Coexistence of CDW with staggered superconductivity in a ferromagnetic material
aa r X i v : . [ c ond - m a t . s up r- c on ] M a r epl draft Coexistence of CDW with staggered superconductivity in a ferro-magnetic material
M. Georgiou , G. Varelogiannis and P. Thalmeier Department of Physics, National Technical University of Athens, GR-15780 Athens, Greece Max Planck Institute for Chemical Physics of Solids, 01187 Dresden, Germany
PACS – Theories and models of superconducting state
PACS – Pairing symmetries (other than s-wave)
Abstract. - In usual superconductivity (SC), the pairs have zero total momentum irrespectiveof their symmetry. Staggered SC would involve, instead, pairs with a finite commensurate totalmomentum, but such exotic states have never been proven to be realized in nature. Here we studyfor the first time the influence of particle-hole asymmetry on the competition of staggered SC withCharge Density Waves (CDW) in a ferromagnetic medium. We obtain unprecedented situationsin which CDW and staggered SC coexist . We also obtain cases of a SC dome near the collapseof a CDW state as well as cascades of transitions that exhibit remarkable similarities with thepressure phase diagram in UGe suggesting that SC in this material may be staggered coexistingand competing with a CDW state. The field of unconventional superconductivity (SC) isan extraordinarilly rich source of challenging problems forfundamental and applied physics. High- T c cuprates, heavyfermion materials, borocarbides and organic SC are exam-ples of unconventional SC. A multitude of unconventionalSC states have been proposed but only few have beenproven to be realized in real material systems. Singletor triplet unconventional SC states considered as realizedso far are characterized by a zero total momentum of thepairs indicating in fact that the superfluid density is ho-mogeneous in momentum space. In the present Letter wesuggest that the surprising ferromagnetic (FM) SC stateof UGe is instead staggered , in which case the pairs have afinite total momentum . We also prove that staggered SCmay coexist with charge density wave (CDW) states ex-plaining features of the pressure phase diagram in UGe and opening a new perspective for the discussion of otherunconventional SC.The unexpected discovery of SC well inside the itiner-ant FM state of UGe under pressure [1] and subsequentlyin ZrZn [2] and URhGe [3] represent a fascinating chal-lenge for our understanding of SC [4–12]. The kind ofunconventional SC involved as well as the complexity ofthe pressure phase diagrams in UGe remain a puzzle. Infact, below T C (0) = 52 K an almost fully polarized FMstate is observed. Applying hydrostatic pressure, FM issuddenly eliminated at p c = 1.5 GPa. Around p c = 1.1 GPa, SC appears and at the optimum p ∗ = 1.2 GPa onefinds T c ( p ∗ ) ≤ C (p ∗ ) ≃
30 K, i.e. 60% of the original T C (0).In addition to SC, presumably another phase is presentinside the FM state below T ∗ (0) = 30 K, and T ∗ ( p ) alsodecreases with pressure until at p ∗ it hits the optimumT c (p) of the SC dome [8, 13]. The nature of the T ∗ phaseis not clear, one possibility that we adopt here is to as-sociate it with a charge density wave (CDW) [1, 8, 14, 15],a scenario supported also by LDA+U calculations [16].This would explain the associated experimentally observedheat-capacity anomalies [17] and jumps in the magneti-zation on crossing the T ∗ phase boundary [13]. Finally,recent NQR experiments suggest that the T ∗ phase sur-vives even below the SC transition dividing the SC domeinto two parts: a low pressure part where SC and the T ∗ order coexist and a high pressure part where there is nosignature of the T ∗ order [18, 19].We explore in this Letter the competition of staggeredSC (i.e. zone boundary SC) with CDW in a strong fer-romagnetic background, a situation that has never beenconsidered before. In fact, all previous theoretical inves-tigations of SC in UGe considered zero momentum (orzone center) SC. Note that the possibility of the relevanceof staggered SC states for some heavy fermion compoundshas first been considered in the past by D.L. Cox andp-1. Georgiou 1 G. Varelogiannis 1 P. Thalmeier 2coworkers [20] in the context of a Ginzburg-Landau ap-proach. Moreover, the SC pairing of spinless (or single-spin) fermions has been introduced in [21] and discussedwithin a Ginzburg-Landau theory. Here we start from amean field BCS-type Hamiltonian H = X k ξ k c † k c k − X k (cid:0) W k c † k c k + Q + h.c. (cid:1) − X k (cid:0) ∆ c † k c †− k + h.c. (cid:1) − X k (cid:0) ∆ Qk c † k c †− k − Q + h.c. (cid:1) (1)The first term describes a 2D tight binding FS whose nest-ing properties with a wave vector ˇQ=( π, π ) are controlledby the ratio of n.n. (t ) and n.n.n. (t ) hopping matrix el-ements. For t /t < SCk,k ′ and V CDWk,k ′ of the itinerant5f-quasiparticles have a purely electronic origin. We mayhave both unconventional SC with zero total pair momen-tum ∆ and at finite pair momentum ∆ Qk . The CDW gapfunction is denoted by W k and like ∆ or ∆ Qk belongs toan irreducible representation of the tetragonal D h group(this is also the approximate symmetry of UGe ).To treat both SC and CDW order parameters in a com-pact manner we introduce a Nambu-type representationusing the spinors Ψ † k = (cid:0) c † k , c − k , c † k + Q , c − k − Q (cid:1) . Accord-ingly we use the tensor products b ρ = (cid:0)b σ ⊗ b I ) and b σ = (cid:0)b I ⊗ b σ ) for the Nambu representation of the Hamilto-nian in Eq. (1). We assume that nesting in the fullyFM polarized band is responsible for the CDW transitionassociated with the T ∗ line in UGe . Pressure reduces T ∗ because it relaxes the nesting conditions. To modelthis effect we write the electron dispersion as a sum ofparticle-hole symmetric terms responsible for nesting andparticle-hole asymmetric terms that represent the devia-tions from nesting: ξ k = γ k + δ k where 2 γ k = ξ k − ξ k + Q and 2 δ k = ξ k + ξ k + Q . When δ k = 0 there is particle-holesymmetry or perfect nesting with wavevector Q . Applica-tion of pressure adds a δ k term in the dispersion in addi-tion to the γ k term already present at zero pressure. Weclassify the SC and CDW order parameters with respectto their behavior under inversion (I) k → − k , translation(t Q ) k → k + Q and time reversal (T) in the charge sector.Instead of the latter we may also use complex conjugation(C) which satisfies the equivalence relations C ≡ -T (∆ k );C ≡ IT (∆ k ) or C ≡ t Q (W k ). These discrete transfor-mations may then be used to classify the possible groupsof competing SC/CDW order parameters. Obviously C isredundant for the three order parameters considered, butwe include it in the notation for clarity.Because the spins are frozen, the q = SC pair statesmay only have odd parity with ∆ − k = − ∆ . Under translation we have both signs ∆ + Q = ± ∆ and un-der C we get (∆ ) ∗ = - (∆ ) T = - ∆ . SC pair stateswith finite momentum may in principle have both par-ities: ∆ Q − k = ± ∆ Qk because the required antisymmetrymay also come from the shift by a lattice vector R withexp( i QR ) = − . On the other hand the t Q transla-tion requires that always ∆ Qk + Q = − ∆ Qk and under C wehave (∆ Qk ) ∗ = (∆ Q − k ) T = - ∆ Q − k . These transformationproperties allow four possible SC order parameters, twoat zone center and two at zone boundary or staggered SC :∆ I −− k , ∆ I − + k , ∆ Q R −− k , ∆ Q I + − k . Here the first in-dex or Q indicates the total momentum of the pair , thesecond index R or I indicates whether the order parame-ter is real or imaginary, the third index ± indicates parityunder inversion I and the last index denotes gap symmetryunder t Q . As mentioned the index I(R) is redundant.For the CDW order parameter both odd and even statesunder I and t Q are allowed so that W − k = ± W k and W k + Q = ± W k may hold. Since C ≡ t Q for this or-der parameter the redundant index R or I is associatedwith the t Q -index ± respectively. As a result we havehere again four different possible (CDW) order param-eters: W R ++ k , W I + − k , W R − + k , W I −− k , where the in-dices have the same meaning as the last three indicesin the SC order parameters. According to the abovesymmetry classification there are sixteen possible pairsof such competing SC/CDW states and only eight ofthem concern staggered SC on which we are interestedhere. Within our formalism we can calculate Green’s func-tions and self-consistent gap equations for each of theseeight cases. As an example relevant for UGe we re-port here for the case of the competition of W I + − k with∆ Q R −− k where the Hamiltonian in spinor representationwith Pauli matrices b σ i and b ρ i is H = P k Ψ † k b Ξ k Ψ † k where b Ξ k = γ k b ρ b σ + δ k b σ + ∆ Q R −− k b ρ b σ − W I + − k b ρ . We notethat if instead of having the competition of W I + − k b ρ with∆ Q R −− k b ρ b σ as above, we had any of the other pairs ofcompeting order parameters, we would just have to replacethe corresponding SC and CDW terms in the above Hamil-tonian. The Green’s functions that result would be modi-fied accordingly. For the W I + − k b ρ and ∆ Q R −− k b ρ b σ orderparameters, the most obvious realization in D h symme-try is a d-wave CDW and p-wave (finite momentum) SCorder parameter given by ∆ Qk = ∆ Q (sin k x + sin k y ) (i.e. E u (1 , W k = W (cos k x − cos k y ) (i.e. B g ). Fromthe Hamiltonians we obtain Green’s functions and thenself-consistent gap equations for both order parameterswhich after analytic summation over the Matsubara fre-quencies take the form of the following system of coupledequations that are reported for the first time here:∆ k = X k ′ V SC k , k ′ ∆ k ′ p δ k ′ + ∆ k ′ h tanh E + ( k ′ )2 T − tanh E − ( k ′ )2 T i (2) W k = X k ′ V CDW k , k ′ W k ′ p γ k ′ + W k ′ h tanh E + ( k ′ )2 T + tanh E − ( k ′ )2 T i (3)p-2oexistence of CDW with staggered superconductivity in a ferromagnetic material E ± ( k ) = q γ k + W k ± q δ k + ∆ k (4)Here the effective potentials V SCk,k ′ , V CDWk,k ′ are separable forthe asummed E u and B g channels. If solutions of thesecoupled SC/CDW gap equations exist they are unique.For uniform order parameters assumed here they also havelower free energy as compared to the normal state [22,23].Eqs. (2,3) account for the following four pairs of compet-ing CDW and zone boundary states: ∆ RQ − with W R ++ ,∆ RQ − with W I + − , ∆ IQ − with W R ++ and ∆ IQ − with W I + − . There is a second system of coupled gap equa-tions that describes the competition of the remaining four pairs of SC and CDW gaps: ∆ RQ − with W R − + , ∆ RQ − with W I −− , ∆ IQ − with W R − + and ∆ IQ − with W I −− :∆ k = X k ′ V SC kk ′ ∆ k ′ (cid:26) B ( k ) + γ k ′ E + ( k ′ ) B ( k ) tanh (cid:20) E + ( k ′ )2 T (cid:21) + B ( k ) − γ k ′ E − ( k ′ ) B ( k ) tanh (cid:20) E − ( k ′ )2 T (cid:21)(cid:27) (5) W k = X k ′ V CDW kk ′ W k ′ (cid:26) B ( k ) + δ k ′ E + ( k ′ ) B ( k ) tanh (cid:20) E + ( k ′ )2 T (cid:21) + B ( k ) − δ k ′ E − ( k ′ ) B ( k ) tanh (cid:20) E − ( k ′ )2 T (cid:21)(cid:27) (6)where E ± ( k ) = vuut ∆ k W k γ k + W k + (cid:20)q γ k + W k ± s B ( k ) γ k + W k (cid:21) (7) B ( k ) = q δ k ′ (cid:0) γ k ′ + W k ′ (cid:1) + γ k ′ ∆ k ′ (8)We have solved selfconsistently the systems of Eqs. (2,3)and (5, 6) for a 2D tight-binding model on a square lattice.In that case, the particle-hole symmetric term correspondsto nearest neighbor hoping γ k = t (cos k x + cos k y ) whileparticle-hole asymmetry is introduced by the next-nearestneighbor hopping terms δ k = t cos k x cos k y . We haveperformed a large number of self consistent calculationsvarying the pairing potentials in the two channels pro-ducing eight maps (two of them reported in figure 1) of all possible transitions induced by particle-hole asymmetry(i.e. by pressure) in the low- T region for all the pairs ofcompeting CDW and staggered SC order parameters thatare possible. To take into consideration the fact that thedifferent CDW and SC gap symmetries involved may cor-respond to different momentum structures for the orderparameters, we have considered the separable potentialsapproximation that allows to search for solutions of a spe-cific momentum structure. Therefore, the axes in Fig.1 arethe amplitudes V CDW and V SC of the pairing interactionsand it is understood that a corresponding form factor hasbeen considered. We have investigated the coexistence oforder parameters in the whole (V CDW , V SC ) plane fromthe moderate coupling to the strong coupling regime sincewe have no microscopic derivation for the effective pairing V CD W V SC CDW CDW SC CD W + S C S C F M CDW+SC SC FMCDW CDW+SCCDWFM F M SC CD W + S C CD W F M CD W + S C CD W V CD W V SC FM CD W SC CDW+SC SC FM
Fig. 1: Maps of the dependence of phase sequences on the ef-fective interactions V CDW and V SC for low temperature. Ar-rows indicate the cascade of phases obtained when t /t growsstarting from zero. The black dots separate regions of differ-ent phase sequences under growing t /t . All phases coexistwith ferromagnetism (FM). The phases indicated as FM, arephases in which there is not any finite SC or CDW order pa-rameter and so only FM is present. Figure a) corresponds tothe competition of ∆ RQ −− with W I −− . Figure b) correspondsto the competition of ∆ RQ −− with W R − + . The potentials arein units of t . strengths. Arrows in Fig. 1 indicate the cascade of phasesobserved when the ratio t /t grows starting from zero.Since we consider a spin polarized background, all statesreported also coexist with FM, and the transitions to theFM state reported at high values of t /t has the meaningof a transition to a state that is only ferromagnetic withno CDW or SC order parameter present. We note in figure1a that in a large portion of the V SC , V CDW parameterspace there is at low temperature a transition from a CDWstate to a state in which CDW and SC coexist. The sametransition is also present over a portion of the parameterspace in the case of figure 1b. We note that the coexis-tence of zone-center SC with CDW has been reported inprevious theoretical studies [22, 27, 28].We now look more closely to the situations in whichCDW and staggered SC may coexist. We report in figure2a the behavior of the CDW+SC state with t /t in acharacteristic example that corresponds to V CDW = 4and V SC = 2 . RQ −− with W R − + at low-T the mapping of which is reported in figure 1b. Weobserve the cascade of transitions from CDW to CDW+SCp-3. Georgiou 1 and G. Varelogiannis 1 and P. Thalmeier 2 / t = 0.9 SC CDW t / t Fig. 2: (Color online): A characteristic example in which devi-ations from nesting induced by t /t lead to coexistence of the W R − + CDW (open circles in red) with the ∆ RQ −− staggeredSC order parameter (filed circles in black) corresponding to V CDW = 4 and V SC = 2 .
5. We observe the cascade of transi-tions from CDW to CDW+SC and then to FM as was alreadyreported in figure 1b in this region of of V CDW and V SC val-ues. In the inset is shown the temperature dependence of bothorder parameters when t /t = 0 . t . at t /t ≈ .
72 and finally to the FM state for t /t > . t /t in the low-T regime whereas the transition fromthe CDW+SC state to the FM state (i.e. the state with noSC or CDW) is first order in t /t . In figure 2b we showthe corresponding transitions with temperature when wetake t /t = 0 .
9. In this case, according to figure 2a,we have indeed in low-T coexistence of CDW and SC.We observe the counter-intuitive behavior that when SCappears as we lower the temperature, the CDW gap growsinstead of being reduced.A particularly interesting cascade of transitions in rela-tion to the observations in UGe , is the one from CDWto CDW+SC to SC and finally to FM. This cascade isobserved over a large portion of the parameter space ofpairing potentials when the ∆ RQ −− and W I −− compete(cf. fig. 1a) and over a smaller portion in the case ofcompetition of ∆ RQ −− with W R − + (fig. 1b). A cascadeof transitions in the low temperature regime that exhibitsamazing similarities with that observed in UGe as a func-tion of pressure is shown in Fig.3. It corresponds to thecompetition of W R − + CDW with the staggered SC orderof the form ∆ RQ −− when V SC = 3 . V CDW = 12 . t /t . Theabove results are in surprising qualitative agreement with t / t T e m pe r a t u r e CDW + FM FMSC + FMCDW + SC + FM
Fig. 3: Evolution with temperature of a cascade of t /t in-duced transitions from CDW to CDW+SC to SC and then toonly FM obtained when V CDW = 12 . V SC = 3 .
5. TheSC and CDW order parameters considered are ∆ RQ −− and W R − + . This phase diagram shows striking similarities withfeatures of the pressure phase diagram of UGe if one supposethat the T x phase corresponds to a W R − + CDW ordering andof course SC to ∆ RQ −− . findings in UGe [18]. In particular, recent NQR resultsindicate that the T ∗ phase coexists indeed with SC over aportion of the SC dome [19] and our results are the firstto provide a theoretical picture for it.Staggered SC states are relevant for magnetic SC [20,24,25] because they are similar to the Fulde-Ferrel states [26]except that the modulation of the superfluid density co-incides with the characteristic wavevector of the CDW. Itappears, therefore, plausible to consider these states in theanalysis of SC in UGe which is observed only in the FMregime because for a staggered SC the FM background isnecessary in the same way as the magnetic field is neces-sary in order to obtain the usual Fulde-Ferrel phase. Sincepressure eliminates the FM state it naturally eliminates si-multaneously the staggered SC state as well. Note finallythat we obtain staggered SC states only over a limiteddome near the collapse of the CDW phase as in figures 2and 3 within a mean field approach without any fluctua-tions involved .In conclusion, we have demonstrated the possibility tohave coexistence of staggered SC with CDW and cascadesof transitions induced by particle-hole asymmetry that re-produce the pressure phase diagram observed in UGe identifying the T ∗ phase as a CDW phase. Such exoticstates may be relevant for other magnetic SC as well. ∗ ∗ ∗ We thank Jacques Flouquet and Modu Saxena for pro-viding experimental references. G.V. acknowledges visitorgrants and hospitality from the MPI CPfS at Dresden.
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