Jackiw-Rebbi Model Research Articles

Integrated terahertz topological valley-locked power divider with arbitrary power ratios

Integrated power dividers (PDs) are essential in terahertz (THz) communication and radar systems, but miniaturization often leads to performance degradation due to fabrication inaccuracies and sharp bends. Topological photonics offers a solution to these issues, yet creating THz power dividers with arbitrary splitting ratios remains challenging. We present a design methodology for on-chip topological THz power dividers with customizable splitting ratios using valley-locked photonic crystals. These crystals feature a tri-layered structure with two distinct valley Chern number layers and an intermediate semimetal layer. Utilizing the Jackiw–Rebbi model, we show that the characteristic impedance of the valley-locked photonic crystals, and thus the power division ratio, can be tuned by adjusting the semimetal layer width. Our approach is validated through simulations and experiments for both equal (1:1) and unequal (4:9) power ratios. This method enables efficient navigation around sharp bends and robust THz on-chip connectivity.

Realization of Jackiw–Rebbi zero-energy modes at photonic crystal domain walls: Emergence of polarization-indiscriminate surface states

The Jackiw–Rebbi model is a relativistic quantum model credited with the theoretical predictions of zero-energy bound states and charge fractionalization prior to the discovery of topological insulators and the fractional quantum Hall effect. In this work, we demonstrate a photonic equivalent of the Jackiw–Rebbi model by resorting to photonic crystal band structure engineering. Specifically, our photonic realization employs two spatial inversion symmetric binary photonic crystals exhibiting complementary signs of differential effective mass parameter (δm) for their second bandgaps. Their concatenation manifests a step discontinuity in the spatial profile of the effective mass parameter, forming a domain wall at the photonic crystal interface. Upon analyzing the reflectance spectra of the concatenated photonic crystal structure, we find a midgap surface state localized at this domain wall. Furthermore, much in agreement with the Jackiw–Rebbi zero-energy solution, the materialized photonic surface state also exhibits a zero-energy character in a differential energy space corresponding to the δm parameter, which has been quantified experimentally. Crucially, the conceived zero-energy mode amounts to the observation of a peculiar surface state with polarization-indiscriminate dispersion that can help realize all-angle polarization neutral optics.

Atomically Thin Current Pathways in Graphene through Kekulé-O Engineering.

We demonstrate that the current flow in graphene can be guided on atomically thin current pathways by the engineering of Kekulé-O distortions. A grain boundary in these distortions separates the system into topologically distinct regions and induces a ballistic domain-wall state. The state is independent of the orientation of the grain boundary with respect to the graphene sublattice and permits guiding the current on arbitrary paths. As the state is gapped, the current flow can be switched by electrostatic gates. Our findings are explained by a generalization of the Jackiw-Rebbi model, where the electrons behave in one region of the system as Fermions with an effective complex mass, making the device not only promising for technological applications but also a test-ground for concepts from high-energy physics. An atomic model supported by DFT calculations demonstrates that the system can be realized by decorating graphene with Ti atoms.

Topological origin of flat bands as pseudo-Landau levels in uniaxial strained graphene nanoribbons and induced magnetic ordering due to electron-electron interactions

Flat bands play a central role in the presence of correlated phases in moir\'e and other modulated two-dimensional systems. In this paper, flat bands are shown to exist in uniaxially periodic strained graphene. Such strain should be produced for example by a substrate. The model is thus mapped into a one-dimensional effective Hamiltonian and this allows to find the conditions for having flat bands, i.e., a long-wavelength modulation only on each one of the bipartite graphene sublattices, while having a tagged strain field between neighboring carbon atoms. The origin of such flat bands is thus tracked down to the existence of topological localized wavefunctions at domain walls separating different regions, each with a nonuniform Su-Schriffer-Hegger model (SSH) type of coupling. Thereafter, the system is mapped into a continuum model allowing to explain the numerical results in terms of the Jackiw-Rebbi model and of pseudo-Landau levels. Finally, the interplay between the obtained flat bands and electron-electron interaction is explored through the Hubbard model. The numerical results within the mean-field approximation indicate that the flat bands induce N\'eel antiferromagnetic and ferromagnetic domains even for a very weak Hubbard interaction, while the repulsive Hubbard interaction results in an effective electron-electron attraction. Thus this paper provides a simple, analytically solvable but realistic model to understand the physical origin of flat bands, pseudo-Landau levels and their effects on the effective electron-electron interaction.

Rotor Jackiw-Rebbi Model: A Cold-Atom Approach to Chiral Symmetry Restoration and Charge Confinement

Understanding the nature of confinement, as well as its relation with the spontaneous breaking of chiral symmetry, remains one of the long-standing questions in high-energy physics. The difficulty of this task stems from the limitations of current analytical and numerical techniques to address non-perturbative phenomena in non-Abelian gauge theories. In this work, we show how similar phenomena emerge in simpler models, and how these can be further investigated using state-of-the-art cold-atom quantum simulators. More specifically, we introduce the rotor Jackiw-Rebbi model, a (1+1)-dimensional quantum field theory where interactions between Dirac fermions are mediated by quantum rotors. Starting from a mixture of ultracold atoms in an optical lattice, we show how this quantum field theory emerges in the long-wavelength limit. For a wide and experimentally-relevant parameter regime, the Dirac fermions acquire a dynamical mass via the spontaneous breakdown of chiral symmetry. Moreover, we study the effect of both quantum and thermal fluctuations, which lead to the phenomenon of chiral symmetry restoration. Finally, we uncover a confinement-deconfinement quantum phase transition, where meson-like fermions fractionalise into quark-like quasi-particles bound to topological solitons of the rotor field. The proliferation of these solitons at finite chemical potentials again serves to restore the chiral symmetry, yielding a clear analogy with the quark-gluon plasma in quantum chromodynamics, where this symmetry coexists with the deconfined fractional charges. Our results show how the interplay between these phenomena could be analyse in realistic atomic experiments.

Photonic Jackiw-Rebbi states in all-dielectric structures controlled by bianisotropy

Electric and magnetic resonances of dielectric particles have recently uncovered a range of exciting applications in steering of light at the nanoscale. Breaking of particle inversion symmetry further modifies its electromagnetic response giving rise to bianisotropy known also as magneto-electric coupling. Recent studies suggest the crucial role of magneto-electric coupling in realization of photonic topological metamaterials. To further unmask this fundamental link, we design and test experimentally one-dimensional array composed of dielectric particles with overlapping electric and magnetic resonances and broken mirror symmetry. Flipping over half of the meta-atoms in the array, we observe the emergence of interface states providing photonic realization of the celebrated Jackiw-Rebbi model. We trace the origin of these states to the fact that local modification of particle bianisotropic response affects its effective coupling with the neighboring meta-atoms which provides a promising avenue to engineer topological states of light.

Electrostatic simulation of the Jackiw-Rebbi zero energy state

We present an analogy between the one dimensional Poisson equation in inhomogeneous media and the Dirac equation in one space dimension with a Lorentz scalar potential for zero energy. We illustrate how the zero energy state in the Jackiw-Rebbi model can be implemented in a simple one dimensional electrostatic setting by using an inhomogeneous electric permittivity and an infinite charged sheet. Our approach provides a novel insight into the Jackiw-Rebbi zero energy state and provides a helpful way to visualize and teach this important quantum field theory model using basic electrostatics.

Dirac Equation and Optical Wave Propagation in One Dimension

We show that the propagation of transverse electric (TE) polarized waves in one‐dimensional inhomogeneous settings can be written in the form of the Dirac equation in one space dimension with a Lorentz scalar potential, and consequently perform photonic simulations of the Dirac equation in optical structures. In particular, we propose how the zero energy state of the Jackiw–Rebbi model can be generated in an optical set‐up by controlling the refractive index landscape, where TE‐polarized waves mimic the Dirac particles and the soliton field can be tuned by adjusting the refractive index.

Charge fractionalization in oxide heterostructures: A field-theoretical model

LaAlO3/SrTiO3 heterostructure with polar and non-polar constituents has been shown to exhibit interface metallic conductivity due to fractional charge transfer to the interface. The interface reconstruction by electron redistribution along the (001) orientation, in which half of an electron is transferred per two-dimensional unit cell to the adjacent planes, resulting in a net transfer of half of the charge to both the interface and topmost atomic planes, has been ascribed to a polar discontinuity at the interface in the polar catastrophe model. This avoids the divergence of the electrostatic potential, as the number of layers are increased, producing an oscillatory electric field and finite potential. Akin to the description of charge fractionalization in quasi–one-dimensional polyacetylene by the field-theoretic Jackiw-Rebbi model with fermions interacting with a topologically non-trivial background field, we show an analogous connection between the polar catastrophe model and the Bell-Rajaraman model, where the charge fractionalization occurs in the soliton free sector as an end effect.

Fractional fermions induced by spatially varying Zeeman fields

We investigate a one dimensional Rashba system in presence of a spatially varying Zeeman field and show that fractional fermions (FFs) originally occurring in Jackiw–Rebbi model can be induced. Chiral symmetry protected FFs, being a characterization of soliton states carrying fractional charge ±e/2, may emerge in the middle of band gaps when the period of the Zeeman field is a multiple of a Rashba dependent quantity. Our results also demonstrate that the shape of the non-uniform field is less important in producing chiral FFs, comparing to its period. In the end, we distinguish chiral FFs from those zero mode FFs breaking chiral symmetry and give several examples.

Fermions embedded in a scalar-vector kink-like smooth potential

The behaviour of massive fermions is analyzed with scalar and vector potentials. A continuous chiral-conjugation transformation decouples the equation for the upper component of the Dirac spinor provided the vector coupling does not exceed the scalar coupling. It is shown that a Sturm-Liouville perspective is convenient for studying scattering as well as bound states. One possible isolated solution (excluded from the Sturm-Liouville problem) corresponding to a bound state might also come into sight. For potentials with kink-like profiles, beyond the intrinsically relativistic isolated bound-state solution corresponding to the zero-mode solution of the massive Jackiw-Rebbi model in the case of no vector coupling, a finite set of bound-state solutions might appear as poles of the transmission amplitude in a strong coupling regime. It is also shown that the possible isolated bound solution disappears asymptotically as the magnitude of the scalar and vector coupling becomes the same. Furthermore, we show that due to the sizeable mass gain from the scalar background the high localization of the fermion in an extreme relativistic regime is conformable to comply with the Heisenberg uncertainty principle.

Some topological states in one-dimensional cold atomic systems

Ultracold atoms trapped in optical lattices nowadays have been widely used to mimic various models from condensed-matter physics. Recently, many great experimental progresses have been achieved for producing artificial magnetic field and spin–orbit coupling in cold atomic systems, which turn these systems into a new platform for simulating topological states. In this paper, we give a review focusing on quantum simulation of topologically protected soliton modes and topological insulators in one-dimensional cold atomic system. Firstly, the recent achievements towards quantum simulation of one-dimensional models with topological non-trivial states are reviewed, including the celebrated Jackiw–Rebbi model and Su–Schrieffer–Heeger model. Then, we will introduce a dimensional reduction method for systematically constructing high dimensional topological states in lower dimensional models and review its applications on simulating two-dimensional topological insulators in one-dimensional optical superlattices.

Probing the topological properties of the Jackiw-Rebbi model with light.

The Jackiw-Rebbi model describes a one-dimensional Dirac field coupled to a soliton field and can be equivalently thought of as a model describing a Dirac field with a spatially dependent mass term. Neglecting the dynamics of the soliton field, a kink in the background soliton profile yields a topologically protected zero-energy mode for the field, which in turn leads to charge fractionalisation. We show here that the model, in the first quantised form, can be realised in a driven slow-light setup, where photons mimic the Dirac field and the soliton field can be implemented–and tuned–by adjusting optical parameters such as the atom-photon detuning. Furthermore, we discuss how the existence of the zero-mode and its topological stability can be probed naturally by studying the transmission spectrum. We conclude by analysing the robustness of our approach against possible experimental errors in engineering the Jackiw-Rebbi Hamiltonian in this optical setup.

One-loop quantum correction to the mass of the supersymmetric Kink in (1 + 1) dimensions using the exact spectra and the phase shifts

We compute the quantum correction to the mass of the kink at the one-loop level in (1 + 1) dimensions with minimal supersymmetry (SUSY). In this paper we discuss this issue from the Casimir energy perspective using phase shifts along with the mode number cutoff regularization method. Exact solutions and, in particular, an exact expression for the phase shifts are already available for the bosonic sector. In this paper we derive analogous exact results for the fermionic sector. Most importantly, we derive a unique and exact expression for the fermionic phase shift, using the exact solutions for the continuum parts of the spectrum and a prescription that we had introduced earlier. We use the strong and weak forms of the Levinson theorem merely for checking the consistency of our phase shifts and results, and not as an integral part of our procedure. Moreover, we find that the properties of the fermionic spectrum, including bound and continuum states, are independent of the magnitude of the Yukawa coupling constant λ, and that the dynamical mass generation occurs at the tree level. These outcomes are all due to SUSY and are in sharp contrast to those from analogous models without SUSY, such as the Jackiw–Rebbi model, where λ is a free parameter. We use the renormalized perturbation theory and find the counterterm which is consistent with supersymmetry. We show that this procedure is sufficient for obtaining the accepted value for the one-loop quantum correction to the mass of the SUSY kink, which is .

Complete spectral analysis of the Jackiw-Rebbi model, including its zero mode

In this paper we present a complete and exact spectral analysis of the $(1+1)$-dimensional model that Jackiw and Rebbi considered to show that the half-integral fermion numbers are possible due to the presence of an isolated self charge conjugate zero mode. The model possesses the charge and particle conjugation symmetries. These symmetries mandate the reflection symmetry of the spectrum about the line $E=0$. We obtain the bound state energies and wave functions of the fermion in this model using two different methods, analytically and exactly, for every arbitrary choice of the parameters of the kink, i.e. its value at spatial infinity ($\theta_0$) and its scale of variations ($\mu$). Then, we plot the bound state energies of the fermion as a function of $\theta_0$. This graph enables us to consider a process of building up the kink from the trivial vacuum. We can then determine the origin and evolution of the bound state energy levels during this process. We see that the model has a dynamical mass generation process at the first quantized level and the zero-energy fermionic mode responsible for the fractional fermion number, is always present during the construction of the kink and its origin is very peculiar, indeed. We also observe that, as expected, none of the energy levels crosses each other. Moreover, we obtain analytically the continuum scattering wave functions of the fermion and then calculate the phase shifts of these wave functions. Using the information contained in the graphs of the phase shifts and the bound states, we show that our phase shifts are consistent with the weak and strong forms of the Levinson theorem. Finally, using the weak form of the Levinson theorem, we confirm that the number of the zero-energy fermionic modes is exactly one.

Majorana zero-modes and topological phases of multi-flavored Jackiw-Rebbi model

Motivated by the recent Kitaev's K-theory analysis of topological insulators and superconductors, we adopt the same framework to study the topological phase structure of Jackiw-Rebbi model in 3+1 dimensions. According to the K-theory analysis based on the properties of the charge conjugation and time reversal symmetries, we classify the topological phases of the model. In particular, we find that there exist $\mathbf{Z}$ Majorana zero-modes hosted by the hedgehogs/t'Hooft-Polyakov monopoles, if the model has a $T^2=1$ time reversal symmetry. Guided by the K-theory results, we then explicitly show that a single Majorana zero mode solution exists for the SU(2) doublet fermions in some co-dimensional one planes of the mass parameter space. It turns out we can see the existence of none or a single zero mode when the fermion doublet is only two. We then take a step further to consider four-fermion case and find there can be zero, one or two normalizable zero mode in some particular choices of mass matrices. Our results also indicate that a single normalizable Majorana zero mode can be compatible with the cancellation of SU(2) Witten anomaly.

Constrained Jackiw-Rebbi model gives McGreevy-Swingle model

We show that the recently considered McGreevy-Swingle model for Majorana fermions in the presence of a 't Hooft-Polyakov magnetic monopole arises when the Jackiw-Rebbi model is constrained to be conjugation self dual.

Topological superconductors as nonrelativistic limits of Jackiw-Rossi and Jackiw-Rebbi models

We argue that the nonrelativistic Hamiltonian of p_x+ip_y superconductor in two dimensions can be derived from the relativistic Jackiw-Rossi model by taking the limit of large Zeeman magnetic field and chemical potential. In particular, the existence of a fermion zero mode bound to a vortex in the p_x+ip_y superconductor can be understood as a remnant of that in the Jackiw-Rossi model. In three dimensions, the nonrelativistic limit of the Jackiw-Rebbi model leads to a "p+is" superconductor in which spin-triplet p-wave and spin-singlet s-wave pairings coexist. The resulting Hamiltonian supports a fermion zero mode when the pairing gaps form a hedgehoglike structure. Our findings provide a unified view of fermion zero modes in relativistic (Dirac-type) and nonrelativistic (Schr\"odinger-type) superconductors.

Dirac sea correction to the topological soliton mass

The Dirac sea contribution to the soliton mass in the Jackiw-Rebbi model of fractional charge is calculated for an arbitrary Yukawa coupling strength G including the effects of the continuum. The correction is found to be positive for all values of G of interest and rapidly increasing for strong coupling. The fermion density of states is shown to be determined by the average transition phase of the corresponding Dirac equation in the soliton background.