--- title: 'Bilayer SSH Model: Glide and Mirror Symmetries' url: https://www.emergentmind.com/topics/bilayer-su-schrieffer-heeger-model type: topic --- # Bilayer SSH Model: Glide and Mirror Symmetries 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 [1604.06292] [2407.05350]. 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 [1604.06292]. Within bilayer generalizations, the term “bilayer SSH model” is not unique. One construction uses two SSH legs offset by \(d/2\) 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 [1604.06292] [2407.05350]. | Variant | Unit cell / basis | Defining symmetry and topology | |---|---|---| | Glided two-leg SSH ladder | \((a_{k\uparrow}^\dagger,b_{k\uparrow}^\dagger,a_{k\downarrow}^\dagger,b_{k\downarrow}^\dagger)\) | Glide reflection; non-Abelian Berry connection and Wilson lines | | Bilayer SSH-like square-root model | \((A\uparrow,B\uparrow,C\uparrow,D\uparrow,A\downarrow,B\downarrow,C\downarrow,D\downarrow)^T\) | 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 \(d/2\) 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 $$ \hat H=\int dx\left[\hat\psi_\sigma^\dagger(x)\hat H_\sigma(x)\hat\psi_\sigma(x)+\Omega(\hat\psi_\uparrow^\dagger\hat\psi_\downarrow+\mathrm{h.c.})\right], $$ with $$ \hat H_\sigma(x)=\hat p^2/(2m)-V_S\cos^2(2\pi x/d)+2V_L\sigma_z\sin(2\pi x/d), $$ where \(V_S>0\) sets the short lattice depth, \(V_L\) the long lattice amplitude, and \(\Omega\) the rf-induced inter-leg coupling [1604.06292]. The tight-binding ladder Hamiltonian in real space is $$ \hat H_L=\sum_j\Big[t_1(a_{j\uparrow}^\dagger b_{j\uparrow}+b_{j\downarrow}^\dagger a_{j+1,\downarrow})+t_2(b_{j\uparrow}^\dagger a_{j+1,\uparrow}+a_{j\downarrow}^\dagger b_{j\downarrow})\Big] +t\sum_j(a_{j\uparrow}^\dagger a_{j\downarrow}+b_{j\uparrow}^\dagger b_{j\downarrow})+\mathrm{h.c.}, $$ where each leg is an SSH chain and, due to the \(d/2\) offset, the roles of \(t_1\) and \(t_2\) exchange between legs. In the Bloch basis \(\Psi_k^\dagger=(a_{k\uparrow}^\dagger,b_{k\uparrow}^\dagger,a_{k\downarrow}^\dagger,b_{k\downarrow}^\dagger)\), the Hamiltonian becomes a \(4\times4\) matrix \(M_k\). It can be block-diagonalized into two \(2\times2\) blocks \(h_{k,\pm}\), $$ h_{k,\pm}= \begin{pmatrix} t\pm (t_1+t_2)\cos(kd/2) & \mp i(t_1-t_2)\sin(kd/2)\\ \pm i(t_1-t_2)\sin(kd/2) & -t\mp (t_1+t_2)\cos(kd/2) \end{pmatrix}, $$ which obey $$ h_{k,\pm}=h_{k+2\pi/d,\mp}. $$ Each block therefore has period \(4\pi/d\), whereas the full Bloch Hamiltonian \(M_k\) has period \(2\pi/d\) [1604.06292]. The symmetry responsible for this structure is the glide operator $$ \hat G=\hat T_{d/2}\hat R, $$ where \(\hat R\) flips spin, \(\uparrow\leftrightarrow\downarrow\), and \(\hat T_{d/2}\) translates by half a lattice spacing. In Bloch form, $$ G^2(k)=e^{ikd}, $$ so the glide eigenvalues are \(\pm e^{ikd/2}\). Because the glide eigenvalue flips sign under \(k\to k+2\pi/d\), bands in the \(+\) sector must connect to bands in the \(-\) sector across the Brillouin zone. With an additional mirror symmetry, the crossing is pinned to \(k=\pm \pi/d\); even without mirror symmetry, a combined inversion and \(AB\) exchange symmetry denoted \(CI\) retains the degeneracies at the Brillouin-zone boundary [1604.06292]. The four-band dispersion has lower energies $$ E_{k,\pm}=-\sqrt{t^2+t_1^2+t_2^2+2t_1t_2\cos(kd)\pm 2t(t_1+t_2)\cos(kd/2)}, $$ with the upper pair given by \(-E_{k,\pm}\). For generic \(k\), the lowest two bands split, except at \(k=\pm\pi/d\), where glide symmetry enforces a degeneracy. At the critical inter-leg coupling $$ t_c=|t_1+t_2|, $$ the lower pair touches the upper pair at \(k=0\), and for \(t>t_c\) a gap reopens between the pairs while the enforced degeneracies at \(k=\pm\pi/d\) remain [1604.06292]. ## 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 \(2\pi/d\); they exchange across \(k=\pm\pi/d\) through the relation \(h_{k,\pm}=h_{k+2\pi/d,\mp}\). The appropriate topological object is therefore the non-Abelian Berry connection for the two-band subspace, $$ A_{mn}(k)=i\langle u_m(k)|\partial_k u_n(k)\rangle, $$ and the associated Wilson line $$ W(k_0\to k_1)=P\exp\left[i\int_{k_0}^{k_1}A(k)\,dk\right]. $$ Its matrix elements encode adiabatic transport under a weak, uniform force \(F\) via Bloch oscillations provided \(w\ll Fd\ll E_G\), where \(w\) is the two-band width and \(E_G\) the gap to higher bands [1604.06292]. For the lowest two periodic Bloch states \(|u_{k,1}\rangle\) and \(|u_{k,2}\rangle\), glide conservation implies that adiabatic transport over one physical Brillouin zone swaps the bands: $$ W(k\to k+2\pi/d)= \begin{pmatrix} 0 & e^{i\phi_-}\\ e^{i\phi_+} & 0 \end{pmatrix} = e^{i\phi_{\mathrm{Zak}}/2} \begin{pmatrix} 0 & e^{-i\phi_r}\\ e^{+i\phi_r} & 0 \end{pmatrix}, $$ where \(\phi_++\phi_-=\phi_{\mathrm{Zak}}\) and \(\phi_r=(\phi_+-\phi_-)/2\). Over two Brillouin-zone traversals, $$ W(k\to k+4\pi/d)=e^{i\phi_{\mathrm{Zak}}}I_2. $$ 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 [1604.06292]. The resulting invariant has two components. The exchange property, described in the source as a nonsymmorphic glide \(Z_2\)-type feature, is robust as long as glide symmetry is preserved and the two-band pair remains gapped from higher bands. The \(U(1)\) part, \(\phi_{\mathrm{Zak}}\in\{0,\pi\}\) mod \(2\pi\), distinguishes phases across the inter-leg transition at \(t_c\). Across this transition, \(\phi_{\mathrm{Zak}}\) changes by \(\pi\), so the \(U(1)\) part of \(W(k\to k+2\pi/d)\) shifts by \(\pi/2\), while the \(SU(2)\) exchange persists continuously [1604.06292]. Open boundary conditions reveal the bulk-boundary correspondence of the \(U(1)\) sector. For \(t<|t_1+t_2|\), there exists a zero-energy end state localized at each end, one per edge, with mixed spin-up and spin-down components. As \(t\) approaches \(t_c\), the edge-state localization length grows and diverges at \(t=t_c\). For \(t>|t_1+t_2|\), the end states disappear. A representative parameter set reported in the source uses \(V_S=8E_R\), \(V_L=4E_R\), and \(\Omega\) chosen such that \(t=0.6\) in units where \(|t_1+t_2|=1.1\), for which edge zero modes exist; for \(t=2.0>1.1\), they vanish [1604.06292]. ## 4. Interaction-driven ferromagnetism and charge fractionalization The glided ladder also supports an interaction-induced mechanism for charge fractionalization. In the flat-band limit \(t_2=t=0\) with \(|t_1|\) dominating, each leg’s lower band consists of localized orbitals $$ c_{j\sigma}^\dagger|0\rangle \propto (a_{j\sigma}^\dagger+b_{j\sigma}^\dagger)|0\rangle, $$ which yields a flat two-fold degenerate ground band labeled by spin. Repulsive onsite interactions, $$ \hat V=U\sum_j\left(\hat n_{j,a\uparrow}\hat n_{j,a\downarrow}+\hat n_{j,b\uparrow}\hat n_{j,b\downarrow}\right),\qquad U>0, $$ lift the degeneracy at half filling and select a ferromagnetic state that fully occupies one leg, producing the two degenerate ground states $$ |G\rangle_1=\prod_j c_{j\uparrow}^\dagger|0\rangle,\qquad |G\rangle_2=\prod_j c_{j\downarrow}^\dagger|0\rangle $$ [1604.06292]. Adding one extra particle to \(|G\rangle_1\) creates two domain walls separating \(\uparrow\)- and \(\downarrow\)-occupied regions. Their confinement depends on which kinetic process dominates. If only inter-leg tunneling \(t\) is present and \(t_2=0\), 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, \(Q=e/2\). If only intra-leg tunneling \(t_2\) is present and \(t=0\), the domain walls are confined, with pair separation of order \(d\) [1604.06292]. With both \(t\) and \(t_2\) finite, projection onto the two-domain-wall subspace gives an effective Hamiltonian $$ \hat H_{\mathrm{eff}} = J\sum_{M,m}\left[\hat D_{M,m}^\dagger \hat D_{M+1/2,m+1}+\hat D_{M,m}^\dagger \hat D_{M-1/2,m+1}\right] + J_2\sum_M\left[\hat D_{M,0}^\dagger \hat D_{M+1,0}+\hat D_{M-1/2,N-1}^\dagger \hat D_{M+1/2,N-1}\right] +\mathrm{h.c.}, $$ with $$ J=t/2,\qquad J_2=|t_2|/2>0. $$ At fixed center-of-mass momentum \(Q\), this reduces to $$ \hat H_Q= 2J\cos(Qd/2)\sum_{m=1}^N\left(\hat D_{Q,m}^\dagger \hat D_{Q,m+1}+\mathrm{h.c.}\right) +2J_2\cos(Qd)\left[\hat D_{Q,0}^\dagger \hat D_{Q,0}+\hat D_{Q,N-1}^\dagger \hat D_{Q,N-1}\right]. $$ The source reports a first-order transition at \(J_2/|J|=2\) between a deconfined phase, with extensive average separation, and a confined phase, with separation \(d/2\). TEBD numerics confirm ferromagnetism for \(|t_2|,|t|\ll U\ll |t_1|\) and the deconfined-to-confined transition [1604.06292]. The fractional charge follows the polarization formula $$ P=\frac{e\phi_{\mathrm{Zak}}}{2\pi}\quad (\mathrm{mod}\ e),\qquad Q=e\,\Delta P\quad (\mathrm{mod}\ e). $$ For domains that differ by \(\Delta P=1/2\), each domain wall carries $$ Q_{\mathrm{dw}}=e/2. $$ Detection is based on the site-resolved density \(n_j\), with background-subtracted profile \(\tilde n_j=n_j-1\). In the deconfined phase, \(\tilde n_j\) matches two hard-core particles’ density. If domain walls are pinned by local potentials \(V_L\) and \(V_R\) at distant wells, the integrated charges satisfy \(\Delta N_L=\Delta N_R=1/2\) with negligible number fluctuations [1604.06292]. ## 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 $$ \Psi(k)=(A\uparrow,B\uparrow,C\uparrow,D\uparrow,A\downarrow,B\downarrow,C\downarrow,D\downarrow)^T. $$ Its Bloch Hamiltonian is $$ H(k)= \begin{pmatrix} h(k) & h_c\\ h_c & h(k) \end{pmatrix}, $$ where $$ h(k)= \begin{pmatrix} 0 & t_a & 0 & t_b e^{-ik}\\ t_a & 0 & t_b & 0\\ 0 & t_b & 0 & t_a\\ t_b e^{ik} & 0 & t_a & 0 \end{pmatrix}, \qquad h_c=t_c I_{4\times4}. $$ In real space, the vertical interlayer coupling \(t_c\) connects corresponding sites on the two layers [2407.05350]. A similarity transformation \(S\) diagonalizes the bilayer Hamiltonian into two decoupled mirror-parity subspaces, $$ S^{-1}HS= \begin{pmatrix} B_1 & 0\\ 0 & B_2 \end{pmatrix}, \qquad B_1=h+t_c I_4,\qquad B_2=h-t_c I_4. $$ The model preserves mirror symmetry along the layer direction, and in this basis the mirror operator becomes $$ S^{-1}M_{2,1}S=\mathrm{diag}(+I_{4\times4},-I_{4\times4}), $$ so \(B_1\) and \(B_2\) carry even and odd mirror parity, respectively [2407.05350]. 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 $$ w=\frac{1}{2\pi}\int_{-\pi}^{\pi}\partial_k \arg[q(k)]\,dk, \qquad q(k)=t_a+t_b e^{-ik}. $$ The nontrivial phase occurs for \(t_a