Transport in extended Kitaev chain with time reversal symmetry breaking and long-range interaction

Abstract: We consider a junction consisting of an extended one-dimensional Kitaev chain which incorporates both time-reversal symmetry (TRS) breaking and long-range interaction, sandwiched between two metallic leads from two sides. In this hybrid device, we study electrical transport under voltage bias for varying strength of the TRS breaking phase. We compare the transport characteristics of long-range type Kitaev chain with that of the short-range Kitaev chain as the strength of the TRS breaking phase varies. We find that the TRS breaking modifies the density of states and localisation/delocalisation property of the eigenstates which in turn affect the transport characteristics. Moreover, we find that the impact of the TRS breaking is not identical for the long-range Kitaev chain and its short-range counterpart. Therefore, noticeable differences in the transport properties can be observed due to the interplay between the TRS breaking and the range of interaction.

• The paper demonstrates that TRS breaking transitions the SRK chain from quantized Majorana zero-bias peaks to suppressed conductance levels.
• It employs a nonequilibrium Green’s function method based on the quantum Langevin equation to evaluate differential conductance and I–V characteristics.
• In LRK chains, long-range interactions induce subgap state splitting and smooth delocalization, contrasting sharply with the SRK response under TRS breaking.

Transport in Extended Kitaev Chains: Effects of Time-Reversal Symmetry Breaking and Long-Range Interaction

Introduction

The study investigates quantum transport in a one-dimensional (1D) Kitaev chain that incorporates both long-range interactions and explicit time-reversal symmetry (TRS) breaking. Using nonequilibrium Green’s function formalism based on the quantum Langevin equation, the work examines a normal-metal–topological-superconductor–normal-metal (N-TS-N) junction, with the Kitaev chain as the superconducting element. The primary focus is the comparative analysis of differential electrical conductance (DEC) and current-voltage ($I$–$V$) characteristics for both short-range (SRK) and long-range (LRK) variants of the Kitaev model as the TRS breaking parameter is tuned across its full range.

Model Framework and Methodology

Hamiltonian Construction

The system Hamiltonian generalizes the canonical Kitaev chain to incorporate:

• Long-range hopping and pairing: Both terms decay algebraically as $l^{-\alpha}$ and $l^{-\beta}$, respectively, where $l$ is the inter-site separation.
• TRS-breaking phase: An explicit phase $\phi_l$ in the hopping terms, with site dependence, breaks time-reversal symmetry except for special values $\phi_l = n\pi$.

This extended Hamiltonian interpolates between the conventional SRK chain (large $\alpha$, $\beta$) and the LRK limit ($\alpha, \beta < 1$).

Transport Formalism

The transport analysis is carried out within the nonequilibrium steady-state regime using the LEGF (Langevin equation–Green's function) method. The chain couples at its ends to metallic reservoirs held at different chemical potentials, enabling the evaluation of steady-state charge current and DEC under finite voltage bias. The model explicitly considers open boundary conditions and attaches tunnel couplings between reservoirs and chain ends.

Differential Electrical Conductance: Majorana Modes and TRS Breaking

In the SRK chain without TRS breaking ($V$0), zero-temperature DEC displays a quantized zero-bias peak (ZBP) with height $V$1—a signature of zero-energy Majorana bound states, consistent with class BDI topological superconductivity and $V$2 classification. This ZBP persists for small $V$3; moderate TRS breaking does not immediately gap out the edge modes, as predicted in earlier theoretical analyses [Degottardi et al., Phys. Rev. B 88, 165111 (2013)].

Upon increasing $V$4 towards $V$5, the TRS breaking drives the system from class BDI to class D, reducing the topological characterization to $V$6. This transition closes the bulk gap, and ZBP disappears—no Majorana signatures persist at maximal TRS breaking. The conductance profile then exhibits non-quantized, equidistant peaks corresponding to the (now delocalized) density of states.

For the LRK chain, even with $V$7, DEC lacks a quantized ZBP regardless of tunnel strength. Long-range interactions split and push edge modes away from zero energy, yielding subgap states with a finite mass [Vodola et al., NJP 18, 015001 (2016)]. For any finite $V$8, these subgap states vanish, and the conductance within the gap region is suppressed. Thus, interplay between long-range physics and TRS breaking qualitatively alters the transport response compared to the short-range regime.

Eigenstate Localization: Inverse Participation Ratio Analysis

The study undertakes a systematic investigation of eigenstate localization via the inverse participation ratio (IPR):

• SRK regime: Bulk state IPR is largely insensitive to $V$9, but edge-state IPRs decrease with increasing TRS breaking, signifying an increased delocalization. At $l^{-\alpha}$0, a novel regime emerges: all eigenstates, bulk and edge, exhibit approximately equal IPRs, indicating homogeneous delocalization/localization characteristics.
• LRK regime: IPR values of all eigenstates decrease monotonically with $l^{-\alpha}$1—bulk and edge states become more delocalized with strong TRS breaking. The long-range nature induces collective hybridization across the chain, eliminating any even-odd effects and leading to a smooth delocalization transition as a function of $l^{-\alpha}$2.

This differential response of localization drives the contrasting transport characteristics between SRK and LRK chains under TRS breaking.

Nonlinear Transport: Current–Voltage Characteristics

The $l^{-\alpha}$3–$l^{-\alpha}$4 response (computed at near-zero temperature) reflects the interplay between localization and the bulk gap:

SRK Chain:

• At $l^{-\alpha}$5, $l^{-\alpha}$6–$l^{-\alpha}$7 shows a rapid initial increase at small bias (due to Majorana edge states), followed by a plateau reflecting the superconducting gap, and saturation beyond the reservoir bandwidth.
• Increasing $l^{-\alpha}$8 smooths and suppresses the initial jump, reduces the plateaux, and lowers the saturation current, reaching a distinctive minimum at $l^{-\alpha}$9.
• The pronounced low-voltage features and current suppression at maximal TRS breaking point to localization-driven transport blockade.

LRK Chain:

• For $l^{-\beta}$0, an initial jump in current is observed, stemming from massive subgap states, but the effect is less robust than for the SRK.
• As $l^{-\beta}$1 increases, the initial jump vanishes and the current grows linearly up to saturation, which now reaches its maximum at $l^{-\beta}$2 due to enhanced delocalization.
• Thus, strong TRS breaking in the LRK enhances transport, in stark contrast to the SRK case, illustrating the nontrivial interplay between the two forms of symmetry breaking.

Implications and Future Directions

This work offers a comprehensive comparative analysis of quantum transport in 1D topological superconductors where both the range of pairing/hopping and explicit TRS breaking are tunable. Some key theoretical and practical implications include:

• Probing symmetry and topology: $l^{-\beta}$3–$l^{-\beta}$4 and DEC measurements can discriminate not just between topological and trivial phases but also between distinct topological classes (BDI vs. D), enabling diagnosis of TRS breaking in engineered nanostructures.
• Long-range interactions: The qualitative changes in conductance spectra and localization with long-range coupling suggest that experimental platforms with tunable interaction ranges (e.g., trapped ions, Rydberg chains) can realize unique transport signatures outside the traditional short-range paradigm.
• Novel phase in SRK: The regime at $l^{-\beta}$5, where all eigenstates in the SRK chain exhibit nearly identical IPRs, constitutes a distinct quantum phase characterized by uniform localization properties; experimental signatures could be sought in engineered quantum wires.
• Derivative observables: The methodology can be extended to analyze other observables sensitive to symmetry breaking and topology such as thermal and spin currents, thermoelectric effects, and shot noise, further broadening the experimental toolbox.

A natural extension is the joint study of thermoelectric and thermal transport, as well as nonequilibrium noise characteristics, in similar geometries—particularly in systems where both disorder and interaction effects can be tuned independently.

Conclusion

The paper provides a detailed analysis of how time-reversal symmetry breaking and long-range interactions modulate the spectral and transport properties of extended Kitaev chains. The results highlight contrasting behaviors in conductance, localization, and non-linear transport between SRK and LRK regimes, especially under maximal TRS breaking. These findings are directly relevant to the ongoing search for topological superconductivity and Majorana signatures in engineered quantum systems, and open new pathways for probing and utilizing symmetry effects in designer quantum materials.

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Reference: "Transport in extended Kitaev chain with time reversal symmetry breaking and long-range interaction" (2606.30483)