Strongly correlated electrons: Analytic mean-field theories with two-particle self-consistency

A two-particle self-consistency is rarely part of mean-field theories. It is, however, essential for avoiding spurious critical transitions and unphysical behavior. We present a general scheme for constructing analytically controllable approximations with self-consistent equations for the two-particle vertices based on the parquet equations. We explain in detail how to reduce the full set of parquet equations so as not to miss quantum criticality in strong coupling. We further introduce a decoupling of convolutions of the dynamical variables in the Bethe-Salpeter equations to make them analytically solvable. We connect the self-energy with the two-particle vertices to satisfy the Ward identity and the Schwinger-Dyson equation and discuss the role of the one-particle self-consistency in making the approximations reliable in the whole spectrum of the input parameters. Finally, we demonstrate the general construction on the simplest static approximation that we apply to the Kondo behavior of the single-impurity Anderson model.

Figure 1

The Bethe-Salpeter equation (6) for the irreducible vertex ${\mathrm{\Lambda }}_{\uparrow \downarrow }$ with the integral kernel, the reducible vertex ${\mathcal{K}}_{\uparrow \downarrow }$ from the the electron-hole channel. We used the four-vector notation, $k=(\mathbf{k},i{\omega }_n)$ for the fermionic and $Q=(\mathbf{Q},i{\nu }_m)$ for the bosonic variables. Both sides have the same external labels and $Q$ is the internal variable. The vertical wavy line is the bare Hubbard interaction.

Figure 2

The reduced Bethe-Salpeter equation (7) for the reducible vertex ${\mathcal{K}}_{\uparrow \downarrow }$ from the the electron-hole channel. Both sides have the same external labels and $k^{\prime \prime }$ is the internal (integration) variable.

Figure 3

Graphical representation of the odd part of the self-energy calculated from the normal part of the irreducible vertex in the electron-hole channel, given by Eq. (11). The odd part of the self-energy is anomalous in that it causes spin flip. It is an order parameter that is nonzero only in the ordered phase.

Figure 4

Schwinger-Dyson equation from which the normal part with even symmetry with respect to the symmetry-breaking field is calculated, given by Eq. (12). The full vertex is the sum of the reducible vertex and irreducible vertex from the electron-hole channel, ${\mathrm{\Gamma }}_{\sigma \overline{\sigma }}={\mathrm{\Lambda }}_{\sigma \overline{\sigma }}+K_{\sigma \overline{\sigma }}$.

Figure 5

Spectral function at $U/\mathrm{\Delta }=8$ at half filling calculated within the static approximation, given by Eq. (35) (Parquet), Ref. [21] (old Parq.), and NRG in energy units $\mathrm{\Delta }$.

Figure 6

Spectral function calculated within the static approximation with the thermodynamic propagator, given by Eq. (37) (noSC), and the full propagator, given by Eq. (43) (SC), in the reduced parquet equations compared with NRG for several values of the interaction in energy units $\mathrm{\Delta }$.

Figure 7

Half width of the Kondo-Suhl quasiparticle peak at half maximum (HWHM) and $Z$ factor calculated with one-particle self-consistency (SC Parquet) and with the thermodynamic propagator (Parquet) compared with the NRG result in energy units $\mathrm{\Delta }$.

Figure 8

Spectral function calculated with the full spectral self-energy, given by Eq. (36) (solid line), and with the asymptotic one, given by Eq. (53) (dashed line), for weak and strong interaction in energy units $\mathrm{\Delta }$.

Figure 9

(a) Real and (b) imaginary parts of the full (solid line) and asymptotic (dashed line) self-energies for weak and strong interaction in energy units $\mathrm{\Delta }$.

Figure 10

Spectral function away from half filling in energy units $\mathrm{\Delta }$, for values of the normalized chemical potential $\mu -\frac{U}{2}=0,-2\mathrm{\Delta }$, and $-4\mathrm{\Delta }$, respectively.

Figure 11

(a) The full spectral function and (b) for spin up in the external magnetic field in energy units $\mathrm{\Delta }$.

Figure 12

Spectral function for spin up compared with the NRG result for several strengths of the magnetic field in energy units $\mathrm{\Delta }$.

Figure 13

Negative logarithm of the dimensionless Kondo scale $a_K$ calculated from its definition, given by Eq. (27) (Parquet), and from the asymptotic form, given by Eq. (54), as a function of (a) the normalized chemical potential at zero magnetic field and (b) the magnetic field at half filling in energy units $\mathrm{\Delta }$. The Bethe-ansatz result (exact) was added.

Figure 14

Irreducible vertex from the electron-hole channel (effective interaction) as a function of (a) the normalized chemical potential and (b) the magnetic field in energy units $\mathrm{\Delta }$. Both full (solid line) and asymptotic (dashed line) solutions from Eqs. (35) and (49), respectively, are plotted.