Critical Density-Wave Vestigial Phases of Commensurate Pair Density Wave

Abstract

The pair-density-wave (PDW) is an exotic pairing state hosting a spatially modulated pairing order parameter, which has attracted great interest. Due to its simultaneously breaking U(1)-gauge and translational symmetries, intriguing vestigial phases which restore only one broken symmetry can emerge at an intermediate temperature regime. Previously, investigations on the vestigial phases of PDW were mainly focused on incommensurate PDW. However, the experimentally observed PDW is usually commensurate, whose vestigial phases have not been systematically investigated. Here we study the vestigial phases of 2D commensurate PDW with $n$-times expanded unit vectors, hosting different numbers of wave vectors. Based on the Ginzburg-Landau theory, we get the low energy effective model Hamiltonian. Subsequent renormalization group (RG) and Monte-Carlo (MC) studies are conducted to obtain the phase diagram and spatial dependent correlation functions. Our RG and MC calculations consistently yield the following result. For $n\le 4$, besides the charge-4e/2e superconductivity, there exists the translational symmetry broken charge-density-wave (CDW) vetigial phase. Intriguingly, for $n\ge 5$, the restore of the translational symmetry with increasing temperature is realized through two successive Berezinskii-Kosterlitz-Thouless transitions. Such a two-step process leads into two critical vestigial phases, i.e. the critical-PDW and the critical-CDW phases, in which the discrete translational symmetry is quasily broken, leading into a power-law decaying density-density correlation even at 2D. Our work appeals for experimental verifications.

Figures (23)

• Figure 1: Directions of $\mathbf{Q}_{\alpha}$, with (a) for $3Q$ PDW, (b) for $2Q$ PDW, respectively.
• Figure 2: Schematic phase diagrams of the commensurate PDW with (a) for $2 \le n \le 4$ and (b) for $n\ge5$. $\rho$, $\mu$ and $T$ denote the superfluid stiffness, the CDW elastic constant and temperature, respectively. The black lines mark the phase boundaries. The dashed line represents a point for the 3Q PDW, while it represents for a solid line for the 2Q PDW.
• Figure 3: (Color online) The correlation functions $\eta_{\theta/\phi_{1}}$ for the $3Q$ PDW are shown for the C-PDW phase in panels (a) and (c), and for the C-CDW phase in panels (b) and (d). Insets of (a, c, d) the log-log plot, and (b) only the y-axis is logarithmic. Further details are provided in the SM SM.
• Figure S1: (Color online) Phase diagrams provided by (a,c) the RG study and (b,d) the MC study, with (a)-(b) for $n=2$ and (c)-(d) for $n=5$ (the $3Q$ PDW state). The white dashed lines in (b,d) mark $\mu/\rho=0.3, 0.63$ and $1$, respectively. The initial values of the coupling parameters in (a,c) are $g_{\theta}=0.1$, $g_{x}=g_{y}=0.1$, $g_\alpha=0.1$ in Eq. (\ref{['eqn:action-SG']}), and in (b,d) are $A=0.22\rho$ in Eq. (\ref{['Hamiltonian_d']}).
• Figure S2: (Color online) Phase diagrams provided by (a,c) the RG study and (b,d) the MC study, with (a)-(b) for $n=2$ and (c)-(d) for $n=5$ (the $2Q$ PDW state). The white dashed lines in (b,d) mark $\mu/\rho=0.15, 0.6$ and $1.6$, respectively. The initial values of the coupling parameters in (a,c) are $g_{\theta}=0.1$, $g_{x}=g_{y}=g_{\frac{1}{2}}^{x}=g_{\frac{1}{2}}^{y}=0.1$, $g_\alpha=0.1$ in Eq. (\ref{['eqn:action-SG']}), and in (b,d) are $A=0.02\rho$ in Eq. (\ref{['Hamiltonian_d4Q']}).