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Physical review. B, Condensed matter

Optical sum rule violation and the kinetic energy change at the phase coherence transition in superconductors with pseudogap

T. K. Kopeć

Published Jan 22, 2003• 7 citations
Publisher

Abstract

We calculate the spectral weights of the optical conductivity of a three-dimensional lattice version of the ``negative-$U''$ Hubbard model of a strongly correlated superconductor in the intermediate to large-$U$ limit. Starting from a tight-binding Fermionic Hamiltonian, we pass to an effective action in terms of the modulus and phase of the superconducting pairing field as the relevant Bosonic variables. Furthermore, we calculate the temperature and U-interaction dependence of the oscillator strength and find that below the pair formation temperature ${T}^{\ensuremath{\star}}$ the optical integral is not conserved, implying the lowering of the kinetic energy at ${T}_{c}$ upon entering the phase coherent superfluid state.

DOI
10.1103/physrevb.67.014520

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We calculate the spectral weights of the optical conductivity of a three-dimensional lattice version of the ``negative-U′′U''U′′ Hubbard model of a strongly correlated superconductor in the intermediate to large-UUU limit. Starting from a tight-binding Fermionic Hamiltonian, we pass to an effective action in terms of the modulus and phase of the superconducting pairing field as the relevant Bosonic variables. Furthermore, we calculate the temperature and U-interaction dependence of the oscillator strength and find that below the pair formation temperature T\ensuremath⋆{T}^{\ensuremath{\star}}T\ensuremath⋆ the optical integral is not conserved, implying the lowering of the kinetic energy at Tc{T}_{c}Tc​ upon entering the phase coherent superfluid state.
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.
The conductivity of the two-dimensional Hubbard model is particularly relevant for high-temperature superconductors. Vertex corrections are expected to be important because of strongly momentum-dependent self-energies. To attack this problem, one must also take into account the Mermin--Wagner theorem, the Pauli principle, and crucial sum rules in order to reach nonperturbative regimes. Here, we use the two-particle self-consistent approach that satisfies these constraints. This approach is reliable from weak to intermediate coupling. A functional derivative approach ensures that vertex corrections are included in a way that satisfies the fff-sum rule. The two types of vertex corrections that we find are the antiferromagnetic analogs of the Maki--Thompson and Aslamasov--Larkin contributions of superconducting fluctuations to the conductivity but, contrary to the latter, they include nonperturbative effects. The resulting analytical expressions must be evaluated numerically. The calculations are impossible unless a number of advanced numerical algorithms are used. These algorithms make extensive use of fast Fourier transforms, cubic splines, and asymptotic forms. A maximum entropy approach is specially developed for analytical continuation of our results. These algorithms are explained in detail in the appendices. The numerical results are for nearest-neighbor hoppings. In the pseudogap regime induced by two-dimensional antiferromagnetic fluctuations, the effect of vertex corrections is dramatic. Without vertex corrections the resistivity increases as we enter the pseudogap regime. Adding vertex corrections leads to a drop in resistivity, as observed in some high-temperature superconductors. At high temperatures, the resistivity saturates at the Ioffe--Regel limit. At the quantum critical point and beyond, the resistivity displays both linear and quadratic temperature dependence and there is a correlation between the linear term and the superconducting transition temperature. A hump is observed in the mid-infrared range of the optical conductivity in the presence of antiferromagnetic fluctuations.
We investigate by means of Dynamical mean-field theory the crossover from BCS superconductivity to Bose-Einstein (BE) condensation of preformed pairs which occurs in the attractive Hubbard model by increasing the attraction strength. Fixing the density to a generic value n=0.75n=0.75n=0.75, we follow the evolution of the two energy scales underlying the superconducting phenomenon, the gap ${\ensuremath{\Delta}}_{0}$, and the superfluid stiffness DS{D}_{S}DS​, which controls the phase coherence. The BCS--BE crossover is clearly mirrored in a change in the hierarchy of these two scales, the smallest of the two controlling the critical temperature. In the whole intermediate-to-strong-coupling region Tc{T}_{c}Tc​ scales with DS{D}_{S}DS​, while TC{T}_{C}TC​ is proportional to ${\ensuremath{\Delta}}_{0}$ only in the BCS regime. This evolution as a function of the interaction qualitatively resembles what happens in the cuprates when the doping is decreased towards the Mott insulator. This continuous change reflects also in the energetic balance at the superconducting transition. While, as it is well known, superconductivity is stabilized by a potential energy gain in the BCS regime, the strong-coupling superconductivity is made stable by a reduction of kinetic energy. Interestingly the intermediate-coupling region, where the maximum Tc{T}_{c}Tc​ is achieved, behaves similarly to the strong-coupling regime, and its gain in kinetic energy is the largest as a function of the coupling. Since the integral of the optical conductivity is proportional to the kinetic energy, the above finding implies that the attractive Hubbard model can account qualitatively for the anomalous behavior of optical spectra around Tc{T}_{c}Tc​, where an increase of spectral weight is observed in under and optimally doped cuprates, while the overdoped samples have a more standard behavior. This qualitative agreement is lost in the normal phase, specifically at strong coupling, calling for the inclusion of strong correlation effects in the theoretical description.
In this paper we address the behavior of the superfluid transition temperature Tc{T}_{c}Tc​ in the attractive Hubbard model. The attractive Hubbard model can be regarded as the generalization of the BCS to Bose-Einstein condensation (BEC) crossover to a lattice and, as such, may have implications for future optical lattice studies. Nevertheless, the BEC limit of the Hubbard model is very different from that of the better-studied Fermi gases, owing to the strong intersite repulsion between pairs. Here we address systematically the effects of pairing fluctuations for all filling fractions over the entire range of attractive interaction strength. A central conclusion of our work is that in a lattice, around half filling, the smooth evolution from the BCS to the BEC limits is interrupted: Tc{T}_{c}Tc​ vanishes when the system approaches the bosonic regime with increasing interaction strength. We suggest that this interruption of crossover may signal a quantum critical transition to another form of superfluid not continuously connected to a BCS-like phase. A simple variational ansatz for an alternate ground state in this more strongly coupled superfluid is presented. A generalization of the (sss-wave) Hubbard model to ddd-wave pairing allows us to address issues of relevance to high-Tc{T}_{c}Tc​ superconductivity. The phase diagram (representing the pairing or pseudogap onset temperature T∗{T}^{*}T∗ and Tc{T}_{c}Tc​) shows that here too one observes a vanishing of Tc{T}_{c}Tc​ when T∗{T}^{*}T∗ becomes sufficiently large. Given this predicted breakdown of the crossover, and given the striking similarity to features of the cuprate phase diagram, we suggest that future experiments on ultracold fermions in optical lattices should not be exclusively limited to the repulsive Hubbard model, but should address the attractive model in order to elucidate features of high-temperature superconductivity.
We provide a fresh perspective on the pseudogap physics for attractive fermions as described by the three-dimensional Hubbard model. The pseudogap in the single-particle spectral function, which occurs for temperatures above the critical temperature Tc{T}_{c}Tc​ of the superfluid transition, is often interpreted in terms of preformed uncondensed pairs. Here we show that the occurrence of pseudogap physics can be consistently understood as due to local excitations which lead to a splitting of the quasiparticle peak for sufficiently large interaction. This effect becomes prominent at intermediate and high temperatures when the quantum-mechanical hopping is incoherent. We demonstrate the absence of the conjectured pairing temperature below which pseudogap physics is expected to occur. Our results are based on approximating the physics of the three-dimensional Hubbard model by dynamical mean-field theory calculations and a momentum independent self-energy. Our predictions can be tested with ultracold atoms in optical lattices with currently available temperatures and spectroscopic techniques.
The properties of a mixture of mutually interacting bound electron pairs and itinerant fermions (the boson-fermion model) on a lattice are further studied. We determine the superconducting critical temperature from a pseudogap phase by applying a self-consistent TTT-matrix approach, which includes the pairing fluctuations and the boson self-energy effects. The analysis is made for a three dimensional cubic lattice with tight-binding dispersion for electrons and for both standard bosons and the case of hard-core bosons. The results describe the BCS-Bose-Einstein condensation crossover with varying position of the bosonic (local pair) level and give a further insight into the nature of resonance superfluidity in the boson-fermion model.