Quantitative functional renormalization group description of the two-dimensional Hubbard model

Using a leading algorithmic implementation of the functional renormalization group (fRG) for interacting fermions on two-dimensional lattices, we provide a detailed analysis of its quantitative reliability for the Hubbard model. In particular, we show that the recently introduced multiloop extension of the fRG flow equations for the self-energy and two-particle vertex allows for a precise match with the parquet approximation also for two-dimensional lattice problems. The refinement with respect to previous fRG-based computation schemes relies on an accurate treatment of the frequency and momentum dependences of the two-particle vertex, which combines a proper inclusion of the high-frequency asymptotics with the so-called truncated unity fRG for the momentum dependence. The adoption of the latter scheme requires, as an essential step, a consistent modification of the flow equation of the self-energy. We quantitatively compare our fRG results for the self-energy and momentum-dependent susceptibilities and the corresponding solution of the parquet approximation to determinant quantum Monte Carlo data, demonstrating that the fRG is remarkably accurate up to moderate interaction strengths. The presented methodological improvements illustrate how fRG flows can be brought to a quantitative level for two-dimensional problems, providing a solid basis for the application to more general systems.

Figure 1

Two-particle vertex decomposed as in Eq. (10) in the different $s$-wave channel contributions at zero bosonic frequency as a function of fermionic frequencies: (a) $1\ell$ including the katanin correction vs (b) multiloop data, for $U=2$ and $1/T=5$.

Figure 2

Antiferromagnetic susceptibility ${\chi }_{\mathrm{AF}}(i\omega =0)$ defined in Eq. (5) as a function of the bare interaction $U$ for $1/T=5$.

Figure 3

Right-hand side of the Schwinger-Dyson equation for the self-energy (12), illustrating the contributions of the different channels, where solid (dashed) lines carry spin up (down). The first diagram on the rhs can be calculated using the convolution theorem and fast-Fourier-transform algorithms. The other contributions can be determined by first combining the two-particle reducible vertices ($\mathrm{\Phi }$) and the bare vertex (dot) through the propagator pair of the corresponding channel (red). Finally, each diagram is closed by a propagator (black) through the direct summation over frequency and momentum.

Figure 4

Illustration of the self-energy flows in (a) the fRG*, restricted to the part with ${\mathrm{\Phi }}_{\overline{ph}}$, and (b) the fRG. The rhs shows two exemplary differentiated diagrams contributing to the self-energy flow at third order, where solid (dashed) lines carry spin up (down) and the diagonal dash symbolizes a scale-differentiated bare propagator. In the fRG*, two bubbles from the same channel (colored) are combined and then closed with the black line. By contrast, in the fRG, the second diagram requires one to insert a $\overline{ph}$ (orange) into a $pp$ (blue) bubble, before closing with the differentiated propagator (black).

Figure 5

(a) Real and (b) imaginary parts of the self-energy as obtained by the conventional fRG (blue) and the fRG* flow (red), together with the respective postprocessed results (dashed lines), for $U=2$ and $1/T=5$. Within the fRG*, the postprocessed (dashed red) ones lie exactly on top of the fRG* flow results (red solid).

Figure 6

Imaginary part of the self-energy at (a) the nodal and (b) the antinodal point, as obtained by the fRG (blue) and fRG* (red), with and without self-energy iterations, for $U=2$ and $1/T=5$. The inset shows the relative difference caused by neglected self-energy iterations.

Figure 7

Frequency dependence of the AF susceptibility ${\chi }_{\mathrm{AF}}\left(i\omega \right)$ as obtained by the conventional fRG (blue) and the fRG* flow (red), with (solid line) and without (dotted line) $\mathrm{\Sigma }$ iteration, for $U=2$ and $1/T=5$.

Figure 8

Antiferromagnetic susceptibility ${\chi }_{\mathrm{AF}}$ as a function of $U$, as obtained by the fRG* (red), the PA (gray), and the DQMC (black), for $1/T=5$. The inset shows the relative difference.

Figure 9

Magnetic susceptibility ${\chi }_{\mathrm{M}}(\mathbf{q},i\omega =0$) as obtained by the fRG* (red), the PA (gray), and the DQMC (black) for $U=2$ and $1/T=5$. The inset shows the correlation length $\xi$ extracted from ${\chi }_{\mathrm{M}}(\mathbf{q},i\omega =0)$ as a function of $U$ for $1/T=5$ (see the text for details of the fitting procedure).

Figure 10

Frequency dependence of the AF susceptibility ${\chi }_{\mathrm{AF}}\left(i\omega \right)$ as obtained by the fRG* (red), the PA (gray), and the DQMC (black) for $U=2$ and $1/T=5$.

Figure 11

(a) Compressibility $\kappa$, (b) charge density wave ${\chi }_{\mathrm{CDW}}={\chi }_{\mathrm{SC},s}$, and (c) superconducting susceptibility ${\chi }_{\mathrm{SC},d}(i\omega =0)$ as a function of $U$, as obtained by the fRG* (red), the PA (gray), and the DQMC (black), for $1/T=5$.

Figure 12

(a) Antiferromagnetic susceptibility ${\chi }_{\mathrm{AF}}$, (b) compressibility $\kappa$, (c) charge density wave ${\chi }_{\mathrm{CDW}}={\chi }_{\mathrm{SC},s}$ (see Appendix pp2), and (d) $d$-wave superconducting susceptibility ${\chi }_{\mathrm{SC}}$ as a function of $U$, as obtained by the fRG* with (solid lines) and without (dashed lines) vertex corrections, for $1/T=5$.

Figure 13

Imaginary part of the self-energy at the (a) first frequency and nodal point, (b) first frequency and antinodal point, (c) second frequency and nodal point, and (d) second frequency and antinodal point as a function of $U$, as obtained by the fRG* (red), the PA (gray), and the DQMC (black), for $1/T=5$.

Figure 14

Imaginary part of the self-energy at (a) the nodal and (b) the antinodal point, as obtained by the fRG* (red), the PA (gray), and the DQMC (black), for $U=2$ and $1/T=5$. The inset shows the relative difference of the fRG* with respect to the PA (gray) and DQMC (black).

Figure 15

(a) Real and (b) imaginary parts of the self-energy, as obtained by the fRG* (red), the PA (gray), and the DQMC (black), for $U=2$ and $1/T=5$.

Figure 16

Double occupancy (DOC) as a function of $U$, as obtained by the fRG* (red), the PA (gray), and the DQMC (black) for $1/T=5$.

Figure 17

Magnetic susceptibility ${\chi }_{\mathrm{M}}(i\omega =0)$ as obtained by the fRG* (red), the PA (gray), and the DQMC (black) for $U=2$, $t^{\prime }=-0.2$, $1/T=5$, and different values of the doping resulting from (a) $\mu =-0.35$, (b) $\mu =-0.7$, (c) $\mu =-1.4$, and (d) $\mu =-2$.

Figure 18

Compressibility $\kappa$, charge density wave ${\chi }_{\mathrm{CDW}}$, and superconducting susceptibility ${\chi }_{\mathrm{SC}}$ in $s$ and $d$ waves as obtained by the fRG* (red), the PA (gray), and the DQMC (black) for $U=2$, $1/T=5$, and different values of the doping $\delta$.

Figure 19

(a) Antiferromagnetic susceptibility ${\chi }_{\mathrm{AF}}$, (b) compressibility $\kappa$, and (c) charge density wave ${\chi }_{\mathrm{CDW}}={\chi }_{\mathrm{SC},s}$ as a function of $U$ obtained by the fRG* for $1/T=5$ and half filling. Postprocessed (closed symbols) and flowing results (open symbols) are shown.

Figure 20

(a) Real and (b) imaginary parts of the self-energy as obtained by the conventional fRG (blue) and fRG* (red), with and without self-energy iteration, respectively, for $U=2$ and $1/T=5$ (at half filling).

Figure 21

Imaginary part of the self-energy at (a) the nodal and (b) the antinodal point, as obtained by fRG (blue) and fRG* flow (red), together with the respective postprocessed results (dashed lines), for $U=2$ and $1/T=5$ (at half filling). Within the fRG*, the postprocessed results (red dashed lines) lie exactly on top of the fRG* flow results (red solid lines).