Introduction

Axions were first introduced as a hypothetical elementary particle through the quantum field theory in high energy physics to solve the strong charge-parity problem1,2,3. Since electrons in the condensed system can also be described in the field operator language, the concept of axions was lately extended to condensed matter physics through the topological field theory4,5, where quasi-particle excitations in condensed system obey the axion electrodynamics, giving birth to the dynamic axion insulator that has sparked a surge of interest across multidisciplinary fields6,7,8,9,10,11. Because of the inherent dynamic axion field, DAIs possess an additional term \({{\mathcal{L}}}_{\theta }=\theta (t)\alpha {\bf{E}}\cdot {\bf{B}}/\pi\) in the Lagrangian4,5, where θ(t) is the time-dependent, massive axion field, α is the fine structure constant, E and B are conventional magnetoelectric field. This Lagrangian introduces a magnetoelectric contribution to Maxwell’s equations, making it possible to realize the axion polaritons or axion instability in condensed systems5,12,13, to drive anomalous magnetoelectric transport14,15, to detect dark matter16,17,18,19, and so on20,21.

Several candidate systems have been predicted to be DAIs14,22,23,24,25,26,27. In particular, recent first-principles calculations suggest that Mn2Bi2Te5 and its family, which possess coexisting antiferromagnetic ordering and topology28, may be materials of this category29,30. Since the axion field in these materials originates from the exchange interaction between topological electrons and the antiparallel magnetic moments, or the Néel vectors, the axion dynamics can then be readily achieved through the magnetic fluctuation that stimulates a time-dependent Néel vector5,29. However, because of the ultrafast dynamics of the axion field, detecting this novel state remains an outstanding challenge, thereby significantly hindering its further exploration and potential applications. Meanwhile, there is also fierce debate about whether the magnitude of the axion mass in condensed matter materials is on the order of eV or meV31,32,33,34.

In this work, we demonstrate that this exotic quantum state can be unambiguously verified by the transport signature of a superconductor-DAI-superconductor Josephson junction. Since DAIs possess spontaneous antiferromagnetic ordering and axion field, an antiferromagnetic resonance (AFMR) can occur in DAI under the driving of a linearly polarized microwave if the microwave frequency ω matches its intrinsic AFMR frequency35,36,37. In this case, the Néel vector, hence the magnetic exchange gap in the DAI, becomes time-dependent, giving rise to a dynamic axion field bound to the AFMR. The quantitative expression for this dynamic axion field turns out to be a sinusoidal function possessing a strikingly doubled frequency ω1 = 2ω. Subsequently, this dynamic axion field leads to a magnetoelectric current in the presence of a static magnetic field, which becomes a harmonic supercurrent when the DAI is sandwiched between two superconductors and simultaneously, populates a coherent phase difference possessing the same doubled frequency ω1 = 2ω. On the other hand, a DC bias V0 can also induce a harmonic phase difference across the Josephson junction, whose frequency is proportional to the applied bias voltage is ω2 = 2eV0. Consequently, when the two frequencies ω1 and ω2 are commensurate with each other, such a superconductor–DAI–superconductor Josephson junction exhibits remarkable doubled Shapiro steps where all odd steps are completely suppressed. Because the magnetoelectric current originates from the axion electrodynamics driven by the AFMR, those Shapiro steps thus not only furnish a fingerprint of DAIs but also provide a method to evaluate the axion mass.

Results

Effective model of the DAI

The generic low-energy effective Hamiltonian for a DAI defined on the basis of \({\psi }_{{\bf{k}}}={(\left\vert {p}_{z}^{+},\uparrow \right\rangle ,\left\vert {p}_{z}^{+},\downarrow \right\rangle ,\left\vert {p}_{z}^{-},\uparrow \right\rangle ,\left\vert {p}_{z}^{-},\downarrow \right\rangle )}^{{\rm{T}}}\) has the form5,29

$${\mathcal{H}}(t)=\mathop{\sum }\limits_{\alpha =1}^{5}{d}_{\alpha }({\bf{k}}){\Gamma }^{\alpha },$$
(1)

where d1,2,3,4,5(k) = [Akx + m5nx(t), Aky, Akz, m0 + Bk2, m5nz(t)] and Γ1,2,3,4,5 = [σx ⊗ sx, σx ⊗ sy, σy, σz, σx ⊗ sz]. Here, A, B, m0 and m5 are system parameters. nx(z)(t) denotes the x-component (z-component) of the Néel vector, which can be time-dependent in the presence of magnetic fluctuations. σx,y,z and sx,y,z are Pauli matrices acting on the orbital and spin spaces, respectively. The lattice wave vectors kx,y,z are defined in the first Brillouin zone with \(k=\sqrt{{k}_{x}^{2}+{k}_{y}^{2}+{k}_{z}^{2}}\). The first four terms in Eq. (1) describe a topological insulator preserving both parity \({\mathcal{P}}={\sigma }_{z}\) and time reversal \({\mathcal{T}}=i{\tau }_{y}{\mathcal{K}}\) (\({\mathcal{K}}\) is the complex conjugation operator) symmetry, whose axion field is quantized to θ = 0 when B = 0 while θ = π otherwise4. The fifth term describes the exchange interaction between the topological electron and the dynamic, antiparallel magnetic moments, which breaks these two symmetries explicitly and thus introduces an additional dynamic part to the static axion field, giving rise to the DAI. The system parameters are generally taken as A = 1, B = −0.5, m0 = 0.1, m5 = 0.1. However, our theory is universal and not limited to these parameters.

Antiferromagnetic resonance in DAIs

Under a linearly polarized microwave, an antiferromagnetic insulator resonates coherently if the frequency of the microwave is identical to its intrinsic AFMR frequency \(\omega =2\pi \gamma \sqrt{{H}_{{\rm {A}}}(2{H}_{{\rm {E}}}+{H}_{{\rm {A}}})}\) given in Supplementary Note 135, where γ = 28 GHz T−1 is the gyromagnetic ratio, HA and HE are the uniaxial anisotropy and the exchange field between two antiparallel magnetic moments m1 and m2. Although the magnetic moment, in this case, individually rotates clockwise (or counterclockwise) on an elliptical orbit, the Néel vector n = m1−m2 oscillates in a purely linear mode as schematically illustrated in Fig. 1a and c, traveling back and forth inside a perpendicular plane normal to \(\hat{y}\)-axis. Consequently, x component of the Néel vector has an explicit form \({n}_{x}(t)={A}_{x}\sin \omega t\) with Ax the AFMR amplitude while z component can, therefore, be analytically expressed as \({n}_{z}(t)=\sqrt{1-{A}_{x}^{2}{\sin }^{2}\omega t}\) since the magnitude of the Néel vector is normalized. Figure 1b and c show the temporal evolutions of nx (b) and nz (c), respectively, at Ax = 0.02, 0.05, 0.1. We see that the periodicity of nz is halved compared to that of nx, indicating that the frequency of nz is multiplied. Since AFMR has been experimentally confirmed in Cr2O336 and MnF237, the same strategy can be conducted to trigger AFMR in antiferromagnetic Mn2Bi2Te5.

Fig. 1: Dynamic axion field induced by antiferromagnetic resonance in DAI.
Fig. 1: Dynamic axion field induced by antiferromagnetic resonance in DAI.
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a Schematic for a dynamic axion insulator driven by a linearly polarized coherent light. The arrows labeled B and E represent the magnetic and electric components of the applied light. The red and blue arrows labeled m1 and m2 represent the antiparallel magnetic moments forming a Néel vector, which oscillates linearly under the driving of the B component of the light. b and c are temporal evolutions of nx and nz under linearly polarized AFMR for three different amplitudes Ax = 0.1, 0.05, 0.02. The black stars mark five representative times ranging from t1 to t5. d Schematics for five representative magnetic configurations during a full circle one by one corresponding to the five times marked by black stars in b and c. The colorful arrows represent the magnetic moments m1 (m2) and θ1 (θ2) denotes their polar angles. The black arrow denotes the direction of motion of the magnetic moment. The tilting angles of the Néel vector are exaggerated for clarity. e Temporal dependence of the dynamic axion field δθ(t) obtained from Eq. (2) in the adiabatic limit at corresponding AFMR amplitudes.

Quantitative expression for the dynamic axion field in the DAI

To establish a quantitative expression for this dynamic axion field driven by the linearly polarized AFMR, we employ the gauge-invariant expression for the axion field in terms of the Chern–Simons 3-form4. While the timescale of the AFMR, hence θ(t), is about seven orders of magnitude larger than the typical electron response time38, the electrons inside the DAI can adjust adiabatically to the instantaneous configuration of the Néel vector, viewing the AFMR as a static magnetic vector almost frozen in time. As a result, we can treat the DAI adiabatically39. As detailed in Supplementary Note 2, this axion field in the adiabatic limit can be calculated by using5

$$\theta =-\frac{1}{4\pi }\int\,d{{\bf{k}}}^{3}\frac{2d+{d}_{4}}{{(d+{d}_{4})}^{2}{d}^{3}}{d}_{5}{\partial }_{{k}_{x}}{d}_{1}{\partial }_{{k}_{y}}{d}_{2}{\partial }_{{k}_{z}}{d}_{3},$$
(2)

where \(d=\sqrt{{\sum }_{i}{d}_{i}^{2}}\) with di presented in Eq. (1). Repeating this calculation at different instants of time gives the time-dependent axion field including a static part and a dynamic one, or θ(t) = θ0 + δθ(t). Figure 1f plots δθ(t) as a function of time at different AFMR amplitudes corresponding to Fig. 1b and c. It turns out that the temporal evolution of δθ(t) is a harmonic function with a doubled frequency compared to the AFMR, that is \(\delta \theta (t)={A}_{\theta }\cos 2\omega t\) where Aθ is the amplitude proportional to \({A}_{x}^{2}\). These mathematical relations, especially the frequency doubling of the dynamic axion field δθ(t), are further confirmed analytically in Supplementary Note 3.

Dynamic magnetoelectric current

Owing to the magnetoelectric term in the Lagrangian \({{\mathcal{L}}}_{\theta }=\theta (t)\alpha {\bf{E}}\cdot {\bf{B}}/\pi\), the Maxwell equation acquires an additional term and thus becomes ∇ ⋅ E = ρ/ϵ−α ∇θ ⋅ B/π, where ϵ is the permittivity and ρ is the charge density20. So a static magnetic field B0 applied to the DAI slab can induce a charge polarization given by Pθ(t) = ϕθ(t)e2/h40, where e is the electron charge, h is the Planck constant, and ϕ = B0S (S the area of the DAI normal to B0) is the magnetic flux penetrating the DAI. This charge polarization should also evolve adiabatically as the time-dependent axion field θ(t), leading to an equivalent magnetoelectric current. At a particular instant of time, the unbalanced charge distribution illustrated in Fig. 2a can be calculated by using the lattice Green’s functions provided in the “Methods” section. Figure 2b displays the unbalanced charge distribution induced by a static magnetic field along \(\hat{z}\)-direction at t1 = 0 for a system with size Ly = 16, Lz = 10 and Ax = 0.1, which corresponds to the charge polarization originating from the static axion term θ0. The charge distributions corresponding to δθ(t) are further plotted in Fig. 2c, ranging from t = 0 to t = π/2ω with an interval of π/10ω after subtracting the background charge distribution displayed in Fig. 2b.

Fig. 2: Doubled magnetoelectric current in DAI.
Fig. 2: Doubled magnetoelectric current in DAI.
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a Schematic for the instant charge polarization induced by a static magnetic field B along \(\hat{z}\)-direction at t1 = 0 and t2 = 2π/5ω, respectively. The charge polarization driven by AFMR oscillates between these two values in a doubled harmonic mode. The yellow and lime spheres represent positive and negative charges, respectively. b Unbalanced charge distribution Q(z) at t1 = 0 for a system with size Lz = 10 and Ly = 16. We use periodic boundary conditions along \(\hat{x}\)-direction. Here, Ax = 0.1 and the magnetic flux inside one unit cell is \({\Phi }_{0}=B{a}_{0}^{2}=0.005{\phi }_{0}\) with ϕ0 = h/e the magnetic flux quanta. c Increment of the charge distribution δQ(z) = Q(z, t)−Q(z, t = 0) at different times ranging from t = 0 to t = 0.5π. ω with an interval of 0.1π/ω at the same parameters as the system in (b). d Temporal evolutions of the increment of the charge polarization δP = [Q(1/2)−Q(−1/2)]/2 obtained in the adiabatic limit from the equilibrium Green’s functions provided in the “Methods” section (red inverted triangle) and the ensuing dynamic magnetoelectric current calculated from the time derivative of δP (blue triangle). The black dashed line shows the anticipated magnetoelectric current obtained from the time derivative of instant charge polarization given by Eq. (2). e Stem plot of the Fourier coefficients In.

Moreover, the red inverted triangle in Fig. 2d shows the instant charge polarization given by δP(t) = [δQ(z = −1/2)−δQ(z = 1/2)]/2, whose temporal evolution is also a harmonic function with a doubled frequency, in agreement with the dynamic axion field shown in Fig. 1e. The time derivative of this instant charge polarization gives the dynamic magnetoelectric current, which is represented by the blue triangle shown in the same figure. The black dashed line is the anticipated magnetoelectric current obtained from the time derivative of the dynamic axion field at Ax = 1 shown in Fig. 1e and is plotted in the same figure as a benchmark. Despite the slight deviation between the two results, which can be ascribed to the finite size effect, both data can be well-fitted by using a double-frequency harmonic function. This quantitative expression of the magnetoelectric current is further verified by the fast-Fourier transform shown in Fig. 2e. Since this harmonic current originates directly from the axion dynamics in DAI driven by the AFMR, it can thus stand as evidence for identifying DAIs. Nevertheless, in view of the high frequency of AFMR, which can even reach terahertz, quantitatively detecting a temporal current with such a high frequency remains a challenge.

Doubled Shapiro steps into the DAI Josephson junction

To detect this fast dynamic magnetoelectric current unique to the axion dynamics in DAI, we consider a Josephson junction consisting of a DAI sandwiched between two identical superconductors, as schematically shown in Fig. 3a. Through Andreev reflection, the dynamic magnetoelectric current could stimulate a coherent phase difference across the junction, potentially synchronizing with the AC Josephson current when subjected to additional DC bias, hence becoming static transport signals termed Shapiro steps. To explore the transport properties of such a DAI Josephson junction, we employ the resistively shunted junction model illustrated in Fig. 3b, where the system is simulated as an effective circuit featuring a resistance R in parallel to the Josephson junction41. This model captures not only the tunneling electron pairs but also the leakage currents through the insulating barrier at the interfaces. The total current then comprises two components: one is the normal supercurrent I0 adjustable by applied DC bias V0 while the other is the time-dependent axion current Iθ(t) induced by a static magnetic field B0. Moreover, this current can be further expressed as the sum of the parallel currents in Kirchhoff’s circuit, whose dynamics can be quantitatively described by the equation of motion42,43

$${I}_{0}+{I}_{\theta }(t)=\frac{V}{R}+{I}_{{\rm {c}}}\sin \psi ,$$
(3)

where I0 + Iθ(t) is the applied current, \(V=\hslash \dot{\psi }/2e\) is the effective voltage with ψ the phase difference across the Josephson junction and Ic is the critical supercurrent. This equation of motion can be solved numerically by using the Runge–Kutta method after substituting the dynamic current \({I}_{\theta }(t)=-{I}_{{\rm {ac}}}\sin 2\omega t\), which is provided in the “Methods” section.

Fig. 3: Transport properties of a DAI Josephson junction.
Fig. 3: Transport properties of a DAI Josephson junction.
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a Schematic for a Josephson junction with a DAI sandwiched between two identical superconductors. A DC bias V0 (with the right superconductor grounded) along with a static magnetic field B0 are applied across the Josephson junction. b The resistively shunted junction model with an electrical resistance R shunted with the Josephson junction I(ψ). I(t) is the total applied current. c and d are static and dynamic washboard potential versus the phase difference ψ. e Time-averaging voltage output as a function of the electron current input (proportional to the DC bias V0) under different Iac (in the unit of 2ℏω/Re). Here, the data are vertically shifted one by one for clarity, and the black dashed lines that mark the boundaries of each Shapiro steps are the guide to the eye. f Shapiro step widths In as a function of Iac. The markers here are obtained from the RSJ model, while the dashed lines of the same color are fitted by using the first kind of Bessel function in Eq. (4).

We first rewrite Eq. (3) as \(\hslash \dot{\psi }/2eR=-\partial U(\psi ,t)/\partial \psi\) where \(U(\psi ,t)=-[{I}_{0}+{I}_{\theta }(t)]\psi -{I}_{{\rm {c}}}\cos \psi\) is the washboard potential. In the absence of dynamic axion current Iθ(t), the static washboard potential for different I0 displayed in Fig. 3c is generally consistent with that in conventional Josephson junctions. When I0 < Ic, the system is trapped in a local minimum, maintaining a constant phase. Increasing the current to I0 = Ic leads to an instability responsible for the emergence of phase oscillation. The static washboard potential apparently obeys a linear Ohmic relation I0 = V/R when I0 ≫ Ic. The dynamic washboard potential for non-vanishing Iθ(t) is presented in Fig. 3d, which oscillates between two critical boundaries represented by the green and magenta lines. This washboard potential exhibits a halved periodicity with T = π/ω, in sharp contrast to conventional T = 2π/ω. During one period, the system remains stable as long as the phase changes 2nπ with n an integer, resulting in a velocity \(\dot{\psi }=2n\pi /T=2n\omega\) in the DAI Josephson junction. As a result, the Shapiro steps emerge when the motion of the system coincides with the dynamics of the washboard potential, manifested as ω0(=2eV0) = 2nω. Indeed, this washboard potential is instructive to understand the behavior of the I-〈V〉 curve.

Figure 3e shows the I−〈V〉 curve with different Iac that is proportional to the magnetic flux ϕ and thus B0. When the magnetic field is absent, Iac = 0 and a typical I–〈V〉 curve for conventional Josephson junctions without any Shapiro steps is restored as the dynamic axion current is vanishing [Iθ(t) = 0]. In the presence of a magnetic field (Iac ≠ 0), we see that the even Shapiro steps emerge at 〈V〉 = ℏω/e while the odd steps disappear completely, in agreement with the halved periodicity in the dynamic washboard potential above. Because the voltage drop inside one Shapiro step is zero, the magnitude, also referred to as the Shapiro spike, can be obtained through the identification of the width of the plateaus at 〈V〉 = ℏω/2e, which is depicted by colorful markers in Fig. 3f as a function of Iac. The result reaffirms that only doubled Shapiro steps, which evolve in perfect accordance with the Bessel functions, are sustained in the DAI Josephson junction.

Phenomenological picture

To understand these doubled Shapiro steps heuristically, we provide a phenomenological analysis following Barone and Paternò’s argument44. When the DAI is intimately connected to two identical superconductors, hence forming a DAI Josephson junction, the magnetoelectric current induced by applied magnetic field B0 leads to an effective bias voltage VDAI(t) = Iθ(t)R across the Josephson junction, where R is the effective resistance originating from the leakage current at the interfaces between the DAI and the superconductors. This voltage, in turn, populates a time-dependent phase difference given by Josephson’s second equation Φθ = ∫ dtVDAI(t)2e/ℏ, which is \({\Phi }_{\theta }=e{V}_{1}\cos 2\omega t/\hslash \omega\) with the effective voltage V1 = IacR. The total phase difference across the junction, combined with an additional DC bias V0, can thus be expressed as \(\Phi =2e{V}_{0}t/\hslash +e{V}_{1}\cos 2\omega t/\hslash \omega\). The Josephson current under such a phase difference thus reads \({I}_{{\rm {s}}}(t)={I}_{{\rm {c}}}{\rm{Im}}\{\exp [i(2e{V}_{0}t/\hslash +e{V}_{1}\cos 2\omega t/\hslash \omega )]\}\) where Ic is the critical supercurrent. Applying the Jacobi–Anger expansion recasts the Josephson current as45

$${I}_{{\rm {s}}}(t)={I}_{{\rm {c}}}\sum _{n}{{\mathcal{J}}}_{n}(e{V}_{1}/\hslash \omega )\sin [(2e{V}_{0}/\hslash +2n\omega )t],$$
(4)

where \({{\mathcal{J}}}_{n}(x)\) is the first kind of Bessel function. Equation (4) manifestly shows that the supercurrent oscillates as a function of time unless at eV0 = −nℏω, where doubled Shapiro steps appear with a magnitude \({I}_{{\rm {s}}}^{n}={I}_{{\rm {c}}}{{\mathcal{J}}}_{n}(e{V}_{1}/\hslash \omega )\). To check the consistency independently, we calculate the Shapiro steps as a function of V1 by using Eq. (4) and superimpose the result as a dotted line in the same figure. We see that the two results agree with each other remarkably well, which demonstrates the consistency and reliability of the obtained results.

Finally, we determine the differential conductance d〈V〉/dI on the I0−Iac plane by using the resistively shunted junction simulation and show the result in Fig. 4. We observe a clear and sharp double Shapiro step pattern, where the differential conductance plateaus appear only near the location I0 = 2nIc. The boundaries of the plateaus marking the width of the doubled Shapiro steps coincide quantitatively with the Bessel function given in Eq. (4). This unique differential conductance pattern provides a fingerprint of the DAIs and, meanwhile, offers a platform to simulate the axion electrodynamics in condensed matter systems.

Fig. 4: Differential conductance of the DAI Josephson junction.
Fig. 4: Differential conductance of the DAI Josephson junction.
Full size image

Differential conductance dV/dI on the I0−Iac plane. I0 is proportional to applied DC voltage V0 while Iac is proportional to applied static magnetic field B0. \({{\mathcal{J}}}_{\alpha }\) represents the first kind of Bessel function of order α.

Discussion

Despite the fact that a microwave is indispensable in both the conventional Josephson junction and the DAI Josephson junction discussed in this work, we stress that the underlying mechanisms of the doubled Shapiro steps are completely different. In conventional Josephson junctions, the microwave plays the role of a driven electric field used to induce a harmonic phase difference resonating with an additional DC bias coherently, resulting in typical Shapiro steps appearing consecutively at 〈V〉 = ℏω/2e. On the contrary, the microwave here is solely used to populate AFMR. The doubled Shapiro steps appear at 〈V〉 = ℏω/e originate from the coherence between the phase difference induced simultaneously from applied DC bias and the dynamic magnetoelectric current, which can thus be further ascribed to the axion dynamics inside DAIs. Note that our proposal is also different from the Witten effect-induced Shapiro steps predicted in ref. 46. In that work, the Josephson current originates from the charge triggered by the magnetic flux penetrating the type-II superconducting lead, which is absent in our work because the magnetic flux is fully expelled from the superconducting lead. In addition, these doubled Shapiro steps can not be ascribed to other existing mechanisms, such as the appearance of Majorana fermions47,48.

On the other hand, performing the Euler–Lagrangian equation to \({{\mathcal{L}}}_{\theta }\) yields the following classical equation of motion for the dynamic axion field16,20

$$\ddot{\delta \theta }-{\nabla }^{2}\delta \theta +{m}_{\theta }^{2}\delta \theta =\alpha {\bf{E}}\cdot {\bf{B}},$$
(5)

where mθ is the axion mass. Although there are ongoing debates regarding the magnitude of axion mass in condensed matter systems31,32,33,34, it is straightforward to see that the dynamic axion field induced by AFMR, \(\delta \theta (t)={A}_{\theta }\cos 2\omega t\), constitutes an analytical solution to Eq. (5) in a uniform system without any electromagnetic field. From this viewpoint, we obtain the axion mass mθ = 2ω. Consequently, the doubled Shapiro steps proposed here also provide a method to detect the axion mass.

In Mn2Bi2Te5, as the anisotropy and the exchange interaction are HA = 0.8 meV and HE = 0.1 meV49, respectively, the intrinsic AFMR frequency is f = ω/2π ≈ 143 GHz. So the AFMR in Mn2Bi2Te5 can be accomplished through a linearly polarized sub-terahertz radiation of frequency 143 GHz, analogous to that in Cr2O336 and MnF237. Moreover, the typical tilting angle of the Néel vector under AFMR is about Ax ~ 1%, in which case the amplitude of the dynamic axion field is Aθ ~ 1.0 × 10−5. For a DAI Josephson junction with size S = 10−6 m2 and typical contact resistance R ~ 10Ω50, the magnitude of the required static magnetic field to observe a sizable doubled Shapiro steps is about B0 ~ 0.02 T. These can be readily achieved within current experimental technology. We thus deem that our proposal can be tested in a Josephson junction with a Mn2Bi2Te5 sandwiched between two conventional superconducting leads.

In summary, we have demonstrated unique doubled Shapiro steps in a superconductor–DAI–superconductor Josephson junction driven by a DC bias and a static magnetic field instead of a microwave. We ascertain that this distinctive phenomenon arises from the frequency doubling of the axion electrodynamics in DAI under the driving of a linearly polarized microwave, or alternatively from the coherent resonance between the axion mass and applied DC bias, which thus provides a fingerprint of the DAIs and can also be utilized to evaluate the mass term of axion field in antiferromagnetic topological insulators.

Methods

Lattice Hamiltonian of the DAI

The low-energy effective Hamiltonian presented in Eq. (1) can be discretized by using the k ⋅ p theory. Performing the substitutions kα=x,y,z = −i∂α → −i(ψi+α−ψi)/(2a0) and \({k}_{\alpha }^{2}=-{\partial }_{\alpha }^{2}\to -({\psi }_{{\bf{i}}+\alpha }+{\psi }_{{\bf{i}}-\alpha }-2{\psi }_{{\bf{i}}})/{a}_{0}^{2}\) with a0 the lattice constant, we can write the effective Hamiltonian as

$${\mathcal{H}}(t)=\sum _{{\bf{i}}}{\psi }_{{\bf{i}}}^{\dagger }{T}_{{\bf{i}}}(t){\psi }_{{\bf{i}}}+\left({\psi }_{{\bf{i}}}^{\dagger }{T}_{x}{\psi }_{{\bf{i}}+x}+{\psi }_{{\bf{i}}}^{\dagger }{T}_{y}{\psi }_{{\bf{i}}+y}+{\psi }_{{\bf{i}}}^{\dagger }{T}_{z}{\psi }_{{\bf{i}}+z}+{\rm{H.c.}}\right),$$
(6)

where H.c. is the shorthand for the Hermitian conjugate, and the hoping matrices take the following form

$$\begin{array}{lll}{T}_{{\bf{i}}}\,\,=\,\left({m}_{0}+6B/{a}_{0}^{2}\right){\Gamma }_{4}+{m}_{5}{n}_{z}(t){\Gamma }_{5}+{m}_{5}{n}_{x}(t){\Gamma }_{1}\\ {T}_{x}\,=\,-iA/(2{a}_{0}){\Gamma }_{1}-B/{a}_{0}^{2}{\Gamma }_{4}\\ {T}_{y}\,=\,-iA/(2{a}_{0}){\Gamma }_{2}-B/{a}_{0}^{2}{\Gamma }_{4}\\ {T}_{z}\,=\,-iA/(2{a}_{0}){\Gamma }_{3}-B/{a}_{0}^{2}{\Gamma }_{4},\end{array}$$
(7)

with \({n}_{x}(t)={A}_{x}\sin \omega t\) and \({n}_{z}(t)=\sqrt{1-{A}_{x}^{2}{\sin }^{2}\omega t}\). Note that in Eq. (6), the translation symmetry along all three spatial directions in the Cartesian coordinate is well preserved.

In the presence of a static magnetic field B = (0, 0, B) along \(\hat{z}\)-direction, the Hamiltonian in Eq. (6) acquires a Peierls phase \({\phi }_{{\bf{ij}}}=2\pi \mathop{\int}\nolimits_{{\bf{i}}}^{{\bf{j}}}{\bf{A}}\cdot {\rm {d}}{\bf{r}}/{\phi }_{0}\) inside one unit cell from site i to j, where A is the vector potential and ϕ0 = h/e is the magnetic flux quanta. In particular, we use the Landau gauge, therefore A = (−yB, 0, 0). The Hamiltonian thus becomes

$${\mathcal{H}}(t)=\sum _{{\bf{i}}}{\psi }_{{\bf{i}}}^{\dagger }{T}_{{\bf{i}}}(t){\psi }_{{\bf{i}}}+\left[{\psi }_{{\bf{i}}}^{\dagger }{T}_{x}{f}_{x}(y){\psi }_{{\bf{i}}+x}+{\psi }_{{\bf{i}}}^{\dagger }{T}_{y}{\psi }_{{\bf{i}}+y}+{\psi }_{{\bf{i}}}^{\dagger }{T}_{z}{\psi }_{{\bf{i}}+z}+{\rm{H.c.}}\right],$$
(8)

where \({f}_{x}(y)={{\rm {e}}}^{-iyB{a}_{0}^{2}/{\phi }_{0}}\).

Green’s functions method for calculating the charge polarization and the magnetoelectric current

Since the AFMR frequency is orders of magnitude smaller compared to the electron response time, we can treat the time-dependent AFMR adiabatically. In this case, the magnetoelectric current induced by a static magnetic field B along \(\hat{z}\)-direction can be obtained from the time derivative of the instant charge polarization. Subsequently, this instant charge distribution parallel to the magnetic field can be expressed as40,51

$$q(z,t)=\frac{e}{2\pi }\sum _{y}\mathop{\int}\nolimits_{\infty }^{{\epsilon }_{f}}{\rm {d}}\epsilon \int\,{\rm {d}}{k}_{x}{\rm{Im}}\{{\rm{Tr}}[{G}^{r}(\epsilon ,t)]\},$$
(9)

where ϵf is the Fermi energy and the Green’s function \({G}^{r}(\epsilon ,t)={[\epsilon +i\eta -{\mathcal{H}}(t)]}^{-1}\) with η the imaginary line width function.

Moreover, Eq. (9) includes only the contribution from the negative electron charge. To obtain the unbalanced charge distribution as well as the instant charge polarization, it is necessary to include the positive background charge originating from ions that compensate for the electron charge, which has the form qb = ∑zq(z, t)/Lz owing to the charge conservation. As a result, the unbalanced charge distribution at time t can finally be expressed as Q(z, t) = q(z, t) + qb.

Fourth-order Runge–Kutta method for solving the equation of motion of the DAI Josephson junction

For convenience, we rewritten Eq. (3) in the form \(\dot{\psi }=f(t,\psi )\) with \(f(t,\psi )=2{I}_{{\rm {c}}}Re({I}_{0}/{I}_{{\rm {c}}}-{I}_{{\rm {ac}}}\sin 2\omega t/{I}_{{\rm {c}}}-\sin \psi )/\hslash\). This is a first-order differential equation that can be numerically solved by using the typical Runge–Kutta method. For a step size h > 0, the phase difference ψ can be obtained iteratively by using the following equations:

$$\begin{array}{lll}\psi ({t}_{n+1})\,=\,\psi ({t}_{n})+\frac{h}{6}({k}_{1}+2{k}_{2}+2{k}_{3}+{k}_{4}),\\\quad\;\, {t}_{n+1}\,=\,{t}_{n}+h,\end{array}$$
(10)

where

$$\begin{array}{lll}{k}_{1}\,=\,f({t}_{n},\psi ({t}_{n})),\\ {k}_{2}\,=\,f({t}_{n}+h/2,\psi ({t}_{n})+h{k}_{1}/2),\\ {k}_{3}\,=\,f({t}_{n}+h/2,\psi ({t}_{n})+h{k}_{2}/2),\\ {k}_{4}\,=\,f({t}_{n}+h,\psi ({t}_{n})+h{k}_{3}).\end{array}$$
(11)

The equation of motion can then be solved numerically by substituting the initial value ψ(t = 0) = 0 and \(\dot{\psi }(t=0)=2{I}_{0}\). The average voltage across the Josephson junction, as a function of the existing parameters I0 and Iac that are adjustable through the manipulation of V0 and B0, respectively, is then given by Josephson’s second equation \(V=\mathop{\int}\nolimits_{t}^{t+T}{{d}}t\hslash \dot{\psi }/2eT\) with the period T = π/ω.