-Continuum limit of bipartite lattices -- The SSH model

TL;DR

The paper advances a spatially continuous, non-local 1D SSH-like model that preserves chiral symmetry and reproduces the SSH bulk spectrum via $E_\pm(k)=\pm\sqrt{v^2+w^2+2vw\cos(a k)}$, with a tunable non-local length $a$ ensuring a periodic Brillouin zone. By expanding the translation operator $T(a)$, the authors connect the non-local model to local approximations $\mathbf{H}^{(n)}$, illustrating how topology emerges and how a true Zak phase is recovered in the non-local limit. A key result is the finite-size behavior: the non-local model supports a flat zero-energy mini-band formed by an infinite countable set of exponentially localized edge states, whose localization is controlled by $a$. The framework offers a general method to construct non-local continuous analogues of bipartite and multipartite lattices and points to experimental challenges for realizing such non-local continuous topological matter.

Abstract

We present a continuous non-local model that faithfully replicates the rich topological and spectral features of the Su-Schrieffer-Heeger (SSH) model. Remarkably, our model shares the SSH models bulk energy spectrum, eigenstates, and Zak phase, hallmarks of its topological character, while introducing a tunable length-scale a quantifying non-locality. This parameter allows for a controlled interpolation between non-local and local regimes. Furthermore, for a specific value of a the exact spectral equivalence to the discrete SSH model is established. Distinct from previous continuous analogues based on Schrödinger or Dirac-type Hamiltonians, our approach maintains chiral symmetry, does not require an external potential and features periodic energy bands. On finite domains, the model supports a flat band with zero energy formed by a countable infinite set of exponentially localized zero-energy edge states of topological origin. Beyond SSH, our method lays the foundation for constructing non-local, continuous analogues of a wide class of bipartite and multipartite lattices, opening new paths for theoretical exploration and new challenges for experimental realization in topological quantum matter.

Figures (4)

• Figure 1: Sketch of the continuous one-dimensional SSH model proposed in the current work. The value of the wave field at $x$ is depending on the corresponding values at $x \pm a$ in the specific manner shown here.
• Figure 2: The spectrum of the non-local continuous model for $w=1$ and $v=0.5$ (in a.u.) shown with red solid lines. We also show the spectra for its approximations obtained through the expansion of the translation operator up to $O(a^n)$ with $n=1$ (blue stars connected with a blue dashed line) and $n=2$ (olive crosses connected with an olive dashed line). Only the region $k a \in [-\pi,\pi]$ is displayed. The horizontal black dashed lines present the infinitely degenerate flat bands, occurring for $n=0$.
• Figure 3: The spectrum of the finite non-local, continuous in space, model $\mathbf{H}_{L}$ for $w_0=1$ and $v_0=0.5$ (in a.u.) shown with blue solid lines. In the same plot we display with red crosses the spectrum of the discrete SSH lattice for the same values of $w_0$, $v_0$, open boundary conditions and number of sites $N=2002$. For illustrative reasons we show every $100$-th eigenenergy of the discrete system. In the inset we magnify the region around $\mathcal{E}_{L}=0$. There is an accumulation of zero energy states in the non-local model in contrast to the discrete SSH lattice where only two states with energy close to zero appear.
• Figure 4: The magnitude of the zero energy eigenstate-components $\psi_s(x)$ ($s=A$, $B$) with $(n_A,m_A)=(3,1),~(n_B,m_B)=(5,2)$ on a logarithmic scale. The parameter values used are $v_0=0.5$, $w_0=1$, $L=10$ and $a=0.2$ all in arbitrary units. The exponential envelope with characteristic exponent ${\frac{1}{a} \ln \vert \frac{w_0}{v_0} \vert} \approx 1.5$ is clearly recognizable. Notice that $\psi_s(x)$ under the action of the operator $\bm{\sigma}_x \mathbf{\Pi}(x)$ (generalized parity) transforms to the function $\tilde{\psi}_s(x)$ with $\tilde{\psi}_A(x)=\psi_B(-x)$ and $\tilde{\psi}_B(x)=\psi_A(-x)$.