We investigate magnetic excitations in the spin-ladder compound Sr 14 Cu 24 O 41 using high-resolution Cu L 3 edge resonant inelastic x-ray scattering (RIXS). Our findings demonstrate that RIXS couples to two-triplon collective excitations. In contrast to inelastic neutron scattering, the RIXS cross section changes only moderately over the entire Brillouin zone, revealing high sensitivity also at small momentum transfers, allowing determination of the two-triplon energy gap as 100 AE 30 meV. Our results are backed by calculations within an effective Hubbard model for a finite-size cluster, and confirm that optical selection rules are obeyed for excitations from this spherically symmetric quantum spin-liquid ground state. DOI: 10.1103/PhysRevLett.103.047401 PACS numbers: 78.70.En, 71.10.Pm, 75.25.+z, 75.30.Ds Collective excitations in strongly correlated electron materials remain a pivotal challenge in contemporary solid state physics. It is widely debated whether magnetic excitations provide the pairing interaction in the hightemperature and unconventional superconductors [1,2]. From that perspective quantum spin systems attract considerable interest. While most such materials, e.g., the cuprate superconductors, exhibit enormous complexity, the two-leg spin ladder is easier to tract theoretically [3][4][5][6]. It consists of two parallel chains (legs) with a transverse (rung) exchange coupling. This system features a singlet ground state and dispersive triplet excitations (triplons), that both have quantum mechanical origin without any classical counterpart. To date, mainly two techniques have been established as momentum-and energy-resolved probes of the dispersion of collective excitations: angleresolved photoelectron spectroscopy and inelastic neutron scattering (INS) for charge and spin degrees of freedom, respectively [7,8]. Because of the latest instrumental improvements [9,10], the energy scale of magnetic exchange is becoming readily accessible for resonant inelastic x-ray scattering (RIXS) [11][12][13][14], which is promising to give information on both, spin and charge degrees of freedom, and in addition is an element-specific technique. Furthermore, RIXS requires only small sample volumes (<0:1 mm 3 ). Recent RIXS studies were performed on long-range ordered magnets with spin-wave excitations [15][16][17].In this Letter, we report a study of the two-leg quantum spin ladder Sr 14 Cu 24 O 41 [18,19] by means of momentumresolved high-resolution RIXS at the Cu L 3 edge. Given that Cu L 3 scattering experiments have been already shown to contain valuable information about the charge degrees of freedom [20], an outstanding question we would like to address here is: how can RIXS provide information on magnetic excitations from a quantum ground state. In the ladder system of Sr 14 Cu 24 O 41 no symmetry breaking occurs-neither in spin-nor in real-space-in contrast to, e.g., a magnetically ordered state, where both symmetries are broken and the direction of the ordered moments dictates the quantization axi...
The magnetic entropy variation for the ordered spin-ice Tb 2 Sn 2 O 7 and the spin-liquid Tb 2 Ti 2 O 7 measured up to ϳ20 K shows that the crystal-field ground state for both systems is a singlet split from the first excited level by a thermal energy of ϳ2 K. It is the superexchange interactions between the Tb 3+ spins which induce the terbium magnetic moments. Our results are analyzed in terms of a tetragonal perturbation for Tb 2 Ti 2 O 7 and this perturbation plus a molecular field for Tb 2 Sn 2 O 7 .Geometrically frustrated magnetic materials are now intensively studied, both theoretically and experimentally. In particular, the pyrochlore crystal structure R 2 M 2 O 7 ͑R stands for a rare-earth ion and M for a nonmagnetic p or d metal͒ built of corner sharing tetrahedra usually inhibits the formation of a collinear long-range magnetic order, leading to a variety of magnetic behaviors, such as spin ices, spin liquids, and complex noncollinear magnetic orders. 1,2Among the pyrochlore compounds, the insulators Tb 2 Ti 2 O 7 and Tb 2 Sn 2 O 7 are probably the less understood systems. They crystallize at room temperature with the cubic space group Fd3m. The local symmetry at a Tb 3+ ion is D 3d . It is important to note that structural distortions below ϳ20 K have been reported from x-ray measurements for Tb 2 Ti 2 O 7 . 3 These distortions are a signature of a transition from cubic to tetragonal structures. However, the phase transition was not observed. Anyhow, the x-ray data show that the rare-earth local symmetry is reduced at low temperature. Neutron diffraction for Tb 2 Sn 2 O 7 shows that the crystal structure is tetragonal in the magnetically ordered state. 4,5 Tb 2 Sn 2 O 7 is a so-called ordered spin-ice system below T c = 0.88͑1͒ K. 4,5 For a given tetrahedron the four spins are close to the ͗111͘ anisotropy axes in a two-in two-out magnetic configuration. In contrast to the classical spin-ice compounds ͑Ho 2 Ti 2 O 7 and Dy 2 Ti 2 O 7 ͒, the magnetic unit cell, which is the crystallographic unit cell, repeats itself. The ordered spin-ice structure is explained within the phenomenological model of Champion et al. 6 which assumes a ferromagnetic first neighbor exchange interaction combined with a finite anisotropy. In fact, a SR experiment has shown that the magnetic ground state is homogeneous ͑no phase separation͒ and dynamical in nature. 5,7 While this has been hotly debated, 8,9 it seems now to be accepted. 10,11 On the other hand, the sibling compound Tb 2 Ti 2 O 7 is classified as a spin-liquid system because it does not display any magnetic order, at least down to 0.05 K, despite the onset of short-range antiferromagnetic correlations developing at ϳ50 K. 12,13 The absence of a long-range magnetic order is a long-standing mystery. [14][15][16] Based on inelastic neutron-scattering measurements, a crystal-field level scheme for the Tb 3+ ions in the two com-pounds of interest here has been proposed assuming the D 3d point symmetry to be valid. 17 The local threefold axis is the crystallographic ax...
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