Isotropic Cooper Pairs with Emergent Sign Changes in Single-Layer High-T Superconductor c J.P. Rodriguez1 6 1Department of Physics and Astronomy, 1 0 2 California State University, Los Angeles, California 90032 l u Abstract J 2 Wemodelasinglelayer ofheavily electron-dopedFeSebyspin-1/2momentsover asquarelattice 2 of iron atoms that include the 3d and 3d orbitals, at strong on-site Coulomb repulsion. Above ] xz yz n o half filling, we find emergent hole bands below the Fermi level at the Brillouin zone center in a half c - metal state characterized by hidden magnetic order and by electron-type Fermi surface pockets at r p u wavenumbersthatdoubletheunitcell alongtheprincipalaxes. Exactcalculations withtwo mobile s . t electrons find evidence for isotropic Cooper pairs that alternate in sign between the electron bands a m and the emergent hole bands. - d n o c [ 2 v 0 6 8 1 0 . 1 0 6 1 : v i X r a 1 Introduction. The discovery of superconductivity in iron-pnictide materials has uncov- ered a new path in the search for high-temperature superconductors[1]. Superconductivity has been observed recently in a single layer of FeSe on a doped SrTiO (STO) substrate[2–4] 3 below critical temperatures as high as 100 K [5]. Electronic conduction originates from the 3d orbitals of the iron atoms, which form a square lattice. Angle-resolved photo-emission spectroscopy(ARPES), in particular, reveals circular electron-type Fermi surface pockets centered at wave numbers (π/a)xˆ and (π/a)yˆ that lie along the principal axes of the iron lattice, where a is the lattice constant[6, 7]. Unlike the case of most iron-pnictide mate- rials, however, ARPES also finds that hole bands centered at zero two-dimensional (2D) momentum lie well below the Fermi level in the case of single-layer FeSe/STO. At low tem- perature, it also finds an isotropic gap at the electron Fermi surface pockets[8], which is confirmed by scanning tunneling microscopy (STM)[9]. The same set of phenomena have been recently observed below critical temperatures in the range 40-50 K at the surfaces of intercalated FeSe[10–12], of alkali-metal dosed FeSe[13–16], and of voltage-gate tuned thin films of FeSe[17, 18]. Comparison with bulk FeSe, which has a much lower critical tempera- ture of 8 K, strongly suggests that the high-temperature superconductivity exhibited above is due to a new 2D groundstate that appears after heavy electron doping. Calculations based on the independent-electron approximation fail to describe the Fermi surfaces in single-layer FeSe/STO[19]. In particular, density-functional theory (DFT) typi- callypredicts thattheholebandscentered atzero 2Dmomentumcross theFermi level[8, 10]. DFTalsofailstoaccountforanearbyMottinsulatorphaseatlowelectrondopinginvoltage- gate tuned thin films of FeSe and in single-layer FeSe/STO[18, 20]. The previous suggests that the limit of strong electron-electron interactions[21, 22] is a better starting point to describe superconductivity in heavily electron-doped FeSe. Below, we propose that the hole bands observed by ARPES below the Fermi level at the Brillouin zone center in a surface layer of FeSe are examples of emergent phenomena. The latter is revealed by both mean-field and exact calculations of the one-electron spectrum in a two-orbital t-J model that includes only degenerate d and d electron bands centered at xz yz wavenumbers (π/a)yˆ and (π/a)xˆ, respectively. Local spin-1/2 moments live on d or- (x±iy)z bitals, on the other hand, which yields isotropic magnetism. Emergent hole bands approach the Fermi level at zero 2D momentum as Hund coupling increases inside of a half metal phase that is characterized by hidden N´eel order per d orbital and by electron-type (x±iy)z 2 || ⊥ || || ⊥ || (A) TRUE SPINWAVE: J > 0, J = 0, J = 0.3 J = J , J = J + 0.1 J 1 1 2 1 2 0 0c 1 2 100 1.5 ||J)1 ( Y G 1 10 R 3d (x+iy)z E N E 0.5 3d (x−iy)z ∆ 1 cSDW 0 (0,0) (1,0) (2,0) (2,1) (2,2) (1,1) (0,0) MOMENTUM (π/2a) ± (B) ONE ELECTRON: t|| = 0, t⊥(x) = +5 J||, t⊥(y) = −5 J|| 1 1 1 1 1 -17 coherent d xz incoherent d xz -18 coherent d ||(J)1 incoherent dyyzz (±) (+) − Y -19 FERMI LEVEL G R -20 E N E -21 -22 (1,1) (0,0) (1,0) (2,0) (2,1) (2,2) (1,1) MOMENTUM (π/2a) FIG. 1: (a) The imaginary part of the transverse spin susceptibility, Eq. 3, in the true spin channel and (b) the imaginary part of the one-electron propagator near half filling, Eq. 6, at site-orbital concentration x = 0.01. Not shown in (b) is intrinsic broadening due to the incoherent contributions in Eq. 5. (See Eq. 7 and ref. [34].) Fermi surface pockets inherited from the above band structure (Fig. 1). Emergent hole bands at wavenumber (π/a)(xˆ + yˆ) are also predicted, but they lie below the former ones in energy. Last, exact calculations of two mobile electrons in the two-orbital t-J model find evidence for isotropic Cooper pairs on both the electron pockets and on the emergent hole bands below the Fermi level as Hund coupling approaches a quantum critical point (QCP) at which commensurate spin-density wave (cSDW) nesting begins. The sign of the Cooper pair wavefunction notably alternates between the electron and hole bands[23, 24]. Local Moment Model. Our starting point is a two-orbital t-J model over the square 3 lattice, where the on-site-orbital energy cost U tends to infinity[25, 26]: 0 H = [ (tα,βc˜† c˜ +h.c.)+Jα,βS S ]+ Jα,βS S hi,ji − 1 i,α,s j,β,s 1 i,α · j,β hhi,jii 2 i,α · j,β P + (J S S +U′n¯ n¯ P . (1) i 0 i,d− · i,d+ 0 i,d+ i,d− Above, S is the spin operatoPr that acts on spin s = 1/2 st(cid:1)ates of d = d and i,α 0 (x−iy)z − d+ = d orbitals α in iron atoms at sites i. Repeated orbital and spin indices in the (x+iy)z hopping and Heisenberg exchange terms above are summed over. Nearest neighbor and next-nearest neighbor Heisenberg exchange across the links i,j and i,j is controlled h i hh ii by exchange coupling constants Jα,β and Jα,β, respectively. Hopping of an electron in 1 2 orbital α to a nearest-neighbor orbital β is controlled by the matrix element tα,β. We adopt 1 the Schwinger-boson (b) slave-fermion (f) representation for the creation operator of the correlated electron at or above half filling[27–29]: c˜† = f† b with the constraint i,α,s i,α i,α,s 2s = b† b +b† b +f† f (2) 0 i,α,↑ i,α,↑ i,α,↓ i,α,↓ i,α i,α enforced at each site-orbital to impose the U limit on electrons with spin s = 1/2. 0 0 → ∞ Finally, J is a ferromagnetic exchange coupling constant that imposes Hund’s Rule, while 0 the last term in (1) represents the energy cost of a fully occupied iron atom. Here n¯ = i,α c˜† c˜ 1 counts singlet pairs at site-orbitals. Last, notice that d e±iθd is s i,α,s i,α,s − ± → ± Pequivalent to a rotation of the orbitals by an angle θ about the z axis. Spin and occupation operators remain invariant under it. Magnetism described by the two-orbital t-J model (1) is hence isotropic. Semi-classical calculations of the Heisenberg model that corresponds to (1) at half filling find a QCP that separates a cSDW at strong Hund coupling from a hidden antiferromagnet at weak Hund coupling when diagonal frustration is present[30]: e.g. Jk > 0, J⊥ = 0, and 1 1 Jk = J⊥ > 0. Here, and represent intra-orbital (d d ) and inter-orbital (d d ) 2 2 k ⊥ ± ± ± ∓ superscripts. The hidden-order magnet shows N´eel spin order per d orbital following the ± k inset to Fig. 1a. Ideal hopping of electrons within an antiferromagnetic sublattice, t = 0 1 andt⊥(xˆ) = t⊥(yˆ) > 0, leavessuchhiddenmagneticorderintactinthesemi-classical limit, 1 − 1 s . Below, we employ a mean-field approximation of (1) and (2) to study this state 0 → ∞ near the QCP. It reveals a half metal with circular Fermi surface pockets at wavenumbers (π/a)xˆ and (π/a)yˆ, for electrons in the d orbital and d orbitals, respectively. yz xz Spin-Fluctuations, One-Electron Spectrum. Following Arovas and Auerbach[27], we first rotate the spins quantized along the z axis on one of the antiferromagnetic sublattices shown 4 in the inset to Fig. 1a by an angle π about the y axis. This decouples the up and down spins between the two sublattices[31]. We next define mean fields that are set by the pat- tern of antiferromagnetic versus ferromagnetic pairs of neighboring spins[27] in the hidden magnetic order: Q = b b , Qk = b b and Q⊥ = b b on the anti- 0 h i,d±,s i,d∓,si 1 h i,d±,s j,d±,si 2 h i,d±,s j,d∓,si ferromagnetic links versus Q⊥ = b† b and Qk = b† b on the ferromagnetic 1 h i,d±,s j,d∓,si 2 h i,d±,s j,d±,si links of the hidden N´eel state. Subscripts 0, 1 and 2 represent on-site, nearest neighbor and next-nearest neighbor links. We add to that list the mean field P⊥ = 1 f† f for 1 2h i,d± j,d∓i nearest-neighbor hopping of electrons across the two orbitals. It has d-wave symmetry. The corresponding mean-field approximation for the t-J model Hamiltonian (1) then has the form H +H , where b f 1 H = Ω (k)[b†(k)b (k)+b ( k)b†( k)]+Ω (k)[b†(k)b†( k)+b ( k)b (k)] b 2 { fm s s s − s − afm s s − s − s } k s XX is the Hamiltonian for free Schwinger bosons, and where H = ε (k)f†(k)f(k) is the f k f Hamiltonian for free slave fermions. Here, k = (k0,k) is the 3-moPmentum for these excita- tions, where the quantum numbers k = 0 and π represent even and odd superpositions of 0 the d and d+ orbitals: d and ( i)d . xz yz − − Enforcing the infinite-U constraint (2) on average over the bulk then results in ideal 0 Bose-Einstein condensation (BEC) of the Schwinger bosons into degenerate groundstates 1/2 at k = 0 and (π,π/a,π/a) in the zero-temperature limit: b = s at large s . (See h i,d±,si 0 0 Fig. 1a and Supplemental Material, Fig. S1.) In such case, all five mean fields among the Schwinger bosons therefore take on the unique value Q = s . This results in diagonal and 0 off-diagonal Hamiltonian matrix elements Ω (k) = (1 x)2s (J +4Jk +4J⊥ fm − 0 0 1 2 4J′′⊥[1 eik0γ (k)] 4Jk[1 γ (k)]) − 1 − 1+ − 2 − 2 Ω (k) = (1 x)2s [J eik0 +4Jkγ (k)+4J⊥eik0γ (k)], afm − 0 0 1 1+ 2 2 for free Schwinger bosons, and the energy eigenvalues ε (k) = 8s t⊥(xˆ)eik0γ (k) for free f − 0 1 1− slave fermions. Above, J′′⊥ = J⊥ 2t⊥(xˆ)P⊥(xˆ)/(1 x)2s , while γ (k) = 1(cosk a 1 1 − 1 1 − 0 1± 2 x ± cosk a) and γ (k) = 1(cosk a + cosk a), with k = k k . Slave fermions in d and y 2 2 + − ± x ± y xz d orbitals lie within circular Fermi surfaces centered at wavenumbers (π/a)yˆ and (π/a)xˆ, yz respectively, with Fermi wave vector k a = (4πx)1/2 at low electron doping per iron orbital, F 5 x 1. (See the inset to Fig. 1b.) The mean inter-orbital electron hopping amplitude is ≪ then approximately P⊥(xˆ) = x/2. 1 The dynamical spin correlation function S S′ is obtained directly from the above h y yi Schwinger-boson-slave-fermion mean field theory. It is given by an Auerbach-Arovas expres- sion at non-zero temperature that is easily evaluated in the zero-temperature limit [26, 32], where ideal BEC of the Schwinger bosons into the degenerate groundstates at 3-momenta k = 0 and (π,π/a,π/a) occurs. It is one half the transverse spin correlator, which under ideal BEC and at large s reads 0 i S(+)S′(−) = (1 x)2s (Ω /Ω )1/2([ω (k) ω]−1 +[ω (k)+ω]−1). (3) k,ω 0 + − b b h i| − − Here, ω = (Ω2 Ω2 )1/2 is the energy dispersion of the Schwinger bosons, and Ω = b fm − afm ± Ω Ω . Figure 1a depicts the imaginary part of the transverse susceptibility (3) in fm afm ± the true spin channel, k = 0, at sub-critical Hund coupling. It reveals a spin gap at cSDW 0 wave numbers (π/a)xˆ and (π/a)yˆ of the form ∆ = (1 x)2(2s )(4J⊥ J )1/2Re(J J )1/2. (4) cSDW − 0 2 − 0c 0 − 0c Here, J = 2(Jk J⊥) 4Jk+(1 x)−2s−12t⊥(xˆ)x is the critical Hund coupling at which − 0c 1 − 1 − 2 − 0 1 ∆ 0. Notice that inter-orbital hopping stabilizes the hidden half metal state. The cSDW → autocorrelator of the hidden spin S S , (3) at k = π, also shows the above spin gap i,d− i,d+ 0 − (4) at cSDW momenta, in addition to a hidden-order Goldstone mode at N´eel wavenumber (π/a)(xˆ+yˆ)[31]. The electronic structure of the hidden half metal state can also be obtained directly from the above Schwinger-boson-slave-fermion mean field theory. In particular, the one-electron propagator is given by the convolution of the conjugate propagator for Schwinger bosons with the propagator for slave fermions in 3-momentum and in frequency. A summation of Matsubara frequencies yields the expression[31] 1 G(k,ω) = 1Ωfm +1 nB[ωb(q−k)]+nF[εf(q)−µ] 2 ωb q−k 2 ω+ωb(q−k)−εf(q)+µ N Xq h(cid:16) (cid:12) (cid:17) (cid:12) + 1Ωfm (cid:12) 1 nB[ωb(q−k)]+nF[µ−εf(q)] . (5) 2 ωb q−k−2 ω−ωb(q−k)−εf(q)+µ (cid:16) (cid:12) (cid:17) i (cid:12) Above, n andn denotetheBose-EinsteinandtheFermi-Diracdistributions, andµdenotes B F (cid:12) the chemical potential of the slave fermions. Ideal BEC of the Schwinger bosons at 3- momenta q k = 0 and (π,π/a,π/a) results in the following coherent contribution to the − 6 electronicspectral functionatzerotemperatureandatlarges : ImG (k,ω) = s πδ[ω+µ 0 coh 0 − ε (k)]. It reveals degenerate electron bands for d and d orbitals centered at cSDW wave f xz yz numbers Q = (π/a)yˆ and Q = (π/a)xˆ , respectively. The electron Fermi surface pockets 0 π at ω = 0 are depicted by the inset to Fig. 1b. At energies below the Fermi level, ω < 0, the remaining contribution is exclusively due to the first fermion term in (5). Inspection of Fig. 1b (solid lines) yields the following expression for it in the limit near half-filling, k a 0, F → at large t/J [33]: π 1 1Ω ImG (k,ω) = x + fm δ[ω +ǫ +ω (q k ,Q k)]. (6) inc ∼ q0X=0,π 2 "2 2 ωb (cid:12)(q0−k0,Qq0−k)# F b 0 − 0 q0 − (cid:12) Figure 1b displays the emergent hole(cid:12) bands predicted above. They lie ǫ + ∆ F cSDW below the Fermi level, with degenerate maxima at k = 0 and (π/a)(xˆ + yˆ). Here, ǫ = (2s )t⊥(xˆ)(k a)2 is the Fermi energy. The emergent hole bands also show intrin- F 0 1 F sic broadening in frequency at zero temperature, which makes them incoherent. Outside the critical region, at large t/J, the broadening is ∆ω k ∇ω . (7) F b Q−k ∼ | | It remains small at the previous maxima[34]. Last, the emergent hole bands predicted by (6) are anisotropic: e.g., the d hole band at zero 2D momentum has mass anisotropy yz m < m . (Cf. ref. [35].) x y | | | | k Adding intra-orbital electron hopping, t > 0, brings the emergent hole bands at 1 wavenumber (π/a)(xˆ+yˆ) down in energy below the ones at zero 2D momentum[35]. This is confirmed by exact calculations of the two-orbital t-J model with one electron more than half filling over a 4 4 lattice of iron atoms under periodic boundary conditions. The pre- × vious Schwinger-boson-slave-fermion description (2) for spin s = 1/2 electrons is exploited 0 to impose strong on-site-orbital Coulomb repulsion. Details are given in ref. [26]. Figure 2a shows the exact spectrum at the QCP, where ∆ 0. The t-J model parameters cSDW → k k coincide with those set by Fig. 1, but with t = 2J , and with Hund coupling tuned to 1 1 k the critical value J = 1.733J . Red states have even parity under orbital swap, P , − 0 1 d,d¯ while blue states have odd parity under it. Notice that the lowest-energy doubly-degenerate states at wave number (π/a)(xˆ + yˆ), which are spin-1/2, lie 0.5Jk in energy above the 1 doubly-degenerate spin-1/2 groundstates at zero 2D momentum. The latter states (purple) move up in energy off the Fermi level set by the groundstates at cSDW momenta as Hund 7 (A) MOBILE ELECTRON AT QCP (B) COOPER PAIR AT PUTATIVE QCP -56 -39 -56.2 -39.2 -56.4 -39.4 P P x y ) -56.6 -39.6 ||J1 ( ∆E vs. −J (J||) Y 0 0 1 RG -56.8 0.6 -39.8 Dx2− y2 E -E vs. −J (J||) E 0.4 D S 0 1 N 0.2 E -57 0 -40 S 2.5 -0.2 2 1.5 1.4 1.6 1.8 1 -57.2 -40.2 0.5 0 0.5 1 1.5 2 2.5 S = 1/2, even -57.4 S = 1/2, odd -40.4 S = 0, even S = 3/2, even S = 0, odd S = 3/2, odd S = 1, even -57.6 S = 5/2, even -40.6 S = 1, odd S = 5/2, odd S = 2, even (1,1) (0,0) (1,0) (2,0) (2,1) (2,2) (1,1) (0,0) (1,0) (2,0) (2,1) (2,2) MOMENTUM (π/2a) MOMENTUM (π/2a) FIG. 2: (a) Low-energy spectrum of two-orbital t-J model, Eq. (1) plus constant 3(N 1)J , 4 Fe − 0 over a 4 4 lattice, with one electron more than half filling. Model parameters coincide with those × k k k listed by Fig. 1, except t = 2J and J = 1.733J . (b) Low-energy spectrum of Eq. (1) plus 1 1 − 0 1 repulsive interactions (see text) plus constant 1(N 2)J , but with two electrons more than half 4 Fe− 0 filling, with J = 2.25Jk, and with U′ = 1J +1000Jk. Some points in spectra are artificially − 0 1 0 4 0 1 moved slightly off their quantized values along the momentum axis for the sake of clarity. coupling falls below the critical value, and they become nearly degenerate with the former states in the absence of Hund’s Rule. This dependence on Hund coupling is demonstrated by the inset to Fig. 2a and by supplemental Fig. S3. The exact low-energy spectrum at sub-critical Hund coupling is therefore consistent with the emergent hole bands obtained by the meanfield approximation, Fig. 1b, but with the hole bands centered at wavenum- ber (π/a)(xˆ + yˆ) pulled down to much lower energy. Last, Fig. 2a shows that the even parity (d ) and odd parity (d ) spin-1/2 groundstates at wavenumber (π/2a)xˆ are nearly xz yz degenerate, which suggests isotropic emergent hole bands at zero 2D momentum near the 8 QCP. Cooper Pairs. Figure 2b shows the spectrum of the same two-orbital t-J model (1), but with two electrons more than half filling. A repulsive interaction has been added to the Heisenberg exchange terms in order to reduce finite-size effects: S S S S + i,α j,β i,α j,β · → · 1n n , equal to 1/2 the spin-exchange operator. Here, n counts the net occupation of 4 i,α j,β i,α holes per site-orbital. Also, the on-site repulsion between mobile electrons in the d+ and d − orbitals, respectively, is set to a large value U′ = 1J +1000Jk. The Schwinger-boson-slave- 0 4 0 1 fermion description of the correlation electron (2) is again employed, with s = 1/2. Details 0 are given in ref. [26]. Last, the ferromagnetic Hund’s Rule exchange coupling constant k is tuned to the critical value J = 2.25J , at which ∆ 0. This is depicted by 0 − 1 cSDW → the dashed horizontal line in Fig. 2b, which shows the degeneracy between the cSDW spin resonance at wavenumber (π/a)xˆ with the hidden-order spin resonance at wavenumber (π/a)(xˆ + yˆ). The former is even (black) under swap of the orbitals, d d+, while − ↔ latter is odd (red) under it. Notice that the groundstate and the second excited state both lie under a continuum of states at zero net momentum. They respectively have even and odd parity under a reflection about the x-y diagonal. We therefore assign S symmetry to the groundstate bound pair and Dx2−y2 symmetry to the excited-state bound pair. The dependence of the energy-splitting between these two states on Hund coupling is shown by the inset to Fig. 2b. It provides evidence for a true QCP in the thermodynamic limit at k J = 2.30J , where the s-wave and d-wave bound states become degenerate. − 0 1 Figure 3 depicts the order parameters for superconductivity of the two bound pair states shown in Fig. 2b: iF(k ,k) = Ψ c˜ (k ,k)c˜ (k , k) Ψ (8) 0 Mott ↑ 0 ↓ 0 Cooper h | − | i times √2, with c˜ (k ,k) = −1/2 e−i(k0α+k·ri)c˜ . Here, Ψ denotes the s 0 N i α=0,1 i,α,s h Mott| k critical antiferromagneticstateofthePcoPrresponding Heisenberg model[30]at−J0c = 1.35J1. (See supplemental Fig. S4.) The groundstate has S symmetry, as expected, but it also alternatesinsignbetweenCooperpairsatelectronFermisurfacepocketsversusCooperpairs at the emergent hole bands. (See Fig. 1b.) Figure 3 also shows that the (second) excited state has Dx2−y2 symmetry, as expected, and that it alternates in sign in a similar way. The present exact results therefore provide evidence for remnant pairing on the emergent hole bands that lie below the Fermi level at zero 2D momentum. Discussion and Conclusions. The electronic structure in single-layer FeSe/STO is qual- 9 |ΨCooper〉 = |Dx2− y2〉 at putative QCP 3d yz 3d xz 0.5 0.5 ) a / π 2 ( y k 0 0 0 0.5 0 0.5 k (2π/a) k (2π/a) x x |Ψ 〉 = |S〉 at putative QCP Cooper 3d yz 3d xz 0.5 0.5 ) a / π 2 ( y k 0 0 0 0.5 0 0.5 k (2π/a) k (2π/a) x x FIG. 3: The complex order parameter for superconductivity, Eq. 8, symmetrized with respect to both reflections about the principal axes. itatively described by the combination of Figs. 1b and 2a. For example, a fit of inelastic neutron scattering data in iron-pnictide superconductors to the true linear spinwave spec- trum Fig. 1a, but at the QCP, yields Jk = 110 meV, J⊥ = 0, and Jk = 40 meV = J⊥ for 1 ∼ 1 2 ∼ ∼ 2 the Heisenberg exchange coupling constants[30]. Hopping parameters set in Figs. 1b and 2a 10