TY - JOUR

T1 - Is the continuum SSH model topological?

AU - Shapiro, Jacob

AU - Weinstein, Michael I.

N1 - Funding Information: We thank Gian Michele Graf for stimulating discussions and Amir Sagiv for help with numerical simulations. J.S. acknowledges the support from the Swiss National Science Foundation (Grant No. P2EZP2_184228) and the support from the Columbia University Mathematics Department and Simons Foundation under Award No. 376319, while a postdoctoral fellow during 2018–2019. M.I.W. was supported, in part, by the National Science Foundation under Grant Nos. DMS-1412560, DMS-1620418, and DMS-1908657 and by the Simons Foundation Math + X Investigator under Award No. 376319. Publisher Copyright: © 2022 Author(s).

PY - 2022/11/1

Y1 - 2022/11/1

N2 - The discrete Hamiltonian of Su, Schrieffer, and Heeger (SSH) [Phys. Rev. Lett. 42, 1698-1701 (1979)] is a well-known one-dimensional translation-invariant model in condensed matter physics. The model consists of two atoms per unit cell and describes in-cell and out-of-cell electron-hopping between two sub-lattices. It is among the simplest models exhibiting a non-trivial topological phase; to the SSH Hamiltonian, one can associate a winding number, the Zak phase, which depends on the ratio of hopping coefficients and takes on values 0 and 1 labeling the two distinct phases. We display two homotopically equivalent continuum Hamiltonians whose tight binding limits are SSH models with different topological indices. The topological character of the SSH model is, therefore, an emergent rather than fundamental property, associated with emergent chiral or sublattice symmetry in the tight-binding limit. In order to establish that the tight-binding limit of these continuum Hamiltonians is the SSH model, we extend our recent results on the tight-binding approximation [J. Shapiro and M. I. Weinstein, Adv. Math. 403, 108343 (2022)] to lattices, which depend on the tight-binding asymptotic parameter λ.

AB - The discrete Hamiltonian of Su, Schrieffer, and Heeger (SSH) [Phys. Rev. Lett. 42, 1698-1701 (1979)] is a well-known one-dimensional translation-invariant model in condensed matter physics. The model consists of two atoms per unit cell and describes in-cell and out-of-cell electron-hopping between two sub-lattices. It is among the simplest models exhibiting a non-trivial topological phase; to the SSH Hamiltonian, one can associate a winding number, the Zak phase, which depends on the ratio of hopping coefficients and takes on values 0 and 1 labeling the two distinct phases. We display two homotopically equivalent continuum Hamiltonians whose tight binding limits are SSH models with different topological indices. The topological character of the SSH model is, therefore, an emergent rather than fundamental property, associated with emergent chiral or sublattice symmetry in the tight-binding limit. In order to establish that the tight-binding limit of these continuum Hamiltonians is the SSH model, we extend our recent results on the tight-binding approximation [J. Shapiro and M. I. Weinstein, Adv. Math. 403, 108343 (2022)] to lattices, which depend on the tight-binding asymptotic parameter λ.

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U2 - 10.1063/5.0064037

DO - 10.1063/5.0064037

M3 - Article

SN - 0022-2488

VL - 63

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

IS - 11

M1 - 111901

ER -