12 where | n 〉 are the Wannier states introduced in Sec.III.B (the Wannier basis). For the sake of completeness, we present the detailed calculations for both approaches. 1. Bloch-basis representation of H . As shown in Sec.II.B, the absence of bound states of H requires that Eq.(18) does not admit of any solution in the complex z plane. Using the expression (50) of the self-energy Σ( z ), Eq.(18) takes the form In this Appendix we derive the expressions of the ma- trix elements G $_{a,a}$( z ) = 〈 a | G ( z ) a 〉 , G $_{k,a}$( z ) = 〈 k | G ( z ) a 〉 , G $_{a,k}$( z ) = 〈 a | ( z ) k 〉 and G k,k ′ ( z ) = 〈 k | G ( z ) k $^{′}$〉 of the re- solvent [Eqs.(6-9) given in the text]. From the identity G ( z )( z − H$_{0}$ − V ) = ( z − H$_{0}$ − V ) G ( z ) = I it follows that Appendix A: Matrix elements of the resolvent ( 1 − κ 2 a 2 κ 2 0 ) z − E$_{a}$ = − i κ 2 a 2 κ 2 0 √ 4 κ 2 0 − z $^{2}$. (B1) 〈 a | G ( z )( z − H$_{0}$ ) a 〉 − 〈 a | G ( z ) V a 〉 = 1 (A1) a G ( z )( z H$_{0}$ ) k a G ( z ) V k = 0 (A2) 〈 k | ( z − H$_{0}$ ) G ( z ) a 〉 − 〈 k | V G ( z ) a 〉 = 0 (A3) 〈 k | G ( z )( z − H$_{0}$ ) k $^{′}$〉 − 〈 k | G ( z ) V k $^{′}$〉 = δ ( k − k $^{′}$) . (A4) We can solve Eq.(B1) by introducing, in place of z , the new complex-valued variable µ defined by Taking into account that Without loss of generality, we may assume Re( µ ) > 0. In fact, the function z ( µ ) defined by Eq.B(2) is invariant for the inversion µ → − µ , so that we may restrict our analysis to the case Re( µ ) > 0. With such a substitution, the square root on the right hand side of Eq.(B1) can be solved analytically, yielding ± 2 iκ$_{0}$ sinh µ . Some care should be taken when choosing the right determination (i.e. sign) of the square root [46]. For Re( µ ) > 0, one obtains z = − κ$_{0}$ [exp( µ ) + exp( − µ )] = − 2 κ$_{0}$ cosh µ. (B2) V | a 〉 = ∫ dkv $^{∗}$( k ) | k 〉 , V | k 〉 = v ( k ) | a 〉 (A5) ( z − E ( k )) G $_{a,k}$( z ) − v ( k ) G $_{a,a}$( z ) = 0 (A7) ( z − E$_{a}$ ) G $_{a,a}$( z ) − ∫ dkv $^{∗}$( k ) G a,k = 1 (A6) 2 ( 1 − κ 2 a 2 κ 2 0 ) cosh µ + E$_{a}$ κ$_{0}$ = − κ 2 a κ 2 0 sinh µ. (B3) After setting ξ = exp( µ ), from Eq.(B3) one obtains Eq.(52) given in the text once cosh µ and sinh µ are expressed in terms of the exponentials exp( ± µ ) = ξ ± $^{1}$. Therefore, if the two roots ξ$_{1}$$_{,}$$_{2}$ of Eq.(52) satisfy the condition | ξ$_{1}$$_{,}$$_{2}$ | ≤ 1, Eq.(B3) does not have roots with Re( µ ) > 0, and hence H does not have bound states. 2. Wannier-basis representation of H . In this ap- proach, we use the tight-binding representation of the Hamiltonian H using the Wannier function basis [Eq.(63)]. Bound states of H correspond in this case to surface states localized near the edge of the truncated lattice of Fig.3. They can be directly determined by looking for a solution to Eqs.(65-67) of the form ( n ≥ 1), where E is the energy of the surface state. The constants µ and A , as well as the dependence of E on µ , are readily determined by substituting Eq.(B4) into Eqs.(65-67). One obtains c$_{n}$ ( t ) = exp[ − µ ( n − 1) − iEt ] , c$_{a}$ ( t ) = A exp( − iEt ) (B4) E = − 2 κ$_{0}$ cosh µ (B5) EA = − κ$_{a}$ + E$_{a}$A. (B7) E = κ$_{0}$ exp( − µ ) − κ$_{a}$A Appendix B: Conditions for a real-valued energy spectrum of the non-Hermitian Hamiltonian from which the following second-order algebraic equa- tions for ξ = exp( µ ) is readily obtained Substitution of Eq.(8) into Eq.(A8) finally yields Eq.(9) given in the text. which can be solved for G a,a and G $_{a,k}$, yielding Eqs.(6) and (7) given in the text. To calculate G $_{k,a}$( z ), we use Eq.(A3) and note that 〈 k | ( z − H$_{0}$ ) G ( z ) a 〉 = 〈 ( z ∗ − H † $_{0}$) k | G ( z ) a 〉 = ( z − E ( k )) G $_{k,a}$( z ) and 〈 k | V G ( z ) a 〉 = 〈 V k | G ( z ) a 〉 = v $^{∗}$( k ) G $_{a,a}$( z ). This yields ( z − E ( k )) G $_{k,a}$( z ) − v $^{∗}$( k ) G $_{a,a}$( z ) = 0, which can be solved for G $_{k,a}$( z ), yielding Eq.(8) given in the text. Finally, the ma- trix element G k,k ′ ( z ) is obtained from Eq.(A4), which can be written in the form ( z − E ( k $^{′}$)) G k,k ′ ( z ) − v ( k $^{′}$) G $_{k,a}$( z ) = δ ( k − k $^{′}$), i.e. G k,k ′ ( z ) = v ( k $^{′}$) G $_{k,a}$( z ) z − E ( k $^{′}$) + δ ( k − k $^{′}$) z − E ( k $^{′}$) . (A8) and that ( z − H$_{0}$ ) | a 〉 = ( z − E$_{a}$ ) | a 〉 , ( z − H$_{0}$ ) | k 〉 = ( z − E ( k )) | k 〉 , Eqs.(A1) and (A2) take the form In this Appendix we derive the necessary and sufficient conditions that ensure a real-valued energy spectrum for the non-Hermitian FFA Hamiltonian H introduced in Sec.III.A. As shown in Sec.II.B, this condition is equivalent to the vanishing of the point-spectrum of H , i.e. to the absence of bound states. The detailed calculations can be performed following two different, though equivalent, approaches. The first one starts from the representation of H in the {| a 〉 , | k 〉} basis (the Bloch basis), whereas the second approach uses a different decomposition of H , namely on the {| a 〉 , | n 〉} basis, ξ 2 + E$_{a}$ κ$_{0}$ ξ + 1 − κ 2 a κ 2 0 = 0 (B8)