Fermi arc reconstruction at the interface of twisted Weyl semimetals
FFermi Arc Reconstruction at the Interface of Twisted Weyl Semimetals
Faruk Abdulla, Sumathi Rao, and Ganpathy Murthy Harish-Chandra Research Institute, HBNI, Chhatnag Road, Jhunsi, Allahabad 211 019, India Department of Physics and Astronomy, University of Kentucky, Lexington, KY 40506-0055
Three-dimensional Weyl semimetals have pairs of topologically protected Weyl nodes, whose pro-jections onto the surface Brillouin zone are the end points of zero energy surface states called Fermiarcs. At the endpoints of the Fermi arcs, surface states extend into and are hybridized with the bulk.Here, we consider a two-dimensional junction of two identical Weyl semimetals whose surfaces aretwisted with respect to each other and tunnel-coupled. Confining ourselves to commensurate angles(such that a larger unit cell preserves a reduced translation symmetry at the interface) enables us toanalyze arbitrary strengths of the tunnel-coupling. We study the evolution of the Fermi arcs at theinterface, in detail, as a function of the twisting angle and the strength of the tunnel-coupling. Weshow unambiguously that in certain parameter regimes, all surface states decay exponentially intothe bulk, and the Fermi arcs become Fermi loops without endpoints. We study the evolution of the‘Fermi surfaces’ of these surface states as the tunnel-coupling strengths vary. We show that changesin the connectivity of the Fermi arcs/loops have interesting signatures in the optical conductivityin the presence of a magnetic field perpendicular to the surface.
I. INTRODUCTION
Weyl semimetals are often described as three di-mensional analogues of graphene , with band-touchingsor nodes at isolated points in the Brillouin zone. Thesenodes are chiral, and can be obtained by separatingthe Dirac nodes of a three dimensional semimetal byeither time-reversal or inversion symmetry break-ing. The low energy excitations about these nodes areWeyl fermions with anisotropic velocities that depend onthe material parameters. Weyl semimetals (WSMs) ex-hibit several novel features such as negative longitudinalmagneto-resistance , anomalous Hall effect , chiral-ity dependent Hall effect , planar Hall effect , etc.Several unconventional features have also been uncov-ered by studying transport across junctions of these Weylsemimetals with other topological and non-topologicalmaterials. For instance, junctions of Weyl semi-metalswith superconductors have also led to new phenom-ena such as chirality dependent Andreev reflection and chirality dependent Josephson effects. Tunnelingconductances across WSM-barrier-WSM junctions havealso been studied with interesting experimentally testableconsequences.
The band topology of the WSM is encoded in themonopole charge or the Chern number of the Berry cur-vature carried by the Weyl node. Hence surfaces in thebulk Brillouin zone (BZ) which enclose only one of thenodes carry Chern number. This leads to surface statescalled Fermi arc (FA) states in the surface BZ joiningthe projections of the Weyl nodes on to the surface BZ.Since the end-points of the FAs are the projections ofthe Weyl nodes, FAs on one surface connect to the FAson the opposite surface through the bulk nodes. In thepresence of a small magnetic field, this gives rise to inter-surface cyclotron orbits which depend on the thickness of the sample. These exotic FA states are the hallmarkof Weyl semimetals and it was their initial experimentalidentification using angle-resolved photoemission spec-troscopy that led to the current explosion in theoreticalinterest in understanding their properties. Morerecently, Shubnikov-de Haas oscillations and the quan-tum Hall effect based on intersurface cyclotron orbits have been seen in Cd As .Our aim in this work is to study the physics thatemerges when two slabs of WSM are twisted with respectto each other and tunnel-coupled. From the analogy tographene bilayers which show interesting effects, in-cluding the emergence of highly correlated states whenthe two layers have a small “magic angle” twist with re-spect to each other, we might expect new physics,both in the bulk of the WSMs and in the interface FAstates.The WSM-WSM junction with no twist was initiallystudied in Refs. 32 and 33 where FA reconstructions werefound when the junction was between WSMs with differ-ent FA connectivities. Coupling of WSMs with small incommensurate twists has also been studied earlier byMurthy, Fertig, and Shimshoni (henceforth MFS), ina perturbative regime of tunnel-coupling. MFS showedthat, due to the effective Moire Brillouin zone that can bedefined in terms of the mismatched lattice wave-vectors,reconstructions of the FAs take place. They also con-jectured that at certain “arcless angles”, at sufficientlystrong tunnel-coupling, the reconstructed surface stateswould consist of Fermi loops totally disconnected fromthe projections of the Weyl points on the surface BZ.In this paper, we extend MFS’s work to arbitrary com-mensurate angles with a reduced lattice translation sym-metry, and thus a larger superlattice unit cell at the inter-face. The presence of true lattice translation symmetry(absent in the work of MFS) allows us to analyze arbi-trary strengths of the tunnel couplings between the twoslabs. We perform a detailed study of the evolution of theFAs as a function of the coupling strength of the tunnel a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b coupling for two sequences of commensurate twist anglesparametrized by a positive integer n , θ n = tan − (1 /n ) and θ n = π/ − (1 /n ) . We unambiguously showthat there exist parameter regimes where all the surfacestates are disconnected from the bulk, like the surfacestates of a topological insulator. We take a detailed lookat the liftoff or detachment transition, where the Fermiarc detaches itself from the Weyl node projection andforms a surface state with a closed Fermi surface. Weanalyze the different ‘geometries’ into which the closedFermi surfaces evolve. Finally, we uncover a duality be-tween strong and weak interface tunnel couplings.The plan of this paper is as follows. In Section II, wedefine our model Hamiltonian and the parameters thatenter it, which include the commensurate twist angle andthe interface hopping matrix. In Section III, we study theevolution of the interface FA states as a function of thetwist parameter and the strength of the tunnel-couplingfor two simple commensurate angles with the smallest su-perlattices at the interface, which display nearly all thephenomena of interest. In Section IV we present the sim-plest model of the liftoff or detachment transition, wherethe FA detaches itself from the Weyl point projection onthe the surface BZ. We end in Section V with conclusions,caveats, a discussion of potential experimental signaturesof the phenomena we uncover, and some promising futuredirections. Many important details of the calculationsfor larger interface superlattices, the symmetries of themodel, the stability of our conclusions to longer rangetunnel-couplings, etc are relegated to a series of appen-dices. II. TWISTED WEYL SEMIMETALS AND THEINTERFACE HAMILTONIANA. Time-reversal symmetry broken model of aWSM and its surface states
To set the notation for the interface states of twotwisted WSMs, we briefly review the derivation of thesurface states for a semi-infinite WSM . We begin witha general two band lattice model of a time reversal sym-metry broken WSM on a cubic lattice. The Hamiltonianin momentum space is H = (cid:88) k c † k { t (cid:48) sin k z σ z + 2 sin k y σ y + 2 σ x (2 + cos k − cos k x − cos k y − cos k z ) } c k . (1)The spectrum has Weyl nodes at ( ± k , , with chiral-ity ± respectively. Here σ µ , µ = ( x, y, z ) are the spinPauli matrices and c k = ( c k ↑ , c k ↓ ) T , are two componentfermions. The hopping amplitude within the x - y planehas been set to unity and t (cid:48) represents the hopping am-plitude along the z direction. We choose our units so thatthe lattice constant a can be set to unity throughout thepaper. In this geometry, we expect Fermi arc states to FIG. 1. a) Two identical semi infinite WSM slabs are tunnelcoupled. The coupling is parametrised by κ . The interface (at z = 0 ) consists of two n = 0 layers, one from the top slab andthe other from the bottom slab. (b) The two slabs are twistedby θ clockwise and anticlockwise by the same angle and thencoupled. For a commensurate twist angle θ , there will be asuperlattice(SL) at the interface (an example is shown in Fig.2a). be present on surfaces which are not normal to the x -axis, that is, those with surface Brillouin zones definedby ( k x , k y ) or ( k x , k z ) .To find the surface states, following Ref. 34, we assumethat the slab is semi-infinite in the z -direction, with lat-tice sites labeled by n = 0 , , . . . and Fourier transform H ( k ) to real space (in the z direction) to get H ( k x , k y ) = (cid:88) n { c † n ( f x σ x + f y σ y ) c n − c † n ( σ x + it (cid:48) σ z ) × c n +1 − c † n +1 ( σ x − it (cid:48) σ z ) c n } (2)where f x = (2 + cos k − cos k x − cos k y ) and f y = sin k y .We have suppressed the k x , k y dependence of all thefermion operators for notational simplicity. Following arotation of the σ matrices by π/ around the x -axis, weget the transformed Hamiltonian to be ˜ H ( k x , k y ) = ∞ (cid:88) n =0 { c † n ( f x σ x + f y σ z ) c n − c † n ( σ x − it (cid:48) σ y ) × c n +1 − c † n +1 ( σ x + it (cid:48) σ y ) c n } , (3)which turns out to be real and hence, more convenient forfurther analysis. As shown in detail in Ref. 34, requiringthe decaying solutions into the bulk to be normalizable,and assuming that < t (cid:48) = sin φ and k < φ gives thedispersion of the surface states as E ( k x , k y ) = 2 f y =2 sin k y and the eigenstates are spin-polarized along the σ z direction. B. Interface between two identical twisted WSMs
In this subsection, we will see how the Fermi arcs oftwo identical WSM slabs, twisted with respect to eachother, get modified and reconstructed when we switch ona tunnel-coupling between them. We consider the inter-face of two identical WSMs as shown in Fig. 1. Both theWSMs are semi-infinite and the layers of the top slab arelabeled by n = 0 , , , .. and the layers of the bottom slabare labeled by n = 0 , − , − , .. . The interface consistsof the zeroth layer of both the slabs, which are tunnel-coupled after twisting the top and bottom slabs aroundthe z-axis by an angle ± θ n in the clockwise direction,analogous to the rotation of individual layers in bilayergraphene. The subscript n indicates that we rotate thetop slab clockwise and the bottom slab anticlockwise un-til the lattice site ( n , of the top layer coincides withthe lattice site ( n , − of the bottom layer, where n is a non negative integer. The twist angle in this caseis clearly θ n = tan − (1 /n ) . This results in a periodicsuperlattice (SL) structure at the interface. We will con-sider only such commensurate twists in this paper.The SL unit cell contains an equal number of latticesites from the top layer and the bottom layer. For agiven twist angle θ n , there are a total N sc = 2 × ( n + 1) lattice sites per SL unit cell when n is even (see Fig.2a for an example when n = 2 ). When n is odd,then total number of lattice sites per SL unit cell are N sc = ( n + 1) . For instance , when n = 0 , , , , and , the explicit number of total lattice sites per SL unitcell are N sc = 2 , , , , and respectively. The rel-ative size (area) of the SL unit cell at the interface (withrespect to the original 2D unit cell) is determined by N sc and it is actually N sc / times larger.In the presence of tunnel-coupling linking the top andbottom slabs, the full Hamiltonian consists of H = H t + H b + κH V . (4)where the t and b refers to the top and bottom slabsand both of them are the same as ˜ H ( k x , k y ) (Eq. 3)except that the slabs are now rotated with respect toeach other and H V is the coupling Hamiltonian (see Fig.1). The overall strength of the coupling is parametrisedby κ . The content of the spin-space matrix in H V willbe described shortly. Since we started with a cubic lat-tice, the planar lattice is a square and after rotation,the primitive lattice vectors of the top layer are givenby a t = (cos θ n , − sin θ n ) , a t = (sin θ n , cos θ n ) andthat of the bottom layer are a b = (cos θ n , sin θ n ) , a b =( − sin θ n , cos θ n ) as shown in Fig. 2a (for n = 2 ). TheHamiltonian is now explicitly given by H γ = (cid:88) k (cid:88) n [2 c † n ( k ) M γ c n ( k ) − c † n +1 ( k ) T c n ( k ) − c † n ( k ) T † c n +1 ( k )] (5)where k here refers to the transverse momentum vector ( k x , k y ) and γ = ( t, b ) refer to the top and bottom layers, and the sum over n goes from n = 0 , , , . . . for the toplayer and from n = 0 , − , − , . . . for the bottom layer.The onsite and hopping matrices are given by M γ = ( f γ σ x + f γ σ z ) , T = ( σ x + it (cid:48) σ y ); (6)with f γ ( k ) = 2 + cos k − cos( k . a γ ) − cos( k . a γ ) and f γ =sin( k . a γ ) As mentioned earlier, the interface consists of the ze-roth layers of the WSM slabs, tunnel coupled by the hop-ping Hamiltonian H V , given by H V = (cid:88) r t (cid:88) r b c † t s ( r t ) V ss (cid:48) ( | r t − r b | ) c b s (cid:48) ( r b ) + h.c, (7)where the lattice sites r t and r b live on the n = 0 layerof the top and bottom slabs respectively. Note that wehave now introduced the labels t and b to distinguish thefermions that live on the top and bottom layers. Wehave also introduced the spin indices s and s (cid:48) since thetunnel coupling is a matrix in spin space. We assumeshort-range hopping so that V ss (cid:48) is nonzero only for siteson the two surfaces with the same 2D coordinates. V ss (cid:48) ( | r t − r b | ) = ( V ) ss (cid:48) δ r t , r b . (8)These are the larger blue dots in Fig. 2a. In Appendix A,we show that the perturbative inclusion of longer-rangehoppings does not qualitatively change the results thatwe obtain in our model.Note that the most general tunneling matrix betweentwo sites can be written as a matrix in spin-space, V = (cid:88) i =0 V i σ i , (9)where V i are complex numbers, σ is the identity matrixand σ i are the three Pauli matrices for i = x, y, z . Thisgives us a eight-parameter space of tunneling matrix el-ements, which is difficult to explore systematically.Fortunately, there is a natural way to restrict thespace of tunnel-couplings. When all tunneling matrixelements between the slabs are set to zero, our model(setting κ = 0 ) enjoys a large number of symmetries,which include unitary, anti-unitary, particle-particle, andparticle-hole type symmetries. For example, it is clearfrom Fig. 2a that, for n = 2 the SL is symmetricabout the positive diagonal with the replacement of sites i = 1 , .. of the upper layer with the corresponding sitesof the lower layer. It is also symmetric along the nega-tive diagonal, with the replacement of the sites ↔ and ↔ between the upper and lower layers. These sym-metries are detailed in Appendix B. In general, not allthe symmetries of the uncoupled model can be satisfiedby the tunneling term. We will assume that the tun-neling conserves the symmetries of rotation by π aroundboth diagonals of the SL unit cell of the surface Brillouinzone for twist angles of the form θ n = π/ − (1 /n ) . -5-4-3-2-1 0 1 2 3 -3 -2 -1 0 1 2 3 b b a t1 a t2 a b1 a b2 R δ t2 δ b2 δ b3 Y X (a) - ππ π π g g k y k x (b) FIG. 2. a) The interface layers for n = 2 . Here the top and bottom slabs are rotated by θ = ± tan − (1 / so that the site (2 , of the n = 0 layer of the top slab coincides with the (2 , − site of the n = 0 layer of the bottom slab. The lattice sitesof the top and bottom slab layers are in red and black colour respectively. The superlattice (SL) sites are in blue and b , b are its primitive vectors.The SL unit cell contains N sc / n + 1) / lattice sites of each slab and they are labelled by α = { , , , , } . (b) Here, the Brillouin zones of the rotated slabs and of the SL lattice are shown. In addition to the first SLBZ (labelled ), the second BZs of the SL are also shown with labels , , and and g , g are the reciprocal lattice vectorsof the SL. For this particular rotation, the SL BZ is N sc / times smaller than the BZ of original lattice. For θ n = tan − (1 /n ) , on the other hand, we will assumesymmetry of rotation by π around the x and y axes. Inboth cases, this leads to a restriction that the tunnelingmatrix be real, and a combination of σ x and σ y only. Itis thus of the form V x σ x + iV y σ y where V x , V y are real.We have verified in Appendix C that perturbations vio-lating these symmetries do not qualitatively change ourconclusions.To physically understand why σ x and/or σ y termsmust be present in order to get a reconstructed Fermiarc at the interface, recall that the surface states of thetop/bottom slabs are spin up/down polarized (see sub-section II A). So keeping solely σ and/or σ z terms cannotlead to any coupling between the unreconstructed Fermiarcs of the top and bottom slabs. III. INTERFACE STATES AND THEIREVOLUTION
In this section, we will study the coupled FA states atthe interface of the two WSMs. These are states that arelocalised on the interface and decay exponentially intothe bulk. The individual FA eigenstates are (details reviewed in Appendix D) (cid:12)(cid:12) Ψ t ( k ) (cid:11) = ∞ (cid:88) n =0 ψ tn ( k ) c † n ( k ) | (cid:105) (10) (cid:12)(cid:12) Ψ b ( k ) (cid:11) = −∞ (cid:88) n =0 ψ bn ( k ) c † n ( k ) | (cid:105) . (11)where ψ tn ( k ) ∈ C ( n ≥ ) and ψ bn ( k ) ∈ C ( n ≤ ) arewavefunctions, living on the n th layer, of the top andbottom slabs respectively. Since translational invarianceis unbroken on the plane, the eigenstates can be labeledby the momenta k = ( k x , k y ) .The two wave-functions ofthe zeroth layer - ψ t ( k ) , ψ b ( k ) - are then matched atthe interface in the presence of the coupling matrix H V given in Eq. 7. As shown in detail in Appendix D, theconditions for the existence of the interface states can beexpressed as a matrix equation M ( E, κ, p x , p y ) A = 0 , (12)where M ( E, κ, p x , p y ) is a (not necessarily hermitian)square matrix of dimension N sc × N sc and A is a col-umn matrix with N sc rows (the factor of 2 is for the spindegrees of freedom). Here, we have used p = ( p x , p y ) torepresent the momenta in the superlattice (SL) BZ. Forinterface localised states to exist, the determinant of thematrix must vanish. Therefore det ( M ( E, κ, p x , p y )) = 0 gives a condition which ( p x , p y ) must satisfy for a given E and κ to have the interface localised states.The setof all such ( p x , p y ) , for E = 0 and fixed κ , yields thereconstructed Fermi arc state at the interface.To proceed further, we need to fix n and explicitly find FIG. 3. The panels show the evolution of the Fermi arcs (FAs) of the interface BZ as a function of the coupling strength. Thepanels in the top row are for n = 0 and those in the bottom row are for n = 1 . (i) Panels (a),(b),(c) and (d) are for n = 0 and a relative twist between the top and bottom slabs of θ = π . The positive (negative) chiral Weyl point projection (WPP)of the top slab and the negative (positive) chiral WPP of the bottom slab coincide at (+( − ) π/ , in the interface BZ. Theparameters are t (cid:48) = sin π/ , k = π/ and V x = 1 , V y = 0 . When κ = 0 , the FAs of both the slabs are just straight lines alongthe x -axis between ( − π/ , π/ . When we switch on the coupling, the FAs evolve with increasing κ as shown in panel (a). Thevertical green lines represent the WPPs. For relatively higher values of κ = 4 , , , the reconstructed FAs are shown separatelyin panels (b), (c) and (d). (ii) Panels (e), (f), (g) and (h) are for n = 1 and for different ranges of κ . The parameter valuesare the same as those for the top panels. Here, the relative twist between the slabs is θ = π/ . The unreconstructed FA forthe bottom slab is along the y-axis and for the top slab, the unreconstructed FA is along the x-axis (shown as black lines inpanel (e)). As a function of the coupling κ , the FAs go through a reconstruction which joins the two positive WPPs and thetwo negative WPPs of the two slabs together and for some values of κ we see Fermi loops disconnected from the WPPs the FAs by solving det ( M ( E, κ, p x , p y )) = 0 . In the nextsubsection, we first consider the simplest cases, n = 0 and n = 1 , with respective relative twist angles θ = π and θ = π/ . In both cases, the two lattices are in reg-istry and the full xy translation symmetry of the originalcubic lattice is present at the interface. In the followingsubsection, we will present results for the n = 2 casein detail, with a larger interface unit cell, and thereforea reduced translation symmetry. The results for othertwists (which are qualitatively similar) are presented inAppendix E. A. Twists with n = and n = To set the stage for studying arbitrary commensurateangles, we first study the case n = 0 . This case wasstudied earlier in Ref. 32 for the special case t (cid:48) = 1 andthe interface hopping being of the same form as the bulkhopping, except with a different strength. Here, we con-sider our more general model, and we consider the evolu-tion of the FAs as a function of the coupling parameter.When the slabs are aligned, the Weyl point projections(WPPs) in the surface BZ of both the top and bottomslabs are at ( ± k , are on top of each other, with thepositively charged chiral WPPs at ( k , and the nega-tively charged chiral WPPs at ( − k , . In this case thereneed not be any states localized at the interface. How- ever, when the two slabs are rotated by a relative angleof θ = π , the position of the positively charged chiralWPP of the top slab coincides, in the surface BZ, withthe negatively charged chiral WPP of the bottom slab.Since the direction of the FA has changed between theslabs, the interface is well defined and there are necessar-ily interface localized states which get reconstructed as afunction of the tunnel coupling. Note that the symmetryof the surface BZ under reflections about the x -axis and y -axis implies that the tunnel-couplings V x and V y in Eq.9 can be chosen to be real and non-zero. For definiteness,we have chosen V x (cid:54) = 0 , although we have checked thattaking a general linear combination of V x and V y doesnot change the result qualitatively. As mentioned above,the reconstructed FAs for E = 0 and for a fixed couplingparameter κ is given by the set of momenta ( p x , p y ) forwhich the determinant of the × matrix M ( E, κ, p x , p y ) vanishes. The results are shown in the panels in the toprow of Fig. 3 for a set of values of κ . At κ = 0 , the FA ofthe individual slabs are just straight lines on the x -axisbetween ( − k , k ) . With the increase in the coupling pa-rameter κ , the FAs evolve as shown in Fig. 3(a). At κ ∼ , in addition to the main FA connecting the WPPs,a pair of freestanding Fermi loops appear. For larger κ ,the freestanding loops disappear and eventually, beyondthe values shown in the graph, we get back the result FIG. 4. The panels show the evolution of the FAs at the interface BZ as a function of the coupling strength for n = 2 . (i) Thepanels in the top row show the evolution in the non-overlapping case where the Weyl point projections (WPPs) of the top (red + ve and − ve signs)and bottom slabs (black + ve and − ve signs) are distinct from each other. The vertical green lines representthe evolution of the WPPs. The parameters chosen are t (cid:48) = sin π/ , and V x = 1 , V y = 0 . (a) Reconstructed FAs are shown forthe coupling parameters in the range κ = (0 , . For κ = 0 , the original FAs of the individual slabs are shown in black. As κ increases, the reconstructed arcs join the + ve chirality WPPs together and the − ve chirality WPPs together (b) The range ofthe coupling is now from κ = 2 to κ = 10 . (c) Shows the change of the sign of curvature of the FAs from positive to negativearound κ = 5 and formation of pairs of closed Fermi loop at the BZ boundary. These loops disappear pairwise after mergingtogether around κ = 5 . . (ii) The panels in the bottom row show the evolution when the WPPs of the top and bottom slabsoverlap. The parameters chosen are t (cid:48) = 1 . , V x = 1 , V y = 0 . . (d) Evolution of the Fermi arcs for small κ . (e) The evolutionshows the detachment of the FAs from the WPPs at around κ ∼ . . Also, the formation of a new small loop passing throughthe WPPS is seen around κ ∼ . (f) A close-up view of the formation of the small closed loop passing through WPPs for the FAs of the decoupled slabs. This is a particularinstance of the duality in Fermi arc reconstruction under κ → /κ which is seen in all cases and which we shalldiscuss in detail later.Next, let us consider the case when n = 1 . This wasalso earlier studied in Ref. 34, but only for weak values ofthe coupling κ . When the slabs are rotated with respectto each other by an angle θ = π/ , the WPPs of one ofthe slabs in the surface BZ is (0 , ± k ) , whereas those ofthe other one are at ( ± k , . As for n = 0 , the latticesare in registry and the reconstructed FAs are given byby a set of momenta ( p x , p y ) for which the determinantof the × matrix M ( E, κ, p x , p y ) vanishes for a given κ and for E = 0 . The results for the reconstructed FAsare shown in the bottom row of panels in Fig. 3. Asexpected, the FAs at zero coupling κ are the FAs of theindividual slabs. For the upper slab, it is a straight lineon the x -axis and for the lower slab, it is a straight line onthe y -axis. For nonzero κ , the reconstructed FAs connectthe positively charged chiral WPPs and the negativelycharged chiral WPPs of the slabs together as shown inFig. 3(e). As κ increases further, the FAs get deformed and a pair of freestanding closed loops form. For evenlarger κ , the freestanding loops disappear and eventually,as in the earlier case, the reconstructed FAs approachesthe zero-coupling result. B. Twist with n = The simplest case where the surfaces of the top andbottom slabs are not in registry occurs for n = 2 ,where the top and bottom slabs are twisted clockwiseand counter-clockwise respectively by the angle θ =tan − (1 / . This brings the lattice point (2 , of thebottom layer of the top slab lie on top of the (2 , − site of the top layer of the bottom slab, as shown inFig. 2a, forming one of the sites of the superlattice(SL). As can be seen from the figure, the SL unit cellcontains 5 lattice points from each of the layers ( N sc =2 × ( n + 1) = 10 lattice sites altogether), and its unitcell is 5 times as large as the original unit cell. The prim-itive lattice vectors of the rotated slabs are now given by a t = (cos θ , − sin θ ) , a t = (sin θ , cos θ ) for the toplayer and a b = (cos θ , sin θ ) , a b = ( − sin θ , cos θ ) for the bottom layer. The superlattice primitive vectorsare given by b = ( √ , and b = (0 , √ and theSL unit cell contain total N sc = 2( n + 1) = 10 latticesites as shown in Fig. 2a. The WPPs in the surfaceBZ of top slab are given by (i) ‘+ (cid:48) at k (cos θ , − sin θ ) and ‘ − (cid:48) at k ( − cos θ , sin θ ) and that of bottom slabare given by (ii) ‘+ (cid:48) at k (cos θ , sin θ ) and ‘ − (cid:48) at k ( − cos θ , − sin θ ) . Note that our model does not spec-ify k - see Eq. 1 - it only says that the Weyl nodes are at ( ± k , , - hence we are free to choose it to be any valueconsistent with the existence of FA states. Without lossof generality, we choose an arbitrary value of k = π/ .The results for this case are shown in Fig. 4. Aswe turn on the coupling parameter κ , the reconstructedFAs connect the + ve chiral WPPs together (and the − vechiral WPPs together). As κ is increased, the curva-ture of the reconstructed FAs changes and flips sign near κ ∼ as shown in Fig. 4(b). This range of coupling κ = (5 . , . is explored further in Fig. 4(c), where a setof four small, closed, freestanding Fermi loops appear atthe corners of the BZ when κ ∼ . . When κ is furtherincreased, they move towards each other, merge, and fi-nally disappear at κ ∼ . . As for n = 0 , , there is aduality in the FA reconstruction between small and large κ ; at large κ we get qualitatively the same FAs as small κ . In Fig. 4(b), one can see that the FAs at κ = 2 , are similar to the FAs at κ = 10 , . Note that, in thiscase, we see that the reconstructed Fermi arcs are alwaysattached to the WPPs.Ref. 34 had conjectured that there might be arcless an-gles in twisted WSMs, where all interface states are dis-connected from the WPPs. Can that occur in our modelwith commensurate twist angles as well? A preconditionfor this is to have WPPs of the same chirality from thetop and bottom slabs overlap in the surface BZ. This canbe achieved by combining a commensurate twist anglewith a suitable choice of k . For example, at n = 2 , werotate the top slab clockwise by an angle θ = tan − (1 / and the bottom slab by the angle ( π/ θ ) anti clock-wise. The SL at the interface is identical to the earliercase, but the primitive lattice vectors are now given by a t = (cos θ , − sin θ ) , a t = (sin θ , cos θ ) for the topslab and a b = ( − sin θ , cos θ ) , a b = ( − cos θ , − sin θ ) for the bottom slab. Now we can choose k = 2 π/ so that the positively charged chiral WPP of both theslabs coincide (and similarly the negatively charged chi-ral WPPs). The positively and negatively charged chiralWPPs are then at k ( ∓ / √ , ∓ / √ in the surface BZ.The reconstructed FAs in this case are shown in thepanels in the bottom row of Fig. 4. Fig. 4(d) shows thesituation for weak-coupling, where the FAs are attachedto the WPPs. Fig. 4(e) reveals that FAs get detachedfrom the WPPs for κ ∼ . and move away from themas κ increases. Subsequently a pair of small closed Fermiloops attached to the WPPs appear at κ ∼ . . Uponincreasing κ they disappear at κ ∼ . (see Fig. 4(f)). Soin the range of κ between . < κ < . , and . < κ < . , the model has FAs which are wholly disconnectedfrom the WPPs and there are no surface states attachedto the WPPs. Beyond κ = 7 . , the results are similarto the case when the couplings are small, because of theduality between large and small couplings.We have also studied the twists with n = 4 and n =5 for completeness. Since the results are qualitativelysimilar to the cases with smaller n , they are relegatedto Appendix E. IV. MODEL FOR A LIFT-OFF TRANSITION
We have seen in the previous section that when theprojections of the Weyl points of positive chirality coin-cide in the surface Brillouin zone (and likewise for thenegative chirality Weyl points), the Fermi arc can de-tach from the WPP at an appropriate coupling strength κ . Because of the κ → /κ duality, the Fermi arc re-attaches again to the WPPs when κ is increased and thereconstructed Fermi arc eventually approaches the un-reconstructed Fermi arc in the limit κ − → . In thissection, our goal is to demonstrate these lift-off and re-attachment transitions in a simple model where there isno twist between the slabs. More specifically, we want toobtain the shape of the segment of the Fermi arc attachedto the WPP at the transition, which, based on empiricalevidence, we believe to be universal.We consider the Hamiltonian given in Eq. 3 in themain text, but with a modified top/bottom-dependent f y term: f y,γ = sin k y + λ γ (cos k x − cos k ) . For the top slab ( γ ≡ t ) we take λ t = λ and for thebottom slab ( γ ≡ b ), λ b = − λ . There is no twist, butthe slabs, though aligned, are not identical even in thebulk except at λ = 0 . We consider short-range hoppingas before (Eq. 8) and the hopping matrix V is taken tobe V = σ x + it (cid:48) σ y .As mentioned before, we need to compute the determi-nant of the matrix M ( E, κ, k x , k y ) which is now a × matrix, in the neighbourhood of the projection of theWeyl point. So we will parametrize the neighbourhoodof the Weyl point projection k = ( k , in the surfaceBZ as (cid:126)k = k ˆ x + (cid:126)q, (cid:126)q = q (cos θ ˆ x + sin θ ˆ y ) , (13)where q = | (cid:126)q | (cid:28) and θ is the polar angle in the two-dimensional surface BZ around the Weyl point. The ma-trix M ( E, κ, k x , k y ) can be explicitly written as M = g ( t )1 ↑ u ( t )1 g ( t )2 ↑ u ( t )2 − κg ( b )1 ↑ − κg ( b )2 ↑ g ( t )1 ↓ u ( t )1 g ( t )2 ↓ u ( t )2 − κg ( b )1 ↓ − κg ( b )2 ↓ κg ( t )1 ↑ κg ( t )2 ↑ − g ( b )1 ↑ u ( b )1 − g ( b )2 ↑ u ( b )2 κg ( t )1 ↓ κg ( t )2 ↓ − g ( b )1 ↓ u ( b )1 − g ( b )2 ↓ u ( b )2 (14) FIG. 5. The lift-off transition (a) The curves denote the values of the coupling κ as a function of θ for the existence of Fermiarcs in the vicinity of the Weyl point projection (WPP) at k = ( π/ , and with V = σ x + it (cid:48) σ y . The parameters chosen are λ = 0 . and t (cid:48) = 0 . . The solutions around θ = π correspond to the Fermi arcs which are in purple( κ = 0 ) and cyan( κ = 28 )in figure (b), whereas the solutions around θ = 0 (or π ) corresponds to the Fermi arcs in green. There are two κ regions ( . < κ < . and . < κ < . ) where there are no Fermi arcs attached to the WPPs. As κ is increased slowly fromzero, the Fermi arc in purple colour in figure (b) evolves until it detaches from the WPPs at κ c ≈ . . Then in the parameterregime, . < κ < . , there are no solutions until κ c ≈ . . In the regime, . < κ < . , the Fermi arc in green evolvesand detaches from the WPPs at the critical value κ c ≈ . . Once again , there are no solutions when . < κ < . .Beyond κ > . we get the dual solution, the Fermi arc in cyan in figure (b) which gradually approaches the zero couplingFermi arc (in purple) as κ → ∞ . where u ( t ) and u ( b ) are determined by Eqs. D3-D4 andthe various g ’s are the components of the spin wavefunc-tion of the top slab Φ t = ( g ( t ) ↑ , g ( t ) ↓ ) T and the bottomslab Φ b = ( g ( b ) ↑ , g ( b ) ↓ ) T . The arguments of the matrix el-ements have been suppressed for notational simplicity.Our strategy is to Taylor expand each u and g to lead-ing order in q. After some algebra, we get the followingleading order expansions for the u’s : u ( t )1 = 1 − qt (cid:48) s t ( θ ) , u ( b )1 = 1 − qt (cid:48) s b ( θ ) ,u ( t )2 = u ( b )2 = 1 − t (cid:48) t (cid:48) (1 + sin k q cos θ ) , (15)where s γ ( θ ) = (1 + λ ) sin k cos θ − λ γ sin k sin 2 θ +sin θ .To avoid singularities, we now divide the whole θ regioninto two sub regions - (i) π/ < θ < π/ and (ii) − π/ <θ < π/ - and choose the spinors appropriately. Theleading order expansions give the following solutions forthe g ’s -In region (i) : g ( t )1 ↓ = sin θ − λ sin k cos θ − sin k cos θ + s t ( θ ) , g ( t )2 ↓ = O ( q ) g ( b )1 ↑ = − sin θ + λ sin k cos θ − sin k cos θ + s b ( θ ) , g ( b )2 ↑ = O ( q ) (16)where we have chosen g ( t )1 ↑ = g ( t )2 ↑ = 1 , g ( b )1 ↓ = g ( b )2 ↓ = 1 .In region (ii): g ( t )1 ↑ = sin θ − λ sin k cos θ sin k cos θ + s t ( θ ) , g ( t )2 ↓ = O ( q ) g ( b )1 ↓ = − sin θ + λ sin k cos θ sin k cos θ + s b ( θ ) , g ( b )2 ↑ = O ( q ) (17)where we have chosen g ( t )1 ↓ = g ( t )2 ↑ = 1 , g ( b )1 ↑ = g ( b )2 ↓ = 1 .We are interested in the existence of Fermi arcs in thevicinity of Weyl point projection. So for q (cid:28) , we canapproximate u ( t )1 = u ( b )1 ≈ and u ( t )2 = u ( b )2 = u ≈ − t (cid:48) t (cid:48) (see Eq. 15). This essentially implies that, very close tothe Weyl point, the polar angle of q , or the angle at whichthe FA attaches to the WPP, is the important parameter.Substituting these expressions in the matrix M , we getthe following determinant vanishing conditions in the tworegions -For region (i) π/ < θ < π/ : ( α t α b ) κ + κ u (cid:0) (1 + u ) − u (1 + α t α b ) (cid:1) + α t α b u = 0 , (18)and for region (ii) − π/ < θ < π/ : κ + κ u (cid:0) (1 + u ) α t α b − u (1 + α t α b ) (cid:1) + 1 u = 0 , (19)where α t = (sin θ − λ sin k cos θ ) / (sin k | cos θ | + s t ( θ )) and α b = (sin θ + λ sin k cos θ ) / (sin k | cos θ | + s b ( θ )) .The solutions to Eqs.18-19 for κ as a function of θ giveus the Fermi arcs attached to the WPP’s. The curvescovered by these solutions are shown in Fig. 5(a) and inthe remaining region, not covered by these lines, there areno Fermi arcs attached to the WPPs. Essentially, if thereexist surface states in the remaining regions, those solu-tions are not coupled to the WPPs and would actuallyform closed Fermi surfaces. However, we do not studysuch solutions here since our aim here was to study thelift-off transition or the limits in the ( κ, θ ) space, whereFermi arcs attached to the WPPs are no longer present. κ = 3.120 κ = 3.1255 κ = 3.1260 p y p x κ = 7.7179 κ = 7.7187 κ = 7.750 p y p x FIG. 6. Singular shape of the Fermi arc near the (a) lift-off and (b) re-attachment transitions for n = 2 (see alsothe reconstructed Fermi arc in Fig. 4(e)). The black dot inthe figure represents the projection of negative chirality Weylnode. We notice that at and near the lift-off (or re-attachment) transition, the shape of the Fermi arc ishighly singular. The two legs that are attached to theWPP have the same slope at the transition. We believethis shape to be universal for all lift-off and re-attachmenttransitions, because in every instance of lift-off or re-attachment where we have “zoomed in” near the WPPat the transition, we find this to be the case. An exam-ple is shown in Fig. 6.
V. CAVEATS, CONCLUSIONS, DISCUSSION,AND OPEN QUESTIONS
In this work we considered two identical semi-infiniteslabs of WSM, twisted by a commensurate angle withrespect to each other, and their free surfaces tunnel-coupled. The constraint that the angle be commensu-rate means that a reduced lattice translation symmetry,defined by a larger superlattice unit cell, is enjoyed bythe Hamiltonian. This has the benefit of allowing us toextend our calculations to arbitrary values of the tunnel-coupling, which allowed us to extend the previous resultsof Murthy, Shimshoni, and Fertig (which were for ar-bitrary small (incommensurate) angles, but perturbativevalues of the tunnel couplings.)At this point it is important to mention several caveatsconcerning the assumptions we have made. Firstly,we focused most of our investigation on the case whenthe tunnel-couplings are ultra-short-range, being nonzeroonly for sites on the two surfaces having the same xy co- ordinates. We did investigate longer-range hoppings asperturbations to this case (Appendix A), but did not un-dertake a study of the most general periodic hopping.Secondly, even with this simplification, the number ofparameters in the hopping matrix is too large to allowsystematic investigation. We therefore chose a particularsubset of symmetries of the model system in the absenceof tunnel-coupling, and imposed this symmetry on thetunnel-couplings as well. Once again, we have checkedperturbatively that adding tunnel-coupling that breakour self-imposed symmetries do not change our resultsqualitatively (Appendix C).Three main results emerge from our work. Firstly, weconfirmed an interesting conjecture from earlier work.MSF considered (among other things) the case whenthe twist angle and the value of k are tuned such thatthe WPPs of the + Weyl nodes of the two slabs overlap(and likewise for the WPPs of the − Weyl nodes). MSFconjectured that at strong enough tunnel-couplings thereconstructed Fermi arcs would detach themselves fromthe WPPs, leaving behind purely interface states, thatis, states with all their spectral weight near the interface.We have confirmed that this appears to be a generic fea-ture when the WPPs overlap. In addition, free-floatingFermi loops far from any WPP appear, expand, contract,and disappear as a function of the strength of the tunnel-coupling.A noteworthy feature of such purely interface states(which they share with the surface states of topo-logical insulators) is that some of them are two-dimensional states that cannot be obtained in a purelytwo-dimensional system of noninteracting electrons. Forexample, consider Fig. 10a, with n = 4 and κ = 5 . .The entire Fermi loop is one connected curve windingaround the SL BZ (periodic boundary conditions applyat the boundaries of the SL BZ). However, it has no insideor outside. There is no notion of the number of statesenclosed inside the Fermi surface.Secondly, we identified a qualitative duality betweenweak and strong tunnel-couplings. This occurs for a veryphysical reason. Let us restrict ourselves to the case whenthe tunnel-couplings are “zero-range”, in the sense thatthe lattice sites of the two layers have to have identical xy coordinates in order for the hopping to be nonzero. Avery strong tunnel coupling κ between the two verticallyaligned sites, considered in isolation, will create a pairof hybridized states of energy of order ± κ . Any tunnel-ing between the slabs must go through these verticallyaligned sites. Thus, the effective tunneling must be oforder t / | κ | , where t is the intra-slab tunneling strength.This shows that the two slabs become essentially isolatedfrom each other as | κ | → ∞ . It may be possible to en-gineer large values of | κ | in WSMs that cleave such thatatoms on one WSM surface are likely to form covalentbonds with atoms on the other WSM surface.Thirdly, we looked at the shape of the reconstructedFermi arcs at the lift-off and re-attachment transitions.We found that they have a very singular shape, as demon-0strated in Fig. 6. The singularity near the WPP alsoseems to be universal, in the sense that all lift-off andre-attachment transitions we have investigated in detailshow the same shape near the WPP.Let us examine potential experimental signatures ofour theoretical results. First we consider the closed Fermicurves of purely two-dimensional interface states, as ex-emplified by Fig. 10a. Upon applying a weak (semiclas-sical) perpendicular orbital magnetic field, wave pack-ets will experience a Lorentz force − e v ( k ) × B where v ( k ) = ∇ k ε ( k ) is the group velocity of the Fermi loopstates. Since the velocity is always perpendicular to theFermi loop, the wave packets semiclassically travel alongthe closed Fermi curves. Under semiclassical quantiza-tion the states along the Fermi curve re-organize them-selves into a set of equally spaced levels, with the thespacing directly proportional to | B | . These levels can beinvestigated by the absorption of electromagnetic wavesof the appropriate frequency.How might one detect lift-off/reattachment transi-tions? A standard technique to look for Fermi arc statespassing through the WPPs is to look at semiclassical or-bits (once again under a weak perpendicular magneticfield) that traverse the Fermi arc on one surface, gothrough the bulk via the Weyl node to the other sur-face, traverse the Fermi arc there and complete the cy-cle.Such intersurface loops can enclose an area, and ex-hibit magneto-oscillations. The period of the cycle de-pends on the thickness of the slab. As usual, semiclassicalquantization will reorganize the closed orbits into a setof equally spaced level, which can be investigated by anelectromagnetic probe.To be more specific on the experimental signatureof the liftoff/re-attachment transitions, let us focus onthe case when the WPPs overlap, and we are at weak-coupling, such that the Fermi arcs are attached to theWPPs. A wavepacket starting on the bottom surfaceof the lower slab will traverse the bulk of the lower slabthrough the Weyl node and reach the interface of the twoslabs. At this point it will split; a part will traverse theFermi arc at the interface, and another part will travelthrough the bulk of the upper slab, traverse the Fermiarc on the top surface of the upper slab, and return tothe interface. Thus, there will be multiple scattering ofthe wavepacket, giving rise to a sequence of periods ofreturn of the wavepacket to the bottom surface of thelower slab. Similarly, there will be a sequence of areasrelevant to magneto-oscillations.Now, if the tunnel-coupling is tuned such that theFermi arcs detach from the WPPs, the interface is inac-cessible to a wavepacket starting on the bottom surfaceof the lower slab. Thus, there is only one period of re-turn for the wavepacket and only one area relevant tomagneto-oscillations.Thus, if an in-situ method (perhaps pressure) canbe found to tune the strength of the tunnel-couplingthrough the lift-off/re-attachment transition, this abruptchange in behavior of the return period and/or magneto- oscillations will be a smoking-gun signature of such atransition.The most important physics left out of our calculationis the effect of disorder. With regard to disorder, de-spite early work indicating the stability of the Weyl nodeagainst weak disorder, a consensus has emerged thatlarge rare regions of strong disorder potential produce anonzero density of states at the Weyl points, destroyingthe WSM even for arbitrarily weak disorder. Similarly,the Fermi arcs get broadened by coupling to bulk disor-der, and conduct dissipatively on short to intermediatelength scales. They get localized at the longest lengthscales, but the chiral velocity persists at the surface. Based on this picture, we can surmise that the Fermiloops traversing the superlattice BZ that we find in ourwork should be detectable as conducting states at all butthe longest length scales. However, the states near theWPPs at the liftoff/re-attachment transitions will be par-ticularly susceptible to disorder, and may be harder todetect via conduction.There are many other interesting open questions, suchas the possibility of interface quantum Hall effects andelectron-electron interactions, which we hope to study infuture work.
ACKNOWLEDGMENTS
SR and GM would like to thank the VAJRA schemeof SERB, India for its support. GM is grateful for par-tial support from the US-Israel Binational Science Foun-dation (Grant No. 2016130), and the hospitality of theInternational Center for Theoretical Sciences, Bangalore,where these ideas were conceived during the workshop onEdge Dynamics in Topological Phases, Dec 2019 - Jan2020.
APPENDICES
The appendices provide details on: (a) The stabilityof FA reconstructions under longer ranged hoppings. (b)The symmetries of the κ = 0 Hamiltonian in the su-perlattice Brillouin zone and the implications for tunnel-couplings. (c) The stability of FA reconstructions underperturbations of tunnelling matrix that break the abovesymmetries. (d) The details of the computation of theFermi arcs at the interface. (e) Some illustrative resultsfor larger values of n , or smaller twist angles. Appendix A: Stability of Fermi arc reconstructionsunder longer ranged hoppings
So far we have discussed results where the hoppingsbetween the top and bottom layers are taken to be ultra-short range, such that only sites of the top and bot-tom slabs with the same 2D coordinates are tunnel-coupled. It is natural to ask what happens to the Fermiarc states at the interface if the range of the hoppings1
FIG. 7. n = 2 : Long ranged couplings. The evolution of reconstructed Fermi arcs as a function of the coupling range parameter ξ is shown for different fixed coupling strengths κ . The parameters are k = 2 π/ , t (cid:48) = 1 . , V x = 1 , V y = 0 . (a) and (b) showthe evolution of the reconstructed Fermi arcs with ξ for κ = 0 . . When ξ is small we recover our previous short-range couplingresults(see Fig. 4d) and it is clear that the results are stable with the increase in the range of the couplings. (c) Here, thecoupling strength is fixed at κ = 4 to examine the stability of the Fermi arc detachment from WPPs. ξ is kept small becausethe effective strength of the coupling gets renormalized to larger value with increasing ξ . Note that in (c), the reconstructedFermi arcs around ‘+’ WPP and ‘-’ WPP are not symmetric for larger values of ξ . Indeed, we do not expect the symmetrybetween the Weyl node projections to exist for a general ξ . were increased to also include the next nearest sites andthe next-next nearest sites and so on. We attempt toanswer this question here by considering the hopping V ss (cid:48) ( | r t − r b | ) as a Gaussian, V ss (cid:48) ( | r t − r b | ) = κ ( V ) ss (cid:48) e − r ξ . (A1)Here ( V ) ss (cid:48) is a constant matrix, κ denotes the strengthof the interaction and ξ is a length scale parameter thatdetermines the range of hopping. Small(large) ξ meansthat the hopping is short(long) ranged. For ξ << , weshould recover our previous results, which was for ultra-short range hopping. Without loss of generality, we studythe case n = 2 . In particular, we consider the case withoverlapping Weyl point projections(WPPs).The results are shown in Fig. 7. We have studiedhow the reconstructed Fermi arcs get modified as a func-tion of the hopping range parameter ξ . We have consid-ered two different coupling strengths κ = 0 . and κ = 4 .Since the effective coupling strength parameter κe − r /ξ gets renormalized to larger value with increasing hop-ping range ξ , we restrict the hopping range parameter tosmaller values for the latter case κ = 4 . We recover ourprevious results of short range hopping for very small ξ << . We find that the short-range hopping resultis stable against longer ranged hoppings unless ξ is toolarge. Appendix B: Symmetries of the κ = 0 Hamiltonianin the superlattice Brillouin zone
In the main text, we have mentioned that the symme-tries of the Hamiltonian of the slabs ( H t + H b ) can bekept intact if we restrict the tunnelling matrix to be ofthe following form V = V x σ x + iV y σ y , where V i ( i = x, y )is a real number. We will show this result here explicitly for a particular value of the twist angle θ n =2 ; however,the result is general and is valid for all n .We will consider the case n = 2 with overlapping Weylpoint projections (WPPs) i.e. for commensurate twistangle of the form θ n = π/ − (1 /n ) . The resultingsuperlattice (SL) is identical to what is shown in Fig.2a (but now with differently oriented lattice vectors a γi ,where γ = t, b and i = 1 , ). To analyse the symmetries,it is convenient to take the Hamiltonian (in Eq. 5) in theposition space in all directions - H γ = (cid:88) n k ) c † s ( n )( σ x ) ss (cid:48) c s (cid:48) ( n ) − (cid:0) c † s ( n + a γ )( σ x ) ss (cid:48) c s (cid:48) ( n ) + h.c. (cid:1) (B1) − (cid:0) c † s ( n + a γ )( σ x − iσ z ) ss (cid:48) c s (cid:48) ( n ) + h.c. (cid:1) − (cid:0) c † s ( n + a γ )( σ x + itσ y ) ss (cid:48) c s (cid:48) ( n ) + h.c. (cid:1) . Here n = n a γ + n a γ + n a γ with n i (i=1, 2, 3) areintegers, and a γi are the primitive lattice vectors. For thetop slab γ ≡ t , n runs over the range (0 , ∞ ) and for thebottom slab γ ≡ b , n runs over the range (0 , −∞ ) .The periodicity at the interface is that of the SL, sowe need to express the operators in terms of the SL sitelabels. So we rewrite the operators as follows: c s ( n ) = c s ( R + δ α , n ) ≡ d αs ( m , n ) for the top slab ( n ≥ ) and c s ( n ) = c s ( R + δ α , n ) ≡ f αs ( m , n ) for the bottom slab( n ≤ ). Here α is the sublattice index and R = m b + m b is the position vector of SL sites (see Fig. 2a). Byinspection, it is clear that there are geometric symmetriesof the model that involve rotation of the lattice by π around both diagonals ( i.e. , the sites in black and redin Fig. 2a get interchanged). We will now see how thesesymmetries are implemented in terms of d αs ( m , n ) and f αs ( m , n ) .First, let us consider the symmetry transformation as-2sociated with the π rotation about m = m diagonal.(i) U d αs ( m , m , n ) U † = ( σ x ) ss (cid:48) f αs (cid:48) ( m , m , n ) U iU † = i, U V U † = σ x V † σ x (B2)where the symmetry is unitarily realised and of theparticle-particle type, and(ii) Q d αs ( m , m , n ) Q † = ( iσ y ) ss (cid:48) f † αs (cid:48) ( m , m , n ) Q iQ † = i, Q V Q † = − σ y V T σ y (B3)where again, the symmetry is unitarily realised, but is ofthe Boguliobov or particle-hole type.Note that the spatial arguments of the fermion opera-tors are not the same on both the sides of the equation -in fact because the spatial transformation involves a ro-tation of the lattice by π about the diagonal, m ↔ m .After lengthy algebraic manipulations, it can be shownthat both the symmetry transformations interchange H t and H b - i.e. , U H t U † = H b and Q H t Q † = H b , andvice-versa. So if we require both the transformationsto be symmetries of the Hamiltonian, then we need thetunnelling matrix to be of the form V = V x σ x + iV y σ y where V i ( i = x, y ) are real numbers (assuming also that V has to be real). This is clear since the transforma-tion (i) will be a symmetry of the total Hamiltonian H = H t + H b + H V , only if the tunnelling matrix V satisfies the condition U V U † = σ x V † σ x = V, (B4)and the transformation (ii) will be a symmetry of H onlyif V obeys the condition Q V Q † = − σ y V T σ y = V. (B5)There are also symmetry transformations associatedwith π rotation around the other diagonal m = − m ofthe SL. In this case we have the following anti-unitarysymmetry transformations:(iii) U d αs ( m , m , n ) U † = ( S ) αα (cid:48) ( σ x ) ss (cid:48) f α (cid:48) s (cid:48) ( − m , − m , n ) U iU † = − i, U V U † = σ x V T σ x (B6)of the particle-particle type and(iv) Q d αs ( m , m , n ) Q † = ( S ) αα (cid:48) ( iσ y ) ss (cid:48) f † α (cid:48) s (cid:48) ( − m , − m , n ) Q iQ † = − i, Q V Q † = − σ y V † σ y . (B7)of the Boguliobov type. Note also that here the geometricsymmetry interchanges m ↔ − m . The matrix in thesublattice index S αα (cid:48) has the following non zero elements S = 1 , S = S = 1 , S = S = 1 . Both the abovetransformations takes H t to H b and vice versa. So the transformation (iii) will be a symmetry of H = H t + H b + H V , if the tunnelling matrix satisfies the condition U V U † = σ x V T σ x = V, (B8)and the transformation (iv) will be a symmetry if Q V Q † = − σ y V † σ y = V. (B9)For a real tunnelling matrix, it is clear from Eq. B8and Eq. B9 that V needs to be of the following form V = V x σ x + iV y σ y , where V x and V y are real for bothsymmetries to be realised. Note that this form of V isidentical to what was needed for the Hamiltonian to besymmetric under the transformations (i) and (ii). So thesymmetry of rotation by π around either diagonal leadsto the same conditions on the tunnelling matrix.For commensurate twist angles of the form θ n =tan − (1 /n ) , the geometric symmetry of the super-lattice(SL) by the rotation π around both the x and y axes willbe a symmetry of the Hamiltonian ( H t + H b ) . For theparticular twist n = 2 , the resulting SL has been shownshown in Fig. 2a.Consider a rotation by π around the x-axis (whichpasses through the centre of the SL unit cell), which takesan SL site ( m , m ) to ( m , − m + 1) . Clearly the lat-tice sites in red get interchanged with the lattice sites inblack. The sites in red (inside the SL unit cell) labelled0, 1, 2, 3 and 4 get mapped to the sites in black labelled0, 4, 1, 2, and 3 respectively (see Fig. 2a). This geo-metric symmetry is realised via the following symmetrytransformations;(i) R x d αs ( m , m , n ) R † x (B10) = W xαα (cid:48) ( σ x ) ss (cid:48) f αs (cid:48) ( m , − m + 1 , n ) R x iR † x = i, R x V R † x = σ x V † σ x (B11)where the symmetry is unitarily realised and of theparticle-particle type, and(ii) S x d αs ( m , m , n ) S † x (B12) = W xαα (cid:48) ( iσ y ) ss (cid:48) f † αs (cid:48) ( m , − m + 1 , n ) S x iS † x = i, S x V S † x = − σ y V T σ y (B13)where again, the symmetry is unitarily realised, but isof the Boguliobov or particle-hole type. Since the sitesin red labelled 0, 1, 2, 3 and 4 get mapped to the sitesin black labelled 0, 4, 1, 2, and 3 respectively, the sym-metric and unitary matrix W xαα (cid:48) has the following nonzero elements : W x = 1 , W x = W x = 1 , W x = W x =1 , W x = W x = 1 , W x = W x = 1 . Here both thetransformations take H t → H b and vice versa. Imposingthe above symmetries on the tunnelling Hamiltonian H V , leads to conditions which are identical to the con-ditions Eqs. B4-B5. Consequently, for a real tunnellingmatrix, V has to be of the same form that we had earlierobtained for symmetry under rotation about the diago-nals, where the twist angle was θ n = π/ tan − (1 /n ) .3Now consider rotation by π about the y-axis (whichagain passes through the centre of the SL unit cell), whichtakes an arbitrary SL site ( m , m ) to ( − m + 1 , m ) .Clearly the sites in red (inside the SL unit cell) labelled0, 1, 2, 3 and 4 get mapped to the sites in black la-belled 0, 2, 3, 4, and 1 respectively (see Fig. 2a). Againthis geometric symmetry is realised through the followinganti-unitary symmetry transformations -(iii) R y d αs ( m , m , n ) R † y (B14) = W yαα (cid:48) ( σ x ) ss (cid:48) f αs (cid:48) ( − m + 1 , m , n ) R y iR † y = − i, R y V R † y = σ x V T σ x (B15)where the symmetry is anti-unitarily realised and of theparticle-particle type, and(iv) S y d αs ( m , m , n ) S † y (B16) = W yαα (cid:48) ( iσ y ) ss (cid:48) f † αs (cid:48) ( − m + 1 , m , n ) S y iS † y = − i, S y V S † y = − σ y V † σ y (B17)where again, the symmetry is anti-unitarily realised, butis of the Boguliobov or particle-hole type. The sym-metric and unitary matrix W yαα (cid:48) has the following nonzero elements W y = 1 , W y = W y = 1 , W y = W y =1 , W y = W y = 1 , W y = W y = 1 . As before, boththe transformations (iii) and (iv) takes H t → H b i.e. R y H t R − y = H b and S y H t S − y = H b and vice versa.Now if we impose the above symmetry transformations(iii) and (iv) on the tunnelling Hamiltonian, we get con-ditions on V, which are exactly identical to the conditionsgiven in Eq. B8-B9.To summarise, for twist angles of the form θ n = tan − (1 /n ) , the geometric symmetries (which lead tosymmetries of H) are the rotations by π about the xand y axes, whereas for twist angles of the form θ n = π/ tan − (1 /n ) , the geometric symmetries (which leadto symmetries of H) are the rotations by π about the di-agonals of SL unit cell. But both the cases lead to thesame conditions on the tunnelling matrix , i.e. , V has beof the form V = V x σ x + iV y σ y (for a real V), where V x and V y are real. Appendix C: Stability of FA reconstruction underperturbations breaking the symmetries of the κ = 0 Hamiltonian
Here, we show that adding perturbations that breakthe symmetries of the κ = 0 Hamiltonian do not changethe results qualitatively. To see this, we consider a par-ticular twist n = 2 with overlapping Weyl point projec-tions. Recall that for V = (cid:80) µ =0 V µ σ µ the symmetriesforced V , V z = 0 and V x , V y real. We will break thesesymmetries by allowing for nonzero V , V z . For smallcouplings κ < , the reconstructed Fermi arcs look verysimilar to those shown in Fig. 4(a). To check whether thelift-off and re-attachment transitions retain the singular - + κ =3.0 κ =4.0 - p y p x - + κ =3.0 κ =4.0 - p y p x κ = 2.900 κ = 2.9062 κ = 2.9064 -- p y p x κ = 7.900 κ = 7.8806 κ = 7.8801 -- p y p x FIG. 8. n = 2 : The reconstructed Fermi arcs (for κ = 3 and4) near the lift-off transitions without perturbations in (a) arecompared with the reconstructed Fermi arcs after adding thesymmetry breaking perturbations in (b). The parameters aregiven by δV = V σ + V z σ z , where V = 0 . and V z = 0 . .The other parameters are the same as in Figs. 4(d-f) and inFig. 6. Note that symmetry of the Fermi arcs about boththe diagonals p x = ± p y is broken in the perturbed case. Notealso that the perturbed Fermi arcs get detached only fromthe -ve chiral WPP, but not from the +ve chiral WPP. (c)and (d) show the singular shapes of the Fermi arcs at andnear the lift-off and re-attachment transitions, respectively,in the presence of the perturbation. It can be seen that thesingular structure of the Fermi arc at and near the lift-off andre-attachment transitions remains intact.( See also Fig .6 forthe unperturbed case.) shape of the Fermi arc close to the transitions, we studythe reconstructed Fermi arcs, close to the transitions,with the symmetry breaking perturbations. The compar-ison with the earlier unperturbed results is shown in Fig.8 and we find that the singular shape of the Fermi arc atand near the lift-off and re-attachment survices addingsymmetry breaking perturbations to the tunnelling ma-trix. With the particular choice of V µ we have made, theFermi arc gets detached only from the negative chiral-ity WPP, but remains attached to the positive chiralityWPP. However, we have confirmed that various choicesof V µ can lead to detachment from either or both of theWPPs. Appendix D: Computation of Fermi Arc States atthe Junction
In this section we will provide the details for the com-putation of the interface localized states. Our computa-tion will closely follow Ref. 34.4
1. Surface states of the slabs
The idea is to look for decaying eigenstates into thebulk for both the slabs. Then we match solutions at theinterface via the coupling term H V to get the interfacelocalised states. Translational invariance is broken in the z -direction, but it remains unbroken in the transversedirections. The eigenstates can, hence, be labeled by themomenta ( k x , k y ) . We proceed with the following ansatzfor the decaying eigenstates, -top slab: | E, k x , k y (cid:105) t = ∞ (cid:88) n =0 (cid:88) s u nt Φ ts c † ns | (cid:105) (D1)bottom slab: | E, k x , k y (cid:105) b = −∞ (cid:88) n =0 (cid:88) s u − nb Φ bs c † ns | (cid:105) (D2)where | u γ | < , for the state | E, k x , k y (cid:105) to be normal-izable and n is the discretised z coordinate and here, γ = ( t, b ) . We then solve the Schrodinger equation, H γ | E, k x , k y (cid:105) γ = E | E, k x , k y (cid:105) γ to obtain ( u γ , Φ γs ) . Asshown in Ref. 34, there are two normalizable solutionsfor u γ for a given E and k = ( k x , k y ) , which are given by ( u γ ) ± = 12 ( ξ γ ± (cid:113) ξ γ − , (D3)where, ξ γ is a solution of the following quadratic equa-tion, (1 − t (cid:48) ) ξ γ − f γ ξ γ − E + 4(( f γ ) + ( f γ ) + t (cid:48) ) = 0 . (D4)For each root of ξ γ , it is obvious from Eq. D3 that the tworoots of u γ obey ( u γ ) + ( u γ ) − = 1 . This implies that oneof the roots of u γ for each root of ξ γ must satisfy | u γ | (cid:54) .The equality holds only when ξ γ is real and − (cid:54) ξ γ (cid:54) .Therefore two roots of ξ γ give two solutions of u γ whichgive normalizable decaying eigenstates solutions for bothslabs, only when ξ γ (which may be complex) lies outsidethe range − (cid:54) ξ γ (cid:54) .A few further steps of algebra suffices to show that thespin wavefunctions Φ γ can be expressed as Φ t = (cid:18) , − (2 f t − E )2 f t − ( u t + 1 /u t ) + t (cid:48) ( u t − /u t ) (cid:19) T (D5) Φ b = (cid:18) (2 f b + E )2 f b − ( u b + 1 /u b ) + t (cid:48) ( u b − /u b ) , (cid:19) T . (D6)We can now construct a general wavefunction for thetop slab as (cid:12)(cid:12) Ψ t ( k ) (cid:11) = ∞ (cid:88) n =0 A t ( k )( u t ( k )) n Φ t ( k ) c † n ( k ) | (cid:105) + A t ( k )( u t ( k )) n Φ t ( k ) c † n ( k ) | (cid:105) = ∞ (cid:88) n =0 ψ tn ( k ) c † n ( k ) | (cid:105) , (D7) and for the bottom slab as (cid:12)(cid:12) Ψ b ( k ) (cid:11) = −∞ (cid:88) n =0 A b ( k )( u b ( k )) − n Φ b ( k ) c † n ( k ) | (cid:105) + A b ( k )( u b ( k )) − n Φ b ( k ) c † n ( k ) | (cid:105) = −∞ (cid:88) n =0 ψ bn ( k ) c † n ( k ) | (cid:105) , (D8)where, ψ tn ( k ) = A t ( k )( u t ( k )) n Φ t ( k ) + A t ( k )( u t ( k )) n Φ t ( k ) ,ψ bn ( k ) = A b ( k )( u b ( k )) − n Φ b ( k ) + A b ( k )( u b ( k )) − n Φ b ( k ) , (D9)and A γ , A γ are unknown constants. Next, our goal isto match the wavefunctions at the interface through thecoupling term H V to fix the constants A γ , A γ .
2. Manipulation of coupling term H V Before proceeding further, we first need to write thecoupling term H V in the transverse momentum space k = ( k x , k y ) . On the interface, the true periodicityis that of superlattice. We label the operators whichlive on the zeroth layer of the top and bottom slabs by d αs ( R ) , and f αs ( R ) respectively, where the superlat-tice sites are given by R = m b + m b , m i ∈ Z . Givena superlattice (SL) site, we then assign a ‘star’ of sitesin both slabs as sublattice sites to this SL site with ‘ α ’representing the sublattice index. Writing the term H V in the SL index, we then get the following form from Eq.7 in the. main paper, H V = (cid:88) R ,α (cid:88) R (cid:48) ,β { d † αs ( R ) V ss (cid:48) ( | R − R (cid:48) + δ tα − δ bβ | ) f βs (cid:48) ( R (cid:48) )+ H.c } , (D10)where we have used c t s ( r t ) = c t s ( R + δ tα ) ≡ d αs ( R ) and c b s ( r b ) = c b s ( R + δ bβ ) ≡ f βs ( R ) . We Fourier transformthe operators, d αs ( R ) = 1 √ N SL (cid:88) p ∈ BZ SL e i p . R ˜ d αs ( p ) f βs ( R ) = 1 √ N SL (cid:88) p ∈ BZ SL e i p . R ˜ f βs ( p ) , (D11)where N SL is total number of SL lattice sites and substi-tute these expressions in Eq. D10, to get the followingcoupling term after simplification, H V = (cid:88) p ∈ BZ SL (cid:88) αβ { ˜ d † αs ( p ) ˜ V αβss (cid:48) ( p ) ˜ f βs (cid:48) ( p ) + H.c } , (D12)where ˜ V αβss (cid:48) ( p ) = (cid:88) R e − i p . R V ss (cid:48) ( R + δ tα − δ bβ ) . (D13)5The δ tα and δ bβ are the relative position vectors of thesublattice sites with respect to the SL unit cell. Sincethe states | Ψ t ( k ) (cid:105) ( k ∈ BZ t ) and (cid:12)(cid:12) Ψ b ( k ) (cid:11) ( k ∈ BZ b ) aredefined where the operators c † ns ( k ) act on the vacuum | (cid:105) , we need to rewrite Eq. D12 in terms of the operators c † t s ( k ) and c † b s ( k ) . So we want to relate the Fouriertransforms of d αs ( R ) and f αs ( R ) with that of c t s ( r t ) and c b s ( r b ) . We recall that d αs ( R ) = c t s ( R + δ tα )= 1 √ N (cid:88) k ∈ BZ t e i k . ( R + δ tα ) c t s ( k )= 1 √ N (cid:88) p ∈ BZ SL N sc / (cid:88) l =1 e i ( p + Q l ) . ( R + δ tα ) c t s ( p + Q l )= 1 √ N SL (cid:88) p ∈ BZ SL e i p . R (cid:112) N sc / N sc / (cid:88) l =1 e i ( p + Q l ) . δ tα c t s ( p + Q l ) . (D14)In going from the second to the third step, we have usedthe fact that the momenta k ∈ BZ t can be decomposedas: k l = p + Q l , where p ∈ BZ SL and Q l = n g + n g , n i ∈ Z . Since g and g are the reciprocal latticevectors of the SL lattice, the following holds Q l . R =2 π × ( integer ) . Here the label l goes over the first andsecond Brillouin zones, since the SL BZ is N sc / timessmaller than the original BZ. From the third to the fourthstep, we use the identity e i Q l . R = 1 and write N = N SL × N sc / . Recall that the number of sites per SL unit cellis given by N sc = 2 × ( n + 1) when n is even and by N sc = ( n +1) when n is odd. In particular when n = 2 (see Fig. 2 in the main paper), there are 5 sites of the toplayer and 5 sites of the bottom layer per SL unit cell and aset of 5 values of Q l = ( , g , g , − g , − g ) . ComparingEq. D14 with the Eq. D11, we get the following relations- ˜ d αs ( p ) = 1 (cid:112) N sc / N sc / (cid:88) l =1 e i ( p + Q l ) . δ tα c t s ( p + Q l ) . (D15)and similarly for ˜ f βs ( p ) , we get ˜ f βs ( p ) = 1 (cid:112) N sc / N sc / (cid:88) l =1 e i ( p + Q l ) . δ bβ c b s ( p + Q l ) . (D16)Now substituting these expression of ˜ d αs ( p ) and ˜ f βs ( p ) in Eq. D12, we finally get the following expression forthe coupling term H V = 1 N sc / (cid:88) p (cid:88) ss (cid:48) N sc / − (cid:88) α,β =0 N sc / (cid:88) l,l (cid:48) =1 { e − i Q l . δ tα + i Q l (cid:48) . δ bβ × c † t s ( p + Q l ) ˜ V αβss (cid:48) ( p ) c b s (cid:48) ( p + Q l (cid:48) ) + H.c } , (D17)where ˜ V αβss (cid:48) ( p ) is given in Eq. D13.
3. Matching conditions
Since | Ψ t ( k l ) (cid:105) and (cid:12)(cid:12) Ψ b ( k l ) (cid:11) are the eigenstates of H t and H b respectively and k l = p + Q l , we can write theeigenstates of H as | Ψ( k l ) (cid:105) = | Ψ t ( k l ) (cid:105) + (cid:12)(cid:12) Ψ b ( k l ) (cid:11) . Notethat we have yet to determine the constants A t , A t and A b , A b , defined in Eq. D9. Solving for H | Ψ( k l ) (cid:105) = E | Ψ( k l ) (cid:105) , gives us a discrete Schrodinger equation fora generic layer n written as h γnn ψ γn ( k l ) + h γn,n − ψ γn − ( k l ) + h γn,n +1 ψ γn +1 ( k l ) = Eψ γn ( k l ) (D18)where γ = ( t, b ) and the different h matrices are obtainedfrom the Eqs. (3) and (4) in the main paper - h γnn = M γ , h γn,n +1 = − T † , h γn,n − = − T . (D19)(i) Next, let us consider γ = t and n = 0 . Now when wesolve H | Ψ( k l ) (cid:105) = E | Ψ( k l ) (cid:105) . In this case, the n = − layer does not exist and is instead replaced by the n = 0 layer of the bottom slab. Hence, the coupling matrix H V will also act on the states | Ψ( k l ) (cid:105) ; this results in thefollowing equation, h t ψ t ( k l ) + (cid:88) l (cid:48) ˜ h ( l, l (cid:48) ) ψ b ( k l (cid:48) ) + h t , ψ t ( k l ) = Eψ t ( k l ) , (D20)where the matrix ˜ h ( l, l (cid:48) ) is given by ˜ h ( l, l (cid:48) ) = κN sc / (cid:88) αβ e ( − i Q l . δ tα + i Q l (cid:48) . δ bβ ) V αβ ( p ) . (D21)Eq. D18 remains valid even for n = 0 (and setting γ = t ),we get h t ψ t ( k l ) + h t , − ψ t − ( k l ) + h t , ψ t ( k l ) = Eψ t ( k l ) , (D22)where h t , − = − T . Now we can combine both Eq. D20and Eq. D22 to finally get the wavefunction matchingcondition at the interface: − T ψ t − ( k l ) + (cid:88) l (cid:48) ˜ h ( l, l (cid:48) ) ψ b ( k l (cid:48) ) = 0 . (D23)(ii) Similarly we can consider γ = b and n = 0 , and getthe second matching condition at the interface: − T † ψ b ( k l ) + (cid:88) l (cid:48) ˜ h † ( l, l (cid:48) ) ψ t ( k l (cid:48) ) = 0 . (D24)Notice that in Eqs.D23 and D24, each term is a × column matrix. There are N sc / number of different k l = p + Q l values. So each equation is essentially a set of N sc equations. Eqs.D23 and D24 together give a set of × N sc equations and there are as many unknown constants( A γ ( k l ) , A γ ( k l ) ) to be determined (see Eq. D9). We can6combine Eq. D23 and Eq. D24 to give a single matrixequation, M ( E, κ, p x , p y ) A = 0 , (D25)where A = ( A t , A t , A b , A b ) T with, A t = ( A t ( l ) , A t ( l ) , ...A t ( l N sc / )) A t = ( A t ( l ) , A t ( l ) , ...A t ( l N sc / )) A b = ( A b ( l ) , A b ( l ) , ...A b ( l N sc / )) A b = ( A t ( l ) , A t ( l ) , ...A t ( l N sc / )) . Non trivial solutions for the constants lead to exponen-tially localised states at the interface, and for non trivialsolutions to exist, we must have det ( M ( E, κ, p x , p y )) = 0 ,for a given E and κ . The determinant vanishing condi-tion gives an equation, which p x and p y must satisfy fora given ( E, κ ) . The set of all such ( p x , p y ) , for E = 0 and fixed κ , yields the reconstructed Fermi arc at theinterface. Appendix E: Twists with n = 4 and n = 5 We have already discussed the results for a non trivialtwist with n = 2 for both the cases of overlapping andnon overlapping WPPs in detail in the main text. Here,for completeness, we consider higher values of n . First,we clarify that the case for n = 3 has not been studiedhere in detail, because it is very similar to the case with n = 2 - in both cases, their superlattices are similar -the SL unit cell contains the same number of lattice sites N sc = 10 and the unit cell is N sc / times larger thanthe original unit cell. So we discuss below, the cases when n = 4 and n = 5 .
1. Twist with n = Here, we shall consider the twist with n = 4 for thecase when the WPPs overlap. We start with a systemwith the slabs aligned and the Fermi arcs lying along the x -axis. Then the top slab is given a twist by an angle θ = tan − (1 / counterclockwise and the bottom slabis twisted counterclockwise by the angle ( π/ − θ ) , suchthat the site (4 , − of top layer lies on top of the site (1 , − of the bottom layer. We choose k = 2 π/ so thatthe positively charged chiral WPP of the top slab lies onthe positively charged chiral WPP of the bottom slaband the negatively charged chiral WPPs also lie on topof each other. The overlapping occurs in the first SL BZafter translating the WPPs by the appropriate reciprocallattice vectors of the SL.The reconstructed Fermi arc is shown in Fig. 9. As inthe n = 2 case, we find a duality in the Fermi arc recon-struction between strong and weak inter-layer coupling FIG. 9. n = 4 with overlapping Weyl point projections inthe surface BZ. In (a) and (b), reconstructed Fermi arcs areshown for couplings in the ranges κ = (0 , and κ = (8 , respectively. The detachment of the FAs from the WPPsand its subsequent reattachment occur in the coupling range κ = (5 , . This region of κ is explored separately in Fig.6.Parameters taken are t (cid:48) = 1 . and V x = 1 , V y = 0 . Theduality between the strong and weak coupling in the FA re-construction is shown in (c) and (d). strengths. There is also a regime of parameters wherethe Fermi arc is detached from the WPPs. The Fermiarc detachment occurs at around κ ∼ and then it againre-attaches at κ ∼ . This is depicted in Fig. 10.
2. Twist with n = The first non trivial case with n odd is when n = 5 .Here we consider the Fermi arc reconstruction for thecase of non overlapping WPPs. We rotate the top slabcounterclockwise and the bottom slab clockwise until thelattice site (5 , − of the top layer and the site (5 , ofthe bottom layer lie on top of each other. Here the angleof rotation is θ = tan − (1 / . We take k = π/ . TheWeyl point projections of positive and negative chiralityare now at k (cos θ , ± sin θ ) and k ( − cos θ , ∓ sin θ ) respectively, where the upper (lower) sign is for the top(bottom) slab. The reconstructed Fermi arcs are shownin Fig. 11. At zero coupling, there are unreconstructedFermi arcs of the individual slabs. As we switch onand increase the coupling, the reconstructed Fermi arcschange their signs of curvature at around κ = 5 (see Fig.11(c) and Fig. 11(d) ). The Fermi arcs deform and splitwhen the coupling is increased slowly beyond κ = 5 . Af-ter splitting, a pair of small closed loops are formed. Thesmall closed loops approach the superlattice BZ bound-7 FIG. 10. The surface BZ for n = 4 in the coupling range κ = (5 . , . is explored, where the FA detaches from the WPPsand then reattaches again.FIG. 11. n = 5 : The panels show the evolution of the reconstructed Fermi arcs with the coupling strength κ . The parametersare k = π/ , t (cid:48) = 0 . , V x = 1 , V y = 0 . The inscribed blue square is the boundary of the superlattice BZ. The Weyl pointprojections of the top (bottom) slab are in red (black) colour. In the first row, the reconstructed Fermi arcs are shown for κ = 0 , , , and 10. Notice the sign change of curvature of Fermi arc for κ = 5 . In the second and third rows, the Fermi arcreconstruction around the coupling κ = 5 has been explored in more detail. 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