Single Dirac Fermion Research Articles

Noninvertible anomalies in SU(N) × U(1) gauge theories

We study 4-dimensional SU(N) × U(1) gauge theories with a single massless Dirac fermion in the 2-index symmetric/antisymmetric representations and show that they are endowed with a noninvertible 0-form {overset{sim }{mathbb{Z}}}_{2left(Npm 2right)}^{upchi} chiral symmetry along with a 1-form {mathbb{Z}}_N^{(1)} center symmetry. By using the Hamiltonian formalism and putting the theory on a spatial three-torus mathbbm{T} 3, we construct the non-unitary gauge invariant operator corresponding to {overset{sim }{mathbb{Z}}}_{2left(Npm 2right)}^{upchi} and find that it acts nontrivially in sectors of the Hilbert space characterized by selected magnetic fluxes. When we subject mathbbm{T} 3 to {mathbb{Z}}_N^{(1)} twists, for N even, in selected magnetic flux sectors, the algebra of {overset{sim }{mathbb{Z}}}_{2left(Npm 2right)}^{upchi} and {mathbb{Z}}_N^{(1)} fails to commute by a ℤ2 phase. We interpret this noncommutativity as a mixed anomaly between the noninvertible and the 1-form symmetries. The anomaly implies that all states in the torus Hilbert space with the selected magnetic fluxes exhibit a two-fold degeneracy for arbitrary mathbbm{T} 3 size. The degenerate states are labeled by discrete electric fluxes and are characterized by nonzero expectation values of condensates. In an appendix, we also discuss how to construct the corresponding noninvertible defect via the “half-space gauging” of a discrete one-form magnetic symmetry.

Hall effect of ferro/antiferromagnetic wallpaper fermions

Nonsymmorphic crystals can host characteristic double surface Dirac cones with fourfold degeneracy on the Dirac points, called wallpaper fermion, protected by wallpaper group symmetry. We clarify the charge and spin Hall effect of wallpaper fermions in the presence of the (anti)ferromagnetism. Based on a four-sublattice model, we construct the effective Hamiltonian of wallpaper fermions coupled with the ferromagnetic or antiferromagnetic moment. Both ferromagnetic and antiferromagnetic moments induce an energy gap for the wallpaper fermions, leading to quantized (spin) Hall conductivity. The ferromagnetic wallpaper fermion induces the Hall conductivity quantized into ${e}^{2}/h$, which is twice that for a single Dirac fermion on the surface of topological insulators. On the other hand, the spin Hall conductivity decays and reaches to be a finite value as the antiferromagnetic coupling increases. We also show that the results above are valid for a general model of wallpaper fermions from symmetry consideration.

Random magnetic field and the Dirac Fermi surface

We study a single two-dimensional Dirac fermion at finite density, subject to a quenched random magnetic field. At low energies and sufficiently weak disorder, the theory maps onto an infinite collection of 1D chiral fermions (associated to each point on the Fermi surface) coupled by a random vector potential. This low-energy theory exhibits an exactly solvable random fixed line, along which we directly compute various disorder-averaged observables without the need for the usual replica, supersymmetry, or Keldysh techniques. We find the longitudinal dc conductivity in the collisionless $\ensuremath{\hbar}\ensuremath{\omega}/{k}_{B}T\ensuremath{\rightarrow}\ensuremath{\infty}$ limit to be nonuniversal and to vary continuously along the fixed line.

Discrete spacetime symmetries, second quantization, and inner products in a non-Hermitian Dirac fermionic field theory

We extend to a non-Hermitian fermionic quantum field theory with PT symmetry our previous discussion of second quantization, discrete symmetry transformations, and inner products in a scalar field theory [arXiv:2006.06656]. For illustration, we consider a prototype model containing a single Dirac fermion with a parity-odd, anti-Hermitian mass term. In the phase of unbroken PT symmetry, this Dirac fermion model is equivalent to a Hermitian theory under a similarity transformation, with the non-Hermitian nature of the model residing only in the spinor structure, whereas the algebra of the creation and annihilation operators is just that of a Hermitian theory.

Experimental signature of the parity anomaly in a semi-magnetic topological insulator

A three-dimensional (3D) topological insulator features a 2D surface state consisting of a single linearly dispersive Dirac cone1–3. Under broken time-reversal symmetry, the single Dirac cone is predicted to cause half-integer quantization of Hall conductance, which is a manifestation of the parity anomaly in quantum field theory1–9. However, despite various observations of quantization phenomena10–15, the half-integer quantization has not been observed because most experiments simultaneously measure a pair of equivalent Dirac cones16 on two opposing surfaces. Here we demonstrate the half-integer quantization of Hall conductance in a synthetic heterostructure termed a semi-magnetic topological insulator, where only one surface state is gapped by magnetic doping and the opposite one is non-magnetic and gapless. We observe half-quantized Faraday and Kerr rotations with terahertz magneto-optical spectroscopy and half-quantized Hall conductance in transport at zero magnetic field. Our results suggest a condensed-matter realization of the parity anomaly4–9 and open a way for studying the physics enabled by a single Dirac fermion. An electron with a linear dispersion relation should contribute half of a quantum of Hall conductance and thereby manifest the parity anomaly. This is demonstrated in a heterostructure of topological insulator materials.

Quantum Hall effect of the topological insulator state of cadmium arsenide in Corbino geometry

The surfaces of a three-dimensional topological insulator each host a single Dirac fermion, which, in a strong magnetic field, contribute to the transverse conductance in integer-and-a-half multiples of the conductance quantum. The direct observation of this extra half integer, the hallmark of the two-dimensional Dirac state, is usually thwarted by the fermion doubling theorem—top and bottom surfaces are not measured independently. Here, we employ a Corbino measurement geometry in which a current, induced by an ac magnetic field, is driven around the ring, as a complementary measurement to the conventional Hall bar geometry. As the device enters the quantum Hall regime, the transverse voltage reaches a series of plateaus when the current is carried by the incompressible bulk states. We compare the results with the corresponding Hall bar measurements.

Condensates and anomaly cascade in vector-like theories

We study the bilinear and higher-order fermion condensates in 4-dimensional SU(N) gauge theories with a single Dirac fermion in a general representation. Augmented with a mixed anomaly between the 0-form discrete chiral, 1-form center, and 0-form baryon number symmetries (BC anomaly), we sort out theories that admit higher-order condensates and vanishing fermion bilinears. Then, the BC anomaly is utilized to prove, in the absence of a topological quantum field theory, that nonvanishing fermion bilinears are inevitable in infrared-gapped theories with 2-index (anti)symmetric fermions. We also contrast the BC anomaly with the 0-form anomalies and show that it is the former anomaly that determines the infrared physics; we argue that the BC anomaly lurks deep to the infrared while the 0-form anomalies are just variations of local terms. We provide evidence of this assertion by studying the BC anomaly in vector-like theories compactified on a small spacial circle. These theories are weakly-coupled, under analytical control, and they admit a dual description in terms of abelian photons that determine the deep infrared dynamics. We show that the dual photons talk directly to the 1-form center symmetry in order to match the BC anomaly, while the 0-form anomalies are variations of local terms and are matched by fiat. Finally, we study the fate of the BC anomaly in the compactified theories when they are held at a finite temperature. The effective field theory that describes the low-energy physics is 2-dimensional. We show that the BC anomaly cascades from 4 to 2 dimensions.

Topological Surface Dirac Fermion in BiTeCl-Based Heterostructures

Inversion asymmetric topological insulator promises great potential to realize many intriguing phenomena such as intrinsic topological [Formula: see text]–[Formula: see text] junctions, pyroelectricity and novel topological magneto-electric effects that require inversion asymmetry. Recently, gapless helical surface Dirac fermions were observed in inversion-asymmetric BiTeCl crystal. However, heavy debate about whether BiTeCl is a topological insulator and controversial experimental results of the electronic structure of BiTeCl exist. In this work, through ab-initio calculations, we discovered that a BiTeCl-dominant BiTeCl/Bi2Te3 heterostructure can give rise to the observed band structures with a single surface Dirac fermion. Interestingly, we further suggest that such a BiTeCl/Bi2Te3 heterostructure could be naturally formed by annealing pure BiTeCl single crystals. Our results not only resolve the discrepancy between the previous experiments on heat treated samples and theoretical investigations, but also point out a pathway to design novel materials with sophisticated properties.

Adventure in Topological Phase Transitions in 3+1 -D: Non-Abelian Deconfined Quantum Criticalities and a Possible Duality

Continuous quantum phase transitions beyond the conventional paradigm of fluctuations of a symmetry-breaking order parameter are challenging for theory. These phase transitions often involve emergent deconfined gauge fields at the critical points as demonstrated in phase transitions between different broken-symmetry states of 2+1-dimensional quantum magnets, as well as those between symmetry-protected topological (SPT) phases. In this paper, we present several examples of deconfined quantum critical points between SPT phases in 3+1-D for both bosonic and fermionic systems. These critical theories can be formulated as non-Abelian gauge theories either in the infrared-free regime or in the conformal window when they flow to the Banks-Zaks fixed points. We explicitly demonstrate several interesting quantum critical phenomena. We describe situations in which the same phase transition allows for multiple universality classes controlled by distinct fixed points. We exhibit the possibility—which we dub “unnecessary quantum critical points”—of stable generic continuous phase transitions within the same phase. We present examples of interaction-driven, band-theory-forbidden, continuous phase transitions between two distinct band insulators. The understanding we develop leads us to suggest an interesting possible 3+1-D field theory duality between SU(2) gauge theory coupled to one massless adjoint Dirac fermion and the theory of a single massless Dirac fermion augmented by a decoupled topological field theory.3 MoreReceived 2 January 2019Revised 2 April 2019DOI:https://doi.org/10.1103/PhysRevX.9.021034Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasCritical phenomenaDualities in field theoryNon-Abelian gauge theoriesSymmetry protected topological statesTopological insulatorsTopological phase transitionCondensed Matter, Materials & Applied PhysicsParticles & Fields

Exotic topology on the surface

Topological Matter Analyzing the spatial symmetries of three-dimensional (3D) crystal structures has led to the discovery of exotic types of quasiparticles and topologically nontrivial materials. Wieder et al. focus on the symmetry groups of 2D surfaces of 3D materials—the so-called wallpaper groups—and find that some of them allow for an additional topological class. This class hosts a single fourfold-degenerate Dirac fermion on the surface of the material and, on the basis of the authors' calculations, is expected to occur in the compound Sr2Pb3. Science , this issue p. [246][1] [1]: /lookup/doi/10.1126/science.aan2802

Topological Sachdev-Ye-Kitaev model

In this Rapid Communication, we construct a large-$N$ exactly solvable model to study the interplay between interaction and topology, by connecting the Sachdev-Ye-Kitaev (SYK) model with constant hopping. The hopping forms a band structure that can exhibit both topologically trivial and nontrivial phases. Starting from a topologically trivial insulator with zero Hall conductance, we show that the interaction can drive a phase transition to a topologically nontrivial insulator with quantized nonzero Hall conductance, and a single gapless Dirac fermion emerges when the interaction is fine tuned to the critical point. The finite temperature effect is also considered, and we show that the topological phase with a stronger interaction is less stable against temperature. Our model provides a concrete example to illustrate the interacting topological phases and phase transitions, and can shed light on similar problems in physical systems.

Dual QED3 at “N F = 1/2” is an interacting CFT in the infrared

We study the fate of weakly coupled dual QED3 in the infrared, that is, a single two-component Dirac fermion coupled to an emergent U(1) gauge field, but without Chern-Simons term. This theory has recently been proposed as a dual description of 2D surfaces of certain topological insulators. Using the renormalization group, we find that the interplay of gauge fluctuations with generated interactions in the four-fermi sector stabilizes an interacting conformal field theory (CFT) with finite four-fermi coupling in the infrared. The emergence of this CFT is due to cancellations in the $\beta$-function of the four-fermi coupling special to "NF = 1/2". We also quantify how a possible "strong" Dirac fermion duality between a free Dirac cone and dual QED3 would constrain the universal constants of the topological current correlator of the latter.

Bosonization and mirror symmetry

We study bosonization in 2+1 dimensions using mirror symmetry, a duality that relates pairs of supersymmetric theories. Upon breaking supersymmetry in a controlled way, we dynamically obtain the bosonization duality that equates the theory of a free Dirac fermion to QED3 with a single scalar boson. This duality may be used to demonstrate the bosonization duality relating an $O(2)$-symmetric Wilson-Fisher fixed point to QED3 with a single Dirac fermion, Peskin-Dasgupta-Halperin duality, and the recently conjectured duality relating the theory of a free Dirac fermion to fermionic QED3 with a single flavor. Chern-Simons and BF couplings for both dynamical and background gauge fields play a central role in our approach. In the course of our study, we describe a chiral mirror pair that may be viewed as the minimal supersymmetric generalization of the two bosonization dualities.

A duality web in [formula omitted] dimensions and condensed matter physics

Building on earlier work in the high energy and condensed matter communities, we present a web of dualities in 2+1 dimensions that generalize the known particle/vortex duality. Some of the dualities relate theories of fermions to theories of bosons. Others relate different theories of fermions. For example, the long distance behavior of the 2+1-dimensional analog of QED with a single Dirac fermion (a theory known as U(1)12) is identified with the O(2)Wilson–Fisher fixed point. The gauged version of that fixed point with a Chern–Simons coupling at level one is identified as a free Dirac fermion. The latter theory also has a dual version as a fermion interacting with some gauge fields. Assuming some of these dualities, other dualities can be derived. Our analysis resolves a number of confusing issues in the literature including how time reversal is realized in these theories. It also has many applications in condensed matter physics like the theory of topological insulators (and their gapped boundary states) and the problem of electrons in the lowest Landau level at half filling. (Our techniques also clarify some points in the fractional Hall effect and its description using flux attachment.) In addition to presenting several consistency checks, we also present plausible (but not rigorous) derivations of the dualities and relate them to 3+1-dimensional S-duality.

Chalker scaling, level repulsion, and conformal invariance in critically delocalized quantum matter: Disordered topological superconductors and artificial graphene

We numerically investigate critically delocalized wavefunctions in models of 2D Dirac fermions, subject to vector potential disorder. These describe the surface states of 3D topological superconductors, and can also be realized through long-range correlated bond randomness in artificial materials like molecular graphene. A frozen regime can occur for strong disorder in these systems, wherein a single wavefunction presents a few localized peaks separated by macroscopic distances. Despite this rarefied spatial structure, we find robust correlations between eigenstates at different energies, at both weak and strong disorder. The associated level statistics are always approximately Wigner-Dyson. The system shows generalized Chalker (quantum critical) scaling, even when individual states are quasilocalized in space. We confirm analytical predictions for the density of states and multifractal spectra. For a single Dirac valley, we establish that finite energy states show universal multifractal spectra consistent with the integer quantum Hall plateau transition. A single Dirac fermion at finite energy can therefore behave as a Quantum Hall critical metal. For the case of two valleys and non-abelian disorder, we verify predictions of conformal field theory. Our results for the non-abelian case imply that both delocalization and conformal invariance are topologically-protected for multivalley topological superconductor surface states.

Discovery of a single topological Dirac fermion in the strong inversion asymmetric compound BiTeCl

BiTeCl is a topological insulator with strong inversion asymmetry, which exhibits bulk charge polarization and pyroelectricity. Such a long-sought topological insulator paves the way for applications involving natural p–n junctions and spintronics.

Orbifold family unification on six dimensions

We study the possibility of family unification on the basis of SU(N) gauge theory on the 6-dimensional space-time, $M^4\times T^2/Z_N$. We obtain enormous numbers of models with three families of SU(5) matter multiplets and those with three families of the standard model multiplets, from a single massless Dirac fermion with a higher-dimensional representation of SU(N), through the orbifold breaking mechanism.

Non-Abelian statistics of vortices with non-Abelian Dirac fermions

We extend our previous analysis on the exchange statistics of vortices having a single Dirac fermion trapped in each core to the case where vortices trap two Dirac fermions with U(2) symmetry. Such a system of vortices with non-Abelian Dirac fermions appears in color superconductors at extremely high densities and in supersymmetric QCD. We show that the exchange of two vortices having doublet Dirac fermions in each core is expressed by non-Abelian representations of a braid group, which is explicitly verified in the matrix representation of the exchange operators when the number of vortices is up to four. We find that the result contains the matrices previously obtained for the vortices with a single Dirac fermion in each core as a special case. The whole braid group does not immediately imply non-Abelian statistics of identical particles because it also contains exchanges between vortices with different numbers of Dirac fermions. However, we find that it does contain, as its subgroup, genuine non-Abelian statistics for the exchange of the identical particles, that is, vortices with the same number of Dirac fermions. This result is surprising compared with conventional understanding because all Dirac fermions are defined locally at each vortex, unlike the case of Majorana fermions for which Dirac fermions are defined nonlocally by Majorana fermions located at two spatially separated vortices.

Numerical computation of the beta function of largeNSU(N)gauge theory coupled to an adjoint Dirac fermion

We use a single site lattice in four dimensions to study the scaling of large N Yang-Mills field coupled to a single massless Dirac fermion in the adjoint representation. We use the location of the strong to weak coupling transition defined through the eigenvalues of the folded Wilson loop operator to set a scale. We do not observe perturbative scaling in the region studied in this paper. Instead, we observe that the scale changes very slowly with the bare coupling. The lowest eigenvalue of the overlap Dirac operator is another scale that shows similar behavior as a function of the lattice coupling. We speculate that this behavior is due to the beta function appoaching close to a zero.

Multifractal nature of the surface local density of states in three-dimensional topological insulators with magnetic and nonmagnetic disorder

We compute the multifractal spectra associated to local density of states (LDOS) fluctuations due to weak quenched disorder for a single Dirac fermion in two spatial dimensions. Our results are relevant to the surfaces of ${\mathbb{Z}}_{2}$ topological insulators such as ${\text{Bi}}_{2}{\text{Se}}_{3}$ and ${\text{Bi}}_{2}{\text{Te}}_{3}$, where LDOS modulations can be directly probed via scanning tunneling microscopy. We find a qualitative difference in spectra obtained for magnetic versus nonmagnetic disorder. Randomly polarized magnetic impurities induce quadratic multifractality at first order in the impurity density; by contrast, no operator exhibits multifractal scaling at this order for a nonmagnetic impurity profile. For the time-reversal invariant case, we compute the first nontrivial multifractal correction, which appears at two loops (impurity density squared). We discuss spectral enhancement approaching the Dirac point due to renormalization, and we survey known results for the opposite limit of strong disorder.