Abstract
We analyze several 1D and 2D topological lattice Hamiltonians using the generalization of Bloch’s theorem developed in Chap. 2. Apart from providing exact solutions for several important models, the analyses of various models in this chapter also serve as illustrations for using the framework of generalized Bloch theorem.
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Notes
- 1.
- 2.
Explicitly, the Jordan–Wigner mapping yields the Hamiltonian \(\widehat {H}_{XY} = -\sum _{j=1}^{N}B_z(\sigma _z^j+1)- \sum _{j=1}^{N-1}\big (J_x\sigma _x^{j}\sigma _x^{j+1} +J_y\sigma _x^{j}\sigma _x^{j+1}\big ),\) where \(\sigma _x^j,\sigma _y^j, \sigma _z^j \) are Pauli matrices for spin j, B z = μ∕2 is the strength of the magnetic field along the z-direction, and J x = t − Δ, J y = t + Δ are coupling strengths along x and y, respectively.
- 3.
If |𝜖〉 is an eigenstate of H with energy 𝜖, satisfying \(\mathcal {S}_1|\epsilon \rangle =|\epsilon \rangle \), then \(\mathcal {S}_2|\epsilon \rangle \) is also an eigenstate with the same energy. Further, \(\mathcal {S}_2|\epsilon \rangle \) is orthogonal to |𝜖〉, as the relation \(\{\mathcal {S}_1,\mathcal {S}_2\}=0\) leads to \(\mathcal {S}_1(\mathcal {S}_2|\epsilon \rangle ) = -(\mathcal {S}_2|\epsilon \rangle )\).
- 4.
Observe that the states corresponding to j = 1, …, N − 2 are related to the basis states Ψj1,± and Ψj2,± as follows: \(\Psi _{j1,\pm } = \tilde {\Psi }_{j,\pm } \pm \tilde {\Psi }_{j+1,\pm }\) and \(\Psi _{j2,\pm } = -\tilde {\Psi }_{j,\pm } \pm \tilde {\Psi }_{j+1,\pm }\).
- 5.
It is worth noting that eigenfunctions may also be obtained by means of Lie-algebraic methods that are in principle applicable beyond quadratic Hamiltonians, see Ref. [33].
- 6.
There are two other solutions of the system in Eq. (3.38), \(z_{1,\pm }=-\frac {1}{2}\big (\xi \pm \sqrt {4+\xi ^2}\big )\). These solutions are excluded because the internal state |u ℓ〉 vanishes identically if evaluated at z ℓ = z 1,±, see Eq. (3.39). Similar remarks apply to the system in Eq. (3.40).
- 7.
Reference [38] used the notation \(P_{B,k_{\|}}\) (\(\equiv Q_{k_{\|}}\)), which however would be confusing in the present content.
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Alase, A. (2019). Investigation of Topological Boundary States via Generalized Bloch Theorem. In: Boundary Physics and Bulk-Boundary Correspondence in Topological Phases of Matter. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-030-31960-1_3
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