--- title: SSHm Model in Asymmetric Topology url: https://www.emergentmind.com/topics/sshm-model type: topic --- # SSHm Model in Asymmetric Topology Searching arXiv for the SSHm model paper and related asymmetric topological systems. The SSHₘ model is a generalized Su–Schrieffer–Heeger chain with an $m$-site unit cell and, in its most basic form, real nearest-neighbor hoppings $\{t_j\}$ that need not be equal. In the formulation developed for asymmetric systems with chiral boundary states, the model is used to show that nontrivial boundary topology can persist even when the full lattice lacks the usual chiral symmetry and, for generic odd $m$ with unequal hoppings, also lacks inversion symmetry. Its central theoretical move is a redefinition of sublattices that exposes “sub-symmetry” protected edge modes, together with a Rice–Mele-like effective Hamiltonian and a quantized normalized Zak phase $\bar Z$ that restores a bulk–edge correspondence for these asymmetric chains [2509.20834]. ## 1. Hamiltonian and lattice structure In real space, the generalized SSH model with an $m$-site unit cell is \[ H_{SSH_m}=\sum_{n=1}^N\Bigl[\sum_{j=1}^{m-1} t_j\,c_{n,j+1}^\dagger c_{n,j}+t_m\,c_{n+1,1}^\dagger c_{n,m}+\mathrm{h.c.}\Bigr], \] where $n=1,\dots,N$ labels unit cells, $j=1,\dots,m$ labels sublattices within a cell, and $t_j\in\mathbb R$ are nearest-neighbor hoppings. In momentum space, the model is written as $H_{SSH_m}=\sum_k C_k^\dagger \mathcal H_m(k) C_k$, with $\mathcal H_m(k)$ an $m\times m$ Bloch Hamiltonian whose off-diagonal structure implements the intracell hoppings $t_1,\dots,t_{m-1}$ and the intercell hopping $t_m e^{\pm ik}$ between the first and last sublattices of neighboring cells [2509.20834]. A defining feature of the SSHₘ setting is asymmetry among the hoppings. For generic odd $m$ and $t_j\neq t_{j'}$, both inversion and the usual chiral symmetry are broken. The model is therefore not a straightforward extension of the conventional two-site SSH chain. The significance of the construction lies precisely in demonstrating that the absence of these global symmetries does not preclude topological boundary phenomena; instead, the topology is recovered through a different organization of the Hilbert space. ## 2. Redefined sublattices and “sub-symmetry” The core mechanism is a unitary redefinition of sublattices near an edge. For left-edge modes, one diagonalizes the $(m-1)\times(m-1)$ block \[ \mathcal H^L_{m-1}=\bigl(\mathcal H_m(0)\bigr)_{j,j'=1,\dots,m-1} \quad\longrightarrow\quad U^\dagger \mathcal H^L_{m-1} U=\mathrm{diag}(V_{1'},\dots,V_{(m-1)'}), \] and defines new orbitals \[ |j'\rangle=\sum_{i=1}^{m-1}U_{i\,j'}\,|i\rangle,\qquad j'=1',\dots,(m-1)'. \] In this basis, the Hamiltonian becomes \[ H_{SSH_m}'=\sum_{j'=1'}^{(m-1)'}\Bigl[H_{j'}^{SSH}+V_{j'}P_{j'}\Bigr], \] where each \[ H_{j'}^{SSH}=\sum_n\Bigl[w_{j'}\,c_{n,m}^\dagger c_{n,j'}+v_{j'}\,c_{n+1,j'}^\dagger c_{n,m}+\text{h.c.}\Bigr] \] is an SSH chain on sublattices $\{j',m\}$ with \[ w_{j'}=t_{m-1}\,\phi_{m-1}^{(j')},\qquad v_{j'}=t_m\,\phi_1^{(j')}, \] and $P_{j'}=\sum_n |n,j'\rangle\langle n,j'|$ [2509.20834]. The important point is that the couplings through $|m\rangle$ break the ordinary chiral symmetry of each effective SSH$_{j'}$ chain, yet preserve a “sub-symmetry” \[ \Gamma_{j'}^{\dagger}\Bigl(\sum_{k'}H_{k'}^{SSH}\Bigr)\Gamma_{j'}\,P_{j'} = -\sum_{k'}H_{k'}^{SSH}P_{j'}, \qquad \Gamma_{j'}=P_{j'}-P_m. \] This guarantees a zero mode of $\sum_{k'}H_{k'}^{SSH}$ living entirely on $j'$, \[ \Bigl(\sum_{k'}H_{k'}^{SSH}\Bigr)|L_{j'}\rangle=0,\qquad P_{j'}|L_{j'}\rangle=|L_{j'}\rangle. \] After the term $V_{j'}P_{j'}$ is restored, the mode energy is shifted to $E=V_{j'}$, and one left-edge mode appears when $|v_{j'}|>|w_{j'}|$. The same logic applied to the block $\{2,\dots,m\}$ yields right-edge modes. A common misconception is that breaking global chiral symmetry eliminates chiral boundary states altogether; the SSHₘ analysis shows, more precisely, that these states can survive as sub-symmetry protected modes after an appropriate redefinition of the sublattice basis. ## 3. Rice–Mele-like effective description A complementary formulation uses an isospectral Schur reduction onto the end sublattices $S=\{1,m\}$. Writing \[ \mathcal H_m(k)= \begin{pmatrix} \mathcal H_{\bar S\bar S} & \mathcal H_{\bar S S}\\ \mathcal H_{S\bar S} & \mathcal H_{SS} \end{pmatrix}, \] the projected sector satisfies \[ \widetilde{\mathcal H}_S(k,E)\,\phi_S=E\,\phi_S, \qquad \widetilde{\mathcal H}_S = \mathcal H_{SS} - \mathcal H_{S\bar S} \bigl(\mathcal H_{\bar S\bar S}-E\bigr)^{-1} \mathcal H_{\bar S S}. \] For $m=3$, this reduction reproduces a two-band Rice–Mele form, \[ \widetilde{\mathcal H}_S(k,E)= \begin{pmatrix} \frac{t_1^2}{E} & \frac{t_1 t_2}{E}+t_3 e^{ik}\\[6pt] \frac{t_1 t_2}{E}+t_3 e^{-ik} & \frac{t_2^2}{E} \end{pmatrix}, \] which hosts chiral boundary states when $E=v_1(E)=t_1$ or $E=v_2(E)=t_2$ [2509.20834]. More generally, the effective Hamiltonian takes the Rice–Mele-like block form \[ \widetilde{\mathcal H}_S(k,E)= \begin{pmatrix} v_1(E)\,I & h(k,E)\\[3pt] h^\dagger(k,E) & v_2(E)\,I \end{pmatrix}, \] with zero modes localized on the two ends of $S$ at $E=v_{1,2}(E)$. This effective description is central because it converts a structurally asymmetric multiband problem into a reduced two-end problem in which the origin of the boundary states becomes explicit. A plausible implication is that the SSHₘ model is less usefully viewed as a perturbation of the conventional SSH chain than as a broader class whose topology is revealed only after energy-dependent reduction. ## 4. Quantized normalized Zak phase Although the full SSHₘ chain lacks spatial inversion in the generic asymmetric case, the reduced Rice–Mele-like Hamiltonian without the $v_{1,2}(E)$ terms, \[ \widetilde{\mathcal H}_S^0(k)= \begin{pmatrix} 0 & h(k,E)\\ h^\dagger(k,E) & 0 \end{pmatrix}, \] always satisfies a mirror symmetry \[ M_x \widetilde{\mathcal H}_S^0(k) M_x=\widetilde{\mathcal H}_S^0(-k). \] If its two-component eigenvectors are written as \[ \psi_S^j(k)= \begin{pmatrix} \phi_A^j(k)\\ \phi_B^j(k) \end{pmatrix}, \] the normalized reduced wavefunctions are defined by \[ \tilde\phi_\sigma^j(k)= \frac{\phi_\sigma^j(k)}{\sqrt{2\langle \phi_\sigma^j|\phi_\sigma^j\rangle}}, \qquad \sigma=A,B, \qquad \tilde\psi_S^j(k)= \binom{\tilde\phi_A^j(k)}{\tilde\phi_B^j(k)}. \] One then has \[ M_x\,\tilde\psi_S^j(k)=e^{i\theta(k)}\,\tilde\psi_S^j(-k), \] and the normalized Zak phase \[ \bar Z_j = i\int_{-\pi}^{\pi}dk\, \bigl\langle \tilde\psi_S^j(k)\big|\partial_k \tilde\psi_S^j(k)\bigr\rangle \in\{0,\pi\} \] remains quantized even though the full chain lacks inversion symmetry [2509.20834]. Physically, $\bar Z_j=\pi$ signals one chiral edge mode emerging in the $j$th gap. The introduction of $\bar Z$ is therefore not merely a reformulation of the ordinary Zak phase; it is a band-topological invariant specifically adapted to the reduced, normalized description of asymmetric systems. This addresses another common misunderstanding: the loss of standard symmetry indicators in the full lattice does not imply the absence of any quantized topological index, only that the appropriate invariant must be defined on the reduced effective problem. ## 5. Bulk–edge correspondence and spectral content With $\bar Z_j\in\{0,\pi\}$ assigned to each bulk band $j$, the bulk–edge correspondence takes a direct form. If $\bar Z_j=0$, there is no new chiral mode in the gap above band $j$; if $\bar Z_j=\pi$, one chiral mode appears in that gap. Equivalently, the total number of left-localized edge states below energy $E$ is \[ N_L(E)=\sum_{j\,:\,E_j2$, a left mode appears at energy $E=t_1=1$ and a right mode appears at $E=t_2=2$. Their wavefunctions decay as $|z|^n$ into the bulk and exhibit a chiral hallmark: the amplitude on one of the three sublattices is exactly zero. At the same transition point, the normalized Zak phases of the lower and middle bands jump from $0$ to $\pi$, in exact agreement with the emergence of the chiral edge states [2509.20834]. The same framework is extended in the source work to more complex asymmetric ladder models and to two-dimensional asymmetric systems. In the 2D BBH3 model, redefined lattices yield new chiral corner states that are independent of any spatial symmetry and exhibit the characteristics of topological bound states in the continuum. The topological invariant can likewise be calculated by introducing $\bar Z$ into 2D, and an acoustic experiment of the BBH3 model is proposed in which chiral corner states are numerically observed. The broader significance claimed for these developments is that models with entirely different structures might share the same topological origins once the appropriate sublattice redefinition is made. Source: https://www.emergentmind.com/topics/sshm-model