For centuries, the scientific discovery process has been based on systematic human observation and analysis of natural phenomena
. Today, however, automated instrumentation and large-scale data ...acquisition are generating datasets of such large volume and complexity as to defy conventional scientific methodology. Radically different scientific approaches are needed, and machine learning (ML) shows great promise for research fields such as materials science
. Given the success of ML in the analysis of synthetic data representing electronic quantum matter (EQM)
, the next challenge is to apply this approach to experimental data-for example, to the arrays of complex electronic-structure images
obtained from atomic-scale visualization of EQM. Here we report the development and training of a suite of artificial neural networks (ANNs) designed to recognize different types of order hidden in such EQM image arrays. These ANNs are used to analyse an archive of experimentally derived EQM image arrays from carrier-doped copper oxide Mott insulators. In these noisy and complex data, the ANNs discover the existence of a lattice-commensurate, four-unit-cell periodic, translational-symmetry-breaking EQM state. Further, the ANNs determine that this state is unidirectional, revealing a coincident nematic EQM state. Strong-coupling theories of electronic liquid crystals
are consistent with these observations.
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
(
) within the halo surrounding Bi
Sr
CaCu
O
vortex cores. We detected numerous phenomena predicted for a field-induced PDW, including two sets of particle-hole symmetric
(
) modulations with wave vectors
and
, 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 CuO
unit-cell periodicity and coexisting with its secondary CDWs.
The quantum condensate of Cooper pairs forming a superconductor was originally conceived as being translationally invariant. In theory, however, pairs can exist with finite momentum Q, thus ...generating a state with a spatially modulated Cooper-pair density. Such a state has been created in ultracold (6)Li gas but never observed directly in any superconductor. It is now widely hypothesized that the pseudogap phase of the copper oxide superconductors contains such a 'pair density wave' state. Here we report the use of nanometre-resolution scanned Josephson tunnelling microscopy to image Cooper pair tunnelling from a d-wave superconducting microscope tip to the condensate of the superconductor Bi2Sr2CaCu2O8+x. We demonstrate condensate visualization capabilities directly by using the Cooper-pair density variations surrounding zinc impurity atoms and at the Bi2Sr2CaCu2O8+x crystal supermodulation. Then, by using Fourier analysis of scanned Josephson tunnelling images, we discover the direct signature of a Cooper-pair density modulation at wavevectors QP ≈ (0.25, 0)2π/a0 and (0, 0.25)2π/a0 in Bi2Sr2CaCu2O8+x. The amplitude of these modulations is about five per cent of the background condensate density and their form factor exhibits primarily s or s' symmetry. This phenomenology is consistent with Ginzburg-Landau theory when a charge density wave with d-symmetry form factor and wavevector QC = QP coexists with a d-symmetry superconductor; it is also predicted by several contemporary microscopic theories for the pseudogap phase.
High T_{c} superconductors show a rich variety of phases associated with their charge degrees of freedom. Valence charges can give rise to charge ordering or acoustic plasmons in these layered ...cuprate superconductors. While charge ordering has been observed for both hole- and electron-doped cuprates, acoustic plasmons have only been found in electron-doped materials. Here, we use resonant inelastic x-ray scattering to observe the presence of acoustic plasmons in two families of hole-doped cuprate superconductors (La_{1.84}Sr_{0.16}CuO_{4} and Bi_{2}Sr_{1.6}La_{0.4}CuO_{6+δ}), crucially completing the picture. Interestingly, in contrast to the quasistatic charge ordering which manifests at both Cu and O sites, the observed acoustic plasmons are predominantly associated with the O sites, revealing a unique dichotomy in the behavior of valence charges in hole-doped cuprates.
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Electron-boson coupling plays a key role in superconductivity for many systems. However, in copper-based high-critical temperature (
) superconductors, its relation to superconductivity remains ...controversial despite strong spectroscopic fingerprints. In this study, we used angle-resolved photoemission spectroscopy to find a pronounced correlation between the superconducting gap and the bosonic coupling strength near the Brillouin zone boundary in Bi
Sr
CaCu
O
The bosonic coupling strength rapidly increases from the overdoped Fermi liquid regime to the optimally doped strange metal, concomitant with the quadrupled superconducting gap and the doubled gap-to-
ratio across the pseudogap boundary. This synchronized lattice and electronic response suggests that the effects of electronic interaction and the electron-phonon coupling (EPC) reinforce each other in a positive-feedback loop upon entering the strange-metal regime, which in turn drives a stronger superconductivity.
The existence of electronic symmetry breaking in the underdoped cuprates and its disappearance with increased hole density p are now widely reported. However, the relation between this transition and ...the momentum-space ($\overrightarrow{\mathrm{k}}$-space) electronic structure underpinning the superconductivity has not yet been established. Here, we visualize the $\overrightarrow{\mathrm{Q}}=0$ = 0 (intra–unit-cell) and $\overrightarrow{\mathrm{Q}}\ne 0$ (density-wave) broken-symmetry states, simultaneously with the coherent $\overrightarrow{\mathrm{k}}$-space topology, for Bi2Sr2CaCu2O8+δ samples spanning the phase diagram 0.06 ≤ p ≤ 0.23. We show that the electronic symmetry-breaking tendencies weaken with increasing p and disappear close to a critical doping pc = 0.19. Concomitantly, the coherent $\overrightarrow{\mathrm{k}}$-space topology undergoes an abrupt transition, from arcs to closed contours, at the same pc. These data reveal that the $\overrightarrow{\mathrm{k}}$-space topology transformation in cuprates is linked intimately with the disappearance of the electronic symmetry breaking at a concealed critical point.
Copper oxide high-TC superconductors possess a number of exotic orders that coexist with or are proximal to superconductivity. Quantum fluctuations associated with these orders may account for the ...unusual characteristics of the normal state, and possibly affect the superconductivity1–4. Yet, spectroscopic evidence for such quantum fluctuations remains elusive. Here, we use resonant inelastic X-ray scattering to reveal spectroscopic evidence of fluctuations associated with a charge order5–14 in nearly optimally doped Bi2Sr2CaCu2O8+δ. In the superconducting state, while the quasielastic charge order signal decreases with temperature, the interplay between charge order fluctuations and bond-stretching phonons in the form of a Fano-like interference increases, an observation that is incompatible with expectations for competing orders. Invoking general principles, we argue that this behaviour reflects the properties of a dissipative system near an order–disorder quantum critical point, where the dissipation varies with the opening of the pseudogap and superconducting gap at low temperatures, leading to the proliferation of quantum critical fluctuations, which melt charge order.X-ray scattering experiments show that the quantum fluctuations associated with charge order take a form that is incompatible with the idea of competition between charge order and superconductivity.
Research on high-temperature superconducting cuprates is at present focused on identifying the relationship between the classic pseudogap phenomenon1,2 and the more recently investigated density wave ...state313. This state is generally characterized by a wavevector Q parallel to the planar CuOCu bonds413 along with a predominantly d-symmetry form factor1416 (dFF-DW). To identify the microscopic mechanism giving rise to this state1729, one must identify the momentum-space states contributing to the dFF-DW spectral weight, determine their particlehole phase relationship about the Fermi energy, establish whether they exhibit a characteristic energy gap, and understand the evolution of all these phenomena throughout the phase diagram. Here we use energy-resolved sublattice visualization14 of electronic structure and reveal that the characteristic energy of the dFF-DW modulations is actually the pseudogap energy 1.
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 et al., 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 et al. Phys. Rev. B 91, 054502 (2015); Wardh and Granath arXiv:2203.08250.
Cuprate superconductors host a multitude of low-energy optical phonons. Using time- and angle-resolved photoemission spectroscopy, we study coherent phonons in ...Bi_{2}Sr_{2}Ca_{0.92}Y_{0.08}Cu_{2}O_{8+δ}. Sub-meV modulations of the electronic band structure are observed at frequencies of 3.94±0.01 and 5.59±0.06 THz. For the dominant mode at 3.94 THz, the amplitude of the band energy oscillation weakly increases as a function of momentum away from the node. Theoretical calculations allow identifying the observed modes as CuO_{2}-derived A_{1g} phonons. The Bi- and Sr-derived A_{1g} modes which dominate Raman spectra in the relevant frequency range are absent in our measurements. This highlights the mode selectivity for phonons coupled to the near-Fermi-level electrons, which originate from CuO_{2} planes and dictate thermodynamic properties.
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