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arXiv:2209.09254 · Physical Review Letters

Pair Density Wave Order from Electron Repulsion

Yiming Wu, P. A. Nosov, Aavishkar A. Patel, S. Raghu

Published Jan 13, 2023• 36 citations
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Abstract

A pair density wave (PDW) is a superconductor whose order parameter is a periodic function of space, without an accompanying spatially uniform component. Since PDWs are not the outcome of a weak-coupling instability of a Fermi liquid, a generic pairing mechanism for PDW order has remained elusive. We describe and solve models having robust PDW phases. To access the intermediate coupling limit, we invoke large-N limits of Fermi liquids with repulsive BCS interactions that admit saddle point solutions. We show that the requirements for long-range PDW order are that the repulsive BCS couplings must be nonmonotonic in space and that their strength must exceed a threshold value. We obtain a phase diagram with both finite temperature transitions to PDW order and a T=0 quantum critical point, where non-Fermi liquid behavior occurs.

DOI
10.1103/physrevlett.130.026001

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A pair density wave (PDW) is a superconductor whose order parameter is a periodic function of space, without an accompanying spatially uniform component. Since PDWs are not the outcome of a weak-coupling instability of a Fermi liquid, a generic pairing mechanism for PDW order has remained elusive. We describe and solve models having robust PDW phases. To access the intermediate coupling limit, we invoke large-N limits of Fermi liquids with repulsive BCS interactions that admit saddle point solutions. We show that the requirements for long-range PDW order are that the repulsive BCS couplings must be nonmonotonic in space and that their strength must exceed a threshold value. We obtain a phase diagram with both finite temperature transitions to PDW order and a T=0 quantum critical point, where non-Fermi liquid behavior occurs.
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We provide a quasi-one-dimensional model which can support a pair-density-wave (PDW) state, in which the superconducting (SC) order parameter modulates periodically in space, with gapless Bogoliubov quasiparticle excitations. The model consists of an array of strongly interacting one-dimensional systems, where the one-dimensional systems are coupled to each other by local interactions and tunneling of the electrons and Cooper pairs between them. Within the interchain mean-field theory, we find several SC states from the model, including a conventional uniform SC state, PDW SC state, and a coexisting phase of the uniform SC and PDW states. In this quasi-1D regime we can treat the strong correlation physics essentially exactly using bosonization methods and the crossover to the 2D system by means of interchain mean-field theory. The resulting critical temperatures of the SC phases generically exhibit a power-law scaling with the coupling constants of the array, instead of the essential singularity found in weak-coupling BCS-type theories. Electronic excitations with an open Fermi surface, which emerge from the electronic Luttinger liquid systems below their crossover temperature to the Fermi liquid, are then coupled to the SC order parameters via the proximity effect. From the Fermi surface thus coupled to the SC order parameters, we calculate the quasiparticle spectrum in detail. We show that the quasiparticle spectrum can be fully gapped or nodal in the uniform SC phase and also in the coexisting phase of the uniform SC and PDW parameters. In the pure PDW state, the excitation spectrum has a reconstructed Fermi surface in the form of Fermi pockets of Bogoliubov quasiparticles.
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.
We present an analytic theory unraveling the microscopic mechanism of instabilities within interacting DDD-dimensional Fermi liquid. Our model consists of a DDD-dimensional electron gas subject to an instantaneous electron-electron interaction of a finite range exceeding the average interparticle distance. Pair, charge, and spin susceptibilities are evaluated via the one-loop renormalization group theory and via the bosonization approach, giving identical results. In case of a repulsive interaction, we identify an intrinsic Fermi-liquid instability towards insulating spin- and charge-density-wave order when the interaction coupling strength reaches a universal critical value. If both electron and hole pockets of the same size are present, the ground state is an excitonic insulator at arbitrarily small repulsive interaction. If the interaction is attractive, the ground state is a singlet non-BCS superconductor with a uniform condensate. In case if both electron and hole Fermi surfaces are present, we predict an instability towards the interpocket pair-density-wave ordering at the critical coupling. This prediction lends strong theoretical support to the pair-density-wave scenario of superconductivity in cuprate materials. Due to its simple and universal nature, presented microscopic mechanism of intrinsic instabilities of interacting DDD-dimensional Fermi liquids constitutes a solid theoretical ground for understanding quantum phase transitions in a variety of quantum materials, from ultraclean semiconductor quantum wells to high-temperature superconductors.
Pair density waves (PDWs) are superconducting states formed by Cooper pairs of electrons containing a nonzero center-of-mass momentum. They are characterized by a spatially modulated order parameter and may occur in a variety of emerging quantum materials such as cuprates, transition-metal dichalcogenides (TMDs), and Kagome metals. Despite extensive theoretical and numerical studies seeking PDWs in a variety of lattices and interacting settings, there is currently no exact mechanism that spontaneously favors a modulated solution of the superconducting order parameter. Here, we study the problem of two electrons subject to an anisotropic attractive potential. We solve the two-body Schr\"odinger wave equation exactly to determine the pair binding energy as a function of the center-of-mass momentum. We find that a modulated (finite momentum) pair is favored over a homogeneous (zero momentum) solution above a critical, intermediate interaction strength. Hence our exact result justifies previous mean-field approximations that obtain modulated ground states at finite but large interactions. Using this insight from the exact two-body solution, we construct a variational many-body wave function and show that the conclusions of the two-body problem are robust in the many-body limit. Our results thus lay the theoretical and microscopic foundation for the existence of PDWs.
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.