Tomonaga-Luttinger Liquid Behavior and Spinon Confinement in YbAlO$_3$

TL;DR

This paper reports on YbAlO$_3$, a rare-earth perovskite that exhibits quantum spin chain behavior, including Tomonaga-Luttinger liquid properties and spinon confinement-deconfinement transitions, advancing experimental understanding of low-dimensional quantum magnets.

Contribution

It demonstrates that YbAlO$_3$ is a rare-earth magnet realizing both Tomonaga-Luttinger liquid behavior and spinon confinement, providing experimental insights into low-dimensional quantum physics.

Findings

YbAlO$_3$ exhibits Tomonaga-Luttinger liquid behavior.

YbAlO$_3$ shows spinon confinement-deconfinement transitions.

Experimental phase diagram reveals different magnetic regimes.

Abstract

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 YbAlO$_3$ 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.

Figures

a). However, we expect that higher harmonics will be induced for this magnetization values producing spin superstructures similar to those emerging from the anisotropic next-nearest-neighbor Ising (ANNNI) model [Elliott1961 ](#bib.bib25); [Fisher1980 ](#bib.bib26); [Selke1988 ](#bib.bib27).

Equations

$$M\approx \frac{M_{\mathrm{s}}}{2}\left(|\cos (\theta -\phi )|+|\cos (\phi +\theta )|\right),$$

$$\mathcal{H}=J\sum _i{\mathbf{S}}_i\cdot {\mathbf{S}}_{i+1}-B_{\mathrm{st}}\sum _i(-1{)}^i{S}_i^z-B_{\mathrm{ex}}\sum _i{S}_i^z,$$

$${\cal L}=\frac{1}{2}\Big{[}c^{-1}(\partial_{\tau}\phi)^{2}+c(\partial_{x}\phi)^{2}\Big{]}-\alpha B_{\rm st}\sin(\beta\phi),\leavevmode\nobreak\ \leavevmode\nobreak\ \beta^{2}=2\pi,\leavevmode\nobreak\ \leavevmode\nobreak\ c=\pi J/2.$$

$$S_n^z=\frac{\beta }{2\pi }{\partial }_x\varphi +(-1{)}^{n}A\sin (\beta \varphi ),$$

$$S_n^z=M+\frac{\beta }{2\pi }{\partial }_x\varphi +A\sin (\beta \varphi +2\pi Mn),$$

$$\sum _{i,j}{J}_{ij}{S}_i^z{S}_j^z\to \sum _{i,j}{J}_{ij}{A}^2\cos [\beta ({\varphi }_i-{\varphi }_j)].$$

$${\omega }^2=v^2{q}_{\parallel }^2+\sum _j{a}_j{\sin }^2({\mathbf{qe}}_j/2).$$

$$\frac{\mathrm{d}M}{\mathrm{d}B}={\tilde{B}}^{\nu (d+z)-2}\phi (\frac{T}{{\tilde{B}}^{\nu z}}),$$

$$\nu z=1.04,$$

$$2-\nu (d+z)=\mathrm{0.51.}$$

$$\frac{\Delta C_{\mathrm{M}}}{T}={\tilde{B}}^{\nu (d-z)}\psi (\frac{T}{{\tilde{B}}^{\nu z}})={\tilde{B}}^{-0.5}\psi (\frac{T}{\tilde{B}}),$$

$$\frac{\Delta C_{\mathrm{M}}}{T}=\frac{C_{\mathrm{M}}(B)}{T}-\frac{C_{\mathrm{M}}(B_{\mathrm{s}})}{T}.$$

Full text

††thanks: These authors contributed equally to this work††thanks: These authors contributed equally to this work††thanks: These authors contributed equally to this work

Tomonaga-Luttinger Liquid Behavior and Spinon Confinement in YbAlO3

L. S. Wu

Neutron Scattering Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA

Department of Physics, Southern University of Science and Technology, Shenzhen 518055, China

wuls@sustc.edu.cn

  

S. E. Nikitin

Max Planck Institute for Chemical Physics of Solids, Nöthnitzer Str. 40, D-01187 Dresden, Germany

Institut für Festkörper- und Materialphysik, Technische Universität Dresden, D-01069 Dresden, Germany

  

Z. Wang

Department of Physics and Astronomy, The University of Tennessee, Knoxville, TN 37996, USA

  

W. Zhu

Westlake Institute of Advanced Study, Hangzhou, 310024, P. R. China

Theoretical Division, T-4 and CNLS, Los Alamos National Laboratory, Los Alamos, NM 87545, USA

  

C. D. Batista

Department of Physics and Astronomy, The University of Tennessee, Knoxville, TN 37996, USA

Shull-Wollan Center, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA

  

A. M. Tsvelik

Condensed Matter Physics and Materials Science Division, Brookhaven National Laboratory, Upton, NY 11973, USA

  

A. M. Samarakoon

Neutron Scattering Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA

  

D. A. Tennant

Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, U.S.A.

Shull-Wollan Center, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA

  

M. Brando

Max Planck Institute for Chemical Physics of Solids, Nöthnitzer Str. 40, D-01187 Dresden, Germany

  

L. Vasylechko

Lviv Polytechnic National University, 79013 Lviv, Ukraine

  

M. Frontzek

Neutron Scattering Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA

  

A. T. Savici

Neutron Scattering Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA

  

G. Sala

Neutron Scattering Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA

  

G. Ehlers

Neutron Technologies Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA

  

A. D. Christianson

Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, U.S.A.

Neutron Scattering Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA

  

M. D. Lumsden

Neutron Scattering Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA

  

A. Podlesnyak

podlesnyakaa@ornl.gov

Neutron Scattering Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA

**ABSTRACT

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.

**

Introduction

The spin $S=1/2$ antiferromagnetic Heisenberg Hamiltonian is one of the simplest models of condensed matter physics containing fractional quantum number excitations. The phase diagram of this model includes both quantum critical (for easy-plane anisotropy) and gapped phases (for the easy-axis anisotropy). Spinons are topological excitations which can be pictured as domain walls between different Néel ground states of the system (this picture is especially appropriate for the moments with easy-axis anisotropy). In view of its exotic physics, it is exciting to find realizations of this model and it is particularly exciting to find them in unexpected places. The three-dimensional metallic system Yb2Pt2Pb is a recent example, where the neutron scattering revealed the existence of one-dimensional spinon continuum Wu2016 ; Classen2018 . The unusual properties of Yb2Pt2Pb were initially attributed to its peculiar crystal structure Kim2013 ; Ochiai2011 , but more recent work Wu2016 demonstrated that the peculiar form of exchange interactions in rare earth ions plays a dominant role. However, the metallic nature of this material complicates the analysis of its magnetic properties.

Here we discuss the insulating analog of Yb2Pt2Pb, namely Yb-based quasi-1D quantum magnet, YbAlO3 Radhakrishna1981 . RKKY interactions are not present because YbAlO3 is an insulator. Despite the strong uniaxial single ion anisotropy of the Yb magnetic moments, our neutron scattering data demonstrates that, unlike Yb2Pt2Pb, YbAlO3 is described by a nearly isotropic (Heisenberg) intrachain interaction, which results in Tomonaga-Luttinger liquid behavior Tomonaga1950 ; Luttinger1963 ; Mattis1965 over a finite window of applied magnetic field values. Below the Néel temperature $T_{\rm N}=0.88$ K, the ordered moments from neighboring chains produce a staggered molecular field that confines the fractional spinon excitations. An additional advantage of YbAlO3 is that the moments can be saturated with a relatively low magnetic field of 1.1 T, enabling exploration of the entire magnetic field-temperature phase diagram.

Results

Magnetization, Single ion anisotropy and Two magnetic sublattices

YbAlO3 crystallizes in an orthorhombically distorted perovskite structure Buryy2010 , with room temperature lattice constants $a=5.126$ Å, $b=5.331$ Å, and $c=7.313$ Å (in conventional $Pbnm$ notation). The local point-group symmetry of the Yb3+ ions splits the eight-fold degenerate $J=7/2$ ($L=3$, $S=1/2$) ground state multiplet ($2J+1=8$) into four doublet states. The ground doublet state $m_{J}=\pm 7/2$ with small admixture of other states is well separated from the first excited levels Radhakrishna1981 . This ensures that the low temperature and low field magnetic properties can be described by a pseudo-spin 1/2 model.

Crystalline electrical fields (CEF) calculations and magnetization measurements reveal strong uniaxial (Ising-like) single-ion anisotropy, with easy-axis in the $ab$-plane. Figure 1a-c shows the field dependence of magnetization, $M$, measured at $T=2$ K. At $B=5$ T, the angle dependent magnetization (Fig. 1d) is well described by the function

[TABLE]

where $\varphi=23.5^{\circ}$ is the tilting angle of the local easy-axis relative to the $a$-axis, $\theta$ indicates the direction of the applied field in the $ab$-plane, and $M_{\rm{s}}=3.8$ $\mu_{\rm{B}}{\rm{/Yb}}$ is the saturation moment Radhakrishna1981 . The effective $g$-factors from magnetization measurements are: $g^{xx}{\simeq}g^{yy}=0.46$, and $g^{zz}=7.6$ Radhakrishna1981 , which are consistent with early studies by Mössbauer effect and electron paramagnetic resonance (EPR) measurements Bonville1978 ; Bonville1980 .

In addition to the $g$-tensor anisotropy, the field dependent magnetization reveals that the coupling between Ising moments with different tilting angles $\pm\varphi$ is weak: only one transition is observed for $B\parallel{a}$, $B\parallel{b}$ (Fig. 1a,b), while two successive transitions are found for $B\parallel$ [110] (Fig. 1c), suggesting that magnetic moments with different easy-axis orientation can be individually tuned.

Model Hamiltonian and zero-field inelastic neutron scattering spectrum

The most important aspect of the Yb-physics is the exchange interaction. Contrary to naive expectations, the Yb Kramers doublets do not behave as classical Ising spins when they interact with each other. The exchange processes between the pseudospins on the same chain include spin flip terms of quantum origin. The reason for this was described in Ref. Wu2016, : the superexchange interaction between total angular momenta of rare earth ions includes matrix elements between all eigenstates, not just between those differing by $\delta J^{z}=\pm 1$. Being projected on the lowest Kramers doublet, this interaction acquires the familiar $S^{a}S^{a}$-form. We remind the reader that spin $S=1/2$ operators $S^{a}$ act on the Kramers doublet and, to a very good approximation, only $S^{z}$ is linearly related to the total angular moment: $J^{z}=g^{zz}S^{z}$ with $g^{zz}\simeq 7.6$ ( $g^{xx}{\simeq}g^{yy}\ll g^{zz}$ ).

The magnetic coupling between different sublattices is dominated by the long-range dipole-dipole interaction, which results in a weak ferromagnetic (FM) Ising-like coupling of type $S_{i}^{z}S_{j}^{z}$ in terms of the effective spins (see Supplementary Note 2). In contrast, the intrachain interaction between neighboring spins is dominated by antiferromagnetic (AFM) super-exchange. This combination leads to an $AxGy$-type AFM ordering Radhakrishna1981 below the Néel temperature $T_{\rm N}=0.88$ K (see Fig. 2a). Above $T_{\rm N}$, the weaker dipole-dipole interchain interaction can be neglected, since thermal fluctuations have destroyed the interchain ordering. In this free-standing 1D chain picture, the elementary excitations are fractional spinons, instead of the conventional magnons ($S=1$) of the magnetically ordered state. As illustrated in Fig. 2b-d, a pair of domain walls can propagate along the chain direction, behaving as free spin 1/2 spinons for $T>T_{\rm N}$. Below $T_{\rm N}$, the staggered interchain molecular field ($B_{\rm st}$) produces a confining potential that increases linearly in the distance between both spinons (Fig. 2e). Consequently, spinons get confined into bound states at temperatures below $T_{\rm N}$ Lake2010 .

In view of these considerations, the magnetic properties of YbAlO3 can be described with an effective 1D spin-1/2 Hamiltonian

[TABLE]

where $x,y$ and $z$ are defined in Fig. 2b, $J$ is the intrachain Heisenberg exchange, $B_{\rm st}$ is the effective staggered molecular field generated below $T_{\rm N}$, and $B_{\rm ex}$ is the external magnetic field applied along the local moments easy $z$-axis. The pseudospin $S=1/2$ operators $S^{a}$ act on the Kramers doublet, consisting of the $m_{J}=\pm 7/2$ states to a good approximation, which explains the $g$-tensor anisotropy. In other words, the magnetic field and the magnetic moment of the neutron can couple only to the diagonal operator $S^{z}$ and the INS spectrum of YbAlO3 is dominated by longitudinal fluctuations: the very weak g-factor along the transverse directions, $(g^{xx}/g^{zz})^{2}\approx 1/273$, renders transverse fluctuations, $S^{xy}(\bm{Q},E)$, ‘hidden’ to neutrons. Thus, the magnetic excitations observed in YbAlO3 only arise from longitudinal fluctuations, as demonstrated by the neutron polarization factor presented in Supplementary Note 4, and by all the simulated spectra shown in Fig. 3c-d. As we will see later, the exchange anisotropy turns out to be surprisingly small in this material. This unexpected situation encountered in an increasing number of compounds Wu2016 ; Rau18 ; Hester18 , has been analyzed theoretically in Wu2016 and later in greater detail in Ref. Rau18 .

Single crystal inelastic neutron scattering measurements have been performed to match the Hamiltonian parameters to experimental observables. The zero-field spectra ($B_{\rm ex}=0$) are presented in Fig. 3a-b at different temperatures (see also Supplementary Note 3). Gapless spinon excitations along the chain direction ($00L$) are observed at 1.0 K ($T>T_{\rm N}$), with a broad two-spinon continuum extending from 0 up to about 0.7 meV in energy transfer (Fig. 3a). No gap in the spin excitation can be resolved in the paramagnetic (PM) state of YbAlO3 at the magnetic Brillouin zone center of $\bm{Q}$=(001), within the instrumental resolution about 0.05 meV. Note that the temperature (1.0 K) is comparable to the instrumental resolution (0.05 meV), both of which are much smaller than the bandwidth of the two-spinon continuum in Fig. 3a, implying that we can safely exclude a scenario where the continuum arises from thermal or instrumental broadening (see also Supplementary Note 5).

In contrast, and in agreement with the analytical result for the spin-1/2 Heisenberg model in a staggered magnetic field Essler97 , a significant gap is observed below the ordering temperature, with a massive triplet mode and multi-spinon continuum extending from about 0.3 to 0.7 meV (Fig. 3b).

As demonstrated in Ref. Essler97, , the staggered field leads to strong confinement of the spinons. In the continuum limit, one can use bosonization and the Hamiltonian is reduced to the sine-Gordon model. The corresponding Lagrangian density is

[TABLE]

and the spin is related to field $\phi(x)$:

[TABLE]

where $A$ is a nonuniversal amplitude. Here the staggered field is determined self-consistently as the one generated by the ordered moments of neighboring chains: $B_{\rm st}=J_{\perp}\langle S^{z}_{\rm st}\rangle$. The exact solution of the sine-Gordon model for this value of $\beta$ yields the spectra $E_{n}(p)=\sqrt{(cp)^{2}+\Delta_{n}^{2}}$ with two excitation branches corresponding to a triplet $n=1$ and singlet $n=2$ with $\Delta_{2}=\sqrt{3}\Delta_{1},\leavevmode\nobreak\ \Delta_{n}\sim(JB_{\rm st}^{2})^{1/3}$. The inverse gap $c/\Delta\sim(J/B_{\rm st})^{2/3}$ is the confinement radius. Since $B_{\rm st}$ is proportional to the order parameter, it vanishes at the transition point and the confinement radius becomes infinite. The triplet excitations with $S=1$ are seen in the dynamical spin susceptibility.

The simulated longitudinal spin excitation spectrum is presented in Fig. 3c-d, using 64-site density matrix renormalization group calculations (DMRG) at $T=0$ with periodic boundary conditions White1992 ; White1993 . The Hamiltonian (2) captures the main features of the experimental data measured at 1 K and 0.05 K for intrachain exchange $J=0.21$ meV, staggered field $B_{\rm st}$=0 (Fig. 3c) and $B_{\rm st}/J=0.27$ (Fig. 3d). It is worth noting that a finite temperature DMRG calculation for the same model Lake13 can account for the deviations between the experimental data and our DMRG result at $T=0$.

The nearly isotropic nature of the intrachain interaction may look surprising if we consider that the Yb3+ ground state doublet is highly anisotropic. However, as was pointed out in Ref. Wu2016, , the exchange interaction of rare earth ions has a form of permutation operator which has matrix elements between all states. Being projected on the lowest Kramers doublet by the crystal field, it acquires the familiar ${\bf S}_{i}\cdot{\bf S}_{i+1}$ form. Apart from Yb2Pt2Pb and YbAlO3 the only known previous example of Heisenberg-like Yb chains is Yb4As3 Kohgi1997 ; Kohgi2001 . It should be noted, however, that the shape of the neutron scattering spectra is not sufficient to determine the magnitude of a possible eaxy-plane exchange anisotropy $\Delta$ that would still be compatible with the observed TLL at zero magnetic field: for an XXZ chain, the transverse (longitudinal) modes for $0.25\lesssim\Delta\lesssim 1$ ($0.5\lesssim\Delta\lesssim 1$) almost do not change Caux2005 ; Caux2012 . Nevertheless, as we explain in a later section, the observation of a free fermion fixed point at the saturation field $B=B_{\rm s}$ excludes the possibility of a significant easy-plane exchange anisotropy.

The temperature evolution of the energy dependent scattering at the AFM zone center $\bm{Q}$=(001) is presented in Fig. 3e. The continuous temperature dependent entropy $S$ in Fig. 3f, suggests a second order transition occurs at $T_{\rm N}$. The gap in spin excitations closes within a narrow temperature $T_{\rm N}\pm 0.1$ K, which is unlikely due to thermal broadening. Figure 3f shows the temperature dependence of the (001) Bragg peak intensity, indicating that static moments build up at the phase transition. Meanwhile, the integrated inelastic neutron intensity shows that about half of the total spectral weight remains inelastic below $T_{\rm N}$ (see Fig. 3f). The temperature dependence of entropy ($S$) is over-plotted in Fig. 3f as well. As expected from the ground state doublets, the full entropy ($R\cdot\rm ln2$) is reached above $T=10$ K, however, only about $0.4R\cdot\rm ln2$ is released at $T_{\rm N}$. This value is consistent with the neutron scattering results: about half of the magnetic moment keeps fluctuating in the ordered state.

Magnetic field-induced quantum phase transitions

With a magnetic field applied along the a-axis, the static AFM order is suppressed at the critical field $B_{\rm c}=0.35$ T in favor of incommensurate magnetic ordering. As entering the IC-AFM phase at $B_{\rm c}$, the spin excitation gap in the magnetic zone center $\bm{Q}=\rm(001)$ closes abruptly (Fig. 4a-c,; Fig. 5e,f and Fig. 6b). Figure 5a shows the field dependence of the magnetic contribution to the specific heat $C_{\rm{M}}/T$ around $B_{\rm{c}}$. The step-like anomaly becomes smaller and broader upon decreasing $T$ and finally evolves into a weak maximum around $B_{\rm{c}}$ (see also Supplementary Note 1). Meanwhile, the field temperature ($B$-$T$) phase line becomes very steep (${\rm d}B/{\rm d}T\rightarrow-\infty$) as $T$ approaches 0 K (Fig. 6c). This indicates that the system is tuned through a first order phase transition at $T=0$ K Flouquet2005 , which is consistent with the neutron scattering observations (Fig. 6b).

For $B>B_{\rm{c}}$, the longitudinal ($S^{zz}(\bm{Q},E)$) and transverse ($S^{xx}(\bm{Q},E)$, $S^{yy}(\bm{Q},E)$) components of the dynamical spin structure factor behave differently Muller1981 . The bosonization formula (4) is modified in the presence of finite magnetization $\langle S^{z}\rangle=M$ :

[TABLE]

Therefore, in the field-induced incommensurate phase, the interchain interaction becomes

[TABLE]

The oscillatory term does not survive after summing over the phases. As we have mentioned above, the spectral gap is suppressed when the magnetic field exceeds the critical value, and the single-chain system becomes critical. However, the interchain coupling remains relevant and leading to long-range magnetic ordering and the corresponding thermodynamic phase transition (see Fig. 5b and Fig. 6c). Below the transition one can expand the cosine in (6) and obtain the quadratic action for $\phi$ so that the interaction leads to a gapless phason mode, which explains a finite low energy spectral weight on Fig. 5f, and a significant $C_{\rm M}/T$ values below the transition:

[TABLE]

At higher energies one can again approximate $\sum_{j}\cos[\beta(\phi_{i}-\phi_{j})]\approx\cos\beta\phi_{i}\sum_{j}\langle\cos\beta\phi_{j}\rangle$ to obtain the effective sine-Gordon model, as it was done in Ref. Essler97, . The gapped $S=1$ excitations of the sine-Gordon model are allowed to decay into low energy phasons and become resonances.

Figure 4a-c and Fig. 6b include the experimental spin excitation spectra of YbAlO3 for different external fields. The low energy features discussed above are obscured in the color pictures since the main spectral weight remains in the single chains. The $\bm{Q}$-cuts show that the incommensurate (IC) zero energy soft modes are symmetrically located around the AF zone center (Fig. 4d-f) Muller1981 . The shift of these IC wave vectors $Q_{L}=1\pm\Delta q_{L}$ is directly proportional to the field-induced magnetization $M$, i.e., $\Delta q_{L}\sim 2\pi M/M_{s}$. As one can see in Fig. 6a, the INS zero energy soft point traced the magnetization curve up to about 1.0 T. Finally, all the magnetic moments become fully polarized above the saturation field $B_{\rm{s}}=1.1$ T. As we have emphasized before, no magnon-like spectra can be observed for $B>B_{\rm{s}}$, because the modes become purely transverse.

For $B_{\rm{c}}<B<B_{\rm s}$, the Yb3+ moments are highly fluctuating even in the ordered phase. Figure 5d-f shows the energy dependent neutron scattering intensity integrated over the first Brillouin zone. The integrated intensity up to 0.8 meV, $S_{\rm{int}}=\iint{S}\left(\bm{Q},E\right){\;}{\rm d}\bm{Q}{\rm d}E$, is then an estimate of the square of the fluctuating moment. In zero field, $S_{\rm{int}}$ at 0.05 K (AFM ordered state) is about $0.37/0.85\simeq 44\%$ (normalized by the intensity at $T=1$ K and $B=0$); while in 0.6 T, it reaches about $0.72/0.85\simeq 85\%$. Similar conclusions are found in the temperature dependent specific heat and entropy as well. The integrated temperature dependent entropy shown in Fig. 5c, reveals that only $10\sim 20\%$ of $R\cdot\ln 2$ is released at the IC-AFM phase transition. These together suggest that large moment fluctuations coexist with a small ordered moment ($10\sim 20\%$ of the full moment) in the IC-AFM phase (Fig. 6c).

Since the interchain molecular field is proportional to the magnitude of the ordered moments, in a first approximation we can ignore it for $B_{\rm c}<B<B_{\rm s}$. This simplification reduces the Hamiltonian (1) to the conventional Heisenberg model in an external field $B_{\rm ex}$. We again use the DMRG calculation to obtain the longitudinal spin structure factor $S^{zz}(\bm{Q},E)$ above $B_{\rm c}$, in various external fields $B_{\rm ex}/J=0.8,1.2,1.6$, as presented in Fig. 4g,h,i. We note that, although a self-consistent treatment of the static moments from neighboring chains is required to obtain a more accurate excitation spectrum, overall qualitative agreement with the experimental data is already achieved at this level of approximation.

It is interesting to note that an $M/M_{\rm s}=1/3$ magnetization plateau appears at 0.75 T, indicating the stabilization of a new magnetic structure. A rather abrupt increase of the magnetic susceptibility is also observed at $M/M_{\rm s}=1/2$, although no obvious magnetization plateau can be resolved.

A key and surprising observation of this work is that YbAlO3 can be modeled by nearly isotropic Heisenberg spin 1/2 chains, despite the very strong $g$-tensor anisotropy. This unusual combination together with the dominant dipolar interchain interactions make YbAlO3 an ideal material for studying the competition between sliding TLL’s and spin density wave ordering. Our measured phase diagram reflects such competition. Each chain behaves as a TLL at temperatures $T_{\rm N}<T\ll J$, while spin density wave ordering (SDW) develops below $T_{\rm N}$. The intrachain Hamiltonian has a divergent longitudinal magnetic susceptibility $\chi(q,\omega)$ at $q=2k_{\rm f}$ and $\omega=0$, where $k_{\rm f}$ is the Fermi wave-vector of the fermionic system associated with the TLL. Given that $k_{\rm f}=\pm(m/2+1/2)\pi$, with $m=M/M_{\rm s}$, $k_{\rm f}$ increases linearly with $M$ between $B_{\rm c}$ and $B_{\rm s}$ ($k_{\rm f}=\pi/2$ for $0\leq B<B_{\rm c}$). The Ising-like interchain dipolar interaction favors a collinear structure with maximum amplitude of the local moment at each site. However, the magnitude of the moment $|\langle S_{i}^{z}\rangle|$ is necessarily modulated for general values of $2k_{\rm f}$, leading to competition between intrachain exchange and the interchain Ising-like dipolar interaction that penalizes any longitudinal modulation of the magnetic moments. The only exceptions are $2k_{\rm f}$ equal to 0 ($\uparrow\uparrow\uparrow$),$\pi$ ($\uparrow\downarrow\uparrow\downarrow$) or $\pm 2\pi/3$($\uparrow\uparrow\downarrow$). The magnitude of the local moments is not modulated for these particular ordering wave vectors, implying that intrachain and interchain interactions can be satisfied simultaneously. Note that the $\uparrow\uparrow\downarrow$ ordering only contains Fourier components $q=\pm 2\pi/3$ plus the field-induced $q=0$ component. The lack of competition for these special cases explains the observed magnetization plateaus at $m=0,1$ and 1/3. A spin density wave with dominant $2k_{\rm f}$ ordering wave-vector is expected for other magnetization values, as it is confirmed by the fact that the ordering wave vector extracted from the INS data tracks the magnetization curve (see Fig. 4a). However, we expect that higher harmonics will be induced for this magnetization values producing spin superstructures similar to those emerging from the anisotropic next-nearest-neighbor Ising (ANNNI) model Elliott1961 ; Fisher1980 ; Selke1988 .

Quantum critical scaling and Universality class

The quantum phase transition at the saturation field is of second order, implying a quantum critical point (QCP) at $B=B_{s}$ and $T=0$. This QCP is a Gaussian fixed point because the effective dimension of the low-energy effective theory is $D=3+1$ (the dynamical exponent is $z=1$ because of the exchange anisotropy). For temperatures higher than the exchange anisotropy and the interchain coupling, we expect a crossover into a regime controlled by a free fermion fixed point ($\nu=1/2$) in dimension $D=1+2$ (the dynamical exponent becomes $z=2$ because the field couples to a conserved quantity in absence of exchange anisotropy) Sachdev1999 ; Zapf2014 .

Figure 7a shows the field dependent magnetization $M$ at different temperatures for $T>0.5$ K. Near the QCP, the magnetic susceptibility scales as

[TABLE]

where $d$ is the spatial dimension and $\tilde{B}=|B-B_{\rm s}|$. The values $d$, $\nu$ and $z$ can be obtained by minimizing the deviations from this scaling behavior (see Supplementary Note 6), giving:

[TABLE]

As shown in Fig. 7b, this set of exponents collapses the measured susceptibility data.

In addition, the field dependent specific heat should obey the following scaling relation Schroder2000 ; Matsumoto2011 ; Bianchi2003 ; Wu2014 :

[TABLE]

where

[TABLE]

The measured temperature dependent $C_{\rm M}/T$ curves, shown in Fig. 7(c), collapse into a single curve for $T\geq 0.3$ K, once they are rescaled with $\nu z=1$ and $d=1$ (see Fig. 7d).

$C_{M}(B)/T$ exhibits a clear two-peak structure in the vicinity of the quantum critical region for each fixed temperature $T\geq 0.3$ K (star markers in Fig. 7e). This is consistent with the expectation of the 1D TLL behavior as discussed in Ref. Blanc2018, . Defining this peak position as the crossover temperature $T^{*}$ into the quantum critical regime, the linear dependence $T^{*}\sim\tilde{B}^{\nu z}$ with $\nu z=1$ for $T\geq 0.3$ K again agrees with the free fermion fix point (Fig. 7e,f). We note that the free fermion fixed point is only compatible with an isotropic exchange interaction because the applied magnetic field is nearly orthogonal to the uniaxial symmetry chain axis (as shown in Fig. 2, the magnetic moments are nearly perpendicular to the chains). Any type of exchange anisotropy would then change the universality class of this QCP to Ising in D=1+1 ($z=\nu=1$). The single-chain Hamiltonian must then be isotropic (Heisenberg) for the model the retain a U(1) symmetry in the presence of the external magnetic field. Based on the above-described analysis and the value of the exchange parameter ($J\approx 0.21$ meV), we can narrow down the range of the exchange anisotropy to $0.88\lesssim\Delta\lesssim 1$.

Discussion

Our results demonstrate that $f$-electron based magnets can provide realizations of various aspects of quasi-one dimensional physics. This is so despite the presence of strong spin-orbit coupling combined with crystal field splitting that produce a ground state Kramers doublet with well separated values of $m_{\rm J}$. Naively one would think about such doublets should behave a as classical (Ising) spins. This unexpected situation, which is appearing in an increasing number of Yb-based compounds Wu2016 ; Rau18 ; Hester18 , enables the study of model Hamiltonians, which have traditionally been regarded as “toy models” due to the combination of interactions that do not appear together. The array of TLL’s coupled by density-density interactions is a clear example of a model that was originally introduced to study the possible existence of sliding TLL phases, but it is difficult to find in real materials due to the very unusual combination of intrachain XXZ exchange and pure Ising interchain interaction. YbAlO3 provides a natural realization of the spin 1/2 version of this model, enabling the observation of a quantum fractional spinon continuum above $T_{\rm N}$ and the transition into a confined (magnetically ordered) low-temperature state characterized by massive excitations. Both, the gapless spinon spectrum and the critical behavior around the saturation field at $T>T_{N}$ unambiguously demonstrate that the intrachain exchange is Heisenberg-like to a very good approximation.

Rau and Gingras Rau18 derived an isotropic super-exchange interaction for Yb-based compounds with edge-sharing octahedra. While it is not yet clear that this result applies directly to our material, it shows that Yb-based compounds can support nearly isotropic super-exchange of the type that we are finding in YbAlO3. Also, note that further experiments such as an electron spin resonance would help quantify the small anisotropies in this system.

Our observations in YbAlO3 suggest that it is worth exploring other members of the rare earth perovskite family RMO3. Compared to most of the $d$-electron based spin chains with critical fields of the order 10-100 T, YbAlO3 can be saturated with a field of order 1 T owing to the much weaker exchange interaction $J$ and large value of the effective $g$-factor. In addition, the crystal structure naturally adapts to existing standard perovskite thin film substrates. These properties can be exploited in future spinon-based research Hirobe2016 for material engineering under easily-accessible laboratory conditions.

Methods

Sample preparation and Thermal property measurements

YbAlO3 single crystals were grown by the Czochralski technique Buryy2010 . Magnetization measurements were carried out using three magnetometers for different temperature ranges: commercial vibration sample magnetometer Quantum Design Magnetic Property Measurement System (MPMS-VSM) for the high-temperature measurements $T=1.8$ – 400 K, MPMS-3 with 3He insert for the temperatures $T=0.5$ – 4 K, and high-resolution capacitive Faraday magnetometer Faraday for the low-temperature range $T=0.05$ – 0.9 K. The specific heat measurements were carried out using the relaxation time method, with a Quantum Design Physical Property Measurements System (PPMS) down to temperatures of 0.36 K and a custom compensated heat-pulse calorimeter with dilution refrigerator insert at MPI CPfS Wilhelm2004 .

Neutron scattering

Neutron scattering measurements were performed at two fixed incident energy of 3.32 meV ($\lambda_{i}=4.97$ Å) and 1.55 meV ($\lambda_{i}=7.26$ Å), with the time-of-flight Cold Neutron Chopper Spectrometer (CNCS) Ehlers2011 ; Ehlers2016 , at the Spallation Neutron Source (SNS) at Oak Ridge National Laboratory. A single crystal of YbAlO3 of about 0.59 g was aligned in the ($0KL$) scattering plane, with magnetic field along the vertical [100] direction. A dilution refrigerator insert that can access temperatures as low as 0.05 K was used. The software packages Dave Dave and MantidPlot Mantid were used for data reduction and analysis.

We applied two different methods to remove the background near $|\mathbf{Q}|=0$ from the direct beam. For zero field data we used an empty can file. For measurements under different fields, we used the high field data at 2 T for background subtraction (almost all the magnetic inelastic signal of the Yb Ising moments is suppressed above the saturation field of 1.1 T). As presented in Fig. 3, Fig. 4 and Fig. 5, the background correction produces some uncertainty within the instrumental resolution of $\pm 0.05$ meV, but it removes the instrumental background for all the other inelastic energy range.

Data Availability

The datasets generated during the current study are available from the corresponding author on reasonable request.

Acknowledgements

We would like to thank G. Loutts for the help with single crystal growth, C. Klausnitzer for the help with magnetization measurements, and I. Zaliznyak, F. Ronning, M. C. Aronson and E-J. Guo for useful discussions. This research used resources at the High Flux Isotope Reactor and Spallation Neutron Source, a DOE Office of Science User Facility operated by the Oak Ridge National Laboratory. Research supported in part by the Laboratory Directed Research and Development Program of Oak Ridge National Laboratory, managed by UT-Battelle, LLC, for the U.S. Department of Energy. This work is partly supported by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES), Materials Science and Engineering Division. This work used resources of the Compute and Data Environment for Science (CADES) at the Oak Ridge National Laboratory, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC05-00OR22725. This research used resources of the Oak Ridge Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract DE-AC05-00OR22725. L.V. acknowledges Ukrainian Ministry of Education and Sciences for partial support under project ’Feryt’. Z.W. and C.D.B. are supported by funding from the Lincoln Chair of Excellence in Physics and by the LANL Directed Research and Development program. S.E.N. acknowledges support from the International Max Planck Research School for Chemistry and Physics of Quantum Materials (IMPRS-CPQM). A.C. is partially supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division. W.Z. is supported by U.S. DOE through the LANL LDRD Program. M.B. would like to thank the DFG for financial support from project BR 4110/1-1.

Author contributions

These authors contributed equally: L. S. Wu, S. E. Nikitin and Z. Wang.

L.S.W and A.P. designed the experiments. S.E.N. and M.B. carried out the low temperature thermal property measurements. Z.W. and W.Z. did the DMRG calculation and A.M.T. and C.D.B. developed theoretical explanations. A.M.S. and D.A.T. performed the classical spin simulation for the paramagnon excitations. L.V. provided single crystals used in this study. L.S.W., A.P., G.E., M.F. and G.S. performed neutron scattering experiments. A.T.S. supported the neutron data analysis. M.D.L., A.D.C., G.E., C.D.B., A.M.T. and A.P. supervised and supported the study. All authors discussed the results and contributed to the writing of the manuscript.

Competing interests

The authors declare no competing interests.

Corresponding authors

Correspondence and requests for materials should be addressed to L. S. Wu and A. Podlesnyak .