Magnetic Response of Majorana Kramers Pairs Protected by Z2 Invariants
aa r X i v : . [ c ond - m a t . s up r- c on ] F e b Journal of the Physical Society of Japan
LETTERS
Magnetic Response of Majorana Kramers Pairs Protected by Z Invariants
Yuki Yamazaki , Shingo Kobayashi , , , and Ai Yamakage Department of Physics, Nagoya University, Nagoya 464-8602, Japan Institute for Advanced Research, Nagoya University, Nagoya 464-8601, Japan Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan RIKEN Center for Emergent Matter Science, Wako, Saitama 351-0198, Japan
On the surface of time-reversal-invariant topological superconductors, Kramers pairs of Majorana fermions with chi-ral and crystalline symmetries exhibit completely uniaxial or octupole anisotropic magnetic response. This paper reportspossible types of magnetic responses of Majorana Kramers pairs with one-dimensional Z invariants defined by crys-talline symmetry. In particular, the general theory predicts a new type of magnetic response where two Majorana Kramerspairs associated with the Z invariant show biaxially (quadrupolar) anisotropic magnetic response, which is a novel typeof response that is rarely observed in conventional and Majorana fermions. The Majorana fermion is a long-sought particle that isits own antiparticle. Some unconventional superconductorswith topological numbers called topological superconductors(TSCs) host Majorana fermions on their surfaces as gaplessAndreev bound states.
The emergent Majorana fermionon the surface follows non-Abelian statistics and is com-pletely stable as long as the superconducting gap remains inthe bulk. These novel properties allow us to apply TSCs tofault-tolerant topological quantum computation with Majo-rana fermions. Many classes of 3D TSCs were discovered based on theconcept of symmetry. Time-reversal symmetry (TRS) pro-tects degenerate gapless states that form a Kramers pair,which is called a Majorana Kramers pair, on the surfaceof 3D time-reversal-invariant TSCs; examples include su-perconducting states in doped topological insulators andDirac semimetals.
In particular, crystalline symmetriesdefine a new type of topological crystalline superconduc-tor (TCSC).
One-dimensional time-reversal-invariant su-perconductors (class DIII) without crystalline symmetry areclassified by the Z topological invariant. In additionto the above Z phase, topological phases in the presenceof crystalline symmetry have been thoroughly explored forreflection,
29, 30) all order-2, nonsymmorphic, and rota-tional
33, 34) symmetries. To study the fundamental nature andpossible applications of these 3D TCSCs, it is necessary to un-derstand how Majorana fermions on 3D TCSCs respond to anexternal field. For instance, a Majorana Kramers pair exhibitsa completely anisotropic (Ising) magnetic response, which isdistinct from the response of conventional (complex) spin-1 / owing to TRS reinforced by crystalline sym-metries, i.e., magnetic symmetry. Previously,
42, 43) we revealed the relation among themagnetic-dipole and magnetic-octupole responses of a Majo-rana Kramers pair; magnetic winding number Z , which coin-cides with the number of Majorana fermions; and irreduciblerepresentation of the superconducting pair potential. By ap-plying this result, one can easily determine which multipolemagnetic response occurs in a 3D TCSC associated with the Z invariant. This method, however, is incomplete: it does notinclude a Majorana Kramers pair associated with Z topologi- cal invariants protected by crystalline symmetry. Recent stud-ies have reported results useful for determining the topologi-cal invariants of 3D TCSCs from the symmetry indicators inthe normal state. In the present paper, we extend thesestudies by systematically elucidating the magnetic responseof Majorana Kramers pairs with one-dimensional invariants ν [ U ] ∈ Z protected by an order-2 symmorphic or nonsym-morphic symmetry U . In the symmorphic case, an energy gapoccurs in the Majorana Kramers pair when an applied mag-netic field breaks the symmetry U . Surprisingly, two Majo-rana Kramers pairs protected by nonsymmorphic symmetryshow a highly anisotropic magnetic response, which is ex-pressed by a quadrupolar-shaped energy gap depending on themagnetic field. E ff ective theory for Majorana Kramers pairs on a sur-face of 3D TSCs. We discuss the magnetic responses of Ma-jorana Kramers pairs by an e ff ective theory for them on a sur-face of 3D TSCs. In general, Z topological invariants cor-respond to the parity of Majorana Kramers pairs thus therecannot exist two or more pairs stably. However, if the sys-tem has a glide plane, there possibly exist two MajoranaKramers pairs.
32, 49)
Therefore, we firstly reveal the depen-dence of the magnetic responses of Majorana Kramers pairson the number of pairs. N Majorana Kramers pairs are de-scribed by Majorana operators γ , · · · , γ N satisfying γ † i = γ i and { γ i , γ j } = δ i j . γ n − and γ n form a Majorana Kramerspair on a surface of 3D TSCs. The time reversals are given by γ n − → γ n and γ n → − γ n − . Then, we obtain the couplingbetween an external field and N Majorana Kramers pairs onthe surface by antisymmetric matrices as J = i γ T A γ , A T = − A , A ∗ = − A . (1)A single Majorana Kramers pair γ = ( γ , γ ) T hosts onlyone operator A = s y , where s i ( i = , x , y , z ) denotes the i thPauli matrix. The operator J is time-reversal-odd (magnetic)and coupled to a magnetic field B . Thus, the Hamiltonianfor a single Majorana Kramers pair under a magnetic field B has the form H MF = f ( B ) s y and the gapped energy spectrum
1. Phys. Soc. Jpn.
LETTERS E M = f ( B ), where f ( B ) is an analytic odd function: f ( B ) = X i ρ i B i + X i , j , k ρ i jk B i B j B k + O (cid:16) B (cid:17) . (2)Here, the coe ffi cients ρ i and ρ i jk depend on the system param-eters.Two Majorana Kramers pairs, on the other hand, can formfour magnetic ( A , A , A , and A ) and two electric ( B and B ) operators. They are represented by A = s y τ , A = s y τ z , A = s τ y , A = s y τ x , B = s z τ y , and B = s x τ y in the basisof ( γ , γ , γ , γ ), where s i acts within a Majorana Kramerspair so that time reversal is given by A → Θ A Θ − with Θ = is y K . The e ff ective Hamiltonian is given by H MF = P i = A i f i ( B ) + P i = B i g i ( B ), where f i ( B ) is an analytic oddfunction that has the same form as Eq. (2) and g i ( B ) is an an-alytic even function of magnetic fields. By diagonalizing theabove matrix, the energy gap is obtained as E M = q f + f + f − q f + g + g ∼ sX i j ρ i j B i B j − sX i j ρ ′ i j B i B j . (3)Consequently, two Majorana Kramers pairs vanish on apply-ing a magnetic field along any direction. However, the in-duced gap shows a highly anisotropic magnetic response, asdemonstrated later.Magnetic responses of Majorana Kramers pairs on a sur-face of 3D TCSCs are constrained by crystalline symmetryin addition to the number of Majorana Kramers pairs dis-cussed above. In the following, we construct a general the-ory for crystalline Z topological phase and discuss the ef-fects of crystalline symmetry on magnetic responses. Then,we show that various anisotropic magnetic responses such asdipole and quadrupole are realized thanks to crystalline sym-metry. Crystalline Z topological phase and magnetic response. In this paper, we focus on Majorana Kramers pairs on a sur-face of 3D TSCs. Note that, as will be shown later, MajoranaKramers pairs protected by crystalline symmetry are charac-terized by a topological number defined in a one-dimensionalsubspace. Hence, the following discussions can be applied totwo-dimensional and one-dimensional systems and nodal su-perconductors. The crystalline symmetry protecting MajoranaKramers pairs which appear on a surface of 3D TCSCs is de-fined below. First of all, we introduce the Hamiltonian andsymmetry operation, following which we classify the possiblemagnetic responses of Majorana Kramers pairs on 3D TCSCs.The BdG Hamiltonian for time-reversal-invariant three-dimensional superconductors has the form H ( k ) = h ( k ) − µ ∆ ( k ) ∆ ( k ) − h ( k ) + µ ! = (cid:2) h ( k ) − µ (cid:3) τ z + ∆ ( k ) τ x , (4)in the basis of ( c k ↑ , c k ↓ , c †− k ↓ , − c †− k ↑ ), where ↑ and ↓ denote theup and down spins, respectively, and the indices for the orbitaland sublattice degrees of freedom are implicit. The Hamil-tonian satisfies TRS, which is expressed as Θ H ( k ) Θ − = H ( − k ); particle–hole C symmetry (PHS), expressed as CH ( k ) C − = − H ( − k ) , C = τ y Θ ; and chiral Γ symmetry,expressed as { Γ , H ( k ) } = , Γ = Θ C = τ y . When the sys- tem is invariant against a symmetry operation g = { R g | τ g } of aspace group, which consists of a rotation / screw axis or reflec-tion / glide plane R g followed by the translation τ g , the Hamil-tonian h ( k ) in the normal state satisfies D † k ( g ) h ( k ) D k ( g ) = h ( g k ), where D k ( g ) is the representation matrix of g and themomentum k is transformed to g k . The pair potential ∆ ( k ),on the other hand, satisfies D † k ( g ) ∆ ( k ) D k ( g ) = χ ( g ) ∆ ( g k ),where χ ( g ) is the character of g for the one-dimensional rep-resentation of the pair potential. Then, the BdG Hamilto-nian is invariant, ˜ D † k ( g ) H ( k ) ˜ D k ( g ) = H ( g k ), for ˜ D k ( g ) = diag[ D k ( g ) , χ ( g ) D k ( g )]. The particle–hole and chiral transfor-mations of the representation matrices depend on the charac-ter: ˜ D †− k ( g ) C ˜ D k ( g ) = χ ( g ) C , (5)˜ D † k ( g ) Γ ˜ D k ( g ) = χ ( g ) Γ , (6)where we use Θ ˜ D k ( g ) Θ − = ˜ D − k ( g ).The square of the representation matrix is given by D k ( g ) = − e − i k · ( R g τ g + τ g ) (7)and classified into D k ( g ) = − D k ( g ) = − g repre-sents twofold rotation and reflection. The twofold screw axisand glide plane also satisfy D k ( g ) = − k · τ g =
0. On theone hand, D k ( g ) = g represents the screw axisand glide plane on the zone boundary with k · τ g = π because τ g can be a half translation vector. We call D k ( g ) = − k ⊥ , k k ), which are components perpendicular and paral-lel to the surface, respectively. Next, k k is fixed to a TRIM,where Majorana Kramers pairs can appear with zero energy,and omitted. The surface symmetry U satisfies[ D k ⊥ ( U ) , h ( k ⊥ )] = . (8)The symmetry can divide the BdG Hamiltonian H into partsin the eigenspaces of ˜ D k ⊥ ( U ) as H = H + ⊕ H − , where the sub-script ± denotes the eigenvalue of ˜ D k ⊥ ( U ). Majorana fermionsin H + and H − do not hybridize with each other. In other words,they are protected by the symmetry U . Note that the represen-tation matrix D k ⊥ ( U ) must be independent of k ⊥ in order toprotect Majorana Kramers pairs, i.e., U is the reflection per-pendicular, glide plane consisting of the reflection perpendic-ular and translation parallel, or twofold rotation perpendicularto the surface. Hereafter, the subscript k ⊥ is omitted.Next, we derive a condition on the representation matrixand character to protect the Z topological phase with Majo-rana Kramers pairs and relate it to the magnetic responses.According to Eq. (5), H ± has particle–hole symmetry for χ ( U ) = D ( U ) = ± D ( U ) are ± ± i for D ( U ) = −
1, respectively. The magneticresponses are classified into three types, (A)–(C), as summa-rized in Table I, which is the main result of this paper.Type (A). For χ ( U ) = − D ( U ) = − H ± breaks TRS(class D). A single Majorana Kramers pair in H = H + ⊕ H − is divided into Majorana fermions in H + and H − , which are
2. Phys. Soc. Jpn.
LETTERSTable I.
Three types, (A)–(C), of magnetic responses of Majorana Kramers pairs. The square of the representation matrix D ( U ) and character χ ( U )of symmetry operation U , which protects Majorana Kramers pairs; symmetry class (Class) and one-dimensional topological invariant (Topo); number ofMajorana Kramers pairs ( E M ( B ) under a magnetic field B are shown. n denotes the direction perpendicular to the mirrorplane when U is a reflection / glide plane or the direction parallel to the rotational axis when U is a twofold rotation. Type k k · τ g D ( U ) χ ( U ) Class Topo E M ( B )A 0 − − Z P i ρ i B i + P i jk ρ i jk B i B j B k with E M ( B n ) = π Z ⊕ Z ∝ n · B C π Z ⊕ Z pP i j ρ i j B i B j − qP i j ρ ′ i j B i B j associated with the Z invariant: ν ± [ U ] = Z π − π dk ⊥ π a ± ( k ⊥ ) mod 2 , (9) a ± ( k ⊥ ) = − i X n ∈ occ h k ⊥ n ±| ∂ k ⊥ | k ⊥ n ±i , (10)where the summation is taken over all occupied states and H ± ( k ⊥ ) | k ⊥ n ±i = E n ± ( k ⊥ ) | k ⊥ n ±i . Because H + and H − areswitched by time reversal, ν [ U ] : = ν + [ U ] = ν − [ U ] holds.A magnetic field along n is perpendicular to the mirror when U is a reflection / glide plane or is parallel to the rotational axiswhen U is a twofold rotation because the applied magneticfield B n keeps the symmetry U . Then, the Z -invariant ν [ U ]remains well-defined under the magnetic field B n and has thesame value as that under zero field. Therefore, we find thatthe energy gap E M is expressed by Eq. (2) as E M = f ( B ),satisfying f ( B n ) = χ ( U ) = D ( U ) =
1, on the other hand, H ± respects TRS (class DIII) and can have a single MajoranaKramers pair, which is associated with ν ± [ U ] = Z π − π dk ⊥ π a ± ( k ⊥ ) mod 2 , (11)with the gauge fixed to Θ | k ⊥ n − ±i = |− k ⊥ n ±i . The wholesystem can be classified into types (B) and (C) as follows.Type (B). This type corresponds to a single MajoranaKramers pair with ( ν + [ U ] , ν − [ U ]) = (1 ,
0) or (0 , U for χ ( U ) = D ( U ) =
1. In reality, it is protected by the magnetic glide-plane symmetry Θ [ U ] = ˜ D ( U ) Θ with Θ [ U ] = −
1, which isregarded as TRS in the whole system with H = H + ⊕ H − . Anenergy gap occurs when the direction n is perpendicular to theglide plane because the applied magnetic field B n breaks themagnetic glide-plane symmetry. The resulting energy gap isproportional to E M ∝ n · B .Type (C). This type corresponds to two Majorana Kramerspairs with ( ν + [ U ] , ν − [ U ]) = (1 , U for χ ( U ) = D ( U ) =
1. Theenergy gap of the Majorana Kramers pairs is given by Eq. (3).
Model Hamiltonian and numerical results.
We finallyverify the type (C) by examining a toy model on a layered 2Dlattice which has two sublattices with the space group
Pmma (No. 51) including a glide plane U on the ( xz ) surface, asshown in Fig. 1. The B u pairing, which is equivalent to p x -wave pairing, satisfies χ ( U ) = D ( U ) = + k x = π andthen the magnetic response possibly is of type (C). Hence, wefocus on the B u pairing for k x = π .Here, we set the lattice constants as 1. The normal part h ( k ) Fig. 1. (a) Top and (b) side views of the lattice structure of the toy model. a represents the primitive translational vector along the x axis. There existtwo sublattices denoted by A (closed circles) and B (open circles). The ( xy )plane (the dashed green line) is the glide plane. Table II.
Parameters of the numerical model. m t t t λ λ ∆ ∆ ′ | B | . . − . . . − . . . . h ( k ) = c ( k ) σ s + t cos( k x / σ ( k x ) s + ( λ s x sin k y + λ s y sin k x ) σ , (12) c ( k ) = m + t cos k x + t cos k y , (13)and the pair potential ∆ ( k ) for the B u pairing is ∆ ( k ) = ∆ σ s z sin k y + ∆ ′ σ ( k x ) s x sin( k x /
2) sin k y , (14)where s and σ denote the Pauli matrices representing the spinand layer degrees of freedom (A and B), respectively. Here,we introduce the modified Pauli matrices σ ( k x ) = e ik x / e − ik x / ! , σ ( k x ) = − ie ik x / ie − ik x / ! , (15)and σ = diag(1 , − λ > ∆ + ∆ ′ , nodes exist at k x = π .Hereafter, we assume the gapped case λ < ∆ + ∆ ′ , whereMajorana Kramers pairs appear. The band structure of the nor-mal state h ( k ) is shown in Fig. 2. The parameters are set tothose listed in Table II. All the bands are twofold degenerateat any momentum owing to inversion and time-reversal sym-metries. Particularly, the bands are fourfold degenerate at the X and S points because of those symmetries and the glide-plane symmetry.The Hamiltonian with the ( xz ) surfaces has the form H ( k x ) = N y X n = c † n ( k x ) ǫ ( k x ) c n ( k x )
3. Phys. Soc. Jpn.
LETTERSFig. 2.
Energy dispersion (left) and Fermi surface (right) of the normalstate h ( k ). + N y − X n = h c † n ( k x ) t y ( k x ) c n + ( k x ) + h . c . i , (16)where N y denotes the number of sites along the y direc-tion. The Fermi energy is set to 0. The system has fourFermi surfaces between the X and S points. The onsiteenergy ǫ ( k x ) and hopping t y ( k x ) are defined by ǫ ( k x ) = ( m + t cos k x ) σ s τ z + t cos( k x / σ ( k x ) s τ z + λ sin k x σ s y τ z and t y ( k x ) = ( t / σ s τ z − i ( λ / σ s x τ z − i ( ∆ / σ s z τ x − i ( ∆ ′ /
2) sin( k x / σ ( k x ) s x τ x , respectively.A magnetic field induces the Zeeman term H Z ( k x ) = N y X n = c † n ( k x ) B · s σ τ c n ( k x ) . (17)The energy spectrum of H ( k x ) + H Z ( k x ) is shown in Fig. 3.Without a magnetic field, i.e., with | B | = ∼ .
4. At k x = π , two Majorana Kramers pairs exist with zero energy onthe ( xz ) surface. Figure 3(b) shows a polar plot of the energygap E M ( B ) with | B | = .
1. Magnetic fields along any directiondestroy the two Majorana Kramers pairs and yield a gap of theorder of ∼ .
12 at maximum. Here, we confirm that Fig. 3(b),which is the magnetic response of type (C), can be reproducedby Eq. (3); E M ( B ) = q ρ xx B x + ρ yy B y + ρ zz B z − q ρ ′ zz B z , (18)where the coe ffi cients are ρ xx = . , ρ yy = . , ρ zz = . , ρ ′ zz = .
709 and all other coe ffi cients are 0. Figure 4shows a polar plot of the function Eq. (18) with | B | = .
1. Bycomparing Figs. 3(b) and 4, one can find that they have almostthe same shape. In conclusion, we verify that the magnetic re-sponse of type (C) appear on the ( xz ) surface of a layered 2Dlattice with the space group Pmma when the pair potential is B u pairing, i.e., the energy gap E M ( B ) mimics the anisotropyof a quadrupole which is allowed within Eq. (3). This behav-ior is entirely di ff erent from those of other Majorana and com-plex fermions. Discussion . We found three types of the magnetic re-sponses in Majorana Kramers pairs with the Z invariants,which depend on the surface symmetry and the numberof Majorana Kramers pairs. In type (A), i.e., the symmor-phic case, there exists only a single Majorana Kramers pair,in which an external magnetic field creates a uniaxiallyanisotropic gap. On the other hand, in types (B) and (C), thecase of nonsymmorphic symmetry, the magnetic response de-pends on the number of Majorana Kramers pairs: (B) a sin-gle Majorana Kramers pair behaves as an Ising spin, whichis the same as the response associated with the Z invariant. (b)(a) Fig. 3. (a) Energy spectrum of Eq. (16) without a magnetic field. Two Ma-jorana Kramers pairs are located at k x = π . They are gapped by an externalmagnetic field. (b) The polar plot of the energy gap | E M | of H + H Z as afunction of B with | B | = . Fig. 4.
The polar plot of the function Eq. (18) with | B | = .
1. The shape isalmost the same as Fig. 3(b). (C) two Majorana Kramers pairs show a biaxially (quadrupo-lar) anisotropic magnetic response, which is a novel typeof response rarely observed in conventional and Majoranafermions.This prediction was verified in a bilayer model with aglide plane (
Pmma ). In fact, our result can be applied tomaterials with any nonsymmorphic space group, such asUCoGe, which has the crystalline symmetry of the spacegroup
Pnma . Previous studies have strongly suggested thatthis material is a ferromagnetic superconductor at ambientpressure
55, 56) and a time-reversal-invariant one at high pres-sure.
A theoretical study reported that, based on an ex-perimental result, the ferromagnetic superconducting statehas A u symmetry of C h and the state can deform into either A u or B u symmetry of D h at high pressure. The B u -pairing(glide-even χ ( G n ) =
1) state hosts two Majorana Kramerspairs on the (0¯11) surface, while the A u -pairing (glide-odd χ ( G n ) = −
1) hosts no Majorana Kramers pair because the sur-face has only G n symmetry satisfying D k x ,π,π ) ( G n ) = χ ( G n ) = B u pairing. Thus, the magnetic response of Majo-rana Kramers pairs on UCoGe is predicted to be of type (C).The gap induced in the Majorana Kramers pairs may be ob-served through surface tunneling spectroscopy under a mag-netic field or with a magnet attached. C , C , and C symmetries, which are beyond the scopeof this paper, might realize new types of magnetic responses.The discussions in this paper suggest that multiple MajoranaKramers pairs can be active against electric perturbations.Therefore, we also need to clarify the Z and Z topologi-cal invariants and the electric responses of multiple MajoranaKramers pairs. These issues will be addressed in a future pa-per. Acknowledgments
This work was supported by Grants-in-Aid for Sci-entific Research on Innovative Areas “Topological Material Science” (GrantNo. JP18H04224) from JSPS of Japan. S. K. was supported by JSPS KAK-4. Phys. Soc. Jpn.
LETTERS
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