Generalized Li-Haldane Correspondence in Critical Free-Fermion Systems

Abstract

Topological phenomena in quantum critical systems have recently attracted growing attention, as they go beyond the traditional paradigms of condensed matter and statistical physics. However, a general framework for identifying such nontrivial phenomena, particularly in higher-dimensional systems, remains insufficiently explored. In this Letter, we propose a universal fingerprint for detecting nontrivial topology in critical free-fermion systems protected by global on-site symmetries. Specifically, we analytically establish an exact relation between the bulk entanglement spectrum and the boundary energy spectrum at topological criticality in arbitrary dimensions, demonstrating that the degeneracy of edge modes can be extracted from the bulk entanglement spectrum. These findings, further supported by numerical simulations of lattice models, provide a universal fingerprint for identifying nontrivial topology in critical free-fermion systems.

Figures (7)

• Figure 1: The boundary energy spectrum for the Hamiltonian (a1) $(\hat{H}_{0}^{\text{1d}} + \hat{H}_{1}^{\text{1d}})/2$, (b1) $(\hat{H}_{1}^{\text{1d}} + \hat{H}_{2}^{\text{1d}})/2$, and (c1) $(\hat{H}_{2}^{\text{1d}} + \hat{H}_{3}^{\text{1d}})/2$, respectively, under open boundary conditions. (a2), (b2) and (c2) are the corresponding bulk entanglement spectrum with an equal bipartition of the chain. Insets provide magnified views of specific regions to highlight the topological degenerate edge modes (red circles). The system size is $L = 800$.
• Figure 2: The energy spectrum of the critical 2D lattice model $(a\hat{H}^{\text{2d}}_{0}+b\hat{H}^{\text{2d}}_{1}+c\hat{H}^{\text{2d}}_{2})/(a+b+c)$ for (a1) $(a,b,c) = (5,6,1)$ and (b1) $(a,b,c) = (1,6,5)$, respectively. The $x$-direction is open while the $y$-direction is periodic. Insets provide zoomed-in views near the zero-energy point. In the trivial case (a1), the color of the branches fades to gray as $k_y$ approaches $0$; the faint residual red/blue hue arises from finite-size effects. (a2) and (b2) are the corresponding bipartite bulk entanglement spectrum. The entanglement cut is taken along the $x$-direction. The color coding indicates the normalized mean real-space position along the $x$-direction of each eigenstate. The blue (red) color represents the left (right) edge modes while the light gray indicates the bulk modes. The system size is $L_{x} = 40$ and $L_{y} = 800$.
• Figure 3: (a) The averaged energy spectrum $\overline{\epsilon_{n}}$ under open boundary conditions and (b) the corresponding bulk entanglement spectrum $\overline{\xi_{n}}$ of the critical 1D model at strong disorder $\delta = 0.5$ . Insets provide magnified views of specific regions to highlight the topological degenerate edge modes (red circles) and logarithmic scalings of the half-chain entanglement entropy that suggest $c_{\text{eff}} = 1.00(1)$ for the clean case and $c_{\text{eff}} \approx \ln{2}$ for the disordered case, respectively. Results are averaged over $10^{4}$ disorder realizations for $L = 400$. (c) The many-body energy spectrum under open boundary conditions and (d) the corresponding bulk entanglement spectrum of the interacting 1D model for $L = 12$ with $U = 0.15$ . The total number of fermions is not fixed in the calculation of the energy spectrum. The inset in (c) gives an schematic for the model, where red solid lines represent the free part of the model and the black dashed lines are interaction terms, as well as the energy splitting within the degenerate edge modes due to the interaction is exponentially small, $\Delta E \equiv E_{4} - E_{1} \sim \exp(-L/l)$ with $l \approx 1.75$ . To resolve this finite-size energy splitting, relatively small system sizes are chosen in our calculations of the spectra. The entanglement scaling in the inset of (d) indicates that the model is critical and belongs to the same universality class of the free model. The bond dimension of the matrix product state used in the simulation is $\chi = 512$ for (c) and $\chi = 1024$ for (d).
• Figure S1: The boundary energy spectrum for the Hamiltonian (a1) $(2\hat{H}_{0}^{\text{1d}} + \hat{H}_{1}^{\text{1d}})/3$, (b1) $(2\hat{H}_{1}^{\text{1d}} + \hat{H}_{2}^{\text{1d}})/3$, (c1) $(2\hat{H}_{2}^{\text{1d}} + \hat{H}_{3}^{\text{1d}})/3$, and (d1) $(\hat{H}_{2}^{\text{1d}} + 2\hat{H}_{3}^{\text{1d}})/3$, respectively. (a2), (b2), (c2), and (d2) are the corresponding half-chain bulk entanglement spectrum. Insets provide magnified views of specific regions to highlight the topological degenerate edge modes (red circles). The system size is $L = 800$.
• Figure S2: The energy spectrum of the gapped 2D lattice model $(a\hat{H}^{\text{2d}}_{0}+b\hat{H}^{\text{2d}}_{1}+c\hat{H}^{\text{2d}}_{2})/(a+b+c)$ for (a1) $(a,b,c) = (4,1,1)$, (b1) $(a,b,c) = (1,4,1)$, (c1) $(a,b,c) = (1,1,4)$, respectively. The $x$-direction is open while the $y$-direction is periodic. (a2), (b2), and (c2) are the corresponding bipartite bulk entanglement spectrum. The entanglement cut is taken along the $x$-direction. The color coding indicates the normalized mean position along the $x$-direction. The blue (red) color represents the left (right) edge modes while the light gray indicates the bulk modes. The system size is $L_{x} = 40$ and $L_{y} = 800$.