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arXiv:2403.00156 · Physical review. B./Physical review. B

Pair density wave order in multiband systems

Nicole Ticea, S. Raghu, Yiming Wu

Published Sep 23, 2024• 10 citations
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Abstract

An indispensable ingredient for pair density wave (PDW) superconductivity is the presence of an attractive pairing interaction at finite momentum. Here, we show how this condition can be met with straightforward electron-density interactions in multiband systems. The electron-density interaction, when projected to the band basis, acquires form factors with nontrivial momentum dependence and thereby exhibits a potential tendency to a finite-momentum pairing instability. By applying a mean field analysis to two simple multiband models, the checkerboard lattice and three-band Hubbard model, we find that PDW order can indeed become the leading instability if a strong nearest-neighbor attraction is present. Moreover, the condition for the transition from a uniform superconductor to a PDW superconductor is shown via a simple quantum geometric argument.

DOI
10.1103/physrevb.110.094515

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An indispensable ingredient for pair density wave (PDW) superconductivity is the presence of an attractive pairing interaction at finite momentum. Here, we show how this condition can be met with straightforward electron-density interactions in multiband systems. The electron-density interaction, when projected to the band basis, acquires form factors with nontrivial momentum dependence and thereby exhibits a potential tendency to a finite-momentum pairing instability. By applying a mean field analysis to two simple multiband models, the checkerboard lattice and three-band Hubbard model, we find that PDW order can indeed become the leading instability if a strong nearest-neighbor attraction is present. Moreover, the condition for the transition from a uniform superconductor to a PDW superconductor is shown via a simple quantum geometric argument.
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 KscK_{sc}Ksc​ 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.
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.
We show, based on microscopic models, that fermionic band insulators with multiple bands and strong interband attraction are generically unstable toward nonzero momentum Cooper pairing leading to a pair density wave (PDW) superfluid state. Our first model considers a band insulating state of fermionic atoms in a three-dimensional cubic optical lattice. We show that this insulator is unstable toward an incommensurate PDW in the vicinity of a Feshbach resonance. Our second model is a two-band tight-binding model relevant to electrons in solids; we show that the insulating state of this model has a PDW instability analogous to the exciton condensation instability in indirect band-gap semiconductors. We discuss relevant experimental signatures of the PDW state.
Abstract Flat-band systems provide an ideal platform for studying strong electron correlations, in which multiple quasi-degenerate charge orders frequently coexist. Motivated by this context, we investigate the pair-density-wave (PDW) superconducting state in a quasi-one-dimensional Creutz lattice featuring two strictly flat bands. Using a renormalized mean-field approach for the strongly correlated t - J model, we obtain self-consistent solutions that demonstrate a PDW phase—characterized by finite-center-of-mass-momentum Cooper pairing—coexisting with charge density wave (CDW) and bond density wave (BDW) orders. Physically, this coexistence phenomenon originates from the perfect band nesting inherent to the flat-band system, which allows for changes in the particle momentum distributions without energy cost. Here, a pure PDW solution without CDW and BDW orders emerges in the underdoped regime when magnetic effects are neglected, while strong Hubbard correlations favor a pure CDW phase. We conclude that a robust PDW phase, accompanied by CDW and BDW orders, occupies the majority of the parameter space. Furthermore, both the PDW order parameter and its critical temperature scale linearly with the superexchange coupling strength—a hallmark of flat-band superconductivity. Overall, our results reveal that a PDW superconducting state can be realized on the Creutz lattice with strong correlations.
We show that the pair-density-wave (PDW) superconducting state emergent in extended Heisenberg-Hubbard models in two-leg ladders is topological in the presence of an Ising spin symmetry and supports a Majorana zero mode (MZM) at an open boundary and at a junction with a uniform d-wave one-dimensional superconductor. Similarly to a conventional finite-momentum paired state, the order parameter of the PDW state is a charge-2e field with finite momentum. However, the order parameter here is a quartic electron operator and conventional mean-field theory cannot be applied to study this state. We use bosonization to show that the 1D PDW state has a MZM at a boundary. This superconducting state is an exotic topological phase supporting Majorana fermions with finite-momentum pairing fields and charge-4e superconductivity.
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.