Kitaev's compass model on the honeycomb lattice realizes a spin liquid whose emergent excitations are dispersive Majorana fermions and static Z_2 gauge fluxes. We discuss the proper selection of physical states for finite-size simulations in the Majorana representation, based on a recent paper by Pedrocchi, Chesi, and Loss [Phys. Rev. B 84, 165414 (2011)]. Certain physical observables acquire large finite-size effects, in particular if the ground state is not fermion-free, which we prove to generally apply to the system in the gapless phase and with periodic boundary conditions. To illustrate our findings, we compute the static and dynamic spin susceptibilities for finite-size systems. Specifically, we consider random-bond disorder (which preserves the solubility of the model), calculate the distribution of local flux gaps, and extract the NMR lineshape. We also predict a transition to a random-flux state with increasing disorder.
An Eu NMR study in the ferromagnetic phase of pure and Gd doped EuO was performed. A complete description of the NMR lineshape of pure EuO allowed for the influence of doping EuO with Gd impurities to be highlighted. The presence of a temperature dependent static magnetic inhomogeneity in Gd doped EuO was demonstrated by studying the temperature dependence of the lineshapes. The results suggest that the inhomogeneity in 0.6% Gd doped EuO is linked to colossal magnetoresistance. The measurement of the spin-lattice relaxation times as a function of temperature led to the determination of the val…
We investigate the impact of two types of disorder, bond randomness and site dilution, on the spin dynamics in the Kitaev model on a honeycomb lattice. The ground state of this model is a canonical quantum spin liquid with spin fractionalization into two types of quasiparticles, itinerant Majorana fermions and localized fluxes. Using unbiased quantum Monte Carlo simulations, we calculate the temperature evolution of the dynamical spin structure factor, the magnetic susceptibility, and the NMR relaxation rate. In the dynamical spin structure factor, we find that the two types of disorder affect…
We present new 29-Si NMR spectra in URu2Si2 for varying temperature T, and external field H. On lowering T, the systematics of the low-field lineshape and width reveal an extra component (lambda) to the linewidth below T_N ~ 17 K not observed previously. We find that lambda is magnetic-field independent and dominates the low-field lineshape for all orientations of H with respect to the tetragonal c axis. The behavior of lambda indicates a direct relationship between the 29-Si spin and the transition at T_N, but it is inconsistent with a coupling of the nuclei to static antiferromagnetic order/…
We report a $^{35}$Cl nuclear magnetic resonance (NMR) study of the diluted Kitaev material $α$-Ru$_{1-x}$Ir$_x$Cl$_3$ ($x=0.1$ and $0.2$) where non-magnetic Ir$^{3+}$ dopants substitute Ru$^{3+}$ ions. Upon dilution, the $^{35}$Cl spectra exhibit unusual large magnetic inhomogeneity, which sets in at temperatures below the Kitaev exchange energy scale. At the same time, the $^{35}$Cl spin-lattice relaxation rate $T_1^{-1}$ as a function of dilution and magnetic field unravels a critical doping of $x_c\approx 0.22$, towards which both the field-induced spin gap and the zero-field magnetic orde…
The incommensurate (IC) spin ordering in quasi-1D edge-shared cuprate NaCu_2O_2 has been studied by ^{23}Na nuclear magnetic resonance spectroscopy in an external magnetic field near 6 Tesla applied along the main crystallographic axes. The NMR lineshape evolution above and below T_N\approx12 K yields a clear signature of an IC static modulation of the local magnetic field consistent with a Cu^{2+} spin spiral polarized in the bc-plane rather than in the ab-plane as reported from earlier neutron diffraction data.
Non-magnetic impurities break a quantum spin chain into finite segments and induce Friedel-like oscillations in the local susceptibility near the edges. The signature of these oscillations has been observed in Knight shift experiments on the high-temperature superconductor YBa$_2$Cu$_3$O$_{6.5}$ and on the spin-chain compound Sr$_2$CuO$_3$. Here we analytically calculate NMR spectra, compare with the available experimental data for Sr$_2$CuO$_3$, and show that the interchain coupling is responsible for the complicated and so far unexplained lineshape. Our results are based on a parameter-free …
In the spin-Peierls compound CuGeO$_{3}$, the commensurate-incommensurate transition concerning the modulation of atomic position and the local spin-polarization is fully monitored at T=0 by the application of an external magnetic field ($H$) above a threshold value $H_{c}\simeq $ 13 Tesla. The solitonic profile of the spin-polarization, as well as its absolute magnitude, has been precisely imaged from $^{65}Cu$ NMR lineshapes obtained for $h=(H-H_{c})/H_{c}$ varying from 0.0015 to 2. This offers a unique possibility to test quantitatively the various numerical and analytical methods developed…
Nuclear magnetic resonance (NMR) measurements of CuO chains of detwinned Ortho-II YBa$_2$Cu$_3$O$_{6.5}$ (YBCO6.5) single crystals reveal unusual and remarkable properties. The chain Cu resonance broadens significantly, but gradually, on cooling from room temperature. The lineshape and its temperature dependence are substantially different from that of a conventional spin/charge density wave (S/CDW) phase transition. Instead, the line broadening is attributed to small amplitude static spin and charge density oscillations with spatially varying amplitudes connected with the ends of the finite l…
A singular evolution toward an insulating phase, shown by 23Na and 13C NMR, has been observed in the superconducting fullerides (NH3)xNaK2C60 for x>1. Unlike most common cases, this insulating phase is non magnetic and 13C spin lattice relaxation shows the presence of a spin gap. These two features suggest that a charge disproportion from C60^(3-) to C60^(2-) and C60^(4-) can drive the system from the metallic to the insulating state. The restoring of the Na+ cation in the center of the octahedral interstice in the insulating phase, as indicated by 23Na and 2H lineshape analysis, confute the c…
The periodic Anderson model is a classic theoretical model for understanding novel physics in heavy fermion systems. Here, we modify it with the Sachdev-Ye-Kitaev interaction, (random all-to-all interaction) thus the resultant model admits an exact solution at large-$N$ (e.g. spin flavor) limit. By analytical field theory arguments and numerical calculations, we establish that the system supports a low-temperature (heavy) Fermi liquid and more interesting a non-Fermi liquid solution at elevated temperature/energy. For physical observable, the latter one contributes a sharp peak at Fermi energy…
We present the synthesis and a detailed investigation of structural and magnetic properties of polycrystalline Ba(TiO)Cu$_{4}$(PO$_{4}$)$_{4}$ (BTCPO) via x-ray diffraction, magnetic susceptibility, heat capacity, and $^{31}$P Nuclear Magnetic Resonance (NMR) measurements. BTCPO has a 2D layered structure with interlinked Cu$_{4}$O$_{12}$ tetramer units. A broad maximum is observed around 16.5 K in our magnetization data accompanied by a sharp anomaly around $T$ = 9.5 K in the heat capacity. An anomaly at $T$ = 9.5 K is also found in the temperature dependence of the $^{31}$P NMR spin-lattice …
Detailed magnetization, specific heat, and $^7$Li nuclear magnetic resonance (NMR) measurements on single crystals of the hyperhoneycomb Kitaev magnet $β$-Li$_2$IrO$_3$ are reported. At high temperatures, {\cred anisotropy of the magnetization is reflected by the different Curie-Weiss temperatures for different field directions}, in agreement with the combination of a ferromagnetic Kitaev interaction ($K$) and a negative off-diagonal anisotropy ($Γ$) as two leading terms in the spin Hamiltonian. At low temperatures, magnetic fields applied along $a$ or $c$ have only a weak effect on the system…
Recent theoretical studies have suggested that Kitaev physics and such effects as formation of a mysterious spin-liquid state can be expected not only in RuCl3 and iridates, but also in conventional $3d$ transition metal compounds. Using DC and AC magnetometry, thermodynamic and $^{23}$Na nuclear magnetic resonance measurements (NMR) we studied such a candidate material Na3Co2SbO6. A full phase diagram of Na3Co2SbO6 in a wide range of magnetic fields and temperatures is presented. The results demonstrate transformation of the antiferromagnetic structure under the external magnetic field, gradu…
We present an NMR study of Na2C60 and K4C60, two compounds that are related by electron-hole symmetry in the C60 triply degenerate conduction band. In both systems, it is known that NMR spin-lattice relaxation rate (1/T1) measurements detect a gap in the electronic structure, most likely related to singlet-triplet excitations of the Jahn-Teller distorted (JTD) C60^{2-} or C60^{4-}. However, the extended temperature range of the measurements presented here (10 K to 700 K) allows to reveal deviations with respect to this general trend, both at high and low temperatures. Above room temperature, 1…
Beta-NMR has been used to study vortex lattice disorder near the surface of the high-Tc superconductor YBCO. The magnetic field distribution from the vortex lattice was detected by implanting a low energy beam of highly polarized 8Li into a thin overlayer of silver on optimally doped, twinned and detwinned YBCO samples. The resonance in Ag broadens significantly below the transition temperature Tc as expected from the emerging field lines of the vortex lattice in YBCO. However, the lineshape is more symmetric and the dependence on the applied magnetic field is much weaker than expected from an…
We investigate the structural and magnetic properties of a Kitaev spin liquid candidate material Ag$_3$LiIr$_2$O$_6$ based on $^7$Li nuclear magnetic resonance line shape, Knight shift and spin-lattice relaxation rate $1/T_1$. The first sample A shows signatures of magnetically ordered spins, and exhibits one sharp $^7$Li peak with FWHM increasing significantly below 14~K. $1/T_1^{stretch}$ of this sample displays a broad local maximum at 40~K, followed by a very sharp peak at $T_N = 9\pm1$~K due to critical slowing down of Ir spin fluctuations, a typical signature of magnetic long range order…
We present an extensive gallium NMR study of the geometrically frustrated kagome bi-layer compound SrCr_(9p)Ga_(12-9p)O_(19) (Cr^3+, S=3/2) over a broad Cr-concentration range (.72<p<.95). This allows us to probe locally the kagome bi-layer susceptibility and separate the intrinsic properties due to the geometric frustration from those related to the site dilution. Our major findings are: 1) The intrinsic kagome bi-layer susceptibility exhibits a maximum in temperature at 40-50 K and is robust to a dilution as high as ~20%. The maximum reveals the development of short range antiferromagnetic c…
We consider the archetypal frustrated antiferromagnet SrCr$_{9p}$Ga$_{12-9p}$O$_{19}$ in its well-known spin-liquid state, and demonstrate that a Cr$^{3+}$ spin $S=3/2$ ion in direct proximity to a pair of vacancies (in disordered $p<1$ samples) is cloaked by a spatially extended spin texture that encodes the correlations of the parent spin-liquid. In this spin-liquid regime, the combined object has a magnetic response identical to a classical spin of length $S/2 = 3/4$, which dominates over the small intrinsic susceptibility of the pure system. This fractional-spin texture leaves an unmistaka…
We report a $^{23}$Na and $^{75}$As nuclear magnetic resonance (NMR) investigation of Na$_{x}$FeAs series ($x=1$, 0.9, 0.8) exhibiting a spin-density wave (SDW) order below $T_{\rm SDW}=45$, 50 and 43 K for $x=1$, 0.9, 0.8, respectively, and a bulk superconductivity below $T_c\approx 12$ K for x=0.9. Below $T_{\rm SDW}$, a spin-lattice relaxation reveals the presence of gapless particle-hole excitations in the whole $x$ range, meaning that a portion of the Fermi surface remains gapless. The superconducting fraction as deduced from the bulk susceptibility scales with this portion, while the SDW…
A prominent feature of quantum spin liquids is fractionalization of the spin degree of freedom. Fractionalized excitations have their own dynamics in different energy scales, and hence, affect finite-temperature ($T$) properties in a peculiar manner even in the paramagnetic state harboring the quantum spin liquid state. We here present a comprehensive theoretical study of the spin dynamics in a wide $T$ range for the Kitaev model on a honeycomb lattice, whose ground state is such a quantum spin liquid. In this model, the fractionalization occurs to break up quantum spins into itinerant matter …
Kitaev's honeycomb spin-liquid model and its proposed realization in materials such as $α$-RuCl$_3$, Li$_2$IrO$_3$ and Na$_2$IrO$_3$ continue to present open questions about how the dynamics of a spin-liquid are modified in the presence of non-Kitaev interactions as well as the presence of inhomogeneities. Here we use $^{23}$Na nuclear magnetic resonance to probe both static and dynamical magnetic properties in single crystal Na$_2$IrO$_3$. We find that the NMR shift follows the bulk susceptibility above 30 K but deviates from it below; moreover below $T_N$ the spectra show a broad distributio…
We report on the results of 23Na NMR in the honeycomb lattice magnet Na2Co2TeO6 which has been nominated as a Kitaev material. Measurements of magnetic shift and width of the NMR line as functions of temperature and magnetic field show that a spin-disordered phase does not appear up to a field of 9 T. In the antiferromagnetic phase just below the Neel temperature TN, we find a temperature region extending down to ~TN/2 where the nuclear spin-lattice relaxation rate 1/T1 remains enhanced and is further increased by a magnetic field. This region crosses over to a low temperature region character…
We have carried out detailed bulk and local probe studies on the hexagonal oxides Ba3MIr2O9 (M=Sc,Y) where Ir is expected to have a fractional oxidation state of +4.5. In the structure, Ir-Ir dimers are arranged in an edge shared triangular network parallel to the ab plane. Whereas only weak anomalies are evident in the susceptibility data, clearer anomalies are present in the heat capacity data. Our 45Sc nuclear magnetic resonance (NMR) lineshape (first order quadrupole split) is symmetric at room temperature but becomes progressively asymmetric with decreasing temperatures. This is suggestiv…
We investigate models of electrons in the Sachdev-Ye-Kitaev class with random and all-to-all electron hopping, electron spin exchange, and Cooper-pair hopping. An attractive on-site interaction between electrons leads to superconductivity at low temperatures. Depending on the relative strengths of the hopping and spin exchange, the normal state at the critical temperature is either a Fermi-liquid or a non-Fermi liquid. We present a large-$M$ (where spin symmetry is enlarged to SU$(M)$) study of the normal state to superconductor phase transition. We describe the transition temperature, the sup…
The 5d-electron honeycomb compound H$_3$LiIr$_2$O$_3$ [K. Kitagawa $et$ $al.$, Nature 554, 341-345 (2018)] exhibits an apparent quantum spin liquid (QSL) state. In this intercalated spin-orbital compound, a remarkable pile up of low-energy states was experimentally observed in specific heat and nuclear magnetic (NMR) spin relaxation. We show that a bond disordered Kitaev model can naturally account for this phenomenon, suggesting that disorder plays an essential role in its theoretical description. In the exactly soluble Kitaev model, we obtain, via spin fractionalization, a random bipartite h…
$^7$Li nuclear magnetic resonance (NMR) and terahertz (THz) spectroscopies are used to probe magnetic excitations and their field dependence in the hyperhoneycomb Kitaev magnet $β$-Li$_2$IrO$_3$. Spin-lattice relaxation rate ($1/T_1$) measured down to 100\,mK indicates gapless nature of the excitations at low fields (below $H_c\simeq 2.8$\,T), in contrast to the gapped magnon excitations found in the honeycomb Kitaev magnet $α$-RuCl$_3$ at zero applied magnetic field. At higher temperatures in $β$-Li$_2$IrO$_3$, $1/T_1$ passes through a broad maximum without any clear anomaly at the Néel tempe…
Fractionalisation -- the breaking up of an apparently indivisible microscopic degree of freedom -- is one of the most counterintuitive phenomena in many-body physics. Here we study its most fundamental manifestation in spin ice, the only known fractionalised magnetic compound in 3D: we directly visualise the $1/r^2$ magnetic Coulomb field of monopoles which emerge as the atomic magnetic dipoles fractionalise. We analyse the internal magnetic field distribution, relevant for local experimental probes. In particular, we present new zero-field NMR measurements which exhibit excellent agreement wi…
We propose a theoretical model for a gapless spin liquid phase that may have been observed in a recent experiment on $\mathrm{H_3Li Ir_2 O_6}$. Despite the insulating and non-magnetic nature of the material, the specific heat coefficient $C/T \sim 1/\sqrt{T}$ in zero magnetic field and $C/T \sim T/ B^{3/2}$ with finite magnetic field $B$ have been observed. In addition, the NMR relaxation rate shows $1/(T_1T) \sim (C/T)^2$. Motivated by the fact that the interlayer/in-plane lattice parameters are reduced/elongated by the hydrogen-intercalation of the parent compound $\mathrm{Li_2 Ir O_3}$, we …
