Bilayer SSH Model: Glide and Mirror Symmetries

• The bilayer SSH model is a one-dimensional system composed of two coupled SSH chains that exhibit novel topological behavior via glide reflection and mirror symmetries.
• The glide-symmetric ladder variant features non-Abelian Berry phases, Möbius band connectivity, and controlled edge-state transitions via inter-leg coupling.
• The mirror-symmetric square-root model inherits SSH topology, enabling finite-energy and embedded boundary states, with experimental realizations in topolectrical circuits.

Searching arXiv for the cited bilayer SSH papers and closely related variants to ground the article in current literature. The bilayer Su–Schrieffer–Heeger (SSH) model denotes a family of one-dimensional coupled-dimer systems built from two SSH-type chains or layers. In the literature considered here, two distinct realizations are especially prominent: a glide-symmetric two-leg SSH ladder formed by coupling two topologically distinct SSH chains shifted by half a lattice spacing, and a mirror-symmetric bilayer SSH-like model obtained by directly coupling two identical square-root SSH chains (Zhang et al., 2016, Guo et al., 2024). Both inherit the dimerized-hopping logic of the single-chain SSH problem, but they differ in symmetry algebra, topological characterization, and boundary phenomenology.

1. Single-chain antecedent and bilayer variants

The single-chain SSH Hamiltonian provides the basic dimerized template,

$$H_{\mathrm{SSH}}=\sum_n \left[t_1 a_n^\dagger b_n+t_2 a_{n+1}^\dagger b_n+\mathrm{h.c.}\right],$$

with two sublattices $A$ and $B$, and alternating nearest-neighbor hoppings $t_1$ and $t_2$. In momentum space,

$H_{\mathrm{SSH}}(k)=d_x(k)\sigma_x+d_y(k)\sigma_y,$

where

$d_x(k)=t_1+t_2\cos k,\qquad d_y(k)=t_2\sin k.$

The Zak phase of a single isolated band switches by $\pi$ when $|t_2|$ crosses $|t_1|$, which marks the conventional SSH topological transition (Zhang et al., 2016).

Within bilayer generalizations, the term “bilayer SSH model” is not unique. One construction uses two SSH legs offset by $A$0 and coupled by spin-changing tunneling, producing a glided two-leg SSH ladder. Another uses two identical four-site square-root SSH chains coupled vertically site by site, producing an eight-site bilayer SSH-like model. The first is controlled by nonsymmorphic glide reflection symmetry and requires a non-Abelian Wilson-line description; the second is controlled by mirror parity and inherits square-root topology from an SSH parent Hamiltonian obtained after squaring (Zhang et al., 2016, Guo et al., 2024).

Variant	Unit cell / basis	Defining symmetry and topology
Glided two-leg SSH ladder	$A$1	Glide reflection; non-Abelian Berry connection and Wilson lines
Bilayer SSH-like square-root model	$A$2	Mirror parity; square-root topology inherited from an SSH parent

2. Glide-symmetric two-leg SSH ladder

In the glide-symmetric realization, spin-up and spin-down atoms experience identical short lattices but opposite dimerizations induced by a spin-dependent long lattice and an rf coupling. Each spin component realizes an SSH chain, and the two chains are shifted by half a lattice spacing $A$3 relative to one another. Identifying the two spin species as legs of a ladder produces a two-leg SSH ladder in which leg exchange is accompanied by a half-translation. The corresponding continuum single-particle Hamiltonian is

$A$4

with

$A$5

where $A$6 sets the short lattice depth, $A$7 the long lattice amplitude, and $A$8 the rf-induced inter-leg coupling (Zhang et al., 2016).

The tight-binding ladder Hamiltonian in real space is

$A$9

where each leg is an SSH chain and, due to the $B$0 offset, the roles of $B$1 and $B$2 exchange between legs. In the Bloch basis $B$3, the Hamiltonian becomes a $B$4 matrix $B$5. It can be block-diagonalized into two $B$6 blocks $B$7,

$B$8

which obey

$B$9

Each block therefore has period $t_1$0, whereas the full Bloch Hamiltonian $t_1$1 has period $t_1$2 (Zhang et al., 2016).

The symmetry responsible for this structure is the glide operator

$t_1$3

where $t_1$4 flips spin, $t_1$5, and $t_1$6 translates by half a lattice spacing. In Bloch form,

$t_1$7

so the glide eigenvalues are $t_1$8. Because the glide eigenvalue flips sign under $t_1$9, bands in the $t_2$0 sector must connect to bands in the $t_2$1 sector across the Brillouin zone. With an additional mirror symmetry, the crossing is pinned to $t_2$2; even without mirror symmetry, a combined inversion and $t_2$3 exchange symmetry denoted $t_2$4 retains the degeneracies at the Brillouin-zone boundary (Zhang et al., 2016).

The four-band dispersion has lower energies

$t_2$5

with the upper pair given by $t_2$6. For generic $t_2$7, the lowest two bands split, except at $t_2$8, where glide symmetry enforces a degeneracy. At the critical inter-leg coupling

$t_2$9

the lower pair touches the upper pair at $H_{\mathrm{SSH}}(k)=d_x(k)\sigma_x+d_y(k)\sigma_y,$0, and for $H_{\mathrm{SSH}}(k)=d_x(k)\sigma_x+d_y(k)\sigma_y,$1 a gap reopens between the pairs while the enforced degeneracies at $H_{\mathrm{SSH}}(k)=d_x(k)\sigma_x+d_y(k)\sigma_y,$2 remain (Zhang et al., 2016).

3. Non-Abelian topology and phase structure in the glided ladder

In the glided ladder, a single-band Zak phase is not well defined over the physical Brillouin zone because the lowest two bands are not separately periodic over $H_{\mathrm{SSH}}(k)=d_x(k)\sigma_x+d_y(k)\sigma_y,$3; they exchange across $H_{\mathrm{SSH}}(k)=d_x(k)\sigma_x+d_y(k)\sigma_y,$4 through the relation $H_{\mathrm{SSH}}(k)=d_x(k)\sigma_x+d_y(k)\sigma_y,$5. The appropriate topological object is therefore the non-Abelian Berry connection for the two-band subspace,

$H_{\mathrm{SSH}}(k)=d_x(k)\sigma_x+d_y(k)\sigma_y,$6

and the associated Wilson line

$H_{\mathrm{SSH}}(k)=d_x(k)\sigma_x+d_y(k)\sigma_y,$7

Its matrix elements encode adiabatic transport under a weak, uniform force $H_{\mathrm{SSH}}(k)=d_x(k)\sigma_x+d_y(k)\sigma_y,$8 via Bloch oscillations provided $H_{\mathrm{SSH}}(k)=d_x(k)\sigma_x+d_y(k)\sigma_y,$9, where $d_x(k)=t_1+t_2\cos k,\qquad d_y(k)=t_2\sin k.$0 is the two-band width and $d_x(k)=t_1+t_2\cos k,\qquad d_y(k)=t_2\sin k.$1 the gap to higher bands (Zhang et al., 2016).

For the lowest two periodic Bloch states $d_x(k)=t_1+t_2\cos k,\qquad d_y(k)=t_2\sin k.$2 and $d_x(k)=t_1+t_2\cos k,\qquad d_y(k)=t_2\sin k.$3, glide conservation implies that adiabatic transport over one physical Brillouin zone swaps the bands:

$d_x(k)=t_1+t_2\cos k,\qquad d_y(k)=t_2\sin k.$4

where $d_x(k)=t_1+t_2\cos k,\qquad d_y(k)=t_2\sin k.$5 and $d_x(k)=t_1+t_2\cos k,\qquad d_y(k)=t_2\sin k.$6. Over two Brillouin-zone traversals,

$d_x(k)=t_1+t_2\cos k,\qquad d_y(k)=t_2\sin k.$7

This is the Möbius connectivity of the glide-symmetric ladder: traversing the physical Brillouin zone sends a state to an orthogonal state in the other band, while only after two traversals does one recover the original state up to the Abelian Zak phase (Zhang et al., 2016).

The resulting invariant has two components. The exchange property, described in the source as a nonsymmorphic glide $d_x(k)=t_1+t_2\cos k,\qquad d_y(k)=t_2\sin k.$8-type feature, is robust as long as glide symmetry is preserved and the two-band pair remains gapped from higher bands. The $d_x(k)=t_1+t_2\cos k,\qquad d_y(k)=t_2\sin k.$9 part, $\pi$0 mod $\pi$1, distinguishes phases across the inter-leg transition at $\pi$2. Across this transition, $\pi$3 changes by $\pi$4, so the $\pi$5 part of $\pi$6 shifts by $\pi$7, while the $\pi$8 exchange persists continuously (Zhang et al., 2016).

Open boundary conditions reveal the bulk-boundary correspondence of the $\pi$9 sector. For $|t_2|$0, there exists a zero-energy end state localized at each end, one per edge, with mixed spin-up and spin-down components. As $|t_2|$1 approaches $|t_2|$2, the edge-state localization length grows and diverges at $|t_2|$3. For $|t_2|$4, the end states disappear. A representative parameter set reported in the source uses $|t_2|$5, $|t_2|$6, and $|t_2|$7 chosen such that $|t_2|$8 in units where $|t_2|$9, for which edge zero modes exist; for $|t_1|$0, they vanish (Zhang et al., 2016).

4. Interaction-driven ferromagnetism and charge fractionalization

The glided ladder also supports an interaction-induced mechanism for charge fractionalization. In the flat-band limit $|t_1|$1 with $|t_1|$2 dominating, each leg’s lower band consists of localized orbitals

$|t_1|$3

which yields a flat two-fold degenerate ground band labeled by spin. Repulsive onsite interactions,

$|t_1|$4

lift the degeneracy at half filling and select a ferromagnetic state that fully occupies one leg, producing the two degenerate ground states

$|t_1|$5

(Zhang et al., 2016).

Adding one extra particle to $|t_1|$6 creates two domain walls separating $|t_1|$7- and $|t_1|$8-occupied regions. Their confinement depends on which kinetic process dominates. If only inter-leg tunneling $|t_1|$9 is present and $A$00, the domain walls are deconfined: the process can repeat without increasing the number of domain walls, allowing arbitrary separation, and each wall carries half the extra charge, $A$01. If only intra-leg tunneling $A$02 is present and $A$03, the domain walls are confined, with pair separation of order $A$04 (Zhang et al., 2016).

With both $A$05 and $A$06 finite, projection onto the two-domain-wall subspace gives an effective Hamiltonian

$A$07

with

$A$08

At fixed center-of-mass momentum $A$09, this reduces to

$A$10

The source reports a first-order transition at $A$11 between a deconfined phase, with extensive average separation, and a confined phase, with separation $A$12. TEBD numerics confirm ferromagnetism for $A$13 and the deconfined-to-confined transition (Zhang et al., 2016).

The fractional charge follows the polarization formula

$A$14

For domains that differ by $A$15, each domain wall carries

$A$16

Detection is based on the site-resolved density $A$17, with background-subtracted profile $A$18. In the deconfined phase, $A$19 matches two hard-core particles’ density. If domain walls are pinned by local potentials $A$20 and $A$21 at distant wells, the integrated charges satisfy $A$22 with negligible number fluctuations (Zhang et al., 2016).

5. Mirror-symmetric bilayer square-root SSH-like model

A different bilayer SSH construction is formed by directly coupling two identical square-root SSH models, each with four sites per unit cell. The bilayer therefore has eight sites per unit cell and a natural basis

$A$23

Its Bloch Hamiltonian is

$A$24

where

$A$25

In real space, the vertical interlayer coupling $A$26 connects corresponding sites on the two layers (Guo et al., 2024).

A similarity transformation $A$27 diagonalizes the bilayer Hamiltonian into two decoupled mirror-parity subspaces,

$A$28

The model preserves mirror symmetry along the layer direction, and in this basis the mirror operator becomes

$A$29

so $A$30 and $A$31 carry even and odd mirror parity, respectively (Guo et al., 2024).

The topology of each parity sector is inherited from the square-root SSH chain. Squaring the single-chain Hamiltonian yields a parent SSH Hamiltonian, up to an overall energy shift and a residual block, and the relevant winding number is

$A$32

The nontrivial phase occurs for $A$33. Since $A$34 and $A$35 differ only by the energy shifts $A$36, the square-root topology is unaffected by $A$37; the bilayer phase boundary remains $A$38, with the gap closing at $A$39 as in the parent SSH model (Guo et al., 2024).

The principal boundary phenomenon in this model is the coexistence of finite-energy boundary states in gaps and embedded in bulk continua. The source fixes $A$40 and $A$41, varies $A$42, and shows that sufficiently large $A$43 can move one set of edge states into the bulk continua of one parity sector while other edge states remain in band gaps of the other sector. For $A$44, the reported spectrum contains “two groups of boundary states embedded in the bulk, while the other groups of boundary states emerge in the band gaps.” Because the mirror-parity decomposition prevents hybridization between states of opposite parity, these in-bulk boundary states remain localized. Mirror-symmetry-preserving next-nearest-neighbor interlayer couplings retain significant end confinements, whereas mirror-symmetry-breaking couplings cause hybridization with the bulk continua and destroy confinement (Guo et al., 2024).

The same work also introduces a decorated SSH-like model with ten sites per unit cell, chiral symmetry

$A$45

and a squared Hamiltonian

$A$46

where $A$47 is the Hamiltonian of the bilayer SSH model with energy shift $A$48. This extension shows how the bilayer SSH parent can generate additional square-root structures, four groups of boundary states, and two zero-energy flat bands associated with sublattice imbalance (Guo et al., 2024).

6. Experimental implementations and conceptual comparisons

The glide-symmetric ladder is formulated for ultracold atoms in a spin-dependent double-well optical lattice. One implementation uses two standing waves to form a short lattice $A$49, a spin-dependent long lattice $A$50, and an rf coupling $A$51. An equivalent description uses a spin-independent lattice dressed by a Raman field modulated as $A$52. In this setting, $A$53 and $A$54 are set by $A$55 and $A$56, the inter-leg coupling $A$57 is set by $A$58, and the glide symmetry is engineered by the $A$59 relative shift between the spin-dependent lattices. Wilson lines are probed by applying a weak force $A$60 to drive Bloch oscillations under the condition $A$61; Ramsey-type interferometric protocols reconstruct $A$62 and $A$63. Edge states are accessed in finite systems with open boundary conditions, and fractionalization is probed by preparing the half-filled ferromagnet, doping by one particle, and imaging $A$64 (Zhang et al., 2016).

The mirror-symmetric bilayer SSH-like model is proposed for topolectrical circuits. The circuit Laplacians $A$65 and $A$66 reproduce the bilayer and decorated SSH-like band structures, with capacitances $A$67, $A$68, and $A$69 implementing $A$70, $A$71, and $A$72, and with

$A$73

Identical grounding capacitors and inductors enforce equal on-site potentials, and edge states as well as in-bulk boundary states are detected through impedance peaks localized at end nodes. The same paper also identifies compatibility with photonic lattices, acoustic metamaterials, and topolectrical circuits (Guo et al., 2024).

Taken together, these constructions show that “bilayer SSH model” refers to more than simple duplication of the single-chain SSH problem. In the glided ladder, nonsymmorphic glide symmetry enforces Brillouin-zone-edge crossings, Möbius band connectivity, and a non-Abelian Wilson-line description, while interactions yield intrinsic $A$74 fractionalization without joining distinct lattices at an interface. In the mirror-symmetric square-root bilayer, interlayer coupling does not move the topological phase boundary but instead shifts mirror sectors in energy, enabling multiple finite-energy boundary states, including states embedded in the bulk continuum. The comparison with other ladders given in the source is correspondingly sharp: Creutz ladders involve $A$75-flux and chiral or particle-hole symmetries, whereas standard bilayer SSH ladders without glide do not enforce band crossings at the Brillouin-zone boundary and can be characterized by independent Zak phases per band (Zhang et al., 2016, Guo et al., 2024).