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arXiv:1007.2433 · Journal of Physics Conference Series

Pair density wave instability and Cooper pair insulators in gapped fermion systems

Predrag Nikolić, A. A. Burkov, Arun Paramekanti

Published Jan 1, 2011• 0 citations
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Abstract

By analyzing simple models of fermions in lattice potentials we argue that the zero-temperature pairing instability of any ideal band-insulator occurs at a finite momentum. The resulting supersolid state is known as "pair density wave". The pairing momentum at the onset of instability is generally incommensurate as a result of phase-space restrictions and relative strengths of interband and intraband pairing. However, commensurate pairing occurs in the strong-coupling limit and becomes a Cooper-channel analogue of the Halperin-Rice exciton condensation instability in indirect bandgap semiconductors. The exceptional sensitivity of incommensurate pairing to quantum fluctuations can lead to a strongly-correlated insulating regime and a non-BCS transition, even in the case of weak coupling as shown by an exact renormalization group analysis.

DOI
10.1088/1742-6596/273/1/012012

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By analyzing simple models of fermions in lattice potentials we argue that the zero-temperature pairing instability of any ideal band-insulator occurs at a finite momentum. The resulting supersolid state is known as "pair density wave". The pairing momentum at the onset of instability is generally incommensurate as a result of phase-space restrictions and relative strengths of interband and intraband pairing. However, commensurate pairing occurs in the strong-coupling limit and becomes a Cooper-channel analogue of the Halperin-Rice exciton condensation instability in indirect bandgap semiconductors. The exceptional sensitivity of incommensurate pairing to quantum fluctuations can lead to a strongly-correlated insulating regime and a non-BCS transition, even in the case of weak coupling as shown by an exact renormalization group analysis.
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 In weakly coupled BCS superconductors, only electrons within a tiny energy window around the Fermi energy, EF, form Cooper pairs. This may not be the case in strong coupling superconductors such as cuprates, FeSe, SrTiO3 or cold atom condensates where the pairing scale, EB, becomes comparable or even larger than EF. In cuprates, for example, a plausible candidate for the pseudogap state at low doping is a fluctuating pair density wave, but no microscopic model has yet been found which supports such a state. In this work, we write an analytically solvable model to examine pairing phases in the strongly coupled regime and in the presence of anisotropic interactions. Already for moderate coupling we find an unusual finite temperature phase, below an instability temperature Ti, where local pair correlations have non-zero center-of-mass momentum but lack long-range order. At low temperature, this fluctuating pair density wave can condense either to a uniform d-wave super- conductor or the widely postulated pair-density wave phase depending on the interaction strength. Our minimal model offers a unified microscopic framework to understand the emergence of both fluctuating and long range pair density waves in realistic systems.
The pseudogap state of cuprate high-temperature superconductors has been often viewed as either a yet unknown competing order or a precursor state to superconductivity. While awaiting the resolution of the pseudogap problem in cuprates, we demonstrate that local pairing fluctuations, vortex liquid dynamics, and other precursor phenomena can emerge quite generally whenever fermionic excitations remain gapped across the superconducting transition, regardless of the gap origin. Our choice of a tractable model is a lattice band insulator with short-range attractive interactions between fermions in the sss-wave channel. An effective crossover between Bardeen-Cooper-Schrieffer (BCS) and Bose-Einstein condensate (BEC) regimes can be identified in any band insulator above two dimensions, while in two dimensions only the BEC regime exists. The superconducting transition is ``unconventional'' (non-pair-breaking) in the BEC regime, identified by either the bosonic mean field or XY\mathit{XY}XY universality class. The insulator adjacent to the superconductor in the BEC regime is a bosonic Mott insulator of Cooper pairs, which may be susceptible to charge density wave ordering. We construct a function of the many-body excitation spectrum whose nonanalytic changes define a sharp distinction between band and Mott insulators. The corresponding ``second-order transition'' can be observed out of equilibrium by driving a Cooper pair laser in the Mott insulator. We explicitly show that the gap for charged bosonic excitations lies below the threshold for Cooper pair breakup in any BEC regime, despite quantum fluctuations. Our discussion ends with a view of possible consequences for cuprates, where antinodal pair dynamics has certain features in common with our simple sss-wave picture.
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.
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.
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.