Hidden Pseudogap and Excitation Spectra in a Strongly Coupled Two-Band Superfluid/Superconductor

We investigate single-particle excitation properties in the normal state of a two-band superconductor or superfluid throughout the Bardeen-Cooper-Schrieffer (BCS) to Bose-Einstein-condensation (BEC) crossover, within the many-body T-matrix approximation for multi-channel pairing fluctuations. We address the single-particle density of states and the spectral functions consisting of two contributions associated with a waekly interacting deep band and a strongly interacting shallow band, relevant for iron-based multiband superconductors and multicomponent fermionic superfluids. We show how the pseudogap state in the shallow band is hidden by the deep band contribution throughout the two-band BCS-BEC crossover. Our results could explain the missing pseudogap in recent scanning tunneling microscopy experiments in FeSe superconductors.

One of the exciting topics in the two-band BCS-BEC crossover is the existence of pseudogaps in the single-particle excitation in the normal state (for reviews discussing the pseudogap, see [7][8][9]). While several experiments for FeSe report signals of pseudogaps and preformed Cooper pairs [39][40][41][42], a recent scanning tunneling spectroscopy (STS) measurement did not observe a pseudogap behavior even in the crossover regime of the BCS-BEC crossover [43]. In addition, a torque magnetometry experiment in the same system indicates weak pairing fluctuations [44]. Theoretically, the screening of pairing fluctuations originating from the two-band configuration with different pairing strengths has been reported [34][35][36][37][38], but the perfect screening observed in the experiment [43] may require a further mechanism for suppressing the pseudogap.
In this article, we resolve this complicated phenomenology by calculating the singleparticle density of states and spectral function throughout the two-band BCS-BEC crossover.
We adopt the many-body T-matrix approximation (TMA) for multi-channel pairing fluctuations in the normal state. We show that the pseudogap occurring in the strongly coupled shallow band is masked by the contribution from the deep band in the total (i.e., summed over the bands) density of state, which is measured in the STS experiment. This masking effect becomes remarkable in the strong-coupling regime for the shallow band due to the overlap of spectral weights in each band. On the other hand, we show that the total spec-tral function relevant for the angular-resolved-photoemission spectroscopy (ARPES) clearly reflects the pseudogap features in the strongly coupled shallow band.

II. FORMALISM
We consider a three-dimensional two-band model for attractive fermions described by where c kσj is the annihilation operator of a fermion with the momentum k, spin σ =↑, ↓ and the band index j (where j = 1 and j = 2 denote the indices of deep and shallow bands, respectively) and ξ kj = k 2 /(2m j ) − µ + E 0 δ j,2 is the single-particle dispersion in the j-band, measured from the chemical potential µ with the energy separation E 0 between the two bands. For simplicity, we take the same effective mass m = m 1 = m 2 in each band. We define a pair-annihilation operator We employ a contact-type interaction. Specifically, the intraband couplings V 11 and V 22 are expressed in terms of the corresponding scattering lengths a 11 and a 22 as [20] where the momentum cutoff k 0 is taken much larger than all other momentum scales.
Superconducting pair-fluctuation effects are incorporated by the two-channel Tmatrix [37,38] wherej denotes the other band with respect to band j and is the lowest order particle-particle correlation function. ν ℓ = 2ℓπT is the bosonic Matsubara frequency. In Eq. (5), f (x) = e x/T + 1 −1 is the Fermi-Dirac distribution function. In the two-channel T -matrix approach, the fermionic self-energy is of the form where In the STS experiment, one observes the tunneling current I occuring via the tunneling where c k ′ σ ′ 0 denotes the annihilation operator of an electron in the weakly-coupled normal metal connected to the sample. For simplicity, we consider the momentum-, spin-, and band-independent tunneling amplitudes t = t 1 = t 2 . The tunneling current is obtained as where V is the bias voltage and e is an electron charge. In Eq. (9), A j (k, ω) is the spectral function given by where δ is an infinitesimal small positive number to generate the retarded Green's function for real frequencies. A r (k ′ , ω) is the spectral function in the reference normal metal. At sufficiently low temperature, we obtain the differential conductance where is the total density of states. N r (0) is the density of states at the Fermi level in the reference metal in the normal state. In this way, one can observe if a pseudogap opens in the total density of states.
In this article, we examine N(ω) at the superconducting (or superfluid) critical temperature T c identified by the Thouless criterion given by The chemical potential µ is determined by the density equation Note that one obtains n = n 1 + n 2 with n j = We take E 0 = 3 5 E F,1 such that the two deep and shallow (occupied) bands are overlapped. We choose the dimensionless coupling parameter in the deep band in the weak-coupling regime as (k F,1 a 11 ) −1 = −2, while the coupling parameter in the shallow band (k F,2 a 22 ) −1 is tuned throughout the BCS-BEC crossover . The dimensionless interband pair-exchange coupling is given bỹ are the Fermi wave-vector and the Fermi energy for the total density n. We take k 0 = 100k F which is a sufficiently large wave-vector cutoff compared to all other momentum scales.

III. RESULTS
First, in Fig. 1 we show the evolution of the critical temperature T c across the BCS-BEC crossover with increasing (k F,2 a 22 ) −1 for three casesṼ 12 = 0,Ṽ 12 = 1, andṼ 12 = 2. In the weak-coupling BCS side (k F,2 a 22 ) −1 < ∼ 0, T c exponentially increases as ∼ exp π 2k F,2 a 22 . The finite pair-exchange couplingṼ 12 gives an enhancement of T c . In the strong-coupling BEC side (k F,2 a 22 ) −1 > ∼ 0, T c approaches to the Bose-Einstein condensation temperature T BEC of tightly bound molecules given by [8,9] T BEC = π m n ζ(3/2) where ζ(3/2) ≃ 2.612 is the Riemann zeta function. This indicates that all the particles in both bands form molecular condensates in the strong-coupling limit. Although the  ∼ 1], the coexistence of large Cooper pairs and small molecules has been discussed within the mean-field [28], NSR [36], and TMA [37,38] approaches. With increasing V 12 , the two-band system undergoes the BCS-BEC crossover even in the case of weak intraband couplings [(k F,2 a 22 ) −1 < ∼ 1]. In this way, one can find that the BCS-BEC crossover is realized by increasing the interaction strength in the shallow band (k F,2 a 22 ) −1 in the present two-band model.
In Fig. 2(a) Fig. 2(a)], N(ω) shows a non-monotonic structure, but not the fully gapped density of states which can be found in the single-band counterpart. To understand these behaviors, we examine the band-selective density of states given by In the inset of Fig. 2, N j (ω) for each band is plotted at (k F,2 a 22 ) −1 = 0, corresponding to the crossover regime in the shallow band. N 2 (ω) clearly exhibits the pseudogap behavior (dip structure around ω = 0) in the shallow band (j = 2) due to the strong pairing fluctuations associated with V 22 . However, the deep band (j = 1) shows the square-root behavior N 1 (ω) ∝ (ω + µ) without the pseudogap signature in the case ofṼ 12 = 0 because the intraband coupling is kept weak. In this regard, the pseudogap structure in the total N(ω) originating from N 2 (ω) is hidden by the square-root contribution of N 1 (ω). Such a situation occurs for larger intraband coupling in the shallow band [e.g. (k F,2 a 22 ) −1 = 0.4 and 0.8 in Fig. 2(a)]. On the other hand, in the case of a finite interband pair-exchange coupling V 12 = 1, shown in Fig. 2(b), one can find a small pseudogap around ω = 0 in N(ω) even in the strong-coupling regime. Furthermore, N(ω) exhibits a large flattened region around the Fermi level (ω = 0). These features can also be understood from the partial density of states N i (ω) as shown in the inset for (k F,2 a 22 ) −1 = 0. The pair-exchange process associated with finite V 12 induces the pseudogap even in the weakly interacting deep band (j = 1).
Hence, one can find two pseudogaps with different sizes in the two bands. The resulting total density of states N(ω) exhibits the small pseudogap originating from N 1 (ω) throughout the BCS-BEC crossover. On the other hand, the large pseudogap in N 2 (ω) is hidden by the contribution of the sizable spectral weight of N 1 (ω).
Finally, we discuss our results for the total spectral function A(k, ω) defined as the sum of the two single-band contributions This is the quantity measured by ARPES experiments. Figure 3 shows where ∆ pg,2 is the pseudogap energy scale induced by strong pairing fluctuations. The backbending curve in the large wave-vector region (p > ∼ k F ) is one of the characteristic features for the pseudogap in the angular resolved photoemission spectroscopy of ultracold Fermi gases and strongly coupled superconductors [10][11][12][13]. Since this curve is not hidden by the contribution from the deep band, it can be regarded as the signature of the pseudogap even in the present two-band system. For the case of strong intraband coupling (k F,2 a 22 ) −1 = 0.4, the pseudogap size becomes large and the lower branch of the Bogoliubov dispersion overlaps with the deep band dispersion. This result indicates that the tightly bound molecules in the shallow band starts dominating the system even in the presence of the cold deep band due to the very strong intraband coupling in the shallow band.

IV. CONCLUSIONS
We have investigated single-particle excitation spectra in a two-band superfluid/superconductor throughout the BCS-BEC crossover and made connections with recent teracting regime for the shallow band is hidden by the contribution of the weakly interacting deep band in the total density of states, which is the quantity measured by the recent STS experiments in FeSe superconductors. On the other hand, the single-particle spectral function consisting of contributions from the two bands, which is relevant to the ARPES measurement, clearly exhibits the signature of the pseudogaps, that is, the Bogoliubov back-bending dispersions. We emphasize that these non-trivial features for the pseudogaps are unique of a two-band fermionic system in which pair-fluctuations interfere in a complex manner, originating screening or amplification phenomena of superfluid/superconductor fluctuations which are absent in the single-band counterpart.