Pole structure of the electronic self-energy with coexistence of charge order and superconductivity

M. Grandadam and C. Pépin

Phys. Rev. B 103, 224507 – Published 3 June 2021

Abstract

We compare the pole structure of the electronic Green's function obtained by cluster dynamical mean-field theory to the results from the fractionalized pair density wave idea. In the superconducting phase, we can consider the system in a state with coexistence of superconducting and charge order. Writing the Green's function in a way analogous to the previously proposed “hidden-fermions” model [Sakai et al., Phys. Rev. Lett. 116, 057003 (2016)] leads to a similar pole structure for the self-energy. The fractionalization of the pair density wave order also describes the pseudogap phase as a superposition of superconducting and charge order fluctuations. Considering a phenomenological lifetime for the particle-particle and particle-hole pairs leads to an electronic spectral function that matches the numerical results.

Received 3 January 2021
Accepted 20 April 2021

DOI:https://doi.org/10.1103/PhysRevB.103.224507

Physics Subject Headings (PhySH)

Superconductivity

Hubbard model Mean field theory

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

M. Grandadam and C. Pépin

Institut de Physique Théorique, Université Paris-Saclay, CEA, CNRS, F-91191 Gif-sur-Yvette, France

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Vol. 103, Iss. 22 — 1 June 2021

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Images

Figure 1
(a) Normal self-energy (red line) and electronic spectral function (green dotted lines) obtained by CDMFT for momenta on the line $k = (0, π) - (π / 2, π / 2)$ . The self-energy shows two isolated poles in the antinodal region at $ω = \pm ω_{0}$ but with a clear asymmetry in weight. In the nodal region the peak at positive energy splits. The pole at negative energy is damped and difficult to follow. (b) Anomalous self-energy obtained by CDMFT for the same momenta. It shows pairs of poles at the same position as in the normal part but with an antisymmetric weight. The same splitting occurs when going from the antinodal to the nodal region. The weight close to $k = (π / 2, π / 2)$ is close to zero, indicating that the coupling to the hidden fermion is specific to the antinodal region.
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Figure 2
Fermi surface for free electrons with dispersion given by $ε_{k} = - 2 t [cos (k_{x}) + cos (k_{y})] - 4 t^{'} cos (k_{x}) cos (k_{y}) - μ$ , where $t = 1$ , $t^{'} = - 0.2$ , and $μ = - 0.56$ . The black arrow indicates the charge order wave vector taken to link different parts of the Fermi surface, while the red line shows the path between $k = (0, π)$ and $k = (π / 2, π / 2)$ where the self energy is computed.
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Figure 3
(a) Normal self-energy in the hidden-fermion model of Ref. [9]. The asymmetry in the antinodal region is well reproduced. As there is only one hidden fermion coupled to electrons, it is not possible to recover the splitting in the nodal region. The weight of the positive energy pole is also maximum for $k = (π / 2, π / 2)$ , in contrast to the numerical results shown in Fig. 2. (b) Normal self-energy obtained while considering hidden fermions due to CDW order with a wave vector along the $x$ axis. Due to the symmetry between $ε_{k + Q} = ε_{k - Q}$ at $k = (0, π)$ , we only have two visible poles with an asymmetry consistent with the CDMFT results. This symmetry is lost when going to the nodal region, and we observe a splitting of the poles. The change of sign of $ε_{k - Q}$ close to $k = (π / 2, π / 2)$ explains the significant weight for the pole at negative energy. We choose here $Q_{0} = 0.27 π$ , $z = 0.22$ , $Δ_{0} = 0.55$ , $V_{0} = 0.5$ , and a numerical broadening factor $i η = 0.03 i$ .
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Figure 4
(a) Electronic spectral function (full line) and self-energy (dotted line) in the pseudogap (red) and superconducting phase (blue) in the hidden-fermion model. Above $T_{c}$ , the self-energy is reduced to a single pole at $ω = ε_{k \pm Q} > 0$ and the resulting electronic spectral function shows two poles. There is no spectral weight at $ω = 0$ but there will be gapless states away from $k_{x} = 0$ , in contrast to experimental observations. (b) Electronic spectral function (full line) and self-energy (dotted line) in the pseudogap phase from a fractionalized PDW at $k = (0, π)$ . Green lines show effects of CDW fluctuations ( $Γ_{1}$ ), while the red lines show the effect of superconducting fluctuations ( $Γ_{2}$ ). Due to the superposition of SC and CDW in the fractionalized PDW idea, the electronic spectral function retains the four poles structure seen in the SC phase. Finite lifetimes for the particle-particle or particle-hole pairs have a much stronger dampening effect in the two poles closest to $ω = 0$ , which could explain the observed two-peak structure in CDMFT or the single occupied band in ARPES above $T_{c}$ . We choose here $Q_{0} = 0.27 π$ , $z = 0.22$ , $| Ψ_{0} | = V_{0} = 0.15$ , $Γ_{1} = Γ_{2} = 0.1$ , and a numerical broadening factor $i η = 0.03 i$ .
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Figure 5
Electronic spectral function at $ω = 0$ for the fractionalized PDW model. The white dotted line indicates the noninteracting Fermi surface, while the black dotted line indicates the surface of zeros of the Green's function. We see that the Fermi surface is washed out in the antinodal region. We choose here $Q_{0} = 0.27 π$ , $z = 0.22$ , $| Ψ_{0} | = 0.15$ , and a numerical broadening factor $i η = 0.03 i$ .
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Figure 6
(a) Anomalous self-energy in the hidden-fermion model of Ref. [9]. We obtain a pair of symmetric peaks with opposite spectral weight in the antinodal region. (b) Anomalous self-energy obtained while considering hidden fermions due to CDW order with a wave vector along the $x$ axis. Due to the symmetry between $ε_{k + Q} = ε_{k - Q}$ at $k = (0, π)$ , we only have two visible poles with opposite spectral weight. This symmetry is lost when going to the nodal region, and we observe a splitting of the poles. The change of sign of $Δ_{k - Q}^{f}$ close to $k = (π / 2, π / 2)$ explains the change of sign in the spectral weight associated to one of the poles. We choose here $Q_{0} = 0.27 π$ , $z = 0.22$ , $Δ_{0} = 0.55$ , $V_{0} = 0.5$ , and a numerical broadening factor $i η = 0.03 i$ as in the main text.
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