Surface states of massive Dirac fermions with separated Weyl nodes
aa r X i v : . [ c ond - m a t . s t r- e l ] J a n Surface states of massive Dirac fermions with separatedWeyl nodes
P. V. Buividovich
Institute of Theoretical Physics, University of Regensburg,D-93053 Germany, Regensburg, Universitätsstraße 31,e-mail:[email protected]
Abstract.
We derive the spectra of surface states for massive Dirac Hamiltonians with either momentum or energy separationbetween the left- and right-handed Weyl nodes. Momentum separation between the Weyl nodes corresponds to the explicitlybroken time-reversal symmetry and the energy separation - to broken parity. Such Hamiltonians provide the simplest modeldescription of Weyl semimetals. We find that the only effect of the energy separation between the Weyl nodes is to decrease theFermi velocity in the linear dispersion relation of the surface states of massive Dirac Hamiltonian. In the case of broken time-reversal symmetry, the spectrum of surface states interpolates in a nontrivial way between the Fermi arc-type and the Diraccone-type dispersion relations. In particular we find that for all values of the mass and the momentum separation between theWeyl nodes the surface states only exist in a strip of finite width in momentum space. We give also some simpler examples ofsurface states in order to make these notes more pedagogical.
Keywords:
Dirac fermions, topological insulators, Weyl semimetals, surface states, Fermi arcs
PACS:
1. INTRODUCTION
Surface states play an extremely important role in the physics of topologically nontrivial states of matter, such as thetopological insulators [1] and the Weyl semimetals [2, 3]. E.g. the nonzero conductivity of the topological insulators issaturated by topologically protected gapless excitations. The hallmark of the Weyl semimetal phase is the emergenceof the Fermi arc in the spectrum of surface states, which is an open line of zero energy surface states joiningthe projections of the bulk Weyl nodes onto the boundary Brillouin zone. A beautiful argument relating 3D Weylsemimetals and 2D Z topological insulators shows that Fermi arcs are also topologically protected [2].The aim of these notes is to provide a simple (and hopefully pedagogical) derivation of the spectrum of surfacestates within the low-energy effective model of topological insulators and Weyl semimetals, which is nothing but thecontinuum Dirac Hamiltonian. In Section (2) we start with a general recipe for computing the spectrum of surfacestates, introducing the wave function of an ideal insulator as the boundary condition for the wave functions of anysurface state. We then consider the simplest example of the surface states of a 3D Z topological insulator, modelledby a Dirac Hamiltonian with a negative mass term.In Section 3 we consider the spectrum of surface states of a massive Dirac Hamiltonian with Weyl nodes which havedifferent energies. In this case the surface states only exist if the Dirac mass term is negative, as for Z topologicalinsulators. The dispersion relation of the surface states is simply the isotropic Dirac cone with the Fermi velocity whichdecreases as the energy separation between the Weyl nodes grows.In Section 4 we study surface states of a Dirac Hamiltonian with the momentum separation between the Weyl nodes.In Subsection 4.1 we consider the simplest case of massless Dirac fermions and demonstrate the existence of theFermi arc in the surface state spectrum. After that we consider massive Dirac fermions and find that the surface statesinterpolate in a nontrivial way between the Fermi arc-like and the Dirac cone-like dispersion relations. In particular,surface states exist only in a finite range of momenta in the surface Brillouin zone for all nonzero values of the Diracmass and the momentum separating the Weyl nodes. When the momentum separation between the Weyl nodes is equalto the Dirac mass, the spectrum of surface states becomes purely one-dimensional. . GENERAL RECIPE FOR THE CALCULATION OF THE SPECTRUM OF SURFACESTATES2.1. Surface states of a Dirac Hamiltonian in the presence of a flat boundary In these notes we will consider the single-particle three-dimensional Dirac Hamiltonians of the following form h = − i a i ¶ i + F ( ~ x ) , (1)where a i = − i g g i = diag ( s i , − s i ) are the Dirac a -matrices, i = x , y , z label spatial coordinates and F ( ~ x ) is someHermitian matrix with two spinor indices which might in general depend on the coordinates. For example, the Diracmass term corresponds to F = m g . The Dirac Hamiltonians of such a general form provide a reasonably accuratelow-energy description of 3D Z topological insulators and Weyl semimetals.In the following we will consider the Hamiltonian (1) as a low-energy description of some crystal which hostsDirac quasiparticles and which fills the infinite half-space z > z = z > z < F do not have any coordinatedependence, and denote F ( ~ x ) ≡ F > for z > F ( ~ x ) ≡ F < for z <
0. We are then interested in the surface states ofthe Hamiltonian (1) confined to the plane z =
0, in a sense that their wave functions decay as e l > z for z → + ¥ and as e l < z for z → − ¥ , with Re l > < l < > v F in front of the derivative term in (1). However, itcan be absorbed into the rescaling of F ( ~ x ) and the energy spectrum, and is not important for the following discussion.Since the system is translationally invariant in the xy plane, we can partially diagonalize the Hamiltonian in theplane wave basis e ik x x + ik y y . We then get the following equation for the eigenstates of h : ( a a k a − i a z ¶ z + F ( z )) Y ( k a , z ) = e ( k a ) Y ( k a , z ) ⇒ ¶ z Y = i a z ( e − a a k a − F > q ( z ) − F < q ( − z )) Y , (2)where small Latin indices are used to label the two transverse coordinates: a = x , y . From the last equation above weimmediately conclude that the coefficients l ≶ which determine the decay of the wave function of the surface stateaway from the surface are the eigenvalues of the constant, space-independent 4 × M ≶ = i a z ( e − a a k a − F ≶ ) . (3)The corresponding eigenvectors are the Dirac spinors y ≶ , for which we assume the normalization | y ≶ | =
1. Thesolution of the equation (2) can be then written in the following general form: Y ( z ) = N > y > e l > z q ( z ) + N < y < e l < z q ( − z ) , (4)The normalization condition for the wave function (4) reads || Y || = | N > | Z − ¥ dz e l > z + | N < | + ¥ Z dz e l < z = | N > | | Re l > | + | N < | | Re l < | = . (5)In addition, in order to satisfy the equation (2) we have to require the continuity of the wave function across theboundary at z =
0, which gives the condition N > y > = N < y < . Taking into account that y ≶ are by definitionnormalized to unity, we conclude that the absolute values of N > and N < should be equal, and that the spinors y > and y < should be equal up to a phase: y > = n y < with | n | = e that the two matrices M ≶ have matchingeigenvectors y > = n y < for which the corresponding eigenvalues l ≶ satisfy Re l > <
0, Re l < > In practice, one is often interested in the surface states which live on the boundary between some topologicallynontrivial material and the vacuum, which can be thought of as an “ideal” insulator. Since we know that in the vacuumthe electrons are described by the Dirac Hamiltonian with a huge (as compared to typical scales in condensed matterhysics) mass of m e ≈ . F = m e g , assuming that the typical electron momenta and energies are much less than m e . Strictly speaking, weshould also take into account that inside the crystal the electron moves with a Fermi velocity v F , and outside - withthe speed of light. At the level of the eigenstate equations (2), however, the difference of the Fermi velocities will onlyresult in a rescaling of m e by v F , which will still be very large as compared to other scales in the problem.We will therefore assume that at z < M < in (3) with positive real part of the eigenvalue, assuming | k | ≪ m e , | e | ≪ m e . Writing the matrix M < and itseigenstate y < = { f < , c < } in the chiral block form, we obtain the following eigenstate equation: i (cid:18) s z − s z (cid:19) (cid:18) e − k / − m e − m e e + k / (cid:19) (cid:18) f < c < (cid:19) = l < (cid:18) f < c < (cid:19) , (6)or, individually for each of the chiral components f < , c < : c < = e − k / + i l < s z m e f < , f < = e + k / − i l < s z m e c < , (7)where we have denoted k / ≡ s x k x + s y k y . Combining the two above equations, we can express l < in terms of e as l < = p m e + k − e . We have taken into account that the real part of l < should be positive and denoted k = q k x + k y .From (7) one can also read off the general form of the eigenstates of M < : y < ( k a ) = N h ( k a ) e − k / + i l < s z m e h ( k a ) ! , (8)where N is the normalization factor and h ( k a ) is an arbitrary normalized two-component Weyl spinor which canhave some momentum dependence.Taking into account the smallness of e and k as compared to m e , we can write l < ≈ m e . We can also neglect e and k / in the second component of (7). We conclude therefore that in the limit k ≪ m e , | e | ≪ m e the eigenstates y < of M < have the following form: y < ( k a ) = √ (cid:18) h ( k a ) i s z h ( k a ) (cid:19) . (9)We call this spinor “the wave function of an ideal insulator”. We see that its form is completely independent of themomentum k , the energy e and the electron mass. By virtue of the continuity of the wave function at the boundaryof material, we conclude that all the surface states of any material bounded by the vacuum should have this structure.The nontrivial properties of the surface states are then encoded in the nontrivial momentum dependence of the Weylspinor h ( k a ) .From now on we will always assume that the nontrivial material occupies the half-space at z >
0, and the vacuum isat z <
0. To shorten the notation, from now on we will also omit the subscript > for the quantities characterizing thewave functions at z > k a from the arguments of the wave functions. In order to illustrate the application of the above formulae on the simplest possible example, in this Subsectionwe consider the surface states of a 3D Z topological insulator, which at low energies can be modelled by the DiracHamiltonian with a negative mass term m [1]. The eigenstate equation for the matrix M has the same form as in (6)with the replacement m e → m , and the corresponding eigenstates read y = N (cid:18) h e − k / + i ls z m h (cid:19) , (10)where l = −√ m + k − e and N is some normalization constant. At the surface of the topological insulator thiswave function should be equal to the wave function (9) of an ideal insulator. These boundary conditions immediatelyead to the following linear equation for h : e − k / + i ls z m h = i s z h . (11)The consistency condition for these linear equations yield the equation from which one can find the energies e of thesurface states: e = k − ( l − m ) ⇒ l = m , e = ± k . (12)Note that for this solution the real part of l is only negative if the Dirac mass m is negative, that is, if our materialhas a nontrivial Z topological index with respect to the vacuum, which we have characterized by the ideal insulatorwave function (9). Of course only the relative sign of the Dirac mass in the vacuum and in the material is important,and the assumption that the Dirac mass in the vacuum is positive is merely a conventional choice [1]. We thus see thatthe dispersion relation of the surface states of a 3D Z topological insulator is simply the Dirac cone with unit Fermivelocity: e = ± k ≡ ± q k x + k y . (13)Substituting now the expressions (12) into (11), we immediately see that the Weyl spinor h is the eigenstate of thespin operator k / : k / h = eh . Since h = (cid:8) h ↑ , h ↓ (cid:9) encodes the spin polarization of the surface states, we conclude thatfor 3D Z topological insulators the spin of the surface states is always aligned with the momentum. This is the famousspin-momentum locking mechanism, which prevents backscattering of surface states at impurities which cannot flipthe spin (e.g. the non-magnetic impurities). Indeed, the back-scattered wave should have opposite orientations of boththe momentum and the spin, but the processes of spin flips are highly suppressed in the absence of magnetization.For completeness, let us also remark that in real 3D topological insulators the spin of the surface states is actuallynot aligned, but rather perpendicular to the momentum. This is simply the consequence of the fact that for real 3D Z topological insulators the kinetic term in the effective Dirac Hamiltonian should be written as k / = k x s y − k y s x (seee.g. [4, 5, 6]). This specific permutation of x and y indices in the Dirac Hamiltonian is a direct reflection of a strongspin-orbital coupling, an important feature of all topological insulators. However, as long as one is interested only inthe energy spectrum of the surface states, one can use the form k / = k x s x + k y s y which is common in high-energyphysics and which only differs from k / = k x s y − k y s x by a redefinition of Pauli s -matrices.
3. SURFACE STATES FOR MASSIVE DIRAC FERMIONS WITH CHIRALITYIMBALANCE
Let us now consider a more nontrivial example of a massive Dirac Hamiltonian for which the left- and the right-handed Weyl nodes have different energies. Such Hamiltonian might serve as a low-energy effective theory of a Weylsemimetal with broken parity [6, 7, 8, 9]. In particular, such a material should support the Chiral Magnetic Effect[10, 8]. Without loss of generality, we can assume that the energies of the Weyl nodes are ± m A , where m A is the chiralchemical potential [10].We now consider the case when the space at z < z > m and the chiral chemical potential m A . In this case the bulk Hamiltonian inmomentum space and the corresponding matrix F in (1) have the form h = (cid:18) k i s i − m A mm − ( k i s i − m A ) (cid:19) , F = (cid:18) − m A mm m A (cid:19) . (14)The bulk energy spectrum which corresponds to such a Dirac Hamiltonian is E s , s (cid:16) ~ k (cid:17) = s r(cid:16) | ~ k | − sm A (cid:17) + m , s , s = ± . (15)The eigenvalue equation for the matrix M in (3) in this case has the form (cid:18) e − k / + m A − m − m e + k / − m A (cid:19) (cid:18) fc (cid:19) = (cid:18) − i s z lf i s z lc (cid:19) , (16)r, in a component-wise form c = e − k / + m A + i s z l m f , f = e + k / − m A − i s z l m c . (17)For simplicity, we assume now that we are interested in the low-energy excitations with e < m . Then the compatibilitycondition for the equations (17) leads to the following two values of l with negative real parts: l s = − r k + (cid:16)p m − e + i sm A (cid:17) , (18)where we assume that √ m − e >
0. We see that in the presence of the chiral chemical potential m A l s acquire somenonzero imaginary part. Moreover, l + and l − are now complex conjugate. Remembering that the wave functions ofthe surface states depend on the “depth” coordinate z as e l z , we conclude that now the wave functions of the surfacestates should exhibit some oscillations as they decay at large z .By a direct substitution one can check that f and c should be proportional to the eigenvectors h s of the operator k / − i s z l s : ( k / − i s z l s ) h s = − i s q l s − k h s = − i s (cid:16)p m − e + i sm A (cid:17) , (19)where in the last line we have used the identity l s − k = (cid:16) √ m − e + i sm A (cid:17) which follows from (18). Up tonormalization h s can be written as h s = { , q s } , q s = i l s + m A − i s √ m − e k x − ik y . (20)Using (17), we can now find the eigenvectors of M which correspond to the eigenvalues (18): y = (cid:18) h s x s h s (cid:19) , x s = e + m A + i s (cid:16)p m − e + i sm A (cid:17) = e + i s p m − e . (21)Since y s , s = ± M with negative real parts, the wave functions of surfacestates can be expressed as some linear combination of y s : Y ( z ) = (cid:229) s = ± c s y s e l s z . In order to satisfy the continuityof the wave function, we have to require that at z = Y ( z ) is equal to the ideal insulator wave function (9). Thisimmediately leads to the following equations: c + h + + c − h − = h , c + x + h + + c − x − h − = i s z h , ⇒ c + ( x + h + − i s z h + ) + c − ( x − h − − i s z h − ) = . (22)Using now the explicit form of h s , we arrive at the following system of equations for c ± : c + ( x + − i ) + c − ( x − − i ) = , c + q + ( x + + i ) + c − q − ( x − + i ) = . (23)These equations are compatible if ( x + − i )( x − + i ) q − = ( x + + i ) ( x − − i ) q + . (24)The real values of e at which this equation is satisfied are the energies of the surface states. Using the explicit form of x s from (21) and q s from (20), after some algebra we can rewrite the above equation as (cid:18) + m √ m − e (cid:19) l + + (cid:18) − m √ m − e (cid:19) l − = ( m + i m A ) . (25)Taking into account that l + and l − are complex conjugate and that m , e , √ m − e and m A are real, one can easily seethat the above equation is equivalent toRe l + = m , Im l + = m A √ m − e m . (26)rom the first equation it becomes obvious again that the surface states only exist if m <
0. This means thatsurface states only exist if the Weyl semimetal is simultaneously also the Z topological insulator. These featuresin fact do not contradict each other, since the dispersion relation (15) features both the Weyl nodes at the energies E = ± q m A + m and the gap of size 2 | m | centered around E =
0. While the conventional ohmic conductivity vanishesat zero temperature because of this gap, the chiral magnetic conductivity is still nonzero due to the existence of Weylnodes [11, 12].In order to find the energies e of surface states, we have now to solve the equation − r k + (cid:16)p m − e + i m A (cid:17) = m + i m A √ m − e m . (27)Squaring both the r.h.s. and the l.h.s. of this equation we finally arrive at the following dispersion relation for thesurface states in the presence of chiral chemical potential: e = ± | k | q + m A m . (28)We see that the dispersion relation at nonzero chiral chemical potential is still the Dirac cone, and the only effect ofthe chiral chemical potential is to decrease the Fermi velocity v F ∼ m √ m + m A . It is interesting to note that the Fermivelocity of the surface states appears to be different (and larger for m A < m ) than the bulk Fermi velocity V F , whichaccording to (15) is equal to V F = (cid:12)(cid:12)(cid:12)(cid:12) ¶¶ k E s , s ( k ) | k = (cid:12)(cid:12)(cid:12)(cid:12) = | m A | q m A + m . (29)
4. SURFACE STATES FOR MASSIVE DIRAC FERMIONS WITH MOMENTUMSEPARATION BETWEEN THE WEYL NODES
In this Section we consider the case of Dirac Hamiltonian with momentum separation between the Weyl nodes.Such a Hamiltonian provides a low-energy effective description of time-reversal breaking Weyl semimetals [2, 6, 7].Physically, momentum separation between the Weyl nodes can be achieved, for example, by magnetic doping of a 3Dtopological insulator [2, 13]. Direct signatures of the momentum separation between the Weyl nodes are the anomalousHall effect [6, 7] as well as the existence of the Fermi arc in the spectrum of surface states - an open line of topologicallyprotected zero energy states which joins the projections of the bulk Weyl nodes onto the surface Brillouin zone [2].We first illustrate the emergence of the Fermi arc states in the simplest case of massless Dirac fermions with Weylnodes at different momenta, and then consider the more general and complicated case of massive Dirac fermions.While similar calculations were presented in [7, 14], here we extend the analysis of the surface states also to thecase when absolute value of the Dirac mass is so large that the Weyl nodes no longer exist and explicitly follow theinterpolation between the Fermi arc-like and the Dirac cone-like dispersion relations.
For massless Dirac fermions with Weyl nodes at ~ k = ± ~ b the bulk Hamiltonian in the momentum space and thecorresponding matrix F in (1) have the form h = (cid:18) s i ( k i − b i ) − s i ( k i + b i ) (cid:19) , F = (cid:18) − b / − b / (cid:19) , (30)where we assume that ~ b = { b , , } . Fermi arcs appear in the spectrum in this case if the boundary of the Weylsemimetal is parallel to the xy plane, that is, exactly in the case which we consider.With F given by (30), the equations for the eigenstates y = { f , c } of M can be written as ( e − k / + b / + i s z l ) f = , ( e + k / + b / − i s z l ) c = . (31)rom these equations, we find two independent solutions with negative real l : l = − q | k − b | − e , f = N (cid:26) , ¯ k − ¯ b e − i l (cid:27) , c = , l = − q | k + b | − e , f = , c = N (cid:26) , − ¯ k + ¯ b e + i l (cid:27) , (32)where N and N are some normalization factors. Taking into account that for the vacuum boundary conditions (9) { f , c } = N { h , i s z h } , we can write the following equation for the general solution which should be representable assome linear combination of y = { f , c } and y = { f , c } : N N k − ¯ b e − i l N − N k + ¯ b e + i l = h ↑ h ↓ i h ↑ − i h ↓ . (33)After some simple algebraic transformations we find the following equation for e : k − b e + i l = k + b e − i l , (34)or, in somewhat more explicit form e − i p | k − b | − e k − b = e + i p | k + b | − e k + b . (35)It is now easy to guess the following solution to this equation: e = − k y , | k x | < | b | . (36)This is the Fermi arc solution. We see that there is a line of zero energy joining the projections of the bulk Weyl nodesonto the surface momentum space. The dispersion relation is effectively one-dimensional in the direction perpendicularto the separation between the Weyl nodes. Moreover, surface states only exist in a finite strip in momentum space with | k x | < | b | . We now consider the more general case of massive Dirac Hamiltonian with the time-reversal breaking terms of theform (30). Correspondingly, the momentum-space bulk Hamiltonian and the matrix F in (1) have the form h = (cid:18) s i ( k i − b i ) mm − s i ( k i + b i ) (cid:19) , F = (cid:18) − b / mm − b / (cid:19) . (37)The bulk dispersion relation of this Hamiltonian is [7]: E s , s (cid:16) ~ k (cid:17) = s s(cid:18)q k x + m + s b (cid:19) + k y + k z , s , s = ± . (38)Physically, such Hamiltonian describes a Z
3D topological insulator with magnetic doping which explicitly breakstime-reversal symmetry. It is important to note that the distance between the Weyl nodes is now 2 √ b − m rather than2 b . Thus the Dirac mass term tends to move the Weyl nodes together and eventually annihilates them if | b | < | m | . Ifthe Dirac mass m is negative, this situation corresponds to the 3D topological insulator for which the magnetic dopingis still small compared to the topological mass gap m .Let us now study the surface states for such a Hamiltonian. Partly this has been done in [7], but here we extend theanalysis also to the case of large negative Dirac masses with m < −| b | in order to see how the Dirac cone dispersionelation (13) typical for the surface states of 3D topological insulators transforms into the Fermi arc-like dispersionrelation (36) typical for Weyl semimetals as the parameter b is tuned across the critical value m = −| b | .The eigenvalue equation for the matrix M now has the form (cid:18) e − k / + b / − m − m e + k / + b / (cid:19) (cid:18) fc (cid:19) = (cid:18) − i s z lf i s z lc (cid:19) , (39)or, in a component-wise form c = e − k / + b / + i s z l m f , f = e + k / + b / − i s z l m c . (40)Substituting the first equation into the second one, we obtain the following equation for f : (cid:0) e + e b / + b + [ k / − i ls z , b / ] − k + l − m (cid:1) f = . (41)We see thus that f should be the eigenstate of the operator D = e b / + [ k /, b / ] − i l [ s z , b / ] , which can be written as thefollowing 2 × = b (cid:18) − ik y e − i le + i l ik y (cid:19) . (42)The eigenvalues of this matrix are d s = s b x q l + e − k y , with s = ±
1. The corresponding eigenvectors have thefollowing form, up to normalization: f s = { , q s } , q s = ik y + s q l + e − k y e − i l = ik y + s p k x + m − b e − i l , (43)where in the last line we have used the explicit expression for l in terms of s (46), given below. Substituting thisexpression for f s into the first equation (40), we also find c s : c s = m (cid:18) ( e + i l ) + ( − k x + ik y + b ) q s ( − k x − ik y + b ) + ( e − i l ) q s (cid:19) (44)Substituting the eigenstates of D f into (41), we obtain the following equation for l : e + b − k + l − m + s b q l + e − k y = . (45)The solutions of this equation with negative real part are given by l s = − s(cid:18)q k x + m − s b (cid:19) + k y − e . (46)The square root in the brackets (inside the outer square root) can in principle have an arbitrary sign. In order to matchthe expression for the component q s of f s given by the last line of (43), we have to set this sign to +
1, assuming that p k x + m >
0. Now l s should be substituted in the expressions (43) and (44) above. It is obvious that l s given by(46) have either zero imaginary or zero real part. In order to find the localized surface states, we should only considerthe solutions with nonzero (and negative) real part.Now we proceed as in Section (3) and represent the wave function Y ( z ) of the surface state as a linear combinationof Y + ( z ) = { f + , c + } e l + z and Y − ( z ) = { f − , c − } e l − z with some coefficients c + and c − . Matching Y ( z = ) to thewave function (9) of an ideal insulator, we arrive at the following equations: c + f + + c − f − = h , c + c + + c − c − = i s z h ⇒ c + ( c + − i s z f + ) + c − ( c − − i s z f − ) = . (47)Taking into account the explicit form of c s and f s given by (43) and (44), we then arrive at the following compatibilitycondition for the above system of equations: ( e + i l + − kq + − im )(( e − i l − ) q − − ¯ k + i q − m ) = ( e + i l − − kq − − im )(( e − i l + ) q + − ¯ k + i q + m ) , (48) - - - ky m = - - - - - ky m = - - - - - ky m = - - - - - ky m = FIGURE 1.
Contour plots of the dispersion relation of the surface states e (cid:0) k x , k y (cid:1) for the Weyl semimetal with a spatialmomentum separation 2 b = . b = .
5) between the Weyl nodes and different values of the Dirac mass m . Red points markthe projections of the bulk Weyl nodes ( k x = ±√ b − m if | b | > | m | , k x = k y =
0) onto the surface momentum space. where we have denoted k = k x − ik y − b . After some algebra and guesswork (guided by the numerical checks and thecalculations of [7]) we solve these equations with respect to e and arrive at the following results for the spectrum ofthe surface states: • At m > | b | no surface states exist. • At −| b | < m < | b | there are surface states for | k x | < √ b − m , which have the simple dispersion relation e = k y .In particular, the open line | k x | < √ b − m , k y = e = k x = ±√ b − m [2]. • At m < −| b | surface states exist for | k x | < m q m b −
1. The dispersion law is the anisotropic Dirac cone: theFermi velocity in the y direction v F y = ¶e¶ k y is always unity, and the Fermi velocity in the x direction v F x = ¶e¶ k x is v F x = q − b m . The Dirac cone is particle-hole symmetric, that is, the energies of surface states always come inpairs ± e . • At m = −| b | the surface states only exist at k x =
0. Their dispersion relation is e = ± k y . Note the appearance ofone more branch of the dispersion relation ( e = − k y ) as compared to the dispersion relation at −| b | < m < | b | .In order to illustrate these results, on Fig. 1 we present the contour plots of the dispersion relation of the surfacestates e ( k x , k y ) for b = . m , both positive and negative. At m < −| b | there are two oppositevalues of e which correspond to the same k x and k y , therefore we show the contour plots only for the branch with e > . CONCLUSIONS In these notes we have given a general recipe for computing the spectrum of surface states for materials in whichquasiparticle excitations are described by Dirac Hamiltonians of the form (1) at low energies. We have also explicitlyderived the spectra of low-energy surface states of 3D topological insulators and Weyl semimetals with broken parityand time-reversal symmetries.In the case when the parity is broken by the energy separation between the Weyl nodes we have found that the onlyeffect of the energy separation is the reduction of the Fermi velocity in the Dirac cone dispersion relation of 2D surfacestates.The spectrum of surface states of massive Dirac Hamiltonian with broken time-reversal symmetry turned out tobe more complicated. Physically, this Hamiltonian describes the magnetically doped 3D topological insulator. If themagnetic doping is sufficiently large, 3D topological insulator turns into a Weyl semimetal with momentum separationbetween the Weyl nodes. We have found that if the magnetic doping is not very large, its effect is to make the Diraccone anisotropic by decreasing the Fermi velocity in the direction of the magnetization. At the same time, the rangeof momenta in which the surface states exist is shrunk to a strip of finite width, which is inversely proportional to themagnetization if the magnetization is small. As the magnetization becomes equal to the topological mass gap, this stripshrinks to a line which is perpendicular to the magnetization, and one branch of the dispersion relation disappears. Asthe magnetization is further increased, the bulk energy spectrum develops two separated Weyl nodes. Correspondingly,the surface states develop the Fermi arc and the dispersion relation becomes effectively one-dimensional: e = k y . Thesurface states still exist in a finite strip with the width being equal to the separation between the bulk Weyl nodes. Thispicture of the evolution of the Dirac cone dispersion relation into the effectively one-dimensional Fermi arc dispersionrelation might be useful for the identification of the critical magnetic doping of 3D topological insulators which leadsto the emergence of a Weyl semimetal phase. Acknowledgements
This work was supported by the S. Kowalevskaja award from the Alexander von Humboldt foundation.
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