Intrinsic cluster-shaped density waves in cellular dynamical mean-field theory

It is well known that cellular dynamical mean-field theory (CDMFT) leads to the artificial breaking of translation invariance. In spite of this, it is one of the most successful methods to treat strongly correlated electrons systems. Here, we investigate in more detail how this broken translation invariance manifests itself. This allows us to disentangle artificial broken translation invariance effects from the genuine strongly correlated effects captured by CDMFT. We report artificial density waves taking the shape of the cluster—cluster density waves—in all our zero temperature CDMFT solutions, including pair density waves in the superconducting state. We discuss the limitations of periodization regarding this phenomenon, and we present mean-field density-wave models that reproduce CDMFT results at low energy in the superconducting state. We then discuss how these artificial density waves help the agreement of CDMFT with high temperature superconducting cuprates regarding the low-energy spectrum, in particular for subgap structures observed in tunneling microscopy. We relate these subgap structures to nodal and antinodal gaps in our results, similar to those observed in photoemission experiments. This fortuitous agreement suggests that spatial inhomogeneity may be a key ingredient to explain some features of the low-energy underdoped spectrum of cuprates with strongly correlated methods. This work deepens our understanding of CDMFT and clearly identifies signatures of broken translation invariance in the presence of strong correlations.

Figure 1

$2\times 2$ tiling realized by the self-energy of the CDMFT lattice Green's function ${\mathrm{G}}^{\text{L}}\left(\tilde{\mathrm{k}}\right)$. This pattern is also the one of bond-centered charge density waves with first-neighbor hoppings $t_{\widehat{\mathbf{x}}}$ and $t_{\widehat{\mathbf{y}}}$, respectively, oscillating according to wave vectors ${\mathrm{Q}}_x=\pi \widehat{\mathrm{x}}$ and ${\mathrm{Q}}_y=\pi \widehat{\mathrm{y}}$. We call the density waves following this pattern cluster density waves (clus.-DW).

Figure 2

(a) Order parameter $\langle {\widehat{t}}_Q\rangle$ of the bond-centered charge density waves (CDW) as a function of doping $p$ in the normal solution (no spontaneously broken symmetry). This order parameter nearly scales with the zero-frequency nearest-neighbor (nn) cluster self-energy $\frac{1}{8}\left|{\mathrm{\Sigma }}_{\text{nn}}^c(z=0)\right|$ (dashed line) and with its high-frequency second moment $\frac{1}{256}{\left|{\omega }^2{\mathrm{\Sigma }}_{\text{nn}}^c\left(i\omega \right)\right|}_{\omega \to \infty }$ (dotted line) where ${\mathrm{\Sigma }}_{\text{nn}}^c\left(z\right)={\mathrm{\Sigma }}_{\mathrm{R}=0,{\mathrm{R}}^{\prime }=\widehat{\mathrm{x}}}^c\left(z\right)$. (b) Order parameters for superconductivity $\langle \widehat{\mathrm{\Delta }}\rangle$ (SC), bond-centered charge density waves $\langle {\widehat{t}}_Q\rangle$ and pair density waves $\langle {\widehat{\mathrm{\Delta }}}_Q\rangle$ (PDW) as a function of doping $p$ in the superconducting solution. Here it is compared to the anomalous nearest neighbor self-energy using $\frac{1}{8}|{\mathrm{\Sigma }}_{\text{nn}}^{c,\text{ano}},\widehat{\mathrm{x}}(z=0)|$ (dashed line) and $\frac{1}{256}{\left|{\omega }^2{\mathrm{\Sigma }}_{\text{nn}}^{c,\text{ano}}\left(i\omega \right)\right|}_{\omega \to \infty }$ (dotted line) scaled to fit in the plot.

Figure 3

For a broadening $\eta =0.02$: (a) Density of states $N\left(\omega \right)$ in the superconducting CDMFT solution at doping $p=0.08$. The superconducting gap presents subgap structures shown by gray arrows. (b) Density of states for coexisting superconductivity and density waves in the mean-field model (20) with mean-field values $D=0.07$ for superconductivity, $B=0.2$ for charge density waves, and $P=0.125$ for pair density waves, at chemical potential $\mu =-0.742$ ($p=0.08$). Subgap structures are similar to those in CDMFT. (c) Density of states for $d$-wave mean-field superconductivity alone, with $B=P=0, D=0.07$ and without subgap structures. (d) Fermi surface (G-periodized spectral weight at $\omega =0$) for the same superconducting CDMFT solution, showing extra nodes at symmetrical position with respect to the Bragg planes of the superlattice (dotted lines). (e) Fermi surface for the mean-field model with superconductivity and density waves (20). Extra nodes are similar to those in CDMFT. (f) Fermi surface for the mean-field model with $d$-wave superconductivity alone, without extra nodes. (g) $\mathrm{k}$-resolved gap (15) for the superconducting CDMFT solution, displaying three sign changes at symmetrical copies of the nodes. (h) $\mathrm{k}$-resolved gap for the mean-field model with superconductivity and density waves, showing a triple sign change similar to those in CDMFT. (i) $\mathrm{k}$-resolved gap for the mean-field model with $d$-wave superconductivity alone, with only one sign change.

Figure 4

(a) Density of state in the superconducting CDMFT solution [same as in Fig. 3 with a closeup of the subgap structure], for three values of Lorentzian broadening $\eta =0.001$, 0.02, and 0.1. (b) G-periodized spectral weight at the energy of the subgap structures $\omega =\pm 0.07$ (left and right), and at the Fermi energy $\omega =0$ (center), for $\eta =0.001$, (c) $\eta =0.02$, and (d) $\eta =0.1$. The dotted lines denote the reduced Brillouin zone boundary. Note that the central panel is the same as Fig. 3. We know that the superconducting gap goes from $\omega =-0.25$ to $\omega =0.25$ from comparison with the density of state in the normal CDMFT solution and a study of the anomalous part of the Green's function as a function of frequency.

Figure 5

Gaps as a function of doping. (a) Pseudogap (PG), superconducting gap (SCg), and subgap structure (SGS) found in the density of states of the normal and superconducting CDMFT solutions for the unperiodized lattice Green's functions. (b) An example of how the gaps are identified for doping $p=0.08$ in the density of states $N\left(\omega \right)$ of the normal solution and (c) in the superconducting solution. The gaps are chosen as the distance between two minima in the second derivative of $N\left(\omega \right)$. (d) Nodal and an antinodal gap identified in the $\mathrm{k}$-dependant spectral weight obtained through G periodization, and following the same doping dependence as the SCg and SGS. (e) The black curve gives the position of the gap in the Brillouin zone, identified as the smallest gap along a given direction $\theta$ defined from $(\pi ,\pi )$. The angle $\theta =0$ corresponds to the nodal direction. (f) Along three directions $\theta$ in $\mathrm{k}$ space (red, green, and golden arrows), results for $A(\mathrm{k},\omega )$ and identification of the smallest gap. (g) Smallest gaps as a function of $\theta$ (black), compared to the expected $d$-wave dependence (dotted). The local maximum near $\theta =0$ is identified as the nodal gap, the maximal value near $\theta =\frac{\pi }{4}$ as the antinodal gap. The gap returns to zero in between, which is consistent with the extra nodes seen in Figs. 3, 4, and 4.

Figure 6

Fortuitous agreement with low temperature STM gaps at various dopings (data from Ref. [83]) showing low-energy subgap structures similar to those found in the gaps of our superconducting CDMFT solutions at zero temperature.

Figure 7

Position of the first Bragg planes (dashed lines) relative to the Fermi surface for half filling (red), $p=0.125$ (black), and $p=0.25$ (blue). For doping $p=0.125$ (black), the position of possible extra nodes is illustrated on the corresponding copies of the Fermi surface (thin lines and dotted line). These copies are given by $t(\mathrm{k}+{\mathrm{Q}}_x), t(\mathrm{k}+{\mathrm{Q}}_y)$, and $t(\mathrm{k}+{\mathrm{Q}}_x+{\mathrm{Q}}_y)$ (a) for ${\mathrm{Q}}_x=(1/4,0)2\pi$ and ${\mathrm{Q}}_y=(0,1/4)2\pi$, as in cuprates (the inset shows a representation of the bidirectional $d$-form-factor density waves) and (b) for ${\mathrm{Q}}_x=(1/2,0)2\pi$ and ${\mathrm{Q}}_y=(0,1/2)2\pi$, as in CDMFT (the inset shows the $2\times 2$ cluster tiling).