Temperature-Driven Gapless Topological Insulator

Miguel Gonçalves, Pedro Ribeiro, Rubem Mondaini, and Eduardo V. Castro

Phys. Rev. Lett. 122, 126601 – Published 28 March 2019

Abstract

We investigate the phase diagram of the Haldane-Falicov-Kimball model—a model combining topology, interactions, and spontaneous disorder at finite temperatures. Using an unbiased numerical method, we map out the phase diagram on the interaction-temperature plane. Along with known phases, we unveil an insulating charge ordered state with gapless excitations and a temperature-driven gapless topological insulating phase. Intrinsic—temperature-generated—disorder is the key ingredient explaining the unexpected behavior. Our findings support the possibility of having temperature-driven topological transitions into gapped and gapless topological insulating phases in mass unbalanced systems with two fermionic species.

Received 6 September 2018
Revised 11 December 2018

DOI:https://doi.org/10.1103/PhysRevLett.122.126601

Physics Subject Headings (PhySH)

Quantum anomalous Hall effect Topological insulators

Disordered systems Strongly correlated systems

Monte Carlo methods

Statistical Physics & ThermodynamicsCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Miguel Gonçalves¹, Pedro Ribeiro^1,2, Rubem Mondaini², and Eduardo V. Castro^1,2,3

¹CeFEMA, Instituto Superior Técnico, Universidade de Lisboa, Avenida Rovisco Pais, 1049-001 Lisboa, Portugal
²Beijing Computational Science Research Center, Beijing 100084, China
³Centro de Física das Universidades do Minho e Porto, Departamento de Física e Astronomia, Faculdade de Ciências, Universidade do Porto, 4169-007 Porto, Portugal

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Vol. 122, Iss. 12 — 29 March 2019

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Images

Figure 1
Phase diagram of the HFKM in the interaction-temperature plane obtained with the Monte Carlo method. Phases at intermediate to high $T$ , outside the CDW: TI for small $U$ , GTI and gapless insulator (GI) for intermediary $U$ , and Mott-like insulating phase (MI) for large $U$ . Phases at low $T$ , inside the CDW phase: Phases with similar features as their high- $T$ counterparts were found, and the suffix “-CDW” was added. The thin (red) and dashed-dotted (blue) curves correspond, respectively, to the topological and CDW phase transitions, and the thick (green) curve bounds the gapless region of the phase diagram. The black squares mark the points used in Figs. 2–2.
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Figure 2
(a) Monte Carlo results of the CDW phase transition along with the small and large $U$ curves obtained with the second-order perturbation theory [59]. The larger sites in the honeycomb unit cell inside the CDW phase represent occupied sites indicating a checkerboard order; the arrows represent the flow of NNN hoppings. The inset shows an example of the usage of the Binder cumulant method to compute the critical temperature $T_{CDW}$ that corresponds to the intersection of the obtained curves for different system sizes, for a case where $U = 5$ . (b)–(d) Density of states for different points in the phase diagram, marked with black squares in Fig. 1: MI-CDW $(U, T) = (2.5, 0.045)$ , GI-CDW (2.5,0.085), TI (1,0.2), GTI (2,0.2), GI (4,0.2), and MI (5,0.2); we use a Lorentzian broadening of width 0.01. (e) Finite-size scaling of the DOS at $E = 0$ for the point (2.5,0.085) used in (b). The DOS ( $E = 0$ ) was computed in an energy window corresponding to 1% of the full bandwidth for $L = 8$ (smallest used system, with volume $V_{0}$ ). This window was reduced proportionally to the system size for larger systems.
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Figure 3
Chern number computed through averages on Monte Carlo configurations of $f$ electrons for different system sizes, with fixed $U$ (a) and $T = 0.2$ (b). The crossing points of these curves were used to obtain the topological transition curve in Fig. 1.
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Figure 4
(a) Standard deviations of the LSS distributions obtained for different energies in the GI [GTI] phase for $(U, T) = (2, 0.1)$ [ $= (3.5, 0.2)$ ] and $L = 14$ . The horizontal (red) line corresponds to $σ / ⟨ s ⟩ = 0.422$ , which is the standard deviation of the Wigner distribution associated to extended states. The two extended states existing in the GTI phase are marked with arrows. (b) [(e)] Finite-size scaling of the IPR with the system volume $V$ for the energies marked with the arrows in (c) [(d)], that shows the IPR for different sizes in the GI-CDW [GI] phase for $(U, T) = (2.5, 0.085)$ [ $= (3.5, 0.2)$ ]. The IPR shown in (c) [(d)] for negative (positive) energies is symmetric in $E$ . The red dashed lines shown in (b),(e) have a unit slope and indicate the scaling $IPR \sim V^{- 1}$ . The colors of the arrows that select specific energies in (c) [(d)] match the corresponding scaling curves in (b) [(e)].
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