Early experimental evidence of a topological quantum state: The signature of the Haldane ground state revealed by scattered neutrons

Neutron scattering played an important role in the experimental exploration of the theoretical ideas of Thouless, Haldane, and Kosterlitz, who won the Nobel Prize in Physics in 2016. This article reviews the signatures of Haldanes predicted topological quantum ground state and their observation in early neutron scattering experiments, which overturned the wisdom of the day.

Separation of magnetic and superconducting behaviour in YBCO6.33 (Tc=8.4 K)

Neutron scattering from high-quality YBa2Cu3O6.33 (YBCO6.33) single crystals with a Tc of 8.4 K shows no evidence of a coexistence of superconductivity with long-range antiferromagnetic order at this very low, near-critical doping of p~0.055. However, we find short-range three dimensional spin correlations that develop at temperatures much higher than Tc. Their intensity increases smoothly on cooling and shows no anomaly that might signify a Neel transition. The system remains subcritical with spins correlated over only one and a half unit cells normal to the planes. At low energies the short-range spin response is static on the microvolt scale. The excitations out of this ground state give rise to an overdamped spectrum with a relaxation rate of 3 meV. The transition to the superconducting state below Tc has no effect on the spin correlations. The elastic interplanar spin response extends over a length that grows weakly but fails to diverge as doping is moved towards the superconducting critical point. Any antiferromagnetic critical point likely lies outside the superconducting dome. The observations suggest that conversion from Neel long-range order to a spin glass texture is a prerequisite to formation of paired superconducting charges. We show that while pc =0.052 is a critical doping for superconducting pairing, it is not for spin order.

Magnetic excitations of spin and orbital moments in cobalt oxide

Magnetic and phonon excitations in the antiferromagnet CoO with an unquenched orbital angular momentum are studied by neutron scattering. Results of energy scans in several Brillouin zones in the (HHL) plane for energy transfers up to 16 THz are presented. The measurements were performed in the antiferromagnetic ordered state at 6 K (well below TN~290 K) as well as in the paramagnetic state at 450 K. Several magnetic excitation modes are identified from the dependence of their intensity on wavevector and temperature. Within a Hunds rule model the excitations correspond to fluctuations of coupled orbital and spin degrees of freedom whose bandwidth is controlled by interionic superexchange. The different <111> ordering domains give rise to several magnetic peaks at each wavevector transfer.

Magnetic field resonantly enhanced free spins in underdoped YBa$_{2}$Cu$_{3}$O$_{6+x}$

Using neutron scattering, we investigate the effect of a magnetic field on the static and dynamic spin response in heavily underdoped superconducting YBa$_{2}$Cu$_{3}$O$_{6+x}$ (YBCO$_{6+x}$) with x=0.33 (T$_{c}$=8 K) and 0.35 (T$_{c}$=18 K). In contrast to the heavily doped and superconducting monolayer cuprates, the elastic central peak characterizing static spin correlations does not respond observably to a magnetic field which suppresses superconductivity. Instead, we find a magnetic field induced resonant enhancement of the spin fluctuations. The energy scale of the enhanced fluctuations matches the Zeeman energy within both the normal and vortex phases while the momentum dependence is the same as the zero field bilayer response. The magnitude of the enhancement is very similar in both phases with a fractional intensity change of $(I/I_{0}-1) sim 0.1$. We suggest that the enhancement is not directly correlated with superconductivity but is the result of almost free spins located near hole rich regions.

Spin dynamics near the critical doping in weakly-superconducting underdoped YBa2Cu3O6.35 (Tc=18K)

Using neutron scattering we have determined the magnetic structure and fluctuations in the YBa2Cu3O6.35 superconductor (Tc=18 K). The long-range ordered collinear spins of the insulating antiferromagnet are replaced by a commensurate central mode arising from slow, isotropically polarized, short-range spin correlations. The inelastic spectrum up to 30 meV is broad in wave vector and commensurate. In contrast to the the resonance peak of higher Tc superconductors, the spins exhibit a single overdamped spectrum whose rate of relaxation decreases on cooling and saturates at 5 meV below 50 K. As the relaxation rate saturates the quasi-static spin correlations grow and become resolution limited in energy. The spin susceptibility at high temperatures illustrates that the dominant energy scale is set by the temperature. At low temperatures, the scale length is geometric and not linked by velocity to dynamic widths. There is no observable suppression of the spin fluctuations or central mode upon the onset of superconductivity. The spins respond not to coherent charge pairs but to hole doping allowing coexistence of glassy short range spin order with superconductivity. Since the physics of the weakly superconducting system YBCO6.35 must connect continuously with that in more strongly superconducting YBCO6.5, we find that neither incommensurate stripe-like spin modulations nor a well-defined neutron spin resonance are essential for the onset with doping of pairing in a high temperature cuprate superconductor.

Orbital and Spin Excitations in Cobalt Oxide

By means of neutron scattering we have determined new branches of magnetic excitations in orbitally active CoO (TN=290 K) up to 15 THz and for temperatures from 6 K to 450 K. Data were taken in the (111) direction in six single-crystal zones. From the dependence on temperature and Q we have identified several branches of magnetic excitation. We describe a model for the coupled orbital and spin states of Co2+ subject to a crystal field and tetragonal distortion.

Antiferromagnetic correlations near the lower edge of superconducting dome in YBCO6+x

Neutron scattering from high-quality YBCO6.334 single crystals with a T$_c$ of 8.4 K shows that there is no coexistence with long-range antiferromagnetic order at this very low, near-critical doping of $sim$0.055, in contrast to claims based on local probe techniques. We find that the neutron resonance seen in optimally doped YBCO7 and underdoped YBCO6.5, has undergone large softening and damping. It appears that the overdamped resonance, with a relaxation rate of 2 meV, is coupled to a zero-energy central mode that grows with cooling and eventually saturates with no change at or below T$_c$. Although a similar qualitative behaviour is found for YBCO6.35, our study shows that the central mode is stronger in YBCO6.334 than YBCO6.35. The system remains subcritical with short-ranged three dimensional correlations.

Evidence for decay of spin-waves above the pseudogap in underdoped YBa2Cu3O6.35

The magnetic spectrum at high-energies in heavily underdoped YBa$_{2}$Cu$_{3}$O$_{6.35}$ (T$_{c}$=18 K) has been determined throughout the Brillouin zone. At low-energy the scattering forms a cone of spin excitations emanating from the antiferromagnetic (0.5, 0.5) wave vector with an acoustic velocity similar to that of insulating cuprates. At high energy transfers, below the maximum energy of 270 meV at (0.5, 0), we observe zone boundary dispersion much larger and spectral weight loss more extensive than in insulating antiferromagnets. Moreover we report phenomena not found in insulators, an overall lowering of the zone-boundary energies and a large damping of $sim$ 100 meV of the spin excitations at high-energies. The energy above which the damping occurs coincides approximately with the gap determined from transport measurements. We propose that as the energy is raised the spin excitations encounter an extra channel of decay into particle-hole pairs of a continuum that we associate with the pseudogap.

Quantum phase transitions in magnetism and superconductivity: emergent spin topology seen with neutrons

Magnetic spins and charges interact strongly in high-temperature superconductors. New physics emerges as layers of copper oxide are tuned towards the boundary of the superconducting phase. As the pseudogap increases the characteristic spin excitation energy decreases. We show that our well-annealed YBa2Cu3O6+x (YBCO6+x) single crystals are orthorhombic and superconducting but not antiferromagnetically ordered. Near the critical concentration for superconductivity for x = 0.35 the spins fluctuate on two energy scales, one a relaxational spin response at ~2 meV and the other a slow central mode that is resolution-limited in energy (<0.08 meV) but broad in momentum. The gradual formation on cooling of a central mode over a range of momenta suggests that the spin ground state from which coherent superconducting pairing emerges may be quantum disordered. We show that YBCO6.35 adopts a homogeneous state that consists of highly-organized frozen sub-critical three-dimensional spin correlations. The continuous spin evolution indicates that a single quantum state occurs in contrast to claims from site-based probes that lightly doped YBCO undergoes a transition to antiferromagnetic Bragg order followed by a sharp transition to a cluster glass phase. For x = 0.35, where Tc = 18 K is reduced to 1/5 of Tcmax, the spin ground state is reached without a sharp transition and consists of short correlations extending over only 8 Angstrom between cells and 42 Angstrom within the planes. Polarized neutrons show the angular spin distribution to be isotropic unlike the AF insulator. Since moment is conserved we interpret this as evidence for hole-induced spin rotations rather than decay.

Dynamic Stripes and Resonance in the Superconducting and Normal Phases of YBa2Cu3O6.5 Ortho-II Superconductor

We describe the relation between spin fluctuations and superconductivity in a highly-ordered sample of YBaCu3O6.5 using both polarized and unpolarized neutron inelastic scattering. The spin susceptibility in the superconducting phase exhibits one-dimensional incommensurate modulations at low-energies, consistent with hydrodynamic stripes. With increasing energy the susceptibility curves upward to a commensurate, intense, well-defined and asymmetric resonance at 33 meV with a precipitous high-energy cutoff. In the normal phase, which we show is gapless, the resonance remains surprisingly strong and persists clearly in Q scans and energy scans. Its similar asymmetric spectral form above Tc=59 K suggests that incoherent superconducting pairing fluctuations are present in the normal state. On cooling, the resonance and the stripe modulations grow in well above Tc below a temperature that is comparable to the pseudogap temperature where suppression occurs in local and low-momentum properties. The spectral weight that accrues to the resonance is largely acquired by transfer from suppressed low-energy fluctuations. We find the resonance to be isotropically polarized, consistent with a triplet carrying ~2.6% of the total spectral weight of the Cu spins in the planes.