Spinon Confinement and a Sharp Longitudinal Mode in Yb$_2$Pt$_2$Pb in Magnetic Fields

TL;DR

This study uses neutron scattering to explore how magnetic fields induce a transition from a gapped, ordered state to a disordered Luttinger-liquid state in Yb2Pt2Pb, revealing spinon confinement and a novel longitudinal mode.

Contribution

It demonstrates the magnetic field-driven transition in Yb2Pt2Pb and provides experimental evidence of spinon confinement and a sharp longitudinal mode, supported by theoretical calculations.

Findings

Magnetic field closes the spin gap and induces a Luttinger-liquid state.

Inter-chain coupling causes spinon confinement similar to quark confinement.

A gapless, dispersive longitudinal mode emerges from confinement.

Abstract

The fundamental excitations in an antiferromagnetic chain of spins-1/2 are spinons, de-confined fractional quasiparticles that when combined in pairs, form a triplet excitation continuum. In an Ising-like spin chain the continuum is gapped and the ground state is N{\'e}el ordered. Here, we report high resolution neutron scattering experiments, which reveal how a magnetic field closes this gap and drives the spin chains in \YPP\ to a critical, disordered Luttinger-liquid state. In \YPP\ 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 matter analog of quark confinement in quantum chromodynamics. Insensitive to transverse fluctuations, our measurements show how a gapless, dispersive longitudinal mode arises from confinement and evolves with magnetic order.

Tables

Table 1. Table S1: Relative strength of the interchain couplings giving the longitudinal interchain mode dispersion.

${\mu }_{0}H\left(T\right)$	${\Delta }_M\left(\mathrm{meV}\right)$	${𝒥}_1^{\mathrm{rel}}$	${𝒥}_2^{\mathrm{rel}}$	${𝒥}_3^{\mathrm{rel}}$	${𝒥}_4^{\mathrm{rel}}$	${𝒥}_5^{\mathrm{rel}}$
1.0	$0.174\pm 0.001$	$1.96\pm 0.09$	$-1.96\pm 0.09$	$-0.882\pm 0.07$	$-0.248\pm 0.06$	$-0.119\pm 0.06$
1.5	$0.167\pm 0.0006$	$0.171\pm 0.03$	$1.01\pm 0.03$	$-0.0433\pm 0.03$	$-0.263\pm 0.03$	$-0.0258\pm 0.03$
1.7	$0.182\pm 0.0007$	$-0.100\pm 0.03$	$0.826\pm 0.03$	$0.190\pm 0.03$	$-0.119\pm 0.03$	$-0.0734\pm 0.03$

Equations

$$\mathcal{H}=\mathcal{J}\sum_{i}\left(S^{x}_{i}S^{x}_{i+1}+S^{y}_{i}S^{y}_{i+1}\right)+\Delta S^{z}_{i}S^{z}_{i+1}-g\mu_{B}\sum_{i}\mbox{$\mathbf{H}$}\cdot\mbox{$\mathbf{S}$}_{i}.$$

$$L=\frac{1}{4\pi}\int dx\Big{[}v^{-1}(\partial_{\tau}\phi)^{2}+v(\partial_{x}\phi)^{2}\Big{]},$$

$$S^z-M=\frac{1}{2\pi }{\partial }_x\varphi +A\sin [\varphi +\pi (1-2M)x],M=⟨S^z⟩,$$

$$L^{\prime }=\sum _{n,m}{A}^2{\mathcal{J}}_m^{\perp }\int dx\cos [\varphi (n,x)-\varphi (m+n,x)].$$

$$E_{\mathrm{mode}}{\left({\mathbf{q}}_{HH}\right)}^2=2\sum _{n=1}^5{\mathcal{J}}_n^{\perp }[1-\cos (2\pi n{\mathbf{q}}_{HH})],$$

$$I\left(E\right)=\frac{A}{1-e^{-E/k_{\mathrm{B}}T}}\left(\frac{\Gamma }{{\left(E-E_{\mathrm{mode}}\right)}^2+{\Gamma }^2}-\frac{\Gamma }{{\left(E+E_{\mathrm{mode}}\right)}^2+{\Gamma }^2}\right)$$

$$S(x-x_0,t)=⟨\psi |e^{i\hat{H}t}{\hat{S}}^z(x)e^{-i\hat{H}t}{\hat{S}}^z(x_0)|\psi ⟩.$$

$$E_{\mathrm{p},\mathrm{h}}\left(\mathbf{q}\right)=\pm {\left(I^2{\sin }^2\left(\pi {\mathbf{q}}_L\right)+{\Delta }_{\mathrm{S}}^2{\cos }^2\left(\pi {\mathbf{q}}_L\right)\right)}^{1/2}$$

Full text

Introduction

The one-dimensional (1D) XXZ Hamiltonian for quantum spin chains given by Eqn. 1 is a cradle of exactly solvable quantum theory models of interacting many-body systems Bethe_Zphys_1931 . The exact solution features purely quantum-mechanical entities and concepts such as fractional excitations and the quantum-critical Luttinger-liquid state Haldane_PRL1980 ; Fadeev_PhysLett1981 ; Polyakov_NucPhys1977 ; Shelton_PRB_1996 ; Lake_NatMat_2005 ; Zaliznyak_NatMat2005 ; Lake_NatPhys_2009 ; Tsvelik_book_2003 . The Hamiltonian considers the components $S_{i}^{\alpha}$ ($\alpha={x,y,z}$) of a spin angular momentum operator, \mbox{\mathbf{S}}_{i} ($S=1/2$) at site $i$ on a 1D chain, with $\mathcal{J}$ a nearest neighbor exchange coupling for $x,y$ spin components, $\Delta$ a uniaxial coupling anisotropy, and $\mathbf{H}$ magnetic field (with $g$ and $\mu_{B}$ the Lande g-factor and Bohr magneton respectively),

[TABLE]

The low energy excitations of this model (1) are spin-1/2 quasiparticles called spinons. In the limit of strong Ising anisotropy, $\Delta\gg 1$, spinons can be visualized as domain walls in an antiferromagnetically ordered ground state of the chain [Fig. 1(A)]. Angular momentum conservation mandates that spinons are always created in pairs, such that each spinon carries a fraction, $\pm 1/2$, of the angular momentum change, $\Delta S^{z}=0,\pm 1$, required to initially introduce the domain walls in an infinite chain. Since moving these domain walls is an energy and angular momentum conserving process, the walls will propagate freely, carrying the quanta of energy, $E$, and linear momentum, $q$, introduced by their creation [Fig 1(A)]. The physics contained in Eqn. 1 leads directly to the separation of the spin from other electronic degrees of freedom, mapping directly onto that of the Luttinger liquid for $-1\leq\Delta\leq 1$ Haldane_PRL1980 ; Lake_NatMat_2005 ; Zaliznyak_NatMat2005 ; Lake_NatPhys_2009 ; Tsvelik_book_2003 ; Giamarchi_book_2004 .

Coupling the chains described by Eqn. 1 leads to new and emergent physics. Analogous to quark confinement in quantum chromodynamicsPolyakov_NucPhys1977 ; Shelton_PRB_1996 ; Lake_NatPhys_2009 , the dimensional crossover from 1D chains to 3D coupled chains leads to quasiparticle confinement, thereby stabilizing long range magnetic order at temperature $T>0$. A new excitation of the longitudinal degree of freedom of the order parameter is predicted when the interchain coupling is weak Lake_NatMat_2005 . These phenomena have been the subject of a considerable amount amount of recent experimental work in XXZ spin chain materials Grenier_PRL_2015 ; Matsuda_PRB_2017 ; Faure_NatPhys_2018 ; Wang_PRB_2015 ; Bera_PRB_2017 . Like a similar longitudinal mode previously observed near the critical point in a system of coupled spin-1/2 dimers  Ruegg_PRL_2008 ; Merchant_NatPhys_2014 , this excitation can be interpreted as a condensed matter analog of the Higgs boson Pekker_ARCMP_2015 . The neutron scattering experiments reported here on the one dimensional rare-earth antiferromagnet Yb2Pt2Pb investigate these fundamental processes in detail, using an external magnetic field as a tuning parameter.

Results

Inelastic Neutron Scattering on Yb2Pt2Pb

Yb2Pt2Pb is a metal with a planar crystal structure where orthogonal pairs of Yb ions form a Shastry-Sutherland lattice (SSL) motif in the tetragonal $\mathbf{a}-\mathbf{b}$ plane Shastry_PB_1981 ; Pottgen_JSSC_1999 ; Kim_PRB_2008 ; Ochiai_JPSJ_2011 ; Shimura_JPSJ_2012 ; Iwakawa_JPSJ_2012 ; Kim_PRL_2013 . High resolution neutron scattering experiments recently showed that the physics of $4f$-orbital overlaps leads to unusual consequences for the magnetism in Yb2Pt2Pb  Wu_Science_2016 . The low energy magnetic excitations are spinons, having a quantum continuum for momentum along the chain direction $\mathbf{q}_{L}$ that can be measured with inelastic neutron scattering, Squires_book_1978 with an excitation bandwidth that is considerably larger than the excitation gap [Fig. 1(B)]. For momenta in the interchain $\mathbf{q}_{HH}$ direction [Fig. 1(C)], the continuum is entirely flat, indicating that the spinons are completely incoherent between the chains.

In zero field, our measurements agree well with time-dependent density matrix renormalization group (tDMRG) calculations White2004 for the XXZ model (1) [Fig. 1(D)], although experiment indicates that the spectral weight is spread throughout the spinon Brillouin zone (BZ) more evenly and to higher energies than these calculations predict, suggesting non-negligible next-neighbor coupling Wu_Science_2016 . Comparisons of our data to theory indicate only a modest anisotropy, $\Delta\sim 2-3$. It is clear that the $XXZ$ Hamiltonian (1) is an appropriate description for Yb2Pt2Pb despite the large and orbitally dominated moment of the Yb ions. Due to their Kramers doublet ground state of almost pure $\ket{J,m_{J}}=\ket{7/2,\pm 7/2}$, the Yb moments have a pseudospin $\mathbf{S}=1/2$ character Wu_Science_2016 ; Miiller_PRB_2016 . Rather than quenching the quantum spin dynamics, the strong Ising magnetic anisotropy imposed by the crystal electric field acting on the $f-$orbital wave function instead singles out the longitudinal excitation channel in two orthogonal sublattices of 1D chains with moments oriented along the (110) and (1̄10) crystal directions.

The essential features of the quantum continuum can be understood by noting that each spinon carries spin $1/2$, and so the angular momentum selection rules dictate that neutron scattering in Yb2Pt2Pb measures two-spinon states where the total spin is zero, with one spinon in each spin state, $\pm 1/2$. In order to describe the boundaries of the two-spinon continuum, it is convenient to adopt the language of particles and holes occupying the fermionic spinon dispersion along the chain direction, $E_{\mathrm{p,h}}=\pm\left(I^{2}\sin^{2}\left(\pi\mathbf{q}_{L}\right)+\Delta_{\mathrm{S}}^{2}\cos^{2}\left(\pi\mathbf{q}_{L}\right)\right)^{1/2}$, $0\leq\mathbf{q}_{L}<1~{}\rm{rlu}$, where $\Delta_{\mathrm{S}}$ is an energy gap brought on by the XXZ anisotropy $\Delta>1$, and $I$ defines the dispersion bandwidth and encodes the coupling $\mathcal{J}$ [Fig. 1(E)] Bethe_Zphys_1931 ; Wu_Science_2016 ; Bougourzi_PRB1998 ; Caux_JStatMech_2006 . In place of electric charge, these particles and holes each carry a half unit of spin angular momentum. The boundaries of the two-spinon continuum are defined by the extremal energy and momentum conserving combinations of one particle and one hole, and they are shown in Fig. 1(B) for both $\Delta=2.6$ and 3.46, the range of values determined in previous work Wu_Science_2016 (see Supplementary Note 1). At zero magnetic field, the chemical potential is in the middle of the gap separating the particle and the hole bands, which describes the antiferromagnetic (AFM) state with zero total spin, $S^{z}=0$. The size of the $T=0$ ordered moment implied by the $XXZ$ anisotropy is consistent with our measurements at $T=0.1$ K, within the precision of our data Wu_Science_2016 . This implies that the interchain coupling responsible for moving the Néel temperature away from $T=0$, the value predicted by the $XXZ$ model, to $T_{\mathrm{N}}=2.07$ K is less than both the intrachain exchange $\mathcal{J}=0.206~{}\mathrm{meV}$ and the spinon gap $\Delta_{\mathrm{S}}=0.095~{}{\rm meV}$, the dominant 1D energy scales. The flat dispersion of the excitations between the chains in zero field [Fig. 1(C)], despite the apparent ladder geometry of the crystal structure, suggests that the effect of interchain interactions on low energy excitations is quenched when $\Delta_{\mathrm{S}}$ is nonzero.

A magnetic field along the $\mathbf{z}$ (110) direction introduces the Zeeman term $-g\mu_{B}H\sum_{i}S^{z}_{i}$ to Eqn. (1), which lowers the chemical potential, $\mu=-g\mu_{B}HS^{z}$. The potential needed to close the energy gap for creating a hole on the spinon dispersion is $\left|\mu\right|=\Delta_{\mathrm{S}}=0.095~{}\mathrm{meV}$. Taking $g=7.3$ Wu_Science_2016 , $\left|\mu\right|=\Delta_{\mathrm{S}}$ corresponds to a critical field of $\mu_{0}H=0.5~{}\rm T$. An abrupt increase in the bulk magnetization is seen at this field when oriented parallel to the magnetic moments of either sublattice at temperatures $k_{B}T<\Delta_{\mathrm{S}}$ [Fig. 2(A)] Shimura_JPSJ_2012 ; Kim_PRL_2013 . On the other extreme of the magnetization, when $\left|\mu\right|>I$, the entire hole band lies above the chemical potential, the field having transformed all holes to particles. Particle-hole pairs can no longer be produced, quenching spinon excitations, and producing a ferromagnetic (FM) state. The saturation field in Yb2Pt2Pb is 2.3 T, precisely the field needed for $\mu=0.485~{}\mathrm{meV}$, the value of $I$ when $\Delta=2.6$, the number obtained by requiring that Eqn. (1) provides best description of the entire $\mu_{0}H=0$ excitation spectrum Wu_Science_2016 . The comparison is less favorable when $\Delta$ is taken to be 3.46, the value derived directly from fitting the lower boundary of the continuum. As the static properties correspond to an integration over all energies, it is not surprising that they are better captured by $\Delta=2.6$ and we adopt this value of $\Delta$ here.

At intermediate fields, $0.5<\mu_{0}H<2.3$ T, the hole band is partially emptied. The chemical potential crosses the hole dispersion at four points in the spinon BZ [Fig. 2(B)], defining a Fermi wavevector $\mathbf{k}_{\mathrm{F}}$ that directly links particle and hole states. There are now eleven unique extremal states made from a single particle and hole, rather than the three that are possible in zero field. The boundaries of the two spinon continuum change dramatically, as the possible extremal states are heavily influenced by the restrictions of the hole energies and momenta and the additional phase space occupied by particles.

That is precisely what is measured in the neutron scattering spectra of Yb2Pt2Pb [Fig. 2(D-F)]. At $\mu_{0}H=1.0$ T, there is strong scattering concentrated at low energies within a range 1\pm 2$$\mathbf{k}_{\mathrm{F}} around the BZ center, with a weaker continuum at higher energies. As the magnetic field is increased, the low-energy spectral weight spreads throughout the zone as $\mathbf{k}_{\mathrm{F}}$ increases, with higher energy pockets of spectral weight bounded by the extremal two spinon states comprising the continuum. While the measured continua are in broad agreement with the results of tDMRG calculations performed on isolated chains [Fig. 2(G-I)], there are marked differences at low energies where interchain interactions are important. The measured lower boundaries are gapped at small energies and also slightly distorted relative to theoretical expectations, with the increased spectral weight indicative of a bound state. This is a direct demonstration of spinon confinement induced by interchain coupling, which is not accounted for in the 1D calculations of Fig. 2(G-I), and also shows how a magnetic field tunes the coexistence of confined and free spinons in Yb2Pt2Pb for $\mu_{0}H>0.5~{}\rm T$.

The schematic picture of spinons and their propagation presented in Fig. 1(A) needs to be modified in the presence of the interchain interactions, where the creation of spinons on one chain leads to frustration of the AFM interactions between the chains. As two spinons separate, the energy of this frustration grows with the number of FM aligned neighbors [Fig. 2(C)]. This provides a linear confining potential, just as quarks are confined by the gluon-mediated strong force in QCD, which also increases with quark separation. When spinons are created with energies above the highest energy level existing in the confining potential introduced by the interchain coupling, these high energy quasiparticles propagate freely within the two spinon continuum, demonstrating the same asymptotic freedom as experienced by unbound quarks Greiner_book_1994 . A spinon bound state is observed in a neutron scattering experiment as excess spectral weight of resolution-limited energy width, which prominently appears near 1\pm 2$$\mathbf{k}_{\mathrm{F}}, the two soft spots around the BZ center, below the quantum continuum Giamarchi_book_2004 .

Perhaps the most interesting aspect of these free and confined excitations is how their dispersions develop in the $\mathbf{q}_{HH}$ direction, perpendicular to the chains. Spinons are continually created in pairs on all chains, and in the absence of interchain coupling they are free to propagate. Order and frustration in coupled chains naturally lead to the bound states, which develop an interchain dispersion that reflects the underlying AFM order. Spinons are generated in registry on adjacent chains, thus minimizing the interchain frustration.

Fig. 3(A) shows a complex phase diagram consisting of several distinct phases that differ markedly in the behavior of the inter-chain dispersion. Most notably, for $0.5\lesssim\mu_{0}H\lesssim 2.3$ T, we observe a new excitation that emerges from the featureless spinon continuum found along $\mathbf{q}_{HH}$ in the gapped zero field Néel phase where the chains are effectively decoupled. This mode resides within the low-energy window of the two spinon bound states, but has a pronounced dispersion in the $\mathbf{q}_{HH}$ (interchain) direction that changes considerably with increasing field [Fig. 3(B-D)]. Remarkably, at 1.0 T the dispersive interchain mode appears nearly gapless, while its intensity is markedly larger than that of the continuum, Figs. 2(D) and 3(B). The mode becomes clearly gapped with increasing fields, while the relative spectral weight of the continuum grows. We model the energy dependence of the scattering at a specific $\mathbf{q}_{HH}$ as a damped harmonic oscillator (DHO) response centered at the mode position and the product of a Lorentzian and step function accounting for the continuum at higher energies, all convolved with the instrument resolution [Fig. 3(E)]. The energy width of the new mode is roughly resolution limited at all fields, and it is always distinguishable from the spinon continuum for fields $\mu_{0}H\leq 1.7~{}\rm T$. The connection of the mode and the confined spinon states can be emphasized by integrating over the energies of the mode and plotting the intensity as a function of momentum along the chains [Fig. 3(E-Inset)]. At all fields, $\approx$85% of the spectral weight is concentrated within the momentum range \mathbf{q}_{L}$$=1\pm 2$$\mathbf{k}_{\mathrm{F}}.

Importantly, this interchain mode is longitudinally polarized. We confirm its longitudinal character by using the fact that neutron scattering cross-section is uniquely sensitive to magnetic fluctuations that are perpendicular to the wave vector transfer Squires_book_1978 . The intensity measured at 4T, in the FM state precisely follows the projection of the scattering wave vector on the (110) direction, revealing fluctuations polarized along the in-plane Ising moments, which are insensitive to magnetic fields [Fig. 3(F)]. When this field-independent contribution is subtracted as a background, the resulting field-dependent intensity [Figs. 2 and 3] does not depend on the wave vector orientation in the scattering plane, indicating magnetic fluctuations polarized along the vertical direction, collinear with the magnetic field Wu_Science_2016 (see Supplementary Note 5). At all fields, our measurements unambiguously probe the longitudinal response. The Ising anisotropy of the $\ket{7/2,\pm 7/2}$ ground state doublet of the Yb moments nearly completely suppresses any transverse magnetic fluctuations from our measurements.

The longitudinal interchain mode changes dramatically over a relatively narrow range of fields as the underlying antiferromagnetic order is weakened and ultimately destroyed. The low temperature $\mathbf{H}-T$ magnetic phase diagram of Yb2Pt2Pb [Fig. 3(A)] has several different AFM ordered phases Ochiai_JPSJ_2011 ; Shimura_JPSJ_2012 ; Iwakawa_JPSJ_2012 ; Kim_PRL_2013 . In zero field, there is a five by five periodicity to the order in the tetragonal $\mathbf{a}-\mathbf{b}$ plane, evidenced by neutron diffraction peaks that index as \mathbf{q}_{HH}$$=0.2~{}\mathrm{rlu} [Fig. 4(A)] Miiller_PRB_2016 . When the gap $\Delta_{\mathrm{S}}$ closes at $\mu_{0}H=0.5~{}\rm T$, those peaks move from \mathbf{q}_{L}$$=1~{}\mathrm{rlu} to incommensurate positions along $\mathbf{q}_{L}$ [Fig. 4(B)], consistent with the longitudinal component of the spin-spin correlation function probed by our neutron scattering measurements being locked to twice the Fermi wave vector Giamarchi_book_2004 . The ordering wave vector follows 2$\mathbf{k}_{\mathrm{F}}$ in turn, connecting to the softest parts of the excitation spectrum. There are several small and abrupt shifts in the ordering wave vector along both $\mathbf{q}_{L}$ and $\mathbf{q}_{HH}$ [Fig. 4(A-F)] that coincide with abrupt jumps in the derivative of the low temperature magnetization [Figs 2(A) and 4(E, F)], which manifest changes in 3D magnetic ordering as the magnetic moments re-arrange to minimize the energy of magnetic dipole interactions. For fields $\mu_{0}H>1.0~{}\rm T$, there is an emergence of a second incommensurate AFM ordered phase that is accompanied by the swift collapse of the original five by five order for $\mu_{0}H>1.2~{}\rm T$ and even a third incommensurate order that persists up to the saturation field [Fig. 4(C-G)]. As the new longitudinal interchain mode is an excitation of the underlying order, it is not surprising that it changes so dramatically between 1.0 and 1.5 T.

For $0.5\lesssim\mu_{0}H\lesssim 1$ T, the $\mathbf{q}_{L}$ component of the magnetic Bragg peak follows the magnetization [Fig. 4E], which reflects the position of the spinon Fermi wavevector, 2$\mathbf{k}_{\mathrm{F}}$ [Fig. 2(B)] (See Supplementary Note 6). Theoretical description of the excitation spectrum in this spin-density-wave (SDW) phase can be obtained by applying bosonization methods for quasi-1D spin-1/2 antiferromagnets Schulz_PRL_1996 ; Essler_PRB_1997 . The low energy sector of such an antiferromagnet is described by the theory of noninteracting bosonic field $\phi$ governed by the Lagrangian,

[TABLE]

where $v\sim J/a$. The expression for the $z$-component of the spins is

[TABLE]

where $A$ is an amplitude with dimension of inverse length. A quantitative description of the interchain dispersion requires knowledge of the relevant couplings for the underlying order. In Yb2Pt2Pb, the interchain interaction is predominantly of dipole-dipole origin. The symmetry of the Yb lattice sites suppresses magnetic dipole interaction between the two orthogonal sublattices, which cancels on the mean field level. On the other hand, due to the Ising nature of Yb magnetic moments the intra-sublattice interactions $\mathcal{J}^{\perp}_{\mathrm{intra}}$ involve only $z$-components of effective spins-1/2 [Fig. 4(H)]. Hence, the interchain coupling can be written as, ${\cal J}_{m}^{\perp}\int dxS^{z}(n,x)S^{z}(m+n,x)$, where $n,n+m$ label different chains. Using (3) and neglecting the marginal terms with derivatives of $\phi$ we get the following contribution to (2),

[TABLE]

Since fluctuations in 3D do not lead to divergencies, we can expand cosines in $L^{\prime}$ around its minimum and get the Lagrangian quadratic in $\phi$. The zero field magnetic structure in Yb2Pt2Pb suggests couplings up to fifth neighbors in the basal plane. We thus obtain a gapless longitudinal “phason” mode with the interchain dispersion,

[TABLE]

which accurately describes the 1.0 T data in Fig. 3B.

The situation is different at 1 T $<\mu_{0}H<$ 2.2 T (region marked by green circles on the phase diagram, Fig. 3A). In this region, the static susceptibility is nonzero and magnetization smoothly increases, but the longitudinal interchain mode is gapped and magnetic Bragg peaks are locked to $q_{L}\approx 0.29$ rlu and do not change their $q_{L}$ positions with field [Fig. 4E]. In principle, this presents a puzzle, which could be resolved by assuming that the transverse components of the effective spins order in a spiral configuration, while the $z$-component has a field independent ordering wave vector. Since neutrons do not register transverse fluctuations in Yb2Pt2Pb, the corresponding Goldstone mode is invisible. In order to describe the gapped longitudinal mode observed in this phase, we add a phenomenological gap term, $\Delta_{\mathrm{M}}^{2}$, to Eq. 5, resulting in a dispersion, $E_{\mathrm{mode}}\left(\mathbf{q}_{HH}\right)^{2}=\Delta_{\mathrm{M}}^{2}\left(1-\frac{1}{3\mathcal{J}^{\perp}_{\mathrm{tot}}}\sum_{n=1}^{5}\mathcal{J}^{\perp}_{n}\cos\left(2\pi n\mathbf{q}_{HH}\right)\right)$, where we normalize the couplings to the total interchain exchange, $\mathcal{J}^{\perp}_{\mathrm{tot}}=\sum_{n=1}^{5}\mathcal{J}^{\perp}_{n}$. We fit this expression to the measured modes by varying the relative couplings $\mathcal{J}^{\perp}_{n}/\mathcal{J}^{\perp}_{\mathrm{tot}}$, resulting in excellent agreement at all fields [Fig. 3(B-D)]. Increasing the field from 1.0 to 1.5 T dramatically changes the relative interchain coupling strengths. The nearest neighbor coupling is reduced by approximately a factor of 11 when the field is increased from 1.0 to 1.5 T, while the next nearest neighbor coupling is reduced by a factor of 2 and changes sign from ferromagnetic to antiferromagnetic. Smaller changes are found in the higher order terms. These changes are consistent with the observation from the magnetic diffraction [Fig. 4] of a tendency towards a weaker, frustrated longitudinal antiferromagnetic order as the applied magnetic field progressively polarizes the moments (See Supplementary Note 7).

It is difficult to visualize the nature of the interchain mode itself, as it is purely quantum mechanical in its origin, with no simple classical analog. In a conventional 3D ordered magnet, these interchain excitations would be transverse spin waves – pseudo-Goldstone modes of an antiferrromagneic order parameter, acquiring a small gap (mass) in the presence of spin anisotropy. The longitudinal polarization reveals that the new excitations observed here in Yb2Pt2Pb, which are separated from $E=0$ with a field-dependent gap $\delta<0.12~{}\mathrm{meV}$, are in fact far more exotic. They represent amplitude excitations of the AFM order parameter, ie the staggered magnetization Ruegg_PRL_2008 ; Merchant_NatPhys_2014 , analogous to the amplitude modes of the superconducting order parameter found in NbSe2 Sooryakumar_PRL_1980 ; Sooryakumar_PRB_1981 and the Higgs boson Pekker_ARCMP_2015 itself. There is a large literature on the subject in the context of the theory of quantum magnets (eg Schulz_PRL_1996 ; Essler_PRB_1997 ; Affleck_PRL_1989 ; Normand_PRB_1997 ), but so far there have been few experiments among weakly coupled chain systems that probe this mode and its dispersion across different regimes of interchain coupling, or its decay into transverse magnons in detail Affleck_PRB_1992 . This damping causes the longitudinal mode to appear as a resonance in the longitudinally polarized continuum rather than the dispersing quasiparticle like excitation observed here in Yb2Pt2Pb Lake_NatMat_2005 ; Lake_PRL_2000 , which often obscures the physics entirely and has even led some to question the assumptions behind the theory Zheludev_PRL_2002 . Recently, some evidence for a sharp longitudinal mode coexisting with the transverse spin waves has been obtained via polarized neutron scattering measurements in a more strongly coupled 2D ladder system Hong_NatPhys_2017 , while other materials with similar XXZ anisotropy to Yb2Pt2Pb tend to confine all spinons into many modes Grenier_PRL_2015 ; Matsuda_PRB_2017 ; Faure_NatPhys_2018 ; Wang_PRB_2015 ; Bera_PRB_2017 . Here we overcome the limitations of such experiments thanks to the tuning parameters of Yb2Pt2Pb, which uniquely single out the longitudinal channel and allow us to clearly identify the dispersing amplitude mode and the deconfined 1D excitations at higher energies.

The physics of Yb2Pt2Pb that suppresses transverse spin waves also protects these longitudinal excitations and allows detailed observation of this mode dispersion and its dependence on an applied magnetic field, which tunes the ordered state across different phases. Accompanying transverse excitations – the spin waves – must exist, but are not observed [Fig. 3(F)]. They are suppressed by a factor ($g_{||}/g_{\perp})^{2}\gtrsim 100$ and can not be measured in this scattering geometry, where even the application of a substantial magnetic field leaves behind the longitudinal continuum from magnetic moments perpendicular to the field. While further experiments are needed in alternate scattering geometries to clearly observe the accompanying transverse excitations, the present results quantify in unique detail an unusual dispersing longitudinal mode, a Higgs-like excitation of a effective spin-1/2 order parameter across different ordered phases induced by magnetic fields in Yb2Pt2Pb.

Methods

Neutron Scattering

The Yb2Pt2Pb sample used in these measurements was the same sample used in Ref. Wu_Science_2016 . The sample consists of approximately 400 co-aligned Yb2Pt2Pb single crystals (total mass $\approx 6~{}\rm g$) mounted to aluminum plates. For all measurements the sample was oriented with the (11̄0) crystal direction vertical leaving the (110) and (001) directions in the horizontal scattering plane, making the scattering plane $(H,H,L)$ in reciprocal space. All momenta are given in reciprocal lattice units (rlu), with 1 rlu given by $2\pi/a=2\pi/7.76~{}\rm{\AA}=0.810~{}\rm{\AA}^{-1}$ along the $H$ direction and $2\pi/c=2\pi/7.02~{}\rm{\AA}=0.895~{}\rm{\AA}^{-1}$ along $L$. The crystallographic unit cell is twice the Yb-Yb near neighbor spacing along the $c$-axis – the relevant spacing for spinons. Therefore, the Brillouin zone for spinons is indexed from 0 to 2 rlu along $(0,0,L)$, rather than the typical 0 to 1 rlu. Notationally, $\mathbf{q}_{HH}$ is parallel to the $(H,H,0)$ direction, with scattering primarily coming from $(H,H,1)$ while $\mathbf{q}_{L}$ is along the $(0,0,L)$ direction.

The inelastic neutron scattering measurements on Yb2Pt2Pb making up the 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, UK Demmel_NIMPRA_2014 . The sample was mounted in a dilution refrigerator inside of a 7 T superconducting magnet with the field in the vertical direction parallel to the (11̄0) crystal direction. OSIRIS is an indirect geometry, time-of-flight neutron spectrometer. The incident neutron beam is a white beam of cold neutrons. The PG002 analyzer was used with a final neutron energy $E_{f}=1.84~{}{\rm meV}$ ($\lambda=6.67~{}{\rm\AA}$). The energy resolution was $\approx 0.03~{}\mathrm{meV}$ at $E=0~{}\mathrm{meV}$. Due to the magnet construction, only scattering into a range of angles $45^{\circ}<2\theta<135^{\circ}$ is permitted. We 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 vanadium standard in the magnet. All data were corrected for the Yb3+ form factor Brown_Xtal_book .

The measurements on OSIRIS were made at $T=0.140~{}\rm K$. A field greater than 2.3 T along the (11̄0) crystal direction polarizes all of the magnetic moments parallel to that direction, while leaving the orthogonal moments along the (110) direction in the horizontal plane unaffected due to the ground state doublet of the Yb ions Wu_Science_2016 . Measurements made at $\mu_{0}H=4~{}\rm T$ can therefore be used as a background for measurements made at $\mu_{0}H<2.3~{}\rm T$, isolating only the lower field scattering contribution from magnetic moments oriented parallel to the field. All neutron scattering results from OSIRIS have a measurement made at $T=0.140~{}\rm K$ and $\mu_{0}H=4~{}\rm T$ subtracted in this fashion. For the nominal zero field measurements, a small bias field of 0.025 T was used to suppress superconductivity in the aluminum sample holder.

In general, inelastic neutron scattering probes the dynamical spin correlation function Squires_book_1978 . Because of the crystal field ground state doublet of the Yb ions in Yb2Pt2Pb we are sensitive only to the longitudinal component of this function (see Supplementary Note 5).

Because our detector coverage does not include an entire Brillouin zone, ($0<\mathbf{q}_{L}<2~{}{\rm rlu}$, $0<\mathbf{q}_{HH}<1~{}{\rm rlu}$) we integrate all data in the scattering plane in the reciprocal space direction orthogonal to the direction being considered. The $\mathbf{q}_{L}$ dependencies show in Figs. 1(B) and 2(D-E) (And Supplementary Fig. 1) integrate the entire measured range of $\mathbf{q}_{HH}$. The same $\mathbf{q}_{HH}$ integration was used to demonstrate the amount of low energy spectral weight in the bound state (main text Fig. 3(E) inset). Similarly, all $\mathbf{q}_{L}$ were integrated to examine the $\mathbf{q}_{HH}$ dependence in Figs. 1(C) and 3(B-D) (and Supplementary Figs. 4 and 5) and in the cuts used to extract the interchain mode dispersion in Fig. 3(E). Although our data from OSIRIS do not cover an entire Brillouin zone, the symmetry of our measurements about both $\mathbf{q}_{L}=1~{}{\rm rlu}$ and $\mathbf{q}_{HH}=0~{}{\rm rlu}$ confirms that our coverage is sufficient.

Inelastic scattering measurements of the polarization factor (main text, Fig. 3(F)) and diffraction measurements (main text, Fig. 4(B-H)) were performed on the Disk Chopper Spectrometer (DCS) at the Center for Neutron Research at the National Institute for Standards and Technology in Gaithersburg, MD, USA Copley_CP_2003 . For these measurements, the same sample was mounted in the same scattering geometry as the OSIRIS experiments, inside of a dilution refrigerator inside of a 10 T vertical superconducting magnet. DCS is a direct geometry spectrometer and $E_{i}=3.27~{}{\rm meV}$ ($\lambda=5.00~{}{\rm\AA}$) was used. The energy resolution was $\approx 0.1~{}\mathrm{meV}$. The temperature of these measurements was $T=0.07~{}\rm K$. Measurements on DCS have a similarly measured background subtracted and the scattering was corrected for the Yb3+ form factor Brown_Xtal_book and neutron absorption using the DAVE software package Azuah_JRN_2009 in the same manner described in the Supplementary Materials for Wu_Science_2016 . The measurement at $\mu_{0}H=4~{}\rm T$ of the polarization factor (main text Fig. 3(F)) is simply the $\mu_{0}H=4~{}{\rm T}$ measurement on its own, with no background subtracted. For the nominal zero field measurements, a small bias field of 0.025 T was used to suppress superconductivity in the aluminum sample holder.

Specific Heat and Magnetization

Specific heat measurements used in the phase diagram shown in Fig. 4(A) (and Supplementary Figs. 2 and 3) were made using a Physical Property Measurement System (PPMS) made by Quantum Designs on a single crystal of Yb2Pt2Pb (note: the identification of the equipment used in the various measurements is not intended to imply recommendation or endorsement by the National Institute of Standards and Technology, nor is it intended to imply that this equipment is necessarily the best available for the purpose.) Measurements for $T<1.8~{}\mathrm{K}$ utilized a PPMS dilution refrigerator insert. The (110) direction was oriented vertically, parallel to the magnetic field. Magnetization measurements for $T<1.8$ K in Figs. 2(A) and 4(A,F) (and Supplementary Fig. 6) were made using a PPMS Hall sensor magnetometer in a dilution refrigerator insert at $T=0.150~{}\rm K$. Magnetization measurements for $T>1.8$ K in Fig. 4(A) were made using a Quantum Designs Magnetic Property Measurement System SQUID magnetometer. The magnetic field for both sets of measurements was along the (110) direction.

Obtaining the Position of the Interchain mode

The interchain mode dispersions shown in Figs. 3(B-D) (and Supplementary Fig. 5) were extracted by integrating a window of $\mathbf{q}_{HH}=0.1~{}{\rm rlu}$ and fitting the resulting cuts of the scattering function as a function of energy $S\left(E\right)$, examples of which are shown in Fig. 3(E) of the main text. The fitting function represents the mode as a damped harmonic oscillator – the product of the Bose population factor with the difference of two Lorentzians, centered at positive and negative energies $E_{\mathrm{mode}}$, with a width $\Gamma$ and amplitude $A$, Eqn. 6.

[TABLE]

This function was added to the product of another Lorentzian and step function each centered at energy $E>E_{\mathrm{mode}}$ representing the continuum. The fit itself was performed to these two functions convolved with a gaussian representing the instrumental resolution.

Time-Dependent Density Matrix Renormalization Group Calculations

The longitudinal component of the dynamical spin structure factor $S(\mathbf{q},\omega)$ was obtained by means of the time-dependent density matrix renormalization group (tDMRG) method White2004 ; Daley2004 ; Feiguin2011vietri ; Feiguin2013a . The approach has been extensively described in the literature and essentially consists of calculating the time-dependent correlation function:

[TABLE]

The operator $\hat{S}^{z}(x_{0}=L/2)$ is applied at the center of the chain and the resulting state is evolved in real-time. At every step, the overlap with the state $\hat{S}^{z}(x)e^{-i\hat{H}t}|\psi\rangle$ is measured and the correlations function in real time and space is recorded. The results are Fourier transformed to frequency and momentum using a properly chosen Hann window that determines the resolution of the final result. In our case, in order to compare to experiments we have used a window of half-width $\Delta t=7$ in units of $1/J$. Only two things differ from conventional calculations: since the parameters considered fall into the Ising phase of the model, which tends to break translational symmetry at finite magnetization, the simulations were carried out using periodic boundary conditions and the time-targeting method with a Krylov expansion of the evolution operator Feiguin2005 . In addition, we plot the results for the operator $(\hat{S}^{z}(x)-m/2)$ (instead of $\hat{S}^{z}(x)$), where $m$ is the magnetization value. The offset allows us to resolve the density fluctuations and eliminates large contributions at low frequencies and momenta Nocera2016 . Surprisingly, due to the low entanglement in the Ising phase, truncation errors smaller than $10^{-5}$ are easily achievable using 300 states, even on a chain of length $L=96$ with periodic boundary conditions.

The color scale for the calculations displayed in Figs. 1(D) and 2(G-I) was determined by normalizing the integral of the $E>0$ portion of the calculation to the measured integral of the inelastic intensity at the the same field.

Acknowledgements

Work at Texas A&M University (W. J. G. and M. C. A) was supported by NSF-DMR-1310008. Work at Brookhaven National Laboratory (I. A. Z and A. M. T) was supported under the auspices of the US Department of Energy, Office of Basic Energy Sciences, under contract DE-SC00112704. Work at Northeastern University (A. E. F.) was supported by the US Department of Energy, Office of Science, Basic Energy Sciences grant number DE-SC0014407. Work at the ISIS neutron source was partly supported by the Science and Technology Facilities Council, STFC. Access to DCS was provided by the Center for High Resolution Neutron Scattering, a partnership between the National Institute of Standards and Technology and the National Science Foundation under Agreement No. DMR-1508249. We thank S. E. Nagler and R. Vega-Morales for helpful discussions about this work.

Author Contributions

This work was designed by W. J. G., L. S. W., I. A. Z., and M. C. A. The sample was synthesized by L. S. W. The neutron scattering measurements at the ISIS neutron source were carried out by W. J. G., I. A. Z., F. D., and M. C. A. The neutron scattering measurements at the NIST Center for Neutron Research were carried out by L. S. W., I. A. Z., Y. Q., J. R. D. C., and M. C. A. The specific heat and magnetization data were measured by M. S. K. All of the data were analyzed by W. J. G. with input from I. A. Z. The tDMRG calculations were performed by A. E. F. The theoretical model was developed by A. M. T. The manuscript was written by W. J. G. with input from all authors.

Data Availability

The data that support the findings of this study are available from the corresponding authors upon reasonable request. Original time-of-flight data is available for the experiments at the ISIS neutron source at

http://dx.doi.org/10.5286/ISIS.E.42580328.

Competing Interests

The authors declare that they have no competing interests.

Supplementary Information

Spinon Confinement and a Sharp Longitudinal Mode in Yb2Pt2Pb in Magnetic Fields

Supplementary Note 1

The observation of a continuum of scattering for all fields $\mu_{0}H<2.3~{}\rm T$ necessarily indicates the presence of multi spinon states. The energy of any two spinon state is given by the sum of the two spinon energies while the wavevector of the two spinon state is given by the vector sum of the momenta of each. The energy and momentum of the two spinon state must conserved with respect to the change in energy and momentum between the incident neutron’s initial and final states. The possible energy and momentum of each spinon is given by the single spinon dispersions

[TABLE]

where $E_{\mathrm{p,h}}$ is the energy for a “particle” and “hole,” i.e. a spin-up spinon and a spin-down spinon. As discussed in the main text, $I=0.485~{}\mathrm{meV}$ and $\Delta_{\mathrm{S}}=0.095~{}\mathrm{meV}$ for Yb2Pt2Pb.

The envelope of the continuum is determined by the extremal two spinon states. In zero field, there are three extremal cases. In one case, if the hole momentum $\mathbf{q}_{\mathrm{h}}=0$, all of the momentum $\mathbf{q}_{\mathrm{2spin}}=\mathbf{q}_{\mathrm{p}}+\mathbf{q}_{\mathrm{h}}$ of the two spinon state will be carried by the particle, $\mathbf{q}_{\mathrm{2spin}}=\mathbf{q}_{\mathrm{p}}$. The energy of the two spinon state $E_{\mathrm{2spin}}$ will given by $E_{\mathrm{2spin}}=E_{\mathrm{h}}\left(0\right)+E_{\mathrm{p}}\left(\mathbf{q}_{\mathrm{2spin}}\right)$. This is degenerate with the case where $\mathbf{q}_{\mathrm{2spin}}=\mathbf{q}_{\mathrm{h}}$ and $\mathbf{q}_{\mathrm{p}}=0$. The third case has the particle and hole sharing the momentum equally, $\mathbf{q}_{\mathrm{p}}=\mathbf{q}_{\mathrm{h}}=\mathbf{q}_{\mathrm{2spin}}/2$, so that the two spinon dispersion is given by $E_{\mathrm{2spin}}=E_{\mathrm{p}}\left(\mathbf{q}_{\mathrm{2spin}}/2\right)+E_{\mathrm{h}}\left(\mathbf{q}_{\mathrm{2spin}}/2\right)$. All other two spinon states have energies that fall between these possibilities. They are sketched in Fig.1(B,D) of the main text.

Application of a field along the $z$ direction large enough for the Zeeman energy $g\mu_{B}HS^{z}$ to close the gap $\Delta_{\mathrm{S}}$ in the single spinon dispersion changes the extremal combinations of two spinons. There are far more possible two spinon states, since there are now more routes to creating particle-hole paris as the hole (or particle, depending on the sign convention) dispersion is emptied with increasing field. The boundaries of the two spinon continuum that are sketched in Fig. 2(D-I) of the main text are arrived at by fixing the particles (holes) at momentum $\mathbf{k}_{\mathrm{F}}$ and 1-$$\mathbf{k}_{\mathrm{F}} and creating excitations with holes (praticles) at $\mathbf{k}_{\mathrm{F}}$ and 1-$$\mathbf{k}_{\mathrm{F}}, fixing the particle to have 0 momentum, fixing the particle to have momentum $\mathbf{q}_{\mathrm{p}}=1$ and allowing the particle and hole to share the momentum equally.

Supplementary Note 2

The spectra for momentum along the chain direction $\mathbf{q}_{L}$ in a magnetic field (main text, Fig. 2(D-F)) are presented in Fig. S1(A-D) without the continuum boundaries. Data at $\mu_{0}H=1.9~{}\rm T$ are also shown. The spectrum at 1.9 T is consistent with 1.5 and 1.7 T – all are in the same antiferromagnetic phase. The 4.0 T data in Fig. S1(E) are the same as in zero field, but with a $\mathbf{q}_{L}$ dependence that reflects the polarization factor of the Ising magnetic moments in the scattering plane, perpendicular to the applied magnetic field.

Supplementary Note 3

The color plot used in the antiferromagnetic phase diagram shown in Fig. 3 of the main text was obtained from the specific heat. We present the same color plot here in Fig. S2 without the phase lines determined from magnetization. Measurements of the specific heat $C_{\mathrm{p}}$ were made at many fixed temperatures as a function of a magnetic field parallel to the crystal (110) direction. An example of such a measurement at $T=0.35~{}\rm K$ is shown in Fig.  S3. To enhance contrast for the color plot, the quantity plotted in the phase digram is the field derivative of the specific heat, ${\rm d}C_{\mathrm{p}}/{\rm d}\mu_{0}H$. An example of the derivative at $T=0.35~{}\rm K$ is shown in Fig. S3. Many such measurements are combined to make Fig. S2, which compares well with the phase diagrams determined previously.  Ochiai_JPSJ_2011 ; Shimura_JPSJ_2012 ; Kim_PRL_2013

Supplementary Note 4

The spectra for momentum along $\mathbf{q}_{HH}$, perpendicular to the chain direction (main text, Fig. 3(B-D)) are shown in Fig. S4(A-D) without the mode position or dispersion fits. Data at $\mu_{0}H=1.9~{}\rm T$ are also shown. The spectrum at 1.9 T is consistent with 1.5 and 1.7 T and is not shown in the main text. The 4.0 T data in Fig. S1(E) are the same as in zero field, but with a $\mathbf{q}_{HH}$ dependence that reflects the polarization factor of the Ising magnetic moments in the scattering plane, perpendicular to the applied magnetic field. In Fig. S5, we show the same spectra with the mode position and dispersion fit over the entire measured range of $\mathbf{q}_{HH}$. The dispersion and fits shown in Fig.3(B-D) of the main text are the same, but are only displayed for $\mathbf{q}_{HH}>0$ in order to show both the mode in the color plot and in the analyzed data simultaneously. The fits themselves were performed over the entire range of $\mathbf{q}_{HH}$.

Supplementary Note 5

The scattering geometry of our measurements orients the (11̄0) crystal direction vertically, parallel to the magnetic field, with the (110) direction in the horizontal scattering plane. Measurements made in a field $\mu_{0}H>2.3~{}\rm T$ will quench all excitations in the Yb sublattice with moments oriented along (11̄0), while having no effect on the perpendicular sublattice. Using such a measurement as a background for lower fields isolates the fluctuations parallel to (11̄0). Plotting the total scattered intensity as a function of $\mathbf{q}_{HH}$ at \mathbf{q}_{L}$$=1 (main text, Fig. 3(F)) demonstrates the wavevector dependence of the scattering as $\mathbf{q}$ is rotated from the $c$-axis, ($\mathbf{q}_{HH}=0$) towards the (110) direction ($\mathbf{q}_{HH}\gg\mathbf{q}_{L}$). We find a constant scattered intensity, and since neutrons only scatter from components of the local magnetization density perpendicular to $\mathbf{q}$, Squires_book_1978 this demonstrates that the fluctuating magnetic moments always have the same projection on the scattering vector and are thus always perpendicular to $\mathbf{q}$. The same data plotted for a fixed field of 4 T show that the scattering precisely follows the geometric projection of the moments in the scattering plane onto the scattering vector, $I\propto\mathbf{q}_{L}^{2}/(2\mathbf{q}_{H}^{2}+\mathbf{q}_{L}^{2})$, providing further confirmation that the fluctuations are purely longitudinal.

Supplementary Note 6

The field dependence of the magnetization matches the field dependence of 2$\mathbf{k}_{\mathrm{F}}$ in 1D systems Giamarchi_book_2004 . In Fig. S6, we show the correspondence between the magnetization measured at $T=0.150~{}\rm K$, below the spinon gap $\Delta_{\mathrm{S}}$ with 2$\mathbf{k}_{\mathrm{F}}$ calculated from where the chemical potential $g\mu_{B}HS^{z}$ crosses the single spinon dispersions, given for particles and holes by Eqn. S1 with $I=0.485~{}{\rm meV}$ and $\Delta_{\mathrm{S}}=0.095~{}{\rm meV}$. The magnetization has several kinks corresponding to metastability in the underlying incommensurate antiferromagnetic order, demonstrated in Fig.4(E,F) of the main text.

Supplementary Note 7

Our description of the longitudinal interchain mode requires fitting the strengths of the relevant interchain couplings, $\mathcal{J}^{\perp}_{n}$, with $n=1$ for nearest neighbor coupling, $n=2$ for next nearest neighbors, etc Schulz_PRL_1996 . The fitting function must be normalized to the total interchain exchange couplings $\mathcal{J}^{\perp}_{\mathrm{tot}}=\sum_{n=1}^{5}\mathcal{J}^{\perp}_{n}$ and include a gap $\Delta_{\mathrm{M}}$. We give the dispersion by $E_{\mathrm{mode}}\left(\mathbf{q}_{HH}\right)^{2}=\Delta_{\mathrm{M}}^{2}\left(1-\frac{1}{3\mathcal{J}^{\perp}_{\mathrm{tot}}}\sum_{n=1}^{5}\mathcal{J}^{\perp}_{n}\cos\left(2\pi n\mathbf{q}_{HH}\right)\right)$, which includes 5 terms in the sum to account for the periodicity of the underlying antiferromagnetic order. Defining $\mathcal{J}^{\mathrm{rel}}_{n}=\mathcal{J}^{\perp}_{n}/\mathcal{J}^{\perp}_{\mathrm{tot}}$, we vary $\mathcal{J}^{\mathrm{rel}}_{n}$ in our fits. The results of the fit, along with the standard deviations of the fits are given in Table S1.