Figure 18: Same as Fig. 10 in Sec. IV.3 for the field $B\\!=\\!8$ T.

###  C.3 Dynamical structure factor

Our Fig. 18 presents the comparison of the experimental and theoretical DSFs
as in Fig. 10 in Sec. IV.3, for the field $B\\!=\\!8$ T.

##  Appendix D Longitudinal field effects

###  D.1 The excitation gap and the band gap

For the magnetic field tilted in the $y_{0}z_{0}$ ($zx$) plane, spin
components $\widetilde{\mathbf{S}}_{i}$ in the local reference frame, which is
tilted by the angle $\theta$ away from the $y_{0}$ axis, are related to the
spin components in the laboratory frame $\mathbf{S}_{i}$ by

$\widetilde{\mathbf{S}}_{i}\\!=\\!\hat{\bf R}_{\theta}\mathbf{S}_{i},$ (90)

where $\hat{\bf R}_{\theta}$ is the rotation matrix

$\displaystyle\hat{\textbf{R}}_{\theta}=\begin{pmatrix}0&-\sin\theta&\cos\theta\\\
1&0&0\\\ 0&\cos\theta&\sin\theta\\\ \end{pmatrix}.$ (91)

The Hamiltonian in the tilted local frame is

$\displaystyle\hat{\cal H}=\sum_{\langle ij\rangle_{n}}\widetilde{\bf
S}_{i}^{\rm T}\widetilde{\bm{J}}_{ij}\widetilde{\bf
S}_{j}-\sum_{i}\widetilde{\bf H}^{\rm T}\widetilde{\bf S}_{i},$ (92)

where $n\\!=\\!1,2$ for the nearest- and next-nearest-neighbor bonds,
respectively. The exchange matrices are

$\displaystyle\widetilde{\bm{J}}_{ij}=\hat{\bf R}_{\theta}^{\rm T}\hat{\bf
J}_{ij}\hat{\bf
R}_{\theta}=\left(\begin{array}[]{ccc}\widetilde{J}_{ij}^{xx}&\widetilde{J}_{ij}^{xy}&\widetilde{J}_{ij}^{xz}\\\
\widetilde{J}_{ij}^{yx}&\widetilde{J}_{ij}^{yy}&\widetilde{J}_{ij}^{yz}\\\
\widetilde{J}_{ij}^{zx}&\widetilde{J}_{ij}^{zy}&\widetilde{J}_{ij}^{zz}\end{array}\right),$
(96)

with the exchange matrices $\hat{\bf J}_{ij}$ in the laboratory frame from
Eqs. (7) and (14), and the rotated field given by

$\widetilde{\bf H}=\mu\hat{\bf R}_{\theta}\hat{\bf g}{\bf B},$ (97)

with ${\bf B}\\!=\\!B_{\perp}\hat{y}_{0}+B_{\parallel}\hat{z}_{0}$ and
$\hat{\bf g}$ being the diagonal g-tensor.

Naturally, the Hamiltonian can be split into the even and odd parts, as
before. At the LSWT level of approximation, we only consider the even part
that reads

$\displaystyle\hat{\cal H}_{\text{even}}=\sum_{\langle ij\rangle_{n}}$
$\displaystyle\Big{(}\widetilde{J}_{ij}^{xx}\widetilde{S}_{i}^{x}\widetilde{S}_{j}^{x}+\widetilde{J}_{ij}^{yy}\widetilde{S}_{i}^{y}\widetilde{S}_{j}^{y}+\widetilde{J}_{ij}^{zz}\widetilde{S}_{i}^{z}\widetilde{S}_{j}^{z}$
(98)
$\displaystyle+\widetilde{J}_{ij}^{xy}\widetilde{S}_{i}^{x}\widetilde{S}_{j}^{y}+\widetilde{J}_{ij}^{yx}\widetilde{S}_{i}^{y}\widetilde{S}_{j}^{x}\Big{)}-\sum_{i}\widetilde{H}^{z}\widetilde{S}_{i}^{z}.$

Importantly, compared to the same LSWT consideration in Sec. III.2, the
diagonal exchanges $\widetilde{J}_{1}^{xx}$ and $\widetilde{J}_{1}^{zz}$ have
now acquired bond-dependent contributions from the staggered $J_{y_{0}z_{0}}$
terms because of the local axes tilt. Therefore, the nearest-neighbor bonds in
the Hamiltonian (98) are not equivalent, the unit cell now contains two spins,
and two species of bosons need to be introduced with the Holstein-Primakoff
transformation that reads

$\widetilde{S}_{\mu l}^{z}=S-n_{\mu l},\quad\widetilde{S}_{\mu
l}^{+}\approx\sqrt{2S}a_{\mu l},$ (99)

where $l$ and $\mu=1,2$ numerate the unit cells of the zigzag chain and the
two sublattices, respectively.

The classical energy, obtained from Eq. (98), is (up to a constant
$S^{2}(J_{y_{0}y_{0}}+J_{2})$),

$\displaystyle\frac{E_{\text{cl}}}{N}=-SH_{\perp}\cos\theta-
SH_{\parallel}\sin\theta-\frac{S}{2}H_{c}\sin^{2}\theta,$ (100)

with $H_{c}$ from Eq. (17). Minimizing Eq. (100) with respect to $\theta$
yields Eq. (47) in Sec. V.

After some algebra, the LSWT Hamiltonian for the two bosonic species can be
written in the matrix form

$\hat{\mathcal{H}}=\frac{1}{2}\sum_{\bf k}\hat{{\bf x}}_{\bf
k}^{\dagger}\hat{{\bf H}}_{\bf k}\hat{{\bf x}}_{\bf
k}^{\phantom{{\dagger}}},\quad\hat{{\bf H}}_{\bf k}=\begin{pmatrix}\hat{{\bf
A}}_{\bf k}&\hat{{\bf B}}_{\bf k}\\\ \hat{{\bf B}}_{\bf k}^{\dagger}&\hat{\bf
A}^{*}_{-{\bf k}}\end{pmatrix},$ (101)

where $\hat{{\bf x}}_{\bf k}^{\dagger}\\!=\\!\big{(}a_{1\bf
k}^{\dagger},a_{2\bf k}^{\dagger},a_{1-{\bf
k}}^{\phantom{{\dagger}}},a_{2-{\bf k}}^{\phantom{{\dagger}}}\big{)}$ are the
bosonic vector operators and the $2\\!\times\\!2$ matrices $\hat{{\bf A}}_{\bf
k}$ and $\hat{{\bf B}}_{\bf k}$ are

$\hat{\textbf{A}}_{\bf k}=\begin{pmatrix}A_{\bf k}&B_{\bf k}\\\ B_{\bf
k}^{*}&A_{\bf k}\end{pmatrix},\ \hat{\textbf{B}}_{\bf k}=\begin{pmatrix}D_{\bf
k}&C_{\bf k}\\\ C_{-{\bf k}}&D_{\bf k}^{*}\end{pmatrix},$ (102)

with

$\displaystyle A_{\bf k}$
$\displaystyle=H_{\perp}c_{\theta}+H_{\parallel}s_{\theta}-2S(J_{y_{0}y_{0}}+J_{2})c^{2}_{\theta}$
$\displaystyle-2S\big{(}J_{z_{0}z_{0}}+J_{2z_{0}}\big{)}s^{2}_{\theta}+S\big{(}J_{2}(1+s^{2}_{\theta})+J_{2z_{0}}c^{2}_{\theta}\big{)}\gamma_{\bf
k}^{(2)},$ $\displaystyle B_{\bf k}$
$\displaystyle=-SJ_{y_{0}z_{0}}s_{2\theta}\bar{\gamma}_{\bf
k}+S\big{(}J_{x_{0}x_{0}}+J_{y_{0}y_{0}}s^{2}_{\theta}+J_{z_{0}z_{0}}c^{2}_{\theta}\big{)}\gamma_{\bf
k}^{(1)},$ $\displaystyle C_{\bf k}$
$\displaystyle=-SJ_{y_{0}z_{0}}s_{2\theta}\bar{\gamma}_{\bf
k}-S\big{(}J_{x_{0}x_{0}}-J_{y_{0}y_{0}}s^{2}_{\theta}-J_{z_{0}z_{0}}c^{2}_{\theta}\big{)}\gamma_{\bf
k}^{(1)},$ $\displaystyle D_{\bf k}$
$\displaystyle=-S\big{(}J_{2}-J_{2z_{0}}\big{)}c^{2}_{\theta}\gamma_{\bf
k}^{(2)},$ (103)

with the shorthand notations $c_{\theta}\\!=\\!\cos\theta$ and
$s_{\theta}\\!=\\!\sin\theta$.

The eigenvalue problem for (101) can be solved analytically by diagonalizing
$(\hat{\bf g}\hat{\bf H}_{\bf k})^{2}$, with the paraunitary matrix $\hat{\bf
g}\\!=\\![1,1,-1,-1]$. Using that $A_{\bf k}\\!=\\!A_{-{\bf k}}$ and $D_{\bf
k}$ are real, $B_{-{\bf k}}\\!=\\!B_{\bf k}^{*}$, and $C_{-{\bf
k}}\\!=\\!C_{\bf k}^{*}$, we get two branches

$\displaystyle\varepsilon_{1,2{\bf k}}=\sqrt{A_{\bf k}^{2}+|B_{\bf
k}|^{2}-|C_{\bf k}|^{2}-D_{\bf k}^{2}\pm 2\sqrt{{\cal R}}},$ (104)
$\displaystyle{\cal R}=\\!A_{\bf k}^{2}|B_{\bf k}|^{2}\\!+\\!|C_{\bf
k}|^{2}D_{\bf k}^{2}\\!-\\!2A_{\bf k}D_{\bf k}{\rm Re}[B_{\bf k}C_{\bf
k}^{*}]\\!+\\!{\rm Im}[B_{\bf k}C_{\bf k}^{*}]^{2}.$

From the spectrum in Eq. (104), one can obtain expressions for the excitation
gap $\Delta_{0}$ and band gap $\Delta_{b}$ using the canting angle $\theta$
calculated numerically from the nonlinear equation (46). For
$H_{\parallel}\\!\ll\\!H_{\perp}\\!=\\!H_{c}$, with the approximate solution
for the canting angle in Eq. (47), after some algebra in Eq. (104), the
excitation gap is

$\displaystyle\Delta_{0}$ $\displaystyle\approx\alpha_{0}H_{\parallel}^{1/3},$
(105) $\displaystyle\alpha_{0}$
$\displaystyle=\left(\frac{2}{H_{c}}\right)^{1/3}\sqrt{\frac{3}{2}}H_{c}\sqrt{1+\frac{2S(J_{x_{0}x_{0}}-J_{y_{0}y_{0}})}{H_{c}}}\
,$

and the band gap

$\displaystyle\Delta_{b}$ $\displaystyle\approx\alpha_{b}H_{\parallel}^{1/3},$
(106) $\displaystyle\alpha_{b}$
$\displaystyle=2S\big{|}J_{y_{0}z_{0}}\big{|}\left(\frac{2}{H_{c}}\right)^{1/3}\sqrt{1+\frac{J_{2}-J_{2z_{0}}}{J_{z_{0}z_{0}}+2J_{2z_{0}}}}\
.$

Neglecting small corrections under the square roots leads to the results in
Eq. (48).

###  D.2 More threshold singularities

The anharmonic cubic coupling in the tilted magnetic field is obtained from
the odd part of the Hamiltonian, similarly to the discussion in Sec. III.1.
Using Eqs. (96) and (97), it can be generally written as

$\displaystyle\hat{\cal H}_{\text{odd}}\\!$ $\displaystyle=\\!\sum_{\langle
ij\rangle_{n}}\widetilde{J}_{ij}^{xz}\Big{(}\widetilde{S}_{i}^{x}\widetilde{S}_{j}^{z}\\!+\\!\widetilde{S}_{i}^{z}\widetilde{S}_{j}^{x}\Big{)}-\sum_{i}\widetilde{H}^{x}\widetilde{S}_{i}^{x}.$
(107)

There are two resulting types of the cubic coupling, one that retains the
staggered structure of such a coupling in the pure transverse field,
$\propto\\!J_{y_{0}z_{0}}$, and the other one originating from the tilted
component of the spin. In the small longitudinal field regime,
$H_{\parallel}\\!\ll\\!H_{\perp}\\!=\\!H_{c}$, the latter is subleading to the
former, $\propto\\!{\cal O}(J_{z_{0}z_{0}}H_{\parallel}^{1/3})$, leaving cubic
anhamonicity unaffected by the field. The secondary component also corresponds
to an unfavorable kinematics for the decays, justifying neglecting it in this
regime.

## References

  * Penc and Läuchli [2011] K. Penc and A. M. Läuchli, Spin Nematic Phases in Quantum Spin Systems, in  _Introduction to Frustrated Magnetism: Materials, Experiments, Theory_, edited by C. Lacroix, P. Mendels, and F. Mila (Springer Berlin Heidelberg, Berlin, Heidelberg, 2011) pp. 331–362.
  * Starykh [2015] O. A. Starykh, Unusual ordered phases of highly frustrated magnets: a review, Rep. Prog. Phys. 78, 052502 (2015).
  * Ross  _et al._ [2011] K. A. Ross, L. Savary, B. D. Gaulin, and L. Balents, Quantum Excitations in Quantum Spin Ice, Phys. Rev. X 1, 021002 (2011).
  * Rau  _et al._ [2018] J. G. Rau, P. A. McClarty, and R. Moessner, Pseudo-Goldstone Gaps and Order-by-Quantum Disorder in Frustrated Magnets, Phys. Rev. Lett. 121, 237201 (2018).
  * Rau and Gingras [2019] J. G. Rau and M. J. Gingras, Frustrated quantum rare-earth pyrochlores, Annu. Rev. Condens. Matter Phys. 10, 357 (2019).
  * Jiang  _et al._ [2023a] S. Jiang, J. Romhányi, S. R. White, M. E. Zhitomirsky, and A. L. Chernyshev, Where is the Quantum Spin Nematic?, Phys. Rev. Lett. 130, 116701 (2023a).
  * Jiang  _et al._ [2023b] S. Jiang, S. R. White, and A. L. Chernyshev, Quantum phases in the honeycomb-lattice ${J}_{1}$–${J}_{3}$ ferro-antiferromagnetic model, Phys. Rev. B 108, L180406 (2023b).
  * Balents [2010] L. Balents, Spin liquids in frustrated magnets, Nature 464, 199 (2010).
  * Yan  _et al._ [2011] S. Yan, D. A. Huse, and S. R. White, Spin-Liquid Ground State of the $S$ = $1/2$ Kagome Heisenberg Antiferromagnet, Science 332, 1173 (2011).
  * Savary and Balents [2016] L. Savary and L. Balents, Quantum spin liquids: a review, Rep. Prog. Phys. 80, 016502 (2016).
  * Knolle and Moessner [2019] J. Knolle and R. Moessner, A Field Guide to Spin Liquids, Annu. Rev. Condens. Matter Phys. 10, 451 (2019).
  * Kitaev [2006] A. Kitaev, Anyons in an exactly solved model and beyond, Ann. Phys. 321, 2 (2006).
  * Nussinov and van den Brink [2015] Z. Nussinov and J. van den Brink, Compass models: Theory and physical motivations, Rev. Mod. Phys. 87, 1 (2015).
  * Hermanns  _et al._ [2018] M. Hermanns, I. Kimchi, and J. Knolle, Physics of the Kitaev Model: Fractionalization, Dynamic Correlations, and Material Connections, Annu. Rev. Condens. Matter Phys. 9, 17 (2018).
  * Jackeli and Khaliullin [2009] G. Jackeli and G. Khaliullin, Mott Insulators in the Strong Spin-Orbit Coupling Limit: From Heisenberg to a Quantum Compass and Kitaev Models, Phys. Rev. Lett. 102, 017205 (2009).
  * Chaloupka  _et al._ [2010] J. Chaloupka, G. Jackeli, and G. Khaliullin, Kitaev-Heisenberg Model on a Honeycomb Lattice: Possible Exotic Phases in Iridium Oxides ${A}_{2}{\mathrm{IrO}}_{3}$, Phys. Rev. Lett. 105, 027204 (2010).
  * Liu and Khaliullin [2018] H. Liu and G. Khaliullin, Pseudospin exchange interactions in ${d}^{7}$ cobalt compounds: Possible realization of the Kitaev model, Phys. Rev. B 97, 014407 (2018).
  * Sano  _et al._ [2018] R. Sano, Y. Kato, and Y. Motome, Kitaev-Heisenberg Hamiltonian for high-spin ${d}^{7}$ Mott insulators, Phys. Rev. B 97, 014408 (2018).
  * Choi  _et al._ [2012] S. K. Choi, R. Coldea, A. N. Kolmogorov, T. Lancaster, I. I. Mazin, S. J. Blundell, P. G. Radaelli, Y. Singh, P. Gegenwart, K. R. Choi, S.-W. Cheong, P. J. Baker, C. Stock, and J. Taylor, Spin Waves and Revised Crystal Structure of Honeycomb Iridate ${\mathrm{Na}}_{2}{\mathrm{IrO}}_{3}$, Phys. Rev. Lett. 108, 127204 (2012).
  * Biffin  _et al._ [2014] A. Biffin, R. D. Johnson, I. Kimchi, R. Morris, A. Bombardi, J. G. Analytis, A. Vishwanath, and R. Coldea, Noncoplanar and Counterrotating Incommensurate Magnetic Order Stabilized by Kitaev Interactions in $\gamma$-${\mathrm{Li}}_{2}{\mathrm{IrO}}_{3}$, Phys. Rev. Lett. 113, 197201 (2014).
  * Johnson  _et al._ [2015] R. D. Johnson, S. C. Williams, A. A. Haghighirad, J. Singleton, V. Zapf, P. Manuel, I. I. Mazin, Y. Li, H. O. Jeschke, R. Valentí, and R. Coldea, Monoclinic crystal structure of $\alpha$-${\mathrm{RuCl}}_{3}$ and the zigzag antiferromagnetic ground state, Phys. Rev. B 92, 235119 (2015).
  * Paddison  _et al._ [2016] J. A. M. Paddison, M. Daum, Z. Dun, G. Ehlers, Y. Liu, M. B. Stone, H. Zhou, and M. Mourigal, Continuous excitations of the triangular-lattice quantum spin liquid YbMgGaO4, Nat. Phys. 13, 117 (2016).
  * Winter  _et al._ [2016] S. M. Winter, Y. Li, H. O. Jeschke, and R. Valentí, Challenges in design of Kitaev materials: Magnetic interactions from competing energy scales, Phys. Rev. B 93, 214431 (2016).
  * Janssen  _et al._ [2017] L. Janssen, E. C. Andrade, and M. Vojta, Magnetization processes of zigzag states on the honeycomb lattice: Identifying spin models for $\alpha$-${\mathrm{RuCl}}_{3}$ and ${\mathrm{Na}}_{2}{\mathrm{IrO}}_{3}$, Phys. Rev. B 96, 064430 (2017).
  * Zhu  _et al._ [2017] Z. Zhu, P. A. Maksimov, S. R. White, and A. L. Chernyshev, Disorder-Induced Mimicry of a Spin Liquid in ${\mathrm{YbMgGaO}}_{4}$, Phys. Rev. Lett. 119, 157201 (2017).
  * Maksimov  _et al._ [2019] P. A. Maksimov, Z. Zhu, S. R. White, and A. L. Chernyshev, Anisotropic-Exchange Magnets on a Triangular Lattice: Spin Waves, Accidental Degeneracies, and Dual Spin Liquids, Phys. Rev. X 9, 021017 (2019).
  * Maksimov and Chernyshev [2020] P. A. Maksimov and A. L. Chernyshev, Rethinking $\alpha$-${\mathrm{RuCl}}_{3}$, Phys. Rev. Res. 2, 033011 (2020).
  * Smit  _et al._ [2020] R. L. Smit, S. Keupert, O. Tsyplyatyev, P. A. Maksimov, A. L. Chernyshev, and P. Kopietz, Magnon damping in the zigzag phase of the Kitaev-Heisenberg-$\mathrm{\Gamma}$ model on a honeycomb lattice, Phys. Rev. B 101, 054424 (2020).
  * Witczak-Krempa  _et al._ [2014] W. Witczak-Krempa, G. Chen, Y. B. Kim, and L. Balents, Correlated Quantum Phenomena in the Strong Spin-Orbit Regime, Annu. Rev. Condns. Matter Phys. 5, 57 (2014).
  * Rau  _et al._ [2016] J. G. Rau, E. K.-H. Lee, and H.-Y. Kee, Spin-Orbit Physics Giving Rise to Novel Phases in Correlated Systems: Iridates and Related Materials, Annu. Rev. Condens. Matter Phys. 7, 195 (2016).
  * Schaffer  _et al._ [2016] R. Schaffer, E. K.-H. Lee, B.-J. Yang, and Y. B. Kim, Recent progress on correlated electron systems with strong spin-orbit coupling, Rep. Prog. Phys. 79, 094504 (2016).
  * Songvilay  _et al._ [2020] M. Songvilay, J. Robert, S. Petit, J. A. Rodriguez-Rivera, W. D. Ratcliff, F. Damay, V. Balédent, M. Jiménez-Ruiz, P. Lejay, E. Pachoud, A. Hadj-Azzem, V. Simonet, and C. Stock, Kitaev interactions in the Co honeycomb antiferromagnets ${\mathrm{Na}}_{3}{\mathrm{Co}}_{2}{\mathrm{SbO}}_{6}$ and ${\mathrm{Na}}_{2}{\mathrm{Co}}_{2}{\mathrm{TeO}}_{6}$, Phys. Rev. B 102, 224429 (2020).
  * Liu  _et al._ [2020] H. Liu, J. Chaloupka, and G. Khaliullin, Kitaev Spin Liquid in $3d$ Transition Metal Compounds, Phys. Rev. Lett. 125, 047201 (2020).
  * Zhong  _et al._ [2020] R. Zhong, T. Gao, N. P. Ong, and R. J. Cava, Weak-field induced nonmagnetic state in a Co-based honeycomb, Sci. Adv. 6, eaay6953 (2020).
  * Yang  _et al._ [2022] H. Yang, C. Kim, Y. Choi, J. H. Lee, G. Lin, J. Ma, M. Kratochvílová, P. Proschek, E.-G. Moon, K. H. Lee, Y. S. Oh, and J.-G. Park, Significant thermal Hall effect in the $3d$ cobalt Kitaev system ${\mathrm{Na}}_{2}{\mathrm{Co}}_{2}\mathrm{Te}{\mathrm{O}}_{6}$, Phys. Rev. B 106, L081116 (2022).
  * Maksimov  _et al._ [2022] P. A. Maksimov, A. V. Ushakov, Z. V. Pchelkina, Y. Li, S. M. Winter, and S. V. Streltsov, Ab initio guided minimal model for the “Kitaev” material ${\mathrm{BaCo}}_{2}$(${\mathrm{AsO}}_{4}{)}_{2}$: Importance of direct hopping, third-neighbor exchange, and quantum fluctuations, Phys. Rev. B 106, 165131 (2022).
  * Halloran  _et al._ [2023] T. Halloran, F. Desrochers, E. Z. Zhang, T. Chen, L. E. Chern, Z. Xu, B. Winn, M. Graves-Brook, M. B. Stone, A. I. Kolesnikov, Y. Qiu, R. Zhong, R. Cava, Y. B. Kim, and C. Broholm, Geometrical frustration versus Kitaev interactions in BaCo2(AsO4)2, Proc. Natl. Acad. Sci. U.S.A. 120, e2215509119 (2023).
  * Yao  _et al._ [2023] W. Yao, Y. Zhao, Y. Qiu, C. Balz, J. R. Stewart, J. W. Lynn, and Y. Li, Magnetic ground state of the Kitaev ${\mathrm{Na}}_{2}{\mathrm{Co}}_{2}{\mathrm{TeO}}_{6}$ spin liquid candidate, Phys. Rev. Res. 5, L022045 (2023).
  * Pfeuty [1970] P. Pfeuty, The one-dimensional Ising model with a transverse field, Ann. Phys. (NY) 57, 79 (1970).
  * Coldea  _et al._ [2010] R. Coldea, D. A. Tennant, E. M. Wheeler, E. Wawrzynska, D. Prabhakaran, M. Telling, K. Habicht, P. Smeibidl, and K. Kiefer, Quantum Criticality in an Ising Chain: Experimental Evidence for Emergent E8 Symmetry, Science 327, 177 (2010).
  * Cabrera  _et al._ [2014] I. Cabrera, J. D. Thompson, R. Coldea, D. Prabhakaran, R. I. Bewley, T. Guidi, J. A. Rodriguez-Rivera, and C. Stock, Excitations in the quantum paramagnetic phase of the quasi-one-dimensional Ising magnet ${\mathrm{CoNb}}_{2}$${\mathrm{O}}_{6}$ in a transverse field: Geometric frustration and quantum renormalization effects, Phys. Rev. B 90, 014418 (2014).
  * Robinson  _et al._ [2014] N. J. Robinson, F. H. L. Essler, I. Cabrera, and R. Coldea, Quasiparticle breakdown in the quasi-one-dimensional Ising ferromagnet CoNb2O6, Phys. Rev. B 90, 174406 (2014).
  * Fava  _et al._ [2020] M. Fava, R. Coldea, and S. A. Parameswaran, Glide symmetry breaking and Ising criticality in the quasi-1D magnet ${\mathrm{CoNb}}_{2}{\mathrm{O}}_{6}$, Proc. Natl. Acad. Sci. U.S.A. 117, 25219 (2020).
  * Woodland  _et al._ [2023a] L. Woodland, D. Macdougal, I. M. Cabrera, J. D. Thompson, D. Prabhakaran, R. I. Bewley, and R. Coldea, Tuning the confinement potential between spinons in the Ising chain compound ${\mathrm{CoNb}}_{2}{\mathrm{O}}_{6}$ using longitudinal fields and quantitative determination of the microscopic Hamiltonian, Phys. Rev. B 108, 184416 (2023a).
  * Woodland  _et al._ [2023b] L. Woodland, I. Lovas, M. Telling, D. Prabhakaran, L. Balents, and R. Coldea, Excitations of quantum Ising chain ${\text{CoNb}}_{2}{\text{O}}_{6}$ in low transverse field: Quantitative description of bound states stabilized by off-diagonal exchange and applied field, Phys. Rev. B 108, 184417 (2023b).
  * Zhitomirsky and Chernyshev [2013] M. E. Zhitomirsky and A. L. Chernyshev, Colloquium: Spontaneous magnon decays, Rev. Mod. Phys. 85, 219 (2013).
  * Maksimov  _et al._ [2016] P. A. Maksimov, M. E. Zhitomirsky, and A. L. Chernyshev, Field-induced decays in XXZ triangular-lattice antiferromagnets, Phys. Rev. B 94, 140407(R) (2016).
  * Chernyshev and Maksimov [2016] A. L. Chernyshev and P. A. Maksimov, Damped Topological Magnons in the Kagome-Lattice Ferromagnets, Phys. Rev. Lett. 117, 187203 (2016).
  * Winter  _et al._ [2017] S. M. Winter, K. Riedl, P. A. Maksimov, A. L. Chernyshev, A. Honecker, and R. Valentí, Breakdown of magnons in a strongly spin-orbital coupled magnet, Nat. Commun. 8, 1152 (2017).
  * Maksimov and Chernyshev [2022] P. A. Maksimov and A. L. Chernyshev, Easy-plane anisotropic-exchange magnets on a honeycomb lattice: Quantum effects and dealing with them, Phys. Rev. B 106, 214411 (2022).
  * Coletta  _et al._ [2013] T. Coletta, M. E. Zhitomirsky, and F. Mila, Quantum stabilization of classically unstable plateau structures, Phys. Rev. B 87, 060407(R) (2013).
  * Coletta  _et al._ [2014] T. Coletta, S. E. Korshunov, and F. Mila, Semiclassical evidence of columnar order in the fully frustrated transverse-field Ising model on the square lattice, Phys. Rev. B 90, 205109 (2014).
  * Cônsoli  _et al._ [2020] P. M. Cônsoli, L. Janssen, M. Vojta, and E. C. Andrade, Heisenberg-Kitaev model in a magnetic field: $1/S$ expansion, Phys. Rev. B 102, 155134 (2020).
  * Khatua  _et al._ [2023] S. Khatua, M. J. P. Gingras, and J. G. Rau, Pseudo-Goldstone Modes and Dynamical Gap Generation from Order by Thermal Disorder, Phys. Rev. Lett. 130, 266702 (2023).
  * Loly [1971] P. D. Loly, The Heisenberg ferromagnet in the selfconsistently renormalized spin wave approximation, J. Phys. C: Solid State Phys. 4, 1365 (1971).
  * Schick  _et al._ [2022] R. Schick, O. Götze, T. Ziman, R. Zinke, J. Richter, and M. E. Zhitomirsky, Ground-state selection by magnon interactions in a fcc antiferromagnet, Phys. Rev. B 106, 094431 (2022).
  * Heid  _et al._ [1995] C. Heid, H. Weitzel, P. Burlet, M. Bonnet, W. Gonschorek, T. Vogt, J. Norwig, and H. Fuess, Magnetic phase diagram of CoNb2O6: A neutron diffraction study, J. Magn. Magn. Mater. 151, 123 (1995).
  * Agrapidis  _et al._ [2018] C. E. Agrapidis, J. van den Brink, and S. Nishimoto, Ordered states in the Kitaev-Heisenberg model: From 1D chains to 2D honeycomb, Sci. Rep. 8, 1815 (2018).
  * Yang  _et al._ [2020a] W. Yang, A. Nocera, T. Tummuru, H.-Y. Kee, and I. Affleck, Phase Diagram of the Spin-$1/2$ Kitaev-Gamma Chain and Emergent SU(2) Symmetry, Phys. Rev. Lett. 124, 147205 (2020a).
  * Yang  _et al._ [2020b] W. Yang, A. Nocera, and I. Affleck, Comprehensive study of the phase diagram of the spin-$\frac{1}{2}$ Kitaev-Heisenberg-Gamma chain, Phys. Rev. Res. 2, 033268 (2020b).
  * Morris  _et al._ [2021] C. M. Morris, N. Desai, J. Viirok, D. Hüvonen, U. Nagel, T. Rõõm, J. W. Krizan, R. J. Cava, T. M. McQueen, S. M. Koohpayeh, R. K. Kaul, and N. P. Armitage, Duality and domain wall dynamics in a twisted Kitaev chain, Nat. Phys. 17, 832 (2021).
  * You  _et al._ [2014] W.-L. You, P. Horsch, and A. M. Oleś, Quantum phase transitions in exactly solvable one-dimensional compass models, Phys. Rev. B 89, 104425 (2014).
  * Iaconis  _et al._ [2018] J. Iaconis, C. Liu, G. Halász, and L. Balents, Spin Liquid versus Spin Orbit Coupling on the Triangular Lattice, SciPost Phys. 4, 003 (2018).
  * Bergman  _et al._ [2007] D. L. Bergman, R. Shindou, G. A. Fiete, and L. Balents, Degenerate perturbation theory of quantum fluctuations in a pyrochlore antiferromagnet, Phys. Rev. B 75, 094403 (2007).
  * Chernyshev and Zhitomirsky [2014] A. L. Chernyshev and M. E. Zhitomirsky, Quantum Selection of Order in an $XXZ$ Antiferromagnet on a Kagome Lattice, Phys. Rev. Lett. 113, 237202 (2014).
  * Zhitomirsky [2015] M. E. Zhitomirsky, Real-space perturbation theory for frustrated magnets: application to magnetization plateaus, J. Phys.: Conf. Ser. 592, 012110 (2015).
  * Fishman  _et al._ [2022] M. Fishman, S. R. White, and E. M. Stoudenmire, The ITensor Software Library for Tensor Network Calculations, SciPost Phys. Codebases , 4 (2022).
  * Elliot  _et al._ [2021] M. Elliot, P. A. McClarty, D. Prabhakaran, R. D. Johnson, H. C. Walker, P. Manuel, and R. Coldea, Order-by-disorder from bond-dependent exchange and intensity signature of nodal quasiparticles in a honeycomb cobaltate, Nat. Commun. 12, 3936 (2021).
  * Zhitomirsky and Chernyshev [1999] M. E. Zhitomirsky and A. L. Chernyshev, Instability of Antiferromagnetic Magnons in Strong Fields, Phys. Rev. Lett. 82, 4536 (1999).
  * Mourigal  _et al._ [2010] M. Mourigal, M. E. Zhitomirsky, and A. L. Chernyshev, Field-induced decay dynamics in square-lattice antiferromagnets, Phys. Rev. B 82, 144402 (2010).
  * Chernyshev and Zhitomirsky [2009] A. L. Chernyshev and M. E. Zhitomirsky, Spin waves in a triangular lattice antiferromagnet: Decays, spectrum renormalization, and singularities, Phys. Rev. B 79, 144416 (2009).
  * Mourigal  _et al._ [2013] M. Mourigal, W. T. Fuhrman, A. L. Chernyshev, and M. E. Zhitomirsky, Dynamical structure factor of the triangular-lattice antiferromagnet, Phys. Rev. B 88, 094407 (2013).
  * Macdougal  _et al._ [2020] D. Macdougal, S. Williams, D. Prabhakaran, R. I. Bewley, D. J. Voneshen, and R. Coldea, Avoided quasiparticle decay and enhanced excitation continuum in the spin-$\frac{1}{2}$ near-Heisenberg triangular antiferromagnet ${\mathrm{Ba}}_{3}{\mathrm{CoSb}}_{2}{\mathrm{O}}_{9}$, Phys. Rev. B 102, 064421 (2020).
  * Pitaevskii [1959] L. Pitaevskii, Properties of the spectrum of elementary excitations near the disintegration threshold of the excitations, Sov. Phys. JETP 9, 830 (1959).
  * Verresen  _et al._ [2019] R. Verresen, R. Moessner, and F. Pollmann, Avoided quasiparticle decay from strong quantum interactions, Nat. Phys. 15, 750 (2019).
  * Chernyshev [2015] A. L. Chernyshev, Strong quantum effects in an almost classical antiferromagnet on a kagome lattice, Phys. Rev. B 92, 094409 (2015).
  * Gochev [1995] I. G. Gochev, Spin-wave interaction effects in the Néel phase of the ${\mathit{J}}_{1}$-${\mathit{J}}_{2}$-${\mathit{J}}_{3}$ model, Phys. Rev. B 51, 16421 (1995).
  * Yang  _et al._ [1997] J. Yang, D.-K. Yu, and J.-L. Shen, A self-consistent approach to the $J_{1}$-$J_{2}$-$J_{3}$ model, Physics Letters A 236, 589 (1997).
  * Chubukov  _et al._ [1994] A. V. Chubukov, S. Sachdev, and T. Senthil, Large-$S$ expansion for quantum antiferromagnets on a triangular lattice, J. Phys.: Condens. Matter 6, 8891 (1994).
  * [80] L. Woodland and R. Coldea, private communication. The spectrum gaps and error bars for 7 T [41] and 9 T [44] where extracted from the multi-dimensional data sets using the fitting procedures described in Ref. [44], correcting the 1D gaps for the interchain effects. The error bars in these two cases are smaller than the sizes of the symbols in Fig. 8. The data for 8 T are adapted from Ref. [43].
  * Zamolodchikov [1989] A. B. Zamolodchikov, Integrals of motion and $S$-matrix of the (scaled) $T=T_{c}$ Ising model with magnetic field, Int. J. Mod. Phys. A 04, 4235 (1989).
  * Rader and Läuchli [2018] M. Rader and A. M. Läuchli, Finite Correlation Length Scaling in Lorentz-Invariant Gapless iPEPS Wave Functions, Phys. Rev. X 8, 031030 (2018).
  * Chernyshev [2005] A. L. Chernyshev, Effects of an external magnetic field on the gaps and quantum corrections in an ordered Heisenberg antiferromagnet with Dzyaloshinskii-Moriya anisotropy, Phys. Rev. B 72, 174414 (2005).
  * [84] Typically, $5$ sweeps and a maximum bond dimension of $m\sim 10$ were needed to ensure good convergence with the truncation error of $\mathcal{O}(10^{-10})$.