Superconductivity in magnetic multipole states
aa r X i v : . [ c ond - m a t . s up r- c on ] J un Superconductivity in magnetic multipole states
Shuntaro Sumita ∗ and Youichi Yanase Department of Physics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan (Dated: October 24, 2018)Stimulated by recent studies of superconductivity and magnetism with local and global brokeninversion symmetry, we investigate the superconductivity in magnetic multipole states in locallynoncentrosymmetric metals. We consider a one-dimensional zigzag chain with sublattice-dependentantisymmetric spin-orbit coupling and suppose three magnetic multipole orders: monopole order,dipole order, and quadrupole order. It is demonstrated that the Bardeen-Cooper-Schrieffer state,the pair-density wave (PDW) state, and the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state arestabilized by these multipole orders, respectively. We show that the PDW state is a topologicalsuperconducting state specified by the nontrivial Z number and winding number. The origin of theFFLO state without macroscopic magnetic moment is attributed to the asymmetric band structureinduced by the magnetic quadrupole order and spin-orbit coupling. I. INTRODUCTION
Emergent phenomena in electron systems lacking in-version symmetry have received a lot of attention inrecent condensed matter physics . In such noncen-trosymmetric systems, antisymmetric spin-orbit cou-pling (ASOC) entangles various internal degrees offreedom: for instance, spin, orbital, sublattice, andmultipole. Recent studies uncovered exotic supercon-ducting and multipole phases induced by thesublattice-dependent ASOC in locally noncentrosymmet-ric systems. In this paper, we clarify nontrivial interplaybetween the superconductivity and the multipole orderby investigating the superconductivity in the magneticmultipole states.Even-parity multipole order has been intensively re-searched mainly in the field of heavy-fermion systems.For instance, the electric quadrupole and magnetic oc-tupole order have been identified in various materials .Furthermore, the electric hexadecapole moment andmagnetic dotriacontapole moment have been proposedas plausible candidates for the hidden order parameterin the heavy-fermion superconductor (SC) URu Si .On the other hand, recent theories pointed out theodd-parity multipole order which may occur in the locallynoncentrosymmetric systems as a result of the antiferroalignment of the even-parity multipole in the unit cell.For instance, the “antiferromagnetic moment” in the unitcell induces a magnetic quadrupole moment , and theantiferro stacking of the local electric quadrupole mo-ment in bilayer Rashba systems is regarded as an electricoctupole order . As a consequence of the spontaneousglobal inversion symmetry breaking, intriguing magneto-electric responses occur in the ferroic odd-parity multi-pole states . Recent experiments detected a signatureof the odd-parity multipoles in Sr IrO . Inspired bythese works, we study exotic superconductivity inducedby the odd-parity multipoles and even-parity multipoles.Intensive theoretical studies in these years have shownthat noncentrosymmetric SCs are platforms of vari-ous nonuniform superconducting states . In the glob-ally noncentrosymmetric systems an infinitesimal mag- netic field stabilizes a Fulde-Ferrell-Larkin-Ovchinnikov(FFLO) state , which is called the helical supercon-ducting state . Agterberg and Kaur discussed thestability of the magnetic-field-induced FFLO (helical)state in Rashba SCs . However, it has been shown thatthe FFLO order parameter is hidden in vortex states .In the locally noncentrosymmetric systems, the pair-density-wave (PDW) state or the complex stripe state may be stabilized, depending on the direction of mag-netic field. These states are not hidden in the vor-tex states, but a magnetic field higher than the Pauli-Chandrasekhar-Clogston limit is required. Reference 3has shown that the PDW state is stable in multilayerSCs having “weak interlayer coupling” and “moderatespin-orbit coupling” when the paramagnetic depairing ef-fect is dominant. Then, the phase of the superconduct-ing order parameter modulates layer by layer. There-fore, the PDW state is an odd-parity superconductingstate although the spin-singlet Cooper pairs lead to thecondensation. Since the odd-parity SC is a platform oftopological superconducting phases , topologically non-trivial properties of the PDW state are implied. Indeed,the PDW state in 2D multilayer systems has been iden-tified as being a crystal-symmetry-protected topologicalsuperconducting state . bt t a c-axis (a) (b) b-axis a(z)-axisb-axis FIG. 1. Crystal structure of the 1D zigzag chain. (a) Projec-tion along the a axis. (b) Projection along the c axis. Blueand red circles represent the a and b sublattices, respectively.The hopping integrals are shown by t and t . The previous theories introduced above discussed thesuperconducting state in the magnetic field. In thispaper, we investigate the superconductivity caused bythe cooperation of various magnetic multipoles andsublattice-dependent ASOC. Since this is an early the-oretical study for those systems, we treat a one-dimensional (1D) zigzag chain (Fig. 1) as a minimalmodel. Indeed, the zigzag chain is a simple crystal struc-ture lacking the local inversion symmetry. Althoughthe superconducting long-range order does not occur instrictly 1D systems because of divergent fluctuations ,we investigate superconducting states with the use ofthe mean-field (MF) theory (Sec. II B) by allowing thelong-range order. This treatment is appropriate for ourpurpose to pave a way to realize exotic superconductiv-ity in a broad range of systems. Indeed, our results arejustified in quasi-1D coupled zigzag chains, and some ofthe results will give new insight on more complicatedthree-dimensional (3D) systems with broken local inver-sion symmetry. a(z)-axis (M) b(x)-axis + a(z)-axis (Q) b(x)-axis + −− + c(y)-axis (D) b(x)-axis +− +− FIG. 2. Magnetic structure in the magnetic (M) monopolestate, (D) dipole state, and (Q) quadrupole state. Projectionalong the c axis is shown in (M) and (Q), meanwhile along the a axis is shown in (D). Black arrows show the ferromagneticor “antiferromagnetic” moments in the unit cell. We show that the PDW state is stabilized in the mag-netic dipole state [Fig. 2, panel (D)] as in the multi-layer systems, while the conventional Bardeen-Cooper-Schrieffer (BCS) state is robust in the magnetic monopolestate [Fig. 2, panel (M)]. Topologically distinct proper-ties of the PDW state are specified by the Z and Z topological invariants. The Majorana end state is as-sociated with nontrivial topological invariants. In thissense, the odd-parity PDW state is regarded as a re-alization of the Kitaev superconducting wire without p -wave Cooper pairs. Ferromagnetic heavy fermion SCs,UGe , URhGe , and UCoGe have crystal structureconsisting of zigzag chains, and they are likely to showthe odd-parity superconductivity. Therefore, these com-pounds are candidates for the PDW state.Furthermore, we show that the FFLO state is sta-ble in the magnetic quadrupole state [Fig. 2, panel(Q)] without macroscopic magnetization. The magneticquadrupole order occurs in several materials. For exam- ple, 1-2-10 compounds such as CeRu Al show magneticquadrupole order in zigzag chains . Because any ex-ternal magnetic field is not required for the FFLO state,the orbital effect harmful for the FFLO state is com-pletely eliminated. Thus, the magnetic quadrupole stateis a good platform realizing the FFLO state which hasbeen searched for more than 50 years .This paper is constructed as follows. In Sec. II A, weintroduce a model for conduction electrons affected by asublattice-dependent ASOC, magnetic multipole order,and an s -wave attractive interaction. Then, we analyzethe model with the use of the MF theory in Sec. II B. Weillustrate the monopole, dipole, and quadrupole order inSec. III A. The symmetry and degeneracy of the bandstructure are elucidated by clarifying the symmetry pro-tection. In Sec. III B, we introduce the order parameterof superconducting states discussed in this paper. Weshow that the BCS state is robust against the magneticmonopole order in Sec. IV. On the other hand, the dipoleorder stabilizes the PDW state as shown in Sec. V A. ThePDW state is identified to be a topological superconduct-ing state in a certain parameter regime (Sec. V B). Sec-tion VI gives the result for the FFLO state induced bythe quadrupole order despite the absence of the exter-nal magnetic field. It is shown that the center-of-massmomentum of Cooper pairs arises from the asymmetricband structure. Finally, a brief summary and discussionare given in Sec. VII. II. MODEL AND FORMULATIONA. Model
First, we introduce a model describing superconductiv-ity coexisting with magnetic order in a 1D zigzag chain, H = X k,s [ ε ( k ) a † ks b ks + h.c.]+ X k,s [ ε ′ ( k ) − µ ][ a † ks a ks + b † ks b ks ]+ α X k,s,s ′ g ( k ) · ˆ σ ss ′ [ a † ks a ks ′ − b † ks b ks ′ ] − X k,s,s ′ [ h a · ˆ σ ss ′ a † ks a ks ′ + h b · ˆ σ ss ′ b † ks b ks ′ ]+ 1 N X k,k ′ ,q V a ( k, k ′ ) a † k + q ↑ a †− k + q ↓ a − k ′ + q ↓ a k ′ + q ↑ + 1 N X k,k ′ ,q V b ( k, k ′ ) b † k + q ↑ b †− k + q ↓ b − k ′ + q ↓ b k ′ + q ↑ , (1)where a ks and b ks are the annihilation operators of elec-trons with spin s = ↑ , ↓ on the sublattices a and b , re-spectively. The wave vector k is directed to the crystal-lographic c axis.The first and second terms are the inter-sublattice andintra-sublattice hopping terms including the chemical po-tential µ , respectively. The kinetic energy ε ( k ) and ε ′ ( k )are obtained by taking into account the nearest- andnext-nearest-neighbor hoppings, ε ( k ) = − t cos k , (2) ε ′ ( k ) = − t cos k. (3)The crystal structure and hopping integrals, t and t ,are illustrated in Fig. 1.The third term is a sublattice-dependent ASOC whichoriginates from the violation of local inversion symme-try . The g vector is approximated as g ( k ) = sin k ˆ z . Wechoose the crystallographic a axis as the quantizationaxis of the spin, namely, ˆ z = ˆ a .The fourth term expresses the molecular field of mag-netic monopole, dipole, and quadrupole order. This termcauses various superconducting phenomena, which aredemonstrated in this paper. We assume that the Neeltemperature T N is much larger than the superconductingtransition temperature T C . In this situation, the fluc-tuation of multipole order is ignorable below T C . Ef-fects of superconductivity on the magnetic order are alsoignorable because the energy scale of superconductivityis much smaller than the magnetic interaction energy.Therefore, our assumption for fixed magnetic order isjustified.In order to study superconductivity in this system, weintroduce an attractive interaction by the fifth and sixthterms in Eq. (1), where N is the number of sites in eachsublattice. For simplicity, we assume s -wave supercon-ductivity by adopting the momentum-independent pair-ing interaction, V a ( k, k ′ ) = V b ( k, k ′ ) = − V. (4)Although the spin-triplet p -wave order parameter is in-duced by the ASOC through either attractive or repulsiveinteraction in the p -wave channel, we neglect the p -waveorder parameter. It has been shown that the admixed p -wave component does not change the phase diagramunless the p -wave attractive interaction is comparable toor larger than the s -wave interaction .The purpose of this paper is to clarify exotic supercon-ducting phases stabilized by the spin-orbit coupling andmagnetic multipole order. For this purpose, we treata “deep” zigzag chain t /t < α/t = 0 . V /t = 1 . B. Mean-field theory
Second, we investigate the superconducting state bymeans of mean-field (MF) theory. The interaction terms are approximated as follows: − VN X k,k ′ ,q a † k + q ↑ a †− k + q ↓ a − k ′ + q ↓ a k ′ + q ↑ + ( a → b ) ≃ X k [∆ ∗ a a − k + q ↓ a k + q ↑ + h.c.] + NV | ∆ a | + ( a → b ) , (5)by introducing the order parameter∆ a = − VN X k ′ h a − k ′ + q ↓ a k ′ + q ↑ i , ∆ b = − VN X k ′ h b − k ′ + q ↓ b k ′ + q ↑ i . (6)Thus, in this paper we assume a single- q state. The con-densation energy is optimized with respect to the center-of-mass momentum q of Cooper pairs. In the BCS stateand PDW state, q = 0 as we introduce in Sec. III B. Wealso examine the q = 0 state corresponding to the FFLOstate . The order parameters of the superconductingstates are summarized in Sec. III B.We here describe the MF Hamiltonian in a matrixform. We define k + ≡ k + q , k − ≡ − k + q , and thevector operatorˆ C † k ≡ ( a † k + ↑ , a † k + ↓ , b † k + ↑ , b † k + ↓ , a k − ↑ , a k − ↓ , b k − ↑ , b k − ↓ ) . (7)Then, we obtain H MF = 12 X k ˆ C † k ˆ H ( k ) ˆ C k + W , (8)with W = − X k ε ′ ( k − ) − µ ] + NV | ∆ a | + NV | ∆ b | . (9)The explicit form of the 8 × H ( k ) is given byˆ H ( k ) = (cid:18) ˆ H ( k + ) ˆ∆ ˆ∆ † − ˆ H T4 ( k − ) (cid:19) , (10)whereˆ H ( k ± ) = ˆ H ( a )2 ( k ± ) − µ ˆ σ ε ( k ± )ˆ σ ε ( k ± )ˆ σ ˆ H ( b )2 ( k ± ) − µ ˆ σ ! , (11)ˆ∆ = a − ∆ a b − ∆ b , (12)ˆ H ( l )2 ( k ± ) = ( ε ′ ( k ± )ˆ σ + α sin k ± ˆ σ z − h a · ˆ σ ( l = a ) ε ′ ( k ± )ˆ σ − α sin k ± ˆ σ z − h b · ˆ σ ( l = b ) . (13)We carry out Bogoliubov transformation with usingthe unitary matrix ˆ U ( k ): H MF = 12 X k ˆ C † k ˆ U ( k ) | {z } ˆΓ † k ˆ U † ( k ) ˆ H ( k ) ˆ U ( k ) | {z } ˆ E ( k ) ˆ U † ( k ) ˆ C k | {z } ˆΓ k + W = 12 X k ˆΓ † k ˆ E ( k )ˆΓ k + W , (14)where ˆ E ( k ) is a diagonal matrix,ˆ E ( k ) = (cid:18) ˆ E ( k ) ˆ0ˆ0 − ˆ E ( k ) (cid:19) . (15)From Eq. (6), the order parameters are obtained by∆ a = − V a N X k ′ Dh ˆΓ † k ′ ˆ U † ( k ′ ) i h ˆ U ( k ′ )ˆΓ k ′ i E = − V a N X k ′ X n =1 h ˆ U † ( k ′ ) i n h ˆ U ( k ′ ) i n f (cid:16)h ˆ E ( k ′ ) i nn (cid:17) , (16)∆ b = − V a N X k ′ X n =1 h ˆ U † ( k ′ ) i n h ˆ U ( k ′ ) i n f (cid:16)h ˆ E ( k ′ ) i nn (cid:17) , (17)where f ( E ) is the Fermi distribution function. Equa-tions (16) and (17) are MF gap equations to be solvednumerically.The Bogoliubov quasiparticle operator ˆΓ † k and energyˆ E ( k ) are expressed with using the indices ( s, l ), where s represents the pseudospin s = ↑ , ↓ and l is the pseudo-sublattice index l = a, b :ˆΓ † k = ( γ † k ↑ a , γ † k ↓ a , γ † k ↑ b , γ † k ↓ b , γ − k ↑ a , γ − k ↓ a , γ − k ↑ b , γ − k ↓ b ) , (18)ˆ E ( k ) = E k ↑ a E k ↓ a E k ↑ b
00 0 0 E k ↓ b . (19)Then, the MF Hamiltonian H MF and free energy Ω areobtained as H MF = X k,s,l E ksl (cid:18) γ † ksl γ ksl − (cid:19) + W , (20)Ω = − β X k,s,l (cid:26) ln (cid:0) e − βE ksl (cid:1) + βE ksl (cid:27) + W , (21)where β = 1 /T is the inverse temperature. III. MAGNETIC MULTIPOLE ORDER ANDEXOTIC SUPERCONDUCTIVITYA. Magnetic and electronic structure in magneticmultipole states
We investigate the superconductivity in three mag-netic multipole states: monopole, dipole, and quadrupolestates. Before going to the main issue, here we introducethe magnetic structure corresponding to the multipoleorder. The symmetry protection on the single-particleband structure is also clarified. Later we attribute theorigin of exotic superconductivity to the unusual bandstructure.First, we illustrate the magnetic structure in Fig. 2.When the magnetic moment is “antiferromagnetic” inthe unit cell and directed along the x axis, two antiferro-magnetic moments are regarded as a magnetic monopole[Fig. 2, panel (M)]. On the other hand, when the anti-ferromagnetic moment is parallel to the z axis, a mag-netic quadrupole moment is induced in the unit cell[Fig. 2, panel (Q)]. It has been shown that the mag-netic quadrupole order is stabilized by the sublattice-dependent ASOC . Indeed, the magnetic structure in1-2-10 compounds resembles magnetic quadrupole or-der . This magnetic structure is also induced bythe electric field applied along the c axis as a resultof the magnetoelectric effect . The magnetic monopoleand quadrupole are odd-parity multipoles leading to thespontaneous global inversion symmetry breaking. Fur-thermore, we also examine the conventional “ferromag-netic” order which is called magnetic dipole order in thispaper [Fig. 2, panel (D)]. The crystal structure of ferro-magnetic SCs UGe , URhGe , and UCoGe is com-posed of coupled zigzag chains . Thus, our study maybe relevant to these ferromagnetic SCs.Next, we clarify the single-particle energy spectrum.The band structure is obtained by the normal partHamiltonian, which is expressed by using the vector op-erator ˆ D † k = ( a † k ↑ , a † k ↓ , b † k ↑ , b † k ↓ ), H (0) = X k ˆ D † k ˆ H ( k ) ˆ D k . (22)Without any loss of generality, we choose the chemicalpotential µ to be zero in ˆ H ( k ) [Eq. (11)]. The itiner-ant magnetic multipole states are studied by taking intoaccount the molecular field h a and h b as follows:( h a , h b ) = ( h AF ˆ x, − h AF ˆ x ) in (M)onopole order( h ˆ y, h ˆ y ) in (D)ipole order( h AF ˆ z, − h AF ˆ z ) in (Q)uadrupole order.(23)Then we show the energy band in Fig. 3. In the ab-sence of the magnetic multipole order, namely, ( h a , h b ) =(0 , E n ( k ) = ε ′ ( k ) ± q ε ( k ) + α sin k. (24)Each band has a twofold degeneracy which arises fromthe spin and sublattice degrees of freedom entangled bythe sublattice-dependent ASOC. This electronic struc-ture is similar to the bilayer Rashba system studied inthe previous study . On the other hand, we obtain thedispersion relation in the magnetic multipole states, E n ( k ) = ε ′ ( k ) ± q ε ( k ) + α sin k + ( h AF ) in (M) ε ′ ( k ) ± q [ ε ( k ) ± h ] + α sin k in (D) ε ′ ( k ) ± q ε ( k ) + ( α sin k − h AF ) in (Q).(25)Table I shows two main features of the band structure:(i) symmetry with respect to the inversion of momentum, k → − k , and (ii) twofold degeneracy. Below, we explainthese features in terms of symmetry in multipole states. TABLE I. Band structure in the magnetic multipole states.(i) Symmetry (ii) Twofold degeneracyMonopole yes yesDipole yes noQuadrupole no yes
First, in the magnetic monopole state, the collinearantiferromagnetic order spontaneously breaks the inver-sion symmetry ( P symmetry) as well as the time-reversalsymmetry ( T symmetry) in spite of the globally cen-trosymmetric crystal structure. However the combined PT symmetry is preserved since the normal part Hamil-tonian H (0) is invariant under the successive operationsof time-reversal and spatial inversion. This combined op-eration satisfies ( PT ) = − R πx ˆ H ( k )( R πx ) − = ˆ H ( − k ) , (26) R πz ˆ H ( k )( R πz ) − = ˆ H ( − k ) in (M) . (27)From Eq. (26) or (27), we understand the symmetric en-ergy dispersion, E n ( k ) = E n ( − k ). Second, in the mag-netic quadrupole state, the band structure preserves atwofold degeneracy owing to the very same reason as themonopole state. However, the quadrupole state is nei-ther invariant under the twofold rotation nor the mirrorreflection with respect to the zx plane which transformsthe wave number k to − k : R πx ˆ H ( k )( R πx ) − = ˆ H ( − k ) , (28) R πz ˆ H ( k )( R πz ) − = ˆ H ( − k ) , (29) M zx ˆ H ( k ) M − zx = ˆ H ( − k ) in (Q) . (30)Thus, all the symmetries protecting the symmetric bandstructure are broken, and indeed, the band structure isasymmetric as shown in Fig. 3, panels (Q-1) and (Q-2).Finally, the band structure is symmetric in the magnetic -4-3-2-1 0 1 2 3 -3 -2 -1 0 1 2 3 (N-1) t / t = 0.5 E n ( k ) -4-3-2-1 0 1 2 3 -3 -2 -1 0 1 2 3 (N-2) t / t = 0.1 -4-3-2-1 0 1 2 3 -3 -2 -1 0 1 2 3 (M-1) t / t = 0.5 E n ( k ) -4-3-2-1 0 1 2 3 -3 -2 -1 0 1 2 3 (M-2) t / t = 0.1 -4-3-2-1 0 1 2 3 -3 -2 -1 0 1 2 3 (D-1) t / t = 0.5 E n ( k ) -4-3-2-1 0 1 2 3 -3 -2 -1 0 1 2 3 (D-2) t / t = 0.1 -4-3-2-1 0 1 2 3 -3 -2 -1 0 1 2 3 (Q-1) t / t = 0.5 E n ( k ) k -4-3-2-1 0 1 2 3 -3 -2 -1 0 1 2 3 (Q-2) t / t = 0.1 k FIG. 3. Band structure of 1D zigzag chain in (N) normalstate, (M) magnetic monopole state, (D) magnetic dipolestate, and (Q) magnetic quadrupole state. The left panelsshow the results for t /t = 0 .
5, while t /t = 0 . h = 0 . h AF = 0 . dipole state since the ferromagnetic order preserves the P symmetry. Because of the violation of the T symmetrythe combined PT symmetry is broken, and therefore, thetwofold degeneracy is lifted.We furthermore show the symmetry protection on theadditional degeneracy at k = ± π . For example, twospinful bands are degenerate at k = ± π in the normalstate. This fourfold degeneracy is protected by the PT symmetry, inversion-glide symmetry PG yz , and mirrorsymmetry M xz . We here prove the fourfold degener-acy at the inversion-glide-invariant momentum k = ± π from relations, ( PG yz ) = − {PG yz , PT } = 0, and {PG yz , M xz } = 0 . Because of the inversion-glide sym-metry, the normal part Hamiltonian at k = ± π is blockdiagonalized and decomposed into the ± i subsectors.The PT symmetry is preserved in each subsector as en-sured by the anticommutation relation between PG yz and PT . Thus, Kramers pairs are formed in each subsector.The anticommutation relation between PG yz and M xz ensures that a Kramers pair in the i subsector is de-generate with another Kramers pair in the − i subsector.Thus, the fourfold degeneracy is protected by symme-try. The mirror symmetry is broken in the monopole andquadrupole states, while the PT symmetry is broken inthe dipole state. Therefore, the fourfold degeneracy islifted in the multipole states.Additional degeneracy is also seen at k = ± π in thedipole state because the normal part Hamiltonian pre-serves the magnetic-glide symmetry T G yz which is a com-bined symmetry of the glide symmetry and the time-reversal symmetry. This antiunitary symmetry ensuresthe extended Kramers theorem proving the degeneratesingle-particle states at k = ± π . The twofold degener-acy is also protected in the magnetic dipole state with h along the x axis. Then, the magnetic-screw symmetry T S πy protects the degeneracy at k = ± π . B. Superconductivity
We here summarize the order parameter of three su-perconducting states which may be stabilized in ourmodel: the BCS state, the PDW state, and the FFLOstate. In the conventional BCS state, Cooper pairshave the zero in-plane center-of-mass momentum, thatis, q = 0. The order parameter is uniform betweensublattices, (∆ a , ∆ b ) = (∆ , ∆). The center-of-mass mo-mentum is also zero in the PDW state. The sign of theorder parameter, however, changes between sublattices,(∆ a , ∆ b ) = (∆ , − ∆). In the FFLO state, the center-of-mass momentum of Cooper pairs is finite (i.e., q = 0). Inthe symmetry considered in this paper, the Cooper paircondensation occurs at a single q , although the double- q state is stable in the conventional FFLO state . There-fore, in real space the order parameter is expressed as∆( y ) = ∆ e iqy , which is usually called “Fulde-Ferrellstate” or “helical state” . As is the case in theBCS state, the order parameter is uniform in sublattices,(∆ a , ∆ b ) = (∆ , ∆). IV. BCS STATE ROBUST AGAINSTMAGNETIC MONOPOLE ORDER
First we discuss the superconductivity coexisting withmagnetic monopole order [see Fig. 2, panel (M)]. Figure 4shows T - h AF phase diagrams for several sets of param-eters. It is shown that the conventional BCS state is stable in the whole phase diagram independently of theparameters µ and t /t . For t /t = 0 . t /t = 0 . µ = 2 is similar to that for µ = 1, although we donot show in Fig. 4. Thus, the BCS state is stable inthe monopole state, irrespective of the number of Fermisurfaces. We confirmed that the in-plane center-of-massmomentum q of the Cooper pair is zero in the whole pa-rameter region. (a) t / t = 0.5, μ = 1.0 T BCS h A F (b) t / t = 0.1, μ = 1.0 T BCS h A F (c) t / t = 0.1, μ = -2.0 T BCS h A F FIG. 4. T - h AF phase diagram in the magnetic monopole statefor (a) t /t = 0 . , µ = 1, (b) t /t = 0 . , µ = 1, and (c) t /t = 0 . , µ = −
2. The BCS state is stable in the wholesuperconducting phase. In the pink shaded area the PDWstate is metastable.
V. PDW STATE BY MAGNETIC DIPOLEORDER
Second, we study the superconductivity in the mag-netic dipole state [see Fig. 2, panel (D)]. This situa-tion is realized when the superconductivity occurs in theferromagnetic metal. Indeed, such ferromagnetic super-conductivity occurs in uranium-based heavy-fermion SCsUGe , URhGe , and UCoGe , which have a zigzagcrystal structure . Although the magnetic dipole mo-ment along the y axis is assumed, the x axis is equivalentto the y axis in the spin space since we consider a purely1D model. Note that both the x and y axes are perpen-dicular to the g vector. A. Phase diagram
Figure 5 shows the T - h phase diagram for t /t = 0 . h region), while the PDW state is stable in a largeparameter range with high spin polarization (large h re-gion). The phase boundary of the BCS and PDW statesis the first-order phase transition line. This phase dia-gram is similar to that obtained in the two-dimensionalbilayer Rashba SCs . h T t / t = 0.1, μ = 1.0, V / t = 1.0 PDW
BCS
FIG. 5. T - h phase diagram in the magnetic dipole state for t /t = 0 . µ = 1. We assume the attractive interaction V /t = 1 .
0. In the pink (cyan) shaded area the PDW (BCS)state is a metastable state.
The mechanism of the PDW state in a spin polarizedstate has been discussed in Ref. 3. When the inter-sublattice hopping is smaller than the spin-orbit cou-pling, a substantial condensation energy is gained inthe PDW state although at zero effective magnetic field( h = 0) it is smaller than the condensation energy ofthe BCS state which gains the inter-sublattice Josephsoncoupling energy. Because the paramagnetic depairing ef-fect is suppressed in the PDW state by the spin-orbitcoupling , at large h the PDW state may be more stablethan the BCS state which is fragile against the param-agnetic effect. The zigzag chain is composed of the “ a sublattice” andthe “ b sublattice,” and thus t is the inter-sublattice hop-ping. When the system has a small t /t and a moderateASOC, the PDW state is stabilized in a large parame-ter regime as shown in Fig. 5. As the inter-sublatticehopping t /t is increased, the PDW state is suppressed.For our choice of parameters, the PDW state is not stablefor t /t > .
7. Thus, in the zigzag chain the PDW statemay be stable even at a moderate t /t . This is partlybecause the inter-sublattice coupling is represented by ε ( k F ) rather than t , and ε ( k ) disappears at k = ± π .As we mentioned in Sec. III A, the fourfold degeneracyat k = ± π is protected by the inversion-glide symmetry PG yz and mirror symmetry. This additional degeneracycomes from the sublattice degree of freedom. Thus, thedisappearance of ε ( ± π ) is ensured by the nonsymmor-phic crystal symmetry. When the Fermi momentum isclose to k = ± π , the PDW state is favored owing to asmall ε ( k F ).As shown in Fig. 5, the PDW state may be stable at µ = 1, where the four energy bands cross the Fermi level[see Fig. 3, panel (D-2)]. Similarly, the PDW state is sta-ble when two or three energy bands have the Fermi sur-face. We confirmed that the in-plane center-of-mass mo-mentum q of the Cooper pair is zero in any case. Whenthe chemical potential is in the vicinity of the band edgeand only one band crosses the Fermi level, however, thePDW state is not stable. B. Topological superconductivity
In this subsection, we show that the PDW state maybe a 1D topological superconducting state specified bythe winding number and the Z invariant. A gaugetransformation, a † k → a † k e ik/ , is carried out so that theBogoliubov-de Gennes (BdG) Hamiltonian is periodic inthe Brillouin zone. This unitary transformation is usefulfor the discussion of topological properties in nonsym-morphic systems .First, we elucidate the winding number. In a ferro-magnetic state with magnetic moment along the x or y axis, the system is invariant under the magnetic mirrorreflection which is a successive operation of time reversal T = i ˆ σ y K and mirror reflection with respect to the xy plane M xy = i ˆ σ z . K is the complex-conjugate opera-tor. Thus, the BdG Hamiltonian derived from Eq. (10)preserves the pseudo-time-reversal symmetry: T ′ ˆ H ( − k ) T ′ † = ˆ H ( k ) , (31)where T ′ = T ′ ˆ0ˆ0 T ′∗ ! , (32)with T ′ = M xy T . Furthermore, the particle-hole sym-metry is implemented in the BdG Hamiltonian: C ˆ H ( − k ) C † = − ˆ H ( k ) , (33)where C = τ x K and τ x is the Pauli matrix in the particle-hole space. Combining the pseudo-time-reversal symme-try with the particle-hole symmetry, we can define thechiral symmetry, { Γ , ˆ H ( k ) } = 0 , (34)with Γ = −CT ′ . The chiral symmetry ensures that the1D winding number ω = 14 πi Z π − π dk Tr (cid:2) ˆ q ( k ) − ∂ k ˆ q ( k ) − ˆ q † ( k ) − ∂ k ˆ q † ( k ) (cid:3) (35)is a Z topological invariant when a finite gap isopen. The 4 × q ( k ) is obtained by carrying outa unitary transformationˆ V ˆ H ( k ) ˆ V † = ˆ0 ˆ q ( k )ˆ q † ( k ) ˆ0 ! , (36)where ˆ V is a unitary matrix which diagonalizes Γ .The BdG Hamiltonian ˆ H ( k ) belongs to the symmetryclass BDI because ( T ′ ) = +1 and C = +1. Therefore,the winding number ω is identified to be an integer topo-logical invariant of the BDI class , ν BDI . Figure 6shows the chemical potential dependence of the wind-ing number together with the energy bands shown inFig. 3, panel (D-2). We obtain a finite winding number, ν BDI = −
1, indicating topologically nontrivial propertieswhen one or three bands cross the Fermi level. Other-wise, the winding number is trivial, ν BDI = 0. -4-3-2-1 0 1 2 3 -3 -2 -1 0 1 2 3 ν BDI = 0 ν BDI = -1 E n ( k ) k FIG. 6. Chemical potential dependence of the winding num-ber ν BDI for t /t = 0 . h = 0 .
40, and ∆ = 0 .
01. Thewinding number is nontrivial, ν BDI = −
1, when the chemi-cal potential lies in the pink shaded region. The blue dashedlines represent the chemical potential at which the windingnumber is ill defined owing to the gap closing.
A nontrivial winding number may ensure the Majo-rana end state according to the index theorem . Indeed,Fig. 7 shows the Majorana end states. The energy spec-trum ε n is obtained in the open boundary condition, and the n -th energy eigenvalue is arranged in ascending or-der ε < ε < · · · . We see the single Majorana end stateprotected by the nontrivial winding number ν BDI = − -0.1-0.05 0 0.05 0.1 (a) µ = -1.00 (four bands) ε n -0.1-0.05 0 0.05 0.1 (b) µ = -1.50 (three bands) -0.1-0.05 0 0.05 0.1 (c) µ = -2.00 (two bands) ε n n -0.1-0.05 0 0.05 0.1 (d) µ = -2.50 (one band) n FIG. 7. Energy spectra in the PDW state with open bound-aries. We assume the ferromagnetic molecular field h = 0 . µ = − .
00, (b) µ = − .
50, (c) µ = − .
00, and (d) µ = − .
50. The number of Fermi surface is 4, 3, 2, and 1. Theother parameters are t /t = 0 . a , ∆ b ) = (0 . , − . This single Majorana end state is robust against per-turbations, even when the magnetic mirror symmetry isbroken. Indeed, the PDW state with ν BDI = − Z invariant in theD class . The parity of the winding number is equiv-alent to the Z invariant, ν , which is explicitly expressedby the Berry phase W [ C ] = 12 π X n ∈ occupied I C dk i h u n ( k ) | ∂ k | u n ( k ) i . (37) C represents a time-reversal-invariant (TRI) closed pathin the Brillouin zone, and | u n ( k ) i is an eigenstateof ˆ H ( k ). Since the BdG Hamiltonian preserves theparticle-hole symmetry, the Berry phase is quantized as e πiW [ C ] = ± . Since the TRI closed path C = { k ∈ [ − π : π ) } is unique in the 1D system, we have a sin-gle Z invariant e πiW [ C ] = ( − ν . In particular, thenormal part Hamiltonian preserves the spatial inversionsymmetry, P ˆ H ( k ) P † = ˆ H ( − k ), and the parity of thegap function is odd, P ˆ∆ P T = − ˆ∆ , in the PDW state.Then, the Z invariant has been evaluated as( − ν = Y n sgn E n (Γ ) sgn E n (Γ ) , (38)where Γ and Γ are the TRI momenta, Γ = 0 andΓ = π . From this representation, the Z invariantis nontrivial when the odd number of bands cross theFermi level. This condition coincides with the situationwith ν BDI = −
1. Thus, the PDW state is identified tobe a 1D Z topological SC in the D class.An intuitive explanation for the topological supercon-ductivity is obtained by looking at the band representa-tion of the BdG Hamiltonian,ˆ U ( k ) † ˆ H ( k ) ˆ U ( k ) ≃ M n =1 E n ( k ) ∆ n ( k )∆ ∗ n ( k ) − E n ( − k ) ! , (39)where ˆ U ( k ) = (cid:16) ˆ U ( k ) ˆ0ˆ0 ˆ U ∗ ( − k ) (cid:17) , and ˆ U ( k ) is a unitarymatrix which diagonalizes ˆ H ( k ). The order parameterin the band basis approximately has the p -wave form,∆ n ( k ) ∼ sin k . In this sense, the situation is similarto the Kitaev chain for the spinless p -wave SC. Al-though the superconductivity is induced by the conven-tional pairing interaction in the s -wave spin-singlet chan-nel, the effective p -wave superconducting state similar tothe Kitaev chain is realized by the inter-sublattice phasemodulation in the order parameter.Topological superconducting phases in 1D noncen-trosymmetric systems have been clarified theoreti-cally , and recently experimental indications for theMajorana state have been obtained in semiconduc-tors and ferromagnetic atomic chains . In contrastto these systems requiring the inversion-symmetry break-ing, our research proposes the centrosymmetric topo-logical superconductivity caused by the spontaneouslyformed odd-parity PDW order parameter.Now we briefly comment on the zero energy end statesin Fig. 7(c). When the two bands cross the Fermi level,we see the two Majorana end states in spite of the trivialwinding number and Z number, ν BDI = ν = 0. Theseend states may be protected by another symmetry. How-ever, the crystal symmetry other than the mirror sym-metry is broken at the boundary. Thus, we leave thetopological protection of these end states for a futurestudy.Finally, we propose two experimental tests to identifythe PDW state. (i) As shown above, the Majorana endstate is generated at the end of the chain in the PDWstate. The Majorana end state may be recognized asa zero bias conductance peak of quasiparticle tunnelingspectroscopy in a normal metal/SC junction . (ii) Inthe external magnetic field, vortices appear in the real 3Dmaterials. Then, the local quasiparticle density of statesin the PDW state is quite different from that in the BCSstate. The zero-energy vortex bound state exists in thePDW state, although it is absent in the BCS state due tothe Zeeman effect . Therefore, the scanning tunnelingmicroscopy/spectroscopy experiments may identify thePDW state by measuring the local density of states. VI. FFLO STATE BY MAGNETICQUADRUPOLE ORDER
Finally we clarify the superconductivity in the mag-netic quadrupole state [see Fig. 2, panel (Q)]. We assume t /t = 0 . A. T - µ phase diagram We address the T - µ phase diagram for two values of h AF in Fig. 8. The Cooper pairs have finite center-of-mass momentum in the whole superconducting phase ow-ing to the asymmetric band structure. The asymmetryresults from the symmetry of magnetic quadrupole state,and therefore, the FFLO state is stable irrespective of theparameters unless the ASOC vanishes. When α = 0, theband structure is symmetric and the BCS state is stablein a large parameter region.Let us discuss the phase diagrams in details. We noticecommon features in Figs. 8(a) and 8(b). Critical temper-ature is rather higher for µ & − µ . −
1. Thisis because the density of states (DOS) is large in thetwo-band region, µ & −
1. The critical temperature isfurthermore enhanced in the vicinity of the band edge( µ ≃ − , − ,
2) because of the large DOS. Figure 8 alsoreveals differences between the “small quadrupole mo-ment region” ( h AF = 0 .
12) and the “large quadrupolemoment region” ( h AF = 0 . q > q <
0) by “FFLO q> ”(“FFLO q< ”). While the center-of-mass momentum q continuously changes in the small quadrupole moment re-gion, the FFLO q< state is separated from the FFLO q> state by the first-order phase transition line in the largequadrupole moment region [Fig. 8(b)]. The negative q in the small µ region comes from the shift of the lowerenergy band to the negative momentum side. The sumof the two Fermi momenta in the lower band is negative.On the other hand, the upper band favors the FFLO q> state, and thus the FFLO q< state competes with theFFLO q> state in the two-band region. As expected,the center-of-mass momentum increases with µ across theLifshitz transition. We show the µ and T dependence of q by color in Fig. 8. We see the continuous change of q inthe small quadrupole moment region [Fig. 8(a)], while weobserve a discontinuous jump at µ ≃ − .
20 in the largequadruple moment region [Fig. 8(b)].0 h AF = 0.12 (a)
0 0.02 0.04 0.06 0.08 0.1 T -3-2-1 0 1 2 -0.1-0.05 0 0.05 0.1 0.15 0.2 μ FFLO q > 0 FFLO q < 0 q
0 0.02 0.04 0.06 0.08 0.1 T -3-2-1 0 1 2 -0.1-0.05 0 0.05 0.1 0.15 0.2 q μ FFLO q > 0 FFLO q < 0 h AF = 0.20(b) FIG. 8. T - µ phase diagram in the magnetic quadrupole statefor (a) h AF = 0 .
12 and (b) h AF = 0 .
20. The center-of-massmomentum of Cooper pairs q is represented by color. Thedashed line shows a first-order phase transition line, whilethe dash-dotted line shows a crossover line. B. Condensation energy and DOS
In order to elucidate what mainly determines the q in the FFLO state, we look at the condensation energy∆Ω = Ω S − Ω N , which is the difference of free energybetween in the superconducting state and in the normalstate. The free energy in the normal state is obtained byjust assuming ∆ a = ∆ b = 0.Figures 9(a) and 9(b) show the condensation energy asa function of q in the small quadrupole moment region.Only one valley appears and its bottom moves to thepositive- q side with increasing µ . Thus, the optimal q which minimizes the condensation energy continuously varies.On the other hand, we find three valleys in the largequadrupole moment region. In Fig. 9(d), the left andright valleys lead to a negative condensation energy, whilethe middle valley shows a positive condensation energyindicating a metastable state. When we decrease thechemical potential to be µ = − . q = − .
026 which adi-abatically changes to the bottom of the middle valley byincreasing µ . This means that the FFLO q< state corre-sponds to the middle valley while the FFLO q> state cor-responds to the right valley. In other words, the center-of-mass momentum q discontinuously changes because thevalley structure appears in the free energy. On the otherhand, the valley structure is hidden and only the mid-dle valley has a local minimum in the small quadrupolemoment region.Next we show the DOS of quasiparticles in order toclarify the superconducting states corresponding to thethree valleys. At both µ = − . µ = 0, the DOSshows a superconducting gap near ω = 0 in the “middle-valley state” [Figs. 9(e) and 9(f)]. The narrower gap at µ = 0 than at µ = − . µ = 0. Indeed, Fig. 9(d)shows that the middle-valley state is metastable and the“right-valley state” is stable. In contrast to the middle-valley state, approximately half of the DOS is residual at ω = 0 in the right-valley state [Fig. 9(f)]. Thus, it is im-plied that although both energy bands contribute to thesuperconductivity in the middle-valley state, the upper(lower) band mainly causes the superconductivity in theright-valley (left-valley) state. In other words, the lowerband is weakly superconducting and gives rise to the largeresidual DOS in the right-valley state. This view is con-sistent with the fact that the center-of-mass momentum q in the right-valley state almost coincides with the sumof the Fermi momentum in the upper band. Thus, theband-dependent FFLO state is stabilized by a large mag-netic quadrupole moment. Quasiparticles on the Fermisurface of the upper band form Cooper pairs, while themismatch of q and distorted lower band suppresses thesuperconducting gap in the lower band. On the otherhand, in the middle-valley state the superconductivityalmost equivalently affects the two bands. Then, q isslightly negative because the distortion of the lower bandis larger than that of the upper band.Finally, we suggest an experimental test for the FFLOstate. The measurement of Josephson current in a FFLOSC/BCS SC junction may identify the single- q FFLOstate. Since Josephson coupling vanishes in this junctiondue to the spatial modulation of the order parameter inthe FFLO SC, the junction should carry a small Joseph-son current. On the other hand, in an applied transverseuniform current in the BCS SC, a peak in the Josephsoncurrent may be found . The peak serves as an indicatorof the FFLO state.1 -0.01-0.008-0.006-0.004-0.002 0-0.4 -0.2 0 0.2 0.4 (a) h AF = 0.12, µ = 1.00 ∆ Ω q -0.01 0 0.01-0.01-0.008-0.006-0.004-0.002 0-0.4 -0.2 0 0.2 0.4 (b) h AF = 0.12, µ = 2.00 ∆ Ω q -0.01 0 0.01 -0.002-0.001 0-0.2 -0.1 0 0.1 0.2 (c) h AF = 0.20, µ = -0.50 q = -0.026 ∆ Ω q (e) ρ S ( ω ) / ρ N ( ) ω q = -0.026 (Middle valley) -0.002-0.001 0-0.2 -0.1 0 0.1 0.2 (d) h AF = 0.20, µ = 0.00 q = 0.086 ∆ Ω q (f) ρ S ( ω ) / ρ N ( ) ω q = -0.016 (Middle valley) q = 0.086 (Right valley) FIG. 9. (a) and (b) The q dependence of the condensation energy ∆Ω = Ω S − Ω N for a small quadrupole moment h AF = 0 . µ = 1 and µ = 2, respectively. (c) and (d) ∆Ω for a large quadrupole moment h AF = 0 .
20 at µ = − . µ = 0,respectively. The red points show the optimal q which minimizes the condensation energy. (e) and (f) Superconducting DOS ρ S ( ω ) normalized by the normal state DOS at the Fermi level ρ N (0) for the parameters in (c) and (d), respectively. VII. SUMMARY AND DISCUSSION
In this paper, we investigated the superconductivity inthe magnetic multipole states. In locally noncentrosym-metric systems with sublattice degree of freedom, notonly the conventional magnetic dipole moment but alsosome odd-parity multipole moments may be polarized.Ferroic multipole states with crystal momentum q M = 0were considered in the 1D zigzag chain as a minimalmodel. Exotic superconducting states were elucidatedas follows.The conventional BCS state is robust against the ex-istence of “antiferromagnetic moments” in the unit cellwhich is regarded as a magnetic monopole. Meanwhile inthe dipole order the odd-parity spin-singlet PDW stateis stabilized. The situation in the latter correspondsto uranium-based heavy-fermion SCs UGe , URhGe ,and UCoGe . It has been thought that the spin-tripletsuperconductivity occurs in these materials. However,our result opens a new possibility that the ferromagneticsuperconductivity in these materials is attributed to thePDW state. From the theoretical point of view, the PDWstate is identified to be a topological superconductingstate when one of the bands is fully spin polarized. Weshowed a nontrivial winding number in the class BDI, aswell as nontrivial Z invariant in the class D. The non-trivial topological numbers ensure the single Majoranaend state.Interestingly, the magnetic quadrupole order combinedwith the spin-orbit coupling makes the band structureasymmetric. As a result of the asymmetric energy band, the FFLO state is stabilized without spin polarization.This finding paves a new way for searches of the FFLOstate . Although previous studies researched SCs witha large Maki parameter , the external magnetic fieldapplied to stabilize the FFLO state induces vorticeswhich may obscure the FFLO state. On the other hand,the FFLO state caused by the magnetic quadrupole or-der is free from the vortex. Thus, a conclusive evidencefor the FFLO state may be obtained by searching the su-perconductivity coexisting with the magnetic quadrupoleorder.The band-dependent properties of the FFLO statewere clarified as follows. When the magnetic quadrupolemoment is small, the upper and lower energy bandsare almost equally superconducting (if they cross theFermi level). Then, the center-of-mass momentum ofthe Cooper pair is small and continuously increases withchemical potential. On the other hand, the center-of-mass momentum discontinuously changes in the largequadrupole moment region. The origin of this first-orderphase transition in the FFLO state is attributed to theband-dependent FFLO superconductivity. While the twobands are almost equally superconducting at small chem-ical potentials, only the upper band mainly causes thesuperconductivity at large chemical potentials. The two-band electronic structure is not an artifact of the 1Dzigzag chain, but is a consequence of the nonsymmorphiccrystal symmetry protecting the band degeneracy at theBrillouin zone boundary. Therefore, the band-dependentFFLO superconductivity may be realized in various non-symmorphic crystals hosting the magnetic quadrupole or-der.2 ACKNOWLEDGMENTS
The authors are grateful to T. Arima, K. Onozawa,M. Sato, S. Takamatsu, Y. Nakamura, T. Hitomi, and A. Daido for fruitful discussions. This work was sup-ported by a “J-Physics” (15H05884) Grant-in-Aid forScientific Research on Innovative Areas from MEXT ofJapan, and by JSPS KAKENHI, Grants No. 24740230,No. 15K051634, No. 15H05745, and No. 16H00991. ∗ [email protected] E. Bauer and M. Sigrist, eds.,
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