it to class $D$ of the ten-fold way which has a Chern number classification.