Exciton Proliferation and Fate of the Topological Mott Insulator in a Twisted Bilayer Graphene Lattice Model

Xiyue Lin, Bin-Bin Chen, Wei Li, Zi Yang Meng, and Tao Shi

Phys. Rev. Lett. 128, 157201 – Published 12 April 2022

Abstract

A topological Mott insulator (TMI) with spontaneous time-reversal symmetry breaking and nonzero Chern number has been discovered in a real-space effective model for twisted bilayer graphene (TBG) at $3 / 4$ filling in the strong coupling limit [1]. However, the finite temperature properties of such a TMI state remain illusive. In this work, employing the state-of-the-art thermal tensor network and the perturbative field-theoretical approaches, we obtain the finite- $T$ phase diagram and the dynamical properties of the TBG model. The phase diagram includes the quantum anomalous Hall and charge density wave phases at low $T$ , and an Ising transition separating them from the high- $T$ symmetric phases. Because of the proliferation of excitons—particle-hole bound states—the transitions take place at a significantly reduced temperature than the mean-field estimation. The exciton phase is accompanied with distinctive experimental signatures in such as in charge compressibilities and optical conductivities close to the transition. Our work explains the smearing of the many-electron state topology by proliferating excitons and opens an avenue for controlled many-body investigations on finite-temperature states in the TBG and other quantum moiré systems.

Received 3 October 2021
Revised 3 January 2022
Accepted 20 March 2022

DOI:https://doi.org/10.1103/PhysRevLett.128.157201

Physics Subject Headings (PhySH)

Excitons Phase diagrams Thermodynamics

Graphene Topological materials

Perturbation theory Tensor network methods

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Xiyue Lin^1,2, Bin-Bin Chen ^3,*, Wei Li^1,4, Zi Yang Meng ^3,†, and Tao Shi^1,‡

¹CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
²School of Physical Sciences, University of Chinese Academy of Sciences, Beijng 100049, China
³Department of Physics and HKU-UCAS Joint Institute of Theoretical and Computational Physics, The University of Hong Kong, Pokfulam Road, Hong Kong, China
⁴School of Physics, Beihang University, Beijing 100191, China

^*bchenhku@hku.hk
^†zymeng@hku.hk
^‡tshi@itp.ac.cn

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Vol. 128, Iss. 15 — 15 April 2022

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Images

Figure 1
The KV model with (a) the cluster charge $Q$ and (b) assisted hopping $T$ terms. (c) Illustration of the formation of exciton between two quasi-particles from the valence and conduction bands. (d) The charge correlations between a central site (the asterisk) and other sites, at intermediate temperature $T = 0.244$ inside the exciton proliferation regime. The vectors $δ_{i}$ are introduced to display the charge correlations in Fig. 3. (e) The scattering $T$ matrix that is the renormalized interaction of electrons by the contributions of the ladder diagrams. (f) The Hartree-Fock-like contribution to the self-energy of the single-particle Green’s function.
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Figure 2
(a) Finite-temperature phase diagram of the KV model, with the red regime being the high- $T$ disorder phase, the intermediate- $T$ orange one where exciton proliferates, the green one being the CDW phase, and the purple one being the QAH phase. The phase transition line between the low- $T$ symmetry breaking and the symmetric phase at intermediate $T$ , as well as the crossover temperatures (white dashed line) between intermediate- to the high- $T$ regimes are determined from analyzing the specific heat $c_{V}$ data. Supporting the phase diagram in panel (a), we map the $α − T$ landscapes of (b) specific heat $c_{V}$ , (c) charge structure factor $C_{n} (M)$ , and (d) current correlation $C_{J} (d)$ at a distance of $d = 6$ to clearly identify the phase boundaries.
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Figure 3
For the case of $α = 0.2$ , (a) shows the specific heat $c_{V}$ data, where the peak at $T_{c} = 0.041$ signifies the thermal phase transition. (b) Current-current correlation at a distance of $d = 6$ , i.e., $C_{J} (d = 6)$ vs $T$ . Here, the XTRG and mean-field, as well as the field-theoretical results (Exciton) are shown. (c) Data collapse of the scaled current correlation $C_{J} (d = 6)$ vs $L^{1 / ν} (T - T_{c}) / T_{c}$ with the Ising critical exponents $β = \frac{1}{8}$ and $ν = 1$ , for the various system sizes. Insets show the $C \cdot L^{2 β / ν}$ versus temperature $T$ , where the crossing point between different curves signifies the transition temperature $T_{c}$ . The left axis of panel (d) shows the particle-hole correlation $g_{2}^{n h} (r)$ correlations with the reference site fixed at the center of the system denoted as the asterisk and $r = d (δ_{1} - δ_{2})$ denoted in Fig. 1. The right axis of panel (d) shows the compressibility $\partial n / \partial μ$ vs $T$ . The dashed lines in panels (b) and (d) indicate the specific heat peak $T_{c} = 0.041$ .
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Figure 4
(a) Quasiparticle spectral function $A_{k} (ω) = A_{k}^{v} (ω) + A_{k}^{c} (ω)$ as the sum of the valence and conducting band spectrum, for $α = 0.2$ at $T = 0.08$ , where the spectral weight distribution gets broadened as compared to the mean-field single-particle dispersion consisted of upper conducting and lower valence bands (white dots). The exciton band lying within the single-particle gap is denoted by green diamonds. The inset depicts the path in the Brillouin zone. Panels (b) and (c) show the valence electron spectral weights at $k = Γ$ and $k = K$ for various temperatures, where the values of $A$ and $ω$ are properly shifted as indicated by the tilted white lines. (d) The gap, defined as twice the peak position of $A_{Γ}^{v} (ω)$ , as a function of temperature $T$ .
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