{'fraction_non_alphanumeric': 0.04897341941487815, 'fraction_numerical': 0.016576513697435003, 'mean_word_length': 4.27585148818656, 'pattern_counts': {'":': 0, '<': 3, '<?xml version=': 0, '>': 2, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 3, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Collective Mode at Lifshitz Transition in Iron-Pnictide Superconductors

We obtain the exact low-energy spectrum of two mobile holes in a t-J model for an isolated layer in an iron-pnictide superconductor. The minimum d xz and d yz orbitals per iron atom are included, with no hybridization between the two. After tuning the Hund coupling to a putative quantum critical point (QCP) that separates a commensurate spin-density wave from a hidden-order antiferromagnet at half filling, we find an s-wave hole-pair groundstate and a d-wave hole-pair excited state. Near the QCP, both alternate in sign between hole Fermi surface pockets at the Brillouin zone center and emergent electron Fermi surface pockets at momenta that correspond to commensurate spin-density waves (cSDW). The dependence of the energy splitting with increasing Hund coupling yields evidence for a true QCP in the thermodynamic limit near the putative one, at which the s-wave and d-wave Cooper pairs are degenerate. A collective s-to-d-wave oscillation of the macroscopic superconductor that couples to orthorhombic shear strain is also identified. Its resonant frequency is predicted to collapse to zero at the QCP in the limit of low hole concentration. This implies degeneracy of Cooper pairs with s, d and s+id symmetry in the corresponding quantum critical state. We argue that the critical state describes Cooper pairs in hole-doped iron superconductors at the Lifshitz transition, where electron bands first rise above the Fermi level. We thereby predict that the s-to-d-wave collective mode observed by Raman spectroscopy in Ba1-xKxFe2As2 at optimal doping should also be observed at higher doping near the Lifshitz transition.

Introduction. The symmetry of the Cooper pairs of electrons in iron-pnictide hightemperature superconductors remains a subject of controversy. A stack of weakly coupled square lattices of iron atoms is the common structural feature in these materials. Early calculations of the electronic band structure within the density functional approximation predicted nested two-dimensional (2D) Fermi surfaces [1] [2], with hole pockets centered at zero 2D momentum, and with electron pockets centered at 2D momenta (π/a)x and (π/a)ŷ.
Here a is the Fe-Fe separation. The predicted Fermi surfaces where later confirmed experimentally by angle-resolved photo-emission spectroscopy (ARPES) [3] [4]. Early calculations based on the above nested electronic structure also predicted an S +− symmetry for the wavefunction of the Cooper pair that alternates in sign between the hole pockets and the electron pockets by virtue of the exchange of spin fluctuations at the nesting vectors [5] [6].
Nearly degenerate Cooper pairs with d-wave symmetry are also predicted to exist because of such spin-fluctuation exchange [7] [8], however. Although experimental support for S +− pairing indeed exists [9], the question of the pairing symmetry in iron-pnictide superconductors remains open.
The recent discovery of superconductivity in single-layer iron selenide is perhaps the greatest challenge for the proposal of S +− Cooper pair symmetry in iron superconductors [10].
ARPES finds the electron pockets common to iron-pnictide superconductors, but it also reveals that the hole bands at zero 2D momentum lie well below the Fermi level [11] [12].
Electronic band-structure calculations typically find that the hole pockets cross the Fermi level [12] [13], on the other hand. This suggests that strong on-site Coulomb repulsion should be taken into account [14][15] [16].
We reveal the nature of a single Cooper pair in a local-moment t-J model for iron superconductors characterized by large on-site Coulomb repulsion. Mobile holes (electrons) roam over a 4 × 4 periodic lattice of spin-1 iron atoms that contain the minimum 3d xz and 3d yz orbitals. The hole bands centered at zero 2D momentum are fixed by the band structure (t), while the electron bands at nesting wave vectors are emergent [17] because of proximity to a commensurate spin-density wave (cSDW) state that is favored by antiferromagnetic frustration (J) [14] [15]. After tuning the Hund's Rule coupling to a quantum critical point (QCP) where the spinwaves at cSDW momenta become degenerate with a hidden-order spinwave at zero 2D momentum [18], we find that the groundstate is a spin-0 Cooper pair with primarily S +− symmetry. It is well separated from a continuum of states, but close by in energy to a D +− pair state. The D +− pair state alternates in sign between the two electron pockets at cSDW momenta. Specifically, the S +− pair and the D +− pair are bonding and anti-bonding superpositions of orbitally ordered (d xz -d xz ) +− and (d yz -d yz ) +− singlet pairs that alternate in sign between the hole and electron pockets. (See Fig. 1d, red versus blue.) The macroscopic superconductor will exhibit an internal Josephson effect between the two species of orbitally ordered Cooper pairs [19] [20][21] [22]. We thereby predict a neutral spin-0 collective mode in the superconducting state that exhibits orbital-order fluctuations. It couples directly to orthorhombic shear [23] [24]. Last, degenerate spin-1 P x and P y pair states are found at the edge of the continuum of states. A spin-1 Leggett mode that carries 2D momentum exists at much lower energy near the S +− groundstate, however, which suggests that these triplet p-wave pair states will move down in energy below the continuum of states on a larger lattice.
Low-Energy Spectrum of Local-Moment Model. The Hamiltonian for the two-orbital t-J model on a square lattice of iron atoms reads [17] Above,c i,α,s = b i,α,s f † i,α is the destruction operator for an electron of spin s, at site i and in orbital α. Also, S i,α = 1 2 s,s′c † i,α,s σ s,s ′c i,α,s ′ is the spin operator in units ofh and n i,α = sc † i,α,sci,α,s measures the net occupation per site-orbital. Repeated orbital and spin indices in the hopping and Heisenberg exchange terms above are summed over. These terms in the Hamiltonian (1) are in turn summed over nearest neighbor and next-nearest neighbor links, i, j and i, j . Double occupancy at a site-orbital is projected out by enforcing the  (1) in the mean-field approximation (refs. [16] and [18]). Heisenberg exchange coupling constants are set to while Hund's Rule coupling −J 0 is critical. Here, the supercripts and ⊥ denote the orbital superscripts d ± d± and d ± d∓, respectively. Shown also (c)-(d) is emergent nesting for the two-orbital t-J model, Eq. (1), with hole bands at zero 2D momentum that are set by hopping matrix elements t (ref. [17]).
an angle δ 0 . The t-J model (1) is therefore invariant under such an arbitrary rotation of the orbitals, which is physically appealing.
The exact low-energy spectrum of a pair of holes that roam over a periodic 4 × 4 square lattice of iron atoms governed by the two-orbital t-J model (1) can be obtained by computer.
First, we replace the Heisenberg exchange terms with the principal spin-exchange part of the interaction, thereby ignoring the short-range hole-hole interaction part: S i,α ·S j,β → 1 2 P i,α;j,β . Also, the total spin along the z axis is constrained to S z = 0. Quantum states are defined by a given spin background over the entire lattice combined with a pair of spin-up and spin-down site-orbitals designated as holes. Fermi statistics of the two slave fermions is accounted for by imposing particle-exchange antisymmetry on the two holes embedded in the spin background. Translation symmetry is also included, and reflection and orbital swap symmetry (d+ ↔ d−) is included for momenta and for hopping parameters that allow it. In the absence of hybridization among the 3d xz and 3d yz holes, for example, including an oddparity reflection about the x axis plus even-parity orbital swap reduces the dimension of the Hilbert space with net momentum (π/2a)x to 601 878 172 states. The ARPACK subroutine library is exploited to obtain low-energy eigenstates via the Lanczos technique [27]. Also, matrix-vector products are accelerated throughout by running parallel OpenMP threads. Figure 2 displays the low-energy spectrum of two holes roaming over a 4 × 4 lattice of spin-1 iron atoms under periodic boundary conditions. The t-J model parameters are set to produce 3d xz and 3d yz hole bands at zero 2D momentum for non-interacting electrons [28] and cSDW spin order when Hund's Rule is obeyed: Here, the notation for the orbital superscripts has been changed: d ± d± → and d ± d∓ → ⊥. Also, the on-site hole-hole repulsion between the d+ and d− orbitals is set to U ′ 0 + (−J 0 /4) = 1000 J ⊥ 1 , while the ferromagnetic Hund's Rule exchange coupling constant is tuned to a critical value of −J 0c = 2.25 J ⊥ 1 . Spinwaves at cSDW momenta become degenerate with a hidden-order spinwave at zero 2D momentum that has odd parity under orbital swap, d+ ↔ d−, at the QCP. (See the horizontal dashed line in Fig. 2.) The hidden-order spinwave signals longrange antiferromagnetic correlations across the d+ and d− orbitals. This becomes possible at weak intra-orbital Heisenberg exchange [15][18], J 1 < J ⊥ 1 , which has been argued to be the case in iron-pnictide superconductors [17]. The same QCP occurs in the corresponding Heisenberg model, with no mobile holes, at a lower critical Hund's Rule coupling of [18] −J 0c = 1.35 J ⊥ 1 . Figures 1a and 1b show the critical spin-wave spectrum of that Heisenberg model, arrived at from a Schwinger-boson mean-field theory analysis instead [16]. Extending the mean-field theory analysis to the two-orbital t-J model with mobile holes (slave fermions) at the QCP yields emergent nesting of hole Fermi surface pockets at zero 2D momentum with electronic structure at cSDW momenta [16]. It is shown by Fig. 1c. Exact calculations for one mobile hole roaming over a 4 × 4 square lattice of spin-1 iron atoms, with the same t-J model parameters studied here, obtain emergent cSDW nesting that is orbitally specific, however [17]. Figure 1d displays it schematically. Last, electron-type dispersion for the emergent bands at cSDW momenta has been argued for on the basis of the critical spin-fluctuation spectrum [17].  3 depicts the two pair wavefunctions: Here, N is two times the number of iron atoms, while the d− and d+ orbitals are enumerated by 0 and 1. Hence, k 0 = 0 and π correspond to the d xz and i×d yz orbitals. Also, Ψ Mott | denotes the critical antiferromagnetic state of the corresponding Heisenberg model [18] at −J 0c = 1.35 J ⊥ 1 . Notice specifically that the groundstate is primarily S +− , with some admixture of S ++ : (cos θ 0 )|S +− + (sin θ 0 )|S ++ , Figure 3 also shows that the second excited state is D +− , which alternates in sign between the emergent electronic structure at cSDW momenta. A superposition of these two Cooper pair states results in a non-stationary state of the form wherehω J ′ = E D − E S . Above, |d 2 xz and |d 2 yz are singlet Cooper pairs restricted to each orbital that alternate in sign between the electron and the hole pockets at the QCP. (See Fig.   1d, red versus blue, and Table I.) In particular, the difference in the probability of finding the Cooper pair (4) in one of the two orbitally-ordered states is cos ω J ′ t. Following refs. [19], [20], and [21], such a mode persists in the macroscopic superconductor when electrons (holes) condense into the dynamical Cooper pair state The number of orbitally ordered pairs and their phase are then canonically conjugate ob- The remaining commutators vanish. An internal Josephson effect between the two species of orbitally ordered Cooper pairs is then predicted by the following hydrodynamic Hamiltonian: where the difference in chemical potential between the two orbitals. Also, V denotes the area of an iron-pnictide layer. Above, χ orb is the susceptibility for orbital order of the Cooper Cooper Pair k 0 = 3d yz k 0 = 3d xz singlet d 2 yz cos k x a @ k y a near 0 0 singlet d 2 xz 0 cos k y a @ k x a near 0 pairs at zero temperature. It attains the normal-state value equal to one half the density of states in the hydrodynamic regime [29]. Also, e J ′ = 2t (s) orb |Ψ xz ||Ψ yz | is the Josephson energy density, where t (s) orb > 0 is the matrix element for internal Josephson tunneling, and where |Ψ xz ||Ψ yz | is the density of Cooper pairs (|Ψ xz | = |Ψ yz |). Notice from (5) that the equilibrium groundstate is S +− at the QCP, and that it shows no orbital order: φ − = 0 and N − = 0 at µ − = 0. The D +− pair state at the QCP, on the contrary, is in unstable equilibrium. Orbital ordering of the Cooper pairs is then governed by standard dynamical equations [20]: and µ(d xz )+µ(d yz ) remain constant. They imply small oscillations in N − and φ − about zero that are 90 • out of phase, at a natural frequency ω J ′ that is related to the internal Josephson tunneling by [21] (hω J ′ ) 2 = 8t (s) orb |Ψ xz ||Ψ yz |/χ orb . Last, orthorhombic shear strain [23] [24] couples to this internal Josepshon effect through the electron-phonon interaction: 1 2 µ − = Ξ u ′ 1 2 (∂u x /∂x−∂u y /∂y), where Ξ u ′ is a deformation potential, and where u is a displacement field for the iron atoms. Level repulsion between the spectrum of orthorhombic phonons and the above neutral spin-0 collective mode is then possible if the Debye frequency ω Db in ironpnictide superconductors is larger than ω J ′ ∼ ∆ 0 /h. (See Fig. 2.) Here, 2∆ 0 is the gap, which is typically comparable tohω Db in iron-pnictide materials [30].
The low-energy spectrum for two holes shown by Fig. 2 also contains a degenerate pair of spin-1 states with no net momentum that lie at the edge of the continuum. One has even parity under a reflection about the x axis and odd parity under a reflection about the y axis.
These parities are reversed for the other. The degenerate two-hole states therefore have P x and P y orbital symmetry. This, along with orbital order, is revealed by the pair wavefunction (3) of the P x state, for example: iF (k 0 , k) ∼ = 0.056 sin(k x a) for k y = 0 and k 0 = d yz , and iF (k 0 , k) ∼ = 0 otherwise. Now notice the spin-1 state at momentum (π/2a)x in Fig. 2 that is nearly degenerate with the groundstate. We propose that the former P states are degenerate triplet Cooper pairs, and that the latter spin-1 state is a corresponding Leggett mode [22].
As Hund's Rule coupling −J 0 increases past the QCP, the degenerate P states lower their energy below the edge of the continuum. The S, D and P Cooper pair states become close in energy while retaining the order shown in Fig. 2: E S < E D < E P . The four Leggett modes at momenta ±(π/2a)x and ±(π/2a)ŷ become the new degenerate groundstates, however.
They are collective spin excitations of a triplet-pair superconductor [22], on the other hand.
We therefore believe that the previous is a finite-size effect, and that the P states are degenerate groundstates on a larger lattice at −J 0 > −J 0c . The P x ±iP y triplet Cooper pairs have the most isotropic gap and exhibit no orbital order. In the folded Brillouin zone, which anticipates the effect of the two inequivalent Fe atoms common to iron superconductors [8], they are notably isotropic on the outer (anti-bonding) Fermi surface pocket centered at the unique cSDW momentum, while they have nodes on the inner (bonding) Fermi surface pocket. Importantly, exact calculations of one hole in the two-orbital t-J model (1) show that the hole bands at zero 2D momentum fall below the Fermi level as −J 0 increases past the critical value −J 0c [17]. The formation of Cooper pairs on the hole bands at zero 2D momentum therefore becomes unlikely.
Discussion. In conclusion, we have shown that the S +− and D +− pair states that have been predicted for iron-pnictide superconductors on the basis of spin-fluctuation exchange [7] are respectively bonding and anti-bonding superpositions of orbitally ordered Cooper pairs that alternate in sign between the hole and electron Fermi surfaces. With the exception of the 3d xy orbital character at the tips of the electron pockets, this claim is roughly borne out for the s-wave and d-wave solutions to the gap equation reported by Graser et al. in ref. [8], which also invokes spin-fluctuation exchange, but at weak on-site Coulomb repulsion. Also, the binding energy of the s-wave groundstate shown in Fig. 2 yields a gap of 2∆ 0 = 0.5 J ⊥ 1 . Taking a value of J ⊥ 1 ∼ 120 meV from a fit to spin-wave spectra in iron-pnictide materials based on the corresponding two-orbital Heisenberg model [18] yields a gap 2∆ 0 ∼ 60 meV, which is roughly consistent with the upper bound for the spin-resonance energy in ironpnictide superconductors [9]. The present study of Cooper pair formation from the localmoment limit further reveals triplet P x ± iP y Cooper pair states that are candidate pairing symmetries for single-layer iron selenide superconductors [12]. (Cf. refs. [31], [32], and [33]). J.P.R. thanks Brent Andersen, Richard Roberts and Timothy Sell for technical help with the use of the virtual shared-memory cluster (Lancer) at the AFRL DoD Supercomputer Resource Center. This work was supported in part by the US Air Force Office of Scientific