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https://doi.org/10.1038/s41467-019-08715-y OPEN confinement and a sharp longitudinal mode in Yb2Pt2Pb in magnetic fields

W.J. Gannon1,8, I.A. Zaliznyak 2, L.S. Wu3,9, A.E. Feiguin4, A.M. Tsvelik2, F. Demmel 5, Y. Qiu6, J.R.D. Copley6, M.S. Kim7 & M.C. Aronson1,8

The fundamental excitations in an antiferromagnetic chain of spins-1/2 are , de-confined fractional quasiparticles that when combined in pairs, form a triplet excitation

1234567890():,; continuum. In an Ising-like chain the continuum is gapped and the is Néel ordered. Here, we report high resolution experiments, which reveal

how a magnetic field closes this gap and drives the spin chains in Yb2Pt2Pb to a critical,

disordered Luttinger-liquid state. In Yb2Pt2Pb the effective spins-1/2 describe the dynamics of large, Ising-like Yb magnetic moments, ensuring that the measured excitations are exclusively longitudinal, which we find to be well described by time-dependent density matrix renormalization group calculations. The inter-chain coupling leads to the confinement of spinons, a condensed analog of confinement in chromodynamics. Insensitive to transverse fluctuations, our measurements show how a gapless, dispersive longitudinal mode arises from confinement and evolves with magnetic order.

1 Department of and Astronomy, Texas A&M University, College Station, TX 77843, USA. 2 and Materials Science Division, Brookhaven National Laboratory, Upton, NY 11973, USA. 3 Quantum Condensed Matter Division, Oak Ridge National Laboratory, Oak Ridge, TN 37830, USA. 4 Department of Physics, Northeastern University, Boston, MA 02115, USA. 5 ISIS Facility, Rutherford Appleton Laboratory, Didcot OX11 0QZ, UK. 6 NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA. 7 Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY 11794, USA. 8Present address: Stewart Blusson Quantum Matter Institute, University of British Columbia, Vancouver, BC V6T 1Z4, Canada. 9Present address: Department of Physics, South University of Science and Technology of China, 518055 Shenzhen, China. Correspondence and requests for materials should be addressed to W.J.G. (email: [email protected])

NATURE COMMUNICATIONS | (2019) 10:1123 | https://doi.org/10.1038/s41467-019-08715-y | www.nature.com/naturecommunications 1 ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-08715-y

he one-dimensional (1D) XXZ Hamiltonian for quantum creation (Fig. 1a). The physics contained in Eq. (1) leads directly Tspin chains given by Eq. (1) is a cradle of exactly solvable to the separation of the spin from other electronic degrees of quantum theory models of interacting many-body sys- freedom, mapping directly onto that of the for tems1. The exact solution features purely quantum-mechanical −1 ≤ Δ ≤ 12,6–10. entities and concepts such as fractional excitations and the Coupling the chains described by Eq. (1) leads to new and quantum-critical Luttinger-liquid state2–9. The Hamiltonian emergent physics. Analogous to quark confinement in quantum considers the components Sα (α = x, y, z) of a spin angular chromodynamics4,5,8, the dimensional crossover from 1D chains = i J momentum operator, Si (S 1/2) at site i on a 1D chain, with a to 3D coupled chains leads to quasiparticle confinement, thereby nearest neighbor exchange coupling for x, y spin components, Δ a stabilizing long range magnetic order at temperature T > 0. A new fi μ uniaxial coupling anisotropy, and H magnetic eld (with g and B excitation of the longitudinal degree of freedom of the order the Lande g-factor and Bohr magneton respectively), parameter is predicted when the interchain coupling is weak6. XÀÁ X These phenomena have been the subject of a considerable H¼J x x þ y y þ Δ z z À μ Á : Si Siþ1 Si Siþ1 Si Siþ1 g B H Si ð1Þ amount amount of recent experimental work in XXZ spin chain i i materials11–15. Like a similar longitudinal mode previously observed near the critical point in a system of coupled spin-1/2 The low energy excitations of this model Eq. (1) are spin-1/2 dimers16,17, this excitation can be interpreted as a condensed quasiparticles called spinons. In the limit of strong Ising aniso- matter analog of the Higgs boson18. tropy, Δ  1, spinons can be visualized as domain walls in an Here, we report neutron scattering experiments on the one antiferromagnetically ordered ground state of the chain (Fig. 1a). dimensional rare-earth antiferromagnet Yb2Pt2Pb to investigate Angular momentum conservation mandates that spinons are these fundamental processes in detail, using an external magnetic always created in pairs, such that each spinon carries a fraction, field as a tuning parameter. ±1/2, of the angular momentum change, ΔSz = 0, ±1, required to initially introduce the domain walls in an infinite chain. Since moving these domain walls is an energy and angular momentum Results conserving process, the walls will propagate freely, carrying the Inelastic neutron scattering on Yb2Pt2Pb.Yb2Pt2Pb is a quanta of energy, E, and linear momentum, q, introduced by their metal with a planar where orthogonal pairs

a e Holes 0.5 a Sz = +1/2 Sz = –1/2 c I 0.0 (meV) E

q1, E1 q2, E2 2Δ –0.5 s

Sz = +1/2 Sz = –1/2

0.0 0.5 1.0 1.5 2.0

qL (rlu) q1, E1 q2, E2 I (q, E) (arb.) 0123 4

b 0 T c 0 T d 1.0 (meV)

E 0.5

0.0 0.0 0.5 1.0 1.5 2.0–1.0 –0.5 0.0 0.5 1.0 0.0 0.5 1.0 1.5 2.0

qL (rlu) qHH (rlu) qL (rlu)

Fig. 1 Spinons on decoupled chains. a An anisotropic AFM spin chain (top). If two spins are interchanged, two domain walls are created between the original antiferromagnetic domain (green) and a new domain (blue) (middle). Those domain walls form the basis for the propagating states carrying energy and momentum quanta (bottom). b The magnetic excitation spectrum and dispersion along the qL direction in reciprocal space measured at T = 0.1 K, summed over −1 ≤ qHH ≤ 1 rlu. The lower boundary of the spectrum (white circles) is shown along with the boundaries of the two spinon continuum obtained by fitting the lower boundary (red lines, Δ = 3.46) and comparing the total measured spectrum to theory (white lines, Δ = 2.6)26. The color scale for parts b–d is shown above part c. Error bars represent one standard deviation. c The magnetic excitation spectrum along the qHH direction in reciprocal space measured at T = 0.1 K, summed over 0 ≤ qL ≤ 2 r.l.u. d The spinon spectrum obtained from tDMRG calculations for the XXZ model Eq. (1) with Δ = 2.6 on the 96-site chain. The continuum boundary (black lines) is the same as that shown in b for Δ = 2.6. e The dispersions of -like (red) and hole- like (black) spinons, symmetric about E = 0 in zero magnetic field, are sketched with the real Δ = 2.6 parameters for Yb2Pt2Pb. The bandwidth parameter I and the spinon gap ΔS are indicated by arrows, with 2ΔS the energy separation between the particle and hole bands at qL = 0, 1, and 2 rlu

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μ = − μ z of Yb ions form a Shastry-Sutherland lattice (SSL) motif in potential, g BHS . The potential needed to close the the tetragonal a−b plane19–25. High resolution neutron energy gap for creating a hole on the spinon dispersion is |μ| = Δ = = 26 μ = Δ scattering experiments recently showed that the physics S 0.095 meV. Taking g 7.3 ,|| S corresponds to a fi μ = of 4f-orbital overlaps leads to unusual consequences for the critical eld of 0H 0.5 T. An abrupt increase in the bulk 26 magnetism in Yb2Pt2Pb . The low energy magnetic excitations magnetization is seen at this field when oriented parallel to the Δ are spinons, having a quantum continuum for momentum along magnetic moments of either sublattice at temperatures kBT < S 23,25 the chain direction qL that can be measured with inelastic (Fig. 2a) . On the other extreme of the magnetization, when neutron scattering27, with an excitation bandwidth that is con- |μ|>I, the entire hole band lies above the chemical potential, the siderably larger than the excitation gap (Fig. 1b). For momenta field having transformed all holes to particles. Particle-hole pairs in the interchain qHH direction (Fig. 1c), the continuum is can no longer be produced, quenching spinon excitations, and entirely flat, indicating that the spinons are completely incoherent producing a ferromagnetic (FM) state. The saturation field in fi μ = between the chains. Yb2Pt2Pb is 2.3 T, precisely the eld needed for 0.485 meV, In zero field, our measurements agree well with time- the value of I when Δ = 2.6, the number obtained by requiring μ = dependent density matrix renormalization group (tDMRG) that Eq. (1) provides best description of the entire 0H 0 calculations28 for the XXZ model Eq. (1) (Fig. 1d), although excitation spectrum26. The comparison is less favorable when Δ is experiment indicates that the spectral weight is spread through- taken to be 3.46, the value derived directly from fitting the lower out the spinon Brillouin zone (BZ) more evenly and to higher boundary of the continuum. As the static properties correspond energies than these calculations predict, suggesting non-negligible to an integration over all energies, it is not surprising that they are next-neighbor coupling26. Comparisons of our data to theory better captured by Δ = 2.6 and we adopt this value of Δ here. Δ – fi μ indicate only a modest anisotropy, ~2 3. It is clear that the At intermediate elds, 0.5 < 0H < 2.3 T, the hole band is XXZ Hamiltonian Eq. (1) is an appropriate description for partially emptied. The chemical potential crosses the hole fi Yb2Pt2Pb despite the large and orbitally dominated moment of dispersion at four points in the spinon BZ (Fig. 2b), de ning a the Yb ions. Due to their Kramers doublet ground state of almost Fermi wavevector kF that directly links particle and hole states. j ; i¼j = ; = i pure J mJ 7 2 ±7 2 , the Yb moments have a pseudospin There are now eleven unique extremal states made from a single S = 1/2 character26,29. Rather than quenching the quantum particle and hole, rather than the three that are possible in zero spin dynamics, the strong Ising magnetic anisotropy imposed field. The boundaries of the two spinon continuum change by the crystal electric field acting on the f–orbital wave function dramatically, as the possible extremal states are heavily influenced instead singles out the longitudinal excitation channel in two by the restrictions of the hole energies and momenta and the orthogonal sublattices of 1D chains with moments oriented additional phase space occupied by particles. along the (110) and (110) crystal directions. That is precisely what is measured in the neutron scattering – μ = The essential features of the quantum continuum can be spectra of Yb2Pt2Pb (Fig. 2d f). At 0H 1.0 T, there is strong understood by noting that each spinon carries spin 1/2, and so scattering concentrated at low energies within a range 1 ± 2kF the angular momentum selection rules dictate that neutron around the BZ center, with a weaker continuum at higher fi scattering in Yb2Pt2Pb measures two-spinon states where the total energies. As the magnetic eld is increased, the low-energy spin is zero, with one spinon in each spin state, ±1/2. In order spectral weight spreads throughout the zone as kF increases, with to describe the boundaries of the two-spinon continuum, it is higher energy pockets of spectral weight bounded by the extremal convenient to adopt the language of particles and holes occupying two spinon states comprising the continuum. While the measured the fermionic spinon dispersion along the chain direction, continua are in broad agreement with the results of tDMRG ÀÁ= – ¼ 2 2ðÞþπ Δ2 2ðÞπ 1 2 ≤ calculations performed on isolated chains (Fig. 2g i), there are Ep;h ± I sin qL Scos qL ,0 qL < 1 rlu, where Δ Δ marked differences at low energies where interchain interactions S is an energy gap brought on by the XXZ anisotropy >1, are important. The measured lower boundaries are gapped at and I defines the dispersion bandwidth and encodes the J 1,26,30,31 small energies and also slightly distorted relative to theoretical coupling (Fig. 1e) . In place of , these expectations, with the increased spectral weight indicative of a particles and holes each carry a half unit of spin angular . This is a direct demonstration of spinon confine- momentum. The boundaries of the two-spinon continuum fi ment induced by interchain coupling, which is not accounted for are de ned by the extremal energy and momentum conserving in the 1D calculations of Fig. 2g–i, and also shows how a magnetic combinations of one particle and one hole, and they are shown fi fi Δ = eld tunes the coexistence of con ned and free spinons in in Fig. 1b for both 2.6 and 3.46, the range of values Yb Pt Pb for μ H > 0.5 T. determined in previous work26 (see Supplementary Note 1). 2 2 0 fi The schematic picture of spinons and their propagation At zero magnetic eld, the chemical potential is in the middle presented in Fig. 1a needs to be modified in the presence of the of the gap separating the particle and the hole bands, which interchain interactions, where the creation of spinons on one describes the antiferromagnetic (AFM) state with zero total z = = chain leads to frustration of the AFM interactions between the spin, S 0. The size of the T 0 ordered moment implied chains. As two spinons separate, the energy of this frustration by the XXZ anisotropy is consistent with our measurements at = 26 grows with the number of FM aligned neighbors (Fig. 2c). This T 0.1 K, within the precision of our data . This implies that provides a linear confining potential, just as are confined the interchain coupling responsible for moving the Néel by the -mediated strong force in QCD, which also increases temperature away from T = 0, the value predicted by the XXZ = with quark separation. When spinons are created with energies model, to TN 2.07 K is less than both the intrachain exchange fi J¼ : Δ = above the highest energy level existing in the con ning potential 0 206 meV and the spinon gap S 0.095 meV, the introduced by the interchain coupling, these high energy dominant 1D energy scales. The flat dispersion of the excitations fi quasiparticles propagate freely within the two spinon continuum, between the chains in zero eld (Fig. 1c), despite the apparent demonstrating the same asymptotic freedom as experienced by ladder geometry of the crystal structure, suggests that the effect of unbound quarks32. A spinon bound state is observed in a neutron interchain interactions on low energy excitations is quenched Δ scattering experiment as excess spectral weight of resolution- when S is nonzero. limited energy width, which prominently appears near 1 ± 2k , A magnetic field alongP the z (110) direction introduces the F À μ z the two soft spots around the BZ center, below the quantum Zeeman term g BH i Si to Eq. (1), which lowers the chemical continuum10.

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a d g 6 1.0 T 3 1.0 1.0 6×10–3 I /T Yb) /T ( B q 2 /Yb)  B , E  ( 3 4 ) (arb.) ( B M /d 0.5 0.5 M (meV) (meV)

d 1 2 E E

0 0 0 0123 0.0 0.0  0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 0H (T) qL (rlu) qL (rlu) b e h Particles 1.5 T Holes 0.5 1.0 1.0 6×10–3 I ( q

0.0 , 4 E (meV) ) (arb.) E  0.5 0.5 (meV) (meV)

E 2 E –0.5 kF 0 0.0 0.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 q (rlu) L qL (rlu) qL (rlu) c f i 1.7 T a c 1.0 1.0 5×10–3 I 4 ( q , E

3 ) (arb.) 0.5 0.5

q , E S = –1/2 S = +1/2 q1, E1 (meV) 2 2 z z 2 (meV) E E 1 0 0.0 0.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 q , E S = –1/2 S = +1/2 q1, E1 2 2 z z qL (rlu) qL (rlu)

Fig. 2 Spinons in a magnetic field. a The magnetic field dependence of the static magnetization at kBT < ΔS [red, right axis, H||(110)] shows several discontinuous jumps (blue and green dashes), corresponding to transitions among different 3D ordered phases that are more clear in dM/dμ0H (black, left axis)23,25. These phases correspond to different ways that magnetic moments arrange into registry minimizing the energy of magnetic dipole interactions z between the Yb moments. b When a magnetic field is applied along the chain direction, the chemical potential μ = −gμBHS (yellow) is lowered, emptying part of the hole band when |μ|>|ΔS|. μ crosses the hole dispersion at four points in the Brillouin zone (black arrows), defining the Fermi wavevector kF. c Two AFM ordered, 1D spin chains (top). If two spins on one chain are interchanged, two domain walls are formed between the original domain (green) and a new one (blue). The new domain frustrates the interchain interaction, represented by the red interchain bonds. This frustration creates a linear potential confining low energy spinons to bound states (bottom). d–f The magnetic excitation spectrum and its dispersion along the qL direction in reciprocal space measured at T = 0.1 K and μ0H = 1.0 T (d), 1.5 T (e), and 1.7 T (f), summed over −1 ≤ qHH ≤ 1 rlu. The dispersions for the extremal combinations of particles and holes are shown (black lines) (See Supplementary Note 2). The spinon bound states are manifest from the enhanced low energy spectral weight around 1 ± 2kF (black arrows). g–i The spinon spectrum computed using tDMRG calculations for the XXZ model Eq. (1) on a 96-site chain with Δ = 2.6 at equivalent chain magnetizations as (d–f), shown on the same color scale. The dispersions for the extremal combinations of particles and holes are also shown as black lines

fi ≲ μ ≲ Perhaps the most interesting aspect of these free and con ned of the inter-chain dispersion. Most notably, for 0.5 0H 2.3 T, excitations is how their dispersions develop in the qHH direction, we observe a new excitation that emerges from the featureless fi perpendicular to the chains. Spinons are continually created in spinon continuum found along qHH in the gapped zero eld pairs on all chains, and in the absence of interchain coupling they Néel phase where the chains are effectively decoupled. This are free to propagate. Order and frustration in coupled chains mode resides within the low-energy window of the two spinon naturally lead to the bound states, which develop an interchain bound states, but has a pronounced dispersion in the qHH dispersion that reflects the underlying AFM order. Spinons are (interchain) direction that changes considerably with increasing generated in registry on adjacent chains, thus minimizing the field (Fig. 3b–d). Remarkably, at 1.0 T the dispersive interchain interchain frustration. mode appears nearly gapless, while its intensity is markedly Figure 3a shows a complex phase diagram consisting of larger than that of the continuum, Figs. 2d and 3b. The mode several distinct phases that differ markedly in the behavior becomes clearly gapped with increasing fields, while the

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a 2.5 d 1.0 1.7 T 0.5 H

0.0 0 2.0  1.0 /d

–0.5 p –3

C 6×10 –1.0 d (J/mol. Yb/K/T) –1.5 I

(K) 1.5 ( q T

4 , E (meV) ) (arb.) E 0.5 1.0 2

0.5 0 0.0 0123 –1.0 –0.5 0.0 0.5 1.0  0H (T) qHH (rlu)

b e 1.0 T 12

) –3 3 9 4×10 1.0 6×10–3 6 (arb. × 10

S 3 I ( q

4 , 0 2 E

(meV) 0.0 0.5 1.0 1.5 2.0 ) (arb.)

E q (rlu) 0.5 L (cnts./std. mon.)

2 S

q = 0.4 rlu 0 0 HH 0.0 0.0 0.5 1.0 1.5 2.0 –1.0 –0.5 0.0 0.5 1.0 E (meV) qHH (rlu) c f 0–4 T 1.5 T E = (0.12, 1) meV 0.75–4 T q = (0.5, 1.5) rlu 1.25–4 T L 2 1.0 1.6–4 T 6×10–3 4 T = 0) I ( q HH , 4 q ( E (meV) ) (arb.) 1 S / E 0.5 S 2

0 0 0.0 0.0 0.5 1.0 1.5 –1.0 –0.5 0.0 0.5 1.0 qHH (rlu) qHH (rlu)

Fig. 3 A coherent, longitudinal interchain mode in Yb2Pt2Pb. a The field-temperature phase diagram of Yb2Pt2Pb deduced from specific heat measurements (see Supplementary Note 3). Symbols representing phase lines of the low field AFM order (red), gapped phase when kBT < ΔS (blue), and second, weaker order (green) are obtained from the magnetic field dependence of the magnetization along the (110) direction at fixed temperatures. Refer to Fig. 2a for an example. Not all of the field-induced AFM phases observed in the magnetization are shown. b–d The magnetic excitation spectrum along the qHH direction of reciprocal space measured at T = 0.1 K and μ0H = 1.0 T (b), 1.5 T (c), and 1.7 T (d), summed over 0 ≤ qL ≤ 2 rlu. The fitted dispersion of the interchain mode is shown (white circles), as well as fits to the expression described in the main text (red lines). The mode dispersion and fits are only shown for qHH 33 > 0 for clarity (see Supplementary Note 4). The continuum boundary appears at 2ΔM (white dashed line). e Example cuts of the energy dependence at qHH = 0.4 rlu measured at B = 1.0 (red) and 1.5 T (green). Fits are described in the text. The mode dispersions shown in b–d are extracted from such fits. (Inset) The integrated intensity of the scattering over the energy of the interchain mode extracted from the fits is shown in parts b–d as a function of momentum qL along the chain. Magnetic fields of 1.0 T (red), 1.5 T (black), and 1.7 T (blue) are offset for clarity. qL = 1±2kF is shown at each field (gold stars). f The polarization of the excitations follows the projection onto the scattering vector q of the magnetic moments contributing to the scattering, indicating that all observed excitations are longitudinal (see Supplementary Note 5). Fields and integration ranges are given in the figure legend; the calculated polarization was averaged within the corresponding qL integration range. Error bars represent one standard deviation and where not visible are smaller than symbol size relative spectral weight of the continuum grows. We model the new mode is roughly resolution limited at all fields, and it is fi fi energy dependence of the scattering at a speci c qHH as a damped always distinguishable from the spinon continuum for elds μ ≤ fi harmonic oscillator (DHO) response centered at the mode 0H 1.7 T. The connection of the mode and the con ned spinon position and the product of a Lorentzian and step function states can be emphasized by integrating over the energies of the accounting for the continuum at higher energies, all convolved mode and plotting the intensity as a function of momentum with the instrument resolution (Fig. 3e). The energy width of the along the chains (Fig. 3(e-Inset)). At all fields, ≈85% of the

NATURE COMMUNICATIONS | (2019) 10:1123 | https://doi.org/10.1038/s41467-019-08715-y | www.nature.com/naturecommunications 5 ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-08715-y spectral weight is concentrated within the momentum range anisotropy of the 7/2, ±7/2 ground state doublet of the Yb = qL 1±2kF. moments nearly completely suppresses any transverse magnetic Importantly, this interchain mode is longitudinally polarized. fluctuations from our measurements. We confirm its longitudinal character by using the fact that The longitudinal interchain mode changes dramatically over neutron scattering cross-section is uniquely sensitive to magnetic a relatively narrow range of fields as the underlying antiferro- fluctuations that are perpendicular to the wave vector transfer27. magnetic order is weakened and ultimately destroyed. The The intensity measured at 4 T, in the FM state precisely low temperature H−T magnetic phase diagram of Yb2Pt2Pb follows the projection of the scattering wave vector on the (Fig. 3a) has several different AFM ordered phases22–25. In zero (110) direction, revealing fluctuations polarized along the in- field, there is a five by five periodicity to the order in the plane Ising moments, which are insensitive to magnetic fields tetragonal a−b plane, evidenced by neutron diffraction peaks 29 (Fig. 3f). When this field-independent contribution is subtracted that index as qHH = 0.2 rlu (Fig. 4a) . When the gap ΔS closes fi μ = = as a background, the resulting eld-dependent intensity [Figs. 2 at 0H 0.5 T, those peaks move from qL 1 rlu to incommen- and 3) does not depend on the wave vector orientation in the surate positions along qL (Fig. 4b), consistent with the long- scattering plane, indicating magnetic fluctuations polarized itudinal component of the spin-spin correlation function probed along the vertical direction, collinear with the magnetic field26 by our neutron scattering measurements being locked to twice fi 10 (see Supplementary Note 5). At all elds, our measurements the Fermi wave vector . The ordering wave vector follows 2kF in unambiguously probe the longitudinal response. The Ising turn, connecting to the softest parts of the excitation spectrum.

–0.2 0 0.2 –0.2 0 0.2

8 1 1.3 a 0.025 T b 0.75 T 1.3 6 ) 2 10 –3 1.2 1.2 4 3 (×10 222 I 4 1.1 1.1 0 11

1.0 1.0 1 (rlu) L

q 1.3 1.3 3

1.2 2 32 1.2 Int. inten. (cnts/std. mon.) 3

4 ) 0.1 1.1 1.1 2 –2 g 1 1.0 1.0 (×10 cd1.2 T 1.7 T I 0123 0  H (T) –0.2 0 0.2 –0.2 0 0.2 0

qHH (rlu) h (001) || (110) qHH 1.4 e 0.4 sat

M || μ

(rlu) 1.2 0.2 (110) 0H / ⊥ L M q 5 M/Msat 1.0 0.0 ⊥ 4 0.2 1 ⊥ intra ⊥ 2 3

(rlu) 3 0.1 ⊥

HH 2

q 4

0.0 f ⊥ 1 0123  0H (T)

Fig. 4 Magnetic order and phase diagram of Yb2Pt2Pb. a–d. Time of flight data with energy transfer E = 0 in the qHH, qL plane, measured at T = 0.1 K in fields of 0.025 T (a), 0.75 T (b), 1.2 T (c), and 1.7 T (d). The magnetic Bragg diffraction peaks in each part of the phase diagram are labeled 1–4. (e, f) The location in reciprocal space along qL (e) and qHH (f) of the magnetic Bragg scattering as a function of the field in the regions plotted in parts a–d. Data shown are an average, symmetrized about qL = 1 and qHH = 0. The open symbols correspond to the peaks labeled 1-4 in panels b–e, with black indicating peak 1, blue peak 2, red peak 3, and green peak 4. Magnetization divided by the saturation magnetization M/Msat as a function of the field along the (110) crystal direction is also shown in panel e (pink line, right axis), demonstrating the initial trend, M/Msat ≈ (qL − 1) and the concurrence of the jumps in magnetization with the abrupt changes in the position of elastic scattering. (g) The intensity of the diffraction peaks in parts b–f. The AFM order that emerges for μ0H > 1 T is considerably weaker than the low field order, while the low field order falls off very abruptly for μ0H > 1.2 T. Symbols are the same as in panels e, f.(h) The zero field magnetic structure and dipole interactions in Yb2Pt2Pb. Yb moments in five unit cells are shown along the diagonal, (110) J ? direction, with the interchain couplings, n , as indicated. The Yb SSL AFM layers are highlighted (blue and green arrows), with the periodicity given by FM pairs every five unit cells (black and red arrows)

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There are several small and abrupt shifts in the ordering wave that the transverse components of the effective spins order in a – fi fi vector along both qL and qHH (Fig. 4a f) that coincide with spiral con guration, while the z-component has a eld indepen- abrupt jumps in the derivative of the low temperature dent ordering wave vector. Since do not register fl magnetization [Figs. 2a and 4e, f), which manifest changes in transverse uctuations in Yb2Pt2Pb, the corresponding 3D magnetic ordering as the magnetic moments re-arrange to Goldstone mode is invisible. In order to describe the gapped minimize the energy of magnetic dipole interactions. For fields longitudinal mode observed in this phase, we add a phenomen- μ Δ2 0H > 1.0 T, there is an of a second incommensurate ological gap term, M, to Eq.P (5), resulting in a dispersion, AFM ordered phase that is accompanied by the swift collapse ð Þ2 = Δ2 ð À 1 5 J ? ð π ÞÞ Emode qHH M 1 J ? n¼1 n cos 2 nqHH , and we 3 tot of the original five by five order for μ0H > 1.2 T and even a third incommensurate order that persists up to the saturation normalizeP the couplings to the total interchain exchange, J ? ¼ 5 J ? fi field (Fig. 4c–g). As the new longitudinal interchain mode is tot n¼1 n .We t this expression to the measured modes J ?=J ? an excitation of the underlying order, it is not surprising that it by varying the relative couplings n tot, resulting in excellent changes so dramatically between 1.0 and 1.5 T. agreement at all fields (Fig. 3b–d). Increasing the field from 1.0 ≲ μ ≲ For 0.5 0H 1 T, the qL component of the magnetic Bragg to 1.5 T dramatically changes the relative interchain coupling peak follows the magnetization (Fig. 4e), which reflects the strengths. The nearest neighbor coupling is reduced by fi position of the spinon Fermi wavevector, 2kF (Fig. 2b) (See approximately a factor of 11 when the eld is increased from Supplementary Note 6). Theoretical description of the excitation 1.0 to 1.5 T, while the next nearest neighbor coupling is spectrum in this spin-density-wave (SDW) phase can be obtained reduced by a factor of 2 and changes sign from ferromagnetic by applying bosonization methods for quasi-1D spin-1/2 to antiferromagnetic. Smaller changes are found in the higher antiferromagnets33,34. The low energy sector of such an antiferro- order terms. These changes are consistent with the observation magnet is described by the theory of noninteracting bosonic field from the magnetic diffraction (Fig. 4) of a tendency towards ϕ governed by the Lagrangian, a weaker, frustrated longitudinal antiferromagnetic order as Z ÂÃthe applied magnetic field progressively polarizes the moments 1 À1 2 2 L ¼ dx v ðÞ∂τϕ þvðÞ∂ ϕ ; ð2Þ (See Supplementary Note 7). 4π x where v ~ J/a. The expression for the z-component of the spins is Discussion 1 fi Sz À M ¼ ∂ ϕ þ A sin½ϕ þ πð1 À 2MÞxŠ; M ¼ hiSz ; ð3Þ It is dif cult to visualize the nature of the interchain mode 2π x itself, as it is purely quantum mechanical in its origin, with no simple classical analog. In a conventional 3D ordered where A is an amplitude with dimension of inverse length. A magnet, these interchain excitations would be transverse spin quantitative description of the interchain dispersion requires waves–pseudo-Goldstone modes of an antiferrromagneic order knowledge of the relevant couplings for the underlying order. parameter, acquiring a small gap (mass) in the presence of In Yb2Pt2Pb, the interchain interaction is predominantly of spin anisotropy. The longitudinal polarization reveals that the dipole-dipole origin. The symmetry of the Yb lattice sites new excitations observed here in Yb2Pt2Pb, which are separated suppresses magnetic dipole interaction between the two ortho- from E = 0 with a field-dependent gap δ < 0.12 meV, are in fact gonal sublattices, which cancels on the mean field level. On far more exotic. They represent amplitude excitations of the the other hand, due to the Ising nature of Yb magnetic AFM order parameter, i.e., the staggered magnetization16,17, J ? moments the intra-sublattice interactions intra involve only z- analogous to the amplitude modes of the superconducting 35,36 18 components of effective spins-1/2R (Fig. 4h). Hence, the interchain order parameter found in NbSe2 and the Higgs itself. J ? zð ; Þ zð þ ; Þ coupling can be written as, m dxS n x S m n x , where n, There is a large literature on the subject in the context of the n + m label different chains. Using (3) and neglecting the theory of quantum magnets (e.g., refs. 33,34,37,38), but so far marginal terms with derivatives of ϕ we get the following there have been few experiments among weakly coupled chain contribution to (2), systems that probe this mode and its dispersion across different X Z regimes of interchain coupling, or its decay into transverse ′ ¼ 2J ? ½ϕð ; ÞÀϕð þ ; ފ: 39 L A m dx cos n x m n x ð4Þ in detail . This damping causes the longitudinal n;m mode to appear as a resonance in the longitudinally polarized continuum rather than the dispersing quasiparticle like fl 6,40 Since uctuations in 3D do not lead to divergencies, we can excitation observed here in Yb2Pt2Pb , which often obscures expand cosines in L′ around its minimum and get the Lagrangian the physics entirely and has even led some to question the 41 quadratic in ϕ. The zero field magnetic structure in Yb2Pt2Pb assumptions behind the theory . Recently, some evidence for suggests couplings up to fifth neighbors in the basal plane. We a sharp longitudinal mode coexisting with the transverse thus obtain a gapless longitudinal “phason” mode with the spin waves has been obtained via polarized neutron scattering interchain dispersion, measurements in a more strongly coupled 2D ladder system42, X5 while other materials with similar XXZ anisotropy to Yb2Pt2Pb ðÞ2¼ J ?½ŠÀ ðÞπ ; ð Þ tend to confine all spinons into many modes11–15. Here Emode qHH 2 n 1 cos 2 nqHH 5 n¼1 we overcome the limitations of such experiments thanks to the tuning parameters of Yb2Pt2Pb, which uniquely single out which accurately describes the 1.0 T data in Fig. 3b. the longitudinal channel and allow us to clearly identify the μ fi The situation is different at 1 T < 0H < 2.2 T (region marked dispersing amplitude mode and the decon ned 1D excitations by green circles on the phase diagram, Fig. 3a). In this region, at higher energies. the static susceptibility is nonzero and magnetization smoothly The physics of Yb2Pt2Pb that suppresses transverse spin waves increases, but the longitudinal interchain mode is gapped and also protects these longitudinal excitations and allows detailed ≈ magnetic Bragg peaks are locked to qL 0.29 rlu and do not observation of this mode dispersion and its dependence on an fi fi change their qL positions with eld (Fig. 4e). In principle, applied magnetic eld, which tunes the ordered state across dif- this presents a puzzle, which could be resolved by assuming ferent phases. Accompanying transverse excitations—the spin

NATURE COMMUNICATIONS | (2019) 10:1123 | https://doi.org/10.1038/s41467-019-08715-y | www.nature.com/naturecommunications 7 ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-08715-y waves—must exist, but are not observed (Fig. 3f). They are sup- same manner described in the Supplementary Materials for26. The measurement at 2 ≳ μ H = 4 T of the polarization factor (main text Fig. 3f) is simply the μ H = 4T pressed by a factor (g||/g⊥) 100 and can not be measured in this 0 0 measurement on its own, with no background subtracted. For the nominal zero scattering geometry, where even the application of a substantial fi fi fi eld measurements, a small bias eld of 0.025 T was used to suppress magnetic eld leaves behind the longitudinal continuum from in the aluminum sample holder. magnetic moments perpendicular to the field. While further experiments are needed in alternate scattering geometries to Specific heat and magnetization. Specific heat measurements used in the clearly observe the accompanying transverse excitations, the phase diagram shown in Fig. 3a (and Supplementary Figs. 2 and 3) were made present results quantify in unique detail an unusual dispersing using a Physical Property Measurement System (PPMS) made by Quantum fi longitudinal mode, a Higgs-like excitation of a effective spin-1/2 Designs on a single crystal of Yb2Pt2Pb (note: the identi cation of the equipment used in the various measurements is not intended to imply recommendation or order parameter across different ordered phases induced by endorsement by the National Institute of Standards and Technology, nor is it magnetic fields in Yb2Pt2Pb. intended to imply that this equipment is necessarily the best available for the purpose.) Measurements for T < 1.8 K utilized a PPMS dilution refrigerator insert. The (110) direction was oriented vertically, parallel to the magnetic field. Methods Magnetization measurements for T < 1.8 K in Figs. 2a and 4e (and Supplementary Neutron scattering. The Yb2Pt2Pb sample used in these measurements was the Fig. 6) were made using a PPMS Hall sensor magnetometer in a dilution refrig- same sample used in ref. 26. The sample consists of approximately 400 co-aligned erator insert at T = 0.150 K. Magnetization measurements for T > 1.8 K in Fig. 3a Yb Pt Pb single crystals (total mass ≈6 g) mounted to aluminum plates. For all 2 2  were made using a Quantum Designs Magnetic Property Measurement System measurements the sample was oriented with the (110) crystal direction vertical SQUID magnetometer. The magnetic field for both sets of measurements was along leaving the (110) and (001) directions in the horizontal scattering plane, making the (110) direction. the scattering plane (H, H, L) in reciprocal space. All momenta are given in reciprocal lattice units (rlu), with 1 rlu given by 2π/a = 2π/7.76 Å = 0.810 Å−1 along the H direction and 2π/c = 2π/7.02 Å = 0.895 Å−1 along L. The crystal- Obtaining the position of the interchain mode. The interchain mode dispersions — shown in Fig. 3b–d (and Supplementary Fig. 5) were extracted by integrating a lographic unit cell is twice the Yb-Yb near neighbor spacing along the c-axis the = fi relevant spacing for spinons. Therefore, the Brillouin zone for spinons is indexed window of qHH 0.1 rlu and tting the resulting cuts of the scattering function as a function of energy S(E), examples of which are shown in Fig. 3e of the main text. from 0 to 2 rlu along (0, 0, L), rather than the typical 0 to 1 rlu. Notationally, qHH is fi – parallel to the (H, H, 0) direction, with scattering primarily coming from (H, H,1) The tting function represents the mode as a damped harmonic oscillator the product of the Bose population factor with the difference of two Lorentzians, while qL is along the (0, 0, L) direction. Γ The inelastic neutron scattering measurements on Yb Pt Pb making up the centered at positive and negative energies Emode, with a width and amplitude A, 2 2 Eq. (6). bulk of the data in this paper were made on the OSIRIS spectrometer at the ! ISIS neutron source at Rutherford Appleton Laboratory in Didcot, Oxfordshire, Γ Γ 43 ðÞ¼ A À ð Þ UK . The sample was mounted in a dilution refrigerator inside of a 7 T IE À = 2 2 6  1 À e E kB T ðÞÀ þΓ2 ðÞþ þΓ2 superconducting magnet with the field in the vertical direction parallel to the (110) E Emode E Emode crystal direction. OSIRIS is an indirect geometry, time-of-flight neutron spectrometer. The incident neutron beam is a white beam of cold neutrons. The This function was added to the product of another Lorentzian and step function fi = λ = fi PG002 analyzer was used with a nal neutron energy Ef 1.84 meV ( 6.67 Å). each centered at energy E > Emode representing the continuum. The t itself was The energy resolution was ≈0.03 meV at E = 0 meV. Due to the magnet performed to these two functions convolved with a gaussian representing the construction, only scattering into a range of angles 45° < 2θ < 135° is permitted. We instrumental resolution. therefore discard the 14 detector channels at the lowest scattering angles and the 3 channels at the highest scattering angles, determined by measurements of a Time-dependent density matrix renormalization group calculations. The 3+ vanadium standard in the magnet. All data were corrected for the Yb form longitudinal component of the dynamical spin structure factor S(q, ω) was 44 factor . obtained by means of the time-dependent density matrix renormalization group The measurements on OSIRIS were made at T = 0.140 K. A field greater than 28,47–49  (tDMRG) method . The approach has been extensively described in the 2.3 T along the (110) crystal direction polarizes all of the magnetic moments literature and essentially consists of calculating the time-dependent correlation parallel to that direction, while leaving the orthogonal moments along the (110) function: direction in the horizontal plane unaffected due to the ground state doublet of the 26 μ = iHt^ ^z ÀiHt^ ^z ð Þ Yb ions . Measurements made at 0H 4 T can therefore be used as a background SxðÞ¼À x ; t hjψ e S ðxÞe S ðÞx jiψ : 7 μ fi 0 0 for measurements made at 0H < 2.3 T, isolating only the lower eld scattering contribution from magnetic moments oriented parallel to the field. All neutron ^zð ¼ = Þ The operator S x0 L 2 is applied at the center of the chain and the resulting scattering results from OSIRIS have a measurement made at T = 0.140 K and ^ state is evolved in real-time. At every step, the overlap with the state ^SzðxÞeÀiHtjiψ μ H = 4 T subtracted in this fashion. For the nominal zero field measurements, a 0 is measured and the correlations function in real time and space is recorded. The small bias field of 0.025 T was used to suppress superconductivity in the aluminum results are Fourier transformed to frequency and momentum using a properly sample holder. chosen Hann window that determines the resolution of the final result. In our case, In general, inelastic neutron scattering probes the dynamical spin correlation in order to compare to experiments we have used a window of half-width Δt = 7in function27. Because of the crystal field ground state doublet of the Yb ions in units of 1/J. Only two things differ from conventional calculations: since the Yb Pt Pb we are sensitive only to the longitudinal component of this function (see 2 2 parameters considered fall into the Ising phase of the model, which tends to break Supplementary Note 5). translational symmetry at finite magnetization, the simulations were carried out Because our detector coverage does not include an entire Brillouin zone, (0 < q L using periodic boundary conditions and the time-targeting method with a Krylov < 2 rlu, 0 < q < 1 rlu) we integrate all data in the scattering plane in the HH expansion of the evolution operator50. In addition, we plot the results for the reciprocal space direction orthogonal to the direction being considered. The q ÀÁ L ^zð ÞÀ = ^zð Þ dependencies show in Figs. 1b and 2d–f (And Supplementary Fig. 1) integrate operator S x m 2 (instead of S x ), where m is the magnetization value. fl the entire measured range of q . The same q integration was used to The offset allows us to resolve the density uctuations and eliminates large con- HH HH 51 demonstrate the amount of low energy spectral weight in the bound state tributions at low frequencies and momenta . Surprisingly, due to the low entan- −5 (main text Fig. 3e inset). Similarly, all q were integrated to examine the q glement in the Ising phase, truncation errors smaller than 10 are easily L HH = dependence in Figs. 1c and 3b–d (and Supplementary Figs. 4 and 5) and in the achievable using 300 states, even on a chain of length L 96 with periodic cuts used to extract the interchain mode dispersion in Fig. 3e. Although our data boundary conditions. – from OSIRIS do not cover an entire Brillouin zone, the symmetry of our The color scale for the calculations displayed in Figs. 1d and 2g i was measurements about both q = 1 rlu and q = 0 rlu confirms that our coverage determined by normalizing the integral of the E > 0 portion of the calculation to the L HH fi is sufficient. measured integral of the inelastic intensity at the the same eld. Inelastic scattering measurements of the polarization factor (main text, Fig. 3f) and diffraction measurements (main text, Fig. 4a–g) were performed on the Disk Data availability Chopper Spectrometer (DCS) at the Center for Neutron Research at the National The data that support the findings of this study are available from the corresponding Institute for Standards and Technology in Gaithersburg, MD, USA45. For these authors upon reasonable request. Original time-of-flight data is available for the measurements, the same sample was mounted in the same scattering geometry as experiments at the ISIS neutron source at https://doi.org/10.5286/ISIS.E.42580328. the OSIRIS experiments, inside of a dilution refrigerator inside of a 10 T vertical superconducting magnet. DCS is a direct geometry spectrometer and Ei = 3.27 meV (λ = 5.00 Å) was used. The energy resolution was ≈0.1 meV. The temperature Received: 16 April 2018 Accepted: 18 January 2019 of these measurements was T = 0.07 K. Measurements on DCS have a similarly measured background subtracted and the scattering was corrected for the Yb3+ form factor44 and neutron absorption using the DAVE software package46 in the

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