Time-dependent Gutzwiller theory of pairing fluctuations in the Hubbard model

We present a method to compute pairing fluctuations on top of the Gutzwiller approximation (GA). Our investigations are based on a charge-rotational invariant GA energy functional which is expanded up to second order in the pair fluctuations. Equations of motion for the fluctuations lead to a renormalized ladder-type approximation. Both spectral functions and corrections to static quantities, such as the ground-state energy, are computed. The quality of the method is examined for the single-band Hubbard model, where we compare the dynamical pairing correlations for $s$- and $d$-wave symmetries with exact diagonalizations. We find a significant improvement with respect to analogous calculations done within the standard Hartree-Fock ladder approximation. The technique has potential applications in the theory of Auger spectroscopy, superconductivity, and cold atom physics.

Figure 1

(Color online) Left panel: Excitation energies for two-particle addition and removal computed for the two-site model with two particles. Right panel: Matrix element of the local (lower curves) and intersite (upper curves) pairing operators. The underlying GA saddle points are paramagnetic (p) (dashed-dotted lines) and antiferromagnetic (dashed lines). The critical value $u=\sqrt{2}-1$ of the corresponding transition is indicated by the vertical dotted line.

Figure 2

(Color online) Double occupancy (left panel) and ground-state energy (right panel) of the two-site Hubbard model computed within HF, BLA, TDGA, and exact diagonalization. Only paramagnetic ground states are considered.

Figure 3

(Color online) Top panel: Sketch of the 18-site cluster and corresponding boundary conditions. Bottom panels: Bond singlet pairing correlations on the 18-site lattice for $U/t=4$ and 10. The bars at the TDGA symbols point to the bare GA value of the singlet correlations. The separation between bond singlets is defined as the shortest distance between their centers. Taking the singlet “$s$” as reference, it should be noted that there exist two parallel singlet configurations “$a$” with distance $R_s-R_a=2$. The singlets “$b$” and “$s$” form a perpendicular configuration.

Figure 4

(Color online) Imaginary part of the (two-particle addition) pairing correlation function in Eq. (37) for on-site pairing (a) and extended $s$-wave (b) and $d$-wave (c) symmetries. Results are for 10 particles on an 18-site cluster and $U/t=10$ for which the underlying mean-field solution in BLA and TDGA is paramagnetic. The deltalike excitations are convoluted by Lorentzians with width $\epsilon =0.2t$. The arrows in the upper panel point to the chemical potentials obtained within the various methods. The insets detail the frequency evolution of the weight at small energies.

Figure 5

(Color online) Main panel: Local part of the effective interaction $V_{ii,ii}$ [see Eq. (24)] for the 18-site cluster. The solid and dashed lines refer to the half-filled cluster with paramagnetic and spin-density wave ground state, respectively. The dashed-dotted line is for the ten-particle system with a paramagnetic ground state. Upper left inset: Effective interaction between pairs on nearest-neighbor sites $V_{ii,jj}$. Lower right inset: Effective interaction $V_{ii,ij}$ between a local pair on site $R_i$ and an intersite pair on the bond defined by $R_i$ and $R_j$. $V_{ii,ij}>(<)0$ for $S_i^z<(>)0$ (solid and dashed lines, respectively).

Figure 6

(Color online) Imaginary part of the (two-particle addition) pairing correlation function [Eq. (37)] for on-site pairing (a) and extended $s$-wave (b) and $d$-wave (c) symmetries. Results are for a half-filled 18-site cluster and $U/t=10$. The underlying mean-field solution in BLA and TDGA is a spin-density wave. The deltalike excitations are convoluted by Lorentzians with width $\epsilon =0.5t$. The insets detail the frequency evolution of the two-particle spectral weight.

Figure 7

(Color online) Ground-state energy of the half-filled $4\times 4$ Hubbard model computed within GA, TDGA in the particle-hole channel, TDGA in the particle-particle channel, and exact diagonalization.