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band. DL is thus expected to occur provided that the lattice is excited in its lowest order band. Figures 2(a) and (b) show a typical spreading (discrete diffraction) of a Gaussian wave packet ψ ( x, 0) = exp( − x $^{2}$/w$^{2}$) in the absence of the external force, for either a broad Gaus- sian beam that excites at normal incidence a few lattice 

sites at z = 0 [2 w = 5 a , Fig.2(a)], and a narrow Gaussian beam that excites at normal incidence a single well of the lattice [2 w = 2 a/ 3, Fig.2(b)]. In both cases, the lowest band of the array is mainly excited, as discussed e.g. in [8]. After application of the sinusoidal force [Figs.2(c-f)], suppression of beam diffraction and self-imaging effects are clearly observed when period and amplitude of forc- ing satisfy the condition Γ ≡ F 0 a/ ( λω ) = 2 . 405 (first zero of Bessel J 0 function), as shown in Figs.2(e) and (f). 

## III. WAVE PACKET DYNAMICS AT THE PT SYMMETRY-BREAKING POINT 

As α is increased to reach and cross the PT symmetry- breaking point α$_{c}$ , gap narrowing till band merging, as- sociated to the appearance of pairs of complex-conjugate eigenvalues, is observed [13]. For instance, for the po- tential defined by Eq.(2), at the transition point the band diagram is given by the free-particle energy dis- persion curve E = λ $^{2}$κ$^{2}$/ (2 n$_{s}$ ), periodically folded in- side the first Brillouin zone [13] (see the dotted curve in Fig.1). In this case, as previously noticed for BOs [17], wave packet transport is deeply modified and can not be described by means of the canonical model (3) in- troduced in the previous section. From a physical view- point, this is related to the highly non-reciprocal behav- ior of Bragg scattering in the crystal and violation of Friedel’s law of Bragg diffraction for crystal inversion [12, 14]. As in Ref. [17], we limit here to consider the dynamical behavior of a broad wave packet in the complex lattice defined by Eq.(2) at α = α$_{c}$ = 1, i.e. V ( x ) = V 0 exp( ik$_{B}$x ), which enables a rather simple ana- lytical and physical analysis. As opposed to Ref. [17], we assume here an ac-like force, namely F ( z ) = F 0 cos( ωz ). Figure 3 shows a few typical examples of wave packet dynamics as obtained by a numerical analysis of Eq.(1) when the lattice is excited at z = 0 with a broad Gaus- sian beam ψ ( x, 0) = exp( − x $^{2}$/w$^{2}$) for a fixed value of the ac modulation period Λ = 2 π/ω and for increas- ing values of the amplitude F 0 . For a small amplitude F 0 , it turns out that the wave packet propagates as if the lattice were absent [Fig.3(a)], following an oscillatory path as predicted by the semiclassical analysis of Eq.(1) with V = 0. Owing to the external force, the mean wave packet momentum varies periodically according to λk ( z ) = ∫ z 0 dξF ( ξ ) = ( F 0 /ω ) sin( ωz ) . As the forcing F 0 reaches and crosses the critical value F$_{c}$ = k$_{B}$λω/ 2, new wave packets, arising from first-order Bragg diffrac- tion, periodically bifurcate from the primary beam at propagation distances z satisfying the Bragg condition k ( z ) = − k$_{B}$/ 2 [see Figs.3(b) and 3(c)]. The mean mo- mentum of the first-order diffracted wave packets differs from that of the primary wave packet by an additional term k$_{B}$ , which explains the refraction of the first-order diffracted beams at the angle θ = dx/dz = k$_{B}$λ/n$_{s}$ ob- served in Figs.3(b) and 3(c). At stronger forcing, namely for F ≥ 3 F$_{c}$ , additional wave packets bifurcate from the 

