Wallpaper Fermions in Topological Crystals

• Wallpaper fermions are symmetry-protected surface quasiparticles defined by 2D wallpaper group symmetries, manifesting as fourfold Dirac points.
• They arise from the interplay of glide reflections, time-reversal symmetry, and spin–orbit coupling, leading to robust band degeneracies on crystal surfaces.
• Their study informs the design of topological phases in magnetic and superconducting systems, with implications for quantized Hall responses and Majorana modes.

Wallpaper fermions are symmetry-protected surface quasiparticles governed by the two-dimensional crystallographic symmetries of a crystal face. In the formulation developed for time-reversal-symmetric, spin–orbit-coupled three-dimensional crystals, a wallpaper fermion is a band degeneracy on a two-dimensional surface enforced solely by the surface’s wallpaper group; in the nonsymmorphic setting this degeneracy becomes a fourfold, point-like, linearly dispersing surface state, often described as a “double Dirac” point. The concept has since branched into several closely related directions: nonsymmorphic Dirac insulators, magnetic wallpaper Dirac fermions and topological magnetic Dirac insulators, magnetically gapped Hall phases, and superconducting descendants with nodal, Majorana, and hybridized surface spectra (1705.01617, Hwang et al., 2022, Mizuno et al., 2023, Yoda et al., 30 Sep 2025, Yoda et al., 12 Mar 2026).

1. Definition and crystallographic setting

A wallpaper fermion is a symmetry-protected band degeneracy on the two-dimensional surface of a time-reversal-symmetric, spin–orbit-coupled three-dimensional crystal, enforced by the surface’s space group among the 17 wallpaper groups. At high-symmetry points and lines of the surface Brillouin zone, surface bands transform in irreducible corepresentations of the wallpaper group; multidimensional corepresentations pinned by the group algebra produce protected degeneracies. In this sense, wallpaper fermions are the surface analogue of symmetry-enforced band degeneracies familiar from three-dimensional space-group theory, but specialized to two-dimensional crystallography (1705.01617).

Within this taxonomy, the symmorphic and nonsymmorphic wallpaper groups behave differently. In the 13 symmorphic wallpaper groups, the stable degeneracies are twofold Kramers pairs or hourglass fermions protected by a single glide, as in $pg$ and $pmg$. By contrast, the two nonsymmorphic groups with two perpendicular glides, $pgg$ and $p4g$, admit a fourfold degeneracy at the corner time-reversal-invariant momentum $M$ of the surface Brillouin zone. These fourfold point-like, linearly dispersing states are the nonsymmorphic Dirac fermions that define the nonsymmorphic Dirac-insulator phase. Later superconducting work uses the narrower phrase “wallpaper fermions” for the fourfold-degenerate two-dimensional surface states on the $(001)$ face of certain nonsymmorphic topological crystalline insulators in wallpaper group $p4g$ (1705.01617, Yoda et al., 30 Sep 2025).

2. Symmetry algebra and effective surface Hamiltonians

The defining algebraic feature of nonsymmorphic wallpaper fermions is the interaction between two perpendicular glide reflections and time reversal. In $pgg$ or $p4g$, along glide-invariant lines such as $\Gamma X$ or $pmg$0, bands carry one-dimensional corepresentations labeled by glide eigenvalues $pmg$1. Along $pmg$2 or $pmg$3, the antiunitary combination $pmg$4 squares to $pmg$5, so all bands are Kramers doubled. At the corner $pmg$6, however, the glides satisfy

$pmg$7

and, with $pmg$8, this algebra enforces a four-dimensional corepresentation. The resulting fourfold degeneracy is therefore not accidental but symmetry-forced (1705.01617).

Near $pmg$9, a minimal $pgg$0 description uses Pauli matrices $pgg$1 on the surface A–B sublattice and $pgg$2 on the effective Kramers spin: $pgg$3 This is a fourfold Dirac cone. In the minimal theory, the two matrices $pgg$4 and $pgg$5 exhaust the $pgg$6-linear terms, and no further linear or quadratic symmetry-allowed term can gap the cone. A later $pgg$7 effective theory written in the four-component basis $pgg$8 gives the most general $pgg$9 Hamiltonian up to second order in $p4g$0 consistent with $p4g$1 and time-reversal symmetry, providing the starting point for superconducting gap-structure analyses (1705.01617, Yoda et al., 30 Sep 2025).

3. Bulk topology, anomalous surfaces, and doubling constraints

The surface fourfold Dirac point is anomalous from a strictly two-dimensional viewpoint. In an isolated two-dimensional crystal, a single fourfold Dirac point would violate a fermion-doubling constraint: the parity-anomaly argument implies that such a spectrum would force a half-quantized spin Hall conductance, inconsistent with $p4g$2 unless another Dirac point is present. The nonsymmorphic Dirac insulator evades this by realizing the unpaired fourfold Dirac point on the boundary of a three-dimensional bulk, where the anomaly is canceled by the opposite surface. This is the sense in which the phase is an exception to a two-dimensional doubling theorem (1705.01617).

The bulk classification can be formulated through the $p4g$3-directed non-Abelian Wilson loop,

$p4g$4

whose eigenvalues $p4g$5 form Wilson bands. For orthorhombic double-glide systems, two $p4g$6 indices $p4g$7 are defined by counting Wilson-band crossings along bent paths through the projected Brillouin zone. An insulating bulk satisfies $p4g$8, leaving eight phases labeled by $p4g$9. In inversion-symmetric settings, the strong $M$0 invariant obeys $M$1, and in $M$2 one may also define diagonal-mirror Chern numbers $M$3 and $M$4 (1705.01617).

The anomalous surface also admits time-reversal-preserving mass domain walls. The only $M$5-even mass at $M$6 is $M$7, which anticommutes with $M$8 and is odd under either glide. Surface regions with $M$9 and $(001)$0 are topologically distinct, and a one-dimensional interface between them hosts a protected helical quantum spin Hall channel. In a separate but related development, generalized Nielsen–Ninomiya theorems were worked out for all 17 wallpaper groups. For $(001)$1 and $(001)$2 with $(001)$3, the minimal total topological-charge magnitude is $(001)$4 in class AIII and $(001)$5 in class AII with $(001)$6 symmetry. When a surface Brillouin zone is not a closed two-torus, this no-go theorem can fail on the surface, and a nonzero sum of surface-node charges signals a nontrivial three-dimensional bulk (1705.01617, Le et al., 2021).

4. Magnetic wallpaper Dirac fermions and topological magnetic Dirac insulators

A magnetic generalization replaces ordinary wallpaper groups by magnetic wallpaper groups. In three-dimensional magnetic crystals whose $(001)$7 surface realizes $(001)$8 or $(001)$9, the little-group algebra at $p4g$0 forces the four surface Bloch states to form a single, fourfold-degenerate Dirac point. These four-component linear crossings are termed magnetic wallpaper Dirac fermions, and when they occur anomalously on the boundary of an insulating bulk the bulk phase is a topological magnetic Dirac insulator (TMDI) (Hwang et al., 2022).

The algebra is explicit. For $p4g$1, the generators satisfy

$p4g$2

while for $p4g$3,

$p4g$4

In either case, starting from a simultaneous eigenstate at $p4g$5, the four states generated by the antiunitary and mirror operations are degenerate and mutually orthogonal, so no twofold splitting is permitted. Around $p4g$6, a minimal $p4g$7 Hamiltonian for $p4g$8 contains four linear kinetic $p4g$9-matrices; the remaining anticommuting $pgg$0-matrices that could serve as masses are forbidden by the two diagonal mirrors. Consequently, no constant mass term is allowed and the cone remains strictly gapless so long as the diagonal mirror symmetries are unbroken (Hwang et al., 2022).

The bulk topology is encoded by a diagonal-mirror Chern number

$pgg$1

On the $pgg$2 face, each diagonal-mirror-invariant line carries a pair of chiral modes with net chirality $pgg$3; the fourfold degeneracy at $pgg$4 forces these chiral branches to reconnect into a fourfold Dirac cone with mirror-resolved slopes $pgg$5 near $pgg$6. In a finite geometry that preserves the diagonal mirror, a nonzero $pgg$7 guarantees at least $pgg$8 propagating hinge modes on each mirror-invariant ridge, furnishing higher-order boundary states. The same work gives a general search recipe: a paramagnetic parent space group can descend to one of 16 magnetic space groups that admit $pgg$9 or $p4g$0 on the $p4g$1 surface, after which Wilson loops on the diagonal mirror planes can be used to identify nonzero $p4g$2 with $p4g$3 (Hwang et al., 2022).

5. Material realizations and Hall responses

First-principles and model studies have identified several representative systems and phases. The following examples were reported as explicit realizations or candidate realizations of wallpaper-fermion physics and its magnetic extension (1705.01617, Hwang et al., 2022).

System	Symmetry or phase	Reported result
Sr$p4g$4Pb$p4g$5	SG 127, $p4g$6	Full bulk gap near $p4g$7, direct gap $p4g$8 at $p4g$9, $\Gamma X$0, single fourfold Dirac cone at $\Gamma X$1
Au$\Gamma X$2Y$\Gamma X$3, Hg$\Gamma X$4Sr$\Gamma X$5 under $\Gamma X$6 strain	SG $\Gamma X$7, hourglass double-glide phase	Full bulk gap opens; $\Gamma X$8 or $\Gamma X$9; Dirac at $pmg$00 buried in bulk bands
Ba$pmg$01In$pmg$02Sb$pmg$03	SG 55, $pmg$04	Narrow bulk gap $pmg$05; $pmg$06; hourglass dispersion along $pmg$07
DyB$pmg$08	MSG 127.392 $pmg$09	Surface MWG $pmg$10; $pmg$11 for $pmg$12, $pmg$13 for slightly smaller $pmg$14; fourfold Dirac cone at $pmg$15
Nd$pmg$16Te$pmg$17Cl$pmg$18O$pmg$19	MSG 129.421 $pmg$20	Surface MWG $pmg$21; robust gap around $pmg$22; $pmg$23; fourfold Dirac cone at $pmg$24

Wallpaper fermions also support quantized Hall responses when gapped by magnetic order. In a four-sublattice surface model, ferromagnetic and antiferromagnetic couplings

$pmg$25

both open a gap $pmg$26 at the wallpaper-fermion Dirac point. In the ferromagnetic case, the surface Hamiltonian splits into two $pmg$27 Dirac blocks, each contributing $pmg$28, so the Hall conductivity becomes

$pmg$29

twice the half-quantized response of a single topological-insulator surface Dirac cone. In the antiferromagnetic case, the charge Hall response vanishes by magnetic-reflection symmetry, but the spin Hall conductivity exhibits a plateau and, as the antiferromagnetic exchange increases, decays to a finite value. The same analysis argues that these conclusions follow from symmetry and therefore hold for a general model of wallpaper fermions (Mizuno et al., 2023).

6. Superconducting descendants

Superconducting wallpaper fermions provide a controlled setting in which nonsymmorphic surface symmetry, BdG topology, and crystalline selection rules can all be made explicit. In a two-dimensional $pmg$30 effective model, exactly six momentum-independent zero-momentum pair potentials $pmg$31 satisfy Fermi statistics and decompose into $pmg$32 irreducible representations. Their gap structures separate into three classes: $pmg$33 are fully gapped; $pmg$34 has four point nodes along $pmg$35 because its reduced gap behaves as $pmg$36; and $pmg$37 have straight line nodes, with $pmg$38 vanishing at $pmg$39 and $pmg$40 vanishing at $pmg$41. The point and generic line nodes are protected by zero-dimensional $pmg$42 invariants in symmetry class BDI, while the line nodes on glide-invariant axes are enforced by a Mackey–Bradley compatibility argument (Yoda et al., 30 Sep 2025).

A second superconducting route couples wallpaper fermions to a ferromagnetic insulator and integrates out magnons. In this heterostructure, BCS pairing at zero center-of-mass momentum yields chiral $pmg$43-wave superconductivity, whereas Amperean pairing with center-of-mass momentum $pmg$44 can produce a parity-mixed state. In the isotropic limit, the BCS solution has only the off-diagonal band component

$pmg$45

while the Amperean channel generically mixes an $pmg$46-wave and a $pmg$47-wave piece,

$pmg$48

The stability of the BCS and Amperean states is governed by the easy-axis anisotropy: when $pmg$49 is small the Amperean $pmg$50-wave channel has the higher $pmg$51, whereas beyond a critical $pmg$52 the zero-momentum chiral $pmg$53-wave instability becomes dominant (Mizuno et al., 9 Mar 2025).

A third development considers bulk superconductivity in a tight-binding model for space group $pmg$54. Four on-site pair potentials are allowed, with irreps $pmg$55. For the $pmg$56 pairing $pmg$57, the one-dimensional magnetic winding number along the $pmg$58 line is

$pmg$59

which implies double Majorana Kramers pairs at the zone center of the surface Brillouin zone. In that phase, wallpaper fermions and two Majorana Kramers pairs coexist and hybridize into a double-twisted surface spectrum, and the surface density of states develops four peaks at $pmg$60 and $pmg$61. The mirror Chern number vanishes, $pmg$62, so the resulting superconducting surface state is mirror-helicity-free (Yoda et al., 12 Mar 2026).

7. Formal classification and related wallpaper-group methods

Wallpaper fermions sit within a broader symmetry-based program that classifies crystalline phases by the same 17 wallpaper groups. In twisted equivariant $pmg$63-theory, the relevant object is

$pmg$64

where $pmg$65 records unitary versus antiunitary operations, $pmg$66 records symmetry versus antisymmetry, and $pmg$67 is the projective twist; for spinful electrons the twist is $pmg$68. This framework applies uniformly to bulk gapful topological crystalline insulators and superconductors, their gapless boundaries and defects, and bulk gapless topological materials. Surface states of three-dimensional systems and fully gapped two-dimensional insulators are thus classified in a unified way by wallpaper groups, both with and without chiral symmetry. For $pmg$69, the class-AIII group contains a free $pmg$70 factor $pmg$71 and a torsion $pmg$72 factor $pmg$73, while the class-A table includes an additional $pmg$74-specific free summand $pmg$75 (Shiozaki et al., 2017).

A complementary wallpaper-group application concerns topological superconductors with surface Majorana Kramers pairs. The central selection rule states that the magnetic coupling $pmg$76 is nonzero only if the irreducible representation $pmg$77 of the bulk gap function matches the irreducible representation $pmg$78 of the multipole operator, equivalently $pmg$79 for all $pmg$80 in the little group. The associated theorem says that, for a single Majorana Kramers pair at a high-symmetry surface point, the leading magnetic operator transforms in the same irreducible representation as the bulk gap function. Applied to wallpaper groups, this yields direct response diagnostics: for example, in $pmg$81 with spin-$pmg$82, $pmg$83 allows only a $pmg$84-dipole response $pmg$85, while in $pmg$86 or $pmg$87 with spin-$pmg$88, $pmg$89 can produce the cubic magnetic octupole $pmg$90. These results do not define wallpaper fermions themselves, but they show how wallpaper-group symmetry supplies a common language for Dirac, Majorana, and multipolar surface phenomena (Kobayashi et al., 2020).