Interplay between pair- and charge-density-wave orders in underdoped cuprates

We analyze the interplay between charge-density-wave (CDW) and pair-density-wave (PDW) orders within the spin-fermion model for the cuprates. We specifically consider CDW order with transferred momenta $(\pm Q,0)/(0,\pm Q)$, as seen in experiments on the cuprates, and PDW order with total momenta $(0,\pm Q)/(\pm Q,0)$. Both orders have been proposed to explain the pseudogap phase in the cuprates. We show that both emerge in the spin-fermion model near the onset of antiferromagnetism. Each order parameter is constructed out of pairs of fermions in “hot” regions on the Fermi surface, breaks $\mathrm{U}\left(1\right)$ translational symmetry, and changes sign when the momenta of the fermions change by $(\pi ,\pi )$. We further show that the two orders are nearly degenerate due to an approximate $\mathrm{SU}\left(2\right)$ particle-hole symmetry of the model. This near degeneracy is similar in origin to that relating conventional $d$-wave superconducting order and bond charge order with momentum $(Q,\pm Q)$. The $\mathrm{SU}\left(2\right)$ symmetry becomes exact if one neglects the curvature of the Fermi surface in hot regions, in which case $\mathrm{U}\left(1\right)$, CDW, and PDW order parameters become components of an $\mathrm{SO}\left(4\right)$-symmetric PDW/CDW “supervector.” We develop a Ginzburg-Landau theory for four PDW/CDW order parameters and find two possible ground states: a “stripe” state in which both CDW and PDW orders develop with either $(\pm Q,0)$ or $(0,\pm Q)$, and a “checkerboard” state, where each order can develop with $(\pm Q,0)$ and $(0,\pm Q)$. We show that the $\mathrm{SO}\left(4\right)$ symmetry between CDW and PDW can be broken by two separate effects. One is the inclusion of Fermi surface curvature, which selects a PDW order immediately below the instability temperature. Another is the overlap between different hot regions, which favors CDW order at low temperatures. For the stripe state, we show that the competition between the two effects gives rise to a first-order transition from PDW to CDW inside the ordered state. We also argue that beyond mean-field theory, the onset temperature for CDW order is additionally enhanced due to feedback from a preemptive breaking of ${\mathbb{Z}}_2$ time-reversal symmetry. We discuss the ground-state properties of a pure PDW state and a pure CDW state, and show in particular that the PDW checkerboard state yields a vortex-antivortex lattice. For the checkerboard state, we considered a situation when both CDW and PDW orders are present at low $T$ and show that at small but finite Fermi surface curvature the presence of both condensates induces a long sought chiral $s+i{d}_{xy}$ superconductivity.

Figure 1

(a) The Fermi surface, Brillouin zone, and magnetic Brillouin zone (dashed line). Hot spots are defined as intersections of the FS with magnetic Brillouin zone. The hot spot pairs 1-2 and 3-4 denotes the CDW/PDW pairings we consider. They are coupled through the antiferromagnetic exchange interaction peaked at momentum $(\pi ,\pi )$, as shown by the dashed arrows. [(b) and (c)] Diagrammatic representation of CDW (b) and PDW (c) instabilities between hot spots (1,2) and (3,4).

Figure 2

The Brillouin zone, the magnetic Brillouin zone (dashed line), and the Fermi surface of a cuprate. The hot spots 1-8 are defined as the intersection of hot spots and the Brillouin zone. We label “bonds” connecting hot spots by $A,B,C,D$ and $-A,-B,-C,-D$. Bonds connecting hot spots 1256 and 3478 are denoted by opposite signs, due to the $d$-form factor.

Figure 3

Diagrammatic representation for fourth-order terms of the CDW/PDW order parameter.

Figure 4

The position dependence of the PDW representation of state II: the contour lines give the magnitude of the gap function while the $(x,y)$ components of the vectors denote the normalized real and imaginary parts of the gap function. Note the vortices and antivortices surrounding the zeros of the gap function, giving a vortex-antivortex lattice.

Figure 5

Diagrammatic representations of the coefficients $\Delta {\beta }_a,\Delta {\beta }_b$. We have associated form factors $f_{A,B}$ and $g_{A,B}$ to each vertex. The momenta are $Q=Q_y=(0,Q)$ and $Q^{\prime }=-Q_x=(-Q,0)$.

Figure 6

(a) A superconducting order parameter ${\Phi }_1$ between hot spots 1 and 6. We label CDW order $\rho$ with a line with an arrow, and PDW and homogeneous superconducting order with lines with double arrows. (c) Diagrams for coupling of ${\Phi }_1$ with $\rho$ and $\phi$. (b) Diagrams for coupling of ${\Phi }_1$ with $\rho$ and $\phi$. Comparing with (b), the minus sign between CDW's and between PDW's cancel out.

Figure 7

The signs of order parameter at hot spots 1-8 for $s$-wave (a) and $d_{xy}$-wave (b) SC.

Figure 8

The dependence of parameter $\sqrt{S_1{S}_2}$ on chemical potential $\mu$. We vary $\mu$ such that the CDW ordering momentum $Q$ varies in the range $(0.3\pi ,0.45\pi )$, as measured by experiments.