Universal anyons at the irradiated surface of topological insulator

Anyons have recently received great attention due to their promising application in topological quantum computation. The best validated system that enjoys the anyonic excitations are the Laughlin states. The quasi-particles in Laughlin states are neither fermions nor bosons but possess the discrete statistical angle θ = π/m, with m being an integer. Here we report a possible realization of the universal Abelian anyons, whose statistical angle can be tuned continuously by external parameters and can take any arbitrary values interpolating θ = 0 and θ = π. The proposed setup is the surface state of a three dimensional topological insulator driven by an amplitude-modulated circularly-polarized light. It is found that the external field leads to a particular Floquet phase, which is a two-spatial-dimensional analogy of the Weyl semimetal phase in the Floquet first Brillouin zone. The chiral anomaly of this phase results in a U(1) Chern-Simons gauge theory with a tunable Floquet Chern number. Owing to this underlying gauge field theory, the irradiated surface of topological insulator constitutes a promising platform for the observation of the universal anyons.

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 points 3 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 T 1 , 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 ( ) t  to be periodic and slowly varying, with the period T 2 . The function ( ) t is generic, and its specific form does not affect the qualitative results below. For purpose of clarity, we assume ( ) = + ( ) q q cos 2 0 1    , with q = ω 2 t and ω 2 = 2π/T 2 . 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", = ( , , ) k k q k x y , where q enters into the Hamiltonian as the introduced "momentum" due to the external periodic field 20 . 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 ± q 0 , with = ( , , ) 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 semimetal 3 . 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, k z is replaced by q, therefore a similar anomalous Hall conductance proportional to q 0 can be expected in the 2D TI surface. Besides, the replacement of k z 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 phase 8 can be constructed any more due to the definition of the braiding of the Floquet states 17 . 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 k y , the Berry phase curvature is defined by the Berry phase gauge field a i (k), and i = x, z. k z is used to denote q for brevity and αk is the Bloch function of the α band. We calculated and plotted the density distribution of f xz (k) on the FFBZ, as is shown in Fig. 2. The first, second and third panel shows f xz (k) for / = − , , we have q 0 = 0, therefore both the two Weyl nodes lie at (0, 0, 0). When / 0 1   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   / 0 1 . 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, θ ∝ ∝ q q 2 0 0 . 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.
Pseudo chiral anomaly. To extract the universal anyons, we study the electromagnetic response theory of the PWSM phase. The external electromagnetic field A μ 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 fermion 5 . 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 effects 21 .
The problem of the chiral anomaly of the conventional WSM is well studied 5,21 . 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 µ ν ρ , , = , , 0 1 2 and e, ħ is set to 1. Eq. (3) is a U(1) C-S theory with a tunable coefficient C, with C = q 0 /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 a μ is the introduced auxiliary field. The above equation exactly coincides with the effective Lagrangian of the v = 1/m FQH 7 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   / 0 1 is not discrete but can be tuned continuously. A variation of the action δ S shows that this state enjoys a tunable QAH effect 5 , with the conductance σ = , we know that flux are attached to point particles, forming quasi-particle excitations 7 . The statistical angle for unit a μ charge is extracted to be the Berry phase θ accumulated during the braiding of the quasi-particles 8 . In this way, we obtain that θ = Cπ = q 0 , in agreement with the conclusion in the last section. As is discussed, C can be continuously tuned by the amplitude   / 0 1 , 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 / 0 1   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 v F is set to 1 for brevity and ( ) = , k k k x y 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 ω 1 = 2π/T 1 lies in the high frequency regime with ω Λ  , it is appropriate to resort to the two-time formalism of the Floquet theory 25 , which leads to . t indicates the fast time and t denotes the slow time. In the two-time formalism, t can be viewed as a parameter 17 . Moreover, since ω 2 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 t , the Hamiltonian is defined in the FFBZ. The Floquet state Since ω 1 lies in the off-resonant high frequency regime, we cannot drop out the term "nω 1 δ n,m ". Instead, a perturbation treatment can be performed 10,13,14 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 ( ) t  , 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, ω Λ  1 and ω Λ  2 should be satisfied. The topological surface states of the realistic material such as Bi 2 Se 3 , Bi 2 Te 3 and Sb 2 Te 3 have been well investigated 26 . Take Sb 2 Te 3 as example, the cutoff energy of the surface Dirac cone is estimated to be Λ ≈ . eV 0 1 26 . So, the frequencies ν ω π = /( ) 2 1 1 and ν ω π = /( ) 2 2 2 should satisfy ν  25THz 1 and ν  25THz 2 . Second, we stress the requirement on the amplitude ( ) q . Taking into account e, ħ and v F , the perturbation treatment in Eq. (8)  (for v 1 = 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 q 0 depend neither on the absolute value of the amplitude nor on the frequencies. It only depends on the ratio of 0  and  1 , which are parameters that can be tuned experimentally. From / = − 1   , 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 v 1 and v 2 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 bundle 27 . 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 insulator 2 , which is very similar to Eq. (3). The only difference is that C is replaced by the first Chern number C 1 . Since C 1 is calculated by the Berry phase defined in terms of the Bloch state 2 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 theory 17,25 , is the Bloch-Floquet state and is the generalized inner product 9 . 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 C 1 = 0 and C 1 = 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 T 2 undergoes two topological phase transitions with the energy gap closes and then reopens. Taking this into account, Eq. (9) can be calculated to be C F = q 0 /π, which exactly equals to the coefficient C in Eq. (3). The identification of C with the Floquet Chern number C F 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 k z 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 . 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 H F , 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 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