Machine learning in electronic-quantum-matter imaging experiments

For centuries, the scientific discovery process has been based on systematic human observation and analysis of natural phenomena¹. 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^2-5. Given the success of ML in the analysis of synthetic data representing electronic quantum matter (EQM)^6-16, the next challenge is to apply this approach to experimental data-for example, to the arrays of complex electronic-structure images¹⁷ 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^18,19 are consistent with these observations.

Localization of electrons due to orbitally ordered bi-stripes in the bilayer manganite La(2-2x)Sr(1+2x)Mn2O7 (x ~ 0.59)

Electronic phases with stripe patterns have been intensively investigated for their vital roles in unique properties of correlated electronic materials. How these real-space patterns affect the conductivity and other properties of materials (which are usually described in momentum space) is one of the major challenges of modern condensed matter physics. By studying the electronic structure of La(2-2x)Sr(1+2x)Mn(2)O(7) (x ∼ 0.59) and in combination with earlier scattering measurements, we demonstrate the variation of electronic properties accompanying the melting of so-called bi-stripes in this material. The static bi-stripes can strongly localize the electrons in the insulating phase above T(c) ∼ 160 K, while the fraction of mobile electrons grows, coexisting with a significant portion of localized electrons when the static bi-stripes melt below T(c). The presence of localized electrons below T(c) suggests that the melting bi-stripes exist as a disordered or fluctuating counterpart. From static to melting, the bi-stripes act as an atomic-scale electronic valve, leading to a "colossal" metal-insulator transition in this material.

Magnon-hole scattering and charge order in Sr14-xCaxCu24O41

The magnon thermal conductivity kappa(mag) of the hole-doped spin ladders in Sr14-xCaxCu24O41 has been investigated at low doping levels x. The analysis of kappa(mag) reveals a strong doping and temperature dependence of the magnon mean free path l(mag), which is a local probe for the interaction of magnons with the doped holes in the ladders. In particular, this novel approach to studying charge degrees of freedom via spin excitations shows that charge ordering of the holes in the ladders leads to a freezing out of magnon-hole scattering processes.

Quantum phase transitions in the cuprate superconductor Bi2Sr2-xLaxCuO6+delta

To elucidate a quantum phase transition (QPT) in Bi(2)Sr(2-x)La(x)CuO(6+delta), we measure charge and heat transport properties at very low temperatures and examine the following characteristics for a wide range of doping: normal-state resistivity anisotropy under 58 T, temperature dependence of the in-plane thermal conductivity kappa(ab), and the magnetic-field dependence of kappa(ab). It turns out that all of them show signatures of a QPT at the 1/8 hole doping. Together with the recent normal-state Hall measurements under 58 T that signified the existence of a QPT at optimum doping, the present results indicate that there are two QPTs in the superconducting doping regime of this material.

Anisotropic electromagnetic response of lightly doped La2-xSrxCuO4 within the CuO2 planes

Using infrared spectroscopy, we show that spin self-organization in untwinned La2-xSrxCuO4 (LSCO) crystals has profound consequences for the dynamical conductivity sigma(omega). The electronic response of CuO2 planes acquires significant anisotropy in the spin ordered state with enhancement of the conductivity along the direction of the diagonal spin stripes by up to a factor of 2. An examination of the anisotropic response indicates that the diagonal spin texture in weakly doped LSCO is also accompanied by the modulation of charge density. The electronic response of the charge stripes is found to be gapless consistent with the hypothesis of the metallic ground state. Our experiments directly show that the striped ordered systems reveal new degrees of freedom not present in ordinary one-dimensional conductors.

High-energy spin dynamics in La1.69Sr0.31NiO4

To test the prediction that the dispersion of the magnetic resonance in superconducting YBa2Cu3O(6+x) is similar to magnons in an incommensurate antiferromagnet, we have mapped out the spin dynamics in a stripe-ordered nickelate, La(2-x)SrxNiO4, with x approximately equal to 0.31, using inelastic neutron scattering. We observe spin-wave excitations up to 80 meV emerging from the incommensurate magnetic peaks with a surprisingly large and almost isotropic spin velocity: variant Planck's over 2 pi c(s) approximately 0.32 eV A. A comparison indicates that the inferred spin-excitation spectrum is not, by itself, an adequate model for the magnetic resonance feature of the superconductor.

Charge ordering fluctuation and optical pseudogap in La(1-x)Ca(x)MnO(3)

Optical spectroscopy was used to investigate the optical gap ( 2 Delta) due to charge ordering (CO) and related pseudogap developments with x and temperature ( T) in La(1-x)Ca(x)MnO(3) ( 0.48< or =x< or =0.67). Surprisingly, we found 2 Delta/k(B)T(CO) is as large as 30 for x approximately 0.5, and decreases rapidly with increasing x. Simultaneously, the optical pseudogap, possibly starting from T* far above T(CO) becomes drastically enhanced near x = 0.5, producing non-BCS T-dependence of 2 Delta with the large magnitude far above T(CO), and systematic increase of T* for x approximately 0.5. These results unequivocally indicate systematically enhanced CO correlation when x approaches 0.5 even though T(CO) decreases.

Electromagnetic response of static and fluctuating stripes in cuprate superconductors

Using infrared spectroscopy, we found that changes in the in-plane charge dynamics attributable to static stripe order in La(1.275)Nd(0.6)Sr(0.125)CuO(4) or superconductivity in La(1.875)Sr(0.125)CuO(4) are confined to energies smaller than 100 cm(-1). An absorption peak in the low- omega conductivity of the Nd-doped compound is suggestive of localization effects due to the reduced dimensionality of static charge stripes. Neither superconductivity nor static stripe ordering has a noticeable effect on the depression of the scattering rate at omega<1000 cm(-1) characteristic of the pseudogap state in other classes of moderately doped cuprates.

Electrical resistivity anisotropy from self-organized one dimensionality in high-temperature superconductors

We investigate the manifestation of stripes in the in-plane resistivity anisotropy in untwinned single crystals of La2-xSrxCuO4 ( x = 0.02-0.04) and YBa(2)Cu(3)O(y) ( y = 6.35-7.0). It is found that both systems show strongly temperature-dependent in-plane anisotropy in the lightly hole-doped region and that the anisotropy in YBa(2)Cu(3)O(y) grows with decreasing y below approximately 6.60 despite the decreasing orthorhombicity, which gives most direct evidence that electrons self-organize into a macroscopically anisotropic state. The transport is found to be easier along the direction of the spin stripes already reported, demonstrating that the stripes are intrinsically conducting in cuprates.

Diamagnetic activity above Tc as a precursor to superconductivity in La2-xSrxCuO4 thin films

Superconductors show zero resistance to electric current, and expel magnetic flux (the Meissner effect) below the transition temperature (Tc). In conventional superconductors, the 'Cooper pairs' of electrons that are responsible for superconductivity form only below Tc. In the unconventional high-Tc superconductors, however, a strong electron correlation is essential for pair formation: there is evidence that some pairs are formed above Tc in samples that have less than the optimal density of charge carriers (underdoped) and an energy gap-the 'pseudogap'-appears to be present. Moreover, excitations that look like the vortices that carry magnetic flux inside the superconducting state have been reported above Tc (refs 6, 7). Although the origin of the pseudogap remains controversial, phase fluctuations above Tc, leading to some form of local superconductivity or local pairing, seem essential. Here we report magnetic imaging (scanning SQUID microscopy) of La2-xSrxCuO4 thin films. Clear quantized vortex patterns are visible below Tc (18-19 K), and we observe inhomogeneous magnetic domains that persist up to 80 K. We interpret the data as suggesting the existence of diamagnetic regions that are precursors to the Meissner state.

From antiferromagnetic order to static magnetic stripes: the phase diagram of (La,Eu)2-xSrxCuO4

The magnetic order of (La,Eu)2-xSrxCuO4 ( x</=0.2) has been investigated with &mgr;SR techniques. In this system a low temperature tetragonal (LTT) structure is present in the entire range of doping and it is possible to follow the evolution from the long range antiferromagnetic state at x = 0 to the static magnetic stripes. We find a nonmonotonic change of the Neel temperature with increasing x and the obtained magnetic phase diagram of the LTT phase resembles the generic phase diagram of the cuprates where the superconductivity is replaced by a second antiferromagnetic phase.

Independent freezing of charge and spin dynamics in La1.5Sr0.5CoO4

We present elastic and quasielastic neutron scattering measurements characterizing peculiar short-range charge-orbital and spin order in the layered perovskite material La1.5Sr0.5CoO4. We find that below T(c) approximately 750 K holes introduced by Sr doping lose mobility and enter a statically ordered charge glass phase with loosely correlated checkerboard arrangement of empty and occupied d(3z(2)-r(2)) orbitals ( Co3+ and Co2+). The dynamics of the resultant mixed spin system is governed by the anisotropic nature of the crystal-field Hamiltonian and the peculiar exchange pattern produced by the orbital order. It undergoes a spin freezing transition at a much lower temperature, T(s) less, similar30 K.

Signature of superfluid density in the single-particle excitation spectrum of Bi(2)Sr(2)CaCu(2)O(8+delta)

We report that the doping and temperature dependence of photoemission spectra near the Brillouin zone boundary of Bi(2)Sr(2)CaCu(2)O(8+delta)exhibit unexpected sensitivity to the superfluid density. In the superconducting state, the photoemission peak intensity as a function of doping scales with the superfluid density and the condensation energy. As a function of temperature, the peak intensity shows an abrupt behavior near the superconducting phase transition temperature where phase coherence sets in, rather than near the temperature where the gap opens. This anomalous manifestation of collective effects in single-particle spectroscopy raises important questions concerning the mechanism of high-temperature superconductivity.

Advances in the physics of high-temperature superconductivity

The high-temperature copper oxide superconductors are of fundamental and enduring interest. They not only manifest superconducting transition temperatures inconceivable 15 years ago, but also exhibit many other properties apparently incompatible with conventional metal physics. The materials expand our notions of what is possible, and compel us to develop new experimental techniques and theoretical concepts. This article provides a perspective on recent developments and their implications for our understanding of interacting electrons in metals.