Pair Density Wave Research Articles

Detection of a Pair Density Wave State in UTe2

Detection of a Pair Density Wave State in UTe2

Pair density wave and loop current promoted by Van Hove singularities in moiré systems

We theoretically show that in the presence of conventional or higher order Van Hove singularities (VHS), the bare finite momentum pairing, also known as the pair density wave (PDW), susceptibility can be promoted to the same order of the most divergent bare BCS susceptibility through a valley-contrasting flux $3\ensuremath{\phi}$ in each triangular plaquette at $\ensuremath{\phi}=\ensuremath{\pi}/3$ and $\ensuremath{\pi}/6$ in moir\'e systems. This makes the PDW order a possible leading instability for an electronic system with repulsive interactions. We confirm that it indeed wins over all other instabilities and becomes the ground state under certain conditions through the renormalization group calculation and a flux insertion argument. Moreover, we also find that a topological nontrivial loop current order becomes the leading instability if the Fermi surface with conventional VHS is perfectly nested at $\ensuremath{\phi}=\ensuremath{\pi}/3$. Similar to the Haldane model, this loop current state has the quantum anomalous Hall effect. If we dope this loop current state or introduce a finite next-nearest-neighbor hopping ${t}^{\ensuremath{'}}$, the chiral $d$-wave PDW becomes the dominant instability. Experimentally, the flux can be effectively tuned by an out-of-plane electric field in moir\'e systems based on graphene and transition metal dichalcogenides.

Pair Density Wave Order from Electron Repulsion.

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.

Chern Fermi pocket, topological pair density wave, and charge-4e and charge-6e superconductivity in kagomé superconductors

The recent discovery of novel charge density wave (CDW) and pair density wave (PDW) in kagomé lattice superconductors AV3Sb5 (A = K, Rb, Cs) hints at unexpected time-reversal symmetry breaking correlated and topological states whose physical origin and broader implications are not understood. Here, we make conceptual advances toward a mechanism behind the striking observations and new predictions for novel macroscopic phase coherent quantum states. We show that the metallic CDW state with circulating loop currents is a doped orbital Chern insulator near van Hove filling. The emergent Chern Fermi pockets (CFPs) carry concentrated Berry curvature and orbital magnetic moment. We find that the pairing of electrons on the CFPs leads to a superconducting state with an emergent vortex-antivortex lattice and the formation of a complex triple-Q PDW. A plethora of correlated and topological states emerge, including a never-before-encountered chiral topological PDW superconductor, a loop-current pseudogap phase, and vestigial charge-4e and charge-6e superconductivity in staged melting of the vortex-antivortex lattice and hexatic liquid crystal. Our findings reveal previously unknown nature of the superconducting state of a current-carrying Chern metal, with broad implications for correlated and topological materials.

Testing for pair density wave order in La1.875Ba0.125CuO4

Charge order is commonly believed to compete with superconducting order. An intertwined form of superconducting wave function, known as pair-density-wave (PDW) order, has been proposed; however, there has been no direct evidence, theoretical or experimental, that it forms the ground state of any cuprate superconductor. As a test case, we consider \lbco\ with $x=1/8$, where charge and spin stripe orders within the CuO$_2$ planes compete with three-dimensional superconducting order. We report measurements of the superconducting critical current perpendicular to the planes in the presence of an in-plane magnetic field. The variation of the critical current with orientation of the field is inconsistent with a theoretical prediction specific to the PDW model. It appears, instead, that the orientation dependence of the critical-current density might be determined by a minority phase of $d$-wave superconductivity that is present as a consequence of doped-charge inhomogeneity.

Titanium doped kagome superconductor CsV3−xTixSb5 and two distinct phases

The vanadium-based kagome superconductor CsV3Sb5 has attracted tremendous attention due to its unexcepted anomalous Hall effect (AHE), charge density waves (CDWs), nematicity, and a pseudogap pair density wave (PDW) coexisting with unconventional strong-coupling superconductivity. The origins of CDWs, unconventional superconductivity, and their correlation with different electronic states in this kagome system are of great significance, but so far, are still under debate. Chemical doping in the kagome layer provides one of the most direct ways to reveal the intrinsic physics, but remains unexplored. Here, we report, for the first time, the synthesis of Ti-substituted CsV3Sb5 single crystals and its rich phase diagram mapping the evolution of intertwining electronic states. The Ti atoms directly substitute for V in the kagome layers. CsV3−xTixSb5 shows two distinct superconductivity phases upon substitution. The Ti slightly-substituted phase displays an unconventional V-shaped superconductivity gap, coexisting with weakening CDW, PDW, AHE, and nematicity. The Ti highly-substituted phase has a U-shaped superconductivity gap concomitant with a short-range rotation symmetry breaking CDW, while long-range CDW, twofold symmetry of in-plane resistivity, AHE, and PDW are absent. Furthermore, we also demonstrate the chemical substitution of V atoms with other elements such as Cr and Nb, showing a different modulation on the superconductivity phases and CDWs. These findings open up a way to synthesise a new family of doped CsV3Sb5 materials, and further represent a new platform for tuning the different correlated electronic states and superconducting pairing in kagome superconductors.

Solvable 3D Kondo Lattice Exhibiting Pair Density Wave, Odd-Frequency Pairing, and Order Fractionalization.

The Kondo lattice model plays a key role in our understanding of quantum materials, but a lack of small parameters has posed a long-standing problem. We present a three-dimensional S=1/2 Kondo lattice model describing a spin liquid within an electron sea. Strong correlations in the spin liquid are treated exactly, enabling a controlled analytical approach. Like a Peierls or BCS phase, a logarithmically divergent susceptibility leads to an instability into a new phase at arbitrarily small Kondo coupling. Our solution captures a plethora of emergent phenomena, including odd-frequency pairing, pair density wave formation and order fractionalization. The ground-state state is a pair density wave with a fractionalized charge e, S=1/2 order parameter, formed between electrons and Majorana fermions.

Interplay between Pair Density Wave and a Nested Fermi Surface.

We show that spontaneous time-reversal-symmetry (TRS) breaking can naturally arise from the interplay between pair density wave (PDW) ordering at multiple momenta and nesting of Fermi surfaces (FSs). Concretely, we consider the PDW superconductivity on a hexagonal lattice with nested FS at 3/4 electron filling, which is related to a recently discovered superconductor CsV_{3}Sb_{5}. Because of nesting of the FSs, each momentum k on the FS has at least two counterparts -k±Q_{α} (α=1, 2, 3) on the FS to form finite momentum (±Q_{α}) Cooper pairs, resulting in a TRS and inversion broken PDW state with stable Bogoliubov Fermi pockets. Various spectra, including (local) density of states, electron spectral function, and the effect of quasiparticle interference, have been investigated. The partial melting of the PDW will give rise to 4×4 and (4/sqrt[3])×(4/sqrt[3]) charge density wave (CDW) orders, in addition to the 2×2 CDW. Possible implications to real materials such as CsV_{3}Sb_{5} and future experiments have been discussed further.

Non-gapless excitation and zero-bias fast oscillations in the LDOS of surface superconducting states

Recently a novel surface pair-density-wave (PDW) superconducting state has been discovered in Barkman et al. (2019) and Samoilenka et al. (2020), which may go through a distinct multiple phase transition (MPT) when the superconductivity fades away from bulk to the boundary (e.g. edges and corners). Based on the Bogoliubov-de Gennes equations for the attractive tight-binding Hubbard modal in a one-dimensional chain, we demonstrate that the surface PDW state has a non-gapless quasiparticle spectrum, which is contrary to the conventional surface superconducting state. Moreover, we find that the MPT is associated with a zero-bias fast oscillating pattern in the LDOS near the surface. Our findings provide a potential experimental clue to identify the surface PDW state.

Enhanced pairing mechanism in cuprate-type crystals

Using a BCS mean-field approach, we show how the interplay between low-momentum optical phonons and Jahn-Teller-type lattice distortions can open an attractive channel that allows the formation of pairs with the corresponding density exhibiting characteristic features of a pair-density wave (PDW). We demonstrate this numerically on a copper-oxide type lattice.

Identification of a nematic pair density wave state in Bi2Sr2CaCu2O8+x

Electron-pair density wave (PDW) states are now an intense focus of research in the field of cuprate correlated superconductivity. PDWs exhibit periodically modulating superconductive electron pairing that can be visualized directly using scanned Josephson tunneling microscopy (SJTM). Although from theory, intertwining the d-wave superconducting (DSC) and PDW order parameters allows a plethora of global electron-pair orders to appear, which one actually occurs in the various cuprates is unknown. Here, we use SJTM to visualize the interplay of PDW and DSC states in Bi2Sr2CaCu2O8+x at a carrier density where the charge density wave modulations are virtually nonexistent. Simultaneous visualization of their amplitudes reveals that the intertwined PDW and DSC are mutually attractive states. Then, by separately imaging the electron-pair density modulations of the two orthogonal PDWs, we discover a robust nematic PDW state. Its spatial arrangement entails Ising domains of opposite nematicity, each consisting primarily of unidirectional and lattice commensurate electron-pair density modulations. Further, we demonstrate by direct imaging that the scattering resonances identifying Zn impurity atom sites occur predominantly within boundaries between these domains. This implies that the nematic PDW state is pinned by Zn atoms, as was recently proposed [Lozano etal., Phys. Rev. B 103, L020502 (2021)]. Taken in combination, these data indicate that the PDW in Bi2Sr2CaCu2O8+x is a vestigial nematic pair density wave state [Agterberg etal. Phys. Rev. B 91, 054502 (2015); Wardh and Granath arXiv:2203.08250].

Ground states of atomic Fermi gases in a two-dimensional optical lattice with and without population imbalance

We study the ground state phase diagram of population balanced and imbalanced ultracold atomic Fermi gases with a short range attractive interaction throughout the crossover from BCS to Bose-Einstein condensation (BEC), in a two-dimensional optical lattice (2DOL) comprised of two lattice and one continuum dimensions. We find that the mixing of lattice and continuum dimensions, together with population imbalance, has an extraordinary effect on pairing and the superfluidity of atomic Fermi gases. In the balanced case, the superfluid ground state prevails the majority of the phase space. However, for relatively small lattice hopping integral $t$ and large lattice constant $d$, a pair density wave (PDW) emerges unexpectedly at intermediate coupling strength, and the nature of the in-plane and overall pairing changes from particle-like to hole-like in the BCS and unitary regimes, associated with an abnormal increase in the Fermi volume with the pairing strength. In the imbalanced case, the stable polarized superfluid phase shrinks to only a small portion of the entire phase space spanned by $t$, $d$, imbalance $p$ and interaction strength $U$, mainly in the bosonic regime of low $p$, moderately strong pairing, and relatively large $t$ and small $d$. Due to the Pauli exclusion between paired and excessive fermions within the confined momentum space, a PDW phase emerges and the overall pairing evolves from particle-like into hole-like, as the pairing strength grows stronger in the BEC regime. In both cases, the ground state property is largely governed by the Fermi surface topology. These findings are very different from the cases of pure 3D continuum, 3D lattice or 1DOL.

Pseudogap phase and fractionalization: Predictions for Josephson junction setup

The pseudogap regime of the underdoped cuprates arguably remains one of the most enigmatic phenomena of correlated quantum matter. Recent theoretical ideas suggest that a pair density wave (PDW) or a "fractionalized PDW" could be a key ingredient for the understanding of the pseudogap physics. These ideas are to be contrasted to the scenario where charge density wave order and superconductivity coexist at low temperatures. In this paper, we present a few tests to compare the two scenarios in a Josephson junction setup. For a PDW scenario, we observe a beat-like structure of AC Josephson current. The additional frequencies for the AC Josephson current appear at the half-odd integer multiple of the standard Josephson frequency. We can extract the modulation wavevector of the PDW state by studying the average Josephson current. Furthermore, the usual sharp Shapiro steps break down. In contrast, these signatures are absent for the simple coexistence of orders. Any detection of such signatures in a similar experimental setup will strongly support the PDW scenario for the pseudogap phase.

Pair-density-wave superconductor from doping Haldane chain and rung-singlet ladder

We report the numerical discovery of a pair-density-wave (PDW) superconductor from doping either (i) a spin-one Haldane chain or (ii) a two-leg ladder in the rung singlet phase in which the doped charges occupy a single leg. We model these systems using a generalized Kondo model. The itinerant electrons are correlated and described by the $t-J$ model, and are further coupled to a spin $1/2$ Heisenberg model through the Kondo coupling $J_K$. When the density of electrons $x$ is one, the Mott insulator is in a Haldane phase or in a rung singlet phase depending on whether $J_K$ is negative or positive. Upon doping, a pair-density-wave with $\mathbf Q=\pi$ can emerge for both signs of $J_K$. In the $J_K \rightarrow -\infty$ limit, the model reduces to the recently proposed type II t-J model and we observes a continuous transition between the PDW superconductor and an unconventional Luttinger liquid phase with doping. We also identify a composite order parameter for the superconductor, which can be understood as a Cooper pair formed by two nearby fermionic spin-polarons. Our model and the predicted PDW phase can be experimentally realized by doping S=1 chains formed by Ni$^{2+}$ in a solid state system or a two-leg ladder of fermionic cold atoms with a potential bias between legs, which preferentially dopes carriers into a single leg.

Majorana fermion arcs and the local density of states of UTe2

$\text{UTe}_2$ is a leading candidate for chiral p-wave superconductivity, and for hosting exotic Majorana fermion quasiparticles. Motivated by recent STM experiments in this system, we study particle-hole symmetry breaking in chiral p-wave superconductors. We compute the local density of states from Majorana fermion surface states in the presence of Rashba surface spin-orbit coupling, which is expected to be sizeable in heavy-fermion materials like UTe$_2$. We show that time-reversal and surface reflection symmetry breaking lead to a natural pairing tendency towards a triplet pair density wave state, which naturally can account for broken particle-hole symmetry.

Emergent space-time supersymmetry at disordered quantum critical points

We study the effect of disorder on the space-time supersymmetry that is proposed to emerge at the quantum critical point of pair density wave transition in $(2+1)$-dimensional (D) Dirac semimetals and $(3+1)$D Weyl semimetals. In the $(2+1)$D Dirac semimetal, we consider three types of disorder, including random scalar potential, random vector potential and random mass potential, whereas the random mass disorder is absent in the $(3+1)$D Weyl semimetal. Via a systematic renormalization-group analysis, we find that any type of weak random disorder is irrelevant due to the couplings between the disorder potential and the Yukawa vertex. The emergent supersymmetry is, thus, stable against weak random potentials. Our paper will pave the way for exploration supersymmetry in realistic condensed-matter systems.

Doping a moiré Mott insulator: A t−J model study of twisted cuprates

We theoretically explore twisted cuprate multilayers, a moir\'e material family where the individual layers are themselves strongly correlated. We study the twisted t-J model, using a slave-boson mean field treatment that is compatible with Mott physics at small doping. Furthermore, we incorporate the interlayer tunneling form-factor dictated by the symmetry of the Cu ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ orbital. Including both these features leads to a phase diagram distinct from earlier weak-coupling treatments that predicted large gap spontaneous topological superconductors. Instead, we find that spontaneous time reversal (T) breaking occurs around twist angle of $\ensuremath{\theta}={45}^{\ensuremath{\circ}}$, but only in a narrow window. Moreover, a nearly gapless superconductor is obtained, whose spectroscopic features parallels that in monolayer cuprates, despite the broken time reversal and reflection symmetries. At smaller $\ensuremath{\theta}$ however, driving an interlayer current can lead to a gapped topological phase. The energy-phase relation of the interlayer Josephson junction displays notable double-Cooper-pair tunneling which dominates around ${45}^{\ensuremath{\circ}}$. The $\ensuremath{\theta}$ dependence of the Josephson critical current and the Shapiro steps are consistent with recent experiments. Utilizing the moir\'e structure as a probe of correlation physics, e.g., the pair density wave state, is discussed.

Charge, bond, and pair density wave orders in a strongly correlated system

The coexistence of multiple quasi-degenerate orders is the hallmark of the strongly correlated materials. Experiments often reveal several spatially modulated orders in the underdoped cuprates. This has come to the forefront with the possible detection of the pair density wave states. However, microscopic calculations often struggle to stabilize such spatially modulating orders as the ground state in the strong correlation limit. This work uses the $t-t^\prime-J$-model with an additional nearest-neighbor repulsion to stabilize spatially oscillating charge, bond, and pairing orders in the underdoped regime. We employ the standard Gutzwiller approach while treating the inhomogeneity for the spatial orders using the self-consistent Hartree-Fock-Bogoliubov methodology. Our calculations reveal that unidirectional bond density states coexisting with charge and pairing modulations can have lower energy than the uniform superconducting state over an extensive doping range. These modulating states vanish monotonically as the modulation wavevector becomes shorter with increased dopings. The finite momentum orders melt upon increasing doping to a vestigial nematic state which breaks the rotational symmetry of the system. The spatial features of the ground state at each doping reveal multiple wavevectors, which potentially drives the incommensuration of charge orders. Interestingly, the spatially modulating states are absent when the strong correlations criteria are relaxed, suggesting that the removal of double occupancy aids the stabilization of density wave orders.

Pair density wave and reentrant superconducting tendencies originating from valley polarization

We investigate superconducting pairing tendencies of a two-dimensional electron fluid with both valley and spin degrees of freedom, both without and in the presence of an in-plane magnetic field. We present suggestive theoretical arguments that spontaneous valley polarization can lead to exotic singlet superconducting tendencies, including pair-density-wave order at zero field and re-entrant superconductivity at high field. We also find a reduced magnetic response in the polarized valley, which allows a finite violation of the Pauli limit. These results are obtained by a mean-field approach to a generalized $t$-$J$ model on a triangular lattice and in the dilute limit. Phenomenological similarities to results of recent magic-angle twisted trilayer graphene experiments are noted.

Little-Parks like oscillations in lightly doped cuprate superconductors

Understanding the rich and competing electronic orders in cuprate superconductors may provide important insight into the mechanism of high-temperature superconductivity. Here, by measuring Bi2Sr2CaCu2O8+x in the extremely underdoped regime, we obtain evidence for a distinct type of ordering, which manifests itself as resistance oscillations at low magnetic fields (≤10 T) and at temperatures around the superconducting transition. By tuning the doping level p continuously, we reveal that these low-field oscillations occur only when p < 0.1. The oscillation amplitude increases with decreasing p but the oscillation period stays almost constant. We show that these low-field oscillations can be well described by assuming a periodic superconducting structure with a mesh size of about 50 nm. Such a charge order, which is distinctly different from the well-established charge density wave and pair density wave, seems to be an unexpected piece of the puzzle on the correlated physics in cuprates.