Tomonaga–Luttinger liquid behavior and spinon confinement in YbAlO3

Low dimensional quantum magnets are interesting because of the emerging collective behavior arising from strong quantum fluctuations. The one-dimensional (1D) S = 1/2 Heisenberg antiferromagnet is a paradigmatic example, whose low-energy excitations, known as spinons, carry fractional spin S = 1/2. These fractional modes can be reconfined by the application of a staggered magnetic field. Even though considerable progress has been made in the theoretical understanding of such magnets, experimental realizations of this low-dimensional physics are relatively rare. This is particularly true for rare-earth-based magnets because of the large effective spin anisotropy induced by the combination of strong spin–orbit coupling and crystal field splitting. Here, we demonstrate that the rare-earth perovskite YbAlO3 provides a realization of a quantum spin S = 1/2 chain material exhibiting both quantum critical Tomonaga–Luttinger liquid behavior and spinon confinement–deconfinement transitions in different regions of magnetic field–temperature phase diagram.

same magnitude, but with opposite signs. In the ordered state, the molecular eld produced by these interactions results into a staggered eld Since the dipole iteration is long range, we have extended this calculation to about forty-eight near neighbors, and the staggered molecular eld saturates to about 0.49 K, which is quite close to the best t B st = 0.27J ∼ 0.66 K used in the DMRG calculation in the main text. With the picture of two sublattices in a staggered eld ( Supplementary Fig. 2c,d), one can clearly see one dimensioanl (1D) Yb antiferromagnetic (AFM) chains along the c-axis emerging from the three dimensional perovskite crystal structure. This observation naturally leads to the 1D Hamiltonian (1) proposed in the main text. AFM wave vector Q = (0, 0, 1) (Supplementary Fig. 3c). At this temperature, the lowest peak of the dispersion along (0K1) can be phenomenologically described as where J is the interchain coupling and E 0 is the gap opened by the staggered eld at L = 1. Energy cuts taken at Q = (0, 0, 1) and (0, 1, 1) are shown in Supplementary Fig. 3d. With tting parameters J = 0.04 meV, and E 0 = 0.32 meV, the above dispersion relation reproduces the peak positions: 0.32 meV at (0, 0, 1) and 0.36 meV at (0, 1, 1). This analysis indicates that the eective interchain interaction is roughly 20% of the intrachain exchange: J /J 0.04/0.21 0.19.

Supplementary Note 4: Polarization Factor and Longitudinal Fluctuations
DC magnetization and CEF studies suggest well separated doublets for the Yb 3+ ion in YbAlO 3 . Since the rst exited CEF levels are high (29 meV ∼ 345 K), the low temperature magnetic properties are dominated by the ground state doublets, which can be eectively described as pseudo-spin S=1/2 states with anisotropic g-tensors: where z is chosen along the local moment easy axis. This Ising-like g-tensor anisotropy does not necessarily lead to anisotropic interactions, but it manifests in the magnetic neutron scattering cross-section [1,2]: Here |F (Q)| 2 is the magnetic form factor of Yb 3+ , δ αβ −Q αQβ is the polarization factor, and S αβ (Q, E) is the dynamical spin structure factor of dierent components. Since 273, transverse contributions to the total magnetic scattering can be neglected and the overall magnetic scattering is dominated by longitudinal uctuations: The magnetic polarization factor can be obtained from the angle between the Yb moments and the a-axis (ϕ = ±23.5 • ): Assuming AFM interactions between nearest-neighbor Yb ions along the c-axis, we can also obtain the static magnetic structure factor where m i are the Yb magnetic moments located at positions r i [3]. The Yb atoms form into a zig-zag chain along the crystal c-axis in the orthorhombic perovskite structure, with small distortions in a and b directions ( Supplementary Fig. 2). The distortion angle along the a-axis is about ∼ 2.2 • , which is negligible for scattering in the (0KL) plane. The larger distortion angle ∼ 9.9 • along the b-aixs introduces an additional wave vector dependence with l ±0.12k in the scattering factor, making the spectral weight around L = 1 more spread at higher values of wave vector K.
The calculated magnetic form factor (|F (Q)| 2 ), polarization factor (1 −Q zQz ) and AF static spin structure factor (S(Q)) are shown in Supplementary Fig. 4a-c. The overall magnetic scattering factor is plotted in Supplementary Fig. 4d. The calculated wave vector dependence of the magnetic scattering can be directly compared to the experimental results, as presented in Supplementary Fig. 5.
The wave vector dependence along the (0K1) direction, measured at temperatures 1 K and 0.05 K, is shown in Supplementary Fig. 5a,d, respectively. The blue lines are the calculated form factor (|F (Q)| 2 ), which changes about 10% along the K direction, while the additional 20% changes are well captured when the polarization factor is included (red lines). The polarization factor is a constant for K = 0 and the intensity variation along the (00L) direction is dominated by the spin structure factor S(Q). As shown in Supple-  Supplementary Fig. 5c,f, where an overall consistency is observed with the calculated pattern shown in Supplementary Fig. 4d. This demonstrates that the magnetic scattering is dominated by longitudinal uctuations along the local easy-axis of the Yb moments.

Supplementary Note 5: Comparison of paramagnon and two-spinon continuum
In the main text, we have shown that the magnetic excitations of YbAlO 3 have a twospinon continuum at T = 1 K (Fig. 3a), right above the Néel temperature T N . The spectrum is consistent with our T = 0 DMRG calculation (Fig. 3c). Furthermore, the broadening of the experimental data can also be captured by a nite temperature tDMRG calculation [4].
While everything is consistent with the two-spinon continuum, it is still interesting to ask if one can obtain the same excitation spectrum from the paramagnon picture. For this comparison, we use the 3-dimensional version of the eective model, with classical spins: where ij denotes the nearest neighbor bonds along the chain, and ij denotes the nearest neighbor bonds in between chains. Following the best t from the main text, we use J =  Fig. 7(a),(b)), while the Landau-Lifshitz simulation produces diuse-like scattering at low energy ( Supplementary Fig. 7(c)), i.e. a simple paramagnon picture does not apply when the temperature is right above T N .

Supplementary Note 6: Quantum critical scaling
The eld dependence of the magnetization is presented in Fig. 7a in the main text.
Following Ref. 5, we dene the deviation from scaling as: As shown in Supplementary Fig. 8a, the minimum deviation is reached for νz = 1.04 (10) which is consistent with a free fermion xed point: ν = 1/2, z = 2 and d = 1.