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arXiv:2608.18199

Chern insulator boundary criticality

Benjamin Moy, Eduardo Fradkin

Published Aug 18, 2026• 0 citations
arXivHTMLPDF

Abstract

We investigate signatures of chirality at Chern insulator transitions in the presence of a boundary. The transition between a trivial insulator and a Chern insulator with Chern number $C=1$ is described by a massless Dirac fermion whose parity anomaly gives a critical Hall conductivity $σ_{xy}=\frac{1}{2}\frac{e^2}{h}$. Using a Dirac mass domain wall construction, we show that the chiral edge mode delocalizes into the bulk at criticality, but the boundary fermion correlation function retains a chiral structure and acquires the scaling dimension of the bulk fermion. We compute current correlation functions and demonstrate that the anomaly of the bulk Hall response is matched by delocalized chiral modes near the boundary. Using only the residual conformal symmetry of a (2+1)d conformal field theory (CFT) in a half-space, we identify parity-odd terms in current and energy-momentum tensor correlation functions that encode these modes and determine their electromagnetic and gravitational anomaly coefficients. Our analysis therefore applies to general time-reversal breaking (2+1)d CFTs, beyond the free Dirac transition. In particular, the analysis of the gravitational anomaly is also applicable to the free Majorana CFT governing the transition between a trivial superconductor and a topological superconductor. We also extend our results to more general Chern number changing transitions and to the transition between a (3+1)d topological insulator and a trivial insulator.

DOI
10.48550/arxiv.2608.18199

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We investigate signatures of chirality at Chern insulator transitions in the presence of a boundary. The transition between a trivial insulator and a Chern insulator with Chern number C=1C=1C=1 is described by a massless Dirac fermion whose parity anomaly gives a critical Hall conductivity σxy=12e2hσ_{xy}=\frac{1}{2}\frac{e^2}{h}σxy​=21​he2​. Using a Dirac mass domain wall construction, we show that the chiral edge mode delocalizes into the bulk at criticality, but the boundary fermion correlation function retains a chiral structure and acquires the scaling dimension of the bulk fermion. We compute current correlation functions and demonstrate that the anomaly of the bulk Hall response is matched by delocalized chiral modes near the boundary. Using only the residual conformal symmetry of a (2+1)d conformal field theory (CFT) in a half-space, we identify parity-odd terms in current and energy-momentum tensor correlation functions that encode these modes and determine their electromagnetic and gravitational anomaly coefficients. Our analysis therefore applies to general time-reversal breaking (2+1)d CFTs, beyond the free Dirac transition. In particular, the analysis of the gravitational anomaly is also applicable to the free Majorana CFT governing the transition between a trivial superconductor and a topological superconductor. We also extend our results to more general Chern number changing transitions and to the transition between a (3+1)d topological insulator and a trivial insulator.
Quantum anomalies arise when symmetries of a classical theory cannot be preserved upon quantization, leading to unconventional topological responses. A prominent example is the parity anomaly of a single two-dimensional Dirac fermion, which enforces a half-quantized Hall response. Anomaly inflow mechanism allows this effect to be observed at the surfaces of three-dimensional topological insulators, however, its realization in a genuinely two-dimensional system has remained elusive. Here we report the observation of a parity-anomalous Hall response at the critical point of a quantum Hall topological phase transition in a synthetic two-dimensional system of ultracold dysprosium atoms. By coupling a continuous spatial dimension to a finite synthetic dimension encoded in atomic spin states, we engineer tunable Chern bands with C = 0 and 1. At the transition, the bulk gap closes at a single Dirac point, where we observe a robust half-quantized Hall drift despite strong non-adiabatic excitations. We show that this response originates from the global structure of the band topology, is protected by an emergent parity symmetry at criticality, and disappears when parity is explicitly broken. Our work establishes synthetic quantum systems as a powerful platform to probe quantum anomalies and their interplay with topology and non-equilibrium dynamics.
The hallmark of two-dimensional chiral topological phases is the existence of anomalous gapless modes at the spatial boundary. Yet, the manifestation of this edge anomaly within the bulk ground-state wavefunction itself remains only partially understood. In this work, we introduce a family of multipartite entanglement measures that probe chirality directly from the bulk wavefunction. Our construction involves applying different permutations between replicas of the ground state wavefunction in neighboring spatial regions, creating "permutation defects" at the boundaries between these regions. We provide general arguments for the robustness of these measures and develop a field-theoretical framework to compute them systematically. While the standard topological field theory prescription misses the chiral contribution, our method correctly identifies it as the chiral conformal field theory partition function on high-genus Riemann surfaces. This feature is a consequence of the bulk-edge correspondence, which dictates that any regularization of the theory at the permutation defects must introduce gapless boundary modes. We numerically verify our results with both free-fermion and strongly-interacting chiral topological states and find excellent agreement. Our results enable the extraction of the chiral central charge and the Hall conductance using a finite number of wavefunction replicas, making these quantities accessible to Monte-Carlo numerical techniques and noisy intermediate-scale quantum devices.
We demonstrate that the fidelity between two states with different Chern numbers $\mathcal{Z}=\mathrm{tr}{\ensuremath{\rho}{\ensuremath{\rho}}^{\ensuremath{'}}}$ serves as a generating theory for an effective conformal field theory (CFT) at the (2+0)(2+0)(2+0)-dimensional temporal interface. $\ensuremath{\rho}$ can be chosen to be a pure trivial insulator, and ${\ensuremath{\rho}}^{\ensuremath{'}}$ can be taken as a pure or dephased Chern insulator density matrix. More specifically, we obtain the following results: (1) Through evaluation of the effective central charge, stiffness, and correlation function (the ``strange correlators''), we demonstrate that the fidelity between a trivial insulator and an insulator with Chern number C=1C=1C=1 maps to a CFT with effective central charge ceff=1{c}_{\mathrm{eff}}=1ceff​=1, while the fidelity between two Chern insulators with Chern numbers $C=\ifmmode\pm\else\textpm\fi{}1$ maps to a CFT with ceff=2{c}_{\mathrm{eff}}=2ceff​=2. (2) The density matrix of the Chern insulator becomes a quantum spin Hall insulator in the doubled Hilbert space, and the dephasing acts as an interaction between the two spin species. (3) In the limit of infinite dephasing the Chern insulator becomes a superconductor in the doubled Hilbert space, featuring the ``strong-weak'' U(1) spontaneous symmetry breaking. An analysis based on the Laughlin wave function and previous studies of the projected wave function of the quantum spin Hall insulator suggest this is a power-law superconductor. (4) With increasing strength of dephasing, the amplitude of the single-particle strange correlator is suppressed, while the Cooper pair strange correlator is enhanced, consistent with the trend of emerging superconductivity.
Recently, it has been shown how topological phases of matter with crystalline symmetry and U(1)U(1)U(1) charge conservation can be partially characterized by a set of many-body invariants, the discrete shift So{\mathcal{S}}_{\mathrm{o}}So​ and electric polarization ${\stackrel{P\vec}{\mathcal{P}}}_{\mathrm{o}}$, where o\mathrm{o}o labels a high-symmetry point. Crucially, these can be defined even with nonzero Chern number and/or magnetic field. One manifestation of these invariants is through quantized fractional contributions to the charge in the vicinity of a lattice disclination or dislocation. In this paper, we show that these invariants can also be extracted from the length and corner dependence of the total charge (mod 1) on the boundary of the system. We provide a general formula in terms of So{\mathcal{S}}_{\mathrm{o}}So​ and ${\stackrel{P\vec}{\mathcal{P}}}_{\mathrm{o}}$ for the total charge of any subregion of the system which can include full boundaries or bulk lattice defects, unifying boundary, corner, disclination, and dislocation charge responses into a single general theory. These results hold for Chern insulators, despite their gapless chiral edge modes, and for which an unambiguous definition of an intrinsically two-dimensional electric polarization has been unclear until recently. We also discuss how our theory can fully characterize the topological response of quadrupole insulators.
The presence of a boundary enriches the nature of quantum phase transitions. However, the boundary critical phenomena in topological superconductors remain underexplored so far. Here, we investigate the boundary criticality in a two-dimensional correlated time-reversal-invariant topological superconductor tuned through a quantum phase transition into a trivial time-reversal-breaking superconductor. Using sign-problem-free determinant quantum Monte Carlo simulations, we chart the quantum phase diagram and reveal the boundary criticalities encompassing ordinary, special, and extraordinary transitions. Additionally, using renormalization group analysis, we compute the boundary critical exponent up to two loops. Remarkably, the simulations and two-loop renormalization group calculations consistently demonstrate that the presence of the boundary Majorana fermion at the special transition gives rise to a new type of boundary Gross-Neveu-Yukawa fixed point. We conclude with a discussion of possible experimental realizations in iron-based superconductors.
The interplay between the curved spacetimes of general relativity and quantum mechanical systems is an active field of research. However, analysis of relativistic gravitation on extended quantum systems remains understudied. To this end, we study here the effects of a general relativistic curved spacetime on the topological phases of the Su-Schrieffer-Heeger model and Kitaev superconducting wire. We find that the topological states remain robust and well localized. In the topological insulator we find that the energy level of the topological state becomes shifted away from zero according to the gravitational redshift, breaking the system's chiral symmetry. In contrast, the Majorana zero mode of the topological superconductor remains at zero energy. Furthermore, within the topological superconductor, we identify the possibility of a gravitationally induced topological phase transition leading to the formation of a domain wall, shifting one of the boundary Majorana zero modes into the bulk.