Two-dimensional superconducting fluctuations in stripe-ordered La1.875Ba0.125CuO4

Pair-Density-Wave Superconductivity: A Microscopic Model on the 2D Honeycomb Lattice

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.

Evidence of striped electronic phases in a structurally modulated superlattice

The electronic properties of crystals can be manipulated by superimposing spatially periodic electric, magnetic or structural modulations. Long-wavelength modulations incommensurate with the atomic lattice are particularly interesting1, exemplified by recent advances in two-dimensional (2D) moiré materials2,3. Bulk van der Waals (vdW) superlattices4-8 hosting 2D interfaces between minimally disordered layers represent scalable bulk analogues of artificial vdW heterostructures and present a complementary venue to explore incommensurately modulated 2D states. Here we report the bulk vdW superlattice SrTa2S5 realizing an incommensurate one-dimensional (1D) structural modulation of 2D transition metal dichalcogenide (TMD) H-TaS2 layers. High-quality electronic transport in the H-TaS2 layers, evidenced by quantum oscillations, is made anisotropic by the modulation and exhibits commensurability oscillations paralleling lithographically modulated 2D systems9-11. We also find unconventional, clean-limit superconductivity in SrTa2S5 with a pronounced suppression of interlayer relative to intralayer coherence. The in-plane magnetic field dependence of interlayer critical current, together with electron diffraction from the structural modulation, suggests superconductivity12-14 in SrTa2S5 is spatially modulated and mismatched between adjacent TMD layers. With phenomenology suggestive of pair-density wave superconductivity15-17, SrTa2S5 may present a pathway for microscopic evaluation of this unconventional order18-21. More broadly, SrTa2S5 establishes bulk vdW superlattices as versatile platforms to address long-standing predictions surrounding modulated electronic phases in the form of nanoscale vdW devices12,13 to macroscopic crystals22,23.

Optically Probing Unconventional Superconductivity in Atomically Thin Bi2Sr2Ca0.92Y0.08Cu2O8+δ

Atomically thin cuprates exhibiting a superconducting phase transition temperature similar to that of the bulk have recently been realized, although the device fabrication remains a challenge and limits the potential for many novel studies and applications. Here, we use an optical pump-probe approach to noninvasively study the unconventional superconductivity in atomically thin Bi2Sr2Ca0.92Y0.08Cu2O8+δ (Y-Bi2212). Apart from finding an optical response due to the superconducting phase transition that is similar to that of bulk Y-Bi2212, we observe that the sign and amplitude of the pump-probe signal in atomically thin flakes vary significantly in different dielectric environments depending on the nature of the optical excitation. By exploiting the spatial resolution of the optical probe, we uncover the exceptional sensitivity of monolayer Y-Bi2212 to the environment. Our results provide the first optical evidence for the intralayer nature of the superconducting condensate in Bi2212 and highlight the role of double-sided encapsulation in preserving superconductivity in atomically thin cuprates.

Designing the stripe-ordered cuprate phase diagram through uniaxial-stress

The ability to efficiently control charge and spin in the cuprate high-temperature superconductors is crucial for fundamental research and underpins technological development. Here, we explore the tunability of magnetism, superconductivity, and crystal structure in the stripe phase of the cuprate La[Formula: see text]Ba[Formula: see text]CuO[Formula: see text], with [Formula: see text] = 0.115 and 0.135, by employing temperature-dependent (down to 400 mK) muon-spin rotation and AC susceptibility, as well as X-ray scattering experiments under compressive uniaxial stress in the CuO[Formula: see text] plane. A sixfold increase of the three-dimensional (3D) superconducting critical temperature [Formula: see text] and a full recovery of the 3D phase coherence is observed in both samples with the application of extremely low uniaxial stress of [Formula: see text]0.1 GPa. This finding demonstrates the removal of the well-known 1/8-anomaly of cuprates by uniaxial stress. On the other hand, the spin-stripe order temperature as well as the magnetic fraction at 400 mK show only a modest decrease under stress. Moreover, the onset temperatures of 3D superconductivity and spin-stripe order are very similar in the large stress regime. However, strain produces an inhomogeneous suppression of the spin-stripe order at elevated temperatures. Namely, a substantial decrease of the magnetic volume fraction and a full suppression of the low-temperature tetragonal structure is found under stress, which is a necessary condition for the development of the 3D superconducting phase with optimal [Formula: see text]. Our results evidence a remarkable cooperation between the long-range static spin-stripe order and the underlying crystalline order with the three-dimensional fully coherent superconductivity. Overall, these results suggest that the stripe- and the SC order may have a common physical mechanism.

Universal Stripe Symmetry of Short-Range Charge Density Waves in Cuprate Superconductors

The omnipresence of charge density waves (CDWs) across almost all cuprate families underpins a common organizing principle. However, a longstanding debate of whether its spatial symmetry is stripe or checkerboard remains unresolved. While CDWs in lanthanum- and yttrium-based cuprates possess a stripe symmetry, distinguishing these two scenarios is challenging for the short-range CDW in bismuth-based cuprates. Here, high-resolution resonant inelastic x-ray scattering is employed to uncover the spatial symmetry of the CDW in Bi2 Sr2 - x Lax CuO6 + δ . Across a wide range of doping and temperature, anisotropic CDW peaks with elliptical shapes are found in reciprocal space. Based on Fourier transform analysis of real-space models, the results are interpreted as evidence of unidirectional charge stripes, hosted by mutually 90°-rotated anisotropic domains. This work paves the way for a unified symmetry and microscopic description of CDW order in cuprates.

Smectic pair-density-wave order in EuRbFe4As4

The pair density wave (PDW) is a superconducting state in which Cooper pairs carry centre-of-mass momentum in equilibrium, leading to the breaking of translational symmetry^1-4. Experimental evidence for such a state exists in high magnetic field^5-8 and in some materials that feature density-wave orders that explicitly break translational symmetry^9-13. However, evidence for a zero-field PDW state that exists independent of other spatially ordered states has so far been elusive. Here we show that such a state exists in the iron pnictide superconductor EuRbFe₄As₄, a material that features co-existing superconductivity (superconducting transition temperature (T_c) ≈ 37 kelvin) and magnetism (magnetic transition temperature (T_m) ≈ 15 kelvin)^14,15. Using spectroscopic imaging scanning tunnelling microscopy (SI-STM) measurements, we show that the superconducting gap at low temperature has long-range, unidirectional spatial modulations with an incommensurate period of about eight unit cells. Upon increasing the temperature above T_m, the modulated superconductor disappears, but a uniform superconducting gap survives to T_c. When an external magnetic field is applied, gap modulations disappear inside the vortex halo. The SI-STM and bulk measurements show the absence of other density-wave orders, indicating that the PDW state is a primary, zero-field superconducting state in this compound. Both four-fold rotational symmetry and translation symmetry are recovered above T_m, indicating that the PDW is a smectic order.

Mechanism for fluctuating pair density wave

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 superconductor or the widely postulated pair-density wave phase depending on the interaction strength. Our minimal model offers a unified framework to understand the emergence of both fluctuating and long range pair density waves in realistic systems.

The breakdown of both strange metal and superconducting states at a pressure-induced quantum critical point in iron-pnictide superconductors

Here we report the first observation of the concurrent breakdown of the strange metal (SM) normal state and superconductivity at a pressure-induced quantum critical point in Ca₁₀(Pt₄As₈)((Fe_0.97Pt_0.03)₂As₂)₅ superconductor. We find that, upon suppressing the superconducting state, the power exponent (α) changes from 1 to 2, and the slope of the temperature-linear resistivity per FeAs layer (A^□) gradually diminishes. At a critical pressure, A^□ and superconducting transition temperature (T_c) go to zero concurrently, where a quantum phase transition from a superconducting state with a SM normal state to a non-superconducting Fermi liquid state occurs. Scaling analysis reveals that the change of A^□ with T_c obeys the relation of T_c ~ (A^□)^0.5, similar to what is seen in other chemically doped unconventional superconductors. These results suggest that there is a simple but powerful organizational principle of connecting the SM normal state with the high-T_c superconductivity.

Emergent superconducting fluctuations in compressed kagome superconductor CsV3Sb5

The recent discovery of superconductivity (SC) and charge density wave (CDW) in kagome metals AV₃Sb₅ (A = K, Rb, Cs) provides an ideal playground for the study of emergent electronic orders. Application of moderate pressure leads to a two-dome-shaped SC phase regime in CsV₃Sb₅ accompanied by the destabilizing of CDW phase. Nonetheless, the nature of this pressure-tuned SC state and its interplay with the CDW are yet to be explored. Here, we perform soft point-contact spectroscopy (SPCS) measurements in CsV₃Sb₅ to investigate the evolution of superconducting order parameter with pressure. Surprisingly, we find that the superconducting gap is significantly enhanced between the two SC domes, at which the zero-resistance temperature is suppressed and the transition is remarkably broadened. Moreover, the temperature-dependence of the SC gap in this pressure range severely deviates from the conventional Bardeen-Cooper-Schrieffer (BCS) behavior, evidencing for strong Cooper pair phase fluctuations. These findings reveal the complex intertwining of the CDW with SC in the compressed CsV₃Sb₅, suggesting striking parallel to the cuprate superconductor La_2-xBa_xCuO₄. Our results point to the essential role of charge degree of freedom in the development of intertwining electronic orders, and thus provide new constraints for theories.

Emergent charge order in pressurized kagome superconductor CsV3Sb5

The discovery of several electronic orders in kagome superconductors AV₃Sb₅ (A means K, Rb, Cs) provides a promising platform for exploring unprecedented emergent physics^1-9. Under moderate pressure (<2.2 GPa), the triple-Q charge density wave (CDW) order is monotonically suppressed by pressure, while the superconductivity shows a two-dome-like behaviour, suggesting an unusual interplay between superconductivity and CDW order^10,11. Given that time-reversal symmetry breaking and electronic nematicity have been revealed inside the triple-Q CDW phase^8,9,12,13, understanding this CDW order and its interplay with superconductivity becomes one of the core questions in AV₃Sb₅ (refs. ^3,5,6). Here, we report the evolution of CDW and superconductivity with pressure in CsV₃Sb₅ by ⁵¹V nuclear magnetic resonance measurements. An emergent CDW phase, ascribed to a possible stripe-like CDW order with a unidirectional 4a₀ modulation, is observed between P_c1 ≅ 0.58 GPa and P_c2 ≅ 2.0 GPa, which explains the two-dome-like superconducting behaviour under pressure. Furthermore, the nuclear spin-lattice relaxation measurement reveals evidence for pressure-independent charge fluctuations above the CDW transition temperature and unconventional superconducting pairing above P_c2. Our results not only shed new light on the interplay of superconductivity and CDW, but also reveal new electronic correlation effects in kagome superconductors AV₃Sb₅.

Observation of Superconducting Collective Modes from Competing Pairing Instabilities in Single-Layer NbSe2

In certain unconventional superconductors with sizable electronic correlations, the availability of closely competing pairing channels leads to characteristic soft collective fluctuations of the order parameters, which leave fingerprints in many observables and allow the phase competition to be scrutinized. Superconducting layered materials, where electron-electron interactions are enhanced with decreasing thickness, are promising candidates to display these correlation effects. In this work, the existence of a soft collective mode in single-layer NbSe₂ , observed as a characteristic resonance excitation in high-resolution tunneling spectra is reported. This resonance is observed along with higher harmonics, its frequency Ω/2Δ is anticorrelated with the local superconducting gap Δ, and its amplitude gradually vanishes by increasing the temperature and upon applying a magnetic field up to the critical values (T_C and H_C2 ), which sets an unambiguous link to the superconducting state. Aided by a microscopic model that captures the main experimental observations, this resonance is interpreted as a collective Leggett mode that represents the fluctuation toward a proximate f-wave triplet state, due to subleading attraction in the triplet channel. These findings demonstrate the fundamental role of correlations in superconducting 2D transition metal dichalcogenides, opening a path toward unconventional superconductivity in simple, scalable, and transferable 2D superconductors.

Unveiling Unequivocal Charge Stripe Order in a Prototypical Cuprate Superconductor

In the cuprates, high-temperature superconductivity, spin-density-wave order, and charge-density-wave (CDW) order are intertwined, and symmetry determination is challenging due to domain formation. We investigated the CDW in the prototypical cuprate La_{1.88}Sr_{0.12}CuO_{4} via x-ray diffraction employing uniaxial pressure as a domain-selective stimulus to establish the unidirectional nature of the CDW unambiguously. A fivefold enhancement of the CDW amplitude is found when homogeneous superconductivity is partially suppressed by magnetic field. This field-induced state provides an ideal search environment for a putative pair-density-wave state.

Phase Coherence of Pairs of Cooper Pairs as Quasi-Long-Range Order of Half-Vortex Pairs in a Two-Dimensional Bilayer System

It is known that the loss of phase coherence of Cooper pairs in two-dimensional superconductivity corresponds to the unbinding of vortex-antivortex pairs with the quasi-long-range order in the order-parameter phase field, described by the Berezinskii-Kosterlizt-Thouless (BKT) transition of a 2D XY model. Here we show that the second-order Josephson coupling can induce an exotic superconducting phase in a bilayer system. By using tensor-network methods, the partition function of the 2D classical model is expressed as a product of 1D quantum transfer operator, whose eigenequation can be solved by an algorithm of matrix product states rigorously. From the singularity shown by the entanglement entropy of the 1D quantum analog, various phase transitions can be accurately determined. Below the BKT phase transition, an interlayer Ising long-range order is established at T_{Ising}, and the phase coherence of both intralayers and interlayers is locked together. For two identical layers, the Ising transition coincides with the BKT transition at a multicritical point. For two inequivalent layers, however, there emerges an intermediate quasi-long-range order phase (T_{Ising}<T<T_{BKT}), where the vortex-antivortex bindings occur in the layer with the larger intralayer coupling, but only half-vortex pairs with topological strings exist in the other layer, corresponding to the phase coherence of pairs of Cooper pairs. So our study provides a promising way to realize the charge-4e superconductivity in a bilayer system.

Topological Doping and Superconductivity in Cuprates: An Experimental Perspective

Hole doping into a correlated antiferromagnet leads to topological stripe correlations, involving charge stripes that separate antiferromagnetic spin stripes of opposite phases. The topological spin stripe order causes the spin degrees of freedom within the charge stripes to feel a geometric frustration with their environment. In the case of cuprates, where the charge stripes have the character of a hole-doped two-leg spin ladder, with corresponding pairing correlations, anti-phase Josephson coupling across the spin stripes can lead to a pair-density-wave order in which the broken translation symmetry of the superconducting wave function is accommodated by pairs with finite momentum. This scenario is now experimentally verified by recently reported measurements on La2−xBaxCuO4 with x=1/8. While pair-density-wave order is not common as a cuprate ground state, it provides a basis for understanding the uniform d-wave order that is more typical in superconducting cuprates.

Possible Evidence for Berezinskii-Kosterlitz-Thouless Transition in Ba(Fe0.914Co0.086)2As2 Crystals

In this study, we measure the in-plane transport properties of high-quality Ba(Fe_0.914Co_0.086)₂As₂ single crystals. Signatures of vortex unbinding Berezinskii–Kosterlitz–Thouless (BKT) transition are shown from both the conventional approach and the Fisher–Fisher–Huse dynamic scaling analysis, in which a characteristic Nelson–Kosterlitz jump is demonstrated. We also observe a non-Hall transverse signal exactly at the superconducting transition, which is explained in terms of guided motion of unbound vortices.

Unusual competition of superconductivity and charge-density-wave state in a compressed topological kagome metal

Understanding the competition between superconductivity and other ordered states (such as antiferromagnetic or charge-density-wave (CDW) state) is a central issue in condensed matter physics. The recently discovered layered kagome metal AV3Sb5 (A = K, Rb, and Cs) provides us a new playground to study the interplay of superconductivity and CDW state by involving nontrivial topology of band structures. Here, we conduct high-pressure electrical transport and magnetic susceptibility measurements to study CsV3Sb5 with the highest Tc of 2.7 K in AV3Sb5 family. While the CDW transition is monotonically suppressed by pressure, superconductivity is enhanced with increasing pressure up to P1 ≈ 0.7 GPa, then an unexpected suppression on superconductivity happens until pressure around 1.1 GPa, after that, Tc is enhanced with increasing pressure again. The CDW is completely suppressed at a critical pressure P2 ≈ 2 GPa together with a maximum Tc of about 8 K. In contrast to a common dome-like behavior, the pressure-dependent Tc shows an unexpected double-peak behavior. The unusual suppression of Tc at P1 is concomitant with the rapidly damping of quantum oscillations, sudden enhancement of the residual resistivity and rapid decrease of magnetoresistance. Our discoveries indicate an unusual competition between superconductivity and CDW state in pressurized kagome lattice.

Two-Dimensional Superconducting Fluctuations Associated with Charge-Density-Wave Stripes in La_{1.87}Sr_{0.13}Cu_{0.99}Fe_{0.01}O_{4}

The presence of a small concentration of in-plane Fe dopants in La_{1.87}Sr_{0.13}Cu_{0.99}Fe_{0.01}O_{4} is known to enhance stripelike spin and charge density wave (SDW and CDW) order and suppress the superconducting T_{c}. Here, we show that it also induces highly two-dimensional superconducting correlations that have been argued to be the signatures of a new form of superconducting order, the so-called pair density wave (PDW) order. In addition, using resonant soft x-ray scattering, we find that the two-dimensional superconducting fluctuation is strongly associated with the CDW stripe. In particular, the PDW signature first appears when the correlation length of the CDW stripe grows over eight times the lattice unit (∼8a). These results provide critical conditions for the formation of the PDW order.

Quantum Monte Carlo study of the Hubbard model with next-nearest-neighbor hopping t': pairing and magnetism

Using the finite-temperature determinant quantum Monte Carlo (DQMC) algorithm, we study the pairing symmetries of the Hubbard Hamiltonian with next-nearest-neighbor (NNN) hopping t' on square lattices. By varying the value of t', we find that the d-wave pairing is suppressed by the onset of t', while the p + ip-wave pairing tends to emerge for low electron density and t' around -0.7. Together with the calculation of the anti-ferromagnetic and ferromagnetic spin correlation function, we explore the relationship between anti-ferromagnetic order and the d-wave pairing symmetry, and the relationship between ferromagnetic order and the p + ip-wave pairing symmetry. Our results may be useful for the exploration of the mechanism of the electron pairing symmetries, and for the realization of the exotic p + ip-wave superconductivity.

Phase Separation and Pairing Fluctuations in Oxide Materials

The microscopic mechanism of charge instabilities and the formation of inhomogeneous states in systems with strong electron correlations is investigated. We demonstrate that within a strong coupling expansion the single-band Hubbard model shows an instability towards phase separation and extend the approach also for an analysis of phase separation in the Hubbard-Kanamori hamiltonian as a prototypical multiband model. We study the pairing fluctuations on top of an inhomogeneous stripe state where superconducting correlations in the extended s-wave and d-wave channels correspond to (anti)bound states in the two-particle spectra. Whereas extended s-wave fluctuations are relevant on the scale of the local interaction parameter U, we find that d-wave fluctuations are pronounced in the energy range of the active subband which crosses the Fermi level. As a result, low energy spin and charge fluctuations can transfer the d-wave correlations from the bound states to the low energy quasiparticle bands. Our investigations therefore help to understand the coexistence of stripe correlations and d-wave superconductivity in cuprates.

Clean 2D superconductivity in a bulk van der Waals superlattice

Advances in low-dimensional superconductivity are often realized through improvements in material quality. Apart from a small group of organic materials, there is a near absence of clean-limit two-dimensional (2D) superconductors, which presents an impediment to the pursuit of numerous long-standing predictions for exotic superconductivity with fragile pairing symmetries. We developed a bulk superlattice consisting of the transition metal dichalcogenide (TMD) superconductor 2H-niobium disulfide (2H-NbS2) and a commensurate block layer that yields enhanced two-dimensionality, high electronic quality, and clean-limit inorganic 2D superconductivity. The structure of this material may naturally be extended to generate a distinct family of 2D superconductors, topological insulators, and excitonic systems based on TMDs with improved material properties.

Preformed Cooper Pairs in Layered FeSe-Based Superconductors

Superconductivity arises from two distinct quantum phenomena: electron pairing and long-range phase coherence. In conventional superconductors, the two quantum phenomena generally take place simultaneously, while in the underdoped high- T_{c} cuprate superconductors, the electron pairing occurs at higher temperature than the long-range phase coherence. Recently, whether electron pairing is also prior to long-range phase coherence in single-layer FeSe film on SrTiO_{3} substrate is under debate. Here, by measuring Knight shift and nuclear spin-lattice relaxation rate, we unambiguously reveal a pseudogap behavior below T_{p}∼60  K in two kinds of layered FeSe-based superconductors with quasi2D nature. In the pseudogap regime, a weak diamagnetic signal and a remarkable Nernst effect are also observed, which indicates that the observed pseudogap behavior is related to superconducting fluctuations. These works confirm that strong phase fluctuation is an important character in the 2D iron-based superconductors as widely observed in high-T_{c} cuprate superconductors.

Using Uniaxial Stress to Probe the Relationship between Competing Superconducting States in a Cuprate with Spin-stripe Order

We report muon spin rotation and magnetic susceptibility experiments on in-plane stress effects on the static spin-stripe order and superconductivity in the cuprate system La_{2-x}Ba_{x}CuO_{4} with x=0.115. An extremely low uniaxial stress of ∼0.1  GPa induces a substantial decrease in the magnetic volume fraction and a dramatic rise in the onset of 3D superconductivity, from ∼10 to 32 K; however, the onset of at-least-2D superconductivity is much less sensitive to stress. These results show not only that large-volume-fraction spin-stripe order is anticorrelated with 3D superconducting coherence but also that these states are energetically very finely balanced. Moreover, the onset temperatures of 3D superconductivity and spin-stripe order are very similar in the large stress regime. These results strongly suggest a similar pairing mechanism for spin-stripe order and the spatially modulated 2D and uniform 3D superconducting orders, imposing an important constraint on theoretical models.

Pair density wave at high magnetic fields in cuprates with charge and spin orders

In underdoped cuprates, the interplay of the pseudogap, superconductivity, and charge and spin ordering can give rise to exotic quantum states, including the pair density wave (PDW), in which the superconducting (SC) order parameter is oscillatory in space. However, the evidence for a PDW state remains inconclusive and its broader relevance to cuprate physics is an open question. To test the interlayer frustration, the crucial component of the PDW picture, we perform transport measurements on charge- and spin-stripe-ordered La_1.7Eu_0.2Sr_0.1CuO₄ and La_1.48Nd_0.4Sr_0.12CuO₄ in perpendicular magnetic fields (H_⊥), and also with an additional field applied parallel to CuO₂ layers (H_∥). We detect several phenomena predicted to arise from the existence of a PDW, including an enhancement of interlayer SC phase coherence with increasing H_∥. These data also provide much-needed transport signatures of the PDW in the regime where superconductivity is destroyed by quantum phase fluctuations.

Among the exotic phases in underdoped cuprates, the evidence of a pair density wave (PDW) remains inconclusive. Here, Shi et al. report transport signatures consistent with the presence of PDW pairing correlations that compete with uniform superconductivity in two underdoped cuprate superconductors.

Atomic-scale electronic structure of the cuprate pair density wave state coexisting with superconductivity

By making a variety of quantitative comparisons between electronic visualization experiments and a theory describing coexisting pair density wave and superconductive states in cuprates, we find striking correspondence throughout. Our model can thus explain the microscopic origins of many key atomic-scale phenomena of the cuprate broken-symmetry state. These observations are consistent with the possibility that a short-range pair density wave (PDW) state coexists with superconductivity below a critical hole density in Bi₂Sr₂CaCu₂O₈, that the charge density wave modulations in cuprates are a consequence of the PDW state, that the cuprate pseudogap is the antinodal gap of the PDW, and that the critical point in the cuprate phase diagram occurs due to disappearance of the PDW.

The defining characteristic of hole-doped cuprates is d-wave high temperature superconductivity. However, intense theoretical interest is now focused on whether a pair density wave state (PDW) could coexist with cuprate superconductivity [D. F. Agterberg et al., Annu. Rev. Condens. Matter Phys. 11, 231 (2020)]. Here, we use a strong-coupling mean-field theory of cuprates, to model the atomic-scale electronic structure of an eight-unit-cell periodic, d-symmetry form factor, pair density wave (PDW) state coexisting with d-wave superconductivity (DSC). From this PDW + DSC model, the atomically resolved density of Bogoliubov quasiparticle states Nr,E is predicted at the terminal BiO surface of Bi₂Sr₂CaCu₂O₈ and compared with high-precision electronic visualization experiments using spectroscopic imaging scanning tunneling microscopy (STM). The PDW + DSC model predictions include the intraunit-cell structure and periodic modulations of Nr,E, the modulations of the coherence peak energy Δpr, and the characteristics of Bogoliubov quasiparticle interference in scattering-wavevector space q-space. Consistency between all these predictions and the corresponding experiments indicates that lightly hole-doped Bi₂Sr₂CaCu₂O₈ does contain a PDW + DSC state. Moreover, in the model the PDW + DSC state becomes unstable to a pure DSC state at a critical hole density p*, with empirically equivalent phenomena occurring in the experiments. All these results are consistent with a picture in which the cuprate translational symmetry-breaking state is a PDW, the observed charge modulations are its consequence, the antinodal pseudogap is that of the PDW state, and the cuprate critical point at p* ≈ 19% occurs due to disappearance of this PDW.

Imaging the energy gap modulations of the cuprate pair-density-wave state

The defining characteristic^1,2 of Cooper pairs with finite centre-of-mass momentum is a spatially modulating superconducting energy gap Δ(r), where r is a position. Recently, this concept has been generalized to the pair-density-wave (PDW) state predicted to exist in copper oxides (cuprates)^3,4. Although the signature of a cuprate PDW has been detected in Cooper-pair tunnelling⁵, the distinctive signature in single-electron tunnelling of a periodic Δ(r) modulation has not been observed. Here, using a spectroscopic technique based on scanning tunnelling microscopy, we find strong Δ(r) modulations in the canonical cuprate Bi₂Sr₂CaCu₂O_8+δ that have eight-unit-cell periodicity or wavevectors Q ≈ (2π/a₀)(1/8, 0) and Q ≈ (2π/a₀)(0, 1/8) (where a₀ is the distance between neighbouring Cu atoms). Simultaneous imaging of the local density of states N(r, E) (where E is the energy) reveals electronic modulations with wavevectors Q and 2Q, as anticipated when the PDW coexists with superconductivity. Finally, by visualizing the topological defects in these N(r, E) density waves at 2Q, we find them to be concentrated in areas where the PDW spatial phase changes by π, as predicted by the theory of half-vortices in a PDW state^6,7. Overall, this is a compelling demonstration, from multiple single-electron signatures, of a PDW state coexisting with superconductivity in Bi₂Sr₂CaCu₂O_8+δ.

Vortex phase diagram and the normal state of cuprates with charge and spin orders

The phase diagram of underdoped cuprates in a magnetic field (H) is key to understanding the anomalous normal state of these high-temperature superconductors. However, the upper critical field (H _c2), the extent of superconducting (SC) phase with vortices, and the role of charge orders at high H remain controversial. Here we study stripe-ordered La-214, i.e., cuprates in which charge orders are most pronounced and zero-field SC transition temperatures T c 0 are lowest. This enables us to explore the vortex phases in a previously inaccessible energy scale window. By combining linear and nonlinear transport techniques sensitive to vortex matter, we determine the T - H phase diagram, directly detect H _c2, and reveal novel properties of the high-field ground state. Our results demonstrate that quantum fluctuations and disorder play a key role as T → 0, while the high-field ground state is likely a metal, not an insulator, due to the presence of stripes.

Anomalies in the pseudogap phase of the cuprates: competing ground states and the role of umklapp scattering

Over the past two decades, advances in computational algorithms have revealed a curious property of the two-dimensional Hubbard model (and related theories) with hole doping: the presence of close-in-energy competing ground states that display very different physical properties. On the one hand, there is a complicated state exhibiting intertwined spin, charge, and pair density wave orders. We call this 'type A'. On the other hand, there is a uniform d-wave superconducting state that we denote as 'type B'. We advocate, with the support of both microscopic theoretical calculations and experimental data, dividing the high-temperature cuprate superconductors into two corresponding families, whose properties reflect either the type A or type B ground states at low temperatures. We review the anomalous properties of the pseudogap phase that led us to this picture, and present a modern perspective on the role that umklapp scattering plays in these phenomena in the type B materials. This reflects a consistent framework that has emerged over the last decade, in which Mott correlations at weak coupling drive the formation of the pseudogap. We discuss this development, recent theory and experiments, and open issues.

Magnetic field-induced pair density wave state in the cuprate vortex halo

High magnetic fields suppress cuprate superconductivity to reveal an unusual density wave (DW) state coexisting with unexplained quantum oscillations. Although routinely labeled a charge density wave (CDW), this DW state could actually be an electron-pair density wave (PDW). To search for evidence of a field-induced PDW, we visualized modulations in the density of electronic states N(r) within the halo surrounding Bi₂Sr₂CaCu₂O₈ vortex cores. We detected numerous phenomena predicted for a field-induced PDW, including two sets of particle-hole symmetric N(r) modulations with wave vectors Q_P and 2Q _P , with the latter decaying twice as rapidly from the core as the former. These data imply that the primary field-induced state in underdoped superconducting cuprates is a PDW, with approximately eight CuO₂ unit-cell periodicity and coexisting with its secondary CDWs.

Photoenhanced metastable c-axis electrodynamics in stripe-ordered cuprate La1.885Ba0.115CuO4

The emergence of superconductivity in high-temperature cuprates arises out of a rich landscape of competing order. For example, stripe order can hoard the electrons needed to form Cooper pairs and establish superconductivity. Intriguingly, the complex interactions of such intertwined orders can be manipulated with light, where nonequilibrium dynamics alter the primacy of one order over another. Following photoexcitation of La_2−xBa_xCuO₄ (x = 0.115) with near-infrared pulses, we observe a long-lived state that exhibits enhanced superconducting correlations well above the equilibrium superconducting transition temperature. Our analysis reveals that this metastable phase arises from a collapse of stripe order, providing an important demonstration of light-directed control in quantum materials.

Quantum materials are amenable to nonequilibrium manipulation with light, enabling modification and control of macroscopic properties. Light-based augmentation of superconductivity is particularly intriguing. Copper-oxide superconductors exhibit complex interplay between spin order, charge order, and superconductivity, offering the prospect of enhanced coherence by altering the balance between competing orders. We utilize terahertz time-domain spectroscopy to monitor the c-axis Josephson plasma resonance (JPR) in La_2−xBa_xCuO₄ (x = 0.115) as a direct probe of superconductivity dynamics following excitation with near-infrared pulses. Starting from the superconducting state, c-axis polarized excitation with a fluence of 100 μJ/cm² results in an increase of the far-infrared spectral weight by more than an order of magnitude as evidenced by a blueshift of the JPR, interpreted as resulting from nonthermal collapse of the charge order. The photoinduced signal persists well beyond our measurement window of 300 ps and exhibits signatures of spatial inhomogeneity. The electrodynamic response of this metastable state is consistent with enhanced superconducting fluctuations. Our results reveal that La_2−xBa_xCuO₄ is highly sensitive to nonequilibrium excitation over a wide fluence range, providing an unambiguous example of photoinduced modification of order-parameter competition.

Observation of two types of charge-density-wave orders in superconducting La2-xSrxCuO4

The discovery of charge- and spin-density-wave (CDW/SDW) orders in superconducting cuprates has altered our perspective on the nature of high-temperature superconductivity (SC). However, it has proven difficult to fully elucidate the relationship between the density wave orders and SC. Here, using resonant soft X-ray scattering, we study the archetypal cuprate La_2-xSr_xCuO₄ (LSCO) over a broad doping range. We reveal the existence of two types of CDW orders in LSCO, namely CDW stripe order and CDW short-range order (SRO). While the CDW-SRO is suppressed by SC, it is partially transformed into the CDW stripe order with developing SDW stripe order near the superconducting T_c. These findings indicate that the stripe orders and SC are inhomogeneously distributed in the superconducting CuO₂ planes of LSCO. This further suggests a new perspective on the putative pair-density-wave order that coexists with SC, SDW, and CDW orders.

To fully elucidate the relationship between density wave orders and superconductivity in high-T_c cuprates remains difficult. Here, the authors reveal two types of charge-density-wave orders and their intertwined relationship with spin-density-wave order and superconductivity in La_2-xSr_xCuO_4.

Superconductor-metal transition in odd-frequency-paired superconductor in a magnetic field

It is generally expected that when a magnetic field destroys superconductivity in two dimensions, the system becomes an insulator. It is shown that there is a type of superconductivity—namely, the one where the wave function of pairs is odd in time—where the result is not an insulator, but a metal with a zero Hall response. It is suggested that the transition recently observed in the striped-ordered high-Tc superconductor La_1.875Ba_0.125CuO₄ may belong to this category.

It is shown that the application of a sufficiently strong magnetic field to the odd-frequency–paired pair-density wave state described in A. M. Tsvelik [Phys. Rev. B 94, 165114 (2016)] leads to formation of a low-temperature metallic state with zero Hall response. Applications of these ideas to the recent experiments on stripe-ordered La_1.875Ba_0.125CuO₄ (LBCO) are discussed.

Phase transition in the cuprates from a magnetic-field-free stiffness meter viewpoint

A method to measure the superconducting (SC) stiffness tensor [Formula: see text], without subjecting the sample to external magnetic field, is applied to La_1.875Sr_0.125CuO₄. The method is based on the London equation [Formula: see text], where J is the current density and A is the vector potential which is applied in the SC state. Using rotor free A and measuring J via the magnetic moment of superconducting rings, [Formula: see text] at T → T_c is extracted. The technique is sensitive to very small stiffnesses (penetration depths on the order of a few millimeters). The method is applied to two different rings: one with the current running only in the CuO₂ planes, and another where the current must cross planes. We find different transition temperatures for the two rings, namely, there is a temperature range with two-dimensional stiffness. Additional low energy muon spin rotation measurements on the same sample determine the stiffness anisotropy at T < T_c.

Tuning from failed superconductor to failed insulator with magnetic field

Do charge modulations compete with electron pairing in high-temperature copper oxide superconductors? We investigated this question by suppressing superconductivity in a stripe-ordered cuprate compound at low temperature with high magnetic fields. With increasing field, loss of three-dimensional superconducting order is followed by reentrant two-dimensional superconductivity and then an ultraquantum metal phase. Circumstantial evidence suggests that the latter state is bosonic and associated with the charge stripes. These results provide experimental support to the theoretical perspective that local segregation of doped holes and antiferromagnetic spin correlations underlies the electron-pairing mechanism in cuprates.

Measuring non-equilibrium dynamics in complex solids with ultrashort X-ray pulses

Strong interactions between electrons give rise to the complexity of quantum materials, which exhibit exotic functional properties and extreme susceptibility to external perturbations. A growing research trend involves the study of these materials away from equilibrium, especially in cases in which the stimulation with optical pulses can coherently enhance cooperative orders. Time-resolved X-ray probes are integral to this type of research, as they can be used to track atomic and electronic structures as they evolve on ultrafast timescales. Here, we review a series of recent experiments where femtosecond X-ray diffraction was used to measure dynamics of complex solids. This article is part of the theme issue 'Measurement of ultrafast electronic and structural dynamics with X-rays'.

Disorder raises the critical temperature of a cuprate superconductor

With the discovery of charge-density waves (CDWs) in most members of the cuprate high-temperature superconductors, the interplay between superconductivity and CDWs has become a key point in the debate on the origin of high-temperature superconductivity. Some experiments in cuprates point toward a CDW state competing with superconductivity, but others raise the possibility of a CDW-superconductivity intertwined order or more elusive pair-density waves (PDWs). Here, we have used proton irradiation to induce disorder in crystals of [Formula: see text] and observed a striking 50% increase of [Formula: see text], accompanied by a suppression of the CDWs. This is in sharp contrast with the behavior expected of a d-wave superconductor, for which both magnetic and nonmagnetic defects should suppress [Formula: see text] Our results thus make an unambiguous case for the strong detrimental effect of the CDW on bulk superconductivity in [Formula: see text] Using tunnel diode oscillator (TDO) measurements, we find indications for potential dynamic layer decoupling in a PDW phase. Our results establish irradiation-induced disorder as a particularly relevant tuning parameter for the many families of superconductors with coexisting density waves, which we demonstrate on superconductors such as the dichalcogenides and [Formula: see text].

Pair Density Wave in the Doped t-J Model with Ring Exchange on a Triangular Lattice

In our previous work [Phys. Rev. Lett. 121, 046401 (2018)PRLTAO0031-900710.1103/PhysRevLett.121.046401], we found a quantum spin liquid phase with a spinon Fermi surface in the two dimensional spin-1/2 Heisenberg model with four-spin ring exchange on a triangular lattice. In this work we dope the spinon Fermi surface phase by studying the t-J model with four-spin ring exchange. We perform density matrix renormalization group calculations on four-leg cylinders of a triangular lattice and find that the dominant pair correlation function is that of a pair density wave; i.e., it is oscillatory while decaying with distance with a power law. The doping dependence of the period is studied. This is the first example where a pair density wave is the dominant pairing in a generic strongly interacting system where the pair density wave cannot be explained as a composite order and no special symmetry is required.

Charge density wave memory in a cuprate superconductor

Although CDW correlations are a ubiquitous feature of the superconducting cuprates, their disparate properties suggest a crucial role for pinning the CDW to the lattice. Here, we report coherent resonant X-ray speckle correlation analysis, which directly determines the reproducibility of CDW domain patterns in La_1.875Ba_0.125CuO₄ (LBCO 1/8) with thermal cycling. While CDW order is only observed below 54 K, where a structural phase transition creates inequivalent Cu-O bonds, we discover remarkably reproducible CDW domain memory upon repeated cycling to far higher temperatures. That memory is only lost on cycling to 240(3) K, which recovers the four-fold symmetry of the CuO₂ planes. We infer that the structural features that develop below 240 K determine the CDW pinning landscape below 54 K. This opens a view into the complex coupling between charge and lattice degrees of freedom in superconducting cuprates.

Magnetic-Field Tuning of Light-Induced Superconductivity in Striped La_{2-x}Ba_{x}CuO_{4}

Optical excitation of stripe-ordered La_{2-x}Ba_{x}CuO_{4} has been shown to transiently enhance superconducting tunneling between the CuO_{2} planes. This effect was revealed by a blueshift, or by the appearance of a Josephson plasma resonance in the terahertz-frequency optical properties. Here, we show that this photoinduced state can be strengthened by the application of high external magnetic fields oriented along the c axis. For a 7 T field, we observe up to a tenfold enhancement in the transient interlayer phase correlation length, accompanied by a twofold increase in the relaxation time of the photoinduced state. These observations are highly surprising, since static magnetic fields suppress interlayer Josephson tunneling and stabilize stripe order at equilibrium. We interpret our data as an indication that optically enhanced interlayer coupling in La_{2-x}Ba_{x}CuO_{4} does not originate from a simple optical melting of stripes, as previously hypothesized. Rather, we speculate that the photoinduced state may emerge from activated tunneling between optically excited stripes in adjacent planes.

Emergence of superconductivity in the cuprates via a universal percolation process

A pivotal step toward understanding unconventional superconductors would be to decipher how superconductivity emerges from the unusual normal state. In the cuprates, traces of superconducting pairing appear above the macroscopic transition temperature Tc, yet extensive investigation has led to disparate conclusions. The main difficulty has been to separate superconducting contributions from complex normal-state behaviour. Here we avoid this problem by measuring nonlinear conductivity, an observable that is zero in the normal state. We uncover for several representative cuprates that the nonlinear conductivity vanishes exponentially above Tc, both with temperature and magnetic field, and exhibits temperature-scaling characterized by a universal scale Ξ0. Attempts to model the response with standard Ginzburg-Landau theory are systematically unsuccessful. Instead, our findings are captured by a simple percolation model that also explains other properties of the cuprates. We thus resolve a long-standing conundrum by showing that the superconducting precursor in the cuprates is strongly affected by intrinsic inhomogeneity.

Lighting up superconducting stripes

Terahertz radiation reveals pair density waves in underdoped cuprate superconductors

Probing optically silent superfluid stripes in cuprates

Unconventional superconductivity in the cuprates coexists with other types of electronic order. However, some of these orders are invisible to most experimental probes because of their symmetry. For example, the possible existence of superfluid stripes is not easily validated with linear optics, because the stripe alignment causes interlayer superconducting tunneling to vanish on average. Here we show that this frustration is removed in the nonlinear optical response. A giant terahertz third harmonic, characteristic of nonlinear Josephson tunneling, is observed in La_1.885Ba_0.115CuO₄ above the transition temperature T_c = 13 kelvin and up to the charge-ordering temperature T_co = 55 kelvin. We model these results by hypothesizing the presence of a pair density wave condensate, in which nonlinear mixing of optically silent tunneling modes drives large dipole-carrying supercurrents.

Charge-Stripe Order and Superconductivity in Ir1-xPtxTe2

A combined resistivity and hard x-ray diffraction study of superconductivity and charge ordering in Ir Ir_1−xPt_xTe₂, as a function of Pt substitution and externally applied hydrostatic pressure, is presented. Experiments are focused on samples near the critical composition x_c ~ 0.045 where competition and switching between charge order and superconductivity is established. We show that charge order as a function of pressure in Ir_0.95Pt_0.05Te₂ is preempted — and hence triggered — by a structural transition. Charge ordering appears uniaxially along the short crystallographic (1, 0, 1) domain axis with a (1/5, 0, 1/5) modulation. Based on these results we draw a charge-order phase diagram and discuss the relation between stripe ordering and superconductivity.

High-temperature charge density wave correlations in La1.875Ba0.125CuO4 without spin-charge locking

Although all superconducting cuprates display charge-ordering tendencies, their low-temperature properties are distinct, impeding efforts to understand the phenomena within a single conceptual framework. While some systems exhibit stripes of charge and spin, with a locked periodicity, others host charge density waves (CDWs) without any obviously related spin order. Here we use resonant inelastic X-ray scattering to follow the evolution of charge correlations in the canonical stripe-ordered cuprate La_1.875Ba_0.125CuO₄ across its ordering transition. We find that high-temperature charge correlations are unlocked from the wavevector of the spin correlations, signaling analogies to CDW phases in various other cuprates. This indicates that stripe order at low temperatures is stabilized by the coupling of otherwise independent charge and spin density waves, with important implications for the relation between charge and spin correlations in the cuprates.

Intertwined Order and Holography: The Case of Parity Breaking Pair Density Waves

We present a minimal bottom-up extension of the Chern-Simons bulk action for holographic translational symmetry breaking that naturally gives rise to pair density waves. We construct stationary inhomogeneous black hole solutions in which both the U(1) symmetry and spatially translational symmetry are spontaneously broken at a finite temperature and charge density. This novel solution provides a dual description of a superconducting phase intertwined with charge, current, and parity orders.

Complementary Response of Static Spin-Stripe Order and Superconductivity to Nonmagnetic Impurities in Cuprates

We report muon-spin rotation and neutron-scattering experiments on nonmagnetic Zn impurity effects on the static spin-stripe order and superconductivity of the La214 cuprates. Remarkably, it was found that, for samples with hole doping x≈1/8, the spin-stripe ordering temperature T_{so} decreases linearly with Zn doping y and disappears at y≈4%, demonstrating a high sensitivity of static spin-stripe order to impurities within a CuO_{2} plane. Moreover, T_{so} is suppressed by Zn in the same manner as the superconducting transition temperature T_{c} for samples near optimal hole doping. This surprisingly similar sensitivity suggests that the spin-stripe order is dependent on intertwining with superconducting correlations.

Remarkable Stability of Charge Density Wave Order in La_{1.875}Ba_{0.125}CuO_{4}

The occurrence of charge-density-wave (CDW) order in underdoped cuprates is now well established, although the precise nature of the CDW and its relationship with superconductivity is not. Theoretical proposals include contrasting ideas such as that pairing may be driven by CDW fluctuations or that static CDWs may intertwine with a spatially modulated superconducting wave function. We test the dynamics of CDW order in La_{1.825}Ba_{0.125}CuO_{4} by using x-ray photon correlation spectroscopy at the CDW wave vector, detected resonantly at the Cu L_{3} edge. We find that the CDW domains are strikingly static, with no evidence of significant fluctuations up to 2 ¾  h. We discuss the implications of these results for some of the competing theories.

Quenched Magnon excitations by oxygen sublattice reconstruction in (SrCuO2)n/(SrTiO3)2 superlattices

The recently discovered structural reconstruction in the cuprate superlattice (SrCuO2)n/(SrTiO3)2 has been investigated across the critical value of n = 5 using resonant inelastic x-ray scattering (RIXS). We find that at the critical value of n, the cuprate layer remains largely in the bulk-like two-dimensional structure with a minority of Cu plaquettes being reconstructed. The partial reconstruction leads to quenching of the magnons starting at the Γ-point due to the minority plaquettes acting as scattering points. Although comparable in relative abundance, the doped charge impurities in electron-doped cuprate superconductors do not show this quenching of magnetic excitations.

Non-equilibrium control of complex solids by nonlinear phononics

We review some recent advances in the use of optical fields at terahertz frequencies to drive the lattice of complex materials. We will focus on the control of low energy collective properties of solids, which emerge on average when a high frequency vibration is driven and a new crystal structure induced. We first discuss the fundamentals of these lattice rearrangements, based on how anharmonic mode coupling transforms an oscillatory motion into a quasi-static deformation of the crystal structure. We then discuss experiments, in which selectively changing a bond angle turns an insulator into a metal, accompanied by changes in charge, orbital and magnetic order. We then address the case of light induced non-equilibrium superconductivity, a mysterious phenomenon observed in some cuprates and molecular materials when certain lattice vibrations are driven. Finally, we show that the dynamics of electronic and magnetic phase transitions in complex-oxide heterostructures follow distinctly new physical pathways in case of the resonant excitation of a substrate vibrational mode.

Magnetic Fluctuations in Pair-Density-Wave Superconductors

Pair-density-wave superconductivity constitutes a novel electronic condensate proposed to be realized in certain unconventional superconductors. Establishing its potential existence is important for our fundamental understanding of superconductivity in correlated materials. Here we compute the dynamical magnetic susceptibility in the presence of a pair-density-wave ordered state and study its fingerprints on the spin-wave spectrum including the neutron resonance. In contrast to the standard case of d-wave superconductivity, we show that the pair-density-wave phase exhibits neither a spin gap nor a magnetic resonance peak, in agreement with a recent neutron scattering experiment on underdoped La_{1.905}Ba_{0.095}CuO_{4} [Z. Xu et al., Phys. Rev. Lett. 113, 177002 (2014)].