Pair density waves in superconducting vortex halos

We analyze the interplay between a $d$-wave uniform superconducting and a pair-density-wave (PDW) order parameter in the neighborhood of a vortex. We develop a phenomenological nonlinear sigma model, solve the saddle-point equation for the order-parameter configuration, and compute the resulting local density of states in the vortex halo. The intertwining of the two superconducting orders leads to a charge density modulation with the same periodicity as the PDW, which is twice the period of the charge density wave that arises as a second harmonic of the PDW itself. We discuss key features of the charge density modulation that can be directly compared with recent results from scanning tunneling microscopy and speculate on the role PDW order may play in the global phase diagram of the hole-doped cuprates.

Figure 1

The numerical solution to the meronlike configuration of the order parameters in the vicinity of an isolated vortex. The PDW order parameter has a uniform phase while the SC one has a $2\pi$ winding. The magnitudes of ${\tilde{\rho }}_{\mathit{Q}}$ and ${\tilde{\rho }}_{2\mathit{Q}}$ (i.e., the period $8a_0$ and period $4a_0$ stripes) are computed from Eqs. (2.5) and (2.6) with coefficients $B_1=B_2=1$.

Figure 2

LDOS with $s$-form factor as a function of position at $V=35\mathrm{mV}$ with a vortex centered at the origin and a halo size $\sim 25a_0$. Note that even if the PDW and SC order parameters we used are rotationally invariant, the envelope of the charge density modulations is elongated along the $x$ direction.

Figure 3

Absolute value of LDOS with $d$-form factor at $V=35\mathrm{mV}$ as a function of position with a vortex centered at the origin and a halo size $\sim 25a_0$. The color code scale is the same with that of Fig. 2.

Figure 4

Top panel: Fourier transform of LDOS with $s$-form factor at $V=35\mathrm{mV}$ for a vortex halo of size $\sim 25a_0$. The peak at $Q=\pi /4$ is visible. Bottom panel: the same data along the $q_y=0$ cut.

Figure 5

Top panel: the Fourier transform of LDOS with $d$-form factor at $V=35\mathrm{mV}$ for a vortex halo of size $\sim 25a_0$. The color code scale is the same with that of Fig. 4. Bottom panel: the same data along $q_y=0$ cut.

Figure 6

Local density of states as a function of position in the neighborhood of an isolated vortex. Notice the $\pi$-phase shift of the $8a_0$ stripes across the vortex center. The lower panel shows the local density along the solid black line in the upper panel. Here, the global phase of the SC order parameter is chosen so that $\mathrm{\Delta }\left(\mathit{r}\right)$ is real along the $x$ axis, and the PDW order parameter is real. The red curve in the bottom panel is a sine function added to help visualize the $\pi$-phase shift.

Figure 7

A putative phase diagram for underdoped cuprates as a function of magnetic field $H$ and temperature $T$ in the absence of quenched disorder. The solid red line marks a transition to a low-temperature/low-field $d$-wave SC vortex crystal phase, and the dotted red line represents a crossover region below which SC fluctuations are significant. Marked by the heavy green line, inside the vortex crystal phase, PDW fluctuations appear inside the SC vortex halo, which are not phase coherent across different vortices but induce a long-range CDW order with primarily $s$-form factor. The explicit analysis of this work applies to the low-field region of this phase. In the high-field/low-temperature region bounded by the heavy blue line, a long-range, unidirectional CDW order develops with a dominantly $d$-form factor. The two CDW phases at low fields and high fields are weakly coupled to each other via a commensurate lock-in. The thin green and blue lines denote a transition into a vestigial nematic phase, with lattice rotational symmetry broken but translational symmetry associated with the CDW order restored.

Figure 8

Numerical solution to the meronlike configuration of the order parameters near a vortex halo in the case in which the phases of PDW and SC order parameters are locked and both wind by $2\pi$. We have used $\overline{\kappa }=0.5\kappa$ and $\lambda =0.5\rho$. The magnitudes of ${\tilde{\rho }}_{\mathit{Q}}$ and ${\tilde{\rho }}_{2\mathit{Q}}$ are computed from Eqs. (2.5) and (2.6) with coefficients $B_1=B_2=1$.

Figure 9

Same as in Fig. 6, but with the global phase choice ${\varphi }_{\text{CDW}}=\pi /2$ and ${\varphi }_{\text{PDW}}=\pi /2$. The lower panel shows the local density along the two cuts shown in the upper panel.

Figure 10

Same as in Fig. 6, but with the global phase choice ${\varphi }_{\text{CDW}}=0$ and ${\varphi }_{\text{PDW}}=0$. The lower panel shows the local density along the two cuts shown in the upper panel.