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Article SPECIAL TOPIC: 2023 January 66 1 ?? 000000 January 11, 2023 April 6,
2023

Magnetism of NaYbS2: From finite temperatures to ground state

<EMAIL_ADDRESS>
Zheng Zhang

Zhuo W Z, Zhang Z, Xie M T, et al

# Magnetism of NaYbS2: From finite temperatures to ground state

Weizhen Zhuo  Zheng Zhang  Mingtai Xie
Anmin Zhang  Jianting Ji  Feng Jin  Qingming Zhang  School of Physical Science
and Technology, Lanzhou University, Lanzhou 730000, China  Beijing National
Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy
of Sciences, Beijing 100190, China

###### Abstract

Rare-earth chalcogenide compounds ARECh2 (A = alkali or monovalent metal, RE =
rare earth, Ch = O, S, Se, Te) are a large family of quantum spin liquid (QSL)
candidate materials. NaYbS2 is a representative member of the family. Several
key issues on NaYbS2, particularly how to determine the highly anisotropic
spin Hamiltonian and describe the magnetism at finite temperatures and the
ground state, remain to be addressed. In this paper, we conducted an in-depth
and comprehensive study on the magnetism of NaYbS2 from finite temperatures to
the ground state. Firstly, we successfully detected three crystalline electric
field (CEF) excitation energy levels using low-temperature Raman scattering
technique. Combining them with the CEF theory and magnetization data, we
worked out the CEF parameters, CEF energy levels, and CEF wavefunctions. We
further determined a characteristic temperature of $\sim$40 K, above which the
magnetism is dominated by CEF excitations while below which the spin-exchange
interactions play a main role. The characteristic temperature has been
confirmed by the temperature-dependent electron spin resonance (ESR)
linewidth. Low-temperature ESR experiments on the dilute magnetic doped
crystal of NaYb_0.1Lu_0.9S2 further helped us to determine the accurate
$g$-factor. Next, we quantitatively obtained the spin-exchange interactions in
the spin Hamiltonian by consistently simulating the magnetization and specific
heat data. Finally, the above studies allow us to explore the ground state
magnetism of NaYbS2 by using the density matrix renormalization group. We
combined numerical calculations and experimental results to demonstrate that
the ground state of NaYbS2 is a Dirac-like QSL.

###### keywords:

rare-earth chalcogenide, anisotropic spin Hamiltonian, quantum spin liquid

75.10.Dg, 76.30.Kg, 05.10.Cc

##  1 Introduction

The quantum spin liquid (QSL) is a novel state that the spins exhibit a high
degree of entanglement. Unlike conventional magnets, these entangled spins do
not form long-range magnetic order even down to zero temperature. This
behavior arises due to the influence of strong quantum fluctuations[1, 2, 3,
4]. Initial research efforts on QSL candidate materials primarily centered on
transition metal compounds.

For example, the copper-based magnetic material ZnCu3(OH)6Cl2[5, 6, 7, 8] with
a Kagome lattice was the first to be extensively studied as a QSL candidate
material. Inelastic neutron scattering (INS) experiments show that its ground
state has a continuous excitation spectrum related to QSL[9]. Organic
materials forming triangular lattice are also promising QSL candidates, such
as κ-(BEDT-TTF)2Cu2(CN)3[10, 11, 12] and EtMe3Sb[Pd(dmit)2]2[13, 14, 15].
Thermal conductivity experiments show that κ-(BEDT-TTF)2Cu2(CN)3 has non-zero
thermal conductivity even as the temperature approaches 0 K. This observation
suggests the presence of itinerant spinon excitations within the material.

In recent years, rare earth materials with triangular lattice have attracted
great attention. Firstly, the triangular lattice inherently possesses a
geometric frustration effect, which provides a key structure foundation for
realizing QSL. Secondly, rare earth ions have strong spin-orbit coupling
(SOC). Based on the Jackeli-Khaliullin scenario[16, 17], anisotropic spin-
exchange interactions can be produced through strong SOC. More importantly,
rare earth ions with an odd number of 4f electrons are protected by time
reversal symmetry, namely Kramers doublets. The Kramers doublets are robust
and less affected by lattice distortions or other structural defects.
YbMgGaO4[18, 19, 20, 21, 22, 23, 24] is the first discovered triangular rare
earth frustrated material satisfying the aforementioned requirements.
Experimental data indicates that YbMgGaO4 exhibits strong magnetic anisotropy.
Crucially, the energy gap between the crystalline electric field (CEF) ground
state and the first excited state in YbMgGaO4 exceeds 30 meV[19], allowing the
doubly degenerate CEF ground state to be mapped to an effective spin-1/2
model. A large amount of experimental evidence, including INS[23, 24], muon
spin relaxation ($\mu$SR)[25], and thermodynamic measurements, consistently
support the ground state of YbMgGaO4 as a gapless QSL state. Therefore,
YbMgGaO4 is considered as a typical QSL material. However, the previously
discussed materials all exhibit limitations to some extent. For example, there
are serious problems of ion vacancies and substitutions in transition metal
magnetic materials, which is related to the valence change of transition
metals. In YbMgGaO4, there is a problem of Mg and Ga disorder, which can cause
charge randomness and possibly affect the ground state magnetism[26].

In recent years, a large family of QSL candidate rare-earth chalcogenide
compounds ARECh2 (A = alkali or monovalent metal, RE = rare earth, Ch = O, S,
Se, Te) has been discovered[27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37].
Within this family, the sub-family NaYbCh2 (Ch = O, S, Se, Te) is particularly
noteworthy, and has the following advantages. Firstly, compared to YbMgGaO4,
the sub-family eliminates the Mg/Ga disorder while maintaining the perfect
triangular lattice. Secondly, through solid-phase reactions, high-quality
polycrystalline or single-crystal samples can be grown[27]. Thirdly, this sub-
family of materials has excellent two-dimensional properties, and can be
achieved with few or single layers through simple mechanical exfoliation.
Fourth, the regulation of magnetic ions or coordination ions can be easily
achieved by doping or replacement, which provides a good platform for in-depth
study of the evolution mechanism of QSL. NaYbO2 is a typical member of this
sub-family. A series of experimental results, including magnetization,
thermodynamic measurements, INS, and $\mu$SR experiments, all point to a spin
disordered ground state for NaYbO2[29, 30]. However, the lack of single
crystal samples of this material hinders a deeper study and understanding of
its ground state magnetism. NaYbSe2 is the most extensively studied in the
sub-family, an important reason being that centimeter-size single crystals can
be obtained through solid-phase reactions. The INS experiments observed
features of a spinon Fermi surface in NaYbSe2[35], which has attracted
widespread attention in the field of frustrated magnetism. Adding to its
intrigue, the energy gap from the valence band to the conduction band of
NaYbSe2 is about 1.9 eV[27], and metallization and even superconductivity can
be achieved by applying high pressure[38, 39]. This has advanced one step
further in revealing the electronic pairing mechanism of QSL and high-
temperature superconductivity. There are also many experimental reports on
NaYbS2[31, 40, 41, 37]. However, some key information about NaYbS2 is not
fully clear yet. Firstly, three CEF energy levels of NaYbS2 have not been
fully determined[31]. A quantitative description of the CEF is lacking in
NaYbS2. Secondly, despite some experimental investigation into the $g$-factor
of NaYbS2[41, 42], further refinement remains necessary. This is particularly
crucial considering the potential influence of the effective field arising
from spin-exchange interactions on the $g$-factor. Another more critical piece
of information is the lack of determination of the spin-exchange interactions
in the low-energy Hamiltonian. These interactions are very important for
studying the ground state magnetism of NaYbS2. There is also a lack of
corresponding numerical calculations for the magnetism of the ground state of
NaYbS2. Therefore, NaYbS2 deserves further and comprehensive study.

In this paper, we conducted an in-depth study on the magnetism of NaYbS2 from
finite temperatures to the ground state. Firstly, we detected the two optical
phonon peaks and three CEF excitation energy levels of NaYbS2 based on
polarized Raman spectroscopy and low-temperature Raman scattering. For the
three CEF excitations, they are at 207 cm-1 ($\sim$ 25.6 meV), 263 cm-1
($\sim$ 32.6 meV), and 314 cm-1 ($\sim$ 38.9 meV), respectively. Secondly, we
obtained the CEF parameters, excitation energy levels, and wave functions by
fitting the temperature-dependent magnetization ($M/H-T$) data of NaYbS2 under
a magnetic field of 0.1 T. In the fitting process, we also confirmed that the
characteristic temperature of NaYbS2 is 40 K. Above the characteristic
temperature, the CEF excitations play important roles in magnetism. Below the
temperature, the spin-exchange interaction dominates. ESR measurements on the
dilute doped sample of NaYb_0.1Lu_0.9S2 yielded precise $g$-factor values of
3.14 in the ab-plane and 0.86 along the c-axis. We also explained the
influence of the effective field contributed by the spin-exchange interaction
on the $g$-factor based on the mean-field (MF) theory. Next, by fitting
$M/H-T$ data under a magnetic field of 14 T below the characteristic
temperature, we obtained the diagonal spin-exchange interactions $J_{\pm}$ and
$J_{zz}$ to be 3.52 K and $-$0.87 K, respectively. The off-diagonal spin-
exchange interactions $J_{\pm\pm}$ and $J_{z\pm}$ were determined to be
$-$0.18 K and 2.66 K, respectively, by fitting the specific heat data in the
low temperature range. The most important thing is that we studied the ground
state magnetism of NaYbS2. We built the ground state phase diagram using the
density matrix renormlization group (DMRG) method. We found that spin-exchange
interactions of NaYbS2 are located in the QSL region of the phase diagram and
the $S\left(\vec{q}\right)$ in reciprocal space exhibits a broadened peak at
the high-symmetry K point. These calculations preliminarily reveal that the
ground state of NaYbS2 is a Dirac-like QSL state, consistent with the neutron
reflections and INS experiment. It also lays the foundation for further
studying the potential topological excitations of the NaYbS2 ground state.

##  2 Materials and Method

The single-crystal NaYbS2, NaLuS2 and NaYb_0.1Lu_0.9S2 were prepared using the
NaCl-flux method[27, 33]. The typical size of a single crystal is 5 $\times$ 5
$\times$ 0.05 mm3. The high quality of these single-crystal samples was
confirmed by X-ray diffraction (XRD) and energy dispersive x-ray spectroscopy
(EDX) [27](see Supporting Information S1 and S2).

$\sim$ 1.75 mg of NaYbS2 single crystals were prepared for M/H-T and magnetic
field-dependent magnetization (M-H) measurements. The anisotropic measurements
along the c-axis and in the ab-plane were performed using a Quantum Design
DynaCool Physical Property Measurement System (PPMS) with vibration sample
meter (VSM) from 1.8 to 300 K under magnetic fields ranging from 0 T to 14 T.

The zero-field heat capacity data of single crystal sample NaLuS2 ($\sim$ 0.93
mg) were performed using PPMS at 1.8 $\sim$ 100 K.

A single crystal of NaYbS2 with size of 5 $\times$ 5 $\times$ 0.05 mm3 was
used for polarized Raman spectra measurements and low-temperature Raman
scattering at 2 K. These data were collected with a Jobin Yvon LabRAM HR
Evolution spectrometer equipped with a liquid-nitrogen-cooled back-illuminated
charge-coupled device detector and a liquid-helium-free cryostat. About 1 mW
of laser power at 473 nm was focused into a spot with a diameter of $\sim$ 5
$\mu$m on the sample surface. The single crystal of NaYbS2 was mechanically
exfoliated to obtain a clean surface before measurement.

ESR measurements were performed on the single crystals of NaYbS2 and
NaYb_0.1Lu_0.9S2 with a Bruker EMX plus 10/12 continuous-wave spectrometer at
X-band frequencies (${f\sim}$ 9.39 GHz). These measurements were carried out
both in the ab-plane and along the c-axis from 1.8 K to 70 K. The angle
rotation ESR experiments were also conducted on the spectrometer from 0∘
(parallel to the ab-plane) to 90∘ (parallel to the c-axis).

We formulated MF approximation for anisotropic spin Hamiltonian and derived
self-consistent equations to simulate $M/H-T$ data. 3 $\times$ 4 spin sites FD
with periodic boundary conditions (PBC) were employed for the specific heat
simulation. We applied the DMRG method[43, 44] to calculate the ground state
phase diagram of anisotropic spin Hamiltonian and ground state magnetism of
NaYbS2. The PBC was employed to get accurate DMRG results. The cluster used in
the DMRG simulations is $L_{x}$ $\times$ $L_{y}$ = 12 $\times$ 12 and $L_{x}$
$\times$ $L_{y}$ = 15 $\times$ 15\. The truncation error was kept $\sim$ 10-5
and performed 20 sweeps to improve the accuracy of the simulations. The
largest bond dimension is $M$ $\sim$ 45000 in the DMRG calculation.

Figure 1: Raman scattering spectra and $M/H-T$ data for NaYbS2. (a) The
polarized Raman scattering spectra of NaYbS2 at 300 K. XX and XY represent
parallel and cross polarization configurations, respectively. (b) The Raman
scattering spectra of NaYbS2 at 2 K. The $M/H-T$ curves in the ab-plane (c)
and along the c-axis (d) under a magnetic field of 0.1 T. The red open circles
and blue solid lines are experimental data and simulation results based on the
CEF theory, respectively.

##  3 Effective magnetic Hamiltonian

The electron configuration of 4$f$ orbital in Yb^3+ is $4f^{13}$. Following
the Hund’s rules, there emerges a spectral term ${}^{2}I_{7/2}$ featuring an
8-fold degeneracy and another spectral term ${}^{2}I_{5/2}$ with a 10-fold
degeneracy after SOC. The energy gap between the two spectral terms is
approximately 1 eV [45], suggesting that transitions between SOC energy levels
can be neglected even at room temperatures. Consequently, the spectral term
${}^{2}I_{7/2}$ denotes the SOC ground state configuration. Considering the
shielding effect of the $6s$ and $5p$ orbitals on the $4f$ electrons, the CEF
environment formed by surrounding $S^{2-}$ ligands with $D_{3d}$ point group
symmetry plays a crucial role in NaYbS2. In this CEF environment, Yb^3+ ions
with an odd number of $4f$ electrons split into four pairs of doubly
degenerate Kramers states, protected by time-reversal symmetry. The INS
experiments on NaYbS2 observed the first CEF excitation at approximately 23
meV ($\sim$ 270 K)[31]. This implies that excitation effects of the CEF cannot
be ignored when the temperature is comparable to the excitation energy. In the
limit of temperatures significantly lower than the first CEF energy level, the
influence of excitations becomes negligible. This allows for the construction
of an effective spin-1/2 model capturing the essential physics. Within this
model, spin-exchange interactions, correlations, and quantum fluctuations take
center stage.

Based on the understanding of the Yb^3+ in NaYbS2 described above, we can
construct a magnetic effective Hamiltonian to describe the magnetism of
NaYbS2. NaYbS2 and NaYbSe2[33, 34] share the same triangular lattice formed by
Yb^3+ ions due to their isostructural crystal structure. Therefore, the
magnetic effective Hamiltonian used to describe the magnetism of NaYbS2 within
the temperature range we studied is represented as follows[46, 34]:

$\displaystyle\hat{H}_{\mathrm{eff}}=$
$\displaystyle\hat{H}_{\mathrm{CEF}}+\hat{H}_{\mathrm{spin}-\text{ spin
}}+\hat{H}_{\text{Zeeman }}$ $\displaystyle=$
$\displaystyle\sum_{i}\sum_{m,n}B_{m}^{n}\hat{O}_{m}^{n}$
$\displaystyle+\sum_{\langle
ij\rangle}\left[J_{zz}S_{i}^{z}S_{j}^{z}+J_{\pm}\left(S_{i}^{+}S_{j}^{-}+S_{i}^{-}S_{j}^{+}\right)\right.$
$\displaystyle+J_{\pm\pm}\left(\gamma_{ij}S_{i}^{+}S_{j}^{+}+\gamma_{ij}^{*}S_{i}^{-}S_{j}^{-}\right)$
$\displaystyle\left.-\frac{iJ_{z\pm}}{2}\left(\gamma_{ij}S_{i}^{+}S_{j}^{z}-\gamma_{ij}^{*}S_{i}^{-}S_{j}^{z}+\langle
i\longleftrightarrow j\rangle\right)\right]$
$\displaystyle-\mu_{0}\mu_{B}\sum_{i}\left[g_{ab}\left(h_{x}S_{i}^{x}+h_{y}S_{i}^{y}\right)+g_{c}h_{c}S_{i}^{z}\right],$
(1)

where $B_{m}^{n}$ denotes the CEF parameters and $O_{m}^{n}$ symbolizes the
Steven operator, which is constructed based on the angular momentum $\hat{J}$
after the SOC. $S_{i}^{\alpha}(\alpha=x,y,z)$ is the effective spin-1/2
operators at site $i$ and $S_{i}^{\pm}=S_{i}^{x}\pm iS_{i}^{y}$ are the non-
Hermitian ladder operators. The nearest neighbor (NN) anisotropic spin-
exchange interactions are denoted by $J_{\pm}$, $J_{zz}$, $J_{\pm\pm}$, and
$J_{z\pm}$. The phase factor $\gamma_{ij}$ is taken as 1,
$\mathrm{e}^{i\frac{2\pi}{3}}$ and $\mathrm{e}^{-i\frac{2\pi}{3}}$ along the
three bonds $\vec{a}_{1},\vec{a}_{2}$ and $\vec{a}_{3}$. The Landé $g$-factors
in the ab-plane and along the c-axis are represented by $g_{ab}$ and $g_{c}$,
respectively.

It is noteworthy that the CEF part and the anisotropic spin-exchange
interactions part in the magnetic effective Hamiltonian do not exist in the
same Hilbert space. The CEF description is particularly effective for
capturing the high-temperature magnetism dominated by CEF excitations.
Conversely, the spin-exchange interaction model is suitable for describing
low-energy spin excitations within the CEF ground state. Therefore, the
analysis of magnetization and thermodynamic data needs to be discussed
separately in two parts. In the following, we would apply the above strategy
to study the magnetism of NaYbS2 from finite temperatures to the ground state.

##  4 CEF and Raman spectra

For the study of CEF excitations in rare-earth materials, INS offers a
powerful capability. This technique can directly probe CEF energy levels that
are independent of momentum transfer $\vec{Q}$. Besides, Raman scattering
provides another valuable tool for investigating CEF excitations. Compared to
INS, Raman scattering offers the advantage of more convenient laboratory-based
measurements of CEF excitations. In fig. 1(b), we show the Raman spectrum of
NaYbS2 at 2 K. From the spectrum, we can clearly observe five excitation peaks
in the range of 100 to 350 cm-1. Among these peaks, we can divide them into
two parts: one part corresponds to the phonon peaks related to optical branch
lattice vibrations, and the other part corresponds to the CEF peaks associated
with CEF excitations. By employing both symmetry analysis and polarization
scattering experiments, we can determine the vibrational modes associated with
the observed phonon peaks. With the spatial symmetry of $R\overline{3}m$,
Raman scattering can detect two phonon modes in NaYbS2, namely $A_{g}$ and
$E_{g}$ modes. The Raman scattering tensors of $A_{g}$ and $E_{g}$ modes tell
us that the $A_{g}$ mode is visible only in the parallel polarization (XX)
configuration, while the $E_{g}$ mode can be observed in both XX and cross
polarization (XY) configurations. The polarized Raman spectrum of NaYbS2 at
300 K are shown in fig. 1(a). The two modes can be clearly identified and
consistent with our symmetry analysis.

Table 1: The fitting results of CEF ${B^{n}_{m}}$ parameters, energy and wave functions for $\mathrm{NaYbS_{2}}$ | ${B^{0}_{2}}$ | ${B^{0}_{4}}$ | ${B^{3}_{4}}$ | ${B^{0}_{6}}$ | ${B^{3}_{6}}$ | ${B^{6}_{6}}$
---|---|---|---|---|---|---
${B^{n}_{m}}$ (meV) | ${-1.409\times 10^{-1}}$ | ${2.302\times 10^{-3}}$ | ${-2.175\times 10^{-1}}$ | ${6.478\times 10^{-4}}$ | ${-5.230\times 10^{-3}}$ | ${1.765\times 10^{-2}}$
Energy (meV) | Wave function |
0.0 | $-0.639(1)\left|-\frac{7}{2}\right\rangle$ | $+0.113(6)\left|-\frac{5}{2}\right\rangle$ | $-0.353(9)\left|-\frac{1}{2}\right\rangle$ | $+0.060(8)\left|\frac{1}{2}\right\rangle$ | $+0.661(5)\left|\frac{5}{2}\right\rangle$ | $-0.109(8)\left|\frac{7}{2}\right\rangle$
0.0 | $-0.109(8)\left|-\frac{7}{2}\right\rangle$ | $-0.661(5)\left|-\frac{5}{2}\right\rangle$ | $-0.060(8)\left|-\frac{1}{2}\right\rangle$ | $-0.353(9)\left|\frac{1}{2}\right\rangle$ | $+0.113(6)\left|\frac{5}{2}\right\rangle$ | $+0.639(1)\left|\frac{7}{2}\right\rangle$
24.449(6) | $-0.038(8)\left|-\frac{7}{2}\right\rangle$ | $-0.366(8)\left|-\frac{5}{2}\right\rangle$ | $-0.598(3)\left|-\frac{1}{2}\right\rangle$ | $+0.613(7)\left|\frac{1}{2}\right\rangle$ | $-0.357(5)\left|\frac{5}{2}\right\rangle$ | $-0.039(8)\left|\frac{7}{2}\right\rangle$
24.449(6) | $-0.039(8)\left|-\frac{7}{2}\right\rangle$ | $+0.357(5)\left|-\frac{5}{2}\right\rangle$ | $-0.613(7)\left|-\frac{1}{2}\right\rangle$ | $-0.598(3)\left|\frac{1}{2}\right\rangle$ | $-0.366(8)\left|\frac{5}{2}\right\rangle$ | $+0.038(8)\left|\frac{7}{2}\right\rangle$
32.290(9) |  |  | $+0.185(3)\left|-\frac{3}{2}\right\rangle$ | $-0.982(7)\left|\frac{3}{2}\right\rangle$ |  |
32.290(9) |  |  | $-0.982(7)\left|-\frac{3}{2}\right\rangle$ | $-0.185(3)\left|\frac{3}{2}\right\rangle$ |  |
39.622(7) | $+0.030(2)\left|-\frac{7}{2}\right\rangle$ | $+0.535(4)\left|-\frac{5}{2}\right\rangle$ | $-0.014(7)\left|-\frac{1}{2}\right\rangle$ | $+0.369(1)\left|\frac{1}{2}\right\rangle$ | $+0.021(3)\left|\frac{5}{2}\right\rangle$ | $+0.758(6)\left|\frac{7}{2}\right\rangle$
39.622(7) | $-0.758(6)\left|-\frac{7}{2}\right\rangle$ | $+0.021(3)\left|-\frac{5}{2}\right\rangle$ | $+0.369(1)\left|-\frac{1}{2}\right\rangle$ | $+0.014(7)\left|\frac{1}{2}\right\rangle$ | $-0.535(4)\left|\frac{5}{2}\right\rangle$ | $+0.030(2)\left|\frac{7}{2}\right\rangle$

In addition to the two phonon peaks, three additional scattering peaks were
detected at 207 $\mathrm{cm^{-1}}$ ($\sim$ 25.6 meV), 263
$\mathrm{cm^{-1}}$($\sim$ 32.6 meV), and 314 $\mathrm{cm^{-1}}$ ($\sim$ 38.9
meV). The scattering peaks at 207 cm-1 and 314 cm-1 are close to the CEF
excitations measured by INS around 23 meV and 39 meV[31]. The scattering peak
at 263 cm-1 closely matches another weak peak observed around 32 meV in INS,
initially attributed to additional peak caused by impurities in the powder
samples[31]. After careful analysis, we can confirm that the scattering peak
at 263 cm-1 is caused by CEF excitation. First, the single crystal samples
prepared for the Raman scattering experiments were obtained using mechanical
exfoliation to ensure clear surface. Therefore, we can rule out the influence
of impurities in the samples. Second, the scattering peak near 263 cm-1 can
still be observed at 300 K despite weakening. The peak position does not shift
significantly with temperature, which is consistent with the rule of CEF
excitation changing with temperature. Third, a weak excitation peak, also
exhibiting momentum independence, was detected by INS at 32 meV[31]. Based on
the analysis of our Raman scattering experiments, we can determine that the
three CEF excitations levels of NaYbS2 are 25.6, 32.6, and 38.9 meV. In
addition, the three CEF energy levels determined by fitting the $M/H-T$ data
of NaYbS2 (see below) are consistent with the CEF excitation energy levels
measured by Raman scattering. Cross-validation of these experimental data
enhances our confidence in using Raman scattering to determine the three CEF
excitation energy levels of NaYbS2.

Meanwhile, we found that the phonon peak of $E_{g}$ mode has a strong
asymmetry, which is mainly caused by the CEF-phonon coupling. We will conduct
a more detailed analysis starting from the microscopic model in our future
work.

Figure 2: The ESR data of NaYbS2 and NaYb_0.1Lu_0.9S2. (a) and (b) ESR
differential spectra of NaYbS2 in the ab-plane and along the c-axis. (e) and
(f) ESR differential of NaYb_0.1Lu_0.9S2 in the ab-plane and along the c-axis.
(c) and (g) Angle-dependent $g$-factor of NaYbS2 and NaYb_0.1Lu_0.9S2 at low
temperature. (d) and (h) Temperature-dependent ESR linewidth of NaYbS2 and
NaYb_0.1Lu_0.9S2 in the ab-plane. The red circles are experimental data and
the blue solid lines are simulation results. Figure 3: $M/H-T$ and $M-H$ data
of NaYbS2. The $M/H-T$ data in the ab-plane (a) and along the c-axis (b) under
a magnetic field of 14 T. The $M-H$ data in the ab-plane and along the c-axis
at 10 K. (e) and (f) Error mapping of $M/H-T$ fitting in the ab-plane and
along the c-axis. (c) and (d) share the same color scale in the right corner.
Red circles are experimental data and the blue solid lines are simulation
results. The intersection marked by the white dashed lines is the optimal
position with minimal error.

##  5 Magnetism of NaYbS2 at finite temperatures

In the previous section, we conducted detailed investigations on the CEF
excitations associated with NaYbS2 based on the low-temperature Raman
scattering experiment, accurately determining the three CEF excitation energy
levels of NaYbS2. This lays foundations for studying the magnetism of NaYbS2
at finite temperatures. In this section, we present a discussion on the
magnetism of NaYbS2 at finite temperatures, including the effect of CEF
excitations and the influence of low-energy spin-exchange interactions on
magnetism.

We first focus on the contribution of CEF excitations to magnetism. At high
temperature ranges, thermal fluctuations primarily influence magnetization
through a series of CEF excitation levels within individual ions. When the
temperature is comparable to the CEF excitation energy levels, the measured
magnetization data are mainly contributed by the CEF excitations. Based on the
CEF energy levels obtained from Raman scattering of NaYbS2 and our measured
$M/H-T$ data, we can simulate the $M/H-T$ data by employing magnetization
calculation formulas[20, 47]. We used a nonlinear least squares method to fit
the contributions of CEF to $M/H-T$ in the ab-plane and along the c-axis. The
CEF parameters obtained from the point charge model were used as initial
values, and three CEF levels were set as constraints. The experimental data of
$M/H-T$ and the simulation results based on the CEF theory are shown in fig.
1(c) and fig. 1(d). A clear discrepancy arises between the simulation results
and experimental data below 40 K both in the ab-plane and along the c-axis.
This indicates that the magnetism is mainly contributed by CEF excitations
above 40 K. In contrast, the magnetism is predominantly governed by the
anisotropic spin-exchange interactions below 40 K. Therefore, we can define 40
K as the characteristic temperature of NaYbS2. Analyzing the temperature-
dependent ESR linewidths can also help us determine the characteristic
temperature associated with CEF excitations. fig. 2(d) and fig. 2(h) show the
temperature-dependent of the linewidth for NaYbS2 and NaYb_0.1Lu_0.9S2
samples, respectively. Both figures show that the ESR linewidth rapidly
broadens with increasing temperature above 40 K. This is consistent with the
results we obtained by simulating the $M/H-T$ data. Furthermore, by fitting
the ESR linewidth above 40 K with the following formula, we can also estimate
the energy gap from the CEF ground state to the first excited state of NaYbS2.

$\mu_{0}\Delta H\simeq\frac{1}{\mathrm{exp}\left(\Delta E/T\right)-1}$ (2)

where $\Delta H$ is the full width at half maximum (FWHM), $\Delta E$ is the
energy gap, and $T$ is the temperature. The fitting results are shown as the
blue solid lines in fig. 2(d) and fig. 2(h), respectively. The $\Delta E$
obtained by fitting estimation is 263 K ($\sim$ 22.67 meV), which is close to
the CEF first excitation energy level measured by Raman scattering. The
characteristic temperature of NaYbS2 is obviously higher than that of NaYbSe2
(25 K)[34]. The primary reason is that NaYbS2 has higher CEF excitation energy
levels compared to NaYbSe2, which fundamentally stems from the stronger
electronegativity of the S^2- anion.

In addition, we also obtained the CEF parameters, excitation energy levels,
and corresponding wave functions during the non-linear fitting process, as
shown in table 1. Based on the fitted CEF parameters, the three CEF excitation
energy levels obtained through diagonalization calculations are 24.45, 32.29,
and 39.62 meV, which are very close to our Raman scattering results. This
further demonstrates the reliability of the CEF-related information obtained
from our fitting and calculations.

In the limit of temperatures significantly lower than the first CEF excitation
energy, the doubly degenerate CEF ground state can be mapped onto an effective
spin-1/2 model. When considering the influence of the magnetic field on the
magnetic moment, two additional key parameters arise: the Landé $g$-factors in
the ab-plane and along the c-axis. The $g$-factor not only reflects the
magnetic anisotropy but also serves as a bridge connecting the CEF ground
state and the effective spin-1/2 model. Therefore, accurately determining the
$g$-factors of NaYbS2 is important. The methods to determine $g$-factors
mainly include two approaches: one method is to obtain the $g$-factors from
the saturation magnetization by applying a magnetic field at low temperature,
and the other is to directly obtain them from the resonance magnetic field in
the ESR spectrum. Both method have their own advantages and disadvantages. For
the saturation magnetization method, there is no need to consider the
influence of effective internal field caused by spin-exchange interactions on
the $g$-factors[48]. However, if the interaction between spins is strong or
exhibits significant paramagnetism, the magnetic moment may require a very
high magnetic field to saturate. For the ESR method, the $g$-factor can be
measured by a weak resonant magnetic field, but it is easily affected by the
internal field. This leads to the $g$-factor measured by ESR to deviate from
the true value. If the $g$-factor of magnetic ions is small, it can also be
influenced by spin-exchange interactions, causing the observed ESR linewidth
to exceed the range of the spectrometer, and thereby limiting the ability to
fully capture the ESR resonance spectrum. This problem exists in the ESR
spectrum along the c-axis of NaYbS2, as shown in fig. 2(b). Therefore, we
propose to determine the accurate $g$-factors by measuring the ESR spectrum of
a dilute magnetic doped sample. In NaYb_0.1Lu_0.9S2, spin-exchange
interactions are almost nonexistent, thus the $g$-factors obtained from ESR
measurements are not affected by spin-exchange interactions. In fig. 2(e) and
fig. 2(f), we respectively present the temperature-dependent ESR resonance
spectra of NaYb_0.1Lu_0.9S2 in the ab-plane and along the c-axis from 1.8 K to
50 K. In dilute magnetic doping samples, issues such as imperfect sample
homogeneity and magnetic ion vacancies are inevitably introduced. The ESR is
highly sensitive to these issues, resulting in some subtle features in fig.
2(e). Based on Lorentzian fitting, we determine the $g$-factors corresponding
to the resonance field as $g_{ab}=3.14$ and $g_{c}=0.86$. By substituting the
determined $g_{ab}$ and $g_{c}$ into the angle-dependent $g$-factor formula
(see Supporting Information S4), we can simulate the experimental measurement
results of the $g$-factor at different angles.

To further check the accuracy of our calculations, we also calculated the
$g$-factors in the ab-plane and along the c-axis based on the CEF ground
state. The expression of the calculation formula is as follows:[33]:

$\displaystyle\begin{gathered}g_{ab-\text{plane}}=g_{j}\left|\left\langle\psi_{\pm}\left|\hat{J}^{\pm}\right|\psi_{\mp}\right\rangle\right|,\\\
g_{c-\text{axis}}=2g_{j}\left|\left\langle\psi_{\pm}\left|\hat{J}^{z}\right|\psi_{\pm}\right\rangle\right|,\end{gathered}$
(5)

where $g_{j}=8/7$ and $|\psi_{\pm}\rangle$ are the Landé $g$-factor of free
Yb^3+ and the CEF ground state. We can obtain the $g_{ab}\sim 3.13$ in the ab-
plane and $g_{c}\sim 0.88$ along the c-axis. This is highly consistent with
the results obtained from the ESR spectrum of NaYb_0.1Lu_0.9S2. On the one
hand, it demonstrates the reliability of the $g$-factors obtained from the
dilute magnetic doped samples. On the other hand, it validates the accuracy of
the CEF parameters obtained through fitting the $M/H-T$. We can compare the
$g$-factors of NaYb_0.1Lu_0.9S2 with those of NaYbS2. We found that the
$g$-factor of NaYb_0.1Lu_0.9S2 is not significantly different from that of
NaYbS2 ($g_{ab}$ = 3.19) in the ab-plane, but the $g$-factor of
NaYb_0.1Lu_0.9S2 is significantly larger than that of NaYbS2 along the c-axis.
The main reasons for this are as follows: In the ab-plane, the spin-exchange
interaction $J_{\pm}$ is antiferromagnetic type (see below), and the internal
field induced by the antiferromagnetic interaction can counteract each other.

Figure 4: Specific heat of NaYbS2 (a) Specific heat and magnetic specific heat
of NaYbS2. The light red squares represent the raw data of the specific heat
of NaYbS2. The red circles represent the magnetic specific heat of NaYbS2. The
green dashed line is the calculated CEF specific heat, and the orange solid
line represents the results of the lattice vibrational specific heat obtained
by fitting the specific heat of NaYbS2 at temperature range of 20 to 30 K. (b)
Magnetic entropy of NaYbS2. (c) $C_{mag}/T$ of NaYbS2 below 10 K and FD
simulation results. The red circles are experimental data, and blue solid line
is FD simulation result.

The ferromagnetic spin-exchange interaction $J_{zz}$ along the c-axis induces
an effective internal field that aligns with the applied magnetic field.
Consequently, the superposition of this internal field and the external field
leads to a smaller $g_{c}$. Based on the MF Hamiltonian along the c-axis, we
can estimate the magnitude of the internal field. The MF Hamiltonian along the
c-axis is represented as follows:

$H_{MF,c-axis}=-g_{c}\mu_{0}\mu_{B}\left(h_{c}-\frac{6J_{zz}m_{c}}{\mu_{B}^{2}g_{c}^{2}}\right)\sum_{i}S_{i}^{z},$
(6)

where $m_{c}$ represents the magnetic moment contributed by the spin-exchange
interaction $J_{zz}$. The relationship between $g_{c}$, resonance frequency,
and resonance magnetic field, ignoring spin-exchange interaction $J_{zz}$, can
be expressed as follows:

$g_{c}=\frac{h\nu}{\mu_{0}\mu_{B}h_{c}},$ (7)

where $h$ is planck’s constant and $\nu$ is resonance frequency. Taking into
account the effect of spin-exchange interaction $J_{zz}$ on the $g$-factor,

$g_{c}^{\prime}=\frac{h\nu}{\mu_{0}\mu_{B}\left(h_{c}-\frac{6J_{zz}m_{c}}{\mu_{B}^{2}g_{c}^{2}}\right)}.$
(8)

By the ratio of Eq. (5) to (6), we can determine the effective magnetic field
of NaYbS2 contributed by the spin-exchange interaction $J_{zz}$ as

$h_{\mathrm{eff}}=\frac{6J_{zz}m_{c}}{\mu_{B}^{2}g_{c}^{2}}=\left(1-\frac{g_{c}}{g_{c}^{\prime}}\right)h_{c}=-\text{0.14
T}$ (9)

The negative sign indicates that the exchange interaction $J_{zz}$ along the
c-axis is ferromagnetic type, aligning with $M/H-T$ data fitting results at
lower temperatures (see below).

So far, we have accurately determined the $g$-factors in the ab-plane and
along the c-axis through ESR experiments on NaYb_0.1Lu_0.9S2. We have also
explained the reasons for the differences in the $g$-factors between
NaYb_0.1Lu_0.9S2 and NaYbS2. Next, we can analyze the magnetization and
specific heat data below the characteristic temperature based on the MF theory
and FD method. By fitting these data, we can extract the spin-exchange
interactions in the spin Hamiltonian, laying the foundation for further
investigation into the ground state magnetism of NaYbS2. Due to the
limitations of the MF spin Hamiltonian, which is suitable for paramagnetic
states, selecting an appropriate magnetic field and a reasonable temperature
range is crucial before fitting and simulating the magnetization data. For
NaYbS2, magnetic fields can induce a series of phase transitions[37]. From the
$M-H$ data of NaYbS2 at 0.8 K and the magnetic field-temperature ($H-T$) phase
diagram, we can see that the magnetic moment gradually saturates as the
magnetic field exceeds 12 T. Meanwhile, $\mu$SR experiments on NaYbS2[40]
reveal that spin fluctuations become insignificant above 8 K. Therefore, we
can confidently select the temperature range of 10 to 30 K under a magnetic
field of 14 T to fit the $M/H-T$ data. In the fitting process, we can extract
the diagonal terms of the spin-exchange interactions $J_{\pm}$ and $J_{zz}$ in
the MF Hamiltonian.

Figure 5: Ground phase diagram, Static spin structure factor $S(\vec{q})$ in
reciprocal space, and magnetic structure. (a) Ground state phase diagram
scaled by energy. (b) Ground state phase diagram scaled by entanglement
entropy. The white dash lines are boundary of different phases. The red
pentagon marks the position of NaYbS2 in the phase diagram. (c) and (e)
$S(\vec{q})$ and magnetic structure of AFM-120∘. (d) and (f) $S(\vec{q})$ and
magnetic structure of stripe-x. (g) and (i) $S(\vec{q})$ and magnetic
structure of stripe-yz. (h) and (j) $S(\vec{q})$ and magnetic structure of
QSL. (c), (d), (g), and (h) share the same color scale in the right.

fig. 3(a) and fig. 3(b) present the $M/H-T$ experimental data and fitting
results, including both in the ab-plane and along the c-axis. From the two
figures, we can see that the simulated results are consistent with the
experimental measurements within the selected temperature range. However, the
simulated results gradually deviate from the experimental data as the
temperature drops below 10 K. The fitting error ($R_{p}$) for the $M/H-T$ data
is presented in fig. 3(e) and fig. 3(f). By searching for the minimum of the
fitting error, we can determine $J_{\pm}$, $J_{zz}$, and the Van Vleck (VV)
paramagnetism at low temperatures. In the ab-plane, we obtained $J_{\pm}$ =
3.52 K, $\chi_{VV}$ = 0.0062 emu Oe-1 mol-1, and $R_{p}$ = 0.19 %. Along the
c-axis, we obtained $J_{zz}$ = -0.87 K, $\chi_{VV}$ = 0.0168 emu Oe-1 mol-1,
and $R_{p}$ = 0.40 %. We double-checked these spin-exchange interactions by
comparing the $M-H$ experimental data with MF simulation results at 10 K. The
$M-H$ data shown in fig. 3(c) and fig. 3(d) are consistent with the simulation
results. From the fitting results, we found that NaYbS2 exhibits
antiferromagnetic-type spin-exchange interaction and a weaker VV paramagnetism
in the ab-plane. In contrast, it exhibits ferromagnetic-type spin-exchange
interaction and a more significant VV paramagnetism along the c-axis. The
paramagnetism also results in nearly linear growth of the magnetic moment
along the c-axis under high magnetic fields[31]. Besides, spin-exchange
interactions can also reflect in the ESR linewidth. As the temperature
decreases, the spin-exchange interactions can lead to broadening of the ESR
linewidth. We can use the following empirical formula to fit the ESR linewidth
of NaYbS2 at low temperature range,

$\mu_{0}\Delta H\simeq\frac{A}{\left(T-T_{\theta}\right)^{p}},$ (10)

where $A$ is the proportionality coefficient, $p$ is the power coefficient,
and $T_{\theta}$ is the Weiss temperature. The Weiss temperature and spin-
exchange interaction $J_{\pm}$ in the ab-plane have the following
relationship, which is $T_{\theta}=-3J_{\pm}$. By substituting the spin-
exchange interaction $J_{\pm}$ into the above formulas to simulate the ESR
linewidth in the ab-plane, it can match the experimental data well (see fig.
2(d)). This further indicates that the broadening of the ESR linewidth of
NaYbS2 at low temperatures is caused by the spin-exchange interaction.

For the other two off-diagonal spin-exchange interactions $J_{\pm\pm}$ and
$J_{z\pm}$, we can obtain them through fitting the zero-field specific heat
data. We select the specific heat of NaYbS2 in the temperature range from 0.3
to 10 K[31]. To subtract the contribution of lattice vibrations to the
specific heat, we adopted the following strategy. Considering that NaYbS2 is
an insulator, when the temperature is significantly greater than the spin-
exchange interactions ($\geq$ 20 K), the contributions to the specific heat of
NaYbS2 are primarily from two sources: lattice vibrations and CEF excitations.
We have determined the CEF parameters, energy levels, and wave functions of
NaYbS2. Therefore, using thermodynamic formulas, we can directly calculate the
contribution of CEF excitations to the specific heat. The calculation results
(green dashed line fig. 4(a)) indicate that the contribution of CEF
excitations to the specific heat does not exceed 0.1 J mol-1 K-1 within the
temperature range of 20 to 30 K. Therefore, we can conclude that the specific
heat is essentially due to lattice vibrations within this temperature range.
To obtain the contribution of lattice vibration to the specific heat of
NaYbS2, we employed the Debye formula to fit the specific heat data of NaYbS2
at the temperature range of 20 to 30 K. The fitting results are shown by the
orange solid line in fig. 4(a). After subtracting the contribution of lattice
vibrations to the specific heat, we obtain the magnetic specific heat of
NaYbS2, as indicated by the red circles in fig. 4(a). By integrating the
magnetic specific heat of NaYbS2 over temperature, we can obtain the magnetic
entropy of NaYbS2, as shown in fig. 4(b). The figure reveals that the magnetic
entropy remains nearly constant above 15 K, yet it has not fully reached
$R$ln2, primarily due to the absence of data for lower temperatures ($\leq$
0.35 K). We also use the Debye formula to fit the specific heat data of NaLuS2
from 1.8 to 20 K (see Supporting Information S7). Next, we can extract off-
diagonal spin interactions from the magnetic specific heat data by selecting
appropriate numerical calculation method. We employ the FD to fit the magnetic
specific heat. The method can preserve off-diagonal spin-exchange
interactions. fig. 4(c) presents the experimental data of $C_{mag}/T$ and the
simulation results obtained for a 3 $\times$ 4 spin lattice with PBC. In
comparison to the experimental data, the simulation results are consistent
with the experimental data above 0.8 K. A noticeable deviation emerges below
0.8 K. The deviation primarily stems from the limitations imposed by the
finite lattice size. Longer-range spin correlations and quantum entanglement
beyond the 3 $\times$ 4 spin lattice are neglected. The off-diagonal spin-
exchange interactions $J_{\pm}$ and $J_{z\pm}$ are determined to be $-$0.18
and 2.66 K, respectively. With all parameters in the anisotropic spin-exchange
interaction Hamiltonian now determined, we are poised for a comprehensive
exploration of NaYbS2’s ground-state magnetism.

Figure 6: Neutron reflections[37] and calculated results of static spin
structure factor $S\left(\vec{q}\right)$ at $\vec{Q}$ = (1/3, 1/3, 0) for
NaYbS2 ground state under different magnetic fields. (a) and (c) The
comparison of the neutron reflections at (1/3, 1/3, 0) between 0 T and 6 T at
0.1 K and the magnetic field-dependence of (1/3, 1/3, 0), respectively. (b)
and (d) The comparison of the $S\left(\vec{q}\right)$ at (1/3, 1/3) between 0
T and 6 T at ground state and the magnetic field-dependence of (1/3, 1/3),
respectively.

##  6 Ground state magnetism

In this section, we employed the DMRG method to investigate the ground state
magnetism of NaYbS2. The PBC was employed to get a accurate matrix product
state (MPS). After benchmarking by selecting clusters of different sizes, we
finally determined that there are no significant size effects for the $L_{x}$
$\times$ $L_{y}$ = 12 $\times$ 12 cluster. In order to determine the location
of NaYbS2 in the phase diagram, we first calculated the ground state phase
diagram. We set $J_{zz}$ as a dimensionless parameter of 1 and fixed the
dimensionless parameters $J_{\pm}/J_{zz}$ = $-$4.05. The parameter range for
$J_{\pm\pm}/J_{zz}$ was chosen to be from $-$2 to 2, and the parameter range
for $J_{z\pm}/J_{zz}$ is from 0 to 6. Eventually, we obtained the two-
dimensional ground state phase diagram. fig. 5(a) and fig. 5(b) respectively
display the ground state phase diagrams scaled by ground state energy and
bipartite entanglement entropy. The phase boundaries marked by white dashed
lines in the ground state phase diagram are determined by bipartite
entanglement entropy, the arrangement of ground state magnetic moments, and
the static spin structure factor $\left\langle S_{i}S_{j}\right\rangle$.
Overall, the ground state phase diagram is in good agreement with existing
results from the classical Monte Carlo simulations[46, 49], exact
diagonalization[49, 50], and DMRG calculations[49, 51, 52]. We can divide the
phase diagram into four regions: stripe-x, stripe-yz, AFM-$120^{\circ}$, and
QSL phase.

fig. 5(e), fig. 5(f), and fig. 5(i) show the periodic arrangement of static
magnetic moments for AFM-$120^{\circ}$, stripe-x, and stripe-yz, respectively.
For the stripe-x phase, the magnetic moments on each point are arranged in
opposite directions along the edge of the triangle lattice. The
$S\left(\vec{q}\right)$ in reciprocal space shows sharp peak located at the
high-symmetry point $M$ in the Brillouin zone (see fig. 5(d)). For the stripe-
yz phase, similar to the stripe-x phase, the magnetic moments are arranged in
opposite directions along the perpendicular direction of the triangular
lattice. The peaks of $S\left(\vec{q}\right)$ are also located at the $M$
points (see fig. 5(g)). The bipartite entanglement entropy can provide us with
more information. Compared to the QSL state and the AFM-$120^{\circ}$ phase,
the entanglement entropy of the two stripe phases is close to zero. This
implies that the quantum fluctuations of these two stripe phases are weak. The
NN spin correlation $\left\langle S_{i}S_{j}\right\rangle$ also exhibits
strong periodicity in the two stripe phases (see Supporting Information S10).
Each spin lattice point forms a negative (blue), negative, and positive (red)
correlation structure with the NN spin lattice points along the $a_{1}$,
$a_{2}$, and $a_{3}$ directions, respectively. AFM-$120^{\circ}$ is not a
trivial quantum state, even though the magnetic moments form a long-range
periodic arrangement in the ab-plane. From the view of entanglement entropy,
the bipartite entanglement of the AFM-$120^{\circ}$ phase is not low,
indicating the presence of strong quantum fluctuations within this regime. The
NN spin correlation $\left\langle S_{i}S_{j}\right\rangle$ also exhibits some
interesting properties (see Supporting Information S10). The bonds with
stronger correlations form a periodic hexagonal structure, while the spin site
located at the center of the hexagonal structure has weak correlations with
the surrounding NN spin sites. This means that these central spins can
generate confined spinon excitations with strong fluctuations. It is possible
to realize a deconfined ”liquid” phase by introducing next-nearest-neighbor or
non-diagonal interactions. More interestingly, AFM-$120^{\circ}$ phase can
simultaneously break the translational symmetry of the triangular lattice and
the U(1) spin rotation symmetry within a certain range of spin-exchange
interactions[53, 54, 55, 56]. The breaking of these two symmetries allows the
spin system to keep strong quantum fluctuations, which is called a spin
supersolid. Both spin wave excitations and continuum excitations can be
observed in INS spectra.

Compared to ordered phases, we are more interested in the QSL phase located in
the central region of the ground state diagram. The phase has the strongest
entanglement entropy in the phase diagram, which means that the spin system
has strong quantum fluctuations due to anisotropic spin-exchange interactions.
The red star marks the position of NaYbS2 in the phase diagram which is
located within the QSL phase region. A series of related experiments,
including neutron diffraction, INS, and $\mu$SR, have confirmed that the
ground state of NaYbS2 is a quantum spin liquid state. We attempted to
understand the ground state of NaYbS2 from numerical calculations. We plotted
the arrangement of magnetic moments in real space and the
$S\left(\vec{q}\right)$ in reciprocal space calculated by DMRG, as shown in
fig. 5(j) and fig. 5(h). From the arrangement of magnetic moments, we found
that the magnetic structure cannot be described by a single magnetic
propagation vector, thus exhibiting the characteristics of spin disorder. Such
a magnetic configuration is more like a distorted AFM-$120^{\circ}$ phase,
which also indicates that off-diagonal spin-exchange interactions play a key
role in the formation of spin disorder. More importantly,
$S\left(\vec{q}\right)$ exhibits a broadened peak at the high-symmetry K
point. One characteristic of Dirac-like spin liquids is the presence of weak
peaks at the K points.[51]. The NN spin correlation $\left\langle
S_{i}S_{j}\right\rangle$ also shows randomness (see Supporting Information
S10). These calculation results can be compared with existing experimental
results. The zero-field neutron reflection of NaYbS2 reveals a weak magnetic
Bragg peak near $\vec{Q}$ = $\left(1/3,1/3,0\right)$ (K point) at 0.1 K[37]
(see fig. 6(a)). At the same temperature, the neutron reflection under 6 T
shows obvious magnetic Bragg peak at $\vec{Q}$ = $\left(1/3,1/3,0\right)$,
indicating that the magnetic field induces a magnetic ordered phase. Based on
the DMRG of 15 $\times$ 15 spin lattice with PBC, we calculated the
$S\left(\vec{q}\right)$ of NaYbS2 under 0 T and 6 T, respectively (see fig.
6(b)). Similar to the experimental data, a weak peak and an obvious peak can
be observed near $\vec{Q}$ = (1/3, 1/3) under 0 T and a 6 T, respectively. The
magnetic field-dependence neutron reflections at (1/3, 1/3, 0) are shown in
fig. 6(c)[37]. With the increasing magnetic field, the peak at (1/3, 1/3, 0)
first increases and then decreases, which is consistent with the DMRG
calculation results (see fig. 6(d)). Considering the neutron diffraction data
was measured using polycrystalline samples of NaYbS2, so the diffraction
intensity might not be sufficient. Besides, the limited data points of neutron
diffraction under different magnetic fields may not fully reflect the trend of
diffraction peaks with magnetic fields, especially the lack of data near
quantum critical points. To better illustrate the series of magnetic field-
induced phase transitions in NaYbS2, we also compare our calculated
$S\left(\vec{q}\right)$ at $\vec{Q}$ = (1/3, 1/3) with the
$\mathrm{d}M/\mathrm{d}H$ of NaYbS2[37]. The $\mathrm{d}M/\mathrm{d}H$ data of
NaYbS2 has a very significant change in the range of 6 to 6.5 T, which
indicates that there is a field-induced phase transition in this magnetic
field range. The peak of the $S\left(\vec{q}\right)$ at $\vec{Q}$ = (1/3, 1/3)
also appears in the range of 6 to 6.5 T, which is consistent with the
$\mathrm{d}M/\mathrm{d}H$ data. Therefore, our computational results
essentially illustrate the evolution of the ground state magnetic structure of
NaYbS2 with the magnetic field. For INS, we can distinguish a cone feature at
$|\vec{Q}|$ $\sim$ 1.24 Å-1 from powder data at 50 mK[37]. This core feature
is also key evidence for the Dirac-like spin liquid. From the perspective of
spin dynamics, the ground state of NaYbS2 also confirms to the Dirac-like spin
liquid[40]. The zero-field $\mu$SR experimental data show that the spin
relaxation rate maintains a finite value even at 0.1 K. However, a weak
magnetic field (about 0.1 T) can cause the spin relaxation rate to drop to
near zero, significantly different from the LF-$\mu$SR spin relaxation rate of
QSL with spinon Fermi surface state[35, 36]. The former can open a spin gap at
the Dirac point by applying a weak magnetic field, while the latter is robust
against the influence of magnetic field. The thermodynamic data of $C_{v}/T$
tends to 0 as the temperature decreases, rather than maintaining a finite
value. All of these numerical calculations and experimental results point to
the ground state of NaYbS2 being a Dirac-like gapless QSL.

The consistency of experimental and computational results also demonstrates
the validity of our chosen low-energy spin Hamiltonian model and the fitted
exchange interactions. In the future, we will conduct more in-depth INS
measurements in single crystal samples of NaYbS2 to study the potential spinon
excitations and topological properties.

##  7 Summary

In summary, we conducted a detailed investigation of the magnetism of NaYbS2
from finite temperatures to ground state. Firstly, we conducted a Raman
scattering on NaYbS2 using Raman spectroscopy techniques. Through polarized
Raman scattering, we identified two phonon excitations with Raman activity,
namely the $E_{g}$ and $A_{1g}$ modes. More importantly, we clearly detected
three CEF excitation energy levels of NaYbS2 at 2 K, which are at 207
$\mathrm{cm^{-1}}$ ($\sim$ 25.6 meV), 263 $\mathrm{cm^{-1}}$($\sim$ 32.6 meV),
and 314 $\mathrm{cm^{-1}}$ ($\sim$ 38.9 meV).

Next, we performed relevant calculations on the contribution of CEF
excitations to the magnetization. By fitting the $M/H-T$ data of NaYbS2 under
a magnetic field of 0.1 T, we can extract the CEF parameters of NaYbS2. Based
on the CEF parameters, we can obtain the excitation energy levels and wave
functions by diagonalizing the CEF Hamiltonian. These energy levels are highly
consistent with the results of Raman scattering, further verifying the
reliability of these parameters we obtained from the fitting. At the same
time, we determined the characteristic temperature of NaYbS2 to be 40 K. Above
the temperature, the CEF excitations play a key role in magnetism, and below
the temperature, the spin-exchange interaction dominates. This is consistent
with the results obtained from the analysis of the ESR linewidths of NaYbS2
and NaYb_0.1Lu_0.9S2. We also studied the $g$-factors in the effective
spin-1/2 model of NaYbS2. Through ESR measurements of a dilute magnetic doped
sample of NaYb_0.1Lu_0.9S2, we determined the accurate $g$-factors in the ab-
plane and along the c-axis to be 3.14 and 0.86, respectively. Through the MF
theory, we can well explain why there are differences in the $g$-factors of
NaYbS2 and NaYb_0.1Lu_0.9S2 along the c-axis.

Before analyzing the ground state magnetism of NaYbS2, we first obtained the
spin-exchange interactions in the low-energy spin Hamiltonian through MF
theory and FD method. By fitting the $M/H-T$ data below the characteristic
temperature, we can obtain the diagonal spin-exchange interactions $J_{\pm}$
and $J_{zz}$ to be 3.52 K and $-$0.87 K, respectively. For the off-diagonal
spin-exchange interactions, $J_{\pm\pm}$ and $J_{z\pm}$, we can obtain them to
be $-$0.18 K and 2.66 K, respectively, by fitting the specific heat data in
the low temperature range using the FD method. We employed the DMRG method to
investigate the ground state of NaYbS2. The calculation results directly point
to the ground state of NaYbS2 being spin disordered. By comparing with the
existing experimental results, we can further confirm that the ground state of
NaYbS2 is a Dirac-like spin liquid. This lays the foundation for us to further
study the topological properties of the ground state magnetism of NaYbS2.

This work was supported by the National Key Research and Development Program
of China (Grant No. 2022YFA1402704), the National Science Foundation of China
(Grant No. 12274186), the Strategic Priority Research Program of the Chinese
Academy of Sciences (Grant No. XDB33010100), and the Synergetic Extreme
Condition User Facility (SECUF). A portion of this work was performed on the
Steady High Magnetic Field Facilities, High Magnetic Field Laboratory, CAS.

The authors declare that they have no conflict of interest.

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We prove the following stronger form of (P1):

######  16.4 Proposition.

For any simple period $A$ over $G_{\alpha}$ we have $[A]_{\alpha}\geq 0.25$
and, consequently, $h_{\alpha}(A)\leq 6$.

###### Proof.

If $\alpha=0$ then $[A]_{0}\geq 1$ by the definition of $[\cdot]_{0}$. Let
$\alpha\geq 1$. Take any $r\geq 1$. Consider a fragmentation $\mathcal{F}$ of
rank $\alpha$ of the cyclic word $(A^{r})^{\circ}$. Assume that $\mathcal{F}$
consists of words $S_{i}$, $i=1,2,\dots,N$ where the first $k$ are subwords of
fragments of rank $\alpha$. By Proposition 15.5α-1 we have $|S_{i}|<4|A|$ for
$i=1,2,\dots,k$. This implies that the cyclic word $(A^{r-4k})^{\circ}$ can be
partitioned into subwords of words in some subset of the remaining $S_{i}$,
$i=k+1,k+2,\dots,N$. Therefore,

$|(A^{r})^{\circ}|_{\alpha}\geq k+\zeta|(A^{r-4k})^{\circ}|_{\alpha-1}.$

Proposition 16.1α-1 says that $(A^{r-4k})^{\circ}$ has at least $r-4k$
disjoint occurrences of a coarsely $B$-periodic word $K$ over $G_{\alpha-1}$
with $\ell_{B}(K)\geq p_{0}$. Then by Lemma 16.3,

$|(A^{r})^{\circ}|_{\alpha}\geq k+5\zeta(r-4k)=0.25r.$

This holds for all $r\geq 1$, so by Definition 12.12 we get $[A]_{\alpha}\geq
0.25$ and hence $h_{\alpha}(A)\leq 6$. ∎

The following lemma is a key tool in the proof of Proposition 16.1. Very
roughly, it corresponds to the statement ‘‘if a word $W$ is periodic with two
simple periods $A$ and $B$ at the same time, and if $|W|\geq 2|A|$, $|W|\geq
2|B|$ then $B$ is a cyclic shift of $A$’’.

######  16.5 Lemma.

Let $\mathsf{L}_{0}$ and $\mathsf{L}_{1}$ be periodic lines in
$\Gamma_{\alpha}$ with simple periods $A$ and $B$ over $G_{\alpha}$,
respectively. Let $\mathsf{S}$ be a coarsely $C$-periodic segment in
$\mathsf{L}_{0}$ where $C$ is another simple period over $G_{\alpha}$,
$\ell_{C}(\mathsf{S})\geq 25$. Assume that there exist coarsely $C$-periodic
segments $\mathsf{T}_{0},\mathsf{T}_{1},\mathsf{T}_{2}$ in $\mathsf{L}_{1}$
such that $\mathsf{T}_{0}<\mathsf{T}_{1}<\mathsf{T}_{2}$ and
$\mathsf{T}_{i}\approx s_{A,\mathsf{L}_{0}}^{i}\mathsf{S}$, $i=0,1,2$.

If $\mathsf{T}_{0}\lesssim s_{B,\mathsf{L}_{1}}^{-1}\mathsf{T}_{1}$ or
$s_{B,\mathsf{L}_{1}}\mathsf{T}_{1}\lesssim\mathsf{T}_{2}$ then, if fact,
$\mathsf{T}_{0}\approx s_{B,\mathsf{L}_{1}}^{-1}\mathsf{T}_{1}$,
$s_{B,\mathsf{L}_{1}}\mathsf{T}_{1}\approx\mathsf{T}_{2}$, words $A$ and $B$
represent conjugate elements of $G_{\alpha}$ and periodic lines
$\mathsf{L}_{0}$ and $\mathsf{L}_{1}$ are parallel.

###### Proof.

Denote $\mathsf{P}_{0}=\mathsf{S}\cup s_{A,\mathsf{L}_{0}}^{2}\mathsf{S}$ and
$\mathsf{P}_{1}=\mathsf{T}_{0}\cup\mathsf{T}_{2}$. Let $\mathsf{S}^{*}$ and
$\mathsf{T}_{i}^{*}$ be stable parts of $\mathsf{S}$ and $\mathsf{T}_{i}$.

The crucial argument is similar to one in the proof of Proposition 13.4.
Denote $\mathcal{P}$ the set of all coarsely $C$-periodic segments
$\mathsf{U}$ in $\Gamma_{\alpha}$ such that $\mathsf{U}\approx
g\mathsf{S}^{*}$ for some $g\in G_{\alpha}$ (i.e. $\mathsf{U}$ and
$\mathsf{S}^{*}$ have the same labels of their periodic bases). We introduce
translations and jumps on the set of coarsely $C$-periodic segments
$\mathsf{U}\in\mathcal{P}$ which occur in $\mathsf{P}_{0}$ or
$\mathsf{P}_{1}$. As in the proof of Proposition 13.4, it will be convenient
to consider two disjoint sets of those $\mathsf{U}\in\mathcal{P}$ which occur
in $\mathsf{P}_{0}$ and in $\mathsf{P}_{1}$. (So formally we introduce the set
$\mathcal{P}_{i}$ $(i=0,1)$ of pairs $(\mathsf{U},\mathsf{P}_{i})$ where
$\mathsf{U}$ occurs in $\mathsf{P}_{i}$; thus
$s_{A,\mathsf{L}_{0}}^{i}\mathsf{S}^{*}$ belongs to $\mathcal{P}_{0}$ and
$\mathsf{T}_{i}^{*}$ belongs to $\mathcal{P}_{1}$ for $i=0,1,2$. For a
coarsely $C$-periodic segment $\mathsf{U}\in\mathcal{P}$, saying ‘$\mathsf{U}$
occurs in $\mathsf{P}_{i}$’ we mean the corresponding element of
$\mathcal{P}_{i}$.)

Let $\mathsf{U},\mathsf{V}\in\mathcal{P}$ be coarsely $C$-periodic segments
each occurring in some $\mathsf{P}_{i}$.

  1. (i)

If $\mathsf{U}$ and $\mathsf{V}$ occur in different paths $\mathsf{P}_{i}$ and
$\mathsf{U}\approx\mathsf{V}$ then $\mathsf{U}$ jumps to $\mathsf{V}$.

  2. (ii)

$\mathsf{U}$ translates to $\mathsf{V}$ in the following cases:

     * $\mathsf{U}$ and $\mathsf{V}$ occur in $\mathsf{P}_{0}$ and $\mathsf{U}\approx s_{A,\mathsf{L}_{0}}^{k}\mathsf{V}$ for some $k\in\mathbb{Z}$; or

     * $\mathsf{U}$ and $\mathsf{V}$ occur in $\mathsf{P}_{1}$ and $\mathsf{U}\approx s_{B,\mathsf{L}_{1}}^{k}\mathsf{V}$ for some $k\in\mathbb{Z}$.

Let $\mathcal{M}$ be a maximal set of pairwise non-(strictly compatible)
segments which can be obtained by these two operations from $\mathsf{S}^{*}$.
Lemma 13.14 implies that $\mathcal{M}$ is a finite set. As in the proof of
Proposition 13.4 we prove the following claim.

  * Claim:

The jump operation is always possible inside $\mathcal{M}$; that is, for any
$\mathsf{U}\in\mathcal{M}$ in $\mathsf{P}_{i}$, $i\in\\{0,1\\}$, there exists
$\mathsf{V}\in\mathcal{P}$ in $\mathsf{P}_{1-i}$ such that
$\mathsf{V}\approx\mathsf{U}$.

To prove the claim, we will apply Lemma 13.1 and do a necessary preparatory
work. Assume that $\mathsf{U}\in\mathcal{M}$ belongs to $\mathsf{P}_{0}$ (the
other case differs only in notation). Let $\mathsf{V}_{0}=\mathsf{S}^{*}$,
$\mathsf{V}_{1}$, $\dots$, $\mathsf{V}_{l}=\mathsf{U}$ be a sequence of
coarsely $C$-periodic segments $\mathsf{V}_{i}\in\mathcal{M}$ such that
$\mathsf{V}_{i+1}$ is obtained from $\mathsf{V}_{i}$ by one of the operations
(i) or (ii). We can assume that $\mathsf{V}_{2j}\to\mathsf{V}_{2j+1}$ are
translations and $\mathsf{V}_{2j+1}\to\mathsf{V}_{2j+2}$ are jumps, so
$l=2k-1$ for some $k$. Under this assumption,
$\mathsf{V}_{2j}\to\mathsf{V}_{2j+1}$ is a translation inside $\mathsf{P}_{0}$
if $j$ is even and inside $\mathsf{P}_{1}$ if $j$ is odd. We then define a
sequence $\mathsf{Y}_{0}$, $\mathsf{Y}_{1}$, $\dots$, $\mathsf{Y}_{k}$ of
paths in $\Gamma_{\alpha}$ ($\mathsf{Y}_{j}$ will be periodic segments with
alternating periods $A$ and $B$) and a sequence $\mathsf{W}_{j}\in\mathcal{P}$
of coarsely $C$-periodic segments in $\mathsf{Y}_{j}$ for $j=0,1,\dots,k-1$
such that $\mathsf{W}_{0}=\mathsf{V}_{1}$ and
$\mathsf{W}_{i}\approx\mathsf{W}_{0}$ for all $i$. For each $j$ we will have
$\mathsf{W}_{j}=f_{j}\mathsf{V}_{2j+1}$ for some $f_{j}\in G_{\alpha}$. The
definition of $\mathsf{Y}_{j}$ and $f_{j}$ goes as follows.

We start with $\mathsf{Y}_{0}=\mathsf{P}_{0}$ and
$\mathsf{W}_{0}=\mathsf{V}_{1}$, so $f_{0}=1$. Assume that $j<k-1$ and
$\mathsf{Y}_{j}$ and $f_{j}$ are already defined. For even $j$,
$\mathsf{V}_{2j}$ translates to $\mathsf{V}_{2j+1}$ inside $\mathsf{P}_{0}$,
so there exists $f_{j+1}\in G_{\alpha}$ of the form
$f_{j}s_{A,\mathsf{P}_{0}}^{t}$ such that $f_{j+1}\mathsf{V}_{2j+1}\approx
f_{j}\mathsf{V}_{2j}$. Thus, $f_{i}\mathsf{P}_{0}$ and $f_{j+1}\mathsf{P}_{0}$
have a common $A$-periodic extension and we take
$\mathsf{Y}_{j+1}=f_{i}\mathsf{P}_{0}\cup f_{j+1}\mathsf{P}_{0}$. Similarly,
for odd $j$ $\mathsf{V}_{2j}$ translates to $\mathsf{V}_{2j+1}$ inside
$\mathsf{P}_{1}$. We take $f_{j+1}\in G_{\alpha}$ of the form
$f_{j}s_{B,\mathsf{P}_{1}}^{t}$ such that $f_{j+1}\mathsf{V}_{2j+1}\approx
f_{j}\mathsf{V}_{2j}$ and take $\mathsf{Y}_{j+1}=f_{i}\mathsf{P}_{0}\cup
f_{j+1}\mathsf{P}_{0}$ inside a common $B$-periodic extension of
$f_{i}\mathsf{P}_{0}$ and $f_{j+1}\mathsf{P}_{0}$. Note that $k$ is odd
because $\mathsf{V}_{2k+1}=\mathsf{U}$ is assumed to occur in
$\mathsf{P}_{0}$. We finally set $\mathsf{Y}_{k}=f_{k-1}\mathsf{P}_{1}$.

We now apply Lemma 13.1 where:

  * •

$S_{j}$ is the set of all coarsely $C$-periodic segments
$\mathsf{V}\in\mathcal{P}$ in $\mathsf{Y}_{j}$.

  * •

$S_{j}$ is pre-ordered by ‘$\lnapprox$’.

  * •

Equivalence is strict compatibility.

  * •

A segment $\mathsf{V}\in\bigcup_{j}S_{j}$ is defined to be stable if
$\mathsf{V}$ is the stable part of some coarsely $C$-periodic segment in
$\mathsf{Y}_{j}$.

  * •

For $a_{j}$, $b_{j}$, $a_{j}^{\prime}$ and $b_{j}^{\prime}$ we take
appropriate translates of $\mathsf{S}^{*}$ and $\mathsf{T}_{i}^{*}$; namely,
$f_{j}\mathsf{S}^{*}$, $f_{j}s_{A,\mathsf{L}_{0}}^{2}\mathsf{S}^{*}$,
$f_{j}\mathsf{T}_{0}^{*}$ and $f_{j}\mathsf{T}_{2}^{*}$ if $j$ is even or
$f_{j}\mathsf{T}_{0}^{*}$, $f_{j}\mathsf{T}_{2}^{*}$, $f_{j}\mathsf{S}^{*}$
and $f_{j}s_{A,\mathsf{L}_{0}}^{2}\mathsf{S}^{*}$ if $j$ is odd, respectively.

  * •

$c_{0}$ is $\mathsf{V}_{1}$.

Note that by Proposition 16.4 we have $h_{\alpha}(C)\leq 6$. Hence the
hypothesis $\ell_{C}(\mathsf{S})\geq 25$ implies $\ell_{C}(\mathsf{V})\geq
13\geq 2h_{\alpha}(C)+1$ for any $\mathsf{V}\in\mathcal{P}$. Condition (ii) of
Lemma 13.1 holds by Lemma 13.14. Conditions (iii) and (iv) of Lemma 13.1 hold
by Lemma 13.16. By the lemma, there exists a coarsely $C$-periodic segment
$\mathsf{V}_{k}\in\mathcal{P}$ in $f_{k-1}\mathsf{P}_{1}$ such that
$\mathsf{V}_{k}\approx f_{k-1}\mathsf{U}$. This gives the required jump
$\mathsf{U}\to f_{k-1}^{-1}\mathsf{V}_{k}$. The claim is proved.

Let $r$ be the number of coarsely $C$-periodic segments
$\mathsf{V}\in\mathcal{M}$ such that and
$\mathsf{K}^{*}\lnapprox\mathsf{V}\lessapprox
s_{A,\mathsf{L}_{0}}\mathsf{K}^{*}$ and let $q$ be the number of coarsely
$C$-periodic segments $\mathsf{V}\in\mathcal{M}$ such that
$\mathsf{T}_{1}^{*}\lnapprox\mathsf{N}\lessapprox
s_{B,\mathsf{L}_{1}}\mathsf{T}_{1}^{*}$ (in other words, $r$ and $q$ are the
numbers of coarsely $C$-periodic segments $\mathsf{V}\in\mathcal{M}$ in one
period $A$ and in one period $B$, respectively). Note that $\gcd(r,q)=1$
because $\mathcal{M}$ is generated by a single segment $\mathsf{S}^{*}$.

We assume first that either $\mathsf{T}_{0}\lessapprox
s_{B,\mathsf{L}_{1}}^{-1}\mathsf{T}_{1}$ or
$s_{B,\mathsf{L}_{1}}\mathsf{T}_{1}\lessapprox\mathsf{T}_{2}$. Since
$\mathcal{M}$ is closed under translations modulo equivalence ‘$\approx$’,
each of these relations implies $q\leq r$ and hence implies the other one. Let
$\mathsf{U}_{0},\mathsf{U}_{1},\dots,\mathsf{U}_{t}$ be all coarsely
$C$-periodic segments in $\mathcal{M}$ belonging to $\mathsf{P}_{0}$ arranged
in their order in $\mathsf{P}_{0}$ (so $\mathsf{U}_{i}$ form a set of
representatives of coarsely $C$-periodic segments in $\mathcal{M}$ modulo
‘$\approx$’). The group $G_{\alpha}$ acts on the set $\mathcal{P}/{\approx}$.
It follows from Corollary 13.9 that the action is free. For equivalence
classes $[\mathsf{U}_{i}]$ of $\mathsf{U}_{i}$ we have

$s_{A,\mathsf{L}_{0}}[\mathsf{U}_{i}]=[\mathsf{U}_{i+r}],\
i=0,1,\dots,t-r\quad s_{B,\mathsf{L}_{1}}[\mathsf{U}_{i}]=[\mathsf{U}_{i+q}],\
i=0,1,\dots,t-q.$

Note also that $t\geq 2r+1$. Applying Lemma 13.2 we get
$s_{A,\mathsf{L}_{0}}=d^{q}$ and $s_{B,\mathsf{L}_{1}}=d^{r}$ for some $d\in
G_{\alpha}$. Since $A$ and $B$ are non-powers we get $q=r=1$ which immediately
implies the conclusion of the proposition.

For the proof, it remains to consider cases $\mathsf{T}_{0}\sim
s_{B,\mathsf{L}_{1}}^{-1}\mathsf{T}_{1}$ and
$s_{B,\mathsf{L}_{1}}\mathsf{T}_{1}\sim\mathsf{T}_{2}$. We consider the case
$s_{B,\mathsf{L}_{1}}\mathsf{T}_{1}\sim\mathsf{T}_{2}$ (the case
$\mathsf{T}_{0}\sim s_{B,\mathsf{L}_{1}}^{-1}\mathsf{T}_{1}$ is symmetric). By
the already proved part, we can assume that $\mathsf{T}_{2}\lnapprox
s_{B,\mathsf{L}_{1}}\mathsf{T}_{1}$. We show that the assumption leads to a
contradiction.

We have $\mathsf{T}_{0}\lnapprox
s_{B,\mathsf{L}_{1}}^{-1}\mathsf{T}_{2}\lnapprox\mathsf{T}_{1}$, so there
exists $\mathsf{T}_{3}\in\mathcal{M}$ such that $\mathsf{T}_{3}\approx
s_{B,\mathsf{L}_{1}}^{-1}\mathsf{T}_{2}$. $\mathsf{T}_{3}$ jumps to some
$\mathsf{S}_{3}\in\mathcal{M}$ in $\mathsf{L}_{0}$ such that
$\mathsf{S}_{3}\sim\mathsf{S}$ and $\mathsf{S}_{3}\lnapprox\mathsf{S}$. Then
$\mathsf{S}_{3}$ translates to $\mathsf{S}_{4}\approx
s_{A,\mathsf{L}_{0}}\mathsf{S}_{3}$ and we have
$\mathsf{S}_{4}\sim\mathsf{S}_{2}$ and
$\mathsf{S}_{4}\lnapprox\mathsf{S}_{2}$. Then $\mathsf{S}_{4}$ jumps to some
$\mathsf{T}_{4}$ in $\mathsf{L}_{1}$ and we can continue the process
infinitely (see Figure 40).

$\mathsf{T}_{0}$
---
$\mathsf{T}_{3}$
---
$\mathsf{T}_{1}$
---
$\mathsf{T}_{4}$
---
$\mathsf{T}_{2}$
---
$s_{B,\mathsf{L}_{0}}\mathsf{T}_{1}$
---
$s_{A,\mathsf{L}_{0}}^{-1}\mathsf{S}$
---
$\mathsf{S}_{3}$
---
$\mathsf{S}$
---
$\mathsf{S}_{4}$
---
$s_{A,\mathsf{L}_{0}}\mathsf{S}$
---
$\mathsf{L}_{1}$
---
$\mathsf{L}_{0}$
---
Figure 40.

∎

###### Proof of Proposition 16.1.

Let $A$ be a suspended period of level $m$ over $G_{\alpha}$,

Assume first that $m=0$. Then by Definition 15.1 and Proposition 12.15 an
$A$-periodic segment $\mathsf{R}$ in $G_{\alpha}$ contains a coarsely
$B$-periodic segment $\hat{\mathsf{T}}$ with $\ell_{B}(\hat{\mathsf{T}})\geq
p_{1}-2h_{\alpha}(B)-2\geq 51$ where $B$ is not conjugate to $A$ in
$G_{\alpha}$. By Lemma 13.12 we have $\hat{\mathsf{T}}\not\sim
s_{A,\mathsf{R}}\hat{\mathsf{T}}$ and $|\hat{\mathsf{T}}|<2|A|$. Let
$\mathsf{T}$ be the stable part of $\hat{\mathsf{T}}$. Since
$h_{\alpha}(B)\geq 2$ by Definition 12.12, we have $|\mathsf{T}|<|A|$ by
Corollary 13.6. Note also that
$\ell_{B}(\mathsf{T})\geq\ell_{B}(\hat{\mathsf{T}})-2h_{\alpha}(B)\geq p_{0}$.
Let $T\eqcirc\operatorname{\mathit{label}}(\mathsf{T})$. We show that $T$ has
the required property (ii) formulated in Proposition 16.1

Let $\mathsf{S}$ be a coarsely $A$-periodic segment in $\Gamma_{\alpha}$ with
$\ell_{A}(\mathsf{S})\geq 4$ and let $\mathsf{P}$ be a periodic base for
$\mathsf{S}$. Denote $t=\ell(\mathsf{S})$. By Remark 12.7 we can assume that
$|\mathsf{P}|\geq t|A|$. Up to placing $\hat{\mathsf{T}}$ in $\Gamma_{\alpha}$
we can assume that $\mathsf{P}$ contains $t-2$ translates $\hat{\mathsf{T}}$,
$s_{A,\mathsf{P}}\hat{\mathsf{T}}$, $\dots$,
$s_{A,\mathsf{P}}^{t-3}\hat{\mathsf{T}}$ of $\hat{\mathsf{T}}$. Using Lemma
10.13(i) (which implies that strictly compatible coarsely periodic segments
are close) and Proposition 12.14 we find disjoint $\mathsf{V}_{i}$
$(i=0,\dots,t-3)$ in $\mathsf{S}$ such that $\mathsf{V}_{i}\approx
s_{A,\mathsf{P}}^{i}\mathsf{T}$. This proves the proposition in the case
$m=0$.

Let $m\geq 1$. The proof consists of two parts. First we provide a
construction of a coarsely $B$-periodic segment $T$ satisfying condition (i)
of Proposition 16.1 and then we prove (ii).

Construction of $T$. According to Definition 15.2, there exists a sequence
$A_{0}$, $A_{1}$, $\dots$, $A_{m}=A$ of simple periods over $G_{\alpha}$ where
$A_{0}$ is suspended of level 0, for each $i\leq m-1$ $A_{i}$ is not conjugate
to $A_{i+1}$ and there are reduced in $G_{\alpha}$ close words
$X_{i}Q_{i}Y_{i}$ and $P_{i+1}\in\operatorname{Per}(A_{i+1})$ where
$Q_{i}\in\operatorname{Per}(A_{i})$ and $|Q_{i}|\geq 4|A_{i}|$. For each $i$,
we consider corresponding close paths
$\mathsf{X}_{i}\mathsf{Q}_{i}\mathsf{Y}_{i}$ and $\mathsf{P}_{i+1}$ in
$\Gamma_{\alpha}$ and place then in such a way that $\mathsf{Q}_{i}$ and
$\mathsf{P}_{i}$ have the common infinite $A_{i}$-periodic extension
$\mathsf{L}_{i}$. We denote also $\mathsf{L}_{0}$ the infinite
$A_{0}$-periodic extension of $\mathsf{Q}_{0}$.

As we proved above, there is a coarsely $B$-periodic segment
$\hat{\mathsf{T}}_{0}$ in $\mathsf{Q}_{0}$ with
$\ell(\hat{\mathsf{T}}_{0})\geq 51$ and the stable part $\mathsf{T}_{0}$
satisfying $\ell(\mathsf{T}_{0})\geq p_{0}$ and $|\mathsf{T}_{0}|<|A|$. Up to
positioning $\hat{\mathsf{T}}_{0}$ in $\mathsf{L}_{0}$ we can assume that
$\mathsf{Q}_{0}$ contains translates
$s_{A_{0},\mathsf{L}_{0}}^{-1}\mathsf{T}_{0}$ and
$s_{A_{0},\mathsf{L}_{0}}\mathsf{T}_{0}$ of $\mathsf{T}_{0}$. In what follows,
if $\mathsf{Z}$ is a coarsely $B$-periodic segment in $\Gamma_{\alpha}$ then
$\mathsf{Z}^{*}$ denotes the stable part of $\mathsf{Z}$. By Lemma 13.12,
$s_{A_{0},\mathsf{L}_{0}}^{t}\mathsf{T}_{0}\not\sim\mathsf{T}_{0}$ for any
$t\neq 0$ and hence
$s_{A_{0},\mathsf{L}_{0}}^{-1}\mathsf{T}_{0}\lnsim\hat{\mathsf{T}}_{0}\lnsim
s_{A_{0},\mathsf{L}_{0}}\mathsf{T}_{0}$. By Proposition 12.14 there are
$\mathsf{T}_{1}$, $\mathsf{U}_{1,1}$ and $\mathsf{W}_{1,1}$ in
$\mathsf{P}_{1}$ such that $\mathsf{T}_{1}\approx\mathsf{T}_{0}$,
$\mathsf{U}_{1,1}\approx s_{A_{0},\mathsf{L}_{0}}^{-1}\mathsf{T}_{0}^{*}$ and
$\mathsf{W}_{1,1}\approx s_{A_{0},\mathsf{L}_{0}}\mathsf{T}_{0}^{*}$.
Application of Lemma 16.5 with $\mathsf{S}:=\mathsf{T}_{0}^{*}$ (note that
$\ell_{B}(\mathsf{T}_{0}^{*})\geq p_{0}-12\geq 27$) gives
$s_{A_{1},\mathsf{L}_{1}}^{-1}\mathsf{T}_{1}\lnsim\mathsf{U}_{1,1}$ and
$\mathsf{W}_{1,1}\lnsim s_{A_{1},\mathsf{L}_{1}}\mathsf{T}_{1}$. In
particular, we have $|\mathsf{T}_{1}|\leq|A_{1}|$. In the case $m=1$ we take
$T:=\operatorname{\mathit{label}}(\mathsf{T}_{1})$.

Assume that $m\geq 2$. We continue a procedure of finding coarsely
$B$-periodic segments $\mathsf{T}_{i}$ in $\mathsf{P}_{i}$. Up to positioning
$\mathsf{Q}_{1}$ in $\mathsf{L}_{1}$ we can assume that $\mathsf{Q}_{1}$
contains both $s_{A_{1},\mathsf{L}_{1}}^{-1}\mathsf{T}_{1}$ and
$s_{A_{1},\mathsf{L}_{1}}\mathsf{T}_{1}$. Using Proposition 12.14 we find
$\mathsf{U}_{2,2}$, $\mathsf{U}_{2,1}$, $\mathsf{W}_{2,1}$ and
$\mathsf{W}_{2,2}$ in $\mathsf{P}_{2}$ such that $\mathsf{U}_{2,2}\approx
s_{A_{1},\mathsf{L}_{1}}^{-1}\mathsf{T}_{1}^{*}$,
$\mathsf{U}_{2,1}\approx\mathsf{U}_{1,1}^{*}$,
$\mathsf{W}_{2,1}\approx\mathsf{W}_{1,1}^{*}$ and $\mathsf{W}_{2,2}\approx
s_{A_{1},\mathsf{L}_{1}}\mathsf{T}_{1}^{*}$. By Lemma 13.15,
$\mathsf{U}_{2,2}\lnsim\mathsf{U}_{2,1}\lnsim\mathsf{W}_{2,1}\lnsim\mathsf{W}_{2,2}$.
We have $\mathsf{U}_{2,1}\approx
s_{A_{0},\mathsf{L}_{0}}^{-1}\mathsf{T}_{0}^{**}$, $\mathsf{W}_{2,1}\approx
s_{A_{0},\mathsf{L}_{0}}\mathsf{T}_{0}^{**}$ and using Proposition 12.14 once
more with $\mathsf{X}:=s_{A_{0},\mathsf{L}_{0}}^{-1}\mathsf{T}_{0}^{**}\cup
s_{A_{0},\mathsf{L}_{0}}\mathsf{T}_{0}^{**}$ and
$\mathsf{Y}:=\mathsf{U}_{2,1}\cup\mathsf{W}_{2,1}$ we find $\mathsf{T}_{2}$ in
$\mathsf{P}_{2}$ such that $\mathsf{T}_{2}\approx\mathsf{T}_{0}$. Application
of Lemma 16.5 gives
$s_{A_{2},\mathsf{L}_{2}}^{-1}\mathsf{T}_{2}\lnsim\mathsf{U}_{2,2}$ and
$\mathsf{W}_{2,2}\lnsim s_{A_{2},\mathsf{L}_{2}}\mathsf{T}_{2}$. In
particular, $|\mathsf{T}_{2}|\leq|A_{2}|$.

Repeating in a similar manner, we find $\mathsf{U}_{m,m}$,
$\mathsf{U}_{m,m-1}$, $\mathsf{W}_{m,m-1}$ and $\mathsf{W}_{m,m}$ in
$\mathsf{P}_{m}$ such that $\mathsf{U}_{m,m}\approx
s_{A_{m-1},\mathsf{L}_{m-1}}^{-1}\mathsf{T}_{m-1}^{*}$,
$\mathsf{U}_{m,m-1}\approx\mathsf{U}_{m-1,m-1}^{*}$,
$\mathsf{W}_{m,m-1}\approx\mathsf{W}_{m-1,m-1}^{*}$, $\mathsf{W}_{m,m}\approx
s_{A_{m-1},\mathsf{L}_{m-1}}\mathsf{T}_{m-1}^{*}$ and
$\mathsf{U}_{m,m}\lnsim\mathsf{U}_{m,m-1}\lnsim\mathsf{W}_{m,m-1}\lnsim\mathsf{W}_{m,m}$.
Then we successively find $\mathsf{U}_{m,m-2}$, $\mathsf{W}_{m,m-2}$,
$\mathsf{U}_{m,m-3}$, $\mathsf{W}_{m,m-3}$, $\dots$, $\mathsf{U}_{m,1}$,
$\mathsf{W}_{m,1}$ such that
$\mathsf{U}_{m,i}\approx\mathsf{U}_{i,i}^{*}\approx
s_{A_{i-1},\mathsf{L}_{i-1}}^{-1}\mathsf{T}_{i-1}^{**}$ and
$\mathsf{W}_{m,i}\approx\mathsf{V}_{i,i}^{*}\approx
s_{A_{i-1},\mathsf{L}_{i-1}}\mathsf{T}_{i-1}^{**}$. Finally, we find
$\mathsf{T}_{m}$ in $\mathsf{P}_{m}$ such that
$\mathsf{T}_{m}\approx\mathsf{T}_{0}$. Application of Lemma 16.5 gives
$s_{A_{m},\mathsf{L}_{m}}^{-1}\mathsf{T}_{m}\lnsim\mathsf{U}_{m,m}$ and
$\mathsf{W}_{m,m}\lnsim s_{A_{m},\mathsf{L}_{m}}\mathsf{T}_{m}$ which implies
$|\mathsf{T}_{m}|\leq|A_{m}|$. We take
$T:=\operatorname{\mathit{label}}(\mathsf{T}_{m})$. This completes the
construction, The whole procedure is schematically shown in Figure 41.

$s_{A_{0},\mathsf{L}_{0}}^{-1}\mathsf{T}_{0}$
---
$\hat{\mathsf{T}}_{0}$
---
$s_{A_{0},\mathsf{L}_{0}}\mathsf{T}_{0}$
---
$s_{A_{1},\mathsf{L}_{1}}^{-1}\mathsf{T}_{1}$
---
$\mathsf{U}_{11}$
---
$\mathsf{T}_{1}$
---
$\mathsf{V}_{11}$
---
$s_{A_{1},\mathsf{L}_{1}}\mathsf{T}_{1}$
---
$s_{A_{2},\mathsf{L}_{2}}^{-1}\mathsf{T}_{2}$
---
$\mathsf{U}_{22}$
---
$\mathsf{U}_{21}$
---
$\mathsf{T}_{2}$
---
$\mathsf{V}_{21}$
---
$\mathsf{V}_{22}$
---
$s_{A_{2},\mathsf{L}_{2}}\mathsf{T}_{2}$
---
$s_{A_{3},\mathsf{L}_{3}}^{-1}\mathsf{T}_{3}$
---
$\mathsf{U}_{33}$
---
$\mathsf{U}_{32}$
---
$\mathsf{U}_{31}$
---
$\mathsf{T}_{3}$
---
$\mathsf{V}_{31}$
---
$\mathsf{V}_{32}$
---
$\mathsf{V}_{33}$
---
$s_{A_{3},\mathsf{L}_{3}}\mathsf{T}_{3}$
---
$\mathsf{L}_{0}$
---
$\mathsf{L}_{1}$
---
$\mathsf{L}_{2}$
---
$\mathsf{L}_{3}$
---
Figure 41.

Note that in $\mathsf{P}_{m}$ we have

$s_{A_{m},\mathsf{L}_{m}}^{-1}\mathsf{T}_{m}\lnsim\mathsf{U}_{m,m}\lnsim\mathsf{U}_{m,m-1}\lnsim\dots\lnsim\mathsf{U}_{m,1}\lnsim\mathsf{T}_{m}\lnsim\mathsf{W}_{m,1}\lnsim\dots\lnsim\mathsf{W}_{m,m}\lnsim
s_{A_{m},\mathsf{L}_{m}}\mathsf{T}_{m}.$

Proof of (ii). Let $\mathsf{S}$ be a coarsely $A_{m}$-periodic segment in
$\Gamma_{\alpha}$ and let $\mathsf{P}$ be a periodic base for $\mathsf{S}$.
Denote $t=\ell_{A_{m}}(\mathsf{S})$. By Remark 12.7 we can assume that
$|\mathsf{P}|\geq t|A|$, so $\mathsf{P}$ contains $t-1$ translates
$\mathsf{T}$, $s_{A,\mathsf{P}}\mathsf{T}$, $\dots$,
$s_{A,\mathsf{P}}^{t-2}\mathsf{T}$ of a coarsely $B$-periodic segment
$\mathsf{T}$ which is a translate of $\mathsf{T}_{m}$ constructed above. By
Proposition 12.14, $\mathsf{S}$ contains coarsely $B$-periodic segments
$\mathsf{Z}_{0}$, $\mathsf{Z}_{1}$, $\dots$, $\mathsf{Z}_{t-2}$ such that
$\mathsf{Z}_{i}\approx s_{A,\mathsf{P}}^{i}\mathsf{T}^{*}$. We claim,
moreover, that for $1\leq i\leq t-3$ there exist $\mathsf{V}_{i}$ in
$\mathsf{S}$ such that $\mathsf{V}_{i}\approx s_{A,\mathsf{P}}^{i}\mathsf{T}$
and $\mathsf{V}_{i}$ are all disjoint. Since
$\ell_{B}(\mathsf{V}_{i})=\ell_{B}(\mathsf{T}_{m})\geq p_{0}$ this will finish
the proof.

Fix an index $k$ in the interval $1\leq i\leq t-3$. Up to positioning
$\mathsf{P}$ and $\mathsf{S}$ in $\Gamma_{\alpha}$ we can assume that
$\mathsf{P}$ and $\mathsf{P}_{m}$ have the common $A_{m}$-periodic extension
$\mathsf{L}_{m}$ and $s_{A,\mathsf{P}}^{k}\mathsf{T}=\mathsf{T}_{m}$. By Lemma
16.5, $s_{A_{m},\mathsf{L}_{m}}^{-1}\mathsf{T}\lnsim\mathsf{U}_{m,m}$ and
$\mathsf{W}_{m,m}\lnsim s_{A_{m},\mathsf{L}_{m}}\mathsf{T}$. Then using
Proposition 12.14 as in the procedure above, we successively find pairs
$(\mathsf{U}_{i},\mathsf{W}_{i})$ for $i=m,m-1,\dots,1$ such that
$\mathsf{Z}_{k-1}\lnsim\mathsf{U}_{m}\lnsim\mathsf{U}_{m-1}\lnsim\dots\lnsim\mathsf{U}_{1}\lnsim\mathsf{Z}_{k}\lnsim\mathsf{W}_{1}\lnsim\dots\lnsim\mathsf{W}_{m}\lnsim\mathsf{Z}_{k+1}$
and $\mathsf{U}_{i}\approx\mathsf{U}_{i,i}^{*}$,
$\mathsf{W}_{i}\approx\mathsf{W}_{i,i}^{*}$ for $i=m,m-1,\dots,1$. Then using
Proposition 12.14 again with
$\mathsf{X}:=s_{A_{0},\mathsf{L}_{0}}^{-1}\mathsf{T}_{0}^{**}\cup
s_{A_{0},\mathsf{L}_{0}}\mathsf{T}_{0}^{**}$,
$\mathsf{Y}:=\mathsf{U}_{1}\cup\mathsf{W}_{1}$ and
$\mathsf{S}=\hat{\mathsf{T}}_{0}$ gives $\mathsf{V}_{k}$ with
$\mathsf{U}_{1}\lnsim\mathsf{V}_{k}\lnsim\mathsf{W}_{1}$ and
$\mathsf{V}_{k}\approx\mathsf{T}_{0}\approx s_{A,\mathsf{P}}^{k}\mathsf{T}$.
The proof is finished. ∎

######  16.6 Proposition.

Let $A\in\mathcal{E}_{\alpha+1}$ and $t\geq 1$ be an integer. Let $P$ be an
$A$-periodic word with $|P|=t|A|$. Then

$\frac{t}{n+t}<\mu(P)<\frac{t}{n-t}+\omega.$

Moreover, for $t\geq 200$ we have also

$0.89\frac{t}{n}<\mu(P)<1.12\frac{t}{n}.$

###### Proof.

Denote $N=|(A^{n})^{\circ}|_{\alpha}$. Recall that $\mu(P)=|P|_{\alpha}/N$. Up
to cyclic shift of $A$, we assume that $P\eqcirc A^{t}$. For the lower bound
on $\mu(P)$ in the first inequality, we observe that the cyclic word
$(A^{n})^{\circ}$ can be covered with $\lceil\frac{n}{t}\rceil$ copies of $P$.
By 4.14, this implies

$N<\left(\frac{n}{t}+1\right)|P|_{\alpha}$

which is equivalent to $\frac{t}{n+t}<\mu(P)$. Similarly, for the upper bound
we observe that $\lfloor\frac{n}{t}\rfloor$ disjoint copies of $P$ can be
placed inside $(A^{n})^{\circ}$. Then again by 4.14,

$N\geq\left\lfloor\frac{n}{t}\right\rfloor(|P|_{\alpha}-1)>\left(\frac{n}{t}-1\right)(|P|_{\alpha}-1)$

which implies by (S1) with $\alpha:=\alpha+1$

$\mu(P)<\frac{t}{n-t}+\frac{1}{N}\leq\frac{t}{n-t}+\omega.$

If $t\geq 200$ then we partition $A^{t}$ into $k$ subwords $A^{t_{i}}$ with
$80\leq t_{i}\leq 120$. We have

$\sum_{i}|A_{t_{i}}|_{\alpha}-(k-1)\leq|P|_{\alpha}\leq\sum_{i}|A_{t_{i}}|_{\alpha}.$

and by the already proved bounds on $\mu(A^{t_{i}})$, for each $i$ we have

$0.94\frac{t_{i}}{n}<\mu(A^{t_{i}})<1.07\frac{t_{i}}{n}+\frac{1}{N}.$

Then

$\mu(P)\geq\sum_{i}\mu(A^{t_{i}})-\frac{k-1}{N}>0.94\frac{t}{n}-\frac{k}{N}.$

By Proposition 16.4, $N\geq 0.25n$. Hence

$\frac{k}{N}\leq\frac{t}{80}\left(\frac{n}{N}\right)\frac{1}{n}\leq
0.05\frac{t}{n}$

and we obtain the required bound $\mu(P)>0.89\frac{t}{n}$. Similarly, for the
upper bound on $\mu(P)$ we get

$\mu(P)\leq\sum_{i}\mu(A^{t_{i}})<1.07\frac{t}{n}+\frac{k}{N}\leq
1.12\frac{t}{n}.$

∎

######  16.7 Corollary.

(P2) implies (S2)α+1.

###### Proof.

By Proposition 16.6, if $P$ is a subword of $A^{n}$ with
$A\in\mathcal{E}_{\alpha+1}$ and $\mu(P)\geq\lambda$ then $|P|\geq t|A|$ where
$t$ satisfies

$\frac{t}{n-t}\geq\lambda-\omega\geq\frac{1}{24}-\frac{1}{480}$

and hence $t>76$. Since $76>p_{1}$, the required implication is
straightforward. ∎

######  16.8 Proposition.

Presentation (15-1) satisfies (P2) and therefore satisfies the iterated small
cancellation condition (S0)–(S3) for all $\alpha\geq 1$.

###### Proof.

Indeed, assume that $\mathsf{L}_{1}$ and $\mathsf{L}_{2}$ are periodic lines
in $\Gamma_{\alpha}$ with periods $A,B\in\mathcal{E}_{\alpha+1}$ respectively.
Let $\mathsf{P}$ and $\mathsf{Q}$ be close subpath of $\mathsf{L}_{1}$ and
$\mathsf{L}_{2}$, respectively, such that $|\mathsf{P}|\geq p_{1}|A|$. If $A$
is conjugate to $B$ in $G_{\alpha}$ then $A=B$ according to Definition 15.3
and the statement follows from Proposition 13.13. If $A$ is not conjugate to
$B$ in $G_{\alpha}$ then $B$ is suspended of level 0 as a simple period over
$G_{\alpha}$ and hence cannot belong to $\mathcal{E}_{\alpha+1}$. ∎

From this point, we may assume that all statements in Sections 5–16 are true
for all values of rank $\alpha$.

######  16.9 Proposition.

Every element of $G$ is conjugate to a power of some $C\in\bigcup_{\alpha\geq
1}\mathcal{E}_{\alpha}$.

###### Proof.

Let $g\in G$. If $g$ has finite order then by Proposition 11.5, $g$ is
conjugate to a power of some $C\in\bigcup_{\alpha\geq 1}\mathcal{E}_{\alpha}$.
We assume that $g$ has infinite order and come to a contradiction.

By Corollary 14.8 we represent $g$ by a word $X$ reduced in $G$ such that for
some $\alpha\geq 1$, $X$ contains no fragments $F$ of rank $\beta\geq\alpha$
with $\mu_{\mathrm{f}}(F)\geq 3\omega$. By our assumption, $X$ has infinite
order in all $G_{\beta}$ for $\beta\geq\alpha$. By Propositions 11.13 and
11.5, $X$ is conjugate in $G_{\alpha}$ to a word of the form $A^{t}$ where $A$
is a simple period over $G_{\alpha}$. Using Proposition 7.13(iii) we conclude
that $X$ is conjugate to $A^{t}$ already in $G_{\alpha-1}$. Then applying
Proposition 8.9 with $\beta:=\alpha,\alpha+1,\dots$ we see that no cyclic
shift of $A$ contains a fragment $K$ of rank $\beta\geq\alpha$ with
$\mu_{\mathrm{f}}(K)\geq 9\omega$ and that $A$ is cyclically reduced in
$G_{\beta}$ for all $\beta>\alpha$. Moreover, by Propositions 8.16(iii) and
8.11, $A$ is strongly cyclically reduced in $G_{\beta}$ for all
$\beta>\alpha$.

Assume that for some $\beta\geq\alpha$, $A$ is conjugate in $G_{\beta}$ to a
power $B^{r}$ of a simple period over $G_{\beta}$. By Proposition 9.16, $A$
and $B^{r}$ are conjugate already in $G_{\alpha}$. Since $A$ is a non-power in
$G_{\alpha}$, we have $r=1$ and then by Propositions 11.13 and 11.5, $A$ is a
non-power in $G_{\beta}$. We showed that $A$ is a simple period over
$G_{\beta}$ for any $\beta\geq\alpha$. But this is impossible because by
Proposition 16.4 we should have $|A|_{\beta}\geq 0.25$ and hence $|A|\geq
0.25\zeta^{-\beta}$ for any $\beta\geq\alpha$. ∎

As an immediate consequence we get:

######  16.10 Corollary.

$G$ satisfies the identity $x^{n}=1$ and therefore is isomorphic to the free
Burnside group $B(m,n)$.

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