Introduction

The Abelian or non-Abelian Chern-Simons (C-S) gauge field action usually serves as the low energy effective response theory of various topological nontrivial state of matters, such as the chiral spin liquid1, the topological insulators (TIs)2 and the Weyl semimetal (WSM) phase3,4,5,6. An interesting application of the C-S gauge field theory is its prediction of the anyonic excitations. As is known, the 1/m fractional quantum Hall state (FQH) possesses the anyons with the statistical angle θ = π/m. Since m is an integer, the value of θ is not universal but discrete. Therefore, it is interesting, both theoretically and experimentally, to answer the question: whether anyons with continuously tunable statistics can be realized. In order to realize the universal tunable anyons, one should search for states that enjoy the Abelian Chern-Simons term with a tunable coefficient7. Since the C-S gauge field theory associated with the WSM phase has a tunable coefficient that is proportional to the distance of the Weyl points5, WSM serves as a promising starting point. However, WSM is defined to be three dimensional material, where the anyons are forbidden8. These arguments suggest that a two-spatial-dimensional version of the WSM phase, whose electro-magnetic response is described by the U(1) Abelian C-S theory with a continuously tunable coefficient, constitutes the promising platform to realize the universal anyons.

In order to search for this special phase, we resort to the non-equilibrium states driven by external fields. Recently, the study on the band topology has been extended to the periodic driving case9, termed as the Floquet phases. It turns out that the Floquet phases may exhibit nontrivial band topology and the utilization of the external field also provides more tunable parameters to achieve novel topological phases10,11,12,13,14,15. For example, the Floquet fractional Chern insulator14, the Floquet topological insulator15, the Floquet Weyl semimetal16 and the Floquet Majorana fermions17,18 are proposed. Moreover, the Floquet-Bloch states of a topological insulator have been observed in experiment19, which shed light on more future applications. The special feature of the Floquet phases is that the periodically driven system can absorb and emit photons, forming Floquet bands denoted by different photon numbers n. This effectively enlarges the “spacetime” by introducing a fictitious dimension to the undriven phase and therefore provides a possibility for the two dimensional (2D) driven system to exhibit the chiral anomaly. This fact motivates us to simulate a 3D WSM by driving a certain 2D phase, where the universal anyons may emerge as the quasi-particle excitations.

In this work, we find that the universal anyons can be realized at the irradiated surface of TI. Our main result is that an amplitude-modulated circularly-polarized light have two effects on the TI surface: (a). Due to the coupling between different Floquet bands, it introduces a fictitious momentum q and enlarges the base manifold of the Hamiltonian, replacing the first Brillouin zone (FBZ) by the Floquet first Brillouin zone (FFBZ)20. (b). It generates an effective inhomogeneous Zeeman field, which is a function of q. These two effects lead to an undiscovered topological state of matter that exhibits the following non-trivial topological properties. First, it enjoys two gapless Weyl points3 in the FFBZ, which are topologically robust to any perturbations. So we term this state the pseudo Weyl semimetal (PWSM) phase. Second, the PWSM possesses the chiral anomaly, which leads to a U(1) Chern-Simons gauge field theory with a tunable coefficient C. This differs from the conventional Chern-Simons action of the integer and fractional quantum Hall state in the sense that it predicts two novel topological behaviors: the universal anyon excitations and the tunable quantum anomalous Hall (QAH) effect.

Results

Quasi-energy band structure

We motivate the description of the details of the model and its analysis by first showing the main results, which are obtained using a combined formalism of functional path integral and the Floquet-Bloch theory. We consider a TI surface driven by an off-resonant, circularly-polarized light with a slowly-modulated amplitude, as is shown in Fig. 1(a). The effect of the off-resonant light, with the period T1, on the TI surface has been studied by ref. 10, where a constant Zeeman gap is opened, leading to the QAH effect. Here we further require the amplitude to be periodic and slowly varying, with the period T2. The function is generic and its specific form does not affect the qualitative results below. For purpose of clarity, we assume  , with q = ω2t and ω2 = 2π/T2.

Figure 1
figure 1

(a) The topological insulator surface driven by an amplitude-modulated circularly-polarized light. (b) The quasi-energy dispersion (with ky = 0) of the driven TI surface. The amplitude-modulated circularly-polarized field drives a single Dirac node into two Weyl nodes that lie in the additional dimension q in the FFBZ, with the location ±q0.

Full size image

Using the two-time Floquet formalism (see below), we obtain the effective Floquet Hamiltonian describing the irradiated TI surface

Here the τ, σ represent the chirality and the band degrees of freedom respectively. In Eq. (1), we use k to denote the three dimensional “momentum”, , where q enters into the Hamiltonian as the introduced “momentum” due to the external periodic field20. The quasi-energy dispersion is shown in Fig. 1(b). As is shown, the amplitude-modulated circularly-polarized light drives the single Dirac cone in the TI surface into two Weyl nodes (with opposite chiralities) in the low-energy window in the FFBZ at ±q0, with and for . It is well known that the gapless nodes in the nondegenerate two band model is robust, since all the three Pauli matrices are used up so that no more mass terms can be added to open up the gap. This robustness is due to topology but not symmetry, leading to a peculiar topological matter of state, termed the Weyl semimetal3. However, different from the conventional 3D WSM, the Weyl nodes here lie in the time axis (q) and they are generated by driving the TI surface, therefore we term this peculiar state the PWSM phase. The normal 3D WSM possesses the semi-quantized anomalous Hall effect, where the conductance is proportional to the distance of the Weyl nodes. Here due to the Floquet theory, kz is replaced by q, therefore a similar anomalous Hall conductance proportional to q0 can be expected in the 2D TI surface. Besides, the replacement of kz by q is nontrivial in the sense that it allows the existence of anyonic excitations. This is because, in this case, no continuous contractible loop that has a vanishing local phase8 can be constructed any more due to the definition of the braiding of the Floquet states17. To validate these expectations, we perform the calculation of the Berry phase curvature and study the chiral anomaly of the PWSM phase.

Berry curvature of the PWSM phase

For any fixed ky, the Berry phase curvature is defined by the Berry phase gauge field ai(k),

with and i = x, z. kz is used to denote q for brevity and is the Bloch function of the α band. We calculated and plotted the density distribution of fxz(k) on the FFBZ, as is shown in Fig. 2. The first, second and third panel shows fxz(k) for , respectively. For , we have q0 = 0, therefore both the two Weyl nodes lie at (0, 0, 0). When is varied from −1 to 1, the Weyl nodes gradually get separated in the FFBZ and finally merge with each other at (0, 0, π). One can check that the q coordinates of the dividing lines between the dark and light areas in Fig. 2 coincide with the locations of the Weyl nodes. As such, the Berry phase curvature has nontrivial values inside and trivial values outside the two Weyl points. This is consistent with the well-known Chern number C of the conventional 3D WSM phase, where C = 1 and C = 0 for areas between and outside the two Weyl nodes respectively. The fact that the PWSM shares the same Chern number distribution with the conventional WSM phase is important, since it allows one to modulate the Berry phase accumulation by tuning the amplitude . Let us consider the braiding of two quasi-particles. In a complete period from q = −π to q = π, the non-trivial Berry phase accumulation only occurs in the topological non-trivial (light) region, while the contribution is zero for the trivial (dark) region. Therefore, the total Berry phase in a complete period is proportional to the width of the light area, that is proportional to the distance of the Weyl points, . In this sense, the amplitude brings about the tunable Berry phase via modulating the separation between Weyl points. This understanding shows that the the time-evolution of the Floquet states can result in the universal anyon statistics (see below) and the anyon here emerges due to completely different reasons from that of the FQH effect. In the following, we strictly prove the existence of the universal anyons by studying the underlying gauge field theory.

Figure 2
figure 2

The density distribution of the Berry phase curvature fxz(k) in the FFBZ (with ky fixed to 0.1).

The first, second, third panel depicts the case where respectively.

Full size image

Pseudo chiral anomaly

To extract the universal anyons, we study the electromagnetic response theory of the PWSM phase. The external electromagnetic field Aμ is minimally coupled to the pseudo Weyl fermions, i.e., Eq. (1). Utilizing the chiral symmetry, we can eliminate the distance of the Weyl fermions and arrive at a massless Dirac fermion5. However, when performing the chiral transformation, great attention must be paid to the Jacobian of the integral measure, which can lead to the chiral anomalies and some observable effects21.

The problem of the chiral anomaly of the conventional WSM is well studied5,21. In the proposed PWSM phase, since all the Dirac matrices are well-defined, a similar chiral anomaly should also exist. With a few modifications, we arrive at the action δ S describing the chiral anomaly (see Method),

where and e, ħ is set to 1. Eq. (3) is a U(1) C-S theory with a tunable coefficient C, with C = q0/π. To make the physical meaning more explicit, it is convenient to make a Hubbard-Stratonovich decomposition, which leads to

where m = 1/C and aμ is the introduced auxiliary field. The above equation exactly coincides with the effective Lagrangian of the v = 1/m FQH7 and it is interesting to observe that the distance of Weyl nodes is the analogy of the filling factor in the FQH state. The key difference here is that is not discrete but can be tuned continuously. A variation of the action δ S shows that this state enjoys a tunable QAH effect5, with the conductance . This is very similar to the semi-quantized Hall conductance in the WSM phase, but the Hall current here resides in the 2D TI surface and can be adjusted by . The Hall conductance is shown by the brown curve in Fig. 3, which varies between 0 and e2/h.

Figure 3
figure 3

The calculated statistical angle θ and the Hall conductance versus the amplitude .

Full size image

Eq. (3) indicates a field theory of anyons. This well-known conclusion is discussed in refs 22, 23, 24. Despite the detailed discussion, the following arguments can show the anyons clearly. From the equation of motion, , we know that flux are attached to point particles, forming quasi-particle excitations7. The statistical angle for unit aμ charge is extracted to be the Berry phase θ accumulated during the braiding of the quasi-particles8. In this way, we obtain that θ = Cπ = q0, in agreement with the conclusion in the last section. As is discussed, C can be continuously tuned by the amplitude , so we arrive at the universal anyonic excitations with tunable θ at the irradiated TI surface, where the anyons are bound states of charge and flux. The interesting universal tunable statistics is shown by the blue curve in Fig. 3. As one varies from −1 to 0 and then to 1, the quasi-particles can evolve from bosons to semions and then to fermions.

The model Hamiltonian

The Hamiltonian describing the driven TI surface reads

where vF is set to 1 for brevity and is the two-dimensional lattice momentum. Since the Dirac cone is a low-energy effective description, an energy cutoff Λ is implicit. As required, we have , . In the following, we focus on the case where ω1 = 2π/T1 lies in the high frequency regime with , (see ref. 10) and ω2 lies in the low frequency regime, with , as required by ref. 20. Due to the presence of the two time scale T1, T2 , it is appropriate to resort to the two-time formalism of the Floquet theory25, which leads to

where . indicates the fast time and t denotes the slow time. In the two-time formalism, can be viewed as a parameter17. Moreover, since ω2 lies in the low frequency regime, we can use the Floquet-Bloch theorem and then drop out the term “i∂t”20, so that for fixed , the Hamiltonian is defined in the FFBZ. The Floquet state and the quasi-energy are determined by . In the frequency ω1 space, the Floquet operator is mapped into, with

Since ω1 lies in the off-resonant high frequency regime, we cannot drop out the term “nω1δn,m”. Instead, a perturbation treatment can be performed10,13,14, which leads to, . Inserting the Hamiltonian of the TI, Eq. (5), we obtain the effective Floquet Hamiltonian describing the irradiated TI surface in the second quantized form,

The Hamiltonian shows that the amplitude-modulated circularly-polarized light has two effects. (a). It enlarges the base manifold of the Hamiltonian, replacing the FBZ by the FFBZ. (b) It generates an effective tunable Zeeman field, which opens and closes the gap of the TI surface. Further assuming a specific function , the Hamiltonian can be reduced to the pseudo Weyl fermions in the low energy window, i.e., Eq. (1).

Now we are going to discuss the experimental applicability of our results. First, and should be satisfied. The topological surface states of the realistic material such as Bi2Se3, Bi2Te3 and Sb2Te3 have been well investigated26. Take Sb2Te3 as example, the cutoff energy of the surface Dirac cone is estimated to be 26. So, the frequencies and should satisfy and . Second, we stress the requirement on the amplitude . Taking into account e, ħ and vF, the perturbation treatment in Eq. (8) is correct only when , where is the maximum value of . We introduce a dimensionless number , with a being the lattice constant of Sb2Te3. Then one can estimate that should satisfy (for v1 = 250THz). So, we do not have particular requirement on the amplitude strength as long as it is not too large. Last, we would like to remark that the splitting of the pseudo Weyl nodes q0 depend neither on the absolute value of the amplitude nor on the frequencies. It only depends on the ratio of and , which are parameters that can be tuned experimentally. From to , the splitting of Weyl nodes can be easily tuned from 0 to π. So, we conclude that the Weyl node separation can be achieved and modulated as long as the condition on the frequencies v1 and v2 are satisfied.

Discussion

Formally, the coupling constant of the topological action (such as C in Eq. (3)) can be identified as a topological invariant in terms of the Berry fiber bundle27. Now we consider the topological invariant and try to reveal the physical essence of the PWSM phase. In order to do so, we recall the effective gauge field theory describing the electromagnetic response of the Chern insulator2, which is very similar to Eq. (3). The only difference is that C is replaced by the first Chern number C1. Since C1 is calculated by the Berry phase defined in terms of the Bloch state2 and the Bloch state is the counterpart of the Bloch-Floquet state in the PWSM phase, we generalize the first Chern number to its Floquet version using the two-time Floquet theory17,25,

where R is a varying parameter, is the Bloch-Floquet state and is the generalized inner product9. If one views the PWSM phase at any fixed parameter t, the PWSM phase would be either a 2D normal insulator or a Chern insulator with the first Chern number C1 = 0 and C1 = 1 respectively, depending on whether t lies in the trivial or non-trivial regime (see Fig. 2). The evolution of the Floquet state in a complete periodic T2 undergoes two topological phase transitions with the energy gap closes and then reopens. Taking this into account, Eq. (9) can be calculated to be CF = q0/π, which exactly equals to the coefficient C in Eq. (3). The identification of C with the Floquet Chern number CF shows that the obtained action Eq. (3) is actually the effective topological field theory describing a Floquet version of the quantum Hall (QH) state.

Now we can conclude that the TI surface driven by an amplitude-modulated circularly-polarized light is a realization of the Floquet QH state, which shows a Weyl semimetal-like dispersion in the low energy window in the FFBZ. This PWSM phase is a close analogy of the normal 3D WSM, with the kz lattice momentum replaced by the time dimension q. A result of this substitution is that no continuous contractible loop that has a vanishing local phase can be constructed any more using the time dimension, therefore the anyons are in principle permitted. This is validated by the associated gauge field theory, since we find that the chiral anomaly of the PWSM phase brings about a 2 + 1D U(1) Chern-Simons theory with a tunable coefficient, which further generates the universal anyonic statistics and the tunable QAH effect. This finding, to our best knowledge, serves as the first theoretical proposal to support the universal anyons in realistic solid state materials.

Method

Combined formalism of functional integral and the Bloch-Floquet theory

In order to study the chiral anomaly of the PWSM phase, we develop a field theory of the periodically driven state. Since we are interested in the electromagnetic response of the PWSM phase, only the U(1) gauge field case needs to be considered. We start from the well-proved observation in ref. 20: A Hamiltonian defined in n − 1D first Brillouin zone (FBZ) is equivalent to a static one defined in nD FFBZ, if it is driven by a field in the low frequency regime (with the period T). The effective static Hamiltonian generally reads, . This equation shows that t can no longer be treated simply as time, it is a “momentum” that enlarges the FBZ to FFBZ. To obtain the electromagnetic response of HF, we take three following steps. (a). By straightforward generalizations of the conventional coherent-state path integral representation, we develop an effective action describing this non-equilibrium state, which reads

In deriving Eq. (10), a new time coordinate t′ is introduced due to the driven field. This treatment is essentially a field theory representation of the Floquet Green’s function method in refs 10,28 and it also agrees with the two-time formalism introduced in ref. 25. (b). We introduce a U(1) gauge field Aμ that couples to the Floquet phase, leading to the action

where we have made a wick rotation to the Euclidean spacetime and set the volume of “real” space to be 1. The driven field leads to a mixed space of lattice coordinate and photon number 20 by introducing an extra dimension (denoted by rn). , with An being an auxiliary component. Aμ is independent on rn since the perturbation field Aμ does not rely on the photon number index of the Floquet state. is the momentum operator of . (c). Eq. (11) can be simplified utilizing the symmetry of S, followed by integrating out the matter fields. Transformation of S may bring about nontrivial Jacobian associated to the integral measure of the partition function , denoted by δ S. Therefore, the one fermion loop effective action for the gauge field can be obtained

where denotes the transformed Hamiltonian. A perturbative treatment of Aμ can further expand Seff, making possible the calculation of the Feynman diagrams order by order. Formally, using both the functional path integral and the Bloch-Floquet theory, we arrive at Eq. (12), which is the effective gauge field theory describing the electromagnetic response of the Floquet phases under our consideration.

Derivation of the pseudo Weyl chiral anomaly

Using the formalism introduced in first section in the Method, the action of the PWSM phase can written as

where we have coupled a U(1) gauge field Aμ to the Floquet phase. Here, as discussed in the last section, A3 is an auxiliary component and Aμ is independent on r3. This action has the chiral symmetry, i.e., S remains unchanged under the transformation . One can show that the action can be simplified to

if θ satisfies . Therefore, the chiral transformation shows that two separate Weyl points can be equivalently shifted into one Dirac node. This observation is physically incorrect. In fact, Eq. (14) only contributes to the first term in Eq. (12) and we have missed the anomalous term coming from the integral measure δ S. The particular transport behavior of the pseudo Weyl semimetal state comes from δ S, therefore in the following we give a detailed derivation of δ S, using the Fujikawa’s method5,21.

By introducing the Dirac matrices with γ0 = τ1, γi = iτ2σi (i = 1, 2, 3) and γ−5 = −iγ0γ1γ2γ3, one can further write the action into,

To calculate the Jacobian of the chiral transformation associated with the functional integral measure. We introduce the infinitesimal chiral transformation , which can be denoted as the operator . From the action Eq. (15), we can introduce a Dirac kernel

where bμ = (0, 0, 0, q0). Then, under the infinitesimal transformation, the Jacobian of the functional integral measure can be calculated, which leads to

where ϕn is the eigenfunction of the Dirac kernel with the eigenvalue ϵn. Since γ5 satisfy , γ5ϕ is also a eigenfunction which corresponds to the eigenvalue −ϵn. Due to the orthogonal condition, we have except for ϵn = 0. This suggests one to pick out the zero energy eigenstates. This can be achieved by introducing a regularization factor21,

with the regularization factor M taking the limit M → ∞. To calculate the sum of n in Eq. (18), we make transformation to the momentum space (FFBZ). Then the total anomalous action can be calculated by further integrating s from s = 0 to s = 1, which leads to

Finally, after expanding the exponential, we arrive at

where . A Wick rotation has been performed to return back to the real time space. Since Aμ is independent on r3, we can integrate r3 in the action. Further integrating by parts and then taking into account , we obtain the action describing the chiral anomaly of the pseudo Weyl semimetal phase.

where μ, v, ρ = 0, 1, 2.

Additional Information

How to cite this article: Wang, R. et al. Universal anyons at the irradiated surface of topological insulator. Sci. Rep. 6, 20075; doi: 10.1038/srep20075 (2016).