Complementary views on electron spectra: From fluctuation diagnostics to real-space correlations

We study the relation between the microscopic properties of a many-body system and the electron spectra, experimentally accessible by photoemission. In a recent paper [O. Gunnarsson et al., Phys. Rev. Lett. 114, 236402 (2015)], we introduced the “fluctuation diagnostics” approach to extract the dominant wave-vector-dependent bosonic fluctuations from the electronic self-energy. Here, we first reformulate the theory in terms of fermionic modes to render its connection with resonance valence bond (RVB) fluctuations more transparent. Second, by using a large-$U$ expansion, where $U$ is the Coulomb interaction, we relate the fluctuations to real-space correlations. Therefore, it becomes possible to study how electron spectra are related to charge, spin, superconductivity, and RVB-like real-space correlations, broadening the analysis of an earlier work [J. Merino and O. Gunnarsson, Phys. Rev. B 89, 245130 (2014)]. This formalism is applied to the pseudogap physics of the two-dimensional Hubbard model, studied in the dynamical cluster approximation. We perform calculations for embedded clusters with up to 32 sites, having three inequivalent $\mathbf{K}$ points at the Fermi surface. We find that as $U$ is increased, correlation functions gradually attain values consistent with an RVB state. This first happens for correlation functions involving the antinodal point and gradually spreads to the nodal point along the Fermi surface. Simultaneously, a pseudogap opens up along the Fermi surface. We relate this to a crossover from a Kondo-type state to an RVB-like localized cluster state and to the presence of RVB and spin fluctuations. These changes are caused by a strong momentum dependence in the cluster bath couplings along the Fermi surface. We also show, from a more algorithmic perspective, how the time-consuming calculations in fluctuation diagnostics can be drastically simplified.

Figure 1

Diagrammatic representation of the self-energy $\mathrm{\Sigma }$ in Eq. (15) (top) in terms of the vertex function $F$ in Eq. (13) (bottom). The quantity $A$ in Eq. (20) is obtained by multiplying by the four “legs” and by summing over the internal Matsubara fermion (${\nu }^{\prime }$) and boson ($\omega$) frequencies. $B,C$, and $D$ in Eq. (22) are obtained by summing over the boson wave vector $\mathbf{Q}$ ($B$), the fermion wave vector ${\mathbf{K}}^{\prime }$ ($C$), or both ($D$).

Figure 2

Schematic picture of an RVB state.

Figure 3

Real parts of $D_{\uparrow \downarrow }(\mathbf{K};\nu ),B_{\uparrow \downarrow }(\mathbf{K},{\mathbf{K}}^{\prime };\nu ),C_{\mathrm{sp}}(\mathbf{K},\mathbf{Q};\nu )$, and $A_{\uparrow \downarrow }(\mathbf{K},{\mathbf{K}}^{\prime },\mathbf{Q};\nu )$ of the four-level model for $\mathbf{K}=(\pi ,0),{\mathbf{K}}^{\prime }=(0,\pi )$, and $\mathbf{Q}=(\pi ,\pi )$ as well as $L\left(\mathbf{K}\right)$. The inset shows the numbering of the sites in Eq. (44). The parameters are $V=-0.1$ eV, $\mathrm{\Delta }U>0,\beta =60{\mathrm{eV}}^{-1}$, and $\nu =\pi /\beta$.

Figure 4

Contributions to the self-energy of the four-level model for $\mathbf{K}=(\pi ,0)$, labeled according to Eqs. (19) and (20) as well as the sum (“Sum”) of these contributions for $\mathrm{\Delta }U=0.03$ (a) and $\mathrm{\Delta }U=-0.03$ (b). The figure illustrates the great importance of the sign of $\mathrm{\Delta }U$ both for the shape of $\mathrm{\Sigma }\left(\nu \right)$ and for the relative contributions of the different terms Eq. (21). The term for $\mathbf{Q}=(\pi ,\pi )$ and ${\mathbf{K}}^{\prime }=(0,\pi )$ gives the main contribution to $\mathrm{\Sigma }$ for $\mathrm{\Delta }U>0$, but is almost zero for $\mathrm{\Delta }U<0$. The parameters are $V=-0.1$ eV, $U=1.6$ eV, and $\beta =60{\mathrm{eV}}^{-1}$.

Figure 5

Real parts of $B_{\uparrow \downarrow }(\mathbf{K},{\mathbf{K}}^{\prime };\nu ),C_{\mathrm{sp}}(\mathbf{K},\mathbf{Q};\nu )$, and $D_{\uparrow \downarrow }(\mathbf{K};\nu )$ for $\mathbf{K}=(\pi ,0)$ and for ${\mathbf{K}}^{\prime }=(0,\pi )$. The figure also shows $\mathrm{Re}{A}_{\uparrow \downarrow }(\mathbf{K},{\mathbf{K}}^{\prime },\mathbf{Q};\nu )$ for $\mathbf{Q}=(\pi ,\pi )$ as well as $L\left(\mathbf{K}\right)$. The parameters of the corresponding DCA calculation are $N_c=4,t=-0.25$ eV, $\beta =60{\mathrm{eV}}^{-1}$, and $\nu =\pi /\beta$. Note that the minor difference between the equivalent values of $B_{\uparrow \downarrow }$ and $C_{\mathrm{sp}}$ visible for $U>1$ is a consequence of the finite numerical precision of the calculation.

Figure 6

DCA spectra for $U=1.0$, 1.1, and 1.2 eV. The parameters are $N_c=4,t=-0.25$ eV, and $\beta =60{\mathrm{eV}}^{-1}$.

Figure 7

$\mathrm{Re}{B}_{\uparrow \downarrow }(\mathbf{K},{\mathbf{K}}^{\prime };\nu )$ and $\mathrm{Re}{D}_{\uparrow \downarrow }(\mathbf{K};\nu )$ for $\mathbf{K}=(\pi ,0)$ (upper panel, red) and $(\pi /2,\pi /2)$ (lower panel, blue) and for ${\mathbf{K}}^{\prime }=\mathbf{K}+(\pi ,\pi )$. The parameters are $N_c=8,t=-0.25$ eV, $\beta =60{\mathrm{eV}}^{-1}$, and $\nu =\pi /\beta$.

Figure 8

DCA spectral functions as a function of $\mathbf{K}$ for $U=1.4$ (a), 1.6 (b), and 1.7 (c). The parameters are $N_c=8,t=-0.25$ eV, and $\beta =60$ (${\mathrm{eV})}^{-1}$.

Figure 9

Spectral functions as a function of $\mathbf{K}$ for $U=0.9$ eV (a), 1.0 (b), and 1.1 eV (c). The parameters are $N_c=32,t=-0.25$ eV, and $\beta =60$ (${\mathrm{eV})}^{-1}$.

Figure 10

$\mathrm{Re}D(\mathbf{K};\nu )$ (left panel), $\mathrm{Re}B(\mathbf{K},{\mathbf{K}}^{\prime };\nu )$ (middle panel), and $L\left(\mathbf{K}\right)$ (right panel) for $\mathbf{K}=(\pi ,0)$ (red solid lines), $\mathbf{K}=(3\pi /4,\pi /4)$ (green long-dashed lines), $\mathbf{K}=(\pi /2,\pi /2)$ (blue short-dashed lines), and ${\mathbf{K}}^{\prime }=\mathbf{K}+(\pi ,\pi )$. The parameters of the corresponding DCA calculation are $N_c=32,t=-0.25$ eV, and $\beta =60$ (${\mathrm{eV})}^{-1}$, and $\nu =\pi /\beta$ is assumed to be much smaller than $\left|U\right|/2$.