Self-energy flows in the two-dimensional repulsive Hubbard model

We study the two-dimensional repulsive Hubbard model by functional renormalization group methods, using our recently proposed channel decomposition of the interaction vertex. The main technical advance of this work is that we calculate the full Matsubara frequency dependence of the self-energy and the interaction vertex in the whole frequency range without simplifying assumptions on its functional form, and that the effects of the self-energy are fully taken into account in the equations for the flow of the two-body vertex function. At Van Hove filling, we find that the Fermi-surface deformations remain small at fixed particle density and have a minor impact on the structure of the interaction vertex. The frequency dependence of the self-energy, however, turns out to be important, especially at a transition from ferromagnetism to $d$-wave superconductivity. We determine non-Fermi-liquid exponents at this transition point.

Figure 1

Graphical representation of the parametrizations (9) and (11) of the interaction vertex with an $s$-wave form factor $f_1(x,y)=1$ in the three interaction channels and a $d$-wave form factor $f_2(x,y)=\cos x-\cos y$ in the pairing channel.

Figure 2

Illustration of momentum discretization for exchange propagators in 12 segments about $\mathbf{p}=(0,0)$ and $\mathbf{p}=(\pi ,\pi )$. Small circles mark the representative momentum associated with each segment. For detection of incommensurate antiferromagnetic (AFM) ordering tendencies, placement of data points along the coordinate axes is advisable.

Figure 3

Scale dependence of the effective saddle-point energy level ${\varepsilon }_{\mathrm{eff}}^{\Omega }(0,\pi )=\varepsilon (0,\pi )-{\Sigma }_{\Omega }[0,(0,\pi )]=-\mu -c_0+2c_2$ in the calculation considering $c_1$ and $c_2$. The flow is calculated at hopping ratio $t_2/t_1=0.45$ and for different values of the chemical potential $\mu /t_1=0,-0.0052,-0.0097,-0.011(*),-0.012,-0.015$ (bottom up). The choice $(*)$ leads to ${\varepsilon }_{\mathrm{eff}}^{{\Omega }_{*}}(0,\pi )=0$.

Figure 4

Flow of hopping corrections in the orthogonal projection scheme (thick lines) vs the local expansion scheme (thin lines). The dashed line corresponds to considering only the linear system from the second-order derivative of $\Sigma$ at the saddle point, the solid curve results from extending this system by two further equations as described in the text. Both schemes agree well except in a small parameter region about $t_2/t_1=0.35$, where the local expansion method misses the flow to interacting VHF. All energy scales are given in units of $t_1$.

Figure 5

Left: Correction parameters $c_1$ to $c_4$ at the stopping scale, displayed by the solid, dashed, dotted, and dashed-dotted lines, respectively. Right: Particle density corresponding to interacting VHF in the orthogonal projection scheme with 2 (solid line) and 4 (dashed line) hopping correction terms. The plot shows the difference to free VHF.

Figure 6

Influence of the moving Fermi line on the flow of the interaction vertex for systems at VHF. Tracing $\Sigma (0,\mathbf{p})$ leads to a small change in the stopping scale compared to neglecting self-energy (dashed line). We have calculated self-energy flows with two as well as with four hopping correction terms; this no longer produces a difference in the stopping scale, the two solid lines are practically indistinguishable. Symbols indicate most dominant ordering tendency: commensurate AFM (filled square), incommensurate AFM (open square), $d$-SC (circle), FM (diamond).

Figure 7

Second-order perturbation theory for the $Z_{\mathbf{p}}$ factor at VHF and momenta $\mathbf{p}=$ $(0,\pi )$, $(x_0,x_0)$, $(0,0)$, $(\pi ,\pi )$ [solid, dashed-dotted, dashed, dotted lines, respectively; $\mathbf{p}=(0,0),(\pi ,\pi )$ almost coincide in the first two plots]. The abscissa $x_0$ provides the intersection point of FS with the BZ diagonal.

Figure 8

Frequency dependence of $Im\Sigma$ for parameter values $t_2=0.355t_1$, $U=3t_1$. The function $\omega \mapsto \frac{1}{\omega }Im\Sigma [\omega ,(0,\pi )]$ is shown for different values of the scale parameter. All energies are measured in units of $t_1$.

Figure 9

Flow of the $Z_{\mathbf{p}}$ factors (inverse quasiparticle weights) for $\mathbf{p}=(0,\pi )$, $(0,0)$, $(\pi ,\pi )$ (solid, dashed, dotted lines, respectively) until reaching the varying stopping scales. The calculations for $t_2/t_1=0.1$ and $0.45$ include feedback of the full frequency and momentum dependencies of $Im\Sigma$. For $t_2/t_1=0.355$, the approximation of the momentum-independent feedback of $Im\Sigma [\omega ,(0,\pi \left)\right]$ is used.

Figure 10

Left: Comparison of stopping scales ${\Omega }_{*}$ at VHF for different vertex parametrizations. From the top down: (i) frequency-dependent vertex disregarding self-energy, (ii) frequency-independent vertex disregarding self-energy, (iii) frequency-dependent vertex including $Im\Sigma [\omega ,(0,\pi \left)\right]$. Dominant ordering tendencies: commensurate AFM (dashed line, filled square), incommensurate AFM (dotted line, open square), $d$-SC (solid line, circle), scattering instability (dashed-dotted line, triangle), FM (short dashes, diamond). Right: Values of largest couplings in the different channels, for setup (iii), at the stopping scale.

Figure 11

Small-frequency asymptotics of the self-energy at the Van Hove points. The discrete data show $Im\Sigma$ at different stages of the flow: $\Omega /t_1={10}^{-5},3\times {10}^{-5},{10}^{-4},3\times {10}^{-4},{10}^{-3},3\times {10}^{-3},{10}^{-2},{10}^{-1}$ (from the top down). In the frequency regime $10\Omega \lesssim \left|\omega \right|\ll t_1$, all curves coincide. This allows us to extract a (non-Fermi-liquid) frequency asymptotics $Im\Sigma [\omega ,(0,\pi )]\simeq {const\left|\omega \right|}^{\alpha }\mathrm{sgn}\omega$ with $\alpha =0.74$ (dashed line). For comparison, the dotted line shows the asymptotics with $\alpha =\frac{2}{3}$, which is not met by the present data.

Figure 12

RG flow of the ferromagnetic exchange function $\omega \mapsto M_{11}(\omega ,\mathbf{0})$ at different stages of the flow: $\Omega /t_1={10}^{-5},3\times {10}^{-5},{10}^{-4},{10}^{-3},{10}^{-2},{10}^{-1},{10}^0$ (from the top down viewed from $\omega /t_1={10}^{-5}$). The dashed line $\sim {\left|\omega \right|}^{-{\alpha }_2}$ represents the estimate for the asymptotic frequency behavior of $M_{11}(\omega ,\mathbf{0})$ with the exponent ${\alpha }_2$ in the region $0.13...0.2$, and the line uses ${\alpha }_2=0.13$.

Figure 13

Flow of the dominant instability in the AFM (left) and FM regimes (right) in different calculation schemes. We compare the flow including $Im\Sigma (\omega ,\mathbf{p})$ (solid and dashed lines) to several setups disregarding self-energy effects: frequency-independent vertex functions (dashed-dotted line) and frequency-dependent vertex functions (dots). The solid line corresponds to the full flow with feed-in of the momentum- and frequency-dependent $Im\Sigma (\omega ,\mathbf{p})$, the dashed one to a feed-in of $Im\Sigma [\omega ,(0,\pi \left)\right]$ only. At $t_2/t_1=0.45$, $Z_{(0,0)}>Z_{(0,\pi )}$ for most of the flow, which leads to the atypical situation that the momentum-independent feedback of $Im\Sigma [\omega ,(0,\pi \left)\right]$ slightly overestimates the true flow instead of underestimating it.

Figure 14

Approximation of momentum-independent feedback of $Im\Sigma (\omega ,\mathbf{q})$ to the flow. The results for several choices of $\mathbf{q}$ [solid line: $\mathbf{q}=(0,\pi )$, dashed line: $\mathbf{q}=(0,0)$, dots: $\mathbf{q}=(\pi ,\pi )$] share three common characteristics compared to neglect of self-energy (uppermost curve): the stopping scale gets (for intermediate $t_2/t_1$ substantially) suppressed; a $d$-SC dominated parameter region is recovered; the scattering process with nonzero frequency exchange is weakened. The flow with full $Im\Sigma$ feed-in is expected to be close to the approximate one with $\mathbf{q}=(0,\pi )$, but with a higher stopping scale. On the other hand, from the discussion in the text, the true stopping scale is supposed to be well below the ones obtained from choices $\mathbf{q}=(0,0)$ or $(\pi ,\pi )$. Color conventions as in Fig. 10.

Figure 15

Imaginary part of the $D_{22}$ exchange function. Left: Flow of its first frequency derivative. Right: The function $\omega \mapsto \frac{1}{\omega }Im{D}_{22}(\omega ,\mathbf{0})$ at the stopping scale ${\Omega }_{*}\approx 0.002t_1$.

Figure 16

Comparison of flows taking into account $Im{D}_{11}$ and $Im{D}_{22}$ or neglecting them. Left: Flow of the leading couplings, line conventions as in Fig. 10. Right: Flow of $Z_{(0,\pi )}$. Both flows are almost indistinguishable.

Figure 17

Overview about the various functional parametrizations for the transfer-frequency dependence of exchange propagators at low RG scales. On the example of the ferromagnetic exchange $\omega \mapsto M_{11}(\omega ,\mathbf{0})$, several Ansätze for the discrete RG data ($+$ symbols) are compared: A single Lorentzian determined from the small-frequency behavior (red dashed line) is appropriate for small frequencies $\left|\omega \right|\ll \Omega$. Likewise, the sum of two Lorentzians (green line in dashes and dots) is a compromise to the different behavior in different frequency regimes and not an accurate parametrization for a large-frequency range. The functional form (41) leaves the frequency exponent in the region of intermediate frequencies free and describes the RG data quite well (blue solid line). It decomposes into the two blue dotted lines. The parameter values are $t_2/t_1=0.341$, $U/t_1=3$.

Figure 18

Effective functional forms from the discretized frequency dependencies obtained with the RG flow for $\omega \mapsto {\omega }^{-1}Im\Sigma [\omega ,(0,\pi )]$ and $\omega \mapsto Re{D}_{22}(\omega ,\mathbf{0})$. The line conventions and further details are given in Fig. 17. The fit for ${\omega }^{-1}Im\Sigma$ uses the additional freedom provided by the Ansatz (42) instead of (41); the cyan dashed-dotted-dotted line shows the fit to the three-parameter Ansatz (43) and captures well the discrete data except for the large-frequency asymptotics. The corresponding fit parameters are given in Tables and . Data at frequency $\omega =0$ (not shown in the logarithmic plots) are very close to the values at the lowest frequencies given in the graphs, in agreement with the Lorentzian form at very small frequencies. The parameter values are $t_2/t_1=0.341$, $U/t_1=3$.