Interplay between Pair Density Wave and a Nested Fermi Surface

We show that spontaneous time-reversal-symmetry (TRS) breaking can naturally arise from the interplay between pair density wave (PDW) ordering at multiple momenta and nesting of Fermi surfaces (FSs). Concretely, we consider the PDW superconductivity on a hexagonal lattice with nested FS at $3/4$ electron filling, which is related to a recently discovered superconductor ${\mathrm{CsV}}_3{\mathrm{Sb}}_5$. Because of nesting of the FSs, each momentum $\mathbf{k}$ on the FS has at least two counterparts $-\mathbf{k}\pm {\mathbf{Q}}_{\alpha }$ ($\alpha =1$, 2, 3) on the FS to form finite momentum ($\pm {\mathbf{Q}}_{\alpha }$) Cooper pairs, resulting in a TRS and inversion broken PDW state with stable Bogoliubov Fermi pockets. Various spectra, including (local) density of states, electron spectral function, and the effect of quasiparticle interference, have been investigated. The partial melting of the PDW will give rise to $4\times 4$ and $(4/\sqrt{3})\times (4/\sqrt{3})$ charge density wave (CDW) orders, in addition to the $2\times 2$ CDW. Possible implications to real materials such as ${\mathrm{CsV}}_3{\mathrm{Sb}}_5$ and future experiments have been discussed further.

Figure 1

First Brillouin zone (BZ) of a hexagonal lattice and nesting feature of PDWs. (a) BZ and the nested FS at $3/4$ filling. The boundary of BZ is in blue, and red lines represent the FS. (b) Each $\mathbf{k}$ on a FS segment along the ${\mathbf{Q}}_{\alpha }$ direction has two counterparts $-\mathbf{k}\pm {\mathbf{Q}}_{\alpha }$ on the FS to form finite momentum ($\pm {\mathbf{Q}}_{\alpha }$) Cooper pairs.

Figure 2

(a) Condensation energy $E_c$ (with an offset $ϵ=0.993$) as a function of ${\theta }_2$ and ${\theta }_3$, where ${\theta }_1=0$ and ${\varphi }_{\alpha }=\pi /2$ have been set [see Eq. (2)]. Maximal $E_c$ occurs at ${\theta }_2={\theta }_3-{\theta }_2\equiv \pm 2\pi /3(\text{mod}\pi )$. (b)–(d) Energy dispersion and Bogoliubov Fermi pockets for the lowest energy state: ${\varphi }_{\alpha }=\pi /2$, ${\theta }_1=0$, ${\theta }_2=2\pi /3$, and ${\theta }_3=-2\pi /3$. (b) $E(\mathbf{k}{)}_1^{\pm }$ around $X$ point that are plotted along $\mathrm{\Gamma }-X-K$. (c) Bogoliubov Fermi pockets at $M$ and $X$ points and their periodic replica due to the PDW. (d) Quasiparticle (hole) pocket around $X$ point.

Figure 3

DOS and LDOS. DOS $\rho (\omega )$ at (a) $k_{B}T/\mathrm{\Delta }=1/60$ and (b) $k_{B}T/\mathrm{\Delta }=1/10$. (c) Wave vectors $\mathbf{q}=\pm {\mathbf{Q}}_{\alpha }\pm {\mathbf{Q}}_{\beta }$ (or their equivalent vectors in first BZ) associated with descendant CDW orders. (d) LDOS $\rho (\mathbf{q},\omega )$ exhibits three types of CDW orders $B$, $C$, and $D$. Here, $\mathrm{\Lambda }=0.1$ and $\mathrm{\Delta }=0.02$ have been chosen.

Figure 4

LDOS modulation $|\delta \rho (\mathbf{q},\omega )|$ for (a) the PDW state and (b) a uniform $s$-wave SC state. Electron spectral function $A(\mathbf{k},\omega )$ for (c) the PDW state and (d) a uniform $s$-wave state. Wave vectors ${\mathbf{q}}_1$ and ${\mathbf{q}}_2$ in (a) and (c) indicate dominant scattering processes connecting two points with large $A(\mathbf{k},\omega )$ (and their symmetric equivalence). Here, $\omega =0.01<\mathrm{\Delta }=0.02$ has been chosen.