Topological phase transitions in multi-component superconductors
TTopological Phase Transitions in Multi-component Superconductors
Yuxuan Wang and Liang Fu Department of Physics and Institute for Condensed Matter Theory,University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, Illinois 61801-3080, USA Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA (Dated: October 26, 2017)We study the phase transition between a trivial and a time-reversal-invariant topological su-perconductor in a single-band system. By analyzing the interplay of symmetry, topology andenergetics, we show that for a generic normal state band structure, the phase transition occursvia extended intermediate phases in which even- and odd-parity pairing components coexist. Forinversion-symmetric systems, the coexistence phase spontaneously breaks time-reversal symmetry.For noncentrosymmetric superconductors, the low-temperature intermediate phase is time-reversalbreaking, while the high-temperature phase preserves time-reversal symmetry and has topologi-cally protected line nodes. Furthermore, with approximate rotational invariance, the system has anemergent U (1) × U (1) symmetry, and novel topological defects, such as half vortex lines bindingMajorana fermions, can exist. We analytically solve for the dispersion of the Majorana fermion andshow that it exhibit small and large velocities at low and high energies. Relevance of our theory tosuperconducting pyrochlore oxide Cd Re O and half-Heusler materials is discussed. Topological superconductivity [1–16] offers a uniqueplatform for studying the interplay between topologi-cal phases of matter, unconventional superconductivity(SC), and exotic quasiparticle and vortex excitations. Inthe presence of time-reversal and inversion symmetry,topological superconductors require an odd-parity orderparameter (e.g. p -wave) [1, 17]. Theoretical studies [1, 7]proposed that Cu x Bi Se , a doped topological insulatorthat becomes superconducting below T c ∼ . Se spontaneously breaks crystal rotational symmetry, onlycompatible with the time-reversal-invariant p -wave pair-ing with the E u symmetry [1, 4]. There is currentlyhigh interest in searching for the topological excitationsin these materials [23–30].In this paper, we study topological phase transitionsin superconductors resulting from the change of pair-ing symmetry from even- to odd-parity. Our study ismotivated by a number of experiments showing thatpairing interactions in even- and odd-parity channelsare of comparable strength in several materials, here-after referred to as multi-component superconductors. Inthe non-centrosymmetric superconductor Li (Pd,Pt) B,the odd-parity spin-triplet and even-parity spin-singletpairing components vary continuously as a function ofthe alloy composition [31–33]. In the pyrochlore oxideCd Re O [34, 35], applying pressure drives phase transi-tions between different superconducting states, accompa-nied by an anomalous enhancement of the upper criticalfield exceeding the Pauli limit [35]. This has been in-terpreted as a transition from spin-singlet to spin-triplet dominated superconductivity. On the theory side, a pair-ing mechanism for odd-parity superconductivity in spin-orbit-coupled systems has been recently proposed [36–38], where the pairing interaction arises from the fluctu-ation of an inversion symmetry breaking order. It wasfound that this interaction is attractive and nearly de-generate [39–41] in the two fully-gapped Cooper channelswith s -wave and p -wave symmetry respectively.The topology of a superconductor depends cruciallyon its order parameter, which is in turn determinedby energetics. Therefore a change of order parame-ter as a function of tuning parameters and tempera-ture can result in a topological phase transition in multi-component superconductors. Furthermore, spontaneoustime-reversal-symmetry breaking can be energetically fa-vored in the transition region, thus changing the symme-try that underlies the classification of topological super-conductors [42]. Both energetics and spontaneous sym-metry breaking need to be taken into account in theoryof topological phase transitions in superconductors.We show that the phase diagram of multi-componentsuperconductors is largely determined by the fermiologyof the normal state, rather than the microscopic pairingmechanism (which is often not exactly known). We findtwo types of phase diagrams for generic Fermi surfaceswith and without inversion symmetry, shown in Fig.1panel (b) and (d). Remarkably, we find that the transi-tion between the s -wave-dominated trivial phase and the p -wave-dominated topological phase is generically inter-rupted by an extended intermediate phase where s -waveand p -wave pairings coexist. For superconductors withinversion symmetry, the intermediate phase is a spon-taneous time-reversal symmetry breaking (TRSB) andinversion symmetry breaking superconducting state with s -wave and p -wave order parameters differing by a fixedrelative phase of ± π/ s + ip state re- a r X i v : . [ c ond - m a t . s up r- c on ] O c t alizes a superconducting analog of axion insulator [47–50], and exhibits thermal Hall conductance on the sur-face. For noncentrosymmetric superconductors [51–53],we predict two intermediate phases in the transition re-gion at different temperatures: a time-reversal-invariantphase at temperatures close to T c , and a time-reversal-breaking phase at low temperature. In particular, thetime-reversal-invariant phase has topologically protected line nodes in the bulk [54, 55].We derive the above results by general considerationsof symmetry, topology and energetics. Important to ouranalysis is an emergent U (1) × U (1) symmetry associ-ated with the two phases of ∆ ± ≡ ∆ s ± ∆ p , where ∆ s and ∆ p are the s -wave and p -wave superconducting orderparameters respectively. In the special case of isotropicFermi surface, the U (1) × U (1) symmetry is exact at thetransition between s -wave and p -wave pairing symme-try, and leads to a direct first-order phase transition be-tween trivial and topological superconductors; see Fig.1 panel (a) and (c). In the general case of supercon-ductors with anisotropic Fermi surfaces and gaps, the U (1) × U (1) symmetry near the topological phase transi-tion is approximate and provides a useful starting pointfor our theory. Moreover, in this regime, half quantumvortices, which corresponds to the winding of one of U (1)phases [56], can appear as topological defects, which bindchiral Majorana modes. We solve for the dispersion of theMajorana mode, and show it has a small velocity at zeroenergy and a large velocity near gap edge.Our theory is largely independent of specific bandstructures or pairing mechanisms, and is potentially ap-plicable to a broad range of materials. At the end ofthis work, we discuss the relevance of our general re-sults for the superconducting phases of pyrochlore ox-ide Cd Re O and half-Heusler compounds, and maketestable predictions. U (1) × U (1) symmetry.— Throughout this work, weassume the system under study has strong spin-orbitcoupling. Then single-particle energy eigenstates in thenormal state generally do not have well-defined spin.Nonetheless, when both time-reversal and inversion sym-metry are present, energy bands remain doubly degener-ate at every momentum k , which we label with pseudo-spin index σ . We choose to work in the manifestly co-variant Bloch basis [57], where the state | k , σ = ±i has the same symmetry property as the spin eigenstate | k , s z = ↑ ( ↓ ) i under the joint rotation of electron’s mo-mentum and spin.As a convenient starting point, we first consider sys-tems with full rotational invariance. In such systems, allthe pairing order parameters can be classified by theirtotal ( J ) angular momentum. We focus on J = 0 pair-ings with a full gap. If inversion symmetry is present,there are two types of J = 0 order parameters, witheven- or odd-parity respectively. The even-parity J = 0pairing has s -wave orbital angular momentum given by (a) Trivial Topological U (1)× U (1)Trivial Topological U (1)× U (1) (c) (b) Trivial TopologicalTRSBTrivial TopologicalTRSB (d) x xT T xT Tx
Gapless FS TopologicalNodal SC xT Broken RotationBroken Inversion Broken Rotation and Inversion
FIG. 1. Schematic phase diagrams near a topological phasetransition in multi-component superconductors with [Panels(a,b)] and without [Panels (c,d)] inversion symmetry, and with[Panels (a,c)] and without [Panels (b,d)] rotational symme-try. In (a,b), the “trivial” phase has an s -wave pairing sym-metry and the “topological” phase is p -wave. In (c,d) with-out inversion symmetry, the topological phase corresponds tothe region where p -wave component is larger. In Panel (c)at the dashed line one of the spin-textured Fermi surface iscompletely gapless, while in panel (d) in the region betweenthe dashed lines the superconducting states have topologicallyprotected line nodes on the Fermi surface. H s = ∆ s c † k iσ y ( c †− k ) T , while the odd-parity J = 0pairing has p -wave orbital angular momentum given by H p = ∆ p c † k (ˆ k · ~σ ) iσ y ( c †− k ) T . This p -wave order param-eter looks similar to that of He- B phase, but the spinquantization axis is rigidly locked to the momentum byspin-orbit coupling here. In both 2D and 3D, ∆ p realizestime-reversal-invariant topological superconductivity inthe DIII class.We now analyze the interplay between s -wave and p -wave pairings. Generically, the free energy is given by F = α | ∆ s | + α | ∆ p | + β | ∆ s | + β | ∆ p | + 4 ¯ β | ∆ s | | ∆ p | + ˜ β (∆ s ∆ ∗ p + ∆ p ∆ ∗ s ) . (1)The temperature-dependent coefficients α , α are deter-mined by the microscopic pairing mechanism. We areinterested in the case when s -wave and p -wave instabil-ities are comparable in strength, i.e., when α ∼ α , sothat tuning some parameters such as pressure or chem-ical composition can drive a phase transition. The in-terplay between s - and p -wave order parameters is con-trolled by the β coefficients only. It is important tonote that, within weak-coupling theory, β ’s do not relyon pairing interactions, and are completely determinedby the normal state electronic structure, as shown fromthe Feynman diagram calculation (for details see [58]).Explicitly evaluating these diagrams, we obtain that β = β = ¯ β = ˜ β ≡ β = 5 ζ (3) / (8 π T N (0)), where N (0) is the density of states, and ζ ( x ) is the Riemannzeta function.The last term in (1) is minimized when the phase differ-ence of the two order parameters at ∆ φ = ± π/
2. Underthis condition, at the phase boundary α = α = α , thefree energy (1) becomes F = α ( | ∆ s | + | ∆ p | ) + β ( | ∆ s | + | ∆ p | ) . (2)This free energy possesses a U (1) × U (1) symmetry [37]associated with the common phase and relative ampli-tude of ∆ s,p . [59] When α = α , the U (1) × U (1) sym-metry is broken, and the free energy is minimized suchthat the pairing channel with higher transition tempera-ture (i.e. smaller α ) completely suppresses the other, andthe phase transition is of first-order. Thus we obtain thephase diagram shown in Fig. 1(a). In a previous work [60]it was reported for a rotational invariant system there isa coexistence phase with both s -wave and p -wave orders.Our results differ here, and as we shall see, to obtain thecoexistence phase it is necessary to break the rotationalinvariance, at least within weak-coupling theory.The emergent U (1) symmetry is a general consequenceof the rotational and inversion symmetry of the assumednormal state electronic structure. To see this more ex-plicitly, it is instructive to divide pseoudo-spin degener-ate states on the Fermi surface into two groups, with helicty χ = ~σ · ˆ k = ± s and∆ p order parameter both correspond to pairing withineach group of helicity eigenstates (which we denote by∆ ± ), with constant gap over the Fermi surface as dic-tated by rotational invariance. The difference of ∆ s and∆ p is that they are even- and odd-combinations of ∆ ± ,i.e., ∆ s,p = (∆ + ± ∆ − ) / √ ± ,the generic free energy (1) can be rewritten as F = α ( | ∆ + | + | ∆ − | ) + δα (∆ + ∆ ∗− + ∆ ∗ + ∆ − )+ β ( | ∆ + | + | ∆ − | ) , (3)where the coefficients α, β for ∆ ± terms are identical dueto inversion symmetry which transforms opposite helicityeigenstates into each other, and δα ≡ ( α − α ) / δα = δα ( x ) controls the rela-tive sign of ∆ ± in the ground state, i.e., whether s -waveor p -wave order is favored. In this form the U (1) × U (1)symmetry is explicit at δα = 0, i.e., the phase bound-ary of s - and p -wave orders. The “second U (1)” can beregarded as a gapless Leggett mode [62] for the relativephase between ∆ ± . Time-reversal symmetry breaking phases.—
In an ac-tual system without full rotational invariance, the U (1) × U (1) symmetry is at best approximate. To see this, westill consider s -wave and p -wave pairing orders, H s =∆ s f s ( k ) c † k iσ y ( c †− k ) T , H p = ∆ p f p ( k ) c † k (ˆ k · ~σ ) iσ y ( c †− k ) T ,where the form factors f s,p ( k ) are positive and evenfunctions of k . For weak-coupling superconductivity, f s,p ( k ) = f s,p (ˆ k ). Since there is no further symmetry + Trivial + - Topological + Nodal + TRSB h i g h T l o w T FIG. 2. Transitions between trivial and topological super-conductor with only time reversal symmetry [the case of Fig.1(d)]. In 3D, at high T the transition occurs via intermediatenodal line (nodal points if 2D) superconducting phases, whileat low T time-reversal symmetry is spontaneously broken. requirement restricting them, in general f s (ˆ k ) = f p (ˆ k ).As a concrete example, we constructed a microscopicmodel [58] (see also [63]) with instabilities towards both s -wave and p -wave orders.By computing the β coefficients [58] in Eq. (1) forgeneric form factors, we find ¯ β = ˜ β and ¯ β < β β . Thisindicates a coexistence phase of s -wave and p -wave or-ders [45]. Thus the first-order transition with U (1) × U (1)symmetry expands into an intermediate phase. Since∆ s and ∆ p differs by a phase π/
2, this s + ip statespontaneously breaks time-reversal symmetry [44–46].Such state in three dimensions has unconventional ther-mal response described by a axion topological field the-ory [48, 60, 64–67], hence can be called an “axion super-conductor”. Phase diagram without inversion symmetry.—
Forspin-orbit-coupled materials without inversion symmetry,the Fermi surface is generally spin-split. With rotationalsymmetry, each spin-split Fermi surface is isotropic andhas a definite helicity χ = ±
1. The free energy, writtenin terms of the order parameters ∆ ± on each of the heli-cal Fermi surfaces, takes a general form F = α + | ∆ + | + α − | ∆ − | + δα (∆ + ∆ ∗− +∆ ∗ + ∆ − )+ β + | ∆ + | + β − | ∆ − | . Atthe phase boundary with δα = 0, the free energy retainsan explicit U (1) × U (1) symmetry. There are two sepa-rate transition temperatures, corresponding to the onsetof ∆ ± respectively. Away from the δα = 0 point, the twoorder parameters are always mixed down to zero tem-perature once either one of them becomes nonzero. Wethus obtain the phase diagram in Fig. 1(c). For a nega-tive (positive) δα , ∆ + and ∆ − take the same (opposite)sign. Switching to ∆ s,p notation, the phase with ∆ ± ofopposite signs has the p -wave pairing component domi-nating over the s -wave pairing. This phase is adiabati-cally connected to the p -wave-only phase in the presenceof inversion symmetry, and hence is topological [68].Finally, with broken rotational symmetry, again thelow-temperature first-order transition with U (1) × U (1)symmetry expands into a time-reversal symmetry break-ing phase [58, 69], as discussed before. At higher temper- kz FIG. 3. The chiral Majorana modes (green arrowed lines)bound to and connecting a pair of half vortices (1 ,
0) and(0 , atures, r ≡ ∆ s / ∆ p is real, and | r | (cid:29) ( (cid:28) )1 corresponds toa fully-gapped trivial (topological) phase. When r ∼ rf s (ˆ k ) = ± f p (ˆ k ), where ± corresponds to two spin-splitFermi surfaces. It can only be satisfied on one of the splitFermi surfaces. The nodes of this intermediate phasehave co-dimension 2 and are isolated points in two dimen-sions and nodal lines in three dimensions. Time-reversalsymmetry further requires that in 3D nodal lines appearin pairs and in 2D nodal points in multiples of four (seeFig. 2) [70]. These nodes are topologically protected bya Z invariant [54, 55], and lead to flat-bands of surfaceAndreev states [71–73]. The nodal lines are gapped uponentering the time-reversal breaking phase. Time-reversalbreaking in nodal line superconductors was obtained inRef. 74, but only for the surface states; here the time-reversal breaking occurs in the bulk. We summarize thephase diagram in Fig. 1(d). Experimental consequences.—
In the time-reversal-breaking phase, e.g. the s ± ip -SC, the surface state canbe thought of as a Majorana cone gapped by the s -wavecomponent [58]. Such a surface state exhibit thermal Halleffect and polar Kerr effect [65, 75].When rotational symmetry (even when approximate)is present, half quantum vortices, i.e. the phase windingof only one of ∆ ± [denoted as ( ± ,
0) and (0 , ± U (1) × U (1)symmetry. The magnetic flux through a half quantumvortex is given by hc/ (4 e ), i.e. half the flux quantum ina superconductor, hence the name. In 2D, the two heli-cal Fermi surfaces with χ = ± π , hence their corresponding half quantum vortex for∆ ± binds a single Majorana zero mode with non-Abelianstatistics [76–78]. This is in contrast with a full vortexin a time-reversal-invariant topological superconductor, which binds two Majorana modes with Abelian statis-tics.In 3D, the half quantum vortex line binds a propa-gating chiral Majorana mode [64, 67]. Furthermore, wefind that the dispersion (cid:15) = (cid:15) ( k z ) of such a chiral Ma-jorana mode exhibit both slow and fast components. In[58] we perturbatively solve the BdG equation for small k z (cid:28) ∆ /v F , and show that the dispersion of the chiralMajorana mode is given by (cid:15) ( k z ) = v M k z where v M ( k z = 0) ≈ (∆ + /µ ) log( µ/ ∆ + ) v F (cid:28) v F . (4)At larger k z ∼ k F , the 2D Fermi surface slice shrinksand the above perturbative result is no longer valid. Thevortex mode becomes higher in energy and merges intothe bulk with a much larger velocity v M ∼ v F . Thereforethe Majorana bound state contains both slow and fastmodes, both of which are chiral. We schematically showsuch a dispersion in the inset of Fig. 3.Given a pair of opposite half quantum vortices, thereexist a pair of chiral Majorana modes on the surface con-necting the two vortices. A (0,1) and (1,0) half quantumvortex pair can be viewed as a vortex/antivortex pair forthe relative phase ∆ ϕ = ϕ + − ϕ − between ∆ ± . Locally,this corresponds to (1 + e i ∆ ϕ ) s + (1 − e i ∆ ϕ ) p symmetry.In the slow-varying spatial limit, across the line where∆ ϕ = π , locally the surface states are described by twoMajorana cones with opposite mass terms [58], shown inFig. 3. The ∆ ϕ = π line acts as a mass domain wall forthe Majorana fermions, and thus support a chiral mode.The chiral Majorana modes bound to the half quantumvortices and the surfaces form a closed contour, shown inFig. 3. This chiral Majorana mode is charge neutral andcan support thermal transport. Relation to materials.—
Our theory can be appliedto systems where even and odd parity superconduct-ing order parameters are intertwined, such as Cd Re O and half-Heusler materials. For Cd Re O [34, 35], theanomalous enhancement in upper critical field H c in-dicates a symmetry change from spin singlet to spintriplet as a function of pressure. Our theory predictsnodal as well as time-reversal-breaking phases near thisregion in the phase diagram. In half-Heusler supercon-ductors YPtBi [79] and LuPtBi [80], order parameterswith a mixed even- and odd-parity pairings have beenproposed [81, 82] to account for penetration depth mea-surements [79]. This microscopic study finds line nodesin a region of mixed-parity phase, consistent with ourgeneral phase diagram for noncentrosymmetric super-conductors presented in Fig. 1(d). Our theory furtherpredicts that the superconducting state with line nodestransitions into a new time-reversal breaking phase uponlowering temperatures. It will be interesting to directlysearch for this time-reversal symmetry low-temperaturephase.This work was supported by the Gordon and BettyMoore Foundation’s EPiQS Initiative through Grant No.GBMF4305 at the University of Illinois (Y.W.) andthe DOE Office of Basic Energy Sciences, Division ofMaterials Sciences and Engineering under Award No.DE-SC0010526 (L.F.). We thank the hospitality ofthe summer program “Multi-Component and Strongly-Correlated Superconductors” at Nordita, Stockholm,where this work was initiated. Y.W. acknowledges sup-port by the 2016 Boulder Summer School for CondensedMatter and Materials Physics through NSF grant DMR-13001648, where part of the work was done. [1] L. Fu and E. Berg, Phys. Rev. Lett. , 097001 (2010).[2] M. Kriener, K. Segawa, Z. Ren, S. Sasaki, and Y. Ando,Phys. Rev. Lett. , 127004 (2011).[3] N. Levy, T. Zhang, J. Ha, F. Sharifi, A. A. Talin, Y. Kuk,and J. A. Stroscio, Phys. Rev. Lett. , 117001 (2013).[4] L. Fu, Phys. Rev. B , 100509 (2014).[5] K. 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Supplemental Material
I. PHASE DIAGRAM FROM GINZBURG-LANDAU THEORYA. With inversion symmetry
In this Section we present the details on the derivation of the four phase diagrams of Fig. 1 of the main text. First,with inversion symmetry, the Ginzburg-Landau (GL) free energy takes the following form F = α | ∆ s | + α | ∆ p | + β | ∆ s | + β | ∆ p | + 4 ¯ β | ∆ s | | ∆ p | + ˜ β (∆ s ∆ ∗ p + ∆ p ∆ ∗ s ) , (S5)and we are interested in regions with α ∼ α . The β coefficients are given by evaluating the Feynman diagrams inFig. S4, where the wavy line and the double line respectively represent ∆ s and ∆ p . Below we compute these diagramsfor cases with and without rotational invariance and obtain the corresponding phase diagrams.
1. With rotational invariance
For rotational invariant system, ∆ s and ∆ p by symmetry have the same uniform form factors (other than the oddparity part coming from the spin texture for p -wave). The β coefficients are distinguished by their spin structuresand symmetry factors: β = β iσ y )( iσ y ) † ( iσ y )( iσ y ) † ] = ββ = β i ˆ k · ~σσ y )( i ˆ k · ~σσ y ) † ( i ˆ k · ~σσ y )( i ˆ k · ~σσ y ) † ] = β ¯ β = β Tr[( i ˆ k · ~σσ y )( i ˆ k · ~σσ y ) † ( iσ y )( iσ y ) † ] = β ˜ β = β i ˆ k · ~σσ y )( iσ y ) † ( i ˆ k · ~σσ y )( iσ y ) † ] = β, (S6)where β = T X m ˆ d k π G ( ω m , k ) G ( − ω m , − k ) = N (0) T X m ˆ d(cid:15) k ( ω m + (cid:15) k ) = 5 ζ (3)8 π T N (0) . (S7)By arguments we have elaborated in the main text, for these values of β ’s the GL free energy has a U (1) × U (1)symmetry when α = α . When α = α the U (1) × U (1) symmetry is lifted and ground state is either s -wave or p -wave. This is shown in Fig. 1(a) of the main text.
2. Without rotational invariance
Without rotational invariance, the s - and p -wave order parameters ∆ s,p are generally of different form factors.Namely, H s =∆ s f s ( k ) × c † k iσ y ( c †− k ) T , H p =∆ p f p ( k ) × c † k ( k · σ ) iσ y ( c †− k ) T . (S8) ¯ ˜ FIG. S4. The diagrams corresponding to the coefficient β ’s. With regards to the β coefficients, the Pauli matrix algebra given in (S6) still holds. The only difference here is thatthe form factors f s ( k ) and f p ( k ) also enters the integral. Since the dominant contribution comes from near the FS,we can safely set | k | = k F in f s,p and reduce them to angular functions f s,p ( θ, φ ). Here ( θ, φ ) are angular coordinateson a 3D Fermi surface, and our analysis below extends straightforwardly to 2D cases. The β coefficients are thusgiven by β = N (0) T X n ˆ d sin θdφ π f s ( θ, φ ) ˆ d(cid:15) ( ω n + (cid:15) ) = β ˆ d sin θdφ π f s ( θ ) β = N (0) T X n ˆ d sin θdφ π f p ( θ, φ ) ˆ d(cid:15) ( ω n + (cid:15) ) = β ˆ d sin θdφ π f p ( θ )¯ β = ˜ β = N (0) T X n ˆ d sin θdφ π f s ( θ, φ ) f p ( θ, φ ) ˆ d(cid:15) ( ω n + (cid:15) ) = β ˆ d sin θdφ π f s ( θ, φ ) f p ( θ ) . (S9)By expanding f s and f p into spherical harmonics components Y ‘m ( θ, φ ), we have β = β ´ f s ( θ, φ ) d sin θdφ/ (4 π ) = β P ‘m S ‘m , β = β ´ f p ( θ, φ ) d sin θdφ/ (4 π ) = β P ‘m P ‘m , and ¯ β = ˜ β = ´ f s ( θ, φ ) f p ( θ, φ ) d sin θdφ/ (4 π ) = β P ‘m S ‘m P ‘m . It is straightforward to verify that¯ β = ˜ β < β β , (S10)In the free energy (S5), the minimization of the term ∆ s ∆ ∗ p dictates that the relative phase between ∆ s and ∆ p is∆ φ = ± π/
2, and thus ∆ s ∆ ∗ p = −| ∆ s | | ∆ p | . After substituting this relation and ¯ β = ˜ β into (S5), we obtain F = α | ∆ s | + α | ∆ p | + β | ∆ s | + β | ∆ p | + 2 ¯ β | ∆ s | | ∆ p | . (S11)Since ¯ β = ˜ β < β β , we find that s -wave and p -wave orders compete but coexist at their phase boundary. Inthe coexisting phase the time-reversal symmetry (TRS) is broken by the choice of ∆ φ = ± π/
2. We show the phasediagram in Fig. 1(b) of the main text.
B. Without inversion symmetry
1. With rotational invariance
When inversion symmetry is broken, the FS splits into two. However when rotational invariance is intact, one canstill define ∆ ± on each of the split FS. The two pairing fields are coupled to FSs with orthogonal helicity χ = ±
1, andthus decouple in the free energy except at quadratic level, which can be induced by pair hopping interaction betweenthe two FS’s. The free energy is thus F = α + | ∆ + | + α − | ∆ − | + δα (∆ + ∆ ∗− + ∆ ∗ + ∆ − ) + β + | ∆ + | + β − | ∆ − | . (S12)This is Eq. (7) of the main text. Note that this form of free energy can also be obtained using ∆ s,p = (∆ + + ∆ − ) / √ F = α | ∆ s | + α | ∆ p | + β | ∆ s | + β | ∆ p | + 4 ¯ β | ∆ s | | ∆ p | + ˜ β (∆ s ∆ ∗ p + ∆ p ∆ ∗ s )+ α (∆ s ∆ ∗ p + ∆ ∗ s ∆ p ) + β ( | ∆ s | + | ∆ p | )(∆ s ∆ ∗ p + ∆ ∗ s ∆ p ) + ¯ β ( | ∆ s | − | ∆ p | )(∆ s ∆ ∗ p + ∆ ∗ s ∆ p ) . (S13)Compared with the inversion symmetric case (S5), terms with linear coupling ∼ ∆ s ∆ ∗ p + ∆ ∗ s ∆ p is allowed due to thebroken inversion symmetry. To evaluate the coefficients one still evaluates the square diagrams, and for the fermionicGreen’s function we use ˆ G ( ω m , k ) =( iω m − (cid:15) k − λ ˆ k · ~σ ) − = iω m − (cid:15) k + λ ˆ k · ~σ [ iω m − (cid:15) k ] − λ , (S14)where λ characterizes the splitting of the FS. The derivation has been performed in Ref. S1 and the resulting freeenergy is identical to (S12). The calculation is rather lengthy and we do not repeat here.This free energy at δα = 0 has an enhanced U (1) × U (1) symmetry, and otherwise ∆ + and ∆ − are either of thesame sign (trivial) or opposite signs (topological) depending on the sign of δα . The corresponding phase diagram isin Fig. 1(c) of the main text.
2. Without rotational invariance
In the absence of both rotational invariance and inversion invariance, the free energy has the most general form, F = α | ∆ s | + α | ∆ p | + β | ∆ s | + β | ∆ p | + 4 ¯ β | ∆ s | | ∆ p | + ˜ β (∆ s ∆ ∗ p + ∆ p ∆ ∗ s )+ α (∆ s ∆ ∗ p + ∆ ∗ s ∆ p ) + β ( | ∆ s | + | ∆ p | )(∆ s ∆ ∗ p + ∆ ∗ s ∆ p ) + ¯ β ( | ∆ s | − | ∆ p | )(∆ s ∆ ∗ p + ∆ ∗ s ∆ p ) (S15)where β β > ¯ β = ˜ β , due to identical arguments as the case with inversion symmetry (it turns out the splitting ofthe FS does not affect the relation [S1]). To simplify the calculation we rescale ∆ p and all the coefficients such that β = β ≡ β , and then we have ¯ β = ˜ β < β = β . We define β ≡ β − ¯ β > α = ( α + α ) / δα = ( α − α ) / F = α ( | ∆ s | + | ∆ p | ) + δα ( | ∆ s | − | ∆ p | ) + β ( | ∆ s | + | ∆ p | ) + ( β − β )(∆ s ∆ ∗ p + ∆ ∗ s ∆ p ) − β | ∆ s | | ∆ p | + α (∆ s ∆ ∗ p + ∆ ∗ s ∆ p ) + β ( | ∆ s | + | ∆ p | )(∆ s ∆ ∗ p + ∆ ∗ s ∆ p ) + ¯ β ( | ∆ s | − | ∆ p | )(∆ s ∆ ∗ p + ∆ ∗ s ∆ p ) . (S16)To connect with previous cases, when inversion symmetry is intact, we have α = 0, β = 0, and ¯ β = 0, and whenrotational invariance is intact, we have β = 0 and ¯ β = 0.Now we analyze the ground states given by this free energy. Note that in (S16), there exist two types of phasecoupling terms, namely ∆ s ∆ ∗ p + ∆ ∗ s ∆ p and (∆ s ∆ ∗ p + ∆ ∗ s ∆ p ) . The interplay between these terms can drive a time-reversal symmetry breaking transition inside the SC phase, as we argued in the main text. However, the directminimization of this free energy is rather lengthy.The physics observed above is more transparent in an alternative approach. To this end, we define auxiliary orderparameters ∆ = (∆ s + ∆ p ) / √ , ∆ = (∆ s − ∆ p ) / √ . (S17)Note that in the absence of rotational symmetry, ∆ , are not uniform SC gaps on helical FS’s with χ = ± ± ), since ∆ s,p have distinct form factors. In terms of ∆ , , the free energy becomes F =( α + α ) | ∆ | + ( α − α ) | ∆ | + δα (∆ ∆ ∗ + ∆ ∗ ∆ )+ (2 β + β − β ) | ∆ | + (2 β − β − β ) | ∆ | + 2 β | ∆ | | ∆ | + ¯ β ( | ∆ | − | ∆ | )(∆ ∆ ∗ + ∆ ∗ ∆ ) + 2 β (∆ ∆ ∗ + ∆ ∗ ∆ ) . (S18)Recall that δα (∆ ∆ ∗ + ∆ ∗ ∆ ) ≡ δα ( | ∆ s | − | ∆ p | ) term distinguishes the onset temperature for s -wave (trivial)and p -wave (topological) SC orders, where δα = δα ( x ) is tuned by a parameter x . To be specific, we assume that δα ( x ) = x − x . On the other hand, upon lowering the temperature, α becomes negative, which drives the SCtransitions. To be specific, we assume that α ( T ) = ( T − T ) (for our purposes the units are unimportant). Withoutloss of generality, we also assume α <
0, which ensures that ∆ component dominates.We now compute the phase diagram in terms of x and temperature T . Upon lowering temperature, supercon-ductivity sets in when one of the eigenvalues of the quadratic form in Eq. (S18) first changes sign, i.e., when α − √ α + δα = 0. Thus the critical temperature of SC is T c ( x ) = T + p ( x − x ) + α . (S19)Below this temperature, in general both ∆ and ∆ are nonzero in the ground state, since the two are linearlycoupled. In this situation, for a generic x and T , calculating the ground state is usually a tedious task. To capturethe essential physics (i.e., to search for a second transition) with relatively simple calculations, we focus on a specialline in the phase space with δα = − ¯ β | ∆ | , ∆ = 0 . (S20)Along this line, the second transition is characterized by when ∆ acquires a nonzero expectation value.One can easily verify that these two conditions are consistent, at least right below T c – substituting (S20) into(S18), the linear coupling between ∆ and ∆ vanishes, and right below T c indeed ∆ = 0 is a local minimum of the(S18). In this case only ∆ is nonzero, and by simple math | ∆ | = s − α + α β + β − β ) . (S21)0 xT T c T TRSB ∆ =0 ∆ ∆ FIG. S5. The phase diagram in the case without rotational invariance and inversion symmetry. The red and blue arrowsdenotes the relative phases of ∆ and ∆ . Substituting (S21) back to (S20), we find that this special line for which ∆ vanishes below T c is given by x = x + ¯ β ( T − T + α )2(2 β + β − β ) , (S22)i.e., a straight line in x - T phase diagram which intersects with T c ( x ) at x = x .When the temperature further lowers, the quadratic coefficient for ∆ also gets negative and ultimately ∆ becomesnonzero. When this happens, the minimization of the last term of (S18) “locks” the relative phase between ∆ and∆ at ∆ ϕ = ± π/
2, the choice of which breaks an additional time-reversal symmetry. In terms of the original orderparameters ∆ s,p = (∆ ± ∆ ) / √
2, we have that | ∆ s | = | ∆ p | but ∆ s,p differ by a fixed phase φ = 2 arctan(∆ / ∆ ),i.e., this is a s + e iφ p state.This temperature is given by α − α − β | ∆ | = 0, which results in T = T + 2 β + β − β β + β − β α , ( α < , (S23)as a secondary transition temperature within the SC state.Away from this line given by (S22), the situation is more complicated, in the sense that right below T c ( x ) the twoorder parameters ∆ , are always mixed. Nevertheless, by continuation, a TRSB transition still exists. The onlydifference is that the presence of both ∆ ∆ ∗ + ∆ ∗ ∆ and ∆ ∆ ∗ + ∆ ∗ ∆ terms dictates that at small ∆ , therelative phase is locked at ∆ φ = 0 (for negative δα ) or ∆ φ = π (for positive δα ), and only deeper into the SC phasethe quartic coupling term becomes important and sets the relative phase at a nontrivial TRSB value ∆ ϕ = ± ϕ . Itis straightforward to verify that in terms of ∆ s,p , this corresponds to a s + ae iφ p state, where a is a real constant.Due to the presence of linear coupling terms which tends to perserve TRS, the TRSB temperature is generally lowerthan T .With these results we can plot the global phase diagram, as shown in Fig. S5. Note that compared with the casewith inversion symmetry, the TRSB phase gets detached from the T c ( x ) line. This is reminiscent of the phase diagramin Ref. S2. Also note that when ∆ = 0, ∆ s = ∆ p . Since ∆ s and ∆ p in the absence of rotational symmetry havedifferent form factors (see next section for more details), the resulting SC state becomes nodal. This is in contrastwith the rotational invariant case, where ∆ s = ∆ p leaves one of the helical FS fully gapless. In this case, even when∆ s = ∆ p but remains close, the SC node survives in a finite but small region of the phase diagram. As we discussedin the main text, the nodes are located on one of the FS’s, and form isolated points in 2D and nodal lines in 3D. Thisphase diagram in Fig. S5 with a nodal region is shown schematically in Fig. 1(d) in the main text. II. A MICROSCOPIC MODEL FOR TIME-REVERSAL SYMMETRY BREAKING PHASE
In the main text and in the previous section, we have used that fact that without rotational invariance, the s -and p -wave form factors are in general different and the resulting GL coefficient relation ensures a phase with brokentime-reversal symmetry. In this section, we explicitly construct a simple 2D model to compute the s - and p -wave formfactors.1 p k pk ( k + k ) · ~ ( p + p ) · ~ FIG. S6. The effective fermion-fermion interaction mediated by parity fluctuations.
In this model, fermions form a spin-degenerate elliptical Fermi surface, and superconductivity is mainly driven byparity fluctuations as proposed in Refs. S3 and S37, namely, U αβ,γδ ( k , k , p , p ) = V [(ˆ k + ˆ k ) · ~σ αβ ][(ˆ p + ˆ p ) · ~σ γδ ] (S24)which is shown diagrammatically in Fig. S6. The specific form of the vertex function of this pairing interactiondecouples fermions with helicity χ = ±
1, therefore the pairing gap on the two helical FS’s can either be of the samesign ( s -wave) or with opposite signs ( p -wave). To lift this degeneracy, we add weak interactions that favors eitherorder. The relative strength of the p -wave favoring and s -wave favoring interactions acts as the tuning parameter x on the horizontal axis of the phase diagrams shown in the main text. Specifically, we include a phonon-mediatedinteraction to favor the s -wave order, and another interaction mediated by ferromagnetic fluctuations [S4, S84] tofavor the p -wave order. The effective interaction mediated by ferromagnetic fluctuation (with magnetic moment σ z )is given by U sf αβ,γδ ( k , k , p , p ) = V sf ( θ − θ ) σ zαβ σ zγδ . (S25)Here θ and θ are Fermi surface angles, and V sf ( θ − θ ) is the (static) correlation function of the spin fluctuationsprojected to the Fermi surface. Admittedly this is only an artificial model, but again, our purpose here is to exemplifya general conclusion.We note that the weak interactions favoring s -wave and p -wave orders generally have different preferences for theSC form factors on a given helical FS. This is due to their different dependences on momentum transfer. In particular,the ferromagnetic fluctuations typically are sharply peaked at zero momentum transfer, thus the form factor of the p -wave order, which it enhances, tends to be more concentrated on FS regions with highest local density of statesto maximize condensation energy. On the other hand, since the spin-fluctuation mediated interaction is repulsive inthe s -wave channel, the form factor of the s -wave order becomes more concentrated on FS regions with lowest localdensity of states to avoid the repulsion.To see this more explicitly, we write down the linear SC gap equations (at T = T c ) for the two helical FS’s,∆ + ( θ ) = log Λ T c ˆ dθ N ( θ ) (cid:20) V cos (cid:18) θ − θ (cid:19) ∆ + + V ph (∆ + + ∆ − ) − V sf ( θ − θ )∆ − (cid:21) ∆ − ( θ ) = log Λ T c ˆ dθ N ( θ ) (cid:20) V cos (cid:18) θ − θ (cid:19) ∆ − + V ph (∆ + + ∆ − ) − V sf ( θ − θ )∆ + (cid:21) , (S26)where we have assumed weak-coupling pairing and we have integrated over frequency and the direction transverse tothe FS. The factor cos [( θ − θ ) /
2] comes from the inner product of spinors aligned with k and k [S1], and in relating∆ + with ∆ − for the V sf term we have used the fact that σ z χ + ( iσ y ) χ T + σ z = − χ − ( iσ y ) χ T − where χ ± = (1 ± ~σ · k ) / N ( θ ) ∝ / | v F ( θ ) | around the FS is not a constant and is peaked at thelong-axis direction. A simple calculation for dispersion E = k x / (2 m ) + k y / (2 m ) shows N ( θ ) = N m m m cos θ + m sin θ (S27)This set of linear gap equations can be easily solved numerically, say, by modeling V sf = V sf(0) cos ( θ − θ ), and weshow the form factors of the leading instabilities (i.e. with largest eigenvalue) are indeed different on a given helicalFS – that s -wave form factor is peaked around θ = 0 , π and the p -wave form factor is peaked around θ = ± π/
2. Weplot the form factors of s -wave and p -wave orders for a given helical FS with χ = 1 in Fig. S7.It is straightforward to verify that for these form factors, we indeed have β + β > β = 2 ˜ β . Applying our generalcriterion obtained in last Section, we indeed have a TRSB phase with s + ip order near the phase boundary of s -waveand p -wave orders, i.e., when the strength of the phonon coupling and the spin-fluctuation coupling are tuned to becomparable.2 θ f o r m f a c t o r ( a r b i t r a r y un i t s ) p -wave s -wave FIG. S7. The form factors for leading s -wave and p -wave orders for the helical Fermi surface χ = 1. The p -wave form factoron the χ = − III. CHIRAL MAJORANA MODES AT THE HALF QUANTUM VORTEX CORES
In this section we derive the chiral Weyl-Majorana modes bound to the core of a half quantum vortex line in 3D.In 3D, the helicity operator χ = ~σ · k is simply the chirality of a Weyl Fermi surface. A half quantum vortex is atopological defect around which only one chiral pairing field winds by 2 π , which is commonly denoted as (0 , ±
1) or( ± , H ( k ) = (cid:18) ~σ · k − µ ∆ e iθ ∆ e − iθ − ~σ · k + µ (cid:19) (S28)where θ is the real space polar angle in the xy plane. Such a Hamiltonian describes a half vortex line (1,0) in z direction. We look for the dispersion E = E ( k z ) of the in-gap modes bound to the vortex line. Our strategy will beto first solve for problem at k z = 0, which is a Majorana bound state, and then treat small k z as a perturbation andobtain the k z dispersion.It is instructive to first analyze the symmetry of Hamiltonian (S28), which strongly restricts the from of the low-energy wave function. First, there is a particle-hole symmetry, given by C = σ y τ y ( τ y is the Pauli matrix in Nambuspace), such that CH T ( − k ) C − = −H ( k ) . (S29)Second, note that (S28) is invariant under θ → θ + α , σ ± → σ ± e ± iα , and τ ± → τ ± e ∓ iα , where σ ± = σ x ± iσ y .The transformation of σ ± is dictated by the ~σ · k coupling, and that of τ ± can be verified explicitly. This rotationalinvariance indicates that one can define a conserved angular momentum j z = ‘ z + σ z / − τ z / . (S30)We expect the in-gap states to have j z = 0. Combining the two symmetry requirement above, we found that a generalform of the eigenstate at k z = 0 is given by χ ( r, θ, k z = 0) = (cid:2) f ( r ) g ( r ) e iθ g ∗ ( r ) e − iθ f ∗ ( r ) (cid:3) T , (S31)where r = x + y . Particle-hole symmetry requires its energy to be zero, and using the fact that ~σ · p = ( e − iθ σ + + e iθ σ − )( − i∂ r ) − ( e − iθ σ + − e iθ σ − )(1 /r ) ∂ θ , i∂ r g − ∆ g ∗ + ig/r + µf = 0 (S32) i∂ r g ∗ + ∆ g + ig ∗ /r + µf ∗ = 0 (S33) i∂ r f ∗ + ∆ f + µg ∗ = 0 (S34) i∂ r f − ∆ f ∗ + µg = 0 . (S35)3 Δμ v M FIG. S8. The Majorana velocity v M (in unit of the Fermi velocity v F ) as a function of the ratio between the SC gap ∆ andchemical potential µ . It is easy to check that Eqs. (S32,S33) and Eqs. (S34,S35) are consistent only when g = (1 + i )¯ g and f = (1 + i ) ¯ f .The equations for ¯ f ( r ) and ¯ g ( r ) are i∂ r ¯ g + i ∆¯ g + i ¯ g/r + µ ¯ f = 0 i∂ r ¯ f + i ∆ ¯ f + µ ¯ g = 0 . (S36)Using the ansatz ¯ f ( r ) = ˜ f ( r ) exp( − ´ r ∆ dr ) and ¯ g ( r ) = ˜ g ( r ) exp( − ´ r ∆ dr ), we have i∂ r ˜ g + i ˜ g/r + µ ˜ f = 0 (S37) i∂ r ˜ f + µ ˜ g = 0 . (S38)Replacing (S38) into (S37) we find that ˜ f satisfies the Bessel equation ∂ r ˜ f + (1 /r ) ∂ r ˜ f + µ ˜ f = 0 and ˜ g can be foundvia Eq. (S38). Using the properties of Bessel functions we have in final form f ( r ) = J ( µr ) exp (cid:20) − ˆ r ∆( r ) dr (cid:21) ( i + 1) g ( r ) = J ( µr ) exp (cid:20) − ˆ r ∆( r ) dr (cid:21) ( i − . (S39)With the knowledge of the wave function (S31), we can treat δ H ( k z ) = k z σ z τ z at a finite but small k z as perturbationand obtain the small- k z dispersion. Simple math shows E ( k z ) = v M k z , where the Majorana velocity v M is given by v M = ´ [ J ( r ) − J ( r )] exp( − r ) dr ´ [ J ( r ) + J ( r )] exp( − r ) dr , (S40)where we have assumed the the SC gap is a constant (at least away from the vortex). The integrals are expressed interms of complete elliptic integrals of first and second kind, K ( x ) and E ( x ), as v M = K ( − µ ∆ ) − ∆ ∆ + µ E ( − µ ∆ ) E ( − µ ∆ ) − K ( − µ ∆ ) . (S41)We plot v M as a function of ∆ /µ in Fig. S8. We see that in the full range v M >
0, which indicates a chiral Majoranamode.In the limit µ = 0 and ∆ /µ → ∞ , we have f = exp[ − ∆ r ]( i + 1) and g = 0, and the wave function is the Fu-Kaneresult [S6] for a superconducting TI surface. In this case one can check that the wave function (S31) is the eigenstatefor δ H ( k z ) (thus the first-order perturbation theory becomes exact), which is consistent with v M = 1 (in units where v F = 1). On the other hand, in the (more physical) limit where ∆ (cid:28) µ , the Majorana velocity is small but stillpositive. Expanding Eq. (S41), we obtain v M ≈ (∆ /µ ) log( µ/ ∆) . (S42)4For the opposite half vortices ( − ,
0) and (0 , IV. SURFACE STATES OF A 3D s + ip SUPERCONDUCTOR
In this section we derive the surface states for a s + ip superconductor, and show that they form a gapped Majoranacone. A similar derivation can also be found in Ref. S7.We assume that the FS is centered around the Γ point, and the Bogoliubov-de Gennes (BdG) Hamiltonian expandedaround Γ point can be written as H = Ψ † ( k ) (cid:18) − µτ z + ∆ p k k · ~στ x + ∆ s τ y (cid:19) Ψ( k ) , (S43)where µ >
0, Ψ( k ) = (cid:0) c ( k ) , iσ y c † T ( − k ) (cid:1) T is the Nambu spinor, σ is spin, and τ ’s are Pauli matrices in Nambu space.Note that ∆ s and ∆ p terms have different Nambu spin structure, indicating their π/ σ z τ x ≡ σ z ⊗ τ x and τ y ≡ I ⊗ τ y ( I is a 2 × z -direction, and model the surface as a domain wall of the chemicalpotential µ ( z ) = µ sgn( z ). Thus, the z < z > H sp = H + H , where H =Ψ † ( k ) (cid:20) − µτ z + ∆ p k ( k z σ z ) τ x (cid:21) Ψ( k ) H s =Ψ † ( k ) (cid:20) ∆ p k ( k x σ x + k y σ y ) τ x + ∆ s τ y (cid:21) Ψ( k ) (S44)The the solution eigenstate of H is standard, and is given byΨ( z ) = Ψ exp (cid:18) − µk ∆ p ˆ z sgn z dz (cid:19) , (S45)which is a surface bound state at z = 0, and σ z τ y Ψ = Ψ . The eigenvalue of H is zero. Using this wave-function,particular its spinor structure, we find that for wave functions of this type, the effective surface Hamiltonian is H surf = ∆ p k ( k x σ y τ z − k y σ x τ z ) + ∆ s σ z , (S46)which is the dispersion of a gapped Majorana cone. [S1] Y. Wang, G. Y. Cho, T. L. Hughes, and E. Fradkin, Phys. Rev. B , 134512 (2016).[S2] S. Maiti and A. V. Chubukov, Phys. Rev. B , 144511 (2013).[S3] V. Kozii and L. Fu, Phys. Rev. Lett. , 207002 (2015).[S4] A. J. Leggett, Rev. Mod. Phys. , 331 (1975).[S5] D. Vollhardt and P. W¨olfle, The superfluid phases of Helium 3 (Taylor & Francis, London, UK, 1990).[S6] L. Fu and C. L. Kane, Phys. Rev. Lett. , 096407 (2008).[S7] P. Goswami and B. Roy, Phys. Rev. B90