---
title: Half-Quantized Hall Metal
url: https://www.emergentmind.com/topics/half-quantized-hall-metal
type: topic
---

# Half-Quantized Hall Metal

Searching arXiv for recent and foundational papers on half-quantized Hall metal, parity anomaly, and semi-magnetic topological insulators.
A half-quantized Hall metal is a metallic or semimetallic state in which the Hall response is pinned to a half-integer multiple of the conductance quantum while the longitudinal channel remains finite, so that $\sigma_{xy}=\frac{\nu}{2}\frac{e^2}{h}$ with $\nu\in\mathbb{Z}$ can coexist with $\sigma_{xx}\neq0$. In the recent literature, the term is used most directly for semi-magnetic topological-insulator thin films and related single-Dirac-cone platforms, where one surface is magnetically gapped, another remains gapless, and the resulting transport is tied to the parity anomaly of a two-dimensional Dirac fermion rather than to the conventional bulk-gap paradigm of the integer quantum Hall effect [2308.04718][2508.19534][2409.15655].

## 1. Defining characteristics

The defining feature of the half-quantized Hall metal is the coexistence of a half-quantized transverse response and metallic longitudinal transport. In the terminology used for disordered magnetic topological insulators, the half-quantized Hall metal (HQHM) is a metallic state in which the Hall conductance is pinned near
\[
\sigma_{xy}=\frac{1}{2}\,\frac{e^2}{h},
\]
while the system still has a finite longitudinal conductivity, $\sigma_{xx}\neq0$ [2508.19534]. This differs from ordinary Hall quantization, which is usually associated with an insulating quantum Hall state or a quantum anomalous Hall insulator.

A complementary formulation emphasizes that the half-quantized Hall phase is not a globally gapped Chern insulator but a topological metallic or semimetallic phase in which the Hall conductance is fixed by the Fermi-surface Berry connection. For a partially filled band,
\[
\sigma_{xy}^{n}=\frac{e^2}{2\pi h}\oint_{\mathrm{C}_{FS}} d\mathbf{l}\cdot\mathbf{A}_{n}(\mathbf{k}),
\]
and the proposed classification is
\[
\sigma_H=\frac{\nu}{2}\frac{e^2}{h}.
\]
The key point is that the relevant protecting symmetries may hold only locally near the Fermi surface, not globally throughout the Brillouin zone; this distinguishes the state from an ordinary two-dimensional metallic ferromagnet with non-quantized anomalous Hall conductivity [2409.15655].

Within semi-magnetic topological-insulator language, the same phase is often described as a “parity anomalous semimetal” or a half-quantized Hall metal because the system remains gapless or metallic at low energy even though its Hall response is pinned to $\pm \frac{1}{2}\frac{e^2}{h}$ [2209.03053].

## 2. Canonical realization in semi-magnetic topological insulators

The canonical platform is a semi-magnetic topological-insulator thin film or heterostructure in which only one surface is magnetized. In one formulation, the top surface is made magnetic by a Zeeman term, opening a Dirac mass gap, while the bottom surface remains gapless; side surfaces also contribute gapless transport channels, so the device is not insulating and therefore has $\sigma_{xx}\neq0$ [2201.12600]. In another formulation, magnetic doping is applied only on the top layers of a topological-insulator slab, producing an asymmetric surface structure in which one surface is magnetically gapped and the other remains gapless and supports a single Dirac cone [2508.19534].

This single gapless Dirac cone is the crucial ingredient. In the finite-field parity-anomaly analysis of a semimagnetic topological-insulator thin film, the low-energy sector is a massless Dirac cone, while the high-energy sector contains a symmetry-breaking $b$-term needed on the lattice to avoid fermion doubling and to realize a single Dirac cone. The model Hamiltonian is written as
\[
H=\begin{pmatrix} M(\Pi^{2}) & v(\Pi_x-i\Pi_y)\\ v(\Pi_x+i\Pi_y) & -M(\Pi^{2}) \end{pmatrix},
\]
with $\Pi_\alpha=-i\hbar\partial_\alpha-eA_\alpha$ and
\[
M(\Pi^2)=f\!\left(k_c^2-\hbar^{-2}\Pi^2\right)m(\Pi^2),\qquad
m(\Pi^2)=-b(\Pi^2-\hbar^2k_c^2).
\]
This is the setting in which the parity anomaly can be realized in condensed matter because the surface hosts a single Dirac cone, unlike graphene with two cones [2308.04718].

A closely related tight-binding description treats a thin film of $(\mathrm{Bi,Sb})_2\mathrm{Te}_3$ with Cr doping only on the top surface. With open boundary conditions along the film thickness, the low-energy spectrum contains a gapless Dirac cone mainly localized on the bottom surface and a gapped Dirac cone mainly localized on the top magnetic surface. The gapless cone remains effectively gapless because its wavefunction is localized away from the magnetic layer, and the resulting state is described as a condensed-matter realization of a single effective gapless Dirac cone with half-quantized Hall conductance [2209.03053].

## 3. Parity anomaly and the origin of the half-quantized response

In this class of systems, the half-quantized Hall response is identified with the parity anomaly of a massless Dirac fermion. For the semimagnetic topological-insulator thin film in a finite magnetic field, the anomalous Hall conductance contains a half-quantized piece,
\[
\sigma_{xy}^{A}=-\frac{e^2}{2h}\left[\operatorname{sgn}(b)+\frac{M(\hbar^2k_F^2)}{|\mu|}\right].
\]
In the parity-symmetric low-energy regime near the Fermi surface, $k_F<k_c$ so that $M(\hbar^2k_F^2)=0$, and therefore
\[
\sigma_{xy}^{A}=-\frac{e^2}{2h}\operatorname{sgn}(b).
\]
At finite field, the Hall conductance is written as
\[
\sigma_{xy}=-\frac{e^2}{2h}\operatorname{sgn}(b)+\sigma_{xy}^{0},
\]
where
\[
\sigma_{xy}^{0}=\frac{e^2}{2h}\left[1-2\sum_{ns}s\,n_F(s\varepsilon_{ns}-s\mu)\right].
\]
The term $-\frac{e^2}{2h}\operatorname{sgn}(b)$ is interpreted as the parity-anomaly term from occupied states far below the Fermi level or, equivalently, from the high-energy regularization sector [2308.04718].

The same analysis relates the anomaly to spectral asymmetry through the Atiyah-Patodi-Singer $\eta$-invariant,
\[
\eta_H=\Omega\frac{eB}{2\pi\hbar}\left[\operatorname{sgn}(\varepsilon_0)-\operatorname{sgn}(b)\right].
\]
The unusual $\operatorname{sgn}(b)$ contribution is the ultraviolet part of the spectrum and is precisely the ingredient that makes the effect a parity anomaly rather than an ordinary Dirac Landau-level effect [2308.04718].

A second route to the same conclusion proceeds from parity symmetry on the Fermi surface. If the Hamiltonian on the Fermi surface is parity invariant, the Berry-phase line integral obeys a half-integer quantization condition, so that
\[
\sigma_H=\frac{\nu}{2}\frac{e^2}{h}.
\]
For a Dirac cone with Berry phase $\pm\pi$, one has $\nu=\pm1$ and therefore
\[
\sigma_H=\pm \frac{1}{2}\frac{e^2}{h}.
\]
This formulation underlies the proposal of a $\frac12\mathbb{Z}$ topological invariant and clarifies why half-quantization can survive in a metallic state when the relevant symmetry is local to the Fermi-surface states [2209.03053][2409.15655].

A significant qualification concerns the lattice realization of the surface response. In a full lattice model with a finite Brillouin zone, a single isolated band cannot carry a standalone half-integer Hall conductance because of the TKNN theorem. One detailed lattice analysis concludes that the gapped and gapless surface bands are characterized by integer-quantized Hall conductance and half-quantized Hall conductance, respectively, and that the exact half-quantized Hall conductance in the lattice model originates from the gapless surface band rather than from an individual gapped surface band [2304.04229].

## 4. Transport mechanisms in a metallic environment

The transport phenomenology of the half-quantized Hall metal is not the standard dissipationless edge-state picture of the integer quantum Hall effect. In the transport theory of a semi-magnetic topological insulator with both gapped and gapless Dirac surfaces, the half-quantized Hall plateau is traced to a half-quantized chiral edge current in the presence of strong dephasing. Using a six-terminal Landauer-Büttiker formulation,
\[
I_p=\frac{e^2}{h}\sum_{q\ne p}\left(T_{qp}V_p-T_{pq}V_q\right),
\]
the edge transmission asymmetry $T_d$ is found numerically to be $+\tfrac12$ on one edge and $-\tfrac12$ on the opposite edge, while the large-size conductance obeys
\[
\sigma_{xy}=\frac{e^2}{h}t_d,\qquad \sigma_{xx}=\frac{e^2}{h}t_n.
\]
Since $t_d=\frac12$, the Hall conductance becomes
\[
\sigma_{xy}=\frac{e^2}{2h}.
\]
In this picture, strong dephasing helps quantization by separating opposite-edge currents in a classical-metal regime rather than by producing an insulating gap [2201.12600].

A related relativistic-quantum-Hall analysis shows that the half-integer Hall conductivity of a single Dirac cone can be understood through dissipative half-integer quantized chiral channels at the interface between a Dirac metal region and a magnetic-field region. The total number of interface channels is
\[
\frac{h}{e}\frac{\partial J_c(E_F)}{\partial E_F}
=
\left\lfloor \frac{(E_F/\epsilon_D)^2}{2}\right\rfloor + \frac{1}{2},
\]
leading, in the Ohmic limit, to
\[
\sigma_{xy}=
\left(
\left\lfloor \frac{(E_F/\epsilon_D)^2}{2}\right\rfloor+\frac12
\right)\frac{e^2}{h}.
\]
The crucial point is that the interface channels are dissipative and that direct observation of the half-integer Hall response requires Ohmic scaling of the longitudinal conductance [2307.13326].

Another formulation replaces localized edge modes by a nonstandard bulk-edge correspondence. In parity anomalous semimetals, the half-quantized Hall effect is realized via bulk massless Dirac fermions, while the required Berry curvature is supplied by massive Dirac fermions separated in momentum space or real space. The resulting equilibrium edge current is not exponentially localized; near a boundary,
\[
j_y^s(x)\propto x^{-3/2}\cos\!\left(2k_F x-\frac{3\pi}{4}\right),
\]
and the integrated response yields
\[
\sigma_H=\frac{e^2}{2h}.
\]
This power-law edge-current profile is a characteristic signature of the metallic, parity-anomalous regime [2202.08493].

## 5. Disorder, weak antilocalization, and crossover structure

Disorder does not simply destroy the half-quantized Hall metal. In disordered semimagnetic topological insulators, the half-quantized Hall effect is reported to be robust in weakly disordered systems because the relevant low-energy states are governed by a single gapless Dirac cone with a $\pi$ Berry phase. The weak-antilocalization correction obeys
\[
\delta\sigma_{qi}\simeq \frac{e^2}{\pi h}\ln\left(\frac{L}{l_e}\right),
\]
and in the single-cone HQHM regime the scaling parameter is $\alpha=1$. The same study identifies an intermediate marginal metal between weak antilocalization and Anderson insulation, with the phase diagram in the $E_F-W$ plane described as HQHM for weak disorder, roughly $W<2.6$ eV, MM for intermediate disorder, roughly $2.6\sim 13.47$ eV, and AI for stronger disorder, $W>13.47$ eV. In the marginal-metal regime the normalized localization length is scale invariant, and the correlation length is fitted by the Berezinskii-Kosterlitz-Thouless-like form
\[
\xi=\exp\left[\frac{b}{\sqrt{(W-W_c)/W_c}}\right],
\]
with fitted exponent $\nu=0.176\pm0.005$ [2508.19534].

Finite magnetic field and disorder together also generate a characteristic crossover. In the semimagnetic topological-insulator thin-film calculation, the zero-field metallic regime exhibits
\[
\sigma_{xy}(0)=-\frac{e^2}{2h}\operatorname{sgn}(b),
\]
whereas at strong field, when the chemical potential lies between Landau levels, the Hall conductance becomes integer quantized as $0$ or $1$ in units of $e^2/h$. If $\mu$ lies in the valence band, the conductance crosses over from $\tfrac12\to1$; if $\mu$ lies in the conduction band, it crosses over from $\tfrac12\to0$. The anomalous half piece persists as an offset from occupied states far below the Fermi level, while disorder broadening controls how clearly the integer plateau emerges [2308.04718].

The robustness issue also appears in other single-cone parity-anomalous semimetals. In an anisotropic Dirac semimetal on a deformed honeycomb lattice with broken time-reversal symmetry, the intrinsic Hall conductivity is
\[
\sigma_{xy}=-\frac{e^2}{2h}\,\mathrm{sgn}(v)\,\mathrm{sgn}(\lambda),
\]
or equivalently
\[
\sigma_{xy}=\frac{e^2}{2h}\bigl[1-\mathrm{sgn}(\delta)\bigr].
\]
Within self-consistent Born analysis, the disorder-corrected response can be approximated as
\[
\sigma_{xy}\approx \frac{e^2}{h}\left(\frac12-4F_0\eta\right),
\]
showing that moderate disorder reduces but does not immediately eliminate the half-integer Hall response, whereas the strong-disorder limit drives $\sigma_{xy}\to0$ [2302.13499].

## 6. Topological classification, neighboring phases, and major distinctions

The half-quantized Hall metal has motivated an extension of topological classification from gapped phases to metallic and semimetallic Hall states. The proposed $\frac12\mathbb{Z}$ invariant is determined by the line integral of the intrinsic anomalous Hall conductance on the Fermi surface and can be protected by local unitary symmetries such as $\sigma_v$ and $C_{2x}$ or by local anti-unitary symmetries such as $T$, $C_{nz}T$, $IT$, $\sigma_hT$, and $S_{nz}T$. A central feature of this framework is that the whole system must break the symmetry globally in order to have a nonzero Hall effect, while the Fermi-surface states can still retain the symmetry locally; this local-symmetry viewpoint is what separates the half-quantized Hall metal from an ordinary metallic ferromagnet [2409.15655].

Several adjacent phases are conceptually close but should not be conflated with the half-quantized Hall metal. Light-induced half-quantized Hall states and axion-insulator or Chern-insulator phases in topological-insulator heterostructures rely on gapped surface Dirac cones and are therefore not metallic in the same sense, even though each gapped surface contributes $\pm\frac{e^2}{2h}$ [2306.03187]. Ferroelectrically switchable half-quantized Hall transport in a MnBi$_2$Te$_4$/Sb$_2$Te$_3$ heterostructure also uses the “one surface gapped, one surface gapless” geometry and supports $\pm e^2/2h$, but the emphasis there is on antiferromagnetism, inversion breaking, and sliding-controlled reversal of the half-quantized Hall conductivity [2507.03985]. By contrast, “metallic quantized anomalous Hall effect without chiral edge states” describes an integer-quantized Hall metal, $\sigma_{xy}=\pm e^2/h$ with $\sigma_{xx}\neq0$, assembled from two anomalous half-quantized Dirac contributions rather than from a single half-quantized one [2308.05963].

A further distinction is needed from “Chern half metal” and “intrinsic Chern half-metal.” In those materials, one spin channel is metallic while the other is topological, or the anomalous Hall conductivity is large and Berry-curvature driven, but the response is not an exactly half-quantized Hall plateau in the strict sense. The terminology therefore overlaps only partially with half-quantized Hall metal physics [1401.5453][2505.00840].

The topic is also entangled with an experimental caution. In quantum anomalous Hall–superconductor hybrids, a half-quantized two-terminal conductance plateau can arise from strong electrical coupling, Andreev processes, and series-resistance or equilibration effects, so that a half-quantized plateau is not, by itself, unique evidence for chiral Majorana fermions. This does not negate half-quantized Hall-metal physics, but it fixes an important interpretive boundary around what a “half-quantized” transport signature can and cannot establish on its own [1904.06463].

Taken together, the contemporary literature defines the half-quantized Hall metal as a parity-anomaly-driven topological metal or semimetal, most naturally realized in semi-magnetic topological-insulator structures with an isolated gapless Dirac cone. Its core signatures are a Hall conductance pinned to $\pm e^2/2h$ or more generally $\nu e^2/2h$, finite longitudinal transport, a decisive role for high-energy regularization or deep occupied states, and a transport mechanism that departs from the conventional dissipationless edge-state picture of insulating Hall phases [2308.04718][2201.12600][2508.19534].

Source: https://www.emergentmind.com/topics/half-quantized-hall-metal