Quasiparticle interference and resonant states in normal and superconducting line nodal semimetals
QQuasiparticle interference and resonant states in normal and superconducting linenodal semimetals
Chandan Setty and Philip W. Phillips
Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois, USA
Awadhesh Narayan
Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois, USA andMaterials Theory, ETH Zurich, Wolfgang-Pauli-Strasse 27, CH 8093 Zurich, Switzerland
We study impurity scattering in the normal and d -wave superconducting states of line nodalsemimetals and show that, due to additional scattering phase space available for impurities on thesurface, the quasiparticle interference pattern acquires an extended character instead of a discretecollection of delta function peaks. Moreover, using the T -matrix formalism, we demonstrate that theconventional behavior of a scalar impurity in a d -wave superconductor breaks down on the surfaceof a line nodal semimetal in the quasi flat band limit. Introduction:
A recent member to the class of topolog-ical states [1–3] of matter include line node semimetals,in which two bands are degenerate over an extended re-gion have gapless excitations [4]. A number of materialshave been proposed to exhibit such line node character-istics [5–14], a list which is growing remarkably rapidly.These proposals, in turn, have inspired numerous theo-retical studies of novel properties of this intriguing bandstructure [15–24].Inducing proximate superconductivity in topologicalstates presents an intriguing playground for exotic formsof superconducting matter [25–29]. Notably, high tem-perature proximity-induced superconductivity has beenrealized on canonical topological insulators bismuth se-lenide and bismuth telluride, using a d -wave cupratesuperconductor [30]. Recent reports of tip-induced su-perconductivity in point node semimetals are an excit-ing new development in exploration of such phenom-ena [31, 32].At the same time, quasiparticle interference has provedto be an important tool in establishing and character-izing the fingerprints of topological matter. Surfacestates of topological insulators have been imaged andtheir spin-momentum locking has been revealed usingscanning tunneling spectroscopy [33–38]. More recently,gapless topological phases of matter, Dirac and Weylsemimetals, have also been studied using scanning tun-neling microscopy, where signatures of Fermi arcs havebeen found [39–43].In this work, motivated by these advancements, weexplore the quasiparticle interference in normal andsuperconducting line node semimetals focusing on bothbulk and surface properties. We show that, unlikein conventional two dimensional metals where nodalsuperconductivity yields point nodes, the surface of aline nodal semimetal gives rise to line nodes. As aconsequence, due to the additional impurity scatteringphase space available within the area of the flat band,the quasiparticle interference pattern on the surface of a line nodal semimetal acquires an extended characterin the Brillouin zone instead of a collection of discretedelta function peaks. Additionally, using the T − matrixformalism, we examine the resonant state energy disper-sions of a single scalar impurity on the surface of a linenodal semimetal with d − wave pairing. Our calculationspoint to a momentum averaged Green function whichcontains a power law type contribution, in addition tothe logarithmic term usually found for nodal supercon-ducting quadratic bands. Such a contribution, unlikethe case of two dimensional electrons with quadraticbands, admits two different under-damped solutions tothe resonant state energies: the first is a broad, lowintensity mode located closer to the continuum thatdisperses toward zero energy in the unitary limit; thesecond is a more intense, sharp, lower energy mode thatdisperses away from zero energy. We argue that firstmode may be challenging to access experimentally whilethe second can be more readily observed. Our resultsalso signal a destruction of zero bias tunneling peaks(in the unitarity limit) on the surface of a line nodalsemimetal with d − wave pairing and could, thereby,motivate future scanning tunneling experiments on linenode semimetals. Toy model for a line nodal semimetal:
To beginwith, we briefly describe a slightly modified version ofthe tight binding toy model put forth in Ref. [44] andstudy some of its bulk and surface properties. Equippedwith a basic understanding of these properties, we go onto study the impurity induced quasiparticle interferencepatterns in both the normal and superconducting statesof the line node semimetal. We take our tight bindingHamiltonian on a square lattice to be of the form (we usethe same notation as in Ref. [44] to make the comparisonexplicit) a r X i v : . [ c ond - m a t . s up r- c on ] A p r FIG. 1. Zero energy local density of states for the line nodesemimetal. Top row: (Left) Bulk and (right) Surface with-out superconductivity. Bottom row: (Left) Bulk and (right)Surface with d − wave superconductivity. ˆ H ( (cid:126)k ) = (cid:34) g ( (cid:126)k (cid:107) ) ν (cid:48)(cid:107) a τ z + (cid:18) ¯ gν (cid:48) a + V (cid:19) τ (cid:35) σ + ν z c sin( ck z ) τ y σ + ˆ H z (1)ˆ H z = (1 − cos( ck z )) ( Z τ τ z σ + Z τ σ ) (2)where τ i , σ i are the Pauli matrices in the orbital andspin basis respectively and a, c are the in-plane andout-of-plane lattice constants. The function g ( (cid:126)k (cid:107) )is defined as g ( (cid:126)k (cid:107) ) = 1 + cos( ak ) − cos( ak x ) − cos( ak y ). We set the parameters to the following val-ues ( Z τ , Z , a, c, k , V )=(0.287 eV, 0.0 eV, 8.26 ˚A, 6.84˚A, 0.206 ˚A − , 0.043 eV) and define ν (cid:48)(cid:107) = ν (cid:107) ak sin( ak ) , ν (cid:48) = ν ak sin( ak ) , ¯ g = 1 + cos( ak ) with ( ν , ν (cid:107) , ν z )=(-0.993 eV˚A ,4.34 eV˚A , 2.5 eV˚A). To explore the flat band surfacestates, we use open boundary conditions along one of thedirections, namely the z axis.In Fig.1, we plot the local density of states (LDOS) inthe bulk and surface of the model described in Eq. 1 atzero energy. The top row shows the LDOS in the bulk(left) and surface (right) in the normal state. In the bulk,there is a continuous contour of Dirac nodes at zero en-ergy which acquires a toroidal structure at non-zero fre-quencies (see Supplemental Material [45]). However, on the surface, a nearly flat band is found which “fills” thebulk contour, the so-called “drumhead states”. In ourdiscussions, we will be most interested in the flat bandlimit of the model where the surface band dispersion isthe smallest energy scale in the problem. At non-zero en-ergy, the flat-band behavior on the surface is absent andthere is little qualitative distinction between the surfaceand the bulk [45]. It is also worthwhile to note that it ispossible to augment the Hamiltonian in Eq. 1 to includeterms which smoothly interpolate between a line nodalsemimetal and a Weyl semimetal (see Supplemental Ma-terial [45]).We now include proximity-induced superconductivityin our setup (see [45] for the details). For the rest ofthe paper, we will model the superconductor in the evenfrequency, orbital and spin singlet pairing channel. Not-ing that fully gapped s − wave superconductors are robustand featureless to scalar impurities due to Anderson’stheorem, interesting impurity effects start to appear withnodal d − wave pairing, which will be the main focus ofthis work. The effect of a d − wave form of the gap onthe bulk and surface LDOS at zero frequency is shownin Fig. 1 (bottom row). The intensity in the bulk (Fig. 1bottom left) is now reduced to four nodal spots corre-sponding to the zeros of the d − wave gap function. Thesenodal points are marked by arrows denoted by (cid:126)Q and (cid:126)Q (cid:48) .However, on the two dimensional surface (Fig. 1 bottomright) a d − wave gap gives rise to line nodes instead ofpoint nodes − a novel feature of the “drumhead” surfacestate that does not occur in usual two dimensional su-perconductors. This would lead to an anomalous scalingof measurable quantities, like the specific heat, leadingto a striking difference which could be readily tested infuture experiments.At non-zero energies, the four nodal points thatexisted in the bulk become slightly extended in mo-mentum space (see Supplemental Material [45]) alongthe diagonals of the Brillouin zone due to the toroidalFermi surface. On the surface, however, when theinduced superconducting gap (∆ µν ) is larger than thechosen energy ( ω = 0 . µν > . except those statesalong the Brillouin zone diagonal. These states thenconverge down to the Fermi level to form line nodesat zero energy, while at non zero energy (less than themaximum value of superconducting gap) they form a“petal” like structure. Impurity scattering:
With the analysis of LDOS inthe normal and superconducting phases at hand, weare now in a position to examine the effect of impurityscattering on line node semimetals. In the presence ofimpurities, the electrons in states with high density atthe same energy can scatter between these states. Thisgives rise to interference patterns which can be measuredusing scanning tunneling methods. Joint density of
FIG. 2. Joint density of states at ω = 0 (panels a,b,c,d) and ω = 0 . d − wave superconducting state. Panels a,e,c and g correspond to thebulk JDOS and b,f,d,h correspond to the surface JDOS. The q x and q y axes on the left half of the figure have the same rangeas panels on the right half. states (JDOS) has proved to be a useful quantity tocompare to experimentally obtained quasiparticle inter-ference patterns and to analyze the possible scatteringprocesses [33]. It can be obtained in a straightforwardmanner by JDOS( (cid:126)q, z ) = (cid:82) DOS( (cid:126)k, z )DOS( (cid:126)k + (cid:126)q, z ) d (cid:126)k .The simplicity of the computation then allows a detailedanalysis of the obtained interference pattern.Fig. 2 shows the JDOS at ω = 0 (Fig. 2 panels a-d )and ω = 0 . ω = 0 in the normal state, both the bulk (panela) and surface (panel b) JDOS show dominant peaksat the Brillouin zone center corresponding to impurityscattering with zero momentum. The ’radius’ of the re-gion with non-zero JDOS intensity for both the casesis about twice that of the vectors (cid:126)Q and (cid:126)Q (cid:48) , as is ex-pected from scattering between these states. However,there are some important features that distinguish thesurface and the bulk JDOS even without induced super-conductivity. First, the intensity of the JDOS is muchlarger on the surface than in the bulk (at zero energy)due to the surface flat band. Second, the JDOS profilein the bulk (Fig. 2(a) ) is quasi-flat away from zero mo-mentum transfer and peaks steeply at zero momentum.On the other hand, the surface JDOS (Fig. 2(b) ) hasa thick cone like feature. This difference is due to theadditional impurity scattering contributions originatingfrom all the momenta within the boundary of the surfaceflat band which is absent in the bulk. In the presence of induced d -wave superconductivity(Fig 2 panels c,d,g,h) at zero energy (panels c and d), thebulk (panel (c)) JDOS profile essentially peaks at ninepoints in the Brillouin zone. These points correspond to (cid:126)q = 0 , ± (cid:126)Q, ± (cid:126)Q (cid:48) , ± ( (cid:126)Q + (cid:126)Q (cid:48) ) , ± ( (cid:126)Q − (cid:126)Q (cid:48) ) which representthe nine different ways to connect the four nodal spotswith themselves and with the rest of the others (seeFig. 1 bottom, left). The surface JDOS (panel (d)) inthe presence of induced superconductivity has additionalintensity within the square bounded by the momentumvectors ± ( (cid:126)Q + (cid:126)Q (cid:48) ) , ± ( (cid:126)Q − (cid:126)Q (cid:48) ). This is entirely a conse-quence of the fact that d − wave superconductivity yieldsline nodes on the surface of a line nodal semimetal in-stead of point nodes (as in the bulk). In such a scenario,all the momentum vectors that lie within the square,correspond to vectors that connect different pointson the X shaped line node (in the DOS appearing inFig. 1 bottom, right) with each other. This is strikinglydifferent from the situation in d -wave superconductivityin materials lacking the “drumhead” states. This couldprove to be an experimentally verifiable signature ofthe surface states of line node semimetals. As discussedbefore, at non-zero energies, there is little differencebetween the bulk (panel e) and the surface (panel f) inthe absence of superconductivity; in fact, the surfacehas a smaller JDOS intensity than the bulk due to theabsence of surface states away from the Fermi level.In the presence of d − wave superconductivity, however,the bulk (panel g) and surface (panel h) JDOS profilesstart to acquire broadened characteristics in accordancewith the LDOS. In such a case, the surface still has agreater intensity than the bulk because surface stateswith momenta along the diagonals disperse all the waydown to zero energy. Impurity resonant states and T -Matrix approximation: Next, we analyze resonant states that may arise aroundthe impurities in line node semimetals.To clarify the no-tation, we briefly outline the T − matrix approximation(for further details refer to [46]). The total electron Greenfunction is written asˆ G ( (cid:126)k, (cid:126)k (cid:48) , ω ) = ˆ G ( (cid:126)k, ω ) δ (cid:126)k,(cid:126)k (cid:48) + ˆ G ( (cid:126)k, ω ) ˆ T ( (cid:126)k, (cid:126)k (cid:48) , ω ) ˆ G ( (cid:126)k (cid:48) , ω ) , (3)where G ( (cid:126)k, (cid:126)k (cid:48) , ω ) and G ( (cid:126)k, ω ) are the total interactingand non-interacting Green functions, and T ( (cid:126)k, (cid:126)k (cid:48) , ω ) isthe T − matrix which contains the physics originatingfrom impurity scattering. For the purposes of this arti-cle, we confine ourselves to scalar potential scatterers;this renders the T − matrix momentum independent andcan be written as ˆ T ( ω ) = (cid:104) ˆ σ − ˆ V ˆ g ( ω ) (cid:105) − ˆ V . Here,we have defined ˆ g ( ω ) = πN (cid:80) (cid:126)k ˆ G ( (cid:126)k, ω ), with N being the density of states at the Fermi level, and thescattering matrix ˆ V given by c ˆ τ . We have also used theparameter c = cot( N U ) as a measure of the strengthof an isotropic scatterer, following Ref. [47], where U is the strength of the impurity scatterer. Therefore, theunitarity limit (large scattering strength, N U → π )corresponds to the case when c → d − wave superconductors fromthe works of Balatsky and Hirschfeld [46–48]. Webegin by writing out the non-interacting Greens’function given as ˆ G ( (cid:126)k, ω ) = (cid:16) ω ˆ σ − ˆ H sc ( (cid:126)k ) (cid:17) − ,where ˆ H sc ( (cid:126)k ) = (cid:15) ( (cid:126)k )ˆ σ + ∆( (cid:126)k ) ˆ σ , (cid:15) ( (cid:126)k ) = αk − µ ( α is a constant and µ is the chemical potential),and ∆( (cid:126)k ) = ∆ cos 2 φ (cid:126)k . In general, the matrixˆ g ( ω ), can be written as ˆ g ( ω ) = (cid:80) i G i ( ω )ˆ σ i wherewe have G i ( ω ) ≡ πN (cid:80) (cid:126)k G i ( ω, (cid:126)k ), G ( ω, (cid:126)k ) = − ωD (cid:126)k , G ( ω, (cid:126)k ) = − ∆( (cid:126)k ) D (cid:126)k , G ( ω, (cid:126)k ) = 0, G ( ω, (cid:126)k ) = − (cid:15) ( (cid:126)k ) D (cid:126)k and D (cid:126)k = ∆( (cid:126)k ) + (cid:15) ( (cid:126)k ) − ω . Given the form of thescattering matrix, ˆ V = c ˆ σ , the condition for theexistence of resonant states is that the determinant of (cid:104) ˆ σ − ˆ V ˆ g ( ω ) (cid:105) must vanish. This translates to G ( ω ) − G ( ω ) + ( c − G ( ω )) = 0 . (4)Our task now is to evaluate these functions for the caseof a d − wave superconductor with a quadratic dispersion in two dimensions. The quantity G ( ω ) is zero since thegap function changes sign across the Brillouin zone andthe φ integral vanishes. Similiary G ( ω ) is zero if weassume particle-hole symmetric bands in two dimensions.Keeping this in mind, we evaluate G ( ω ) for quadraticbands and, in the limit ω (cid:28) ∆ , it can be shown that[47] G ( ω ) (cid:39) − ωπ ∆ (cid:20) log (cid:18) ω (cid:19) − i π (cid:21) . (5)The condition for the existence of a resonant state (ap-pearing in Eq. 4) with frequency Ω (whose real and imag-inary parts are denoted by Re(Ω) and Im(Ω)) simply re-duces to G (Ω) = ± c . The only under-damped solutionto this equation as a function of c has two importantfeatures to which one needs to pay attention (Fig 3 leftpanel): (i) both the real and imaginary parts of Ω go tozero in the unitarity limit ( c → c , the real part of Ω is slightly larger than theimaginary part of Ω. This is the regime where the reso-nant state is reasonably well defined, and above this valueof c , the state is heavily damped. We wish to comparethis result to the dispersive properties of an impurity onthe surface of a line nodal semimetal with a d − wave pair-ing in the quasi-flat band limit. To do so, we choose thenormal state density of states profile as a Lorenztian ofthe form ρ ( (cid:15) ) = γ/πγ + (cid:15) with a width γ that peaks at theFermi level. The energy scale γ can be chosen to be thesmallest among all other energy scales in the problem(bandwidth W , pairing amplitude ∆ and frequency ω ).Similar to the previous case of a quadratic dispersion, wehave G ( ω ) and G ( ω ) to be zero. To calculate G ( ω ) forthe surface of a line nodal semimetal, we substitute forthe Lorentzian density of states profile into the momen-tum integral. In the limit of W (cid:29) ∆ (cid:29) ω (cid:29) γ , weobtain (see Supplemental material for details [45]) G ( ω ) LNS (cid:39) − γ ξ ∆ (cid:20) ξ + 12 log (cid:18) ξ (cid:19) + i π (cid:21) , (6)where we have defined ξ ≡ ω ∆ . This form of G ( ω ) LNS bears some similarities to the ones we derived in Eq. 5;however, the crucial difference in Eq. 6 is the appearanceof an additional term ξ due to the presence of the quasi-flat band. This power law term has important conse-quences to the resonant state energies (see Fig. 3). Unlikethe two dimensional electron case with quadratic bands,the condition G ( ω ) = ± c admits two under-damped so-lutions (Ω , ), one for each sign. The real and imaginaryparts of these solutions are shown in the center panel ofFig. 3. While Ω is weakly undamped only for small c ,Ω remains sharp for all values of c . Moreover, the real FIG. 3. Comparison of the real and imaginary parts of the resonant state energies obtained by solving G (Ω) = ± c . (Left)Without the ξ term in Eq. 6. This is similar to the case of a d -wave superconductor with a quadratic dispersion. (Center)The case corresponding to the quasi-flat band where there are two solutions Ω , admissible. There is a regime for small c (leftshaded) where both the resonances − though well defined − are broad and have low spectral intensity; hence, they are challengingto observe experimentally. In the opposite limit (right shaded), the Ω solution no longer holds due to weakening of the flatband approximation. (Right) Corresponding DOS vs energy plots. Note that for these values of c , Ω is damped. parts of Ω and Ω disperse in opposite directions in theunitary regime. Note, however, that the dispersion of thereal part of Ω cannot go on to zero energy in the weakscattering (or large c ) limit. It is reasonable to expectthis as there should be no in-gap resonant states whenthe scattering strength goes to zero. Our result is con-sistent with this expectation since for large values of c (shaded region on the right in Fig. 3, center panel), Ω becomes comparable to γ , and the quasi-flat band ap-proximation weakens and eventually breaks down. Onthe other hand, in the unitarity limit c < ∼ . and Ω approach a relatively large fraction( Ω∆ ∼ .
8; compare this to the quadratic band case inFig. 3 left most panel, where it goes to zero energy) ofthe maximum gap value. This proximity to the contin-uum, coupled with the fact that the peak intensities goto zero for large impurity scattering, makes it experi-mentally challenging to observe this mode. Therefore,there is an optimal window of the scattering strengthswhere the resonance occurs predominantly due to quasi-flat band effects and, at the same time, is experimentallyobservable (see Fig. 3, right panel). Finally, there is ex-pected to be little spatial variation of the peak intensityon different sites close to/ at the impurity [49, 50] due tolack of spatial dynamics in a quasi-flat band system.
Summary:
To conclude, we studied the effect of nodal d − wave pairing in the bulk and on the surface of a linenodal semimetal, and determined the role of impuritiesthrough the joint density of states, which could be mea-sured via quasiparticle interference experiments. We ob-served that, unlike conventional two-dimensional metalswhere nodal superconductivity yields point nodes, thesurface of a line nodal semimetal gives rise to line nodes.As a consequence, due to the additional impurity scatter-ing phase space available within the area of the flat band,the JDOS pattern on the surface of a line nodal semimetalacquires an extended character in the Brillouin zone in- stead of a collection of discrete delta function peaks. Us-ing the T − matrix formalism, we also examined resonantstate energy dispersions of a single scalar impurity on thesurface of a line nodal semimetal with d − wave pairing.Our results demonstrated that the momentum averagedGreen function contains a power law type contribution inaddition to the logarithmic term usually found for nodalsuperconducting quadratic bands. Such a contributionadmits two different under-damped solutions to the res-onant state energies, unlike the case of two dimensionalelectrons with quadratic bands where there is only oneunder-damped solution. The first solution is a broad, lowintensity mode located closer to the continuum that dis-perses toward zero energy in the unitary limit; the secondis a more intense, sharp, lower energy mode that dispersesaway from zero energy. We argued that first mode maybe challenging to access experimentally while the secondcan be more readily observed. Our results also signal adestruction of zero bias tunneling peaks (in the unitar-ity limit) on the surface of a line nodal semimetal with d − wave pairing. Looking forward, it could be interest-ing to explore impurity effects in Josephson junctions online node semimetal surfaces, analogous to investigationson helical metals [51]. We are hopeful that our findingswould motivate scanning tunneling spectroscopic experi-ments on line node semimetals. Acknowledgments:
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SUPPLEMENTAL MATERIAL
Bogoliubov-de Gennes equations:
For the inclusion of proximity induced superconductivity and the joint densityof states in our setup appearing in the main text, and for the purposes of fixing our notation, we provide a basicintroduction to the Bogoliubov-de Gennes (BdG) equations. For the superconducting state with periodic boundaryconditions along all the three directions, we use the BdG Hamiltonian in momentum space given by [52] H BdG = (cid:88) (cid:126)kµν (cid:16) c † (cid:126)k ↑ µ c − (cid:126)k ↓ µ (cid:17) ˆ H BdG ( (cid:126)k ) (cid:32) c (cid:126)k ↑ ν c †− (cid:126)k ↓ ν (cid:33) (7)ˆ H BdG ( (cid:126)k ) = (cid:32) H ( (cid:126)k ) µν ∆ µν ∆ † µν − H ( − (cid:126)k ) µν (cid:33) , (8)where c (cid:126)kσν and c † (cid:126)kσν are the annihilation and creation operators for electrons in orbital ν , momentum (cid:126)k and spin σ and ∆ µν is the induced superconducting gap. In the presence of a surface, when we have open boundary conditionsalong the z − direction we can only Fourier transform along the k x and k y directions (together denoted as (cid:126)k (cid:107) ). In sucha scenario one can generically write the BdG eigenvalue equation asˆ H BdG ( (cid:126)k (cid:107) , z ) | ψ n ( (cid:126)k (cid:107) , z ) (cid:105) = E n ( (cid:126)k (cid:107) , z ) | ψ n ( (cid:126)k (cid:107) , z ) (cid:105) (9)where ˆ H BdG is the BdG Hamiltonian Fourier transformed only along the (cid:126)x (cid:107) direction, | ψ n ( (cid:126)k (cid:107) , z ) (cid:105) are the BdGwavefunctions, and E n ( (cid:126)k (cid:107) , z ) are the corresponding eigenvalues with a band index n . Effect of a perturbation and non-zero frequency:
In this section of the Supplemental Material, we studyhow it is possible to think of a point node semimetal as being the limiting case of a line nodal semimetal. It is possibleto augment the Hamiltonian in Eq. 1 of the main text to include the following term which smoothly interpolatesbetween a line nodal semimetal and a Weyl semimetalˆ H (cid:48) ( (cid:126)k ) = δ sin( ak y ) τ x σ , (10)where δ controls the strength of the perturbation, σ i and τ i are the Pauli matrices in the spin and orbital basis.Practically, it has been suggested that such a perturbation could be induced by light [19–22]. For the followingdiscussion of the effect of such a perturbation term, refer to Fig. 4. At zero energy and a finite value of a perturbationparameter (chosen to be δ = 0 . ω = 0 . > . except those states along the Brillouin zone diagonal. These states then converge down to the Fermi level toform line nodes at zero frequency, while at non zero frequency (less than the maximum value of superconductinggap) they form a “petal” like structure shown in Fig. 2. Bound state calculations:
We begin by recalling and expanding details of the T -matrix approximation that wasused in the main text. The total Greens’ function is written asˆ G ( (cid:126)k, (cid:126)k (cid:48) , ω ) = ˆ G ( (cid:126)k, ω ) δ (cid:126)k,(cid:126)k (cid:48) + ˆ G ( (cid:126)k, ω ) ˆ T ( (cid:126)k, (cid:126)k (cid:48) , ω ) ˆ G ( (cid:126)k (cid:48) , ω ) . (11)Here G ( (cid:126)k, (cid:126)k (cid:48) , ω ) and G ( (cid:126)k, ω ) are the total interacting and non-interacting Greens functions and T ( (cid:126)k, (cid:126)k (cid:48) , ω ) is the T -matrix which contains all the information about the impurity scattering. For the purposes of this article, we confine FIG. 4. Plots of the local density of states for a modified model (described in the main text) based on Ref. [44] but with anon-zero value of the perturbation parameter which converts the line nodal semimetal into a Weyl semimetal. (Left to right)Bulk density of states at zero energy, surface density of states at zero energy, bulk density of states at non-zero energy andsurface density of states at non-zero energy. We have chosen the value of ω = 0 . δ = 0 . ω = 0 . δ = 0. ourselves to scalar potential scatterers; this makes the T -matrix independent of momentum. Under this condition,the T -matrix becomes ˆ T ( ω ) = ˆ V + ˆ V ˆ g ( ω ) ˆ V + ˆ V ˆ g ( ω ) ˆ V ˆ g ( ω ) ˆ V + ... (12)= (cid:104) ˆ σ − ˆ V ˆ g ( ω ) (cid:105) − ˆ V .
Here, we have defined ˆ g ( ω ) = 12 πN (cid:88) (cid:126)k ˆ G ( (cid:126)k, ω ) , (13) N is the density of states at the Fermi level, and the scattering matrix ˆ V is given by c ˆ σ . We have also used theparameter c = cot( N U ) as a measure of the strength of an s -wave scatterer as done in Ref. [47], where U is thestrength of the impurity scatterer; therefore, the unitarity limit (large scattering strength, N U → π ) correspondsto the case when c → Superconducting state with quadratic dispersion in two dimensions (D=2) and a d − wave pairing form: Forthe sake of comparison with the case of the quasi-flat band, we revisit the calculation of Ref. [47] for the energy ofthe in gap bound state in a nodal, single band, d − wave superconductor. We begin by writing out its non-interactingGreens function given as ˆ G ( (cid:126)k, ω ) = (cid:16) ω ˆ σ − ˆ H ( (cid:126)k ) (cid:17) − , (14)where ˆ H ( (cid:126)k ) = (cid:15) ( (cid:126)k )ˆ σ + ∆( (cid:126)k ) ˆ σ and σ i are the Pauli matrices (henceforth, we will absorb the chemical potential µ into (cid:15) ( (cid:126)k ) and keep it to be non-zero, in general. It will be explicitly shown where important). The Greens’ functionis explicitly evaluated as ˆ G ( (cid:126)k, ω ) = 1∆( (cid:126)k ) + (cid:15) ( (cid:126)k ) − ω − (cid:16) (cid:15) ( (cid:126)k ) + ω (cid:17) − ∆( (cid:126)k ) − ∆( (cid:126)k ) (cid:16) (cid:15) ( (cid:126)k ) − ω (cid:17) . (15)We now proceed to evaluate the condition for the existence of a bound state when the Greens’ function is a matrix.In general, the matrix ˆ g ( ω ), can be written asˆ g ( ω ) = 12 πN (cid:88) (cid:126)k ˆ G ( (cid:126)k, ω ) = (cid:88) i G i ( ω )ˆ σ i = (cid:18) G ( ω ) + G ( ω ) G ( ω ) G ( ω ) G ( ω ) − G ( ω ) (cid:19) , (16)where the functions G i ( ω ) ≡ πN (cid:80) (cid:126)k G i ( ω, (cid:126)k ), and G ( ω, (cid:126)k ) = − ωD (cid:126)k , G ( ω, (cid:126)k ) = − ∆( (cid:126)k ) D (cid:126)k , G ( ω, (cid:126)k ) = − (cid:15) ( (cid:126)k ) D (cid:126)k and D (cid:126)k =∆( (cid:126)k ) + (cid:15) ( (cid:126)k ) − ω . Given the form of the scattering matrix, ˆ V = c ˆ σ , the condition for the existence of bound statesis that the determinant of (cid:16) ˆ σ − ˆ V ˆ g ( ω ) (cid:17) must vanish. As discussed in the main text, this condition is given by G ( ω ) − G ( ω ) + ( c − G ( ω )) = 0 . (17)Our task now is to evaluate these functions for the case of a d − wave superconductor with a quadratic dispersion in D = 2. The function G ( ω ) is zero since the gap function changes sign across the Brillouin zone and the φ integralvanishes. For G ( ω ), we write G ( ω ) = 12 πN (cid:88) (cid:126)k − (cid:15) ( (cid:126)k ) D (cid:126)k = 12 πN (cid:18) L π (cid:19) (cid:90) W − W d(cid:15)dφ α (cid:34) − (cid:15) ∆ φ + (cid:15) − ω (cid:35) . (18)Here, we have chosen a dispersion of the form (cid:15) ( (cid:126)k ) = αk − µ and ∆ φ = ∆ cos 2 φ . From now on, we set the totalbandwidth as 2 W and a chemical potential ( µ = E f ∼ W ) close to or at half filling. As it can be seen, in the 2D casefor a quadratic band, the chemical potential does not play a role. The integrand appearing above is anti-symmetricin (cid:15) and, hence, G ( ω ) = 0. Next we calculate G ( ω ) given by G ( ω ) = 12 πN (cid:88) (cid:126)k − (cid:15) ( (cid:126)k ) D (cid:126)k = 12 πN (cid:18) L π (cid:19) (cid:90) W − W d(cid:15)dφ α (cid:34) − ω ∆ φ + (cid:15) − ω (cid:35) , (19)where N is the total 2D density of states at the Fermi level and is given by L πα . Performing the (cid:15) integral andsubstituting for N yields G ( ω ) = − π (cid:90) π dφ (cid:113) ∆ cos φω − , (20)where we have substituted ∆ φ for the d -wave order parameter and ∆ is the pairing amplitude. The φ integral canbe performed easily to give G ( ω ) = − π ω (cid:112) ∆ − ω K (cid:18) ∆ ∆ − ω (cid:19) , (21)where K ( x ) is the elliptic K function. Since we are looking for in gap bound states, we study the case where ω (cid:28) ∆ .A series expansion of the elliptic K function is well known in this limit and G ( ω ) reduces to G ( ω ) (cid:39) − ωπ ∆ (cid:20) log (cid:18) ω (cid:19) − i π (cid:21) . (22)0The condition for the existence of a bound state for this case simply reduces to G (Ω) = ± c as discussed in the maintext. Superconducting state with a quasi-flat band and a d − wave pairing form: Here we aim to model the surfaceof a line-nodal semimetal and find its bound state properties. We choose a density of states profile as a Lorenztianof the form ρ ( (cid:15) ) = γ/πγ + (cid:15) , with a width γ , that peaks at the Fermi level. The energy scale γ can be chosen to bethe smallest among all other energy scales ( W, ∆ , ω ) in the problem, as discussed in the main text. Just like theprevious case, we have G ( ω ) = 0 due to the d − wave sign change in the Brillouin zone. Moreover, the chosen densityof states profile is even in (cid:15) , G ( ω ) is also zero. To calculate G ( ω ), we substitute for the Lorentzian density of statesprofile into the momentum integral. Noting that the density of states at the fermi level diverges as γ − , we get G ( ω ) = γ π (cid:90) W − W dφd(cid:15)γ + (cid:15) (cid:34) − ω ∆ φ + (cid:15) − ω (cid:35) . (23)The (cid:15) integral can be performed to give G ( ω ) = γ π (cid:90) π ωdφγ arctan (cid:16) Wγ (cid:17) − γ √ ∆ φ − ω arctan (cid:18) W √ ∆ φ − ω (cid:19) γ + ω − ∆ φ . (24)In the limit of large W (compared to the rest of the energy scales, γ, ∆ , ω , with ∆ > ω ), the integral reduces to G ( ω ) (cid:39) γ π (cid:18) ωγ (cid:19) (cid:90) π π − πγ √ ∆ φ − ω − ∆ φ + γ + ω dφ. (25)This integral can be performed and cast in terms of the function EllipticPi (Π( x, y )), i.e. G ( ω ) (cid:39) γ π (cid:18) ωγ (cid:19) − πγ Π (cid:16) ∆ ∆ − γ − ω , ∆ ∆ − ω (cid:17)(cid:112) ∆ − ω ( γ + ω − ∆ ) . (26)We are interested in the limit where ∆ (cid:29) ω (cid:29) γ . In this limit, G ( ω ) reduces to G ( ω ) (cid:39) − γ ξ ∆ (cid:20) ξ + 12 log (cid:18) ξ (cid:19) + i π (cid:21) , (27)where we have defined ξ ≡ ω ∆ . This expression for G ( ω ) has been used in the main text in obtaining Fig. 3. Normal state with a quasi-flat band (Line nodal semi-metal surface):
We have not disscussed or summarizedthe normal state bound state properties in our manuscript as several works have already studied this in detail (SeeRef. [46] and references therein); however, we want to briefly state the result for the case of the quasi-flat band. Tostudy the case of the flat band surface of a line nodal semi-metal without superconductivity we follow the sameprocedure as before. To this end, we consider the case of µ = 0 with the same Lorentzian density of states profile weused in the superconducting case. We obtain (cid:88) (cid:126)k G ( (cid:126)k, ω ) = (cid:90) W ηd(cid:15)π ( (cid:15) + η ) ( ω − (cid:15) ) . (28)This integral can be performed without difficulty. In the limit of η → − | ω | )as | ω |(cid:39) | U | . (29)Thus the bound state energy goes linearly with the strength of the impurity scatterer compared to quadratic for D = 1 and exponential for D = 2 [46].1 Model for the line nodal semimetal:
For the sake of completeness, we have provided the bulk bandstructure plots, bulk density of states, and the surface bands in the flat band and quasi-flat band limits in figures 3and 4 of this Supplemental Material.
FIG. 6. Plots of the bulk line nodal bands (left) and bulk DOS (right) for the two band model appearing in the main text.FIG. 7. Energy dispersion along the k x axis with k y set to zero on the surface of the line nodal semimetal with open boundaryconditions along the z axis. (Left) Quasi-flat ’drum head’ shaped surface band near the Fermi energy with Z = − . eV .(Right) Fully flat surface band with Z = 0 ..