---
title: Chiral Fermions Explained
url: https://www.emergentmind.com/topics/chiral-fermions
type: topic
---

# Chiral Fermions Explained

Chiral fermions are fermionic degrees of freedom whose left- and right-handed components transform differently under Lorentz or gauge symmetries, or whose low-energy band structure carries a definite momentum-space topological charge. The term therefore has several technically distinct uses: relativistic Weyl fields, lattice fermions with exact Ginsparg–Wilson symmetry, gauge-chiral matter, massless quasiparticles in topological materials, chiral modes bound to defects, and emergent extensions such as rank-2 chiral fermions. Across these settings, chirality is associated with a decomposition into handed components, a nontrivial index or anomaly, helicity selection, or quantized Berry-curvature flux.

## 1. Relativistic chirality, helicity, and gauge representations

For a massless Dirac field, left- and right-handed components are defined by
\[
\Psi_{L,R}=P_{L,R}\Psi,\qquad P_{L,R}=\frac12(1\pm\gamma_5).
\]
In massless QCD, these sectors decouple:
\[
\mathcal L_{\rm QCD}
=\bar\Psi_LD\Psi_L+\bar\Psi_RD\Psi_R+\mathcal L_{\rm gauge}.
\]
For \(N_f\) massless flavors, the classical flavor symmetry is
\[
U(N_f)_L\otimes U(N_f)_R.
\]
The axial \(U(1)\) factor is anomalous, while the non-singlet symmetry is expected to break spontaneously as
\[
SU(N_f)_L\otimes SU(N_f)_R\longrightarrow SU(N_f)_{L+R}.
\]
For two flavors this is locally equivalent to \(O(4)\to O(3)\), with pions as the associated Nambu–Goldstone bosons [1007.0285].

For massless fermions, chirality is Lorentz invariant and coincides with helicity. Helicity is the projection of spin along momentum; with the convention
\[
\hat h=\frac{2\mathbf S\cdot\mathbf p}{|\mathbf p|},
\]
the helicity eigenvalues are \(\lambda=\pm1\). A massless left-chiral Weyl field has negative helicity, whereas a massless right-chiral Weyl field has positive helicity. For massive fields, chirality and helicity differ: a massive left-chiral field contains both helicities, and helicity is frame-dependent [1607.03916].

Gauge chirality is a separate concept. A four-component fermion is chiral under a gauge symmetry if its left- and right-handed components transform differently. In a two-component description, left-handed Weyl fields with charges $q_i$ can form a bare mass term only when paired charges satisfy
\[
q_i+q_j=0.
\]
A spectrum without such pairings is chiral. A nonzero $U(1)$ charge also forbids a Majorana mass before symmetry breaking, since $\psi_i\psi_i$ has charge $2q_i$ [1507.00916].

The Standard Model realizes gauge chirality through hypercharge. Written entirely in terms of left-handed Weyl fields, one generation transforms as
\[
Q=(3,2)_{1/6},\qquad u^c=(\bar3,1)_{-2/3},\qquad d^c=(\bar3,1)_{1/3},
\]
\[
L=(1,2)_{-1/2},\qquad e^c=(1,1)_1.
\]
There are no elementary vector-like pairs within this spectrum, so fermion masses arise through Yukawa couplings to the Higgs field after electroweak symmetry breaking.

For a massive spinor, the phases of the scalar and pseudoscalar mass components can be parameterized by
\[
m e^{i\gamma^5\beta}
=m\cos\beta+i m\gamma^5\sin\beta.
\]
A separate phase $\alpha$ can be introduced through a chiral change of gamma-matrix basis,
\[
\gamma^\mu_\alpha
=e^{-i\gamma^5\alpha/2}\gamma^\mu e^{i\gamma^5\alpha/2},
\]
which leaves the Lorentz generators unchanged. The relative phase $\alpha-\beta$ determines the scalar and pseudoscalar bilinears and, in the proposed formalism, generates the three regular Lounesto classes. Class 1 occurs for generic $\alpha-\beta$, class 2 for $\alpha-\beta=0,\pi$, and class 3 for $\alpha-\beta=\pi/2,3\pi/2$ [2108.09647].

## 2. The lattice obstruction and exact modified chiral symmetry

In the continuum, massless chiral symmetry requires
\[
\{D,\gamma_5\}=0.
\]
A direct lattice implementation encounters fermion doubling. The naïve lattice Dirac operator is
\[
D_{\rm n}(p)=i\sum_\mu\gamma_\mu\sin p_\mu.
\]
Although it anticommutes with $\gamma_5$, $\sin p_\mu$ vanishes at both $p_\mu=0$ and $p_\mu=\pi$. In $d$ dimensions this produces $2^d$ light species: one physical fermion and $2^d-1$ doublers.

The Nielsen–Ninomiya theorem formalizes the obstruction under assumptions including translational invariance, locality, a suitable Hermiticity condition, and exact naïve chiral symmetry. Wilson fermions remove doublers with
\[
D_{\rm W}=D_{\rm n}-\frac12\Delta,
\]
where the lattice Laplacian term vanishes near $p=0$ but gives cutoff-scale masses to modes near Brillouin-zone corners. The price is explicit chiral-symmetry breaking, additive quark-mass renormalization, and the need to tune a negative bare mass to reach the massless limit [1007.0285].

The Ginsparg–Wilson construction modifies rather than abandons chiral symmetry. In one normalization,
\[
\{D^{-1},\gamma_5\}=\gamma_5
\]
is equivalent to
\[
\boxed{\{\gamma_5,D\}=D\gamma_5D}.
\]
More generally,
\[
\{\gamma_5,D\}=aD\gamma_5D
\]
or
\[
\{\gamma_5,D\}=\frac1\rho D\gamma_5D.
\]
The relation becomes the continuum anticommutation relation when $aD\to0$. It evades the Nielsen–Ninomiya obstruction because it is not the naïve continuum relation at finite lattice spacing, while permitting a local, doubler-free Dirac operator.

With $\gamma_5$-Hermiticity,
\[
D^\dagger=\gamma_5D\gamma_5,
\]
the Ginsparg–Wilson relation implies
\[
D+D^\dagger=D^\dagger D.
\]
Defining $A=D-1$ gives
\[
A^\dagger A=1,
\]
so $A$ is unitary. The eigenvalues of $D=A+1$ lie on
\[
|\lambda-1|=1,
\]
or, with parameter $\rho$,
\[
|\lambda-\rho|=\rho.
\]
This circular spectrum is characteristic of Ginsparg–Wilson fermions and prevents the additive mass shift associated with Wilson fermions.

The overlap operator provides an explicit realization. With the Hermitian Wilson kernel
\[
H_{\rm W}=\gamma_5(D_{\rm W}-\rho),
\]
one may write
\[
D_{\rm ov}
=\rho\left[1+\gamma_5\,\operatorname{sign}(H_{\rm W})\right].
\]
The sign function projects the Wilson kernel onto a Ginsparg–Wilson solution. The operator is local for sufficiently smooth or admissible gauge fields, while locality can deteriorate on very coarse lattices; problems were noted for lattice spacings roughly larger than $0.17$ fm [1007.0285].

The exact lattice chiral transformation is
\[
\delta\bar\Psi
=\varepsilon\bar\Psi\left(1-\frac12D\right)\gamma_5,
\qquad
\delta\Psi
=\varepsilon\gamma_5\left(1-\frac12D\right)\Psi.
\]
The fermionic action is invariant at nonzero lattice spacing:
\[
\delta(\bar\Psi D\Psi)=0.
\]
The path-integral measure is not invariant; its Jacobian reproduces the axial anomaly. Thus the formulation retains exact non-singlet chiral symmetry while preserving anomalous singlet physics.

In Hamiltonian lattice gauge theory, the corresponding conserved charge is
\[
\boxed{
\hat Q=\hat\psi^\dagger\gamma^5
\left(1-\frac R2\hat D\right)\hat\psi
}.
\]
For the overlap Hamiltonian,
\[
[\hat H,\hat Q]=0,
\]
whereas the naïve charge
\[
\hat Q_{\rm naive}=\hat\psi^\dagger\gamma^5\hat\psi
\]
does not generally commute with the Hamiltonian. Simultaneous energy and chiral-charge eigenstates satisfy
\[
\frac{R^2}{4}\varepsilon_n^2+q_n^2=1.
\]
A chiral chemical potential is introduced as
\[
\hat H_{\mu_5}=\hat H-\mu_5\hat Q,
\]
which shifts the energy by $-\mu_5q_n$. For dynamical gauge fields, the fermionic charge can still be violated because the gauge-field Hamiltonian does not commute with the link-dependent $\hat Q$; this provides a Hamiltonian realization of anomaly-induced chirality production [2305.18934].

## 3. Index, topology, defects, and chiral modes

The Ginsparg–Wilson relation supplies a lattice index theorem. Zero modes satisfy
\[
D\psi_0=0,\qquad \gamma_5\psi_0=\pm\psi_0.
\]
The index is
\[
\nu=n_+-n_-,
\]
or equivalently
\[
\boxed{
\nu=\operatorname{Tr}\left[\gamma_5\left(1-\frac12D\right)\right]
}.
\]
With explicit lattice spacing, this becomes
\[
\nu=\operatorname{Tr}\left[\gamma_5\left(1-\frac{aD}{2}\right)\right].
\]
The index defines the lattice topological charge directly from fermion zero modes rather than from a smoothing prescription. The topological susceptibility is
\[
\chi_t=\frac{\langle\nu^2\rangle}{V}.
\]
A quoted quenched continuum result is
\[
\chi_t^{1/4}\simeq191(5)\ {\rm MeV},
\]
supporting the Witten–Veneziano interpretation of the large $\eta'$ mass as partly arising from topological fluctuations [1007.0285].

The same index structure appears in finite-matrix compactifications. In an IIB matrix model on
\[
M^4\times T^6,\qquad T^6=T_1^2\times T_2^2\times T_3^2,
\]
unitary matrices represent gauge fields on noncommutative tori. Block-diagonal backgrounds generate gauge factors $U(p^a)$, while off-diagonal blocks transform as bifundamentals,
\[
\psi^{ab}\sim(p^a,\overline{p^b}).
\]
Relative magnetic fluxes $q_l^a-q_l^b$ generate nonzero indices for these blocks. The Ginsparg–Wilson operator obeys
\[
\gamma D_{\rm GW}+D_{\rm GW}\widehat\gamma=0,
\]
with
\[
\widehat\gamma=\frac{H}{\sqrt{H^2}},
\qquad
H=\gamma(1-\epsilon D_{\rm W}).
\]
Its index is
\[
\operatorname{index}(D_{\rm GW})
=\frac12\operatorname{Tr}(\gamma+\widehat\gamma).
\]
For factorized fluxes on $T^6$,
\[
\boxed{
\operatorname{index}(\psi^{ab})
=p^ap^b\prod_{l=1}^3(q_l^a-q_l^b)
}.
\]
The complete adjoint has zero total index, but individual bifundamental blocks can be chiral. A four-block configuration gives nonzero sectors with index magnitude three and gauge group
\[
U(3)\times U(2)\times U(1)^3
\simeq SU(3)\times SU(2)\times U(1)^5.
\]
The index produces three generations in Standard-Model-like representations. Hypercharge is a linear combination of the five Abelian charges, while the other Abelian gauge bosons are expected to become massive through additional dynamics not developed in the construction [1011.1015].

Defects can also bind chiral modes. A winding-one axion string has
\[
\phi=f(r)e^{i\theta},\qquad \oint d\alpha=2\pi.
\]
In the continuum it supports one normalizable chiral mode localized near the string core, with radial profile
\[
\Psi(r)\sim e^{-f_\infty r}
\]
and a $1+1$-dimensional dispersion such as
\[
E=-p_1.
\]
Naïve lattice discretization produces four longitudinal Brillouin-zone corners, whose chiralities alternate, yielding zero net chirality. Wilson-like terms lift the doublers differently. For a symmetric crossed-wall configuration and $R=a$, the net chiral content changes across
\[
am=2,\quad4,\quad6:
\]
\[
N_{\rm chiral}=
\begin{cases}
-1,&0<am<2,\\
+2,&2<am<4,\\
-1,&4<am<6,\\
0,&am>6.
\end{cases}
\]
These changes coincide with two-dimensional phase transitions where effective masses close on surfaces passing through the defect core [2207.01640].

The defect anomaly is canceled by bulk inflow. A negative-chirality string mode has
\[
\partial_a j^a_{\rm string}=-\frac{1}{2\pi}E_1.
\]
The bulk Goldstone–Wilczek current is
\[
J^\mu_{\rm bulk}
=\frac{1}{8\pi^2}\epsilon^{\mu\nu\lambda\rho}
F_{\nu\lambda}\partial_\rho\alpha.
\]
For winding one,
\[
\int_0^{2\pi}r\,d\theta\,J_r=\frac{E_1}{2\pi},
\]
which exactly cancels the string anomaly. The lattice one-loop current changes at the same parameter values as the localized spectrum, demonstrating anomaly inflow across the lattice topological phases.

## 4. Chiral quasiparticles in topological materials

In condensed matter, a Weyl fermion is a twofold band crossing whose Berry curvature has monopole flux
\[
C=\frac{1}{2\pi}\oint_S\boldsymbol\Omega\cdot d\mathbf S.
\]
For a simple Weyl node, $C=\pm1$. The total Berry charge in a periodic Brillouin zone vanishes, and surface Fermi arcs connect projections of nodes with compensating charge.

Chiral crystals support higher-fold crossings. Multifold fermions include three-, four-, and sixfold degeneracies protected by crystal symmetries. A threefold spin-1 node has ideal spectrum
\[
E=\hbar v_Fk,\quad0,\quad-\hbar v_Fk
\]
and Chern numbers
\[
(C_1,C_2,C_3)=(-2,0,2).
\]
A spin-$3/2$ fourfold node has
\[
(C_1,C_2,C_3,C_4)=(-3,-1,1,3)
\]
at rotationally invariant points. Sixfold nodes can realize two copies of the threefold sequence [1806.09642].

Crystal chirality is the absence of inversion, mirror, and roto-inversion symmetries. Sixty-five crystallographic space groups are chiral. Their symmetry representations can enforce three-, four-, and sixfold crossings at high-symmetry momenta without requiring conventional band inversion. The corresponding higher-charge nodes generate long surface Fermi arcs and helicoid-like surface dispersions [1810.09970].

The relation between real-space and momentum-space chirality was demonstrated by comparing CoSi and RhSn, which belong to space group $P2_13$ but have opposite structural handedness. In the spin-orbit-free description, CoSi has charges
\[
C_\Gamma=+2,\qquad C_R=-2,
\]
whereas RhSn has
\[
C_\Gamma=-2,\qquad C_R=+2.
\]
Angle-resolved photoemission spectroscopy observed bulk structures consistent with calculations, surface Fermi arcs, and opposite helical surface-band winding. The work established that reversing structural chirality reverses the chiral charges and surface-state handedness [1912.02383].

Multifold topology generates response functions that differ from those of Weyl nodes. The gyrotropic magnetic effect,
\[
j_i(\omega)=\alpha_{ij}(\omega)B^j(\omega),
\]
depends on orbital magnetic moments as well as Berry curvature. A band can have zero Berry curvature but finite gyrotropic response because orbital magnetic moments contain different interband energy denominators. The circular photogalvanic effect,
\[
\frac{dj_i}{dt}
=\beta_{ij}(\omega)
[\mathbf E(\omega)\times\mathbf E^*(\omega)]^j,
\]
can become quantized when optically activated resonant surfaces are closed and unwanted transitions are blocked. In the ideal linear model, the response is determined by Chern-number sums; realistic quadratic corrections, remote bands, additional pockets, and finite scattering limit exact quantization [1806.09642].

Chiral quasiparticles can persist beyond perfect crystals. In amorphous chiral metals, crystal momentum is replaced by pseudo-momentum, and topology is extracted from disorder-averaged Green functions and Wilson loops. Long-range order in local structural chirality, rather than positional order, stabilizes Weyl, double-Weyl, and multifold nodes. The amorphous Wilson-loop winding is $C=\pm1$ for Kramers–Weyl fermions and $C=\pm2$ for a threefold multifold model. Near equal fractions of opposite local handedness, the Wilson spectrum becomes nonsmooth, signaling a topological transition or gap closing [2512.19791].

A related proposed extension is the rank-2 chiral fermion. Its two-dimensional dispersion is
\[
E(\mathbf k)=\hbar v\xi k_xk_y-\mu.
\]
It has no net ordinary one-dimensional chirality, but supports a mixed charge-momentum anomaly:
\[
\partial_\mu\mathcal J_a^\mu
=\chi_2\sigma_{ab}
\frac{e\Lambda^2}{4\pi^2}E_b.
\]
The reciprocal anomaly is
\[
\partial_\mu j^\mu
=\chi_2\frac{e\Lambda^2}{4\pi^2}\mathcal E_{xy}.
\]
These modes occur as anomalous surfaces of a three-dimensional time-reversal-invariant Weyl semimetal whose four Weyl nodes form a momentum-space quadrupole. Bulk anomaly inflow cancels the surface nonconservation [2102.08959].

## 5. Semiclassical dynamics, vortical media, and cosmology

A semiclassical Weyl fermion with Berry connection $\mathbf a(\mathbf p)$ has action
\[
S_{\rm chiral}
=\int\left[
(\mathbf p+e\mathbf A)\cdot\dot{\mathbf x}
-(|\mathbf p|+e\phi)
-\mathbf a(\mathbf p)\cdot\dot{\mathbf p}
\right]dt.
\]
For helicity $s=1/2$, the Berry curvature is
\[
\boldsymbol\Theta(\mathbf p)
=s\frac{\mathbf p}{|\mathbf p|^3}.
\]
The resulting equations contain anomalous velocity terms,
\[
\dot{\mathbf x}
=\frac{\mathbf p}{|\mathbf p|}
+e\mathbf E\times\boldsymbol\Theta
+e\boldsymbol\Theta(\dot{\mathbf x}\cdot\mathbf B),
\]
and the effective symplectic mass
\[
m^*=1+e\mathbf B\cdot\boldsymbol\Theta.
\]
The position coordinates are noncommutative:
\[
\{x^i,x^j\}
=-\frac{\epsilon^{ijk}\Theta_k}
{1+e\mathbf B\cdot\boldsymbol\Theta}.
\]

The Berry term can be derived from the classical spin two-form of a massless spinning particle. The unreduced system has Wigner–Souriau translations and a three-dimensional kernel. Imposing the enslavement condition
\[
\mathbf s=s\widehat{\mathbf p}
\]
reduces the spin variables and produces the Berry curvature. Ordinary boosts do not preserve this gauge choice, so the reduced chiral model realizes Poincaré symmetry in a momentum-dependent or twisted form [1406.0718].

Electromagnetic coupling breaks the Wigner–Souriau shift symmetry. For minimal coupling, the gyromagnetic ratio is $g=0$, and constant crossed fields yield Hall-type trajectories with
\[
|\dot{\mathbf r}|>1
\]
and divergence as $\mathbf p\cdot\mathbf B\to0$. These superluminal motions indicate the pathological character of the minimally coupled classical model. For $g=2$ in a constant magnetic field, the equations instead give
\[
\frac{d\mathbf r}{dt}=\frac{\mathbf p}{\mathcal E},
\]
with momentum and spin precessing at the same angular velocity. The trajectory is a helix or spiral on a cylinder.

In a vortical QED or QCD plasma, chiral kinetic theory predicts a polarization correction transverse to both vorticity and momentum:
\[
\Delta\mathcal P^i\propto
\epsilon^{ijk}\omega_jp_k.
\]
The effect has opposite signs for left- and right-handed fermions and cancels in an unpolarized ensemble. For a momentum-anisotropic probe population it can generate a transverse vector current,
\[
\Delta\mathbf J_V\propto\boldsymbol\xi\times\boldsymbol\omega.
\]
The derivation uses spinor Wigner functions and the Kadanoff–Baym equation rather than an explicit scalar Berry-curvature variable. The effect is interaction-induced, depends on soft Coulomb or magnetic scattering, and is damped by collisional factors [2008.03862].

Helicity-asymmetric production of chiral fermions can source chiral gravitational waves. In an expanding universe,
\[
h_{\mu\nu}''+2\mathcal Hh_{\mu\nu}'-\nabla^2h_{\mu\nu}
=\frac{2}{m_p^2}T_{\mu\nu}.
\]
A helicity imbalance produces a parity-odd stress tensor and unequal circular tensor correlators:
\[
M_{++}\neq M_{--}
\quad\Longrightarrow\quad
\langle h_+h_+\rangle\neq\langle h_-h_-\rangle.
\]
In a pseudoscalar inflation model, the parameter
\[
\vartheta=-\frac{C\dot\phi}{fH}
\]
controls helicity-selective production. For $\vartheta<0$, $|\vartheta|\gg1$, and $m\ll H$, negative-helicity modes are enhanced. The resulting spectrum is nearly scale invariant and slightly red. For one species,
\[
\Delta\chi\simeq35\frac{H^2}{m_p^2},
\]
and for $H\lesssim10^{13}$ GeV the estimated chirality is $\Delta\chi\lesssim10^{-8}$ [1607.03916].

## 6. Model building, computation, and experimental control

Chiral gauge theories require anomaly cancellation. For a $U(1)$ theory of left-handed Weyl fields,
\[
\sum_iq_i=0,\qquad \sum_iq_i^3=0.
\]
A concrete anomaly-free dark-sector spectrum contains
\[
3(+1)+2(-2)+2(-3)+3(+4)+(-5)=0
\]
and
\[
3(+1)^3+2(-2)^3+2(-3)^3+3(+4)^3+(-5)^3=0.
\]
A scalar $\phi$ with charge $+1$ breaks $U(1)_\nu$. The resulting spectrum contains three massive Dirac $\chi_i$, two massive Dirac $N_i$, and one massless chiral $\nu^c$ per dark family. Accidental global symmetries stabilize the lightest $\chi$ as a dark-matter candidate. A heavy vector-like mediator generates the dimension-five operator
\[
\frac{\kappa_L\kappa_i}{\Lambda}(1_i^+L)(H\phi),
\]
and hence small Dirac neutrino masses
\[
m_i^D=\frac{\kappa_L\kappa_i vv_\phi}{\Lambda}.
\]
The construction requires at least four mostly sterile neutrino states when two independent active-neutrino masses are generated [1507.00916].

Overlap fermions are computationally expensive because applying the matrix sign function requires approximations or spectral methods. A contour-integral formulation avoids explicit low-mode calculation:
\[
\operatorname{sign}(H_W)
=P_+(H_W)-P_-(H_W),
\]
with
\[
P_\pm(H_W)
=\oint_{\Gamma_\pm}\frac{dz}{2\pi i}
(zI-H_W)^{-1}.
\]
The method reduces sign-function application to shifted linear systems,
\[
(zI-H_W)\mathbf y=\mathbf x,
\]
which can be distributed over contours and quadrature points. On an $8^4$ lattice, approximately $N_{\rm qtot}\gtrsim500$ contour divisions gave sign-function violation
\[
\sigma\lesssim10^{-14},
\]
while the implementation exhibited strong scaling to 120 CPU processes [220.009?]. The reported study is arXiv:2204.01583.

In one spatial dimension, the overlap and Wilson operators coincide for the relevant parameters:
\[
D_{\rm overlap}=D_{\rm W}(m=0).
\]
This permits local Wilson quantum circuits to simulate a theory with the modified conserved chiral charge. In a $Z_2$ gauge model, the fermionic charge commutes with the fermion Hamiltonian but not with the gauge Hamiltonian, so dynamical gauge fields produce time-dependent chirality. The construction has been proposed for quantum-computer studies of anomaly-related dynamics, although the demonstrated $Z_2$ model does not possess the continuum limit of QED or QCD [2305.18934; 2309.10507].

Chirality can also be manipulated optically. In CoSi, circularly polarized light produces a Floquet correction that shifts rather than gaps multifold crossings. For a spin-1 node and a double Weyl node, the shifts scale as
\[
\delta_k\propto\frac{\gamma\chi_kA^2}{\Omega},
\]
where $\gamma$ is light helicity and $\chi_k$ is the crossing chirality. Opposite-charge crossings at $\Gamma$ and $R$ move in opposite directions. The drive breaks time reversal transiently and can generate anomalous Hall signals detectable through time-resolved Kerr or Faraday spectroscopy [2408.03115].

Other proposed probes include circular photogalvanic and gyrotropic magnetic effects, chiral magnetic photocurrents, helical magnetic effects, quantum oscillations, strain-induced currents, and ultrafast terahertz emission. In Weyl systems, the chiral anomaly,
\[
\dot\rho_5
=\frac{e^2}{2\pi^2}\mathbf E\cdot\mathbf B
-\frac{\rho_5}{\tau_V},
\]
generates a chiral imbalance that can produce the chiral magnetic effect,
\[
\mathbf j_{\rm CME}
=\frac{e^2}{2\pi^2}\mu_5\mathbf B.
\]
Strain and torsion act as effective gauge fields with
\[
\partial_\mu j^\mu
=\frac{\chi}{4\pi^2}\mathbf E_{\rm eff}\cdot\mathbf B_{\rm eff}.
\]
These effects are mechanism-dependent and require suitable symmetry, disorder, scattering, chemical-potential, and frequency regimes [2112.13749].

Chiral Majorana fermions constitute another condensed-matter realization. A chiral Majorana mode is a unidirectional self-conjugate edge excitation,
\[
\gamma(x)=\gamma^\dagger(x),\qquad
\{\gamma(x),\gamma(x')\}=\frac{\delta(x-x')}{2},
\]
with Hamiltonian
\[
H_M=-i\hbar v_F\int dx\,\gamma(x)\partial_x\gamma(x).
\]
It is an Abelian propagating fermion, not automatically a non-Abelian Majorana zero mode. In a quantum anomalous Hall insulator–topological-superconductor junction, propagation and recombination of Majorana components can implement the same unitary transformation as localized Majorana-mode braiding. A Corbino geometry realizes
\[
U_{\rm Corbino}=ZH\,R_{\phi_G}\,ZH,
\]
with gate phase
\[
\phi_G=\frac{eV_Gl_G}{\hbar v_F}.
\]
Coherent propagation produces conductance oscillations, whereas incoherent mixtures give a constant half-quantized conductance. The architecture provides a braid-equivalent single-qubit operation but does not establish complete universal, fault-tolerant quantum computation [1712.06156].

Chirality therefore denotes a family of related but nonidentical structures: Lorentz representations, gauge charges, helicity, lattice-modified axial symmetry, topological index, Berry monopole charge, defect-bound propagation, structural handedness, or nonequilibrium optical response. The common mathematical themes are projection onto handed sectors, spectral flow, anomaly cancellation, and quantized topological data, while the physical interpretation depends on whether the fermion is elementary, lattice-regularized, a quasiparticle, a defect mode, or an emergent crystalline or amorphous excitation.

Source: https://www.emergentmind.com/topics/chiral-fermions