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arXiv:2512.12746

Proximity effects and a topological invariant in a Chern insulator connected to leads

Satyam Sinha, Rekha Kumari, Junaid Majeed Bhat, Abhishek Dhar, R. Shankar

cond-mat.mes-hallcond-mat.othercond-mat.stat-mechquant-phSubmitted Dec 14, 2025• Announced Dec 16, 2025
v1 · Dec 14, 2025v2 · Aug 26, 2026v3 · Aug 27, 2026
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Abstract

The observed robustly quantized Hall conductance in quantum Hall systems and Chern insulators (CI) have so far been understood in terms of the topology of isolated systems, which are not coupled to leads. It is assumed that the leads act as inert reservoirs that simply supply/absorb electrons to/from the sample. Within a model of a CI coupled to leads with a cylindrical geometry, we show that this is not true. In the proximity of the CI, the edge current leaks into the leads, with the Hall conductance quantized only if this novel proximity effect is taken into account. For a special choice of leads, we identify the conductance with a topological invariant of the system, in terms of the winding number of the phase of the reflection coefficients of the scattering states.

Comments
14 pages, 4 figures

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The observed robustly quantized Hall conductance in quantum Hall systems and Chern insulators (CI) have so far been understood in terms of the topology of isolated systems, which are not coupled to leads. It is assumed that the leads act as inert reservoirs that simply supply/absorb electrons to/from the sample. Within a model of a CI coupled to leads with a cylindrical geometry, we show that this is not true. In the proximity of the CI, the edge current leaks into the leads, with the Hall conductance quantized only if this novel proximity effect is taken into account. For a special choice of leads, we identify the conductance with a topological invariant of the system, in terms of the winding number of the phase of the reflection coefficients of the scattering states.
Integer and fractional Chern insulators exhibit a nonzero quantized anomalous Hall conductivity due to a spontaneous breaking of time reversal symmetry. To identify nontrivial topology in their time-reversal symmetric many-body spectra, we introduce many-body Euler numbers as a counterpart to many-body Chern numbers. Exemplarily, we perform calculations in a topological Hubbard model that can realize Chern and fractional Chern insulating phases. Furthermore, we lay out a classification scheme to realize different topological phases in interacting systems using symmetry indicators in analogy to topological band theory.
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We investigate the transport properties of Chern insulators following a quantum quench between topological and nontopological phases. Recent works have shown that this yields an excited state for which the Chern number is preserved under unitary evolution. However, this does not imply the preservation of other physical observables, as we stressed in our previous work. Here we provide an analysis of the Hall response following a quantum quench in an isolated system, with explicit results for the Haldane model. We show that the Hall conductance is no longer related to the Chern number in the postquench state, in agreement with previous work. We also examine the dynamics of the edge currents in finite-size systems with open boundary conditions along one direction. We show that the late-time behavior is captured by a Generalized Gibbs Ensemble, after multiple traversals of the sample. We discuss the effects of generic open boundary conditions and confinement potentials.
Chern insulators are band insulators which exhibit a gap in the bulk and gapless excitations in the edge. Detection of Chern insulators is a serious challenge in cold atoms since the Hall transport measurements are technically unrealistic for neutral atoms. By establishing a natural correspondence between the time-reversal invariant topological insulator and the quantum anomalous Hall system, we show for a class of Chern insulators that the topology can be determined by only measuring Bloch eigenstates at highly symmetric points of the Brillouin zone. Furthermore, we introduce two experimental schemes, including the spin-resolved Bloch oscillation, to carry out the measurement. These schemes are highly feasible under realistic experimental conditions. Our results may provide a powerful tool to detect topological phases in cold atoms.
Driven Floquet systems can realize topological phases with no static counterparts. This so-called anomalous Floquet topology breaks the bulk-boundary correspondence based on the Chern number. The number of edge modes in each band gap is instead determined by another integer index, a winding number, which is calculated from the time evolution operator of the bulk states within one driving period. While topological Chern numbers can be inferred from real-space measurements, it is not straightforward to obtain information about the anomalous Floquet topological phase from bulk measurements. Here we consider two-band models and show that the area enclosed during a Floquet period by an initially localized particle signals the presence of an anomalous phase when it approaches half the unit cell area. In general, we show that for dispersionless dynamics during the micromotion, the enclosed area is quantized and an exact proportionality relation exists between the area and the winding number. Direct detection of anomalous topology in real space could be realized in several quantum simulation platforms, and could be useful for systems with disorder or interactions. Building on the connection between area and winding number, we also show a way to realize arbitrarily high winding numbers.
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.