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arXiv:2210.16324 · Physical review. B./Physical review. B

Triplet pair density wave superconductivity on the π -flux square lattice

Daniel J. Shaffer, Luiz H. Santos

Published Jul 18, 2023• 24 citations
arXivHTMLPDF

Abstract

Pair-density waves (PDWs) are superconducting states that spontaneously break translation symmetry in systems with time-reversal symmetry (TRS). Evidence for PDWs has been seen in several recent experiments, as well as in the pseudogap regime in cuprates. Theoretical understanding of PDWs has been largely restricted to phenomenological and numerical studies, while microscopic theories typically require strong coupling or fine tuning. In this work, we provide a symmetry-based mechanism under which PDWs emerge as a weak-coupling instability of a two-dimensional TRS metal. Combining mean-field and renormalization-group analyses, we identify a weak-coupling instability towards a triplet PDW realized in the $\ensuremath{\pi}$-flux square lattice model with on-site repulsion and moderate nearest-neighbor attraction when the Fermi level crosses Van Hove singularities at 1/4 and 3/4 fillings. This PDW is protected by the magnetic translation symmetries characteristic of Hofstadter systems, of which the $\ensuremath{\pi}$-flux lattice is a special time-reversal-symmetric case.

DOI
10.1103/physrevb.108.035135

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Pair-density waves (PDWs) are superconducting states that spontaneously break translation symmetry in systems with time-reversal symmetry (TRS). Evidence for PDWs has been seen in several recent experiments, as well as in the pseudogap regime in cuprates. Theoretical understanding of PDWs has been largely restricted to phenomenological and numerical studies, while microscopic theories typically require strong coupling or fine tuning. In this work, we provide a symmetry-based mechanism under which PDWs emerge as a weak-coupling instability of a two-dimensional TRS metal. Combining mean-field and renormalization-group analyses, we identify a weak-coupling instability towards a triplet PDW realized in the \ensuremathπ\ensuremath{\pi}\ensuremathπ-flux square lattice model with on-site repulsion and moderate nearest-neighbor attraction when the Fermi level crosses Van Hove singularities at 1/4 and 3/4 fillings. This PDW is protected by the magnetic translation symmetries characteristic of Hofstadter systems, of which the \ensuremathπ\ensuremath{\pi}\ensuremathπ-flux lattice is a special time-reversal-symmetric case.
We study the phase diagram of the extended Hubbard model on a two-dimensional square lattice, including on-site (UUU) and nearest-neighbor (VVV) interactions, at weak couplings. We show that the charge-density wave phase that is known to occur at half filling when 4V>U4V>U4V>U gives way to a dxy{d}_{xy}dxy​-wave superconducting instability away from half filling, when the Fermi surface is not perfectly nested, and for sufficiently large repulsive VVV and a range of on-site repulsive interaction UUU. In addition, when nesting is further suppressed and in the presence of a nearest-neighbor attraction, a triplet time-reversal breaking (px+ipy)({p}_{x}+i{p}_{y})(px​+ipy​)-wave pairing instability emerges, competing with the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ pairing state that is known to dominate at fillings just slightly away from half. At even smaller fillings, where the Fermi surface no longer presents any nesting, the (px+ipy)({p}_{x}+i{p}_{y})(px​+ipy​)-wave superconducting phase dominates in the whole regime of on-site repulsions and nearest-neighbor attractions, while dxy{d}_{xy}dxy​ pairing occurs in the presence of on-site attraction. Our results suggest that zero-energy Majorana fermions can be realized on a square lattice in the presence of a magnetic field. For a system of cold fermionic atoms on a two-dimensional square optical lattice, both an on-site repulsion and a nearest-neighbor attraction would be required, in addition to rotation of the system to create vortices. We discuss possible ways of experimentally engineering the required interaction terms in a cold atom system.
Abstract A pair-density-wave (PDW) is a superconducting state with an oscillating order parameter. A microscopic mechanism that can give rise to it has been long sought but has not yet been established by any controlled calculation. Here we report a density-matrix renormalization-group (DMRG) study of an effective t-J-V model, which is equivalent to the Holstein-Hubbard model in a strong-coupling limit, on long two-, four-, and six-leg triangular cylinders. While a state with long-range PDW order is precluded in one dimension, we find strong quasi-long-range PDW order with a divergent PDW susceptibility as well as the spontaneous breaking of time-reversal and inversion symmetries. Despite the strong interactions, the underlying Fermi surfaces and electron pockets around the K and K′{K}^{\prime}K′ K ′ points in the Brillouin zone can be identified. We conclude that the state is valley-polarized and that the PDW arises from intra-pocket pairing with an incommensurate center of mass momentum. In the two-leg case, the exponential decay of spin correlations and the measured central charge c ≈ 1 are consistent with an unusual realization of a Luther-Emery liquid.
Pair-density wave (PDW) is a long-sought exotic state with oscillating superconducting order without external magnetic field. So far it has been rare in establishing a 2D microscopic model with PDW long-range order in its ground state. Here, we propose to study PDW superconductivity in a minimal model of spinless fermions (or spin-polarized electrons) on the honeycomb lattice with nearest-neighbor and next-nearest-neighbor interaction V_{1} and V_{2}, respectively. By performing a state-of-the-art density-matrix renormalization group study of this t-V_{1}-V_{2} model at finite doping on six-leg and eight-leg honeycomb cylinders, we show that the ground state exhibits PDW ordering (namely quasi-long-range order with a divergent PDW susceptibility). Remarkably this PDW state persists on the wider cylinder with 2D-like Fermi surfaces. To the best of our knowledge, this is probably the first controlled numerical evidence of PDW in systems with 2D-like Fermi surfaces.
A pair density wave (PDW) is a superconducting (SC) state with a spatially modulated order parameter. Although much is known about the properties of the PDW state, its realization in microscopic models with divergent susceptibility has been challenging. Here we report a density-matrix renormalization group study of a three-band Hubbard model (also known as the Emery model) for cuprates on long two-leg square cylinders. Upon light doping, we find that the ground state of the system is consistent with that of a PDW state with mutually commensurate and power-law SC, charge (CDW), and spin (SDW) density wave correlations. The SC correlations are dominant between neighboring Cu sites with ddd-wave pairing symmetry. Interestingly, we find that the near-neighbor interactions, especially the near-neighbor attractive Vpd{V}_{pd}Vpd​ interaction between neighboring Cu and oxygen sites, can notably enhance the SC correlations while simultaneously suppressing the CDW correlations. For a modestly strong attractive Vpd{V}_{pd}Vpd​, the SC correlations become quasi-long-ranged with a divergent PDW susceptibility.
Motivated by exploring superconductivity in multi-orbital systems, we study two orbital models of spinful fermions representing (px,pyp_x,p_ypx​,py​) or (dxz,dyz)d_{xz}, d_{yz})dxz​,dyz​) orbitals on the square lattice. For minimal interorbital ttt-JJJ or ttt-VVV on-site interactions, a random phase approximation uncovers regimes of instability towards incommensurate dxyd_{xy}dxy​ pair density wave (ddd-PDW) superconductivity with driven by interband pairing. We study the competition of PDW order with uniform nodal dxyd_{xy}dxy​ pairing states and magnetic and charge density wave (CDW) instabilities. At strong coupling, we derive an effective hard-core Cooper pair Hamiltonian which we study using a bosonic Gutzwiller ansatz to reveal a period-222 PDW over a wide range of fillings as well as a checkerboard CDW at quarter-filling. Our results apply to correlated multi-orbital materials with quasi-1D bands, Hubbard models on the square-octagon lattice, and atomic fermions in ppp-orbitals. Our work highlights the role of the orbital content and multiband Fermi surfaces in stabilizing interband PDW states.
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 $t\text{\ensuremath{-}}J\text{\ensuremath{-}}V$ 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 intrapocket pairing.