Topological insulator realization induced by fermionic interaction through BF mediators

Abstract: This paper demonstrates that ordinary fermions, interacting via a BF mediator, form a renormalized structure with non-trivial topological properties. We explore the analogy between the renormalized fermion and a subset of real three-dimensional topological insulators (TIs) in the vicinity of their single Dirac cones. Using the typical magnitude of TI lattice and gaps, we constrain model parameters. The topological structure induced by radiative corrections enters a class of modified Dirac equations, ensuring the existence of helical gapless near-boundary modes. We derive an effective potential for inter-quasi-particle interactions that incorporates finite-size effects in the axial direction, revealing signatures of spin-orbit coupling and time-reversal invariance. Remarkably, particles with different spins do not interact via the characteristic spin-orbit coupling, and a non-central spin-dependent force arises. We also address the behavior of the system in the phase transition associated with the thin-film limit.

• The paper shows that one-loop corrections from a BF-mediated interaction generate a momentum-dependent fermion mass whose sign reversal meets the Z₂ topological-insulator criterion for coupling strengths from 3×10⁻⁴ to 83.3 eV⁻¹.
• The model uses auxiliary fields and transverse source constraints to define consistent propagators, while a physical lattice cutoff makes its predictions an effective, non-renormalizable description of materials such as Bi₂Se₃.
• Finite-size analysis finds that thin films become topologically trivial as dimensional reduction changes the self-energy, consistent with surface-state hybridization and thickness-driven transitions observed in topological-insulator films.

Overview and motivation

The paper by de Gracia, da Rocha, Nogueira, and de Barcelos constructs a quantum field-theoretic model in which ordinary Dirac fermions interact through a four-dimensional BF-type mediator — an antisymmetric rank-two bosonic field $\mathcal{B}_{\mu\nu}$ coupled to a vector gauge field $A_\mu$ via $$\varepsilon_{\mu\nu\alpha\beta}\,\partial^\alpha A^\beta\,\mathcal{B}^{\mu\nu}$$ (2608.16660). The fermionic sources are a vector current $J^\mu = e\bar\Psi\gamma^\mu\Psi$ and a spin pseudo-tensor $$T_{\mu\nu} = \frac{g}{4}\bar\Psi\gamma_5[\gamma^\nu,\gamma^\mu]\Psi$$. The central claim is that the radiatively corrected (renormalized) fermion acquires a non-trivial topological character, making it a continuum analogue of a three-dimensional topological insulator (TI) near its single Dirac cone, in the class of Bi$_2$Se$_3$ and Bi$_2$Te$_3$ materials. The vector field plays the role of an effective electromagnetic channel while the pseudo-tensor encodes a spin channel; unlike quantum Hall phenomenology, parity and time-reversal invariance are preserved, which is the prerequisite for a TI description. The authors are explicit that the model is not perturbatively renormalizable: the coupling $g$ is dimensionful, and the cutoff $A_\mu$0 is physical, set by the inverse of a typical inter-site lattice distance ($A_\mu$1, the largest primitive vector of the quintuple layers of Bi$A_\mu$2Se$A_\mu$3). The renormalization conditions are therefore used not to absorb infinities but as on-shell prescriptions fixing physical parameters.

Auxiliary fields, constraints, and propagators

A key structural ingredient is the set of auxiliary (Lagrange-multiplier) fields $A_\mu$4, $A_\mu$5, and $A_\mu$6, which convert the constrained first-order system into a well-defined second-class structure with proper Dirac brackets, and which project all physical amplitudes onto the transverse sector of the sources. The tensor source decomposes as $A_\mu$7; a unit-Jacobian field redefinition removes the longitudinal part of $A_\mu$8, so that only the transverse sector contributes at all orders. This is not a gauge-fixing of the original theory but the definition of a new interacting theory whose constraints exclude longitudinal source sectors from all observables — an important subtlety, since $A_\mu$9-type gauges would leave uncanceled longitudinal propagator contributions.

Finite size in the axial direction is implemented by compactifying $$\varepsilon_{\mu\nu\alpha\beta}\,\partial^\alpha A^\beta\,\mathcal{B}^{\mu\nu}$$0 with periodic boundary conditions, converting the $$\varepsilon_{\mu\nu\alpha\beta}\,\partial^\alpha A^\beta\,\mathcal{B}^{\mu\nu}$$1 integral into a discrete sum $$\varepsilon_{\mu\nu\alpha\beta}\,\partial^\alpha A^\beta\,\mathcal{B}^{\mu\nu}$$2. The resulting mixed propagator is manifestly transverse, and its static limit contains the factor $$\varepsilon_{\mu\nu\alpha\beta}\,\partial^\alpha A^\beta\,\mathcal{B}^{\mu\nu}$$3, which controls all finite-width physics. The thin-film limit $$\varepsilon_{\mu\nu\alpha\beta}\,\partial^\alpha A^\beta\,\mathcal{B}^{\mu\nu}$$4 is treated via the small-argument expansion of $$\varepsilon_{\mu\nu\alpha\beta}\,\partial^\alpha A^\beta\,\mathcal{B}^{\mu\nu}$$5, with delta-like contact terms discarded as incompatible with the long-distance continuum picture.

Radiatively induced topological structure

The one-loop fermion self-energy, computed in dimensional regularization and re-expressed via a physical cutoff through the Veltman–Passarino procedure, generates a momentum-dependent mass gap

$$\varepsilon_{\mu\nu\alpha\beta}\,\partial^\alpha A^\beta\,\mathcal{B}^{\mu\nu}$$6

Applying the topological Hamiltonian criterion of Wang, Yan, and Zhang — the $$\varepsilon_{\mu\nu\alpha\beta}\,\partial^\alpha A^\beta\,\mathcal{B}^{\mu\nu}$$7 invariant being the product of signs of the momentum-dependent gaps at time-reversal-invariant momenta, equivalently $$\varepsilon_{\mu\nu\alpha\beta}\,\partial^\alpha A^\beta\,\mathcal{B}^{\mu\nu}$$8 — the system is topological when $$\varepsilon_{\mu\nu\alpha\beta}\,\partial^\alpha A^\beta\,\mathcal{B}^{\mu\nu}$$9, using a gap of order $J^\mu = e\bar\Psi\gamma^\mu\Psi$0 eV and $J^\mu = e\bar\Psi\gamma^\mu\Psi$1. The perturbative regime, requiring radiative corrections smaller than $J^\mu = e\bar\Psi\gamma^\mu\Psi$2, imposes $J^\mu = e\bar\Psi\gamma^\mu\Psi$3, yielding a wide admissible window. The authors quantify the sensitivity to the cutoff choice: switching from a Bi$J^\mu = e\bar\Psi\gamma^\mu\Psi$4Se$J^\mu = e\bar\Psi\gamma^\mu\Psi$5 to a Bi$J^\mu = e\bar\Psi\gamma^\mu\Psi$6Te$J^\mu = e\bar\Psi\gamma^\mu\Psi$7 lattice scale changes the logarithmic factor from 44.90 to 44.61, a mere $J^\mu = e\bar\Psi\gamma^\mu\Psi$8 effect, so the topological lower bound is robust against material-dependent cutoff variation. The self-energy develops an imaginary part above the threshold $J^\mu = e\bar\Psi\gamma^\mu\Psi$9 (finite quasiparticle lifetime), but vanishes near the mass shell, so the quadratic expansion around the renormalized gap is well-defined — the standard low-energy truncation used for TIs near a high-symmetry point.

The authors candidly note that the Lorentz-invariant loop calculation is a technical simplification: real TIs have quasiparticle velocities $$T_{\mu\nu} = \frac{g}{4}\bar\Psi\gamma_5[\gamma^\nu,\gamma^\mu]\Psi$$0, and Lorentz/rotational symmetry-breaking corrections will modify dispersion parameters quantitatively. The argument that the topological invariant is preserved under smooth deformations as long as the bulk gap does not close is asserted rather than proven for the specific anisotropic case, resting on the general continuity of the topological Hamiltonian classification. The induced structure belongs to the class of modified Dirac theories (with a $$T_{\mu\nu} = \frac{g}{4}\bar\Psi\gamma_5[\gamma^\nu,\gamma^\mu]\Psi$$1 term in the gap, coefficient $$T_{\mu\nu} = \frac{g}{4}\bar\Psi\gamma_5[\gamma^\nu,\gamma^\mu]\Psi$$2) that guarantee gapless helical boundary modes with spin–momentum locking.

Thin-film dimensional reduction and loss of topology

In the $$T_{\mu\nu} = \frac{g}{4}\bar\Psi\gamma_5[\gamma^\nu,\gamma^\mu]\Psi$$3 limit the $$T_{\mu\nu} = \frac{g}{4}\bar\Psi\gamma_5[\gamma^\nu,\gamma^\mu]\Psi$$4 Kaluza–Klein mode dominates, producing an effective dimensional reduction with a self-energy scaling as $$T_{\mu\nu} = \frac{g}{4}\bar\Psi\gamma_5[\gamma^\nu,\gamma^\mu]\Psi$$5 — a square-root, not a linear-in-$$T_{\mu\nu} = \frac{g}{4}\bar\Psi\gamma_5[\gamma^\nu,\gamma^\mu]\Psi$$6, correction. This places the renormalized fermion outside the class of modified Dirac equations carrying non-trivial topology: the thin film is topologically trivial. The authors connect this to experimentally observed thickness-driven transitions in HgTe quantum wells (critical thickness $$T_{\mu\nu} = \frac{g}{4}\bar\Psi\gamma_5[\gamma^\nu,\gamma^\mu]\Psi$$7 nm) and in Bi$$T_{\mu\nu} = \frac{g}{4}\bar\Psi\gamma_5[\gamma^\nu,\gamma^\mu]\Psi$$8Te$$T_{\mu\nu} = \frac{g}{4}\bar\Psi\gamma_5[\gamma^\nu,\gamma^\mu]\Psi$$9/Bi$_2$0Se$_2$1 films of one to ten quintuple layers, where hybridization between opposite surface states destroys the topological phase. They are careful to state that the effective continuum framework cannot predict the exact critical thickness, which depends on microscopic parameters beyond its scope; a field-theoretic estimate identifies the transition with the condition that the cutoff fall below the axial energy scale $_2$2.

Inter-quasiparticle potential and spin–orbit structure

At tree level, within the Born approximation and the regime $_2$3, $_2$4, the Fourier transform of the boundary-scattering amplitude yields a long-range potential (up to contact terms)

$_2$5

exhibiting a spin–orbit coupling proportional to $_2$6. Two results stand out. First, quasiparticles with opposite spins do not interact via the characteristic spin–orbit term at all. Second, same-spin particles on the same boundary experience an attractive central interaction plus a non-central spin-dependent force that exerts opposite axial torques; rotational equilibrium is achieved when both particles reach equal momentum orientation ($_2$7 in the center-of-mass frame). The authors emphasize that this tree-level potential does not by itself derive the helical configuration, but generates constraints that can include it — a deliberately modest claim consistent with the boundary modes being properties of the one-loop-renormalized solution rather than of the tree-level potential.

Finite-width effects are incorporated through $_2$8, evaluated via a Lipschitz–Hankel identity into an image-charge-like series. This breaks rotational symmetry down to residual plane rotations. In the thin-film limit the potential is enhanced by $_2$9 and decays more slowly, and for quasi-particles at opposite boundaries separated by a planar projection $_3$0, the spin–orbit contribution cancels exactly. The authors flag this result as approximate — it relies on the $_3$1 expansion — and interpret it conservatively as evidence for the breakdown of the effective boundary-state description rather than as a definitive prediction of a physical singularity. The cancellation is consistent with the independent loop-level finding of triviality in the $_3$2 limit.

Limitations and open questions

The paper is explicit about several caveats. The theory is non-renormalizable and valid only as a continuum limit of a lattice system below a physical cutoff; all quantitative bounds on $_3$3 inherit the chosen values of $_3$4 and the gap. The Lorentz-invariant treatment is a simplification whose quantitative deviations at physical velocities $_3$5 are not computed. The thin-film spin–orbit cancellation is established only within the long-distance approximation and may not survive in the exact theory. The claim that finite-size corrections do not alter the topological number for large $_3$6 is argued qualitatively — the $_3$7-dependent piece of the self-energy is finite and functionally distinct — but a threshold $_3$8 at which such contributions could change the invariant is not determined. Material-dependent critical thicknesses remain outside the effective description. Open questions include the computation of the topological invariant including full finite-size self-energy corrections, the extension beyond the Born approximation for the boundary potential, and whether the spin–orbit cancellation at $_3$9 persists beyond the $_2$0-expansion regime.

Conclusion

The paper demonstrates that a BF-mediated interaction between ordinary fermions, supplemented by constraint-imposing auxiliary fields, generates through one-loop corrections a renormalized Dirac structure with a momentum-dependent gap satisfying the $_2$1 topological criterion, within the perturbative window $_2$2 for Bi$_2$3Se$_2$4-inspired parameters. The resulting boundary phenomenology — a spin–orbit interparticle potential in which opposite-spin quasiparticles decouple from the characteristic coupling, with equilibrium favoring aligned momenta — is compatible with helical spin–momentum locking, and the thin-film limit consistently exhibits topological trivialization through two complementary mechanisms, in qualitative agreement with observed thickness-driven transitions. The analysis remains an effective, cutoff-dependent continuum description, and its quantitative predictions are correspondingly bounded by that framework.