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arXiv:cond-mat/0202084 · Physical review. B, Condensed matter

Pseudogap and spectral function from superconducting fluctuations to the bosonic limit

Andrea Perali, P. Pieri, Giancarlo Calvanese Strinati, Claudio Castellani

Published Jul 3, 2002• 205 citations
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Abstract

The crossover from weak to strong coupling for a three-dimensional continuum model of fermions interacting via an attractive contact potential is studied above the superconducting critical temperature ${T}_{c}.$ The pair-fluctuation propagator, the one-loop self-energy, and the spectral function are investigated in a systematic way from the superconducting fluctuation regime (weak coupling) to the bosonic regime (strong coupling). Analytic and numerical results are reported. In the strong-coupling regime, where the pair fluctuation propagator has bosonic character, two quite different peaks appear in the spectral function at a given wave vector, a broad one at negative frequencies and a narrow one at positive frequencies. The broad peak is asymmetric about its maximum, with its spectral weight decreasing by increasing coupling and temperature. In this regime, two crossover temperatures ${T}_{1}^{*}$ (at which the two peaks in the spectral function merge in one peak) and ${T}_{0}^{*}$ (at which the maximum of the lower peak crosses zero frequency) can be identified, with ${T}_{c}\ensuremath{\ll}{T}_{0}^{*}<{T}_{1}^{*}.$ By decreasing coupling, the two-peak structure evolves smoothly. In the weak-coupling regime, where the fluctuation propagator has diffusive Ginzburg-Landau character, the overall line shape of the spectral function is more symmetric and the two crossover temperatures approach ${T}_{c}.$ The analysis of the spectral function identifies specific features which allow one to distinguish by ARPES whether a system is in the weak- or strong-coupling regime. Connection of the results of our analysis with the phenomenology of cuprate superconductors is also attempted.

DOI
10.1103/physrevb.66.024510

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The crossover from weak to strong coupling for a three-dimensional continuum model of fermions interacting via an attractive contact potential is studied above the superconducting critical temperature Tc.{T}_{c}.Tc​. The pair-fluctuation propagator, the one-loop self-energy, and the spectral function are investigated in a systematic way from the superconducting fluctuation regime (weak coupling) to the bosonic regime (strong coupling). Analytic and numerical results are reported. In the strong-coupling regime, where the pair fluctuation propagator has bosonic character, two quite different peaks appear in the spectral function at a given wave vector, a broad one at negative frequencies and a narrow one at positive frequencies. The broad peak is asymmetric about its maximum, with its spectral weight decreasing by increasing coupling and temperature. In this regime, two crossover temperatures T1∗{T}_{1}^{*}T1∗​ (at which the two peaks in the spectral function merge in one peak) and T0∗{T}_{0}^{*}T0∗​ (at which the maximum of the lower peak crosses zero frequency) can be identified, with Tc\ensuremath≪T0∗<T1∗.{T}_{c}\ensuremath{\ll}{T}_{0}^{*}<{T}_{1}^{*}.Tc​\ensuremath≪T0∗​<T1∗​. By decreasing coupling, the two-peak structure evolves smoothly. In the weak-coupling regime, where the fluctuation propagator has diffusive Ginzburg-Landau character, the overall line shape of the spectral function is more symmetric and the two crossover temperatures approach Tc.{T}_{c}.Tc​. The analysis of the spectral function identifies specific features which allow one to distinguish by ARPES whether a system is in the weak- or strong-coupling regime. Connection of the results of our analysis with the phenomenology of cuprate superconductors is also attempted.
The BCS-BEC crossover is studied in a systematic way in the broken-symmetry phase between zero temperature and the critical temperature. This study bridges two regimes where quantum and thermal fluctuations are, respectively, important. The theory is implemented on physical grounds, by adopting a fermionic self-energy in the broken-symmetry phase that represents fermions coupled to superconducting fluctuations in weak coupling and to bosons described by the Bogoliubov theory in strong coupling. This extension of the theory beyond mean field proves important at finite temperature, to connect with the results in the normal phase. The order parameter, the chemical potential, and the single-particle spectral function are calculated numerically for a wide range of coupling and temperature. This enables us to assess the quantitative importance of superconducting fluctuations in the broken-symmetry phase over the whole BCS-BEC crossover. Our results are relevant to the possible realizations of this crossover with high-temperature cuprate superconductors and with ultracold fermionic atoms in a trap.
The two-dimensional attractive Hubbard model is studied in the weak-to-intermediate-coupling regime by employing a nonperturbative approach. It is shown that this approach is in quantitative agreement with Monte Carlo calculations for both single-particle and two-particle quantities. Both the density of states and the single-particle spectral weight show a pseudogap at the Fermi energy below some characteristic temperature T∗,{T}^{*},T∗, also in good agreement with quantum Monte Carlo calculations. The pseudogap is caused by critical pairing fluctuations in the low-temperature renormalized classical regime $(\ensuremath{\Elzxh}\ensuremath{\omega}<{k}_{B}T)$ of the two-dimensional system. With increasing temperature the spectral weight fills in the pseudogap instead of closing it, and the pseudogap appears earlier in the density of states than in the spectral function. Small temperature changes around T∗{T}^{*}T∗ can modify the spectral weight over frequency scales much larger than temperature. Several qualitative results for the s-wave case should remain true for d-wave superconductors.
We consider a lattice version of the ``negative-U′′U''U′′ Hubbard model in three dimensions, to study the crossover from a superconducting phase transition (dominated by phase fluctuation) to a regime where the amplitude of the superconducting order parameter controls the critical temperature. Starting from a fermionic Hamiltonian, we give a microscopic derivation of the effective action in terms of the relevant physical fields: the modulus and phase of the superconducting order parameter. Furthermore, by employing to the resulting coarse--grained quantum XY\mathrm{XY}XY model the phase fluctuation algebra between number and phase operators (given by the Euclidean group E2),{E}_{2}),E2​), we map the ``phase-only'' action onto a solvable quantum spherical model. We establish the self--consistent theory (involving both fermionic and bosonic degrees of freedom), and calculate the superconducting phase coherence transition temperature Tc{T}_{c}Tc​ as a function of the coupling strength ∣U∣,|U|,∣U∣, exhibiting a maximum where the behavior crosses over from BCS to Bose-Einstein (BE) condensation. We examine a pseudogap in the normal state which emerges naturally as a precursor of superconductivity at the second characteristic temperature Tg{T}_{g}Tg​ due to a state with bound pairs but without long--range phase coherence. Furthermore, we demonstrate that the reduced gaps of the model $2{\ensuremath{\Delta}}_{0}{/k}_{B}{T}_{g}\ensuremath{\sim}4$ is almost independent on the pairing strength ∣U∣/t,|U|/t,∣U∣/t, whereas $2{\ensuremath{\Delta}}_{0}{/k}_{B}{T}_{c}$ increases with increasing ∣U∣|U|∣U∣---in striking analogy with the phenomenology of the high--temperature cuprate superconductors when interpreted in terms of the BCS-BC crossover scenario.
One-particle spectral properties in the normal phase of the two-dimensional attractive Hubbard model are investigated in the weak-coupling regime using the non-self-consistent T-matrix approximation. The corresponding equations are evaluated numerically directly on the real-frequency axis. For temperatures sufficiently close to the superconducting transition temperature, a pseudogap in the one-particle spectral function is observed, which can be assigned to the increasing importance of pair fluctuations.
We generalize the standard attractive Hubbard, having an on-site pair creation operator Qi = ai|ai|, to one with n components Qiα = aiα|aiα|, a = 1, 2, ... , n.In the limit n -> co we obtain the Ginzburg-Landau functional.On this basis we explore the crossover from weak (BCS) to strong coupling (Bose-Einstein condensation) superconductivity.The associated self-consistent equations for the Ginzburg-Landau parameters are similar to those of the T-matrix approach.The evolution of the band structure with increasing interaction strength is studied and correlated with the behavior of the pair propagator and the transition temperature.We find that the pairing interaction creates a new band which moves downwards in energy as the interaction strength increases and separates into a lower Hubbard band when the interaction strength becomes comparable to the band width.In the strong coupling regime, a third band with small spectral weight is also found in between the lower and upper Hubbard bands.
The pseudogap phenomenon is a hallmark of strongly interacting Fermi systems, from high-temperature superconductors to ultracold atomic gases, yet its precise origin remains debated. Here we calculate the spectral function and rf spectra of ultracold atomic gases across the BCS-BEC crossover to quantitatively investigate the pairing mechanism of the pseudogap. We advance our pairing fluctuation theory by incorporating particle-hole fluctuations, which renormalize the effective interaction in the particle-particle channel. To achieve quantitative accuracy, we employ a full numerical convolution for the pair susceptibility and self-energy, moving beyond previous analytic pseudogap approximations. This convolution approach automatically captures two critical effects: (i) the full spectral broadening of fermions due to finite pair lifetime, and (ii) the previously neglected pair-hole scattering effect, which manifests as a substantial Hartree energy. We calculate the spectral function, and use rf spectral intensity maps and energy distribution curves to determine the quasiparticle dispersion. From these, we extract the pseudogap ΔΔΔ, Hartree energy, and chemical potential, mapping their evolution across the crossover. Our results show that the pseudogap emerges continuously as the system moves from the BCS regime toward BEC. Furthermore, the pair spectral function reveals that pairs become diffusive at energies above 2ΔΔΔ, indicating that the pair lifetime is governed by virtual binding and unbinding processes. Our calculations achieve quantitative agreement with recent experiments across the BCS-BEC crossover, including at unitarity, providing strong support for a pairing-based origin of the pseudogap as described by our pairing fluctuation theory.