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the anti-Hermitian strength of H 0 . The spectrum of H 0 turns out to be real-valued for α < α$_{c}$ , where α$_{c}$ ≥ 0 corresponds to the transition from unbroken to broken PT symmetry. Numerical studies generally show that for α < α$_{c}$ the spectrum is composed by bands separated by gaps like in an ordinary crystal, whereas for α ≥ α$_{c}$ band merging is observed with the appearance of pairs of complex-conjugate eigenvalues [13, 16]. For instance, for the potential defined by 

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one has α$_{c}$ = 1 [13]. 

In this section we consider the unbroken symmetry phase, i.e. the case α < α$_{c}$ . As recently shown in Ref.[17], for α < α$_{c}$ the motion of a Bloch particle in pres- ence of an external dc force F can be described fol- lowing the same lines as in ordinary crystals by ex- panding the field ψ ( x, z ) as a superposition of Bloch- Floquet eigenfunctions φ$_{n}$ ( x, κ ) = u$_{n}$ ( x, κ ) exp( iκx ) of H 0 , where the wave number κ varies in the first Bril- louin zone, i.e. − k$_{B}$/ 2 ≤ κ < k$_{B}$/ 2, k$_{B}$ = 2 π/a is the Bragg wave number, n is the band index, and u$_{n}$ ( x, κ ) is the periodic part of the Bloch-Floquet eigenfunc- tion. After setting ψ ( x, z ) = ∑ n ∫ dκc$_{n}$ ( κ, z ) φ$_{n}$ ( κ, z ) and assuming normalized eigenfunctions such that ∫ dxφ ∗ n ′ ( − x, − κ $^{′}$) φ$_{n}$ ( x, κ ) = D $_{n}$δ$_{n,n}$ ′ δ ( κ − κ $^{′}$) with D n = ± 1, the evolution equations for the spectral coefficients c$_{n}$ ( κ, z ) read [17] 

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where E$_{n}$ ( κ ) is the energy of φ$_{n}$ ( x, κ ) [with E$_{n}$ ( − κ ) = E$_{n}$ ( κ )] and X$_{n,l}$ ( κ ) ≡ (2 πi/a ) ∫ a 0 dxu ∗ $_{n}$( − x, − κ ) ∂$_{κ}$u$_{l}$ ( x, κ ). The off-diagonal elements X$_{n,l}$ ( n ̸ = l ) in Eq.(3) are responsible for inter- band transitions, i.e. Zener tunneling (ZT). If bands n and l are separated by a large gap and the ac force F ( z ) is small enough such that | F X$_{n,l}$ ( κ ) | ≪ | E$_{n}$ ( κ ) − E$_{l}$ ( κ ) | in the entire Brillouin zone, ZT is negligible as in ordinary lattices and one can make the single-band approximation by setting X$_{n,l}$ ≃ 0 for n ̸ = l in Eq.(3). In the single-band approximation one thus obtains 

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where we omitted, for the sake of simplicity, the band in- dex n and set i Φ( κ ) ≡ D $_{n}$X$_{n,n}$ ( κ ). Because of the sym- metry of V ( x ), Re( u$_{n}$ ( k, x )) and Im( u$_{n}$ ( k, x )) have well defined and opposite parity under the inversion x → − x ; this implies that Φ( κ ) is a real-valued function of κ , van- ishing for a real potential (i.e. for α = 0). As previ- ously shown in [17], when a dc force F is applied to the crystal, from Eq.(4) it follows that the energy spectrum is described by a complex-valued Wannier-Stark ladder. The non-reality of the energy spectrum comes from the extra-term Φ( κ ) in Eq.(4) and is physically due to the 

fact that the external dc force F breaks the PT sym- metry of the full Hamiltonian H = H 0 − F x . For an ac-like force with period Λ = 2 π/ω , because of the z - periodicity of the Hamiltonian its energy spectrum is re- placed by a quasi-energy spectrum. Moreover, in the single-band approximation DL corresponds to a complete collapse of the quasi-energy band like in an ordinary crys- tal [2]. According to Floquet’s theorem of periodic sys- tems, the quasi-energy E ( κ ) for the n -th lattice band can be readily calculated by looking for a solution to Eq.(4) of the form c ( z, κ ) = a ( z, κ ) exp[ − i E ( κ ) z/λ ] with a ( z + Λ , κ ) = a ( z, κ ). One obtains 

where we have set κ ′ ≡ κ − k (Λ) + k ( z ) and k ( z ) = (1 /λ ) ∫ z 0 dξF ( ξ ). Let us assume that the ac forcing F ( z ) is an odd function with respect to some point z 0 , i.e. that F ( z − z 0 ) = − F ( z 0 − z ) for some z 0 in the oscil- lation cycle. This condition is satisfied, for instance, for the important case of an harmonic (e.g. sinusoidal or cosinusoidal) ac driving force, originally considered by Dunlap and Kenkre [1] and that will be assumed in the following. Owing to this additional temporal symmetry on the driving force, which is absent for the BO prob- lem [17], a real-valued quasi-energy spectrum for the non-Hermitian time-periodic Hamiltonian H 0 − F ( z ) x is obtained. In fact, in this case the imaginary term on the right hand side of Eq.(5) vanishes after integra- tion because F ( z ) and Φ( κ $^{′}$) have opposite parity for the inversion ( z − z 0 ) → − ( z − z 0 ). The quasi-energy spectrum is thus real-valued and its expression takes the usual form as in a conventional crystal. DL corre- sponds to a collapse of the quasienergy band E ( κ ), i.e. to d E ( κ ) /dκ = 0. For most driving fields like a sinusoidal field, DL can be attained exactly solely in the NNTB approximation, i.e. when the band shape E ( κ ) is sinu- soidal [3], E ( κ ) = E 0 − Δ cos( κa ). In this case, assum- ing for the sake of definiteness a sinusoidal ac-like force F ( z ) = F 0 cos( ωz ), the explicit form of the quasi-energy reads [2] 

Band collapse, leading to DL, is thus attained when J 0 ( F 0 a/λω ) = 0 [1, 2, 7]. 

To check the correctness of the analysis, we investigated DL for the complex crystal V ( x ) = V 0 [cos(2 πx/a ) + iα sin(2 πx/a )] in the unbroken PT symmetry phase ( α < 1) by a direct numerical analysis of Eq.(1) using a pseudo- spectral split-step method with absorbing boundary con- ditions. Figure 1 shows a typical example of band dia- gram below the phase transition point ( α = 0 . 3), numeri- cally computed by a spectral analysis of the Hamiltonian H 0 . Note that the lowest band of the array is with ex- cellent accuracy approximated by a sinusoidal curve and turns out to be separated by a large gap from the second 

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