Floquet topological superconductors with many Majorana edge modes: topological invariants, entanglement spectrum and bulk-edge correspondence

One-dimensional Floquet topological superconductors possess two types of degenerate Majorana edge modes at zero and $\pi$ quasieneriges, leaving more room for the design of boundary time crystals and quantum computing schemes than their static counterparts. In this work, we discover Floquet superconducting phases with large topological invariants and arbitrarily many Majorana edge modes in periodically driven Kitaev chains. Topological winding numbers defined for the Floquet operator and Floquet entanglement Hamiltonian are found to generate consistent predictions about the phase diagram, bulk-edge correspondence and numbers of zero and $\pi$ Majorana edge modes of the system under different driving protocols. The bipartite entanglement entropy further show non-analytic behaviors around the topological transition point between different Floquet superconducting phases. These general features are demonstrated by investigating the Kitaev chain with periodically kicked pairing or hopping amplitudes. Our discovery reveals the rich topological phases and many Majorana edge modes that could be brought about by periodic driving fields in one-dimensional superconducting systems. It further introduces a unified description for a class of Floquet topological superconductors from their quasienergy bands and entanglement properties.


I. INTRODUCTION
Floquet topological phases of matter appear in systems with time-periodic modulations.
When applied to a pristine and noninteracting lattice model, periodic driving fields could mainly generate three effects. At high frequencies, the driving field could break the symmetry of the originally static system and induce gap closing/reopening transitions in its spectrum, especially around gapless points like Dirac cones [7]. This effect can usually be taken into account by implementing perturbative expansions over the driving frequency [9][10][11][12]. When the driving amplitude and frequency of the field are comparable with other energy scales of the system, more drastic changes could appear in its spectral and topological properties. First, the Floquet bands of both the bulk and edge states could develop crossings and windings around the whole quasienergy Brillouin zone, yielding anomalous Floquet topological metals [22,29,33] and insulators [18,21,24] without any static analogies. Second, spatially non-decaying, long-range couplings could be induced between different lattice sites, yielding Floquet systems with large topological invariants [17,25], many topological edge states [19,20] and multiple topological phase transitions [26][27][28]. By designing suitable driving protocols, the last effect has been utilized to obtain Floquet topological insulators with arbitrarily large winding numbers and arbitrarily many dispersionless/chiral edge and corner states [19, 25-28, 30, 31]. It has also been employed to realize Floquet Chern insulators with large Chern numbers in both theory [23] and experiments [44]. However, much less is known regarding this effect in Floquet superconducting systems [51][52][53][54][55]. Specially, could we also generate Floquet superconductors with arbitrarily large topological numbers and unboundedly many Majorana edge modes with the help of driving-induced long-range couplings? A systematic exploration of this problem could not only expand the usage of Floquet engineering in superconducting setups, but also provide a rich source of Majorana edge modes that might be used to realize more complicated operations in boundary time crystals and Floquet quantum computing [48][49][50].
In this work, we uncover the richness of phases and transitions in one-dimensional (1D) Floquet topological superconductors. We begin with an overview of the Ising-Kitaev chain [56] regarding its symmetries, topological properties, Majorana edge modes, entanglement spectrum (ES) and entanglement entropy (EE) in Sec. II. Next, we introduce two representative models of Floquet Kitaev chains by adding time-periodic kicks to the hopping or pairing amplitudes of the system in Sec. III. Methods of characterizing the topological and entanglement features of these Floquet superconducting models in terms of their winding numbers, edge states, ES and EE will also be discussed. In Sec. IV, we systematically explore the phases and transitions in our kicked Kitaev chains by investigating the topological properties of their Floquet bands and Floquet entanglement Hamiltonians. The emergence of Floquet superconducting phases with large topological invariants and many Majorana zero/π edge modes will be revealed from the topological phase diagrams, quasienergy spectrum and ES for each model. In Sec. V, we summarize our results and discuss potential future directions.

II. KITAEV CHAIN
In this section, we recap the Kitaev chain (KC) model, which describes a 1D superconducting wire with p-wave pairings [56]. We first introduce the model and obtain its bulk spectrum under the periodic boundary condition (PBC). We next introduce a winding number to characterize its topological phases and establish the topological phase diagram. This is followed by the presentation of the spectrum of KC under the open boundary condition (OBC) and the discussion of the correspondence between its bulk winding number and Majorana edge modes. Finally, in terms of the ES, EE and entanglement winding number, we characterize the topological properties of KC from the entanglement perspective. A collection of the quantities introduced in this section is summarized in Table I. The second quantized Hamiltonian of KC takes the form of Here c † n (c n ) creates (annihilates) a spinless fermion on the lattice site n. µ is the chemical potential. J is the nearest-neighbor hopping amplitude. ∆ is the superconducting pairing amplitude. For a chain of length L and under the PBC, we can perform the Fourier transformationĉ n = 1 √ L k e iknĉ k to find the expression ofĤ in momentum space aŝ is the Nambu spinor operator and the quasimomentum k is defined in the first Brillouin zone (BZ) (−π, π]. The Bloch Hamiltonian where σ y and σ z are Pauli matrices. The energy dispersion of H(k) takes the form which could become gapless at k = 0 (BZ center) and k = π (BZ edge) if µ = −J and µ = J, respectively. H(k) possesses the time-reversal symmetry T = K with T H(k)T −1 = H(−k), the particle-hole symmetry C = σ x K with CH(k)C = −H(−k) and the chiral symmetry It thus belongs to the symmetry class BDI [57,58]. Each of its topological phases can be characterized by an integer-quantized winding number, which is defined as [59] w =ˆπ Here S is the chiral symmetry operator and the projector |ε + (k)⟩ and |ε − (k)⟩ are the eigenstates of H(k) with energies ε(k) and −ε(k), respectively.
In Fig. 1(a), we show the winding number w versus J and µ at ∆ = 1, which defines the topological phase diagram of the Kitaev chain. We find |w| = 1 for |J| > |µ| and w = 0 for |J| < |µ|. In the former case, the coupling between Majorana modes in adjacent unit cells is stronger then their coupling within each unit cell. The Kitaev chain then belongs to the topologically nontrivial phase, supporting two Majorana zero modes at its two ends under the OBC. In the latter case, the intracell coupling between adjacent Majorana modes is stronger, and the Kitaev chain belongs to the trivial phase without Majorana edge states.
To see the bulk-edge correspondence of the KC more clearly, we consider its spectrum under the OBC. Applying the transformation between two fermionic bases {ĉ n ,ĉ † n } and we arrive at the following BdG self-consistent equations The complex coefficients u jn and v jn satisfy the normalization condition L j=1 (|u jn | 2 + |v jn | 2 ) = 1. The system Hamiltonian is assumed to be diagonal in the basis {f j ,f † j }. Eqs. (8) and (9) then follow by evaluating the commutator [Ĥ,ĉ n ] forĤ in its diagonal formĤ = L j=1 E jf † jf j and in the form of Eq. (1) separately, and then letting them to be equal [61]. Note in passing that the redundant index j has been dropped in the final equations. The spectrum obtained by solving the BdG equations (8) and (9) under the OBC is shown in Fig. 1(b). We observe a pair of degenerate edge modes at E = 0 (n 0 = 2) in the topological nontrivial regime (|J| > |µ|) of the bulk with |w| = 1 [see also Fig. 1(a)], which verifies the bulk-edge correspondence n 0 = 2|w| of the KC.
The topological nature of KC can also be revealed by its bipartite ES and EE. The Eqs. (8) and (9) can be equivalently generated by a synthetic Hamiltonian of spin-1/2 fermionŝ Hereĉ † n ≡ (ĉ † n↑ ,ĉ † n↓ ) andĉ † nσ creates a fermion with spin σ (=↑, ↓) on the lattice site n. Expressing a general state as |ψ⟩ = n (u nĉ † n↑ + v nĉ † n↓ )|∅⟩ and inserting it into the eigenvalue equationĤ|ψ⟩ = E|ψ⟩, we arrive at the Eqs. (8) and (9), where |∅⟩ represents the vacuum state with no particles. At half-filling, decomposing the system under the PBC (with a ring geometry) into two parts A and B of equal lengths L/2 and tracing out all the degrees of freedom belonging to the subsystem B, we obtain the reduced density matrix of the subsystem A asρ A = 1 Z e −Ĥ A . The eigenspectrum {ξ j |j = 1, ..., L} of entanglement Hamiltonian H A defines the ES, and the von Neumann EE is given by S = −Tr(ρ A lnρ A ). According to Ref. [62], both the ES and EE can be expressed in terms of the eigenvalues {ζ j |j = 1, ..., L} of the system's single particle correlation matrix C mn = ⟨ĉ † mĉ n ⟩ (m, n ∈ A) as Due to the one-to-one correspondence between ξ j and ζ j , we will also refer to the set {ζ j |j = 1, ..., L} as the ES. In Fig. 1(c), we show the ES of the Kitaev chain. Two entanglement eigenmodes at ζ = 1/2 [counted by the N = 2 in Fig. 1(d)] are found only in the topological nontrivial regime (|J| > |µ|). Each of them is localized around the spatial entanglement cuts between the two subsystems and yields a largest contribution ∆S = ln 2 to the EE.
Discontinuous changes in the number of these maximally entangled eigenmodes are observed at J = ±µ, i.e., the topological transition points predicted by the winding number w of the Kitaev chain. They are also signified by the non-analytic cusps in the EE of Fig. 1   The set {ζ j } includes all the eigenvalues of the correlation matrix C mn = ⟨ĉ † mĉn ⟩ restricted to the subsystem A. The meanings of L ′ , Tr ′ , S A , N A and Q are discussed below Eq. (13).

III. PERIODICALLY KICKED KITAEV CHAIN
In this section, we introduce the models of Floquet topological superconductors that will be studied in this work. After presentating the Hamiltonians and Floquet operators of these models, we discuss how to characterize their quasienergy spectra and topological properties under different boundary conditions. We further introduce the definitions of ES, EE and real-space winding number for the Floquet entanglement Hamiltonian, which allow us to reveal the topological properties and bulk-edge correspondence of Floquet topological superconductors from the entanglement viewpoint. A set of key quantities introduced in this section is summarized in Table II.
We now add time-periodic modulations to the KC in Eq. (1). We consider adding δ-kicks periodically to the pairing or hopping amplitudes. The resulting model is thus described by the Hamiltonian orĤ where δ(t) ≡ ℓ∈Z δ(t − ℓ) and we have assumed the driving period T = 1. We will refer to these periodically kicked Kitaev chains as PKKC1 [Ĥ I (t)] and PKKC2 [Ĥ II (t)] for brevity.
Their corresponding Floquet operators, which describe the time evolution of the system over a complete driving period (e.g., from t = ℓ + 0 − to t = ℓ + 1 + 0 − ) are given bŷ U s =T e −i´1 0Ĥ s(t)dt for s = I, II, whereT performs the time ordering. They take the explicit Under the PBC, we can again perform Fourier transformations from position to momentum representations and obtainĤ for s = I, II, where The corresponding Floquet operators in the Nambu basisΞ † k = ĉ † kĉ −k are thus given by We see that both of them can be expressed in the following piecewise form which does not possess any apparent symmetries. To determine the symmetry class of a 1D Floquet system, we can first transform it into a pair of symmetric time frames [63][64][65]. For the U(k) in Eq. (23), these time frames can be found by splitting the time duration of its kick or free evolution part by half. The resulting Floquet operators in these time frames are It is clear that U(k) and U 1,2 (k) are related by unitary transformations, which implies that they share the same quasienergy spectrum. In the symmetric time frames, one can verify , and the particle-hole symmetry C = σ x K with CU α (k)C −1 = U α (−k) for both the PKKC1 and PKKC2. Therefore, the system described by U(k) also belongs to the BDI symmetry class [63][64][65]. It further possesses the same set of symmetries as the original KC. Each Floquet topological phase of the PKKC1 or PKKC2 can then be characterized by a pair of integer topological winding numbers For our U(k), the (w 0 , w π ) can be obtained from the winding numbers of U 1 (k) and U 2 (k) in the symmetric time frames. Referring to the Eq. (5), we define Here S = σ x is the chiral symmetry operator and the projector With these w α (α = 1, 2), we can express the topological invariants of U(k) as As will be seen shortly, the (w 0 , w π ) provide us with complete topological characterizations for the Floquet superconducting phases in both the PKKC1 [Eq. (16)] and PKKC2 [Eq. (17)].
Under the OBC, we can identify the presence of Floquet Majorana edge modes from the quasienergy spectra and eigenstates ofÛ I andÛ II for the two kicking protocols. Starting with the Eq. (14) or (15) and performing the BdG transformation as presented in Eq. (7), we can also obtain the BdG self-consistent equations for the kicking systems. For theĤ I (t) [Eq. (14)] they are given by For theĤ II (t) [Eq. (15)] these equations are Here E = E(t) denotes the instantaneous eigenvalue of the corresponding set of equations.
Further derivation details of Eqs.
The eigensystem of U can then be obtained by solving the eigenvalue equation to an eigenstate of U with the quasienergy E = 0 (E = ±π) and a localized profile around the edges of the chain. When the system undergoes a topological phase transition accompanied by the closing of its quasienergy gap at E = 0 (E = ±π), we expect to see a quantized change in the number n 0 (n π ) of Floquet Majorana zero (π) modes. As will be seen in the next section, the winding numbers w 0 and w π in Eq. (28) For the PKKC2, such a Hamiltonian is found to be [see Eqs. (31) and (32)] Integrating the corresponding Schrödinger equations over a driving period, both the Floquet operators ofĤ I (t) andĤ II (t) can be written in the piecewise form of which can be further expressed in two symmetric time frames [63][64][65] aŝ The bipartite ES and EE of the system can then be obtained following the recipe of Ref. [66] (see also Ref. [67]). In short, we first divide the whole lattice of PKKC1 or PKKC2 under the PBC into two half-chains A and B of equal lengths. Next, we assume that initially the system is at half-filling, i.e., all the Floquet eigenstates ofÛ α with quasienergies E ∈ [−π, 0) are filled (α = 1, 2). Tracing out all the degrees of freedom belonging to the subsystem B, we obtain the reduced density matrix of subsystem A in the time frame α asρ α HereĤ α A is the Floquet entanglement Hamiltonian of subsystem A, whose eigenvalues define the ES in the αth time frame. For either the PKKC1 or PKKC2 and in the time frame α, the ES {ξ j α |j = 1, ..., L} and EE S α are given by [66] ξ j Here {ζ j α |j = 1, ..., L} represent the eigenvalues of the single-particle correlation matrix Its matrix elements are given by (C α ) mn = ⟨ĉ † mĉ n ⟩ (m, n ∈ A), where the average is taken over the half-filled initial state [66]. {|ϕ j α ⟩} are the eigenvectors of C α . We will also refer to the set {ζ j α } as the ES ofρ α A in the time frame α. Moreover, the topological properties of Here the definitions of S A , N A , L ′ and Tr ′ are the same as those introduced in Eq. (13). The projector Q α in the time frame α (= 1, 2) is defined as The bulk-edge correspondence of chiral symmetric 1D Floquet systems, in either insulating or superconducting phases, can then be established with the help of the ES and the (W 1 , W 2 ) from the quantum entanglement viewpoint [66].
In the next section, we will see that even though the PKKC1 and PKKC2 share the same set of symmetries with the static KC, the driving fields allow Majorana edge modes, topological phases and phase transitions that are significantly different from and much richer than those expected in the static KC to appear in these Floquet models. The topological properties, entanglement nature and bulk-edge correspondence of these intriguing nonequilibrium phases will then be revealed.

IV. RESULTS
We now investigate the topological and entanglement features of the two variants of periodically kicked KC introduced in Eqs. (14) and (15). In each case, we first identify the topological phase diagram of the system under the PBC. Next, we consider the Floquet spectrum of the system under the OBC and build the correspondence between the bulk topological invariants and the numbers of Floquet Majorana edge modes at zero and π quasienergies. Finally, we extract the signatures of topological phases, phase transitions and bulk-edge correspondence of the system from its ES and EE under the PBC. We will see that the physical properties of both models are drastically modified by driving fields.
We find that there are two Floquet bands, whose quasienergies are symmetric with respect to E = 0. Moreover, as E(k) is defined modulus 2π, these bands are also symmetric with respect to E = ±π. The quasienergy spectrum could then possess gaps around E = 0 and E = π. When one or both of these quasienergy gaps close, we may encounter a phase transition in the system. Requiring that E(k) = 0 (π), we find cos(µ+J cos k) cos(∆ sin k) = 1 (−1), yielding the unified phase boundary equation with p, q ∈ Z under the condition that |∆| ≥ |pπ| and |µ ± J| ≥ |qπ|. That is, the Floquet spectrum of our PKKC1 will become gapless at E = 0 or π once its parameters satisfy where the E(k) is given by Eq. (44). It is clear that we will have  Figs. 3(c) and 3(d). We observe that with the increase of J or ∆, the system could indeed undergo a series of gap-closing transitions at E = 0 and E = π. Moreover, many Floquet Majorana edge modes emerge at these quasienergies following the transitions. In each gapped phase, a simple counting reveals that the numbers (n 0 , n π ) of Floquet Majorana zero and π edge modes are related to the winding numbers (w 0 , w π ) in Fig. 2 through the relations n 0 = 2|w 0 | and n π = 2|w π |, which describe the bulk-edge correspondence of our PKKC1. Notably, we find that both the numbers of zero and π Floquet Majorana edge modes could increase monotonically with the increase of J or ∆. Therefore, we can in principle obtain arbitrarily many Majorana zero and π modes in the limit L → ∞ by tuning a single parameter of the system, which is not achievable for the static KC. These abundant Majorana modes may provide more room for the realization of Floquet topological quantum computation [49].
We now investigate the topology and bulk-edge correspondence of the PKKC1 from the entanglement perspective. Referring to Eqs. (34) and (36), we can identify theĤ a andĤ b of our PKKC1 asĤ Plugging them into the expressions ofÛ 1 andÛ 2 in Eqs. (37) and (38), we obtain the Floquet operators of the system described byĤ I (t) [Eq. (34)] in two symmetric time frames. Considering a spatially bisected, half-filled lattice under the PBC and following the procedure discussed in the last section, we can obtain the bipartite ES ζ α and EE S α ofÛ α in the symmetric time frame α for α = 1, 2. Their combination could generate a complete topological characterization for the PKKC1. In Fig. 4, we show the changes of (ζ 1 , ζ 2 ) with respect to J and ∆ for two typical situations. We observe that in both cases, the configuration of either ζ 1 or ζ 2 changes discontinuously around ζ = 1/2 whenever the full system described byÛ I [Eq. (16)] undergoes a topological phase transition as observed in Fig. 2. A direct counting and (48) into Eq. (36) and following the steps from Eqs. (37) to (43). The results show that the winding numbers W 1 and W 2 also get integer jumps at the transition points, confirming their topological nature from an entanglement viewpoint. Moreover, W 1 and W 2 remain quantized within each topological phase, and they are related to the numbers of ζ = 1/2 entanglement eigenmodes through the relations We refer to the Eq. for Floquet topological insulators in one-dimension with chiral symmetry. Finally, we can write down the correspondence between the entanglement winding numbers (W 1 , W 2 ) and the numbers of Floquet Majorana edge modes (n 0 , n π ) in the quasienergy spectrum, i.e., Therefore, the ES and EE could also provide us with sufficient information to identify topological phase transitions, characterizing different topological phases and describing the bulk-edge correspondence of 1D Floquet topological superconductors. We will further confirm this view by investigating the Floquet KC under a different kicking protocol in the following subsection. for ∆ = π/2. In each panel, every region with the same color corresponds to a phase in which w 0 or w π takes the same integer value, as can be figured out from the shared color bar of the four panels.
in Eqs. (16) and (17). We can infer this directly from the bulk dispersion relations in Eqs. (44) and (B1), where long-range hopping and pairing terms in real-space are expected upon inverse Fourier transformations. In the meantime, we also observe many topological nontrivial phases and transitions in the regime |J| < |µ|, where the static KC is topologically trivial. Therefore, time-periodic kickings applied to either the hopping or pairing amplitudes of KC could significantly alter its topological properties.
In Fig. 7  numbers of these Majorana edge modes (n 0 , n π ) and their changes versus (J, ∆) are found to be precisely captured by the winding numbers (w 0 , w π ) in the bulk topological phase diagram (Fig. 6) through the bulk-edge correspondence It is expected that these relations for the PKKC1 and PKKC2 are identical, as both models are in one-dimension with the same set of protecting symmetries and they are thus characterized by the same type of topological winding numbers. Meanwhile, we find a series of topological phase transitions with the increase of J and ∆, after which more and more Floquet Majorana zero and π edge modes tend to appear in the spectral gaps. Therefore, similar to the case of PKKC1, we can obtain Floquet superconducting phases with arbitrarily many Majorana edge modes in principle in the PKKC2. In Fig. 8, we present the ES of the PKKC2 in two symmetric time frames within the same parameter ranges of Fig. 7. Similar to the case of PKKC1, we observe eigenmodes in the ES with ζ = 1/2 for the system in both time frames. These eigenmodes are localized around the entanglement cuts between the subsystems A and B, with each of them contributing a ∆S = ln 2 to the EE in Fig. 9. Furthermore, the numbers of these entanglement edge modes (N 1 , N 2 ) are found to undergoing quantized changes whenever a topological phase transition happens with the change of J or ∆, as shown in Fig. 9. A simple counting suggests the following relations between the numbers of Floquet Majorana edge modes ofÛ II and the numbers of its entanglement edge modes with ζ = 1/2, i.e., max(N 1 , N 2 ) = n 0 + n π , min(N 1 , N 2 ) = |n 0 − n π |.
We have systematically verified these relations for both the PKKC1 and PKKC2 in our numerical calculations (see also Appendix C), which implies that they represent a general set of equalities between the spectral and entanglement topology of 1D, chiral symmetric Floquet topological superconductors.
Finally, we show in Fig. 9 Fig. 6, we further notice that W α = w α for α = 1, 2. Therefore, the winding numbers Theoretically, even though the topological and entanglement bulk-edge correspondences in Eqs. (49), (50), (51) and (52) are extracted from numerical calculations, they are expected to hold in any 1D Floquet topological insulators and superconductors in the symmetry classes AIII and BDI [15], which are both characterized by the integers (w 0 , w π ) in Eq. (28). With slight modifications, these relations should also be applicable to 1D Floquet models in the symmetry class CII, which are characterized by 2Z × 2Z topological invariants [66]. We thus expect that our identifications about the entanglement winding numbers, the topological winding numbers and the numbers of edge modes are valid in generic 1D gapped Floquet topological phases protected by chiral symmetry, which go beyond the models considered in the present work. Feshbach resonances [68], synthetic spin-orbit couplings [69], or the orbital degrees of freedom in association with s-wave interactions [70]. The Floquet driving could be introduced through the periodic modulation of magnetic fields [71], laser fields [72] or Raman-coupling amplitudes [73]. The δ-kicking may then be implemented via a short-pulsed modulation with a large amplitude. Replacing the δ-kicking on the hopping or pairing amplitude with time-periodic quenching could yield the same Floquet superconducting phases upon suitable rescaling of system parameters. The topological winding numbers of our system may be extracted from the mean chiral displacements of initially localized wavepackets in each symmetric time frame [74,75]. Finally, the Floquet Majorana zero and π edge modes may be detected by applying spatially resolved rf spectroscopy and in situ imaging techniques [46]. Note in passing that the ranges of system parameters for us to observe phases with large winding numbers and many Floquet Majorana modes are essentially available under current experimental conditions [46][47][48]. Therefore, we believe that the realization of our models and the detection of their topological properties should be within reach in near-term experiments.
In future work, it is interesting to check the transport properties [76] and the robustness of Floquet superconducting phases found here to disorder [77] and many-body interactions.
Possible experimental probes of the ES, EE and entanglement winding numbers in Floquet systems also deserve to be explored.
Meanwhile, we can compute the commutator [Ĥ s (t),ĉ n ] directly with theĤ s (t) in Eq. (14) or (15) and theĉ n in Eq. (7), yielding where ∆(t) ≡ ∆δ(t) for s = I and J(t) ≡ Jδ(t) for s = II. For the Eqs. (A1) and (A2) to be equal, the coefficients in front off † j andf j must be separately identical. Therefore, after dropping the redundant band index j, we obtain the BdG self-consistent equations in the fermionic basis for the PKKC1 and PKKC2, as shown in Eqs.   or | sin(r 1 ) sin(r 2 )| = 0. In the former case, we must have ∆ sin k = 0, yielding the gapless condition cos(J ± µ) = 0 for k = 0, π. In the latter case, we will have cos(r 1 ) cos(r 2 ) = ±1 when the quasienergy gap closes. Putting together, we find the equations that describing all possible phase boundaries in the parameter space of the PKKC2, i.e., or for p, q ∈ Z assuming |J| ≥ |pπ| and ∆ 2 + µ 2 ≥ |qπ|.