Feedback of superconducting fluctuations on charge order in the underdoped cuprates

Metals interacting via short-range antiferromagnetic fluctuations are unstable to sign-changing superconductivity at low temperatures. For the cuprates, this leading instability leads to the well-known $d$-wave superconducting state. However, there is also a secondary instability to an incommensurate charge-density wave, with a predominantly $d$-wave form factor, arising from the same antiferromagnetic fluctuations. Recent experiments in the pseudogap regime of the hole-doped cuprates have found strong evidence for such a charge-density wave order and, in particular, the predicted $d$-wave form factor. However, the observed wave vector of the charge order differs from the leading instability in Hartree-Fock theory, and is that of a subleading instability. In this paper, we examine the feedback of superconducting fluctuations on these different charge-density wave states, and find that over at least a small temperature window, they prefer the experimentally observed wave vector.

Figure 1

Generic Fermi surface for hole-doped cuprates, with the filled states shaded in blue. Solid circles, numbered $1,...,8$, represent hot spots where the Fermi surface intersects the magnetic Brillouin zone for a $(\pi ,\pi )$ SDW. The purple arrows represent the CDW-a wave vectors, which closely resemble the experimentally observed wave vector, while the green arrows represent the CDW-b wave vectors, which arise as the leading instability in a HF calculation. The Fermi velocity, $(v_x,v_y)$, is shown at the hot spot 1. The wave vectors of CDW-a are $(\pm Q_0,0)$ and $(0,\pm Q_0)$, while those of CDW-b are $(Q_0,\pm Q_0)$. The computation in the present paper applies for a range of values of $Q_0$, and not just when it is equal to the separation between hot spots as shown above; for different $Q_0$ we have to consider a corresponding set of 8 points around the Fermi surface, and our computations proceed unchanged.

Figure 2

Plot of the smallest eigenvalue (a) ${\lambda }_S/{\chi }_0$ and (b) ${\lambda }_C/{\chi }_0$ as a function of $Q_x$ and $Q_y$. We use a band structure with $t_1=1.0,t_2=-0.32,t_3=0.128,\mu =-1.11856$ (corresponding to the same Fermi surface used in Ref. [31]). The other parameters are $\xi =2,T=0.1,\delta =1/4$, and $L=20$. For the pair-density wave, the global minimum is located at $\mathbf{Q}=0$ while for the charge order, it is located at $\mathbf{Q}=(Q_0,Q_0)$ (CDW-b). In (b), note the valleys extending from $(Q_0,Q_0)$ to $(Q_0,0)$ and $(0,Q_0)$ (CDW-a)—the latter are local, but not global, minima.

Figure 3

Feynman diagrams representing the various 4-point functions that contribute to different terms in $S_{eff}$. The solid internal lines carry different hot-spot indices $\left(\mu ,\nu ,\rho ,\delta \right)$ depending on the term being evaluated. The dashed, dotted, and double lines represent ${\Phi }^a,{\Phi }^b$, and $\Psi$, respectively. The individual diagrams are labeled (a) $I_{\mu \nu \rho \delta }$, (b) $J_{\mu \nu \rho \delta }$, (c) $K_{\mu \nu \rho \delta }$, (d) $L_{\mu \nu \rho \delta }$, (e) $M_{\mu \nu \rho \delta }$, (f) $N_{\mu \nu \rho \delta }$, (g) $P_{\mu \nu \rho \delta }$, (h) $Q_{\mu \nu \rho \delta }$.

Figure 4

Feynman diagram representing the 3-point function, $Y_{\mu \nu \rho }$, between ${\Phi }^a$ (dashed lines) and ${\Phi }^b$ (dotted line). The solid internal lines carry different hot-spot indices $\left(\mu ,\nu ,\rho \right)$.

Figure 5

Contracting the $\Psi$ fields (solid double lines) in (a) $M_{\mu \nu \rho \delta }$, (b) $N_{\mu \nu \rho \delta }$ renormalize the $|{\Phi }^a{|}^2$ term, and (c) $P_{\mu \nu \rho \delta }$, (d) $Q_{\mu \nu \rho \delta }$ renormalize the $|{\Phi }^b{|}^2$ term.

Figure 6

Absolute values of diagrams contributing to $u_a$ and $w_a$ as a function of temperature $T$ for (a), (b) different values of $\kappa$ but fixed $\alpha =10.0$. Other parameters are $\Lambda =2.0$ and $v_x=0.5$. Note the almost perfect agreement of the analytical result for $\kappa =0$ (dashed green line) with the numerical results for small curvature at low temperatures.

Figure 7

Absolute values of diagrams contributing to $u_b$ and $s_b$ as a function of temperature $T$ for (a)–(c) different values of $\kappa$ but fixed $\alpha =10.0$ and (d)–(f) different values of $\alpha$ but fixed $\kappa =0.05$. (a) ${\tilde{J}}_{1515}$, (b) ${\tilde{P}}_{2626}$, (c) ${\tilde{Q}}_{2626}$. (d)–(f) plot the same diagrams scaled with $\alpha$. Other parameters are $\Lambda =2.0$ and $v_x=0.5$. The vertical black dotted lines represent the approximate temperature where the computation in the presence of a nonzero $\kappa$ starts deviating from the one with $\kappa =0$. Note that in panels (a)–(c), the green dashed curves (representing the analytical results) overlap almost perfectly with the blue solid curves for $\kappa =0$.

Figure 8

Absolute values of diagrams contributing to $s_a$ as a function of temperature $T$ for (a) ${\tilde{M}}_{1262}$, (b), ${\tilde{N}}_{2615}$, (c) ${\tilde{N}}_{2651}$ for different values of $\kappa$ but fixed $\alpha =10.0$. Other parameters are $\Lambda =2.0$ and $v_x=0.5$. Note the almost perfect agreement of the analytical result for $\kappa =0$ (dashed green line) with the numerical results for small curvature at low temperatures.

Figure 9

Absolute value of ${\tilde{Y}}_{261}$ as a function of temperature and comparison with $\mathcal{Y}$ (a) at fixed $\alpha =10$, (b) at fixed $\kappa =0.1$.