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arXiv:cond-mat/0405239 · Physical Review B

Linear response theory around a localized impurity in the pseudogap regime of an anisotropic superconductor: Precursor pairing versus d-density-wave scenario

Natascia Andrenacci, G. G. N. Angilella, H. Beck, R. Pucci

Published Jul 1, 2004• 7 citations
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Abstract

We derive the polarizability of an electron system in (i) the superconducting phase, with $d$-wave symmetry, (ii) the pseudogap regime, within the precursor pairing scenario, and (iii) the $d$-density-wave (dDW) state, characterized by a $d$-wave hidden order parameter, but no pairing. Such a calculation is motivated by the recent proposals that imaging the effects of an isolated impurity may distinguish between precursor pairing and dDW order in the pseudogap regime of the high-${T}_{c}$ superconductors. In all three cases, the wave-vector dependence of the polarizability is characterized by an azimuthal modulation, consistent with the $d$-wave symmetry of the underlying state. However, only the dDW result shows the fingerprints of nesting, with nesting wave vector $\mathbf{Q}=(\ensuremath{\pi},\ensuremath{\pi})$, albeit imperfect, due to a nonzero value of the hopping ratio ${t}^{\ensuremath{'}}∕t$ in the band dispersion relation. As a consequence of nesting, the presence of hole pockets is also exhibited by the $(\mathbf{q},\ensuremath{\omega})$ dependence of the retarded polarizability.

DOI
10.1103/physrevb.70.024507

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We derive the polarizability of an electron system in (i) the superconducting phase, with ddd-wave symmetry, (ii) the pseudogap regime, within the precursor pairing scenario, and (iii) the ddd-density-wave (dDW) state, characterized by a ddd-wave hidden order parameter, but no pairing. Such a calculation is motivated by the recent proposals that imaging the effects of an isolated impurity may distinguish between precursor pairing and dDW order in the pseudogap regime of the high-Tc{T}_{c}Tc​ superconductors. In all three cases, the wave-vector dependence of the polarizability is characterized by an azimuthal modulation, consistent with the ddd-wave symmetry of the underlying state. However, only the dDW result shows the fingerprints of nesting, with nesting wave vector Q=(\ensuremathπ,\ensuremathπ)\mathbf{Q}=(\ensuremath{\pi},\ensuremath{\pi})Q=(\ensuremathπ,\ensuremathπ), albeit imperfect, due to a nonzero value of the hopping ratio t\ensuremath′∕t{t}^{\ensuremath{'}}∕tt\ensuremath′∕t in the band dispersion relation. As a consequence of nesting, the presence of hole pockets is also exhibited by the (q,\ensuremathω)(\mathbf{q},\ensuremath{\omega})(q,\ensuremathω) dependence of the retarded polarizability.
We study the electronic structure within a system of phase-decoupled one-dimensional superconductors coexisting with stripe spin and charge-density-wave order. This system has a nodal Fermi surface (Fermi arc) in the form of a hole pocket and an antinodal pseudogap. The spectral function in the antinodes is approximately particle-hole symmetric contrary to the gapped regions just outside the pocket. We find that states at the Fermi energy are extended whereas states near the pseudogap energy have localization lengths as short as the interstripe spacing. We consider pairing which has either local ddd-wave or sss-wave symmetry and find similar results in both cases, consistent with the pseudogap being an effect of local pair correlations. We suggest that this state is a stripe-ordered caricature of the pseudogap phase in underdoped cuprates with coexisting spin-, charge-, and pair-density-wave correlations. Lastly, we also model a superconducting state which (1) evolves smoothly from the pseudogap state, (2) has a signature subgap peak in the density of states, and (3) has the coherent pair density concentrated to the nodal region.
The response to a local strong nonmagnetic impurity in the pseudogap phase is examined in two distinctly different scenarios: phase fluctuation (PF) of pairing field and d-density-wave (DDW) order. In the PF scenario, the resonance state is generally double peaked near the Fermi level, and is abruptly broadened by vortex fluctuations slightly above the transition temperature. In the DDW scenario, the resonance is single peaked and remains sharp up to gradual intrinsic thermal broadening, and the resonance energy is analytically determined to be at minus of the chemical potential.
The normal state of strongly coupled superconductors is characterized by the presence of ``preformed'' Cooper pairs well above the superconducting critical temperature. In this regime, the electrons are paired, but they lack the phase coherence necessary for superconductivity. The existence of preformed pairs implies the existence of a characteristic energy scale associated with a pseudogap. Preformed pairs are often invoked to interpret systems where some signatures of pairing are present without actual superconductivity, but an unambiguous theoretical characterization of a preformed-pair system is still lacking. To fill this gap, we consider the response to an external pairing field of an attractive Hubbard model, which hosts one of the cleanest realizations of a preformed pair phase, and a repulsive model where sss-wave superconductivity cannot be realized. Using dynamical mean-field theory to study this response, we identify the characteristic features which distinguish the reaction of a preformed pair state from a normal metal without any precursor of pairing. The theoretical detection of preformed pairs is associated with the behavior of the second derivative of the order parameter with respect to the external field, as confirmed by analytic calculations in limiting cases. Our findings provide a solid test bed for the interpretation of state-of-the-art calculations for the normal state of the doped Hubbard model in terms of ddd-wave preformed pairs and, in perspective, of nonequilibrium experiments in high-temperature superconductors.
The pseudogap state of cuprate high-temperature superconductors has been often viewed as either a yet unknown competing order or a precursor state to superconductivity. While awaiting the resolution of the pseudogap problem in cuprates, we demonstrate that local pairing fluctuations, vortex liquid dynamics, and other precursor phenomena can emerge quite generally whenever fermionic excitations remain gapped across the superconducting transition, regardless of the gap origin. Our choice of a tractable model is a lattice band insulator with short-range attractive interactions between fermions in the sss-wave channel. An effective crossover between Bardeen-Cooper-Schrieffer (BCS) and Bose-Einstein condensate (BEC) regimes can be identified in any band insulator above two dimensions, while in two dimensions only the BEC regime exists. The superconducting transition is ``unconventional'' (non-pair-breaking) in the BEC regime, identified by either the bosonic mean field or XY\mathit{XY}XY universality class. The insulator adjacent to the superconductor in the BEC regime is a bosonic Mott insulator of Cooper pairs, which may be susceptible to charge density wave ordering. We construct a function of the many-body excitation spectrum whose nonanalytic changes define a sharp distinction between band and Mott insulators. The corresponding ``second-order transition'' can be observed out of equilibrium by driving a Cooper pair laser in the Mott insulator. We explicitly show that the gap for charged bosonic excitations lies below the threshold for Cooper pair breakup in any BEC regime, despite quantum fluctuations. Our discussion ends with a view of possible consequences for cuprates, where antinodal pair dynamics has certain features in common with our simple sss-wave picture.
Motivated by the effect of symmetry breaking in cuprates superconductors YBa[Formula: see text]Cu[Formula: see text]O[Formula: see text], we employ the renormalized mean-field theory to study the presence of uniform superconducting and charge-ordered states in two anisotropic t-J-U models, either with hopping strength anisotropy or antiferromagnetic interaction anisotropy. In the case of uniform superconducting state, compared with the isotropic t-J-U model with only [Formula: see text]-wave superconducting state, there is an additional s-wave superconducting state in the model with hopping strength anisotropy. Meanwhile, the hopping anisotropy may enhance the critical Coulomb interaction [Formula: see text] at the Mott insulator to the Gossamer superconductor transition point, and strong hopping anisotropy may weaken the superconducting state. In the case of a charge-ordered state, hopping anisotropy may suppress the amplitude of the charge density waves and pair density waves, which originate from local Coulomb interactions. These results indicate that the effects of hopping anisotropy and local Coulomb interactions are competitive. Moreover, the antiferromagnetic interaction anisotropy only weakly suppresses the superconducting gap and density wave amplitude. Our results show that the t-J-U model with hopping anisotropy is qualitatively consistent with experimental superconducting pair symmetry and charge density waves in the YBa[Formula: see text]Cu[Formula: see text]O[Formula: see text] system.
In light of the new experimental and theoretical important developments in high-Tc{T}_{c}Tc​ superconductivity, we revisit the fermionic hot-spot model relevant to the phenomenology of the cuprates. We extend previous results by means of a complete two-loop order renormalization group (RG) framework. Here, we explicitly study the effect of the charge-density-wave (CDW) order parameter with a ddd-wave form factor with the experimentally observed modulation $(\ifmmode\pm\else\textpm\fi{}{Q}_{0},0)$ and $(0,\ifmmode\pm\else\textpm\fi{}{Q}_{0})$ at the infrared-stable nontrivial fixed point obtained previously for this model. Additionally, we proceed to investigate also the so-called pair-density-wave (PDW) order that was recently proposed in the literature as a possible candidate for the ``hidden'' order to describe the pseudogap phase observed in underdoped cuprates. We confirm that although the above two ordering tendencies are also found to be nearly degenerate both at one-loop and two-loop RG orders and linked by an emergent SU(2) pseudospin symmetry, they turn out to be subleading for weaker couplings in the present model to antiferromagnetism, ddd-wave bond-density wave (BDW) order with modulation along Brillouin zone diagonals $(\ifmmode\pm\else\textpm\fi{}{Q}_{0},\ifmmode\pm\else\textpm\fi{}{Q}_{0})$, and ddd-wave singlet superconductivity (SSC). However, as we increase the strength of the initial coupling towards moderate values, we do capture a tendency for the entangled PDW/CDW order to become leading compared to BDW/SSC in the model, which suggests that the former composite order might be indeed a viable concept to describe some cuprate superconductors at high temperatures in the underdoped regime, as has been recently alluded to by many authors in the literature.