Generalized Falicov-Kimball models

Xiao-Hui Li, Zewei Chen, and Tai Kai Ng

Phys. Rev. B 100, 094519 – Published 12 September 2019

Abstract

In this paper we extend the Falicov-Kimball model (FKM) to the case where the “quasiparticles” entering the FKM are not “ordinary” fermions. As an example we first discuss how the FKM can be generalized to the case with spin-dependent hopping. Afterward, we discuss several cases where the quasiparticles entering the FKM are Majorana fermions [extended Majorana-Falicov-Kimball model (MFKM)]. Two examples of the extended MFKM are discussed in detail: (i) a $p$ -wave BCS superconductor on a bipartite lattice and (ii) a BCS-Anderson model. We also discuss the most general forms of extended MFKM, including a brief discussion of the case where the Majorana fermions represent spins but not real fermion particles.

Received 2 April 2019

DOI:https://doi.org/10.1103/PhysRevB.100.094519

Physics Subject Headings (PhySH)

Majorana fermions Quantum phase transitions Topological phases of matter

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Xiao-Hui Li^*, Zewei Chen, and Tai Kai Ng^†

Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China

^*xlibb@connect.ust.hk
^†phtai@ust.hk

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Vol. 100, Iss. 9 — 1 September 2019

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Images

Figure 1
(a) Solitonic excitation properties. The solid curve corresponding to the label on the left is the solitonic excitation energy $Δ E / t_{1 ↑}$ . The dashed curve corresponding to the label on the right is the spin carried by the excitation $Δ S / ℏ$ . The dash-dotted curve corresponding to the label on the right is the charge carried by the excitation $Δ C$ . (b) The corresponding quasiparticle excitation spectrum. The dashed horizontal line describes the chemical potential $μ = 0$ . The other parameters for both panels are set as $ϕ = \frac{π}{2}, t_{2 ↑} = 0.2 t_{1 ↑}$ .
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Figure 2
Chern number phase diagrams as a function of $U / t_{1}$ and $δ$ . We set $ϕ = \frac{π}{2}$ and $t_{2, σ} = 0.2 t_{1, σ}$ .
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Figure 3
BCS-Hubbard model on a square lattice, with the blue dots representing the $A$ sublattice and the red dots representing the $B$ sublattice [14]. The hopping term $t$ is uniform along all the nearest-neighbor bonds, i.e., $t_{A \to B} = t_{B \to A}$ , while the pairing potential has a staggered form, $Δ_{A \to B} = - Δ_{B \to A}$ . $U$ represents the Hubbard on-site interaction.
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Figure 4
The solitonic excitation in the BCS-Hubbard model. The solid curve corresponding to the left label is the solitonic excitation energy $Δ E$ . The dash-dotted curve corresponding to the right label is the spin $Δ m_{y}$ carried by the excitation. The charge carried by the excitation is always zero.
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Figure 5
(a) The ground-state properties as a function of interaction strength $U$ for $μ = 0.3 t$ and $B = 0$ ( $μ > B$ , the uniform $D_{i}$ case). The solid curve represents the total electron density of the system $ν$ , the dashed curve represents the pairing correlation $Δ_{G} / t$ , and the dotted curve represents the magnetization along the $y$ direction $S_{y G} / 0.5 ℏ$ . (b) The ground-state properties as a function of interaction strength $U$ for $μ = 0$ and $B = 0.3 t$ ( $μ < B$ , the staggered $D_{i}$ case). The solid curve represents the total electron density of the system $ν$ . The dashed curve represents the staggered pairing correlation ${\bar{Δ}}_{G} / t$ , and the dotted curve represents the staggered magnetization along the $y$ direction ${\bar{S}}_{y G} / 0.5 ℏ$ . The other parameters are set as $V = 0.6 t, N = 50$ .
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Figure 6
(a) The ground-state properties as a function of interaction for the uniform $D_{i}$ case ( $μ > B$ , here $B = 0$ and $μ = 0.3 t$ ). The solid curve represents the filling factor of the system $ν$ . The dashed curve represents the pairing correlation $Δ_{G} / t$ , and the dotted curve represents the magnetization along the $y$ direction $S_{y G} / 0.5 ℏ$ . (b) The ground-state properties as a function of interaction for the staggered $D_{i}$ case ( $μ < B$ , here $μ = 0$ and $B = 0.3 t$ ). The solid curve represents the filling factor of the system $ν$ . The dashed curve represents the staggered pairing correlation ${\bar{Δ}}_{G} / t$ , and the dotted curve represents the staggered magnetization along the $y$ direction ${\bar{S}}_{y G} / 0.5 ℏ$ . The other relevant parameters for both panels are set as $V = 0.6 t, N = 50$ .
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