Electronic correlations stabilizing time-reversal broken chiral superconductivity in single-trilayer TiSe 2
R. Ganesh, G. Baskaran, Jeroen van den Brink, Dmitry V. Efremov
aa r X i v : . [ c ond - m a t . s up r- c on ] M a r Electronic correlations stabilizing time-reversal broken chiral superconductivity insingle-trilayer TiSe R. Ganesh, G. Baskaran,
2, 3
Jeroen van den Brink,
1, 4 and Dmitry V. Efremov Institute for Theoretical Solid State Physics, IFW-Dresden, D-01171 Dresden, Germany The Institute of Mathematical Sciences, C.I.T. Campus, Chennai 600 113, India Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada N2L 2Y5 Department of Physics, TU Dresden, D-01062 Dresden, Germany (Dated: January 23, 2018)Bulk TiSe is an intrinsically layered transition metal dichalcogenide (TMD) hosting both super-conducting and charge density wave (CDW) ordering. Motivated by the recent progress in preparingtwo-dimensional TMDs, we study these frustrated orderings in single trilayer of TiSe within a renor-malization group approach. We establish that a novel state with time-reversal symmetry brokenchiral superconductivity can emerge from the strong competition between CDW formation and su-perconductivity. Its stability depends on the precise strength and screening of the electron-electroninteractions in two-dimensional TiSe . PACS numbers:
Introduction
Transition metal dichalcogenides(TMDs) with the chemical formula MX , where M is atransition metal from groups IV-VI (Ti, Zr, Hf, V, Nb,Ta etc. ) and X is a chalcogen element (Se, S, Te), areemerging as a new class of two-dimensional materialswith high potential for nanoelectronics applications [1–4]. The intense research activity in this field is inspiredby the graphene boom, which was sparked by thepossibility of manufacturing a purely two-dimensionalmaterial with high carrier mobility. TMDs consist ofstacked X-M-X trilayers which, just as graphene, havehexagonal symmetry. These trilayers are held togetherby weak van der Waals forces, which allows exfoliationof the individual trilayers and the deposition of theselayers onto various substrates [5].Interestingly, the plethora of phenomena that occurin TMDs is even more multifarious than in graphene.Metallic TMDs not only have a generic instability to-wards the formation of different types of charge densitywaves (CDWs), but some of them also host supercon-ductivity (SC). Moreover, due to the presence of tran-sition metal elements electron-electron interactions canplay a significant role. From the point of view of su-perconductivity, this is a highly interesting mix of in-gredients. It is well known that the competition of SCwith density-waves in the presence of electronic corre-lations may lead to unconventional superconducting or-der, particularly in lower-dimensional systems. Exam-ples are d -wave pairing in quasi-2D cuprate superconduc-tors [6], s + − pairing in layered iron-pnictides [7, 8] and p -wave triplet pairing in Sr RuO [9]. Sr RuO is par-ticularly interesting as superconductivity is characterizedby a chiral order parameter that spontaneously breakstime-revisal symmetry [10], a property it shares with justa few other very low temperature SCs, e.g. UPt [11] and(TMTSF) PF [12]. Ordering which breaks time rever-sal has also been discussed in the context of cuprates[13] π /34 π /3 + ++ + ++- TRI s+- order AnticlockwiseClockwiseChiral, TRB superconductivity
FIG. 1: Unconventional superconducting orders in a singletrilayer of TiSe . From left to right: colour map representingsuperconducting phase, Time Reversal Invariant (TRI) s + − ordering, and clockwise and anticlockwise variants of chiral,time reversal broken (TRB) ordering. and Na x CoO .yH O[14, 15]. Vortices in these chiral SCsharbor Majorana fermions [16] which may constitute thebuilding blocks needed for future topological quantumcomputing technologies, robust against decoherence [17].Here we focus on the frustrated superconductivity inTiSe , which in bulk form is a layered semi-metal with aCDW transition at ∼ T c ≈
4K [19]. In the bulk material, the su-perconducting order parameter is nodeless [19]. Using arenormalization group approach, we focus on the case ofa single trilayer of TiSe and show that it has excitingordering phenomena. In this case, melting of the CDWphase gives way to one of two possible superconductingground states, both of which are unconventional. Thefirst is a time-reversal invariant (TRI) state with s + − pairing while the other corresponds to time-reversal bro-ken (TRB), chiral SC, see Fig. 1. Their relative stabil-ity depends on the precise strength and screening of theelectron-electron interactions in 2D trilayer of TiSe ontop of its substrate. Effective Lagrangian and couplings
A TiSe trilayerhas an elegant band structure. We have performed ab initio calculations using FPLO[20, 21] to find the fermisurfaces. In line with previous reports[22], we find (i) twohole-like pockets around the Γ point which are nearly de-generate, and (ii) three electron-like pockets, one aroundeach M point in the Brillouin zone. In the 3D case,these bands become elongated along the Z axis and formdistorted cylinders – the 3D material has an additionalspherical pocket around the Γ point.Due to approximate nesting between electron andhole bands, there are logarithmic singularities in bothparticle-particle and particle-hole channels. In order totreat these on an equal footing, we use renormalizationgroup (RG) analysis to establish the low energy cou-plings. In previously studied cases with nesting and on-site repulsion such as cuprates, pnictides and graphene,RG flow gives low energy couplings that are conduciveto SDW order [23–25]. Here however, CDW order arises although the microscopic interactions are repulsive. Wewill show that this comes about via a special umklapp-process that is allowed by the hexagonal symmetry ofTiSe which strongly renormalizes the particle-hole andparticle-particle channels.In the following RG analysis, we approximate the bandstructure as follows. We merge the two hole pocketsaround the Γ-point and give it the band index 0. Withthe three electron pockets around the M -points, we asso-ciate the indices α = 1 , ,
3. The electron and hole pock-ets are approximately nested, so that there are nine dif-ferent scattering processes allowed by momentum conser-vation (see Supplementary Material for a diagrammaticrepresentation). As the Fermi surfaces have small radii,these couplings can be taken as independent of the pre-cise initial and final momenta. The system is describedby the Lagrangian: L = ψ † ( ∂ τ − ε k ) ψ + X α =0 ψ † α ( ∂ τ − ǫ αk ) ψ α − n U ( ψ † ψ † ψ ψ + ψ † ψ † ψ ψ + cyclic exchange) (1)+ 12 U ψ † ψ † ψ ψ + X α =1 (cid:2) U ψ † ψ † α ψ α ψ + U ψ † ψ † α ψ ψ α + 12 U ( ψ † ψ † ψ α ψ α + h.c. ) + 12 U ψ † α ψ † α ψ α ψ α (cid:3) + 12 X α = β (cid:2) U ψ † α ψ † β ψ β ψ α + U ψ † α ψ † β ψ α ψ β + U ψ † α ψ † α ψ β ψ β (cid:3)o We have implicitly assumed the spin structure σσ ′ σ ′ σ , i.e. : U ψ † ψ † α ψ ψ α = P σσ ′ U ψ † σ ψ † σ ′ α ψ σ ′ ψ σα . Fornested hole and electron pockets the dispersions reduceto ( − ) ǫ k ≈ ǫ k + M = ( k x + k y ) / m − µ . The interactions U , U and U are allowed umklapp processes depictedin Fig. 2. We emphasize that U has no analogue inother multi-band systems considered within RG recently,neither in pnictides [24, 26] nor in graphene [25, 27]. It isallowed by the hexagonal band structure, as the three Mmomenta add to zero. We later show that precisely thisprocess drives CDW order in TiSe as opposed to SDWorder in the pnictides or in graphene.RG flow proceeds by integrating out excitations abovea floating cutoff scale. Due to approximate nest-ing, the electron-hole polarization bubble ( | Π el − h | ∝ N log(Λ / max { T, µ d } )) has the same logarithmic diver-gence as particle-particle bubble ( C h − h = C el − el ∝ N log(Λ /T )). Treating both on an equal footing, we useconventional one-loop RG approach keeping only parquet diagrams. The flow of couplings is given by:˙ u = u + u − u , ˙ u = − u − u + 2 u u , ˙ u = u { u − u − u − u − u } , ˙ u = − u − u , ˙ u = − u − u − u , ˙ u = u { u − u + u − u − u } , ˙ u = 2 u − u − u , ˙ u = − u u , ˙ u = − u + 2 u − u u − u . (2)The derivative is with respect to RG time t = log ( W/E ),where W is the bandwidth and E is the floating RGscale. In addition, we have scaled the interaction ampli-tudes U i by the DOS at the Fermi level N i ( u i ≡ N i U i ).The derivation for u is illustrated in the SupplementaryMaterial; others can be derived similarly. These par-quet equations are valid for the energy E & µ . Belowthis energy, density wave channels and superconductiv-ity decouple and the flow has to be modified[28]. In thisletter, we consider small Fermi pockets, thus neglectingthe change of flow at E ∼ µ . G M K Γ G M K Γ G M K Γ M M M M M M M M M Γ Γ Γ
FIG. 2: Representative umklapp scatterings allowed by thegeometry of TiSe . They arise from the symmetry propertiesof the M points, viz., ~M + ~M + ~M = 0 and 2 ~M i ≡ The leading instabilities
To investigate the leading in-stabilities we introduce infinitesimal test vertices in theparticle-hole and particle-particle channels: δ L CDW = X α =1 ρ (0) cα σ ηη ′ ψ † η ψ αη ′ ,δ L SDW = X α =1 ρ (0) sα σ xηη ′ ψ † η ψ αη ′ ,δ L SC = ∆ (0)0 iσ yηη ′ ψ † η ψ † η ′ + X α =1 ∆ (0) α iσ yηη ′ ψ † αη ψ † αη ′ , where σ and σ α are the identity and the Pauli matri-ces respectively. We suppose implicit summation overthe spin index. Writing the gap equation for each or-der, we identify a corresponding ‘effective vertex’ as afunction of u α couplings (see Supplementary Material).Within this analysis in the framework of the linear ap-proximation, the CDW and SDW orders at each M pointdecouple. Furthermore, at each M point, both CDW andSDW order parameters decouple into two parts which wedesignate ‘real’ and ‘imaginary’. They obey correspond-ingly ( ρ rc/s,α ) ∗ = + ρ rc/s,α and ( ρ ic/s,α ) ∗ = − ρ ic/s,α . Theeffective vertices for real and imaginary SDW and CDWorders are given by Γ SDWreal/imag = u ± u , Γ CDWreal/imag = u ∓ u − u . At lower temperatures, multiple-Q orderingmay appear due to interaction between modes. Indeed,such 3Q-ordering has been observed in 3D TiSe [29].In the superconducting channel, our Fermi surface ge-ometry couples the order parameters on individual pock-ets. In accord with symmetry considerations, we get foureigenmodes of superconductivity: (i) s ++ conventionalsuperconductivity, characterized by real order parame-ters on the central pocket (∆ = ∆ Γ ) and the pocketsaround M-points (∆ = ∆ = ∆ = ∆ M ), both havingthe same sign ( sign (∆ Γ ) = sign (∆ M )). (ii) s + − with realorder parameters having different signs on the central andM-pockets, i.e., ( sign (∆ ) = − sign (∆ M )), as shown inFig. 1. It is analogous to the order parameter proposedfor the recently discovered Fe-based superconductors. (iii& iv) chiral superconductivity, which breaks time rever-sal symmetry. At the level of linearized gap equations,the central pocket is completely decoupled. There aretwo degenerate solutions, corresponding to clockwise and t = log (W/E) CDWI - SCchiral
CDWI - SCs+-
SDWI - SCs++
SDWR
FIG. 3: RG flow of effective vertices. We have used bareinteractions estimated assuming intra-atomic Coulomb inter-actions: u (0)1 = u (0)2 = u (0)3 = 0 . u (0) , u (0)4 = 1 . u (0) , u (0)5 = u (0)7 = u (0)8 = u (0)9 = 0 . u (0) and u (0)6 = 0 . u (0) ,taking u (0) = 0 . anticlockwise winding of the phase of the order parame-ters, shown in Fig. 1. One of the two solutions is given by∆ = e i π/ ∆ = e − i π/ ∆ = ∆ M . A similar phase hasbeen proposed in highly doped graphene[25, 27, 30, 31].The effective vertices are given by − Γ SCs ++ = − ( u + u +2 u ) / − sign ( u ) R , − Γ SCs + − = − ( u + u + 2 u ) / sign ( u ) R , − Γ SCchiral = − u + u , where we have denoted R = p u + ( u − u − u ) / g orbitals and Se p orbitals to DOS inthe Fermi pockets. From ab initio calculations, we findthe orbital contributions to states in each Fermi pocketto be N Γ T i ∼ . N Γ Se ∼ . N MT i ∼ .
75 and N MSe ∼ . u (0)1 = u (0) ( N Γ T i N MT i + N Γ Se N MSe ), u (0)4 = u (0) ( { N Γ T i } + { N Γ Se } ),where u (0) is a parameter capturing the strength of theCoulomb interaction. Using these values, we find thatthe largest effective vertex corresponds to real SDW or-der Γ SDWreal ∼ (0 . u (0) . Superconducting channels dropout as their effective vertices are repulsive. Mean fieldtreatment thus predicts SDW order; however, RG flowmodifies the couplings and changes the preferred order-ing. Fig. 3 shows the RG flow of effective vertices startingfrom these bare interactions – chiral SC ultimately dom-inates. Fixed points in the RG flow
The flow of couplingsgiven by Eqs. 2 is governed by three fixed points, whereinall couplings scale with with one diverging quantity. We ζ (Screening ) - M u m k l app p r o c e ss , u ( ) FIG. 4: The basins of the three fixed points. The dashed linedepicts u bare ( ζ ). The bare vertices are u (0)1 /ζ = u (0)2 = u (0)3 =0 . u (0) /ζ , u (0)4 = 1 . u (0) , u (0)5 /ζ = u (0)7 /ζ = u (0)8 = u (0)9 =0 . u (0) /ζ . The precise location of basin boundaries weaklydepends on u (0) . rewrite the interactions as u i = b i u with u > b = −
1, with all other cou-plings negligible b i = 0. At this fixed point, the largesteffective vertices correspond to both real and imaginarysolutions of CDW order Γ CDWreal = Γ
CDWimag .(ii) “Chiral SC fixed point”: b = − b / >
0, whileother b ’s vanish. The largest effective vertex then corre-sponds to chiral SC.(iii) “ s + − fixed point”: b = − b = p / b =1 / b , b = − /
3. The couplings b , b , b , b vanish.The leading vertex is s + − SC.If ordering in TiSe were driven by phonons, we wouldexpect to see CDW order and s ++ superconductivity.In contrast, electronic correlations under RG flow giveCDW order and s + − or chiral superconductivity. Theabsence of s ++ superconductivity can be traced to theflow equation for u in Eqs. 2. The sign of u cannotchange under RG flow and always remains positive, thusfavouring s + − pairing over s ++ . Phase diagram in RG scheme
RG flow depends onthe initial, bare couplings which we estimate using abinitio data for the orbital DOS. Building upon this, weintroduce two free parameters, u and ζ , to characterizethe bare couplings. The parameter u is simply the barevalue of the u coupling; we use it as a parameter in orderto emphasize the key role of the u process. The secondparameter ζ models the momentum dependence of thescreened Coulomb interaction. The low energy scatteringprocesses fall into two classes: small and large ( ∼ M ) mo-mentum transfer. The latter are reduced by the factor ζ .For example, we have u (0)1 = u (0) ( N Γ T i N MT i + N Γ Se N MSe ) and u (0)2 = u (0) ( N Γ T i N MT i + N Γ Se N MSe ) /ζ . For strong screening,we expect local interactions and momentum-independent interactions, giving ζ ∼
1. For weak screening, ζ > u = 0. When the barevalue of u is zero, RG flow cannot generate a finite u value (see Eqs. 2). Without u (along the u = 0 line),we do not approach the CDW fixed point or the chiralSC fixed point. To estimate the ‘microscopic’ value of u , we use the same reasoning as with the other bareparameters to obtain u bare ( ζ ) = { ( N Γ Se ) / ( N MSe ) / +( N Γ T i ) / ( N MT i ) / } /ζ . As u involves large momentumtransfer, it is scaled down by ζ . This choice of u placesus in the basins of CDW and s + − fixed points. However,for some ζ values, the microscopic parameters lie veryclose to the border of the CDW basin. Taken together,our results suggest that the ground state of 2D TiSe could have CDW order, chiral superconductivity or s + − pairing. Discussion
Our results for TiSe should be comparedwith the pnictides wherein Coulomb interactions lead to s + − SC order which competes with SDW order. Ul-timately, this difference in behaviour stems from u ,the umklapp process allowed by the geometry of the Mpoints. CDW order requires a negative value of u at thefixed point, whereas the bare Coulombic value of u ispositive. However, a non-zero u reduces the value of u under RG flow (see Eq. 2) to negative values.The chiral superconducting state breaks time reversalsymmetry with many interesting consequences. Previ-ous studies have highlighted the presence of fractionalvortices and domain walls in such a superconductingstate [32]. If the central pocket is indeed decoupled asshown in Fig. 1, this pocket may undergo pairing witha different transition temperature. Instead, the centralpocket could possess d + id order! This possibility is alsofavoured by ab initio calculations which show that eachM pocket is dominated by a single Ti t g orbital. Thecentral pocket shows a strong angular dependence in t g orbital character consistent with d + id pairing. Summary
We have analyzed the competing phases intwo dimensional hexagonal structures which allow specialumklapp processes. In systems with Coulomb repulsion,these processes give rise to CDW order instead of SDW.This CDW state competes with chiral and s + − supercon-ductivity and not s ++ superconductivity expected froma phonon mechanism. We have focused on two dimen-sional TiSe with two small hole pockets around Γ pointand electron pockets around the M-points, but it is inter-esting to note that 2D TiS actually has a similar bandstructure. Layered 3D TiSe has an additional sphericalhole pocket around the Γ point – our RG group analysisis still valid, but for high energies when the bands are2D-like. While nodeless superconductivity has been seenin 3D TiSe [19, 33], our results call for a more detailedexamination of the nature of superconductivity, in partic-ular in exfoliated layers of this materials with nanoscopicthicknesses. [1] M. Xu, T. Liang, M. Shi, and H. 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B , 085431 (2010), URL http://link.aps.org/doi/10.1103/PhysRevB.81.085431 .[32] T. Yanagisawa, Y. Tanaka, I. Hase, and K. Yamaji, Jour-nal of the Physical Society of Japan , 024712 (2012),URL http://jpsj.ipap.jp/link?JPSJ/81/024712/ .[33] E. Morosan, H. W. Zandbergen, B. S. Dennis, J. W. G.Bos, Y. Onose, T. Klimczuk, A. P. Ramirez, N. P. Ong,and R. J. Cava, Nature Physics , 544 (2006). r X i v : . [ c ond - m a t . s up r- c on ] M a r Supplementary Material:Electronic correlations stabilizing time-reversal broken chiralsuperconductivity in single-trilayer TiSe R. Ganesh, G. Baskaran,
2, 3
Jeroen van den Brink,
1, 4 and Dmitry V. Efremov Institute for Theoretical Solid State Physics,IFW-Dresden, D-01171 Dresden, Germany The Institute of Mathematical Sciences,C.I.T. Campus, Chennai 600 113, India Perimeter Institute for Theoretical Physics,Waterloo, Ontario, Canada N2L 2Y5 Department of Physics, TU Dresden, D-01062 Dresden, Germany (Dated: Version 4, 18 Nov 2013, compiled January 23, 2018) K s G M K Γ Γ FIG. S1: Left: Fermi surfaces of two dimensional TiSe slab obtained from ab initio calculations.Right: Approximate Fermi surface geometry considered for RG calculations. u α α u αα u ααu u αα αα u α βγ α = β = γ u αα u βα αβ u α βα β α = βα = β α = β β β FIG. S2: The nine quartic couplings present in the low energy theory. Solid lines represent electronsthat live on the central pocket. Dashed lines with coefficients α = 1 , , I. BAND STRUCTURE AND ALLOWED COUPLINGS
The Fermi surfaces obtained from ab initio calculations are depicted in Fig. S1a. Fig. S1bshows the approximate Fermi surface structure that we have used in the RG calculations.This configuration allows for nine different interaction processes, shown in Fig. S2. Thesecouplings are included in the Lagrangian defined in the main text.
II. PARQUET DIAGRAMS FOR RG FLOW
One-loop RG proceeds by integrating out higher energy excitations over a floating energyscale. For a ‘renormalizable’ system, this process preserves the structure of low energy scat-terings allowing a systematic treatment. The conditions for renormalizability are met in ourtwo dimensional system which has a constant density of states as we take our Fermi surfacesto be circular. Assuming perfect nesting between electron and hole pockets, both particle-2 u u u α α ∼ ++ αα u u α αα u u α αβ = α (2) αα α αα β = α FIG. S3: Parquet diagrams for the RG flow of coupling u , representing inter-pocket scatteringwithin the α pocket. particle and particle-hole bubbles contribute in the RG process, leading to a logarithmiccorrection to low energy scatterings.Going from 2D TiSe to 3D TiSe , the quasi-circular bands elongate along the axis to formdistorted cylinders. However, the 3D material has one extra band which crosses the Fermilevel – a spherical band around the Γ point. This band ruins renormalizability by introducinga strong energy-dependence into the density of states at the Fermi level. However, we stillexpect our results to apply to 3D TiSe where the central band may not play a crucial role.RG flow equations derived from parquet diagrams are given in Eqs. 2 in the main text.As an illustration, the derivation of the u equation is shown diagrammatically in Fig. S3 III. GAP EQUATIONS AND EFFECTIVE VERTICES
To test SC susceptibility above T c , we introduce external pairing fields ∆ (0) α . As shownin Fig. S4, they are dressed by interactions giving rise to response fields ∆ ∆ ∆ ∆ = I × + C u u u u u u u u u u u u u u u u − ∆ (0)0 ∆ (0)1 ∆ (0)2 ∆ (0)3 . Here, C = T P ω m R d k (2 π ) G /α ( k , ω m ) G /α ( − k , − ω m ) represents the particle-particle or hole-hole bubble (their values are assumed to be equal due to nesting). The Green’s functions G ( k , ω m ) and G α ( k , ω m ) are on the central pocket and on the M α pocket ( α = 1 , , = ∆ (0)0 +∆ +∆ α u ∆ = ∆ (0)1 +∆ u +∆ u +∆ β
11 11 11 11 11 u αα u ββ FIG. S4: Gap equations for superconducting orders. Top: dressing of a pairing field in the centralpocket. The index α has to be summed over all three M pockets. Bottom: dressing of a pairingfield acting on the M pocket. The index β has to be summed over { , } , the other two M pockets. respectively. These equations are diagonalized by the transformation ∆ clock.chiral ∆ anti − cl.chiral ∆ s ++ ∆ s + − = e i π/ e i π/ e i π/ e i π/ λ + λ − ∆ ∆ ∆ ∆ . ∆ clock.chiral and ∆ anti − cl.chiral represent degenerate, clockwise and anti-clockwise, order parameters ofchiral superconductivity.In the above equation, λ ± = { ( u − u − u ) / ± R } /u ; where R is as defined in the maintext. Provided u >
0, the ∆ s + − solution satisfies ( sign (∆ M ) = − sign (∆ Γ )), correspondingto s + − pairing. The ∆ s ++ solution also has the correct symmetry for s ++ order. However,if u <
0, the characters of the solutions are swapped; the effective vertices of s ++ and s + − pairing are interchanged. However, the flow equation for u is such that it can never changesign. As the bare Coulombic coupling is positive, u always remains positive.In terms of these new SC order parameters, the dressed response fields simplify to∆ clock./anti − cl.chiral = { C · Γ SCchiral } − ∆ clock./anti − cl. (0) chiral , ∆ s + − = { C · Γ SCs + − } − ∆ (0) s + − , ∆ s ++ = { C · Γ SCs ++ } − ∆ (0) s ++ . (S1)Upon cooling to the critical temperature, provided Γ SC <
0, the quantity within braces willvanish; the response field can then be non-zero even when the external field is infinitesimal.This condition gives T c . The order with the highest Γ will have the highest T c .The effective vertices for CDW and SDW order can be derived similarly. The gap equa-tions for CDW and SDW order are shown diagrammatically in Fig. S5. These diagrams give4 ρ (0) cα ρ cα + ρ cα α α α + ρ ∗ cα u u ρ sα = ρ (0) sα ~σ ~σ + ρ sα ~σα α α αu + ρ ∗ sα ~σ αu α αα u αα + ρ ∗ cα + ρ cα u ααα FIG. S5: Diagrammatic representation of the gap equations for CDW and SDW orders. linearly decoupled order parameters ( ρ rc/s ) ∗ = ( ρ rc/s ) and ( ρ ic/s ) ∗ = − ( ρ ic/s ) for both CDWand SDW. We will designate them as ‘real’ and ‘imaginary’ order parameters. We extractthe effective vertices from the linearized gap equation; i.e., approaching from high temper-atures, we assume that ordering sets in at some transition temperature. This requires thefollowing condition,1= − T r,iCDW Γ CDW ( r,i ) X ω m Z d k (2 π ) G ( k , ω m ) G α ( k + M i , ω m ) , − T r,iSDW Γ SDW ( r,i ) X ω m Z d k (2 π ) G ( k , ω m ) G α ( k + M i , ω m ) , where r and i denote real and imaginary parts of the order parameters. T iSDW and T rSDW are transition temperatures for the order parameters ρ rs and ρ is respectively. The effectivevertices for CDW order are given Γ CDWreal = u − u − u and Γ CDWimag = u + u − u . Thevertices for SDW order are Γ SDWreal = u + u and Γ SDWimag = u − u . IV. ROLE OF THE CHEMICAL POTENTIAL
At each step of RG flow, we integrate out excitations that lie between E and E + dE ,where E is the floating RG scale. We track the resulting correction to low energy scatter-ing processes that involve quasiparticles near the Fermi surface. As E decreases from thebandwidth down to the Fermi energy, the flow of couplings separates into two regimes. Athigher energies E > µ (we measure energies from the Fermi level), when integrating out5 t = log (W/E)
CDWI - SCchiral
CDWR - SCs+-
SDWI - SCs++
SDWR
FIG. S6: Flow of effective vertices for bare interactions estimated as in main text, with screeningparameter ζ = 1 .
07. The bare interactions are u (0)1 /ζ = u (0)2 = u (0)3 = 0 . u (0) /ζ , u (0)4 = 1 . u (0) , u (0)5 /ζ = u (0)7 /ζ = u (0)8 = u (0)9 = 0 . u (0) /ζ and u (0)6 = 0 . u (0) /ζ , with u (0) = 0 . higher energy excitations, we can approximate the total incoming momentum to be zero.This RG group flow is captured by parquet diagrams. The particle-particle and particle-hole channels are coupled due to umklapp processes U and U . The renormalization flowis captured by parquet diagrams leading to the Eqs. 2 in the main text. At low energiesbelow µ , the polarization correction depends on the incoming momentum of the scatteringprocess. The two channels decouple – flow is be described by separate ladder diagrams foreach channel[1]. Thus, flow according to parquet diagrams can be cut off at µ . For example,Fig. S6 shows RG flow which ultimately goes to the s + − SC fixed point. However, duringmost of the duration of RG flow, the effective vertex for imaginary CDW order dominates.When the floating RG scale reaches µ , CDW order may have the largest effective vertexeven though the ultimate fixed point corresponds to s + − SC order. Below µ , the effectivevertices for CDW and superconducting orders will evolve according to different sets of lad-der diagrams. It is then possible that CDW order may set in at a higher temperature, thuspre-empting superconductivity. For large values of ζ & .
6, we find that SDW order maypre-empt superconductivity. 6
1] S. Maiti and A. V. Chubukov, Phys. Rev. B , 214515 (2010), URL http://link.aps.org/doi/10.1103/PhysRevB.17.1839 ..