This article discusses fluctuating order in a quantum disordered phase proximate to a quantum critical point, with particular emphasis on fluctuating stripe order. Optimal strategies are derived for extracting information concerning such local order from experiments, with emphasis on neutron scattering and scanning tunneling microscopy. These ideas are tested by application to two model systems---an exactly solvable one-dimensional (1D) electron gas with an impurity, and a weakly interacting 2D electron gas. Experiments on the cuprate high-temperature superconductors which can be analyzed using these strategies are extensively reviewed. The authors adduce evidence that stripe correlations are widespread in the cuprates. They compare and contrast the advantages of two limiting perspectives on the high-temperature superconductor: weak coupling, in which correlation effects are treated as a perturbation on an underlying metallic (although renormalized) Fermi-liquid state, and strong coupling, in which the magnetism is associated with well-defined localized spins, and stripes are viewed as a form of micro phase separation. The authors present quantitative indicators that the latter view better accounts for the observed stripe phenomena in the cuprates.

How to detect fluctuating stripes in the high-temperature superconductors

Angle-resolved photoemission experiments reveal evidence of an energy gap in the normal state excitation spectrum of the cuprate superconductor Bi2Sr2CaCu2O8+delta. This gap exists only in underdoped samples and closes around the doping level at which the superconducting transition temperature Tc is a maximum. The momentum dependence and magnitude of the gap closely resemble those of the dx2-y2 gap observed in the superconducting state. This observation is consistent with results from several other experimental techniques, which also indicate the presence of a gap in the normal state. Some possible theoretical explanations for this effect are reviewed.

https://science.sciencemag.org/content/sci/273/5273/325.full.pdf

Excitation Gap in the Normal State of Underdoped Bi2Sr2CaCu2O8+δ

Inelastic neutron scattering experiments have been carried out on YBa2Cu3O6+x single crystals in order to perform a systematic investigation of the spin dynamics in the various typical regimes : the pure and doped AF-states (x = 0.15, 0.37), the weakly doped metallic and superconducting state (x= 0.45, 0.51) and the heavily doped metallic state with the Tc= 60 K (x = 0.69) and Tc = 90 K (x = 0.92) phases. The major results are the strong effect of hole doping on the AF-order, the persistence of dynamical AF-correlations in the metallic state and the observation of an energy gap in the spin excitation spectrum at low temperatures in superconducting samples.

Neutron scattering study of the YBa2Cu3O6+x system

In materials with strong local Coulomb interactions, simple defects such as atomic substitutions strongly affect both macroscopic and local properties of the system. A nonmagnetic impurity, for instance, is seen to induce magnetism nearby. Even without disorder, models of such correlated systems are generally not soluble in 2 or 3 dimensions, and so few exact results are known for the properties of such impurities. Nevertheless, some simple physical ideas have emerged from experiments and approximate theories. Here, we first review what we can learn about this problem from 1D antiferromagnetically correlated systems. We then discuss experiments on the high Tc cuprate normal state which probe the effect of impurities on local charge and spin degrees of freedom, and compare with theories of single impurities in correlated hosts, as well as phenomenological effective Kondo descriptions. Subsequently, we review theories of impurities in d-wave superconductors including residual quasiparticle interactions, and compare with experiments in the superconducting state. We argue that existing data exhibit a remarkable similarity to impurity-induced magnetism in the 1D case, implying the importance of electronic correlations for the understanding of these phenomena, and suggesting that impurities may provide excellent probes of the still poorly understood ground state of the cuprates.

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Defects in correlated metals and superconductors

Magnetic susceptibility and inelastic neutron scattering experiments have been performed in the nearly ideal one-dimensional Heisenberg antiferromagnet with spin one, Ni(C2H8N2)2NO2ClO4. The experimental results are consistent with the recent theoretical predictions for a quantum energy gap between the ground state and the first excited states.

http://iopscience.iop.org/article/10.1209/0295-5075/3/8/013/pdf

Presumption for a Quantum Energy Gap in the Quasi-One-Dimensional S?=?1 Heisenberg Antiferromagnet Ni(C2H8N2)2NO2(ClO4)

Inelastic neutron scattering experiments have been carried out on YBa2Cu3O6+x single crystals to investigate the spin dynamics in the pure AF-state (x = 0.15, TN = 410 K), in the doped AF-state (x = 0.37, TN = 180 K), in the metallic weakly doped superconducting state (x = 0.45, Tc = 37 K; x = 0.51, Tc = 47 K) and in the Tc = 60 K superconducting phase (x = 0.69, Tc = 59 K). The main results are summarized: the two dimensional character of the spin-wave spectrum in the AF-state, strong renormalization of low-energy excitations and overdamping of in-plane spin fluctuations in the doped AF-state, dynamical AF correlations in the superconducting state, evidence for a pseudo-gap induced by superconductivity and existence of superconducting correlations above Tc.

Investigation of the spin dynamics in YBa2Cu3O6+x by inelastic neutron scattering

We have performed elastic- and inelastic-neutron-scattering experiments on a single crystal of the quasi-one-dimensional system CuGe0.993Si0.007O3. Our experimental data show unambiguously that both quasi-gapless antiferromagnetic fluctuations and gapped magnetic fluctuations, associated with a dimerized phase, coexist in this material below a spin-Peierls transition temperature TSP ≈ 9.2 K. When the temperature is lowered below TN ≈ 4 K, part of the system undergoes a new phase transition towards long-range antiferromagnetism, coexisting with the dimerized non-magnetic phase. We will describe the temperature and wave vector dependences of magnetic responses in these two phases, which can be understood as the superposition of two independent spectra.

https://iopscience.iop.org/article/10.1209/0295-5075/32/7/007/pdf

Coexistence of Dimerization and Antiferromagnetism in Si-Doped CuGeO3

We have investigated the spin dynamics of the spin-Peierls system CuGe${\mathrm{O}}_{3}$ by inelastic neutron scattering. The measurements have been performed as a function of wave vector, temperature, and magnetic field, for fields perpendicular to the chain axis (up to 10 T). Our neutron results confirm the occurrence of a crystalline distortion below ${T}_{\mathrm{SP}}\ensuremath{\approx}14.2$ K, corroborated by the appearance of superlattice peaks indexed with a commensurate wave vector ${\mathbf{k}}_{\mathrm{SP}}$=(\textonehalf{},0,\textonehalf{}). At low temperature, the spin-excitation spectrum exhibits a well defined energy gap $\ensuremath{\Delta}\ensuremath{\approx}2 \mathrm{meV}\ensuremath{\approx}1.66 k{T}_{\mathrm{SP}}$ at the antiferromagnetic point ${\mathbf{k}}_{\mathrm{AF}}$=(0,1,\textonehalf{}), distinct from ${\mathbf{k}}_{\mathrm{SP}}$. The experimental results for $T\ensuremath{\ll}{T}_{\mathrm{SP}}$ are quantitatively understood from an alternating-exchange Hamiltonian with an exchange parameter $(2{J}_{1})\ensuremath{\approx}10.6$ meV and an alternation parameter $\ensuremath{\alpha}=\frac{{J}_{2}}{{J}_{1}}\ensuremath{\approx}0.92$, despite the existence of sizable interchain couplings both along the $a$ and $b$ axes ($\frac{{J}_{a}^{\ensuremath{'}}}{{J}_{1}}\ensuremath{\approx}0.011$ and $\frac{{J}_{b}^{\ensuremath{'}}}{{J}_{1}}\ensuremath{\approx}0.11$, respectively). We present the temperature dependence of the spin dynamics on both sides of ${T}_{\mathrm{SP}}$. Our results confirm the persistence of a pseudogap in the excitation spectrum for q=${\mathbf{k}}_{\mathrm{AF}}$, but only in the immediate vicinity of the spin-Peierls transition temperature. The pseudogap vanishes rapidly with $T$, following decreasing size of dimerized segments. The stronger inelasticity observed above ${T}_{\mathrm{SP}}$ at q=${\mathbf{k}}_{\mathrm{SP}}$ has been attributed to an effect of the dispersion of magnetic excitations due to interchain couplings ${J}_{a}^{\ensuremath{'}}$ and ${J}_{b}^{\ensuremath{'}}$. Under a magnetic field applied perpendicular to the chain axis, the excited triplet is split into three components, with gap values depending linearly on the field. The spin-Peierls transition temperature ${T}_{\mathrm{SP}}$ decreases with increasing field, following a quadratic dependence on $H$, in agreement with theoretical as well as other experimental results. We have determined the field dependence of the dimerization peak intensities up to 10 T, which show very small changes. In particular, no trace of pretransitional incommensurability could be detected, at least up to $0.77{H}_{c}$.

Inelastic-neutron-scattering investigation of the spin-peierls system cugeo3

We have investigated the spin dynamics of the nearly ideal one-dimensional S=1 antiferromagnet Ni(${\mathrm{C}}_{2}$${\mathrm{H}}_{8}$${\mathrm{N}}_{2}$${)}_{2}$${\mathrm{NO}}_{2}$${\mathrm{ClO}}_{4}$ (NENP) by inelastic neutron scattering. The measurements have been performed as a function of both the wave vector (in the vicinity of the antiferromagnetic point q=\ensuremath{\pi}) and the magnetic field, for field configurations parallel (up to 5 T) or perpendicular to the chain axis (up to 10 T). Our experimental results at low temperature are in good quantitative agreement with most theoretical predictions (both analytical and numerical) established for the spin-1 Haldane-gap system in presence of finite orthorhombic anisotropy: nonmagnetic singlet ground state, three gaps in the excitation spectrum corresponding to the excited triplet (${\mathrm{\ensuremath{\Delta}}}_{\mathit{y}}^{0}$\ensuremath{\approxeq}1.05 meV, ${\mathrm{\ensuremath{\Delta}}}_{\mathit{x}}^{0}$\ensuremath{\approxeq}1.23 meV, ${\mathrm{\ensuremath{\Delta}}}_{\mathit{z}}^{0}$\ensuremath{\approxeq}2.5 meV), finite correlation lengths at T=0 (${\ensuremath{\xi}}_{\mathit{x}\mathit{y}}$/d\ensuremath{\approxeq}8,${\ensuremath{\xi}}_{\mathit{z}}$/d\ensuremath{\approxeq}4) and unconventional field dependencies. The evolution with field of the various modes and their respective polarization have been determined experimentally and are compared to recent field-theory treatments of the S=1 anisotropic antiferromagnetic chain together with numerical calculations on finite-size systems. We present results concerning the field dependencies of the correlation lengths, both in field parallel and perpendicular to the chain axis, which are not fully understood from the existing theories.

Inelastic-neutron-scattering study of the spin dynamics in the Haldane-gap system Ni(C2H8N2)2NO2ClO4.

We have performed magnetic-susceptibility measurements and Cu nuclear-magnetic-resonance studies on single crystals of the quasi-one-dimensional antiferromagnet CuGe1-xSixO3, with 0 ≤ x ≤ 0.1. The Si doping has a drastic effect on the spin-Peierls temperature TSP which decreases at TSP (x)/TSP (0) 1-50x at low Si concentrations. For x ≥ 0.005, three-dimensional antiferromagnetic order with an easy axis along the Cu chains takes place below a well-defined Neel temperature TN. A qualitative interpretation of the phase diagram (T, x) is given.

https://iopscience.iop.org/article/10.1209/0295-5075/30/8/006/pdf

L. P. Regnault

Papers

Investigation of the spin dynamics in YBa2Cu3O6+x by inelastic neutron scattering

Coexistence of Dimerization and Antiferromagnetism in Si-Doped CuGeO3

Inelastic-neutron-scattering investigation of the spin-peierls system cugeo3

Inelastic-neutron-scattering study of the spin dynamics in the Haldane-gap system Ni(C2H8N2)2NO2ClO4.

Competition between Spin-Peierls Phase and Three-Dimensional Antiferromagnetic Order in CuGe1-xSixO3