Type-II Quantum Spin Hall Phase

• Type-II QSH phase is a two-dimensional topological insulator characterized by even spin-Chern invariants and edge transport protected by spin symmetry rather than time-reversal symmetry.
• It appears in systems like twisted bilayer WSe₂ and InAs/GaSb, where multiple helical edge pairs result in nearly quantized conductance plateaus measured in units of e²/h.
• Robust against first-order spin mixing via spin-U(1) quasisymmetry, its stability hinges on preserving a conserved spin component despite broken time-reversal symmetry.

The Type-II quantum spin Hall (QSH) phase is a nonconventional two-dimensional topological insulating phase in which edge transport is governed by spin-Chern topology and spin-selective symmetry protection rather than solely by the conventional time-reversal-symmetric $\mathbb{Z}_2$ classification. In one common recent nomenclature, it denotes an even spin-Chern QSH state with $\mathbb{Z}_2=0$, exemplified by the experimentally observed double QSH phase in twisted bilayer WSe$_2$, where two Kramers pairs of helical edge states appear at $\nu=4$ with $|C_s|=2$ (Kang et al., 2024). In a broader usage, it also includes QSH states that persist under explicit time-reversal-symmetry breaking provided a conserved or quasi-conserved spin $U(1)$ survives, as in the magnetic-field-stabilized Mott QSH state in twisted WSe$_2$ (Jin et al., 17 May 2026). A symmetry-based formulation identifies spin $U(1)$ quasisymmetry as the mechanism that suppresses first-order spin mixing, allowing nearly quantized spin Hall response and weakly gapped or effectively gapless edge transport even when the conventional $\mathbb{Z}_2$ index is trivial or inapplicable (Liu et al., 2024).

1. Terminology and scope

The expression “Type-II QSH” is not used uniformly across the literature. In moiré WSe$_2$, “double QSH” is stated to correspond exactly to “Type-II QSH,” with Type-I reserved for the conventional single-pair QSH insulator (Kang et al., 2024). In the spin-$\mathbb{Z}_2=0$0-quasisymmetry framework, Type-II denotes an even spin-Chern phase with trivial $\mathbb{Z}_2=0$1 index, robust nearly quantized spin Hall conductance, and multiple helical edge pairs protected against first-order spin mixing (Liu et al., 2024). In the correlated moiré literature, the label is extended to a QSH phase that survives explicit time-reversal-symmetry breaking because $\mathbb{Z}_2=0$2 remains conserved, even when the bulk gap is interaction-driven rather than single-particle in origin (Jin et al., 17 May 2026).

A different usage appears in recent work on unconventional magnetism, where “type-II QSHI” is defined by spin-dependent band inversions at distinct momenta and by spin-chiral rather than helical boundary modes; in that terminology, stacking is additive in spin Chern number and does not trivialize as in the $\mathbb{Z}_2=0$3 case (Tan et al., 7 Aug 2025). By contrast, in the earlier InAs/GaSb literature, “Type II” in the title refers to the staggered semiconductor band alignment of the heterostructure, not to a Type-I/Type-II QSH taxonomy (0801.2831).

Context	Meaning of “Type-II”	Representative source
Moiré WSe$\mathbb{Z}_2=0$4	Double QSH with two Kramers pairs, $\mathbb{Z}_2=0$5	(Kang et al., 2024)
Spin-$\mathbb{Z}_2=0$6 quasisymmetry	Even spin-Chern, $\mathbb{Z}_2=0$7 QSH	(Liu et al., 2024)
Correlated moiré WSe$\mathbb{Z}_2=0$8	TRS-broken, spin-conserved Mott QSH	(Jin et al., 17 May 2026)
Unconventional magnetism	Spin-chiral, high-$\mathbb{Z}_2=0$9 stacked phase	(Tan et al., 7 Aug 2025)
InAs/GaSb “Type II semiconductors”	Staggered band alignment	(0801.2831)

This suggests that “Type-II QSH” should be read operationally rather than nominally: the relevant questions are which symmetry is protecting the edge modes, which topological invariant is quantized, and whether the boundary supports one or multiple spin-filtered channel pairs.

2. Topological invariants, edge counting, and symmetry protection

When a spin component is conserved, the topological classification is integer-valued. For spin-resolved Chern numbers $_2$0 and $_2$1,

$_2$2

and the number of helical edge pairs is

$_2$3

In this limit, the ideal two-terminal edge conductance is

$_2$4

so a single-pair QSH phase has $_2$5, whereas a double QSH phase has $_2$6 (Kang et al., 2024).

Once exact spin conservation is lost, the relevant construction uses the occupied-band projector

$_2$7

the projected spin operator

$_2$8

and the spectral projectors $_2$9 onto its positive- and negative-eigenvalue subbundles. One then defines

$\nu=4$0

with $\nu=4$1 the Chern numbers of the projected-spin subspaces (Liu et al., 2024). In the TRS-broken Kane–Mele analysis, the spin Chern numbers $\nu=4$2 remain quantized as long as the bulk gap and the spectral gap of the projected spin operator stay open; this is precisely the regime in which a TRS-broken QSH phase persists before transitioning to a quantum anomalous Hall phase at a bulk-gap closing (Yang et al., 2011).

The main obstruction to higher-pair QSH phases is spin mixing. Multiple helical pairs are generically unstable in real materials because generic spin mixing gaps the edge states. The spin-$\nu=4$3-quasisymmetry framework isolates the mechanism by which realistic systems can evade this instability: one decomposes the low-energy Hamiltonian into a spin-preserving part $\nu=4$4 and a spin-mixing perturbation $\nu=4$5, with

$\nu=4$6

but within the relevant low-energy subspace

$\nu=4$7

First-order spin mixing is therefore eliminated, so deviations from exact quantization appear only at second order through virtual mixing with remote bands (Liu et al., 2024). In moiré transition-metal dichalcogenides, the closely related physical statement is that an Ising spin axis strongly suppresses spin mixing and thereby stabilizes phases with $\nu=4$8 (Kang et al., 2024).

3. Canonical models and early platforms

A key historical platform is the inverted InAs/GaSb/AlSb quantum well. InAs/GaSb is a prototypical Type-II heterostructure in which the GaSb valence-band maximum lies above the InAs conduction-band minimum by about $\nu=4$9. In the quantum-well geometry, the lowest conduction-like subband $|C_s|=2$0 is localized in InAs and the lowest heavy-hole-like subband $|C_s|=2$1 in GaSb. As the well thicknesses increase, $|C_s|=2$2 drops and $|C_s|=2$3 rises until they invert; at fixed GaSb thickness $|C_s|=2$4, the critical InAs thickness is $|C_s|=2$5. Hybridization between the spatially separated subbands opens a finite-$|C_s|=2$6 gap $|C_s|=2$7, and a BHZ-type low-energy model supplemented by bulk inversion asymmetry (BIA) and structural inversion asymmetry (SIA) captures the resulting QSH phase and its gate tunability (0801.2831).

The later InAs/GaSb bilayer experiments established the magnetic robustness expected of a spin-Chern-protected QSH phase. Wide conductance plateaus of $|C_s|=2$8 persisted to $|C_s|=2$9 applied in-plane magnetic field, and no bulk-gap closing was observed up to $U(1)$0 perpendicular field while the Fermi level remained inside the bulk gap. The reported phenomenology was interpreted as first evidence for a QSH insulator protected by a spin Chern invariant rather than by time-reversal symmetry alone (Du et al., 2013).

A complementary model perspective comes from the Kane–Mele Hamiltonian with intrinsic SOC, Rashba SOC, and a uniform exchange field. In that setting, the TRS-broken QSH phase with $U(1)$1 persists for exchange field $U(1)$2, where

$U(1)$3

At $U(1)$4 the bulk gap closes and a transition to a quantum anomalous Hall phase occurs; adding a staggered sublattice potential likewise drives a transition to an ordinary insulator only through bulk-gap closure (Yang et al., 2011). This established explicitly that a QSH-like phase can survive broken time-reversal symmetry if the spin-Chern topology remains well defined.

Another route is interaction-enabled rather than SOC-enabled. In twisted bilayer graphene under perpendicular magnetic field and interlayer bias, the “pseudo-QSH” phase consists of helical spin-polarized edge states formed when effectively decoupled layers are tuned to $U(1)$5 and $U(1)$6. The bulk gap is interaction-induced in the zero Landau level, and protection relies on spin conservation and the absence of interlayer backscattering rather than on time-reversal symmetry (Finocchiaro et al., 2016). This route is conceptually close to later TRS-broken moiré realizations.

4. Double QSH in twisted bilayer WSe$U(1)$7

The clearest experimental realization of Type-II QSH in the even-spin-Chern sense is twisted bilayer WSe$U(1)$8 at moiré hole filling $U(1)$9. The platform uses twist angles $_2$0–$_2$1 degrees, with devices D1 at $_2$2 and D2 at $_2$3, in a dual-gated Hall-bar geometry where the out-of-plane electric field $_2$4 tunes the interlayer potential difference. The topmost moiré valence bands derive from monolayer $_2$5 valley states and are strongly spin-split by approximately $_2$6–$_2$7, with spin–valley locking generated by a uniform Ising spin–orbit field. For twist angles above approximately $_2$8–$_2$9, the top three moiré valence bands in the $U(1)$0 valley have Chern numbers $U(1)$1, $U(1)$2, and $U(1)$3 in descending order, while the $U(1)$4 valley carries the opposite values. With spin–valley locking, these become spin-contrasting Chern bands (Kang et al., 2024).

At $U(1)$5, filling the first band gives $U(1)$6 and a single QSH phase. At $U(1)$7, filling the first two bands gives $U(1)$8 and a double QSH phase, which the paper states corresponds exactly to Type-II QSH. In this device, $U(1)$9 labels the number of filled topmost spin-Chern bands, so $\mathbb{Z}_2$0 at $\mathbb{Z}_2$1, with ideal

$\mathbb{Z}_2$2

Filling	Phase	Topological count	Key transport
$\mathbb{Z}_2$3	Single QSH	$\mathbb{Z}_2$4, $\mathbb{Z}_2$5	$\mathbb{Z}_2$6
$\mathbb{Z}_2$7	Double QSH / Type-II	$\mathbb{Z}_2$8, $\mathbb{Z}_2$9	$_2$0

The bulk is insulating at both fillings. Penetration capacitance shows incompressible peaks at $_2$1 and $_2$2 around $_2$3, with bulk charge gaps of approximately $_2$4 and $_2$5, respectively. Despite that bulk insulation, local transport exhibits nearly quantized resistance plateaus near $_2$6 and $_2$7, with measured peaks about $_2$8–$_2$9 higher than the ideal values at $\mathbb{Z}_2=0$00. Identical resistances on opposite sides of the Hall bar confirm uniformity over the $\mathbb{Z}_2=0$01 voltage-probe separation (Kang et al., 2024).

The field dependence isolates the protecting symmetry. The plateaus at $\mathbb{Z}_2=0$02 and $\mathbb{Z}_2=0$03 are nearly independent of out-of-plane field $\mathbb{Z}_2=0$04 from $\mathbb{Z}_2=0$05 to $\mathbb{Z}_2=0$06, but conductance is strongly suppressed by in-plane field $\mathbb{Z}_2=0$07 from $\mathbb{Z}_2=0$08 to $\mathbb{Z}_2=0$09. At $\mathbb{Z}_2=0$10, the normalized conductance $\mathbb{Z}_2=0$11 saturates to about $\mathbb{Z}_2=0$12 at $\mathbb{Z}_2=0$13 and $\mathbb{Z}_2=0$14 at $\mathbb{Z}_2=0$15; at $\mathbb{Z}_2=0$16, $\mathbb{Z}_2=0$17 near $\mathbb{Z}_2=0$18 and $\mathbb{Z}_2=0$19 near $\mathbb{Z}_2=0$20. The transport gap induced by $\mathbb{Z}_2=0$21 grows linearly at small field and saturates above approximately $\mathbb{Z}_2=0$22 to about $\mathbb{Z}_2=0$23 at $\mathbb{Z}_2=0$24 and $\mathbb{Z}_2=0$25 at $\mathbb{Z}_2=0$26, whereas the bulk gaps are insensitive to $\mathbb{Z}_2=0$27 up to $\mathbb{Z}_2=0$28. This is consistent with the statement that $\mathbb{Z}_2=0$29 preserves the Ising spin axis and therefore does not gap the helical edges, while $\mathbb{Z}_2=0$30 breaks spin conservation, allows spin mixing, and opens an edge gap (Kang et al., 2024).

Nonlocal transport further supports the edge interpretation. Large nonlocal resistances appear around $\mathbb{Z}_2=0$31, with maxima of about $\mathbb{Z}_2=0$32 at $\mathbb{Z}_2=0$33 and $\mathbb{Z}_2=0$34 at $\mathbb{Z}_2=0$35, whereas the background nonlocal signal remains below or equal to $\mathbb{Z}_2=0$36 even where local resistance reaches roughly $\mathbb{Z}_2=0$37 in compressible regimes. For the geometry with current $\mathbb{Z}_2=0$38 and voltage $\mathbb{Z}_2=0$39–$\mathbb{Z}_2=0$40, ideal edge-only Landauer–Büttiker values are $\mathbb{Z}_2=0$41 at $\mathbb{Z}_2=0$42 and $\mathbb{Z}_2=0$43 at $\mathbb{Z}_2=0$44; the smaller measured values are attributed to finite bulk conduction and a finite edge coherence length comparable to or shorter than the $\mathbb{Z}_2=0$45 probe separation. The local plateaus persist up to approximately $\mathbb{Z}_2=0$46, while the nonlocal signal decreases monotonically with increasing temperature (Kang et al., 2024).

5. TRS-broken and correlated Type-II QSH in moiré WSe$\mathbb{Z}_2=0$47

A broader recent usage of the term appears in the observation of a Mott QSH insulator in twisted WSe$\mathbb{Z}_2=0$48. In that work, the main device has twist angle $\mathbb{Z}_2=0$49, and the correlated state occurs at $\mathbb{Z}_2=0$50, corresponding to half filling of the second moiré valence band. The state requires an out-of-plane magnetic field $\mathbb{Z}_2=0$51–$\mathbb{Z}_2=0$52, so time-reversal symmetry is explicitly broken, yet transport exhibits the same resistance plateau as the single-particle QSH state at $\mathbb{Z}_2=0$53, indicating the same number of helical edge channels. The paper states that these operational criteria match a Type-II QSH phase: TRS is broken, the bulk gap is interaction-driven, and helical edge transport survives because spin conservation protects it (Jin et al., 17 May 2026).

At $\mathbb{Z}_2=0$54, $\mathbb{Z}_2=0$55, and $\mathbb{Z}_2=0$56, the local resistances are $\mathbb{Z}_2=0$57 at $\mathbb{Z}_2=0$58, $\mathbb{Z}_2=0$59 at $\mathbb{Z}_2=0$60, and $\mathbb{Z}_2=0$61 at $\mathbb{Z}_2=0$62, with $\mathbb{Z}_2=0$63 at all three fillings. Over the range $\mathbb{Z}_2=0$64, the $\mathbb{Z}_2=0$65, $\mathbb{Z}_2=0$66, and $\mathbb{Z}_2=0$67 plateaus remain essentially constant. The $\mathbb{Z}_2=0$68 plateau emerges above approximately $\mathbb{Z}_2=0$69–$\mathbb{Z}_2=0$70 and saturates to approximately $\mathbb{Z}_2=0$71 for $\mathbb{Z}_2=0$72, while the $\mathbb{Z}_2=0$73 plateau is nearly field-independent across $\mathbb{Z}_2=0$74 (Jin et al., 17 May 2026).

The temperature dependence distinguishes the correlated Type-II regime from the single-particle one. The $\mathbb{Z}_2=0$75 and $\mathbb{Z}_2=0$76 plateaus remain robust from $\mathbb{Z}_2=0$77 up to $\mathbb{Z}_2=0$78, whereas the $\mathbb{Z}_2=0$79 plateau remains near $\mathbb{Z}_2=0$80 only for $\mathbb{Z}_2=0$81 and decreases rapidly above that, reaching about $\mathbb{Z}_2=0$82 at $\mathbb{Z}_2=0$83. Hall measurements identify a characteristic scale $\mathbb{Z}_2=0$84, corresponding to

$\mathbb{Z}_2=0$85

which sets the Mott-gap scale for the $\mathbb{Z}_2=0$86 Type-II QSH state (Jin et al., 17 May 2026).

The supporting edge signatures are parallel to the single-particle case. Pronounced nonlocal resistance appears at $\mathbb{Z}_2=0$87 in both nonlocal geometries once $\mathbb{Z}_2=0$88, while at the trivial Mott state $\mathbb{Z}_2=0$89 the ratio $\mathbb{Z}_2=0$90. A strong negative in-plane magnetoconductance is observed near $\mathbb{Z}_2=0$91 and $\mathbb{Z}_2=0$92, with device conductance suppressed by nearly $\mathbb{Z}_2=0$93 under $\mathbb{Z}_2=0$94. The stated interpretation is that $\mathbb{Z}_2=0$95 suppresses bulk kinetic transport via orbital cyclotron effects, enhances Coulomb effects, and opens a Mott gap at half filling, while the helical edge states remain protected by spin $\mathbb{Z}_2=0$96 because the Ising-like SOC keeps $\mathbb{Z}_2=0$97 approximately conserved (Jin et al., 17 May 2026).

Under this usage, Type-I and Type-II are distinguished less by edge-pair multiplicity than by the origin of the bulk gap and the status of time-reversal symmetry: the Type-I state is a band QSH phase at $\mathbb{Z}_2=0$98, whereas the Type-II state is a magnetic-field-stabilized, interaction-gapped Mott QSH phase at $\mathbb{Z}_2=0$99 with the same effective helical channel count.

6. Crystalline, stacked, and alternative generalizations

Recent work extends Type-II QSH beyond moiré WSe$_2$00 into crystalline and magnetic settings with higher spin Chern number. In altermagnetic multilayers, the relevant protecting structure is the combination of horizontal mirror symmetry $_2$01 and the antiunitary symmetry $_2$02. Because $_2$03 and $_2$04, the system admits a mirror-sector decomposition with mirror eigenvalues $_2$05, and the appropriate invariant is a mirror–spin Chern number

$_2$06

For the bilayer, Wilson-loop calculations yield $_2$07 and hence two pairs of gapless helical edge states; for the trilayer, $_2$08 and three pairs. The spin Hall conductance is exactly quantized as

$_2$09

so the plateau scales linearly with layer number. First-principles calculations identify Fe$_2$10Se$_2$11O bilayers and trilayers as candidate realizations (Chen et al., 5 Aug 2025).

A related stacking proposal in unconventional magnetism starts from a monolayer “type-II QSHI” with $_2$12 and argues that AA-stacking with weak interlayer coupling and interlayer ferromagnetic alignment produces a bilayer with $_2$13 rather than a trivial insulator. In that formulation, the bilayer hosts two pairs of topological edge states with opposite chirality and polarization coexisting at the boundary, and the quantized spin Hall conductance doubles from $_2$14 in the monolayer to $_2$15 in the bilayer. First-principles calculations propose bilayer Nb$_2$16SeTeO as a candidate high-spin-Chern realization (Tan et al., 7 Aug 2025).

Another distinct usage appears in the Archimedean-lattice literature, where “type-II” refers to the parent semimetallic dispersion rather than to spin-Chern taxonomy. The oblique $_2$17 lattice hosts an overtilted type-II Dirac crossing in the absence of SOC, and intrinsic SOC gaps that crossing to produce a $_2$18 QSH phase with helical edge states. In that context, a “type-II-derived QSH” means a time-reversal-invariant QSH insulator obtained by gapping a type-II Dirac cone, not an even-spin-Chern or TRS-broken state (Lima et al., 2019).

Several common misconceptions therefore require qualification. Type-II QSH does not necessarily mean “QSH in a Type-II semiconductor,” because in InAs/GaSb that phrase originally referred to staggered band alignment rather than to the topological class (0801.2831). It does not always mean “two helical pairs,” because some recent work uses the label for a TRS-broken Mott QSH state with only one helical pair (Jin et al., 17 May 2026). It also does not always mean “helical” in the narrow Kramers-pair sense, because some authors define a type-II QSHI through spin-chiral edge structure and momentum-separated spin-resolved band inversions (Tan et al., 7 Aug 2025). This suggests that the most stable description of the subject is invariant-first: a Type-II QSH phase is best specified by the protecting symmetry, the quantized integer invariant that survives beyond the ordinary $_2$19 setting, and the resulting edge-channel multiplicity and spin structure in the concrete platform under discussion.