Exceptional Non-Hermitian Phases in Disordered Quantum Wires
EExceptional Non-Hermitian Phases in Disordered Quantum Wires
Benjamin Michen, Tommaso Micallo, and Jan Carl Budich ∗ Institute of Theoretical Physics , Technische Universit¨at Dresden andW¨urzburg-Dresden Cluster of Excellence ct.qmat , , Germany (Dated: February 26, 2021)We demonstrate the occurrence of nodal non-Hermitian (NH) phases featuring exceptional de-generacies in chiral-symmetric disordered quantum wires, where NH physics naturally arises fromthe self-energy in a disorder-averaged Green’s function description. Notably, we find that at leasttwo nodal points in the clean Hermitian system are required for exceptional points to be effectivelystabilized upon adding disorder. We identify and study experimental signatures of our theoreti-cal findings both in the spectral functions and in mesoscopic quantum transport properties of theconsidered systems. Our results are quantitatively corroborated by numerically exact simulations.The proposed setting provides a conceptually minimal framework for the realization and study oftopological NH phases in quantum many-body systems.
Introduction.—
Novel phases of matter combining theparadigm of topological insulators and semimetals [1–4]with the notion of effective non-Hermitian (NH) Hamil-tonians describing dissipative systems [5, 6] have becomea broad frontier of current research [7–27]. With appli-cations ranging from the field of classical robotic meta-materials [28, 29] to the realm of quantum many-bodysystems [30–34], various microscopic mechanisms of dis-sipation have been found capable of inducing such NHtopological phases (see Ref. [27] for an overview).A prominent example in the context of quantum ma-terials is provided by impurity scattering in disorderedsystems. Beyond simply giving a finite life-time to theBloch states of a clean solid, disorder potentials exhibit-ing a non-trivial structure in internal degrees of freedomsuch as spin may induce a NH self-energy Σ with a com-plex matrix structure. In particular, if Σ does not com-mute with the free Hamiltonian H , topologically sta-ble NH semimetal phases may be stabilized, as has beenpredicted in both two-dimensional and three-dimensionalsystems by means of perturbative calculations [35–37].Generally speaking, in NH systems the notion of band-touching points such as Dirac- and Weyl-nodes is gener-ically replaced by exceptional points (EPs) [38–40], i.e.degeneracies at which the NH Hamiltonian becomes non-diagonalizable.Here, we report on the discovery of topologically sta-ble exceptional NH phases in disordered one-dimensional quantum systems with chiral symmetry. To this end, westudy the effective NH Hamiltonian H e ( k ) = H ( k ) + Σ( k, ω = 0) , (1)associated with the disorder averaged Green’s function[41] at the Fermi Energy ( ω = 0) that gives rise to theNH self-energy Σ. While we find that a single Hermitiannodal point in H cannot split into a pair of stable EPsto leading order, impurity scattering between two band-touching points is shown to yield a NH band structurewith four EPs (see Fig. 1c for an illustrating example).We study the physical properties of this intriguing NH (a)(b)(c) FIG. 1: (a) Illustration of the translation-invariant modelHamiltonian H (see Eq. (2)). (b) Illustration of the dis-order term V (see Eq. (3)). V consists of random amplitudeon-site and nearest-neighbour terms. (c) Complex spectrumof the effective NH Hamiltonian H e (see Eq. (1)) of a disor-dered system microscopically described by the total Hamilto-nian H = H + V (see Eqs. (2-3)). Exact numerical results areshown in blue and red, perturbative results within first bornapproximation in purple and yellow. Two pairs of exceptionalpoints connected by Fermi arcs are visible (see also inset). Pa-rameters are w = − i , α = 0 . s = 1 . v = − . i . nodal phase. Specifically, basic observables such asspectral functions and the linear response conductancein a simple two-terminal mesoscopic quantum transportsetting [42] are compared to a conventional disordered a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b phase without EPs, thus pinpointing experimentallyaccessible differences between the inequivalent NHphases. Our results are qualitatively derived within aperturbative approach and quantitatively corroboratedby means of numerically exact simulations. Microscopic model.—
We propose and study a micro-scopic two-band model in one spatial dimension (1D) thatis described by the Hamiltonian H = H + V , consistingof a translation-invariant part H (see Fig. 1a for an il-lustration) that is perturbed by a random disorder term V (see Fig. 1b). Specifically, H = N − (cid:88) j =1 w (cid:16) ψ † j +1 ,B ψ j,A + ψ † j,B ψ j +1 ,A (cid:17) + h . c ., (2)where length is measured in units of the lattice con-stant and energy in terms of the complex nearest neigh-bor hopping chosen as w = − i , the field-operators ψ j,A ( B ) annihilate a fermion in unit cell j on sublat-tice A (B) (cf. Fig. 1a). Assuming periodic bound-ary conditions, the corresponding Bloch Hamiltonianin reciprocal space takes the form H ( k ) = d R ( k ) · σ with d R ( k ) = ( − cos( k ) , − cos( k ) , σ z H † ( k ) σ z = − H ( k ), where the stan-dard Pauli-Matrices σ act on the A-B sublattice pseudo-spin. The random disorder term V is similarly structured(cf. Fig. 1b) and has the real space representation V = N − (cid:88) j =1 a j (cid:20) s (cid:16) ψ † j,A ψ j,B + h . c . (cid:17) + v (cid:16) ψ † j +1 ,B ψ j,A + ψ † j,B ψ j +1 ,A (cid:17) + h . c . (cid:21) , (3)where s ∈ R , v ∈ C , and the amplitudes { a j } are uncorre-lated and drawn from the uniform distribution on the realinterval [ − α, α ], i.e. the overall disorder strength is pa-rameterized by α . It is worth noting that the correlationin amplitude between the on-site and nearest-neighborterms that share the same random amplitude a j is im-portant for the formation of EPs. Below, we demonstratethat a nodal NH phase with EPs or a gapped phase arisesdepending on the parameters s and v in (3).To obtain the effective Hamiltonian H e ( k ) = H ( k ) + Σ( k, ω = 0) (cf. Eq. 1), we calculatethe retarded Green’s function in frequency space,i.e. the Fourier transform of the propagator G R k,k (cid:48) ( t − t (cid:48) ) = − i Θ( t − t (cid:48) ) (cid:104) |{ c k ( t ) , c † k (cid:48) ( t (cid:48) ) }| (cid:105) withrespect to ( t − t (cid:48) ), where c † k ( t ) = ( c † k,A ( t ) , c † k,B ( t )) denotesthe Heisenberg picture A-B sublattice spinor of creationoperators in reciprocal space. Upon averaging over theimpurity amplitudes, translational invariance is restored,thus rendering the resulting disorder averaged Green’sfunction G R , av k block diagonal in momentum space [41].From G R , av k ( ω ) = [ ( ω + iη ) − H ( k ) − Σ( k, ω )] − , where η > k, ω ) and the effectiveHamiltonian H e ( k ), respectively. (a)(b)(c) FIG. 2: (a) Spectrum of H e ( k ) in the conventional phase.The exact numerical result is plotted in blue and red, theFBA result is shown in purple and yellow for comparison.Parameters are α = 0 . s = 1 . v = − .
5. (b) Exact numer-ical result for the spectral function A ( k, ω ) of the exceptionalphase. Parameters are α = 0 . s = 1 . v = − . i . (c) Exactnumerical result for the spectral function A ( k, ω ) of the con-ventional phase. Parameters are α = 0 . s = 1 . v = − . Disorder-induced exceptional degeneracies.—
To inves-tigate the occurrence of disorder-induced EPs in ourmodel system (2-3), it is convenient to represent the ef-fective NH Hamiltonian (1) in the form H e ( k ) = d ( k ) σ + d ( k ) · σ , (4)where d = d R + i d I with d R , d I ∈ R is the complexgeneralization of a Bloch vector, and d ∈ C is a com-plex energy shift. The degeneracy points of the complexspectrum E ± = d ± (cid:112) d R − d I + 2 i d R · d I of H e gener-ically represent EPs and rely on two real conditions thatamount to a vanishing real and imaginary part under thelatter square-root function. Due to chiral symmetry, d must be purely imaginary and the imaginary part of theaforementioned EP conditions, namely d R · d I = 0, is al-ways satisfied. The only remaining condition for an EPis then given by d R = d I (5)In our model system, Eq. (5) is typically satisfied atpairs of momenta where the Hermitian part of H e is closeto an ordinary diagonalizable nodal point (i.e. close to | d R | = 0), and an anti-Hermitian term ∼ iσ z associatedwith the symmetry allowed z -component of d I occurs inthe self-energy (cf. Eq. (1)). In this sense, the self-energyΣ induced by the disorder term (3) may split Hermitiangap-closing points into pairs of EPs. However, as we an-alytically derive further below, for this mechanism to beeffective at least a pair of Hermitian degeneracy pointsmust be present in the unperturbed spectrum, thus lead-ing to a minimum of four EPs in the complex energyband-structure of H e .Our numerically exact results on the complex spec-trum of the disorder-averaged effective Hamiltonian H e (see Eq. (1)) are compared to first Born approximation(FBA) calculations in Fig. 1c and Fig. 2a, respectively.For the parameter choice s = 1 . v = − . i in Eq. (3),a nodal NH phase featuring two pairs of EPs, at whichboth the real and the imaginary part of the effective en-ergy become degenerate, is stabilized (see Fig. 1c). Theunderlying disorder-free Hamiltonian H (see Eq. (2)) hasordinary band crossings at k = ± π/
2. We note that thedistance between the EPs occurring around each of thosecrossings at weak disorder continuously increases withthe overall disorder strength α . Furthermore, a so calledNH Fermi-arc, i.e. a continued two-fold degeneracy inthe real part of the spectrum of H e , connects the EPs.The splitting of this degeneracy from the EPs exhibits acharacteristic square-root dispersion, in contrast to theat least linear splitting of degeneracies in the Hermitianrealm.By contrast, for the parameter choice s = 1 . v = 0 .
5a gapped phase without EPs is observed (see Fig. 2a).There, anti-crossings open around the nodal points ofthe unperturbed Hamiltonian H , and the correspondingenergy gap increases with the overall disorder strength α . Experimental signatures.—
From the analysis of theeffective Hamiltonian H e (see Eq. (1)) that providesphenomenological insights about the quasi-particle ex-citations close to the Fermi energy, we now turn toidentifying more immediately observable characteristicsof the disorder-induced exceptional NH phase. Specif-ically, we start by comparing the spectral function A ( k, ω ) = − G R , av k,k ( ω )]) for the exceptional phase(see Fig. 2b) to the conventional phase in (see Fig. 2c).Most notably, the spectral function of the exceptional phase peaks at zero energy along the Fermi arcs (cf.Fig. 1c), whereas the spectral function of the conven-tional phase almost vanishes at zero energy and ex-hibits two ridges to the right and left, which is indica-tive of the spectral gap in Fig. 2a. We note that theexceptional phase may also readily be distinguished fromthe disorder-free system which is hallmarked by band-crossings at isolated points rather than extended Fermi-arcs. FIG. 3: Energy-dependent two-terminal conductance of a dis-ordered system (cf. Eqs. (2-3)) with N = 400 sites in the ex-ceptional and conventional phase (see plot legend). The resultis averaged over 100 disorder realizations. The parameters inthe exceptional phase are α = 0 . s = 1 . v = − . i and inthe ordinary phase α = 0 . s = 1, v = − .
5. Inset: Ampli-tude of the zero-energy conductance peak (averaged over theenergy interval [-0.015, 0.015]) in the exceptional phase as afunction of system size in a semi-log plot along with an expo-nential fit σ ( l ) = σ e − ln(2) l/l with σ = 0 . e /h , l = 147 . As a second experimental signature, we study thelinear-response conductance in a simple two-terminalmesoscopic quantum transport setting [42]. The ther-mal reservoirs (leads) attached to both ends of the sys-tem are modeled with the same dispersion as the un-perturbed Hamiltonian H . In Fig. ?? , we compare thetransmission as a function of energy for the exceptionaland the gapped phase through a system with 400 sitesat disorder strength α = 0 .
3. The exceptional phaseshows a pronounced peak around zero energy whereasthe gapped phase exhibits a strong suppression of trans-port in that energy region. This constitutes a quali-tative difference between the two inequivalent disorder-induced NH phases occurring in our model system. Yet,the transport properties of the exceptional phase are dis-tinct from the metallic disorder-free phase that wouldexhibit a conductance of 2 e /h that does not decay withsystem size. By contrast, due to the small but finiteimaginary part of the zero-energy states constituting theFermi-arcs (cf. Fig. 1c), the zero-energy conductance ofthe exceptional phase exhibits a slow exponential decaywith system size (see inset of Fig. ?? ). The peak am-plitude for each system length is taken as the average ofthe transmission over the energy interval [-0.015, 0.015].An exponential fit reveals a decay length of about 150sites. Our transport simulations are performed using theKwant library [44]. General analysis of disorder-induced EPs. –
Our un-perturbed model Hamiltonian H (see Eq. (2)) featurestwo band-crossing points at the momenta k = ± π/
2. It isnatural to ask whether this represents a minimal settingor whether it is also possible to split a single nodal pointinto a pair of EPs. We now derive on general groundsthat indeed at least two nodal points are required to ob-tain EPs from leading order disorder scattering in one-dimensional systems. This result is independent of themicroscopic details of H , especially its number of bands.Our analysis is based on perturbation theory in FBA, andqualitatively agrees with all our data from exact numer-ical simulations of two banded model systems.The mainsteps of our argumentation are as follows.We first derive an analytical expression for the anti-Hermitian (AH) part of the self-energy in FBA. In thecase of disorder with random amplitudes, the self-energycorrection in FBA generally takes the formΣ( k, ω ) = (cid:104) a (cid:105) f V k,k + 1 N (cid:104) a (cid:105) f (cid:88) p V k,p G R, p ( ω ) V p,k , (6)where (cid:104) ... (cid:105) f denotes the expectation value with respectto the probability distribution f of the amplitudes { a j } , V k,p the matrix-valued disorder scattering vertex in recip-rocal space, and G R, p ( ω ) the free retarded Green’s func-tion. As any matrix, the self-energy can be uniquely de-composed into a Hermitian and an AH part as Σ( k, ω ) =Σ H ( k, ω )+Σ AH ( k, ω ). In the continuum limit of Eq. (13),the AH part can be written as [43]Σ AH ( k, ω ) = − i (cid:104) a (cid:105) f n (cid:88) m =1 (cid:88) k m ∈ K m n (cid:89) j =1 j (cid:54) = m ω − E j ( k m ( ω )) × | E (cid:48) m ( k m ( ω )) | V k,k m ( ω ) adj[ ω − H ( k m ( ω ))] V k m ( ω ) ,k , (7)where K m is the set of momenta at which the band E m ( k )of the the free Hamiltonian H ( k ) intersects the energy ω , n the number of bands, and adj[...] denotes the ad-jugate matrix. Roughly speaking, this equation tells usthat there is a term contributing to Σ AH ( k, ω ) for eachmomentum at which a band crosses the energy ω .Using this intuition, we now investigate a system witha single Dirac point at zero energy. We assume thattwo bands E l and E l +1 cross at some momentum k t with a slope v = | E (cid:48) l ( k t ) | = | E (cid:48) l +1 ( k t ) | , where the lat-ter equality sign follows from chiral symmetry. Wethen take the limit ω → E l ( k t + q ) = − vq , E l +1 ( k t + q ) = vq and calculateΣ AH ( k, ω = 0) = lim q → Σ AH ( k, ω = vq ). In the vicin-ity of the nodal point, we can effectively describe the system as a two-band model by projecting it onto thesubspace spanned by the eigenstates of the two crossingbands | E l ( k ) (cid:105) and | E l +1 ( k ) (cid:105) . Marking the projected op-erators by a tilde, the linearized effective model reads as (cid:101) H ( k t − q ) = vqσ z . The projection of the self-energydirectly at the nodal point k t is then found to take theform (cid:101) Σ AH ( k t , ω = 0) = − i v (cid:104) a (cid:105) f ( (cid:101) V k t ,k t ) . (8)The matrix Γ that induces the chiral symmetry byΓ H ( k )Γ − = − H ( k ) can be projected onto the sub-space as well. We find that (cid:101) Γ( k t ) (cid:101) V k t ,k t ( (cid:101) Γ( k t )) − = − (cid:101) V k t ,k t , which implies that (cid:101) V k t ,k t cannot contain σ ifwe represent it as (cid:101) V k t ,k t = d σ + d · σ . In conclusion,( (cid:101) V k t ,k t ) only contains σ and so does (cid:101) Σ AH ( k t , ω = 0)according to Eq. (19).From this projected two-banded form, we can easily seethat the occurrence of EPs is impossible in this setting.If the nodal point of H was to be split into a pair of EPs,they would have to be connected by a Fermi-arc with apurely imaginary energy gap. As discussed above, thespectrum of a Matrix d σ + ( d R + i d I ) · σ is given by E ± = d ± (cid:112) d R − d I + 2 i d R · d I , and hence d R · d I = 0as well as d R − d I < (cid:101) H contains σ z , (cid:101) Σ AH ( k t , ω = 0) would thenhave to contain iσ x or iσ y and no iσ z to create a Fermi-arc around momentum k t in the spectrum of (cid:101) H e ( k ) = (cid:101) H ( k ) + (cid:101) Σ( k, ω = 0). As we just saw, (cid:101) Σ AH ( k t , ω = 0) hasa trivial matrix structure, containing only iσ and thusrendering a Fermi arc impossible.To sum up, the AH self-energy contribution from dis-order cannot lead to any finite splitting of a single nodalpoint into a pair of EPs within FBA. However, we stressthat if a second nodal point is present, a scattering pro-cess between the two points can split them into four EPs[43], as is the case in our model (2-3). There, for weakto moderate disorder strengths, the resulting exceptionalNH phase is correctly captured within FBA up to smallquantitative deviations (see Fig. 1c). Concluding discussion.—
In this work, we haveexplored the possibility of EPs in disordered one-dimensional systems. To this end, we have presented atwo-banded model with two nodal points that can enteran exceptional NH phase through scattering between thenodal points, given a suitable chosen random disorderwith up to nearest neighbor terms. The spectral prop-erties and transport capabilities of the exceptional NHphase are studied and compared to a conventional NHphase, where the most striking feature of the exceptionalphase is an enhanced conductance for energies close tothe EPs. Finally, a main finding that is valid beyond thestudied model system is that a single nodal point cannotsplit into EPs by potential scattering to leading order.To put our findings into a broader context, we wouldlike to stress a key difference to interaction-induced ex-ceptional NH phases, where EPs result from inter-particlescattering rather than potential scattering [34]. There, afinite life-time of quasi-particles that is closely connectedto an anti-Hermitian part of the self-energy at the Fermi-surface in thermal equilibrium typically requires finiteTemperature. By contrast, all calculations discussed inthis work were performed at zero temperature, thus high-lighting that disorder-induced NH physics does not relyon the phase-space for scattering processes provided bythermal excitations.
Acknowledgments.—
We would like to thank EmilBergholtz for discussions. We acknowledge financialsupport from the German Research Foundation (DFG)through the Collaborative Research Centre SFB 1143,the Cluster of Excellence ct.qmat, and the DFG Project419241108. Our numerical calculations were performedon resources at the TU Dresden Center for InformationServices and High Performance Computing (ZIH). ∗ Electronic address: [email protected][1] M. Z. Hasan and C. L. Kane,
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Supplementary Material for “Exceptional Non-Hermitian Phases in Disordered Quantum Wires”
Here, we provide the details on the claims and arguments in the main text. First, we present the derivation of theperturbation series building up on chapter 12 of [S1]. Second, the derivation of the general analytical expression forthe anti-Hermitian (AH) part of the self-energy in first Born approximation (FBA) is given. Third, we elaborate onthe application to a model with a single nodal point and the projection of the AH part of the self-energy as well asthe chiral symmetry onto the subspace of the two touching bands. Finally, we argue how scattering between twonodal points can lead to EPs and why the correlation of the on-site (OS) and nearest-neighbour (NN) terms in therandom disorder is necessary.
Perturbation Theory
In this section we will use imaginary timeor rather imaginary frequency iω to shorten the notation. For the finalresult we will replace iω → ω + iη . Consider a system consisting of a simple free Hamiltonian H plus some perturbation V , i.e. H = H + V . If H has been solved and the free retarded Green’s function (GF) G ( iω ) = [ iω − H ] − infrequency space is known, a peturbation series for the full retarded GF G ( iω ) = [ iω − H ] − = [ iω − H − V ] − arises by self-inserting the Dyson equation as G ( iω ) = G ( iω ) + G ( iω ) V G ( iω ) + G ( iω ) V G ( iω ) V G ( iω ) + ... (9)If H describes a translationally invariant tight-binding model with n internal degrees of freedom on each site, thefree GF G k,k (cid:48) ( iω ) = δ k,k (cid:48) G ( k, iω ) becomes block-diagonal in the Bloch basis of H . The blocks of the free GF read G ( k, iω ) = [ iω − H ( k )] − , where H ( k ) is the n x n -Bloch-Hamiltonian.In our case, the perturbation consists of the same type of impurity at each site but with a random amplitude a j drawn from a normalized probability distribution f ( a ), so the impurity part of the Hamiltonian in second quantizationreads V = N site (cid:88) j =1 a j (cid:20)
12 Ψ † j V OS Ψ j + Ψ † j +1 V NN Ψ j + Ψ † j +2 V NNN Ψ j + ... + h . c . (cid:21) = N site (cid:88) j =1 (cid:88) k,k (cid:48) a j e − ij ( k − k (cid:48) ) N sites c † k (cid:104) V OS + ( V NN e − ik + V † NN e ik (cid:48) ) + ( V NNN e − i k + V † NNN e i k (cid:48) ) + ... (cid:105)(cid:124) (cid:123)(cid:122) (cid:125) V k,k (cid:48) c k (cid:48) . (10)The Ψ † j are n -spinors of creators for on-site states and the c † k = √ N site (cid:80) j e ijk Ψ † j are n -spinors of creators for theBloch-states. The series from Eq. (9) leads to an expression for the blocks of the full GF G k,k (cid:48) ( iω ) = δ k,k (cid:48) G ( k, iω ) + N site (cid:88) j =1 G ( k, iω ) a j e − ij ( k − k (cid:48) ) N sites V k,k (cid:48) G ( k (cid:48) , iω )+ (cid:88) q N site (cid:88) j ,j =1 G ( k, iω ) a j e − ij ( k − q ) N sites V k,q G ( q, iω ) a j e − ij ( q − k (cid:48) ) N sites V q,k (cid:48) G ( k (cid:48) , iω ) + ... To obtain the effective Hamiltonian, this expression is averaged over the impurity amplitudes { a j } by calculating G avk,k (cid:48) ( iω ) = (cid:104) G k,k (cid:48) ( iω ) (cid:105) f = (cid:90) d a f ( a ) (cid:90) d a f ( a ) ... (cid:90) d a N site f ( a N site ) G k,k (cid:48) ( iω ) . (11)In Eq. (11) we encounter terms of the form (cid:104) (cid:80) N site j ,...j m =1 a j a j ...a j m e (cid:80) ml =1 q l j l (cid:105) f . We can group the sums into thosewhere all scattering vectors q ∈ Q = { q , q , ..., q m } are connected to one, two, three and so on impurities. Thenotation | Q r | simply indicates the number of elements in the subset Q r ⊂ Q . (cid:104) N site (cid:88) j ,...j m =1 a j a j ...a j m e (cid:80) ml =1 q l j l (cid:105) f = (cid:104) N site (cid:88) h =1 ( a h ) m e (cid:80) q ∈ Q qh (cid:105) f + (cid:104) (cid:88) ∪ r =1 Q r = Q N site (cid:88) h =1 N site (cid:88) h =1 h (cid:54) = h ( a h ) | Q | ( a h ) | Q | e (cid:80) q ∈ Q1 q h e (cid:80) q ∈ Q2 q h (cid:105) f + (cid:104) (cid:88) ∪ r =1 Q r = Q N site (cid:88) h =1 N site (cid:88) h =1 h (cid:54) = h N site (cid:88) h =1 h (cid:54) = h ,h ( a h ) | Q | ( a h ) | Q | ( a h ) | Q | × e (cid:80) q ∈ Q1 q h e (cid:80) q ∈ Q2 q h e (cid:80) q ∈ Q3 q h (cid:105) f + ... Now we need to introduce a small error of the order N site by letting the h sums run unrestricted, e.g. (cid:80) N site h =1 h (cid:54) = h → (cid:80) N site h =1 .The average doesn’t act on the exponentials and the distribution of the amplitudes is uncorrelated, so we can pull theaverage of the amplitudes out of the sums. Performing the sums gives a delta function for all momenta connected tothe same impurity and we arrive at (cid:104) N site (cid:88) j ,...j m =1 a j a j ...a j m e (cid:80) ml =1 q l j l (cid:105) f = N site (cid:104) a m (cid:105) f δ , (cid:80) q ∈ Q q + ( N site ) (cid:88) ∪ r =1 Q r = Q δ , (cid:80) q ∈ Q1 q (cid:104) a | Q | (cid:105) f (cid:104) a | Q | (cid:105) f δ , (cid:80) q ∈ Q2 q + ( N site ) (cid:88) ∪ r =1 Q r = Q (cid:104) a | Q | (cid:105) f (cid:104) a | Q | (cid:105) f (cid:104) a | Q | (cid:105) f δ , (cid:80) q ∈ Q1 q δ , (cid:80) q ∈ Q2 q δ , (cid:80) q ∈ Q3 q + ... Bearing this result in mind, we can express Eq. (11) in terms of Feynman diagrams. It is given by the sum over alltopologically different diagrams of the form G avk,k ( iω ) = (cid:104) a (cid:105) f V k,k k + (cid:104) a (cid:105) f V k,k kk + (cid:104) a (cid:105) f V k,q V q,k qq − k k − qkk + (cid:104) a (cid:105) f V k,k (cid:104) a (cid:105) f V k,k k kk + ..., (12)which obey simple Feynman rules. The solid-lined propagators with momentum k denote a matrix-valued free GF G ( k, iω ). The dashed propagators denote an also matrix-valued factor V q L ,q R , where q L is the momentum leavingthe vertex of the dashed and the two solid propagators to the left and q R the momentum joining it from the right.A vertex of m dashed propagators obtains the m -th moment of the distribution (cid:104) a m (cid:105) f as a prefactor. The dashedpropagators formally carry the momentum q R − q L and all momenta joining a vertex of multiple dashed propagatorsadd up to zero. A sum N sites (cid:80) q over all momenta inside a closed loop is implied.Now the series can be rearranged by defining the self-energy Σ( k, iω ) as the sum of all irreducible diagrams thatcannot be separated by cutting a single propagator G avk,k ( iω ) = k + Σ kk + Σ Σ kkk + ... = G ( k, iω ) + G ( k, iω )Σ( k, iω ) (cid:2) G ( k, iω ) + G ( k, iω )Σ( k, iω ) G ( k, iω ) + ... (cid:3) = G ( k, iω ) + G ( k, iω )Σ( k, iω ) (cid:2) G avk,k ( iω ) (cid:3) . Because the free Green’s function is given by G ( k, iω ) = [ iω − H ( k )] − , we finally obtain G avk,k ( iω ) = [ − ( H ( k ) + Σ( k, iω ))] − . As was said, the self-energy consists of all irreducible diagrams and assumes the formΣ( k, iω ) = (cid:104) a (cid:105) f V k,k (cid:104) a (cid:105) f V k,q V q,k qq − k k − q + (cid:104) a (cid:105) f (cid:104) a (cid:105) f V k,q V q,q V q,k qq − k k − qq + ... In FBA we only consider the first two diagrams of the series. The real-time frequency space GF and self-energy canbe found through analytical continuation to the real axis, which simply amounts to replacing iω → ω + iη with someinfinitesimal regularization η = 0 + . An Analytical Expression for the AH Part of the Self-Energy
The disorder-induced self-energy correction in FBA assumes the formΣ( k, ω + iη ) = (cid:104) a (cid:105) f V k,k + 1 N site (cid:104) a (cid:105) f (cid:88) q V k,q G ( ω + iη, q ) V q,k , = (cid:104) a (cid:105) f V k,k + (cid:104) a (cid:105) f π (cid:90) π − π d qV k,q G ( ω + iη, q ) V q,k , (13)where we have assumed the continuum limit N sites (cid:80) q → π (cid:82) π − π d q in the second line. Since hermiticity requires that V † k,q = V q,k , we have V † k,k = V k,k and ( V k,q G ( ω + iη, q ) V q,k ) † = V k,q G ( ω + iη, q ) † V q,k , so the self-energy can onlyobtain an AH part if the free retarded Green’s function G ( ω + iη, q ) is non-Hermitian. Because of that, we willdecompose G into a Hermitian and an AH part as the first step.Labeling the adjugate matrix as adj[...] and the eigenenergies/bands of H ( q ) as { E j ( q ) } , the Green’s function canbe written as G ( ω + iη, q ) = ( ( ω + iη ) − H ( q )) − = 1det[ ( ω + iη ) − H ( q )] adj[ ( ω + iη ) − H ( q )]= 1 n (cid:89) j =1 ( ω + iη − E j ( q )) adj[ ( ω + iη ) − H ( q )] . η → = n (cid:89) j =1 (cid:18) ω − E j ( q ) − iπδ ( ω − E j ( q )) (cid:19) adj[ ω − H ( q )] . (14)In the last line we used the Dirac identity lim η → iη + f ( x ) = f ( x ) − iπδ ( f ( x )). If we now consider values of ω suchthat two bands never cross it at the same momentum q , all products containing more than one δ -function from Eq.(14) vanish. We are left with G ( ω, q ) = n (cid:89) j =1 (cid:18) ω − E j ( q ) (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) / det[ ω − H ( q )] − iπ n (cid:88) m =1 δ ( ω − E m ( q )) n (cid:89) j =1 j (cid:54) = m (cid:18) ω − E j ( q ) (cid:19) adj[ ω − H ( q )]= ( ω − H ( k )) − − iπ n (cid:88) m =1 δ ( ω − E m ( k )) n (cid:89) j =1 j (cid:54) = m (cid:18) ω − E j ( k ) (cid:19) adj[ ω − H ( q )] . (15)The free Green’s function has been decomposed into a Hermitian and an AH part. By substituting δ ( ω − E m ( q )) = (cid:80) k m ∈ K m δ ( q − k m ( ω )) | E (cid:48) m ( k m ( ω )) | , with K m denoting the set of momenta at which the band E m ( q ) intersects ω , and plucking Eq.(15) into Eq. (13), the AH self-energy contribution is readily obtained asΣ AH ( k, ω ) = − i (cid:104) a (cid:105) f n (cid:88) m =1 (cid:88) k m ∈ K m n (cid:89) j =1 j (cid:54) = m ω − E j ( k m ( ω )) 1 | E (cid:48) m ( k m ( ω )) | V k,k m ( ω ) adj[ ω − H ( k m ( ω ))] V k m ( ω ) ,k . (16)0 Application to a Single Nodal Point and Projection Onto the Subspace
FIG. 4: Linearization around k t . Now we will apply the findings from the previous sec-tion to a model with a single nodal point at ω = 0. Weassume that two bands E l and E l +1 cross with a slope v = | E (cid:48) l ( k t ) | = | E (cid:48) l +1 ( k t ) | at some momentum k t . As weare interested in an effective Hamiltonian or respectively theself-energy correction at ω = 0, we take the limit ω → E l ( k t + q ) = − vq , E l +1 ( k t + q ) = vq and calculating Σ AH ( k, ω = 0) = lim q → Σ AH ( k, ω = vq ).The bands intersect ω = qv at k t ± q , which is schematicallydepicted in Fig. 4.Under these assumptions, Eq. (16) tells us thatΣ AH ( k, qv ) = − i (cid:104) a (cid:105) f n (cid:89) j =1 j (cid:54) = l,l +1 qv − E j ( k t − q ) qv − E l +1 ( k t − q ) 1 | E (cid:48) l ( k t − q ) | V k, ( k t − q ) adj[ qv − H ( k t − q )] V ( k t − q ) ,k − i (cid:104) a (cid:105) f n (cid:89) j =1 j (cid:54) = l,l +1 qv − E j ( k t + q ) qv − E l ( k t + q ) 1 | E (cid:48) l +1 ( k t + q ) | V k, ( k t + q ) adj[ qv − H ( k t + q )] V ( k t + q ) ,k = − i (cid:104) a (cid:105) f n (cid:89) j =1 j (cid:54) = l,l +1 qv − E j ( k t − q ) qv v V k, ( k t − q ) adj[ qv − H ( k t − q )] V ( k t − q ) ,k − i (cid:104) a (cid:105) f n (cid:89) j =1 j (cid:54) = l,l +1 qv − E j ( k t + q ) qv v V k, ( k t + q ) adj[ qv − H ( k t + q )] V ( k t + q ) ,k . From this, we see thatΣ AH ( k,
0) = lim q → Σ AH ( k, vq )= − i v (cid:104) a (cid:105) f n (cid:89) j =1 j (cid:54) = l,l +1 − E j ( k t ) V k,k t lim q → (cid:18) qv adj[ qv − H ( k t − q )])+ 12 qv adj[ qv − H ( k t + q )] (cid:19) V k t ,k . (17)Next, we use the unitary transformation U k to the eigenbasis of H ( k ) and insert an identity of the form U † ( k t − q ) U ( k t − q ) into the first adjugate from Eq. (17). Further we use that adj[ AB ] = adj[ B ]adj[ A ] and obtain12 qv adj[ qv − H ( k t − q )] = 12 qv adj[ U † ( k t − q ) U ( k t − q ) ( qv − H ( k t − q )) U † ( k t − q ) U ( k t − q ) ]= 12 qv adj[ U ( k t − q ) ]adj[ qv − U ( k t − q ) H ( k t − q ) U † ( k t − q ) ]adj[ U † ( k t − q ) ] . q it follows trivially that U ( k t − q ) H ( k t − q ) U † ( k t − q ) = E ( k t − q ) . . . qv − qv . . . E n ( k t − q ) , where the red box encloses the matrix elements belonging to the subspace spanned by the two touching bands E l and E l +1 . Thusly, qv adj[ qv − H ( k t − q )] takes the form12 qv adj[ U ( k t − q ) ]adj ( qv − E ( k t − q )) . . . qv . . . ( qv − E n ( k t − q )) adj[ U † ( k t − q ) ]= 12 qv adj[ U ( k t − q ) ] n (cid:89) j =1 j (cid:54) = l,l +1 ( qv − E j ( k t − q )) qv . . . 0 adj[ U † ( k t − q ) ]=adj[ U ( k t − q ) ] n (cid:89) j =1 j (cid:54) = l,l +1 ( qv − E j ( k t − q )) . . . 0 adj[ U † ( k t − q ) ] . The second line becomes obvious if we remember the element-wise definition of the adjugate matrix as(adj[ A ]) i,j = ( − i + j M j,i , where the minor M j,i of A denotes the determinant of the matrix that is obtainedby deleting row j and column i of A.The second term qv adj[ qv − H ( k t + q )] in Eq. (17) can be treated in the same fashion to read=adj[ U ( k t + q ) ] n (cid:89) j =1 j (cid:54) = l,l +1 ( qv − E j ( k t + q )) . . . 0 adj[ U † ( k t + q ) ] . With these results, Eq. (17) becomes2Σ AH ( k,
0) = − i v (cid:104) a (cid:105) f V k,k t adj[ U k t ] . . . 0 adj[ U † k t ] V k t ,k . = − i v (cid:104) a (cid:105) f V k,k t U † k t . . . 0 U k t V k t ,k , (18)which can be used to calculate the projection (cid:101) Σ AH ( k t ,
0) of the AH self-energy contribution onto the subspace of thecrossing bands at k t . Let (cid:101) V k t ,k t denote the subspace-projection of V k t ,k t such that U k t V k t ,k t U † k t = . . . . . . ... ...... ... . . . . . . . (cid:101) V k t ,k t Then Eq. (18) yields U k t Σ AH ( k t , U † k t = − i v (cid:104) a (cid:105) f U k t V k t ,k t U † k t . . . 0 U k t V k t ,k t U † k t = − i v (cid:104) a (cid:105) f . . . . . . ... ...... ... . . . . . . , (cid:101) V k t ,k t and thus we arrive at the expression used in the main text (cid:101) Σ AH ( k t ,
0) = − i v (cid:104) a (cid:105) f ( (cid:101) V k t ,k t ) . (19)Lastly in this section, we elaborate on the projection of the chiral symmetry onto the subspace. The chiralsymmetry in general indicates that there exists some matrix Γ acting on the internal degrees of freedom suchthat ⊗ Γ( H + V ) ⊗ Γ − = − ( H + V ), where is the N site × N site identity matrix. This implies thatΓ H ( k )Γ − = − H ( k ) and Γ V k,k Γ − = − V k,k .Due to the chiral symmetry, the eigenvalues come in positive and negative pairs, which we denote here by E + j ( k )and E − j ( k ) = − E + j ( k ). We can transform H ( k ) into its eigenbasis, where we sort the eigenvectors in pairs of positiveand negative eigenvalue. The transformed Hamiltonian is denoted by a hat and readsˆ H ( k ) = diag[ E +1 ( k ) , E − ( k ) , ..., E + l ( k ) , E − l ( k ) , ..., E + n ( k ) , E − n ( k )]3The pair E + l , E − l is again the one that touches at zero energy. Next, we consider the transformation of Γ into thisbasis, denoted by ˆΓ( k ) . Since we only performed a unitary basis transformation, ˆ H ( k )ˆΓ( k ) = − ˆΓ( k ) ˆ H ( k ) stillholds and thus ˆ H ( k )ˆΓ( k ) | E + j ( k ) (cid:105) = − ˆΓ( k ) ˆ H ( k ) | E + j ( k ) (cid:105) = − E + j ( k )ˆΓ( k ) | E + j ( k ) (cid:105) for any j = 1 , ..., n . Apparently,ˆΓ( k ) | E + j ( k ) (cid:105) is an eigenstate of ˆ H ( k ) with eigenvalue − E + j ( k ). As long as the eigenstates are non-degenerate, thereis only one such eigenstate, so ˆΓ( k ) | E + j ( k ) (cid:105) = λ − j | E − j ( k ) (cid:105) with some λ − j ∈ C and similarly ˆΓ( k ) | E − j ( k ) (cid:105) = λ + j | E + j ( k ) (cid:105) with some λ + j ∈ C . In our basis, the eigenstates of ˆ H ( k ) are simply unit vectors and the ± -pairs are situated ontop of each other, for example | E +1 ( k ) (cid:105) = (1 , , , ..., T , | E − ( k ) (cid:105) = (0 , , , ..., T and so on. Thereby, ˆΓ( k ) must be2 × γ j,x ( k ) σ x + γ j,y ( k ) σ y for each energy pair. We writeˆΓ( k ) = diag[ γ ,x ( k ) σ x + γ ,y ( k ) σ y , ..., γ l,x ( k ) σ x + γ l,y ( k ) σ y (cid:124) (cid:123)(cid:122) (cid:125) (cid:101) Γ( k ) , ..., γ n,x ( k ) σ x + γ n,y ( k ) σ y ] . (20)The block belonging to the l -th energy pair (the one that touches at zero energy) is exactly the projection of Γ ontothe subspace, which we consistently denote by (cid:101) Γ( k ). Up to now, we only considered values of k with no degeneracyin the spectrum. However, for a physically sensible Hamiltonian H ( k ) degeneracies should only appear at isolatedpoints in the parameter space, so ˆΓ( k ) cannot deviate from the block form of Eq. (20) at these points due tocontinuity. Of course, ˆΓ − ( k ) is also block-diagonal.Now we can show that (cid:101) Γ( k ) (cid:101) V k,k ( (cid:101) Γ( k )) − = − (cid:101) V k,k for any k . If we transform the scattering vertex V k,k into theeigenbasis of H ( k ), the chiral symmetry remains and ˆΓ( k ) ˆ V k,k ˆΓ − ( k ) = − ˆ V k,k . We write this in terms of matricesand mark the blocks belonging to the subspace of | E ± l ( k ) (cid:105) by a red box.ˆΓ( k ) ˆ V k,k ˆΓ − ( k ) = . . . ... ...... ... . . . . . . . . . ... ...... ... . . . . . . . . . ... ...... ... . . . = . . . . . . ... ...... ... . . . . . . = − ˆ V k,k = − . . . . . . ... ...... ... . . . . . . . (cid:101) Γ( k ) (cid:101) V k,k ( (cid:101) Γ( k )) − (cid:101) Γ( k ) (cid:101) V k,k ( (cid:101) Γ( k )) − (cid:101) V k,k Comparing the first and second line shows that indeed (cid:101) Γ( k ) (cid:101) V k,k ( (cid:101) Γ( k )) − = − (cid:101) V k,k . Scattering Between Two Nodal Points and Necessity of the Correlation Between OS- and NN-Terms in theRandom Disorder
Consider a general model H = H + V with two internal degrees of freedom and a chiral symmetry such that σ z Hσ z = − H , similar to the model studied in the main text. This implies the symmetry σ z H e ( k, w ) † σ z = − H e ( k, − w )for the effective Hamiltonian H e ( k, w ) = H ( k ) + Σ( k, w ) emerging from the averaged GF description, so σ z H † e ( k, σ z = − H e ( k,
0) and thus d x , d y ∈ R and d , d z ∈ i R if we parametrize H e ( k,
0) = d σ + d · σ .The spectrum of a Matrix d σ + ( d R + i d I ) σ with d R , d I ∈ R is given by E ± = d ± (cid:112) d R − d I + 2 i d R · d I and exceptional points occur if d R − d I = 0 and d R · d I = 0 is satisfied simultaneously while d R , d I (cid:54) = 0. Inconclusion, an iσ z contribution in the self-energy Σ( k,
0) is required to open an exceptional point in the spec-trum of H e ( k,
0) in our chirally symmetric model. Using Eq. (19), we already discussed in the main text thatsuch a term is impossible directly at the band touching point k t of a model with only a single nodal point atzero energy. However, if H ( k ) exhibits two nodal points at momenta k t and k t , where the bands cross withslopes v and v , a “scattering” process between the two points can create an iσ z term, as we will show in the following.4Due to the symmetry constraints, the impurity matrix elements from Eq. (10) take the form V k,k (cid:48) = α k,k (cid:48) σ x + β k,k (cid:48) σ y .We can apply Eq. (18) to both points separately and directly read off the result for the AH part of the self-energy inFBA Σ AH ( k,
0) = − i v (cid:104) a (cid:105) f V k,k t1 V k t1 ,k − i v (cid:104) a (cid:105) f V k,k t2 V k t2 ,k = (cid:18) − i v (cid:104) a (cid:105) f ( | α k,k t1 | + | β k,k t1 | ) − i v (cid:104) a (cid:105) f ( | α k,k t2 | + | β k,k t2 | ) (cid:19) σ (cid:18) iv (cid:104) a (cid:105) f (Im[ α k,k t1 β k t1 ,k ] + iv (cid:104) a (cid:105) f Im[ α k,k t2 β k t2 ,k ] (cid:19) σ z . Due to hermiticity, V † k,k (cid:48) = V k (cid:48) ,k must always hold true and thus the iσ z term created by the nodal point at k t vanishes in the self-energy Σ AH ( k t ,
0) at k t , since Im[ α k t1 ,k t1 β k t1 ,k t1 ] = 0. This agrees fully with the argumentspresented this far. On the other hand, the iσ z term created by the second nodal point k t can persist. If we assumeOS and NN impurities such that V k,k (cid:48) = (cid:16) V OS x + (cid:16) V NN x e − ik + V ∗ NN x e ik (cid:48) (cid:17)(cid:17) σ x + (cid:16) V OS y + (cid:16) V NN y e − ik + V ∗ NN y e ik (cid:48) (cid:17)(cid:17) σ y = (cid:16) V OS x + | V NN x | (cid:16) e − i ( k +∆ x ) + e i ( k (cid:48) +∆ x ) (cid:17)(cid:17) σ x + (cid:16) V OS y + | V NN y | (cid:16) e − i ( k +∆ y ) + e i ( k (cid:48) +∆ y ) (cid:17)(cid:17) σ y , we get Im[ α k t1 ,k t2 β k t2 ,k t1 ] = V OS x | V NN y | (sin ( k t + ∆ x ) − sin ( k t + ∆ x )) + V NN x V OS y (sin ( k t + ∆ x ) − sin ( k t + ∆ x )),which is non-zero in general. The contribution to the AH self-energy from the second nodal point can open an EP atthe first nodal point in k -space and vice versa, which is exactly what happens in our model from the main text.Finally, we discuss why the correlation in amplitude between the OS and NN terms is necessary. Suppose thatthere were two different kinds of impurities with independent random amplitudes { a I j } and { a II j } , such that V fromEq. (10) takes the form V = N site (cid:88) j =1 (cid:88) k,k (cid:48) e − ij ( k − k (cid:48) ) N sites c † k (cid:0) a I j V I k,k (cid:48) + a II j V II k,k (cid:48) (cid:1) c k (cid:48) . The perturbation theory can be generalized easily to this case. The perturbation series for the full GF remains thesame as in Eq. (9) and we average the blocks G k,k (cid:48) ( iω ) over the distributions f I and f II of the amplitudes similarlyto Eq. (11) G avk,k (cid:48) ( iω ) = (cid:104) G k,k (cid:48) ( iω ) (cid:105) f I ,f II = (cid:90) d a I1 d a II1 f I ( a I1 ) f II ( a II1 ) (cid:90) d a I2 d a II2 f I ( a I2 ) f II ( a II2 ) ... (cid:90) d a I N site d a II N site f I ( a I N site ) f II ( a II N site ) G k,k (cid:48) ( iω ) . This expression contains terms of the form (cid:104) (cid:80) N site j ,...j m =1 a I j a II j ...a II j m e (cid:80) ml =1 q l j l (cid:105) f I ,f II . Again, we treat these terms bygrouping them into terms where all scattering vectors q ∈ Q = { q , q , ..., q m } are connected to one, two, three and soon impurities. The variables m I , m II denote the total number of impurities of type I, II (so m I + m II = m ), and m I ,r , m II ,r denote the number of impurities of type I, II to which the momenta from the subset Q r ⊂ Q are connected.5 (cid:104) N site (cid:88) j ,...j m =1 a I j a II j ...a II j m e (cid:80) ml =1 q l j l (cid:105) f I ,f II = (cid:104) N site (cid:88) h =1 ( a I h ) m I ( a II h ) m II e (cid:80) q ∈ Q qh (cid:105) f I ,f II + (cid:104) (cid:88) ∪ r =1 Q r = Q N site (cid:88) h =1 N site (cid:88) h =1 h (cid:54) = h ( a I h ) m I , ( a II h ) m II , ( a I h ) m I , ( a II h ) m II , e (cid:80) q ∈ Q1 q h e (cid:80) q ∈ Q2 q h (cid:105) f + (cid:104) (cid:88) ∪ r =1 Q r = Q N site (cid:88) h =1 N site (cid:88) h =1 h (cid:54) = h N site (cid:88) h =1 h (cid:54) = h ,h ( a I h ) m I , ( a II h ) m II , ( a I h ) m I , ( a II h ) m II , ( a I h ) m I , ( a II h ) m II , × e (cid:80) q ∈ Q1 q h e (cid:80) q ∈ Q2 q h e (cid:80) q ∈ Q3 q h (cid:105) f + ... We introduce a small error of the order N site by letting the h sums run unrestricted, e.g. (cid:80) N site h =1 h (cid:54) = h → (cid:80) N site h =1 and thentake the average to obtain (cid:104) N site (cid:88) j ,...j m =1 a I j a II j ...a II j m e (cid:80) ml =1 q l j l (cid:105) f I ,f II = N site (cid:104) ( a I ) m I (cid:105) f I (cid:104) ( a II ) m II (cid:105) f II δ , (cid:80) q ∈ Q q + ( N site ) (cid:88) ∪ r =1 Q r = Q (cid:104) ( a I ) m I , (cid:105) f I (cid:104) ( a II ) m II , (cid:105) f II δ , (cid:80) q ∈ Q1 q × (cid:104) ( a I ) m I , (cid:105) f I (cid:104) ( a II ) m II , (cid:105) f II δ , (cid:80) q ∈ Q2 q + ( N site ) (cid:88) ∪ r =1 Q r = Q (cid:104) ( a I ) m I , (cid:105) f I (cid:104) ( a II ) m II , (cid:105) f II δ , (cid:80) q ∈ Q1 q (cid:104) ( a I ) m I , (cid:105) f I (cid:104) ( a II ) m II , (cid:105) f II × δ , (cid:80) q ∈ Q2 q (cid:104) ( a I ) m I , (cid:105) f I (cid:104) ( a II ) m II , (cid:105) f II δ , (cid:80) q ∈ Q3 q + ... This result can be used to devise a similar graphical representation of the averaged Green’s function as in Eq. (12).After resummation, the self-energy is obtained as the sum over all irreducible diagrams of the form6Σ( k, iω ) = (cid:104) a I (cid:105) f I V I k,k (cid:104) a II (cid:105) f II V II k,k (cid:104) a I (cid:105) f I (cid:104) a II (cid:105) f II V I k,k V II k,k qq − k k − q + (cid:104) a II (cid:105) f II (cid:104) a I (cid:105) f I V II k,k V I k,k qq − k k − q + (cid:104) ( a I ) (cid:105) f I V I k,k V I k,k qq − k k − q + (cid:104) ( a II ) (cid:105) f II V II k,k V II k,k qq − k k − q + (cid:104) ( a I ) (cid:105) f I (cid:104) a II (cid:105) f II V I k,k V II k,k V I k,k qq − k k − qq + ..., (21)which obey simple Feynman rules again. The solid-lined propagators with momentum k denote a matrix-valued freeGF G ( k, iω ). The dashed propagators carry an also matrix-valued factor V I q L ,q R or V II q L ,q R , where q L is the momentumleaving the vertex of the dashed and the two solid propagators to the left and q R the momentum joining it from theright. A vertex of m I dashed propagators with V I q L ,q R and m II dashed propagators with V II q L ,q R obtains a prefactor of (cid:104) ( a I ) m I (cid:105) f I (cid:104) ( a II ) m II (cid:105) f II . The dashed propagators formally carry the momentum q R − q L and all momenta joining avertex of multiple dashed propagators add up to zero. A sum N sites (cid:80) q over all momenta inside a closed loop is implied.It is only necessary to consider distributions f with vanishing first moment, as we can always redefine H = H + V = (cid:88) k c † k H ( k ) c k + N site (cid:88) j =1 (cid:88) k,k (cid:48) e − ij ( k − k (cid:48) ) N sites c † k (cid:0) a I j V I k,k (cid:48) + a II j V II k,k (cid:48) (cid:1) c k (cid:48) . = (cid:88) k c † k (cid:0) H ( k ) + (cid:104) a I (cid:105) f I V I k,k + (cid:104) a II (cid:105) f II V II k,k (cid:1)(cid:124) (cid:123)(cid:122) (cid:125) H new0 c k + N site (cid:88) j =1 (cid:88) k,k (cid:48) e − ij ( k − k (cid:48) ) N sites c † k (cid:0) a I j V I k,k (cid:48) + a II j V II k,k (cid:48) (cid:1) c k (cid:48) − (cid:88) k,k (cid:48) δ k,k (cid:48) c † k (cid:0) (cid:104) a I (cid:105) f I V I k,k (cid:48) + (cid:104) a II (cid:105) f II V II k,k (cid:48) (cid:1) c k (cid:48) = (cid:88) k c † k H new0 c k + N site (cid:88) j =1 (cid:88) k,k (cid:48) e − ij ( k − k (cid:48) ) N sites c † k (cid:0)(cid:0) a I j − (cid:104) a I (cid:105) f I (cid:1) V I k,k (cid:48) + (cid:0) a II j − (cid:104) a II (cid:105) f II (cid:1) V II k,k (cid:48) (cid:1) c k (cid:48) . Then, the self-energy from Eq. (21) is apparently given by the sum of the self energies emerging from the perturba-tions V I k,k (cid:48) and V II k,k (cid:48) alone plus mixed terms starting at fourth order ( ∝ (cid:104) ( a I ) (cid:105) f I (cid:104) ( a II ) (cid:105) f II ).As was shown in the first part of this section, we require OS and N N terms in a single perturbation to create an iσ z contribution in the FBA self-energy correction to our chirally symmetric two-banded model. In conclusion, if we7split this perturbation into two uncorrelated perturbations with only OS and only N N terms, an iσ z -term cannotemerge in FBA. We could only hope for EPs from higher-order effects, which is unlikely and beyond the scope ofthis analysis anyway. This also agrees with the results from the full numerics, where EPs could only be observed insystems with correlated OS and N N terms in the perturbation. ∗ Electronic address: [email protected][S1] H. Bruus and K. Flensberg,