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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November 2022
Research Article|
November 14 2022
Is the continuum SSH model topological?
Special Collection:
Mathematical Aspects of Topological Phases
Jacob Shapiro
;
Jacob Shapiro
a)
(Conceptualization, Data curation, Formal analysis, Funding acquisition, Investigation, Methodology, Project administration, Resources, Software, Supervision, Validation, Visualization, Writing – original draft, Writing – review & editing)
1
Department of Physics, Princeton University
, Princeton, New Jersey 08544, USA
a)Author to whom correspondence should be addressed: jacobshapiro@princeton.edu; Present address: Department of Mathematics, Princeton University, Princeton, New Jersey 08544, USA
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Michael I. Weinstein
Michael I. Weinstein
(Conceptualization, Data curation, Formal analysis, Funding acquisition, Investigation, Methodology, Project administration, Resources, Software, Supervision, Validation, Visualization, Writing – original draft, Writing – review & editing)
2
Department of Applied Physics and Applied Mathematics, Department of Mathematics, Columbia University
, New York, New York 10027, USA
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a)Author to whom correspondence should be addressed: jacobshapiro@princeton.edu; Present address: Department of Mathematics, Princeton University, Princeton, New Jersey 08544, USA
Note: This paper is part of the Special Topic on Mathematical Aspects of Topological Phases.
J. Math. Phys. 63, 111901 (2022)
Article history
Received:
July 19 2021
Accepted:
October 24 2022
Citation
Jacob Shapiro, Michael I. Weinstein; Is the continuum SSH model topological?. J. Math. Phys. 1 November 2022; 63 (11): 111901. https://doi.org/10.1063/5.0064037
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