Pair-density-wave phase of strongly interacting electrons on the triangular lattice: A variational Monte Carlo study

TL;DR

This work addresses the existence and nature of pair-density-wave order in a strongly interacting electron system on a triangular lattice by analyzing an effective t-J-V model derived from the Holstein-Hubbard framework. Using large-scale variational Monte Carlo with Gutzwiller projection, the authors identify valley-polarized PDW ground states: an s-wave PDW at low electron density and a nematic d-wave PDW at intermediate densities and phonon frequencies, both with a center-of-mass pairing momentum set by the lattice valleys. The PDW states arise from valley polarization and intra-pocket pairing, and the phase diagram also features CDW and 120° AFM order at higher densities, with half-filling returning the 120° AFM state. The results suggest PDW phases can emerge in 2D strongly correlated systems with significant electron-phonon coupling and may be relevant to moiré materials and heavy-fermion systems, while also highlighting finite-size and geometry effects in comparing with DMRG studies.

Abstract

A robust theory of the mechanism of pair density wave (PDW) superconductivity (i.e. where Cooper pairs have nonzero center of mass momentum) remains elusive. Here we explore the triangular lattice $t$-$J$-$V$ model, a low-energy effective theory derived from the strong-coupling limit of the Holstein-Hubbard model, by large-scale variational Monte Carlo simulations. When the electron density is sufficiently low, the favored ground state is an s-wave PDW, consistent with results obtained from previous studies in this limit. Additionally, a PDW ground state with nematic d-wave pairing emerges in the intermediate range of electron densities and phonon frequencies. For these s-wave and d-wave PDWs arising in states with spontaneous breaking of time-reversal and inversion symmetries, PDW formation derives from valley-polarization and intra-pocket pairing.

Figures (8)

• Figure 1: (a) 2D triangular lattice where the arrows indicate the direction of the orbital currents in the valley-polarized ground states. (b) The first Brillouin zone is contained within the solid lines with $\pmb \Gamma$, $\pmb K$, $\pmb K'$ points marked. The red circles indicate the occupied valleys in the valley-polarized state. (c) Phase diagrams of the effective $t$-$J$-$V$ model in the limit of large system size with various electron densities, $n=N_e/N$. Here $\omega_{\text{D}}$ is phonon frequency in the Holstein-Hubbard model. The specific parameters used in the calculation are listed in Table \ref{['tab:parameter']}. The PDW ordering vector is $\pmb Q=2\pmb K$. Abbreviations: sPDW = s-wave pair density wave; nematic dPDW = d-wave PDW plus broken lattice rotational symmetry.
• Figure 2: Properties of the lowest-energy variational sPDW state for the model with $\omega_{\text{D}}/|t|=2$ and filling $n=1/12$. (a) The occupation number in momentum space $n(\pmb{k})=\sum_\sigma \langle c^\dag_{\pmb{k}\sigma}c_{\pmb{k}\sigma}\rangle$ as a function of $\pmb k$ in the first Brillouin zone (represented by the red dashed line) is plotted in color scale; it is clear that the electrons are valley polarized. (b) The static SC structure factor $|S^{\text{SC}}_{\pmb{a}_1,\pmb{a}_1}(\pmb{k})|$ plotted in color scale; (c) The static spin structure factor $S(\pmb{k})$ plotted in color scale.
• Figure 3: Properties of the dPDW state with $\omega_{\text{D}}/|t|=3$, $n=5/12$. (a) The occupation number in momentum space $n(\pmb{k})$; (b) The static SC structure factor $|S^{\text{SC}}_{\pmb{a}_1,\pmb{a}_1}(\pmb{k})|$; (c) The static spin structure factor $S(\pmb{k})$. The red dotted box represents the first Brillouin zone.
• Figure 4: (a) The density structure factor $C(\pmb k)$ of the CDW order with $\omega_{\text{D}}/|t|=2.5$, $n=0.5$; (b) The static spin structure factor $S(\pmb{k})$ of the $120^{\circ}$-AFM order with $\omega_{\text{D}}/|t|=2.5$, $n=0.94$. The red dotted box represents the first Brillouin zone.
• Figure S1: Properties of PDW states with various phonon frequencies and electron densities. (a) and (d): The occupation number in momentum space $n(\pmb{k})$; (b) and (e): The static SC structure factor $|S^{\text{SC}}_{\pmb a_1, \pmb a_1}(\pmb{k})|$; (c) and (f): The static spin structure factor $S(\pmb{k})$. The left three figures [(a)$\sim$(c)] are for sPDW with setting $\omega_{\text{D}}/|t|=2$ and $n=1/6$, and right three figures [(d)$\sim$(f)] are for dPDW with setting $\omega_{\text{D}}/|t|=3$ and $n=1/4$. The red dotted box represents the first Brillouin zone.