Coexistence of Charge-Density-Wave and Pair-Density-Wave Orders in Underdoped Cuprates

We analyze incommensurate charge-density-wave (CDW) and pair-density-wave (PDW) orders with transferred momenta $(\pm Q,0)/(0,\pm Q)$ in underdoped cuprates within the spin-fermion model. Both orders appear due to an exchange of spin fluctuations before magnetic order develops. We argue that the ordered state with the lowest energy has nonzero CDW and PDW components with the same momentum. Such a state breaks $C_4$ lattice rotational symmetry, time-reversal symmetry, and mirror symmetries. We argue that the feedback from CDW/PDW order on fermionic dispersion is consistent with ARPES data. We discuss the interplay between the CDW/PDW order and $d_{x^2-y^2}$ superconductivity and make specific predictions for experiments.

Figure 1

The Brillouin zone, the Fermi surface, and the hot spots. We label bonds connecting hot spots as $A$, $B$, $C$, $D$, $a$, and $b$. (Inset) The structure of the mixed CDW/PDW state in one of the hot regions.

Figure 2

Fermionic dispersion in the antinodal region in the presence of the mixed CDW/PDW order. (Panel (a)) The dispersion in the presence of the CDW/PDW order for various $k_x$’s ($k_x=\pi$ corresponds to the cut along the Brillouin zone boundary). (Panel (b)) The spectral function. Thin line on both panels is the bare dispersion. (Panel (c)) Experimental data from Ref. [9] for various $k_x$’s, for comparison. The experimental data have been taken below $T_c$ and show a gapped dispersion in a wider range of $\pi -k_x$.

Figure 3

The phase diagram. The transition into the CDW/PDW state is weakly first order and the superconducting $T_c$ drops by a finite amount upon entering into the coexistence region. In the region labeled as “preemptive,” discrete $C_4$ and time-reversal or mirror symmetries are broken but continuous U(1) translational symmetry (associated with the locking of the common phases of ${\rho }_A$ and ${\rho }_B$) remains unbroken [23]. In the shaded region, Mott physics develops and the onset temperature of charge ordering shrinks.