4 growing term being ∼ t ρ where ρ is the (maximal) dif- ference between the algebraic and geometric multiplicity of defective eigenvalues. In our case, ρ = 1 because the maximal algebraic multiplicity of any eigenvalue of H ( q ) is 2; moreover, since H ( q ) may have defective eigenvalues only for q = 0 or q = − π/a , secular growing terms may appear solely when the wave number k of the exciting plane wave is an integer multiple of k$_{B}$/ 2. The absence of secular growing terms in c$_{l}$ ( t ) for any excitation wave number k = nk$_{B}$/ 2 ( n = 0 , ± 1 , ± 2 , ... ) implies that the matrix J is diagonal, which ensures the lack of defective eigenvalues of H and thus of spectral singularities of H according to Theorem 1. FIG. 1: (color online) (a) Schematic of Bragg scattering of a plane wave off a complex crystal. (b) Band structure of the the complex crystal V ( x ) = V 0 exp( ik B x ). The circles mark the spectral singularities inside the continuous spectrum. x (a) (b) t a 0 0 GLYPH<c112>GLYPH<c47> a GLYPH<c45>GLYPH<c112>GLYPH<c47> a 0 c$_{l}$ 0 c$_{l}$ E E 1 E 2 E 3 E 4 E 5 ( ) t 0 the free-particle energy dispersion curve E = k $^{2}$, period- ically folded inside the first Brillouin zone [see Fig.1(b)]. The eigenvalues are distinct for q = 0 , − π/a . At the crossings of the folded parabolas of Fig.1(b), i.e. at q = 0 and at q = − π/a , one has E$_{−}$$_{α}$ ( q ) = E$_{α}$ ( q ) and E$_{1}$$_{−}$$_{α}$ ( q ) = E$_{α}$ ( q ), respectively, i.e. the eigenvalues coa- lesce in pairs and become defective. Therefore, the con- tinuous spectrum of H , E ≥ 0, contains a sequence of spectral singularities at E$_{n}$ = ( nk$_{B}$/ 2) $^{2}$, n = 1 , 2 , 3 , ... [see Fig.1(b)], which spoil the completeness of the Bloch- Floquet eigenfunctions. The defective nature of degen- erate eigenvalues at q = 0 and q = − π/a can be readily proven by direct calculation of the eigenvectors $_{w}$( α ) of H . Note that, as wave scattering from complex potential barriers enables a finite number of spectral singularities in the continuous spectrum [12, 17], in our example the number of spectral singularities is countable but infinite. According to Theorem 3, a secular growth of Bragg diffraction pattern for a plane wave that excites the crys- tal at normal incidence (or tilted by an angle which is an integer multiple of the Bragg angle) provides a distinctive signature of the appearance of spectral singularities at the PT symmetry-breaking transition point λ = λ$_{c}$ = 1. However, in any experimental setting aimed to observe such a secular growth, the wave that excites the crystal is always spatially limited or truncated, and it is thus of major relevance to investigate the impact of spectral sin- gularities on the evolution of a wave packet with a broad- ened angular spectrum, an issue which was not consid- ered in previous works by Berry. Here we investigate the Bragg diffraction of a wave packet with a broadened mo- mentum distribution, ψ ( x, 0) = ∫ dkF ( k ) exp( ikx ), and show that spectral broadening leads to a saturation of the secular growth of scattered waves. Such a saturation behavior is basically due to the fact that spectral sin- gularities are of measure zero (they are a countable set of points embedded in the continuous energy spectrum E ≥ 0). For the sake of simplicity, we consider a shallow lattice and a wave packet with a narrow spectrum F ( k ) centered at k = − k$_{B}$/ 2 of width Δ k ≪ k$_{B}$ . Following the same lines detailed in the proof of Theorem 3, one can show that the diffraction pattern ψ ( x, t ) can be written as the interference of different wave packets describing which has been recently considered to highlight unusual diffraction and transport properties of complex optical lattices [6, 25]. In this case, λ$_{c}$ = 1 [6] and at the sym- metry breaking point one has V ( x ) = V$_{0}$ exp( ik$_{B}$x ), a po- tential which is amenable for an analytical study [20, 22]. For this potential, H has a block diagonal form, namely H n,m = ( n + k$_{B}$q ) $^{2}$δ$_{n,m}$ + V$_{0}$δ$_{m,n}$$_{+1}$ , and its eigenval- ues are simply the elements on the main diagonal, i.e. E$_{α}$ ( q ) = ( q + αk$_{B}$ ) 2 ( α = 0 , ± 1 , ± 2 , ... ). This means that, as previously noticed [6, 22, 25], the band structure of the PT potential V = V$_{0}$ exp( ik$_{B}$x ) coincides with V$_{R}$ ( x ) = V$_{0}$ cos(2 πx/a ) , V$_{I}$ ( x ) = V$_{0}$ sin(2 πx/a ) , (5) Let us specialize the previous results to the case of a PT -symmetric lattice, for which V ( − x ) = V $^{∗}$( x ). Let V$_{R}$ ( x ) and λV$_{I}$ ( x ) be the real and imaginary parts of the potential, respectively, where λ ≥ 0 measures the anti- Hermitian strength of H . The spectrum of H is real for λ ≤ λ$_{c}$ , where λ$_{c}$ ≥ 0 defines the symmetry break- ing point. According to the previous analysis, complex- conjugate pairs of eigenvalues for H ( q ) should appear as λ is increased from below to above λ$_{c}$ . The typical scenario that describes symmetry breaking in a finite-dimensional PT matrix is the appearance of an exceptional point via the merging of two real eigenvalues into a single real and defective eigenvalue at λ = λ$_{c}$ (see, for instance, [7, 29]). We note that the existence of such a branching point for a certain class of non-hermitian matrices as a control pa- rameter is varied was proven in a rather generally way in Ref.[30] (see also [7]). We may thus conjecture that for a PT complex crystal symmetry-breaking is accompa- nied by the appearance of spectral singularity, which arise from defective eigenvalues of H at q = 0 or q = − π/a . This scenario is in agreement with numerical or analyt- ical results obtained from band computation of specific complex periodic potentials (see, for instance, [6, 8]). As an example, let us consider the PT crystal defined by III. BRAGG SCATTERING IN PT COMPLEX CRYSTALS AND WAVE PACKET DYNAMICS