Exciton collective modes in a bilayer of axion insulator $\text{MnBi}_2 \text{Te}_4$

Abstract

We investigate the emergence of an exciton condensate and associated collective modes in a bilayer configuration of $\text{MnBi}_2\text{Te}_4$, an antiferromagnetic topological insulator and van der Waals material, recognized for hosting axion physics. Utilizing a minimal low-energy Hamiltonian for the two layer system which is gapped by the intrinsic Néel order, we first employ mean-field theory to establish the conditions for exciton condensation. Our analysis identifies a nonzero, spin-singlet exciton order parameter which is tuned by external displacement field, temperature, and Coulomb attraction. Beyond the mean-field, we explore collective mode fluctuations in the uncondensed phase via many-body perturbation theory and the random phase approximation. From this, we derive the exciton spectral function which allows for a direct comparison between theoretical prediction and experimental observation. We detail how the softening of the collective mode peak is a function of the competition between interlayer detuning and thermal fluctuations. This work elucidates how the unique topological and magnetic environment of $\text{MnBi}_2\text{Te}_4$ offers a tunable platform for the realization and manipulation of exciton condensates and the corresponding collective excitations. Our findings contribute to understanding the interplay of topology and bosonic condensates, which could inspire application in optically accessing topological properties, dissipationless transport, and gate-tunable optoelectronics.

Figures (4)

• Figure 1: Crystal structure of $\text{MnBi}_2\text{Te}_4$ with space group $R\bar{3}m{\rm :}H$ with (a) showing the isotropic view. A single septuple layer is highlighted, with the AFM magnetic moments of Mn atoms denoted with vectors. (b) Schematic of the configuration of the two layers of $\text{MnBi}_2\text{Te}_4$, where each sheet indicates a single septuple layer of material. The oppositely spin-polarized Dirac cones on each layer illustrate that this material is a topological insulator. The two layers are connected by an external displacement field, D. The AFM ordering on each layer is indicated by Néel vector, $N_z$, which points along the $z$-direction. The layer operators are denoted as $\tau^{\pm}$ which correspond to taking an electron from the lower layer to the upper layer, and vice versa. The electrostatic Coulomb interaction, $V$, arises due to chemical potential imbalance between the upper and lower layers, which is driven by the displacement field.
• Figure 2: Upper panel (a)-(c): Mean-field phase diagram of interlayer exciton order parameter $\Delta_0$: color encodes $|\Delta_0|/N_z$ obtained by solving the self-consistent gap equations derived from the Hubbard--Stratonovich decoupling of the interlayer Coulomb term from Eqs. \ref{['eq:S_eff_minimize']}-\ref{['eq:G_action']}. Axes are the dimensionless displacement field $D / N_z$ and temperature $T / N_z$; panels (a)-(c) compare the interaction strengths $V N_z / v_F^2 = 0.285,0.293,$ and $0.301$. At moderate values of $V N_z / v_F^2$, a finite $\Delta_0$ appears only once $D$ exceeds a threshold $D_c(T)$ which creates a population imbalance of electrons (upper layer) and holes (lower layer). The critical line $T_c(D)$ rises with both $D$ and $V$ as the interlayer particle-hole susceptibility grows, while the overall energy scale of $\Delta_0$ and $T_c$ is set by the Néel order $N_z$, which fixes the Dirac gap and available phase space for pairing. All energies are normalized by $N_z$, and $v_F$ is the Fermi velocity. Lower panel (e)-(g): exciton spectral function $A_{00}(\Omega, \boldsymbol{q}=0)$, in the uncondensed phase ($T>T_c$). Each subfigure shows two values of displacement field $D$ (solid vs. dashed curves) at fixed interaction strengths (a) $V N_z/v_F^2{=}0.285$, (b) $0.293$, (c) $0.300$, and compares two temperatures (red vs. blue curves). Note that as we consider higher D values, we must also increase the temperature so that we stay outside of the condensate dome shown in the phase diagram above each subfigure. The displacement field and temperature for each subfigure are indicated on the phase diagram plot immediately above it.
• Figure 3: Many-body band structure for constant $T/N_z = 0.15$ and $V N_z / v_F^2 = 0.285$. Band evolution as D is scanned across the mean field phase diagram from the uncondensed into the condensed phase. At small displacement field $D/N_z = 0.3$, the bilayer $\text{MnBi}_2\text{Te}_4$ spectrum consists of two gapped Dirac cones separated by the Néel order $N_z$. As D increases through the condensate region of Fig. \ref{['fig:PhaseDiagram_SpectralFunction']}(a)-(c), hybridization mediated by the exciton order parameter $\Delta_0$ drives a band inversion, signaling a possible topological phase transition. In this regime the condensate may be identified as an exciton insulator, where the excitonic gap supplants the single-particle Dirac mass.
• Figure 4: Dependence of the interlayer Coulomb interaction V on the critical exciton condensation temperature $T_c$. The solid black curve delineates where the self-consistent mean-field solution yields a finite exciton order parameter $\Delta_0$ at zero displacement field $D=0$. The red dashed line indicates an interaction strength V such that no exciton condensation can take place. The purple dashed line indicates a threshold voltage, defining the minimal interaction strength required to overcome thermal and single-particle energy scales for exciton binding. Physically, larger $V$ enhances interlayer electron–hole attraction, raising $T_c$ and stabilizing the condensate, whereas for $V < V_c$ only incoherent excitonic fluctuations persist.