A "negative" route to pair density wave order

Abstract

Pair density waves (PDW) are novel forms of superconducting states that exhibit periodically modulated pairing. A remaining challenge is to elucidate how intrinsic PDW order can emerge robustly in strongly correlated electrons. Here we propose that PDW is prone to form in strongly coupled multiband superconductors simply with interband Cooper pairing between electrons from oppositely dispersing bands. This scenario is heuristically motivated by the observation that uniform interband pairing in such systems would exhibit negative superfluid weight -- a signature of an instability towards pairing modulation, implying that PDW emerges naturally in the true ground state. Using large-scale density-matrix-renormalization-group calculations with finite-size scaling analysis, we demonstrate this PDW mechanism in a minimal model with strong interband attractions. Our simulations reveal power-law superconducting correlations characterized by incommensurate modulations. The exponent $K_{sc}$ of the power-law PDW correlation decreases systematically with increasing ladder width, confirming a genuine long-range PDW order in the 2D limit. Our study therefore demonstrates a promising route to robust PDW states in multiband systems.

Figures (4)

• Figure 1: (Color online) (a) Illustration of an effective self-Josephson effect in a superconductor, generated by transfer of Cooper pairs. Superfluid weight is a measure of this Josephson coupling. Cooper pairing is depicted by wiggly lines encircled by an ellipse, with one electron (blue sphere) from band-$i$ with wavevector $\boldsymbol{\mathrm{k}}$ and another (orange sphere) from band-$j$ with $\bar{\boldsymbol{\mathrm{k}}}=-\boldsymbol{\mathrm{k}}$. The transfer of the Cooper pair is facilitated by the motion of the two electrons, described respectively by their velocities $V^i_{\boldsymbol{\mathrm{k}}}$ and $V^j_{\bar{\boldsymbol{\mathrm{k}}}}$. (b) Interband Cooper pairing between two oppositely dispersing bands. (c) Free energy as a function of the superconducting phase modulation wavevector $\mathbf q$. The two curves respectively describe the two scenarios where uniform pairing (i.e. $\mathbf q=0$) would exhibit positive (upper curve) and negative (lower curve) superfluid weight.
• Figure 2: (a) The band structure of the bilayer model Eq. \ref{['eq:hamk']} with $(t_1,t_2)=(-1,1.2)t$, and the chemical potential $(\mu_1,\mu_2)$ are chosen so that the electron filling on the two bands are $x_1=0.4$ and $x_2=0.7$, respectively. The inset shows the Fermi surfaces. The right panel shows the variation of the free energy with modulation wavevector $\mathbf q=(q_x,0)$ for (b) $s$-wave and (c) $d$-wave states, and with the pairing amplitude held fixed.
• Figure 3: (a) Envelope of superconducting correlations $|\Phi(\mathbf{r})|$ as a function of distance $r_x$ for width $L_y=2,3,4$, with power-law fits $|\Phi(\mathbf{r})| \sim r_x^{-K_{sc}}$. (b) Normalized superconducting correlations $f(\mathbf{r}) = \Phi(\mathbf{r}) \cdot r_x^{K_{sc}}$ for $L_y=4$, revealing the spatial modulation characteristic of a pair-density-wave. Positive (negative) values are represented by orange (blue) circles, with areas proportional to their magnitudes. The reference point is marked by a black square. The exponent $K_{sc}$ used for normalization is determined from the power-law fits in (a). (c) Fourier transform of the normalized superconducting correlation data. Only half of the Brillouin zone is shown, because $|\Tilde{\Phi}(\mathbf{q})| = |\Tilde{\Phi}(-\mathbf{q})|$.
• Figure 4: (a-b) Intraband density correlation $|D(r)|$ for $L_y = 2,3,4$ with power-law fits $|D(r)| \sim r_x^{-K_c}$. The dashed lines are fitting lines. (c-d) Density structure factor $|S(\mathbf{k})|$ in the reduced Brillouin zone ($k_x \in [0,\pi]$) exploiting $|S(\mathbf{k})| = |S(\mathbf{-k})|$. Data at $k_y = \pi/2$ and $k_y = 3\pi/2$ (blue curves) are scaled 3$\times$ for visual clarity. To minimize boundary effects, we exclude edge data in the Fourier analysis. Primary CDW peaks are labeled $K_1$ to $K_{6}$.