# Contributions of the 4-D Chern-Simons Electrodynamics to the Molecular
Biochirality

A. C. L. Santos<EMAIL_ADDRESS>Universidade Estadual do Ceará,
Faculdade de Educação, Ciências e Letras de Iguatu, Av. Dario Rabelo, s/n,
CEP: 63.500-000, Iguatu, CE, Brazil  C. R. Muniz<EMAIL_ADDRESS>Universidade Estadual do Ceará, Faculdade de Educação, Ciências e Letras de
Iguatu, Av. Dario Rabelo, s/n, CEP: 63.500-000, Iguatu, CE, Brazil  L. T.
Oliveira<EMAIL_ADDRESS>Universidade Estadual do Ceará, Faculdade
de Educação, Ciências e Letras de Iguatu, Av. Dario Rabelo, s/n, CEP:
63.500-000, Iguatu, CE, Brazil  J. T. Souza<EMAIL_ADDRESS>Universidade Estadual do Ceará, Faculdade de Educação, Ciências e Letras de
Iguatu, Av. Dario Rabelo, s/n, CEP: 63.500-000, Iguatu, CE, Brazil

###### Abstract

In this paper we analyze the physical basis of the molecular biochirality,
which is responsible by a Parity Violation Energy Difference (PVED) in some
organic molecules (e.g. amino acids, which occur as levogyrous enantiomers in
Nature). We investigate the role of the 4-D Chern-Simons theory in the origin
of this PVED, particularly of the Carrow-Field-Jackiw electrodynamics, which
violates both Lorentz and parity symmetries, by considering the terrestrial
scenario through the Earth magnetic field influence.

Key words: Biochirality, Chern-Simons theory, Carrol-Field-Jackiw
electrodynamics.

##  I Introduction

At XIX Century, Louis Pasteur discovered a singular peculiarity of the nature.
It at times produces dextrogyrous (which rotates the plane of polarization of
light to the right) or levogyrous (which rotates the plane of polarization of
light to the left) molecules, when these are synthesized in biological
metabolism, the so-called biochirality. In some fundamental molecules for the
development of life, the biochirality is lead to the extreme (via
autocatalysis mechanism), becoming homochirality. For example, there are
practically only L-amino acids in proteins and D-sugars in DNA and RNA gal .

The origin of this feature of nature is still misunderstood. However, some
theoretical proposals have been launched, among them the contributions from
fundamental forces producing a PVED (Parity Violation Energy Difference)
between the enantiomers. In the weak interaction, which is known to violate
the parity symmetry, the PVED is found in the interaction between electron-
WIMPS, electron-neutrino, nucleon-neutrino, nucleon-electron. Regarding the
gravitational interaction, PVED arises from Chern-Simons Theory, Loop Quantum
Gravity (LQG) and modified Leitner-Okubo gravitational potentials leitner ;
barguenotudela .

##  II PVED IN THE CARROL-FIELD-JACKIW ELECTRODYNAMICS

Modified Maxwell’s equations can be obtained from the Lagrangean

$\mathcal{L}=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}+\frac{1}{2}K^{\mu}A^{\nu}\epsilon_{\mu\nu\rho\sigma}F^{\rho\sigma},$
(1)

in the context of a $4D$ Chern-Simons electromagnetic theory - the Carrol-
Field-Jackiw Electrodynamics [8], which is gauge but not Lorentz and parity
invariant, resulting in $(c=1)$:

$\displaystyle\nabla\cdot\textbf{E}$ $\displaystyle=$ $\displaystyle
4\pi\rho-\textbf{K}\cdot\textbf{B},$ (2) $\displaystyle\nabla\cdot\textbf{B}$
$\displaystyle=$ $\displaystyle 0,$ (3) $\displaystyle\nabla\times\textbf{E}$
$\displaystyle=$
$\displaystyle\displaystyle-\frac{\partial\textbf{B}}{\partial t},$ (4)
$\displaystyle\nabla\times\textbf{B}$ $\displaystyle=$ $\displaystyle
4\displaystyle\pi\textbf{J}-K_{0}\textbf{B}+\textbf{K}\times\textbf{E}+\frac{\partial\textbf{E}}{\partial
t}.$ (5)

Adopting a coordinate system in which the background quadrivector
$K^{\mu}=(K_{0},\textbf{K})$ has only the temporal component $K_{0}$, we can
write equation (5) as:

$\displaystyle\nabla\times\textbf{B}=4\displaystyle\pi\textbf{J}-K_{0}\textbf{B}+\frac{\partial\textbf{E}}{\partial
t},$ (6)

and by considering $\textbf{B}=\nabla\times\textbf{A}$, we get:

$\displaystyle\displaystyle\nabla\times(\textbf{B}+K_{0}\textbf{A})$
$\displaystyle=$ $\displaystyle
4\pi\textbf{J}+\frac{\partial\textbf{E}}{\partial t},$ (7)
$\displaystyle\nabla\times\textbf{\~{B}}$ $\displaystyle=$ $\displaystyle
4\pi\textbf{J}+\frac{\partial\textbf{E}}{\partial
t};\;\;\text{with}\;\;\textbf{\~{B}}=\textbf{B}+K_{0}\textbf{A}.$ (8)

The Maxwell’s equations can now be written in terms of B̃ by taking the
Coulomb gauge, $\nabla\cdot{\bf A}=0$, as well as static electromagnetic
fields, in the vacuum. On the other hand, there is a difference in the
potential energy associated with the interaction between the magnetic dipole
moment $\mu_{a}$ of the molecules L and R and the redefined magnetic field,
given by:

$\displaystyle
U_{R}=-\displaystyle{\bf\mu}_{a}\cdot\textbf{\~{B}}=-({\bf\mu}_{a}\cdot\textbf{B}+K_{0}{\bf\mu}_{a}\cdot\textbf{A}).$
(9)

By realizing a parity operation, one gets:

$\displaystyle
U_{L}=-{\bf\mu}_{a}\cdot\textbf{B}+K_{0}{\bf\mu}_{a}\cdot\textbf{A},$ (10)

and the PVED will be, therefore

$\displaystyle\Delta U=U_{L}-U_{R}=2K_{0}{\bf\mu}_{a}\cdot\textbf{A}.$ (11)

Assuming a spherical body with magnetic dipole moment given by m, we have that
the potential vector $A(r)$ will be

$\displaystyle\textbf{A}(r)=\frac{\mu_{0}}{4\pi}\textbf{m}\times\frac{\textbf{r}}{r^{3}},$
(12)

so that

$\displaystyle\Delta
U=\displaystyle\frac{\mu_{0}}{2\pi}\mu_{a}K_{0}\frac{|\textbf{m}\times\textbf{r}|}{r^{3}}\cos{\theta},$
(13)

where $\theta$ is the angle between $\mu_{a}$ and A.

##  III PVED IN A TERRESTRIAL SCENARIO

From equation (13), an upper limit for PVED will be:

$\displaystyle\Delta U=\frac{\mu_{0}}{2\pi}\mu_{a}K_{0}\frac{m}{r^{2}}.$ (14)

For a terrestrial scenario, taking the upper limit for $K_{0}=5.0\times
10^{-11}$ m-1, which is based on geomagnetic arguments provided in carrol ,
and considering $\mu_{a}\cong 5\mu_{\textbf{B}}=4.6\times 10^{-23}$ JT-1,
$m=8\times 10^{22}$ Am2, and $r=6.2\times 10^{6}$ m, we have that the maximum
PVED will be

$\displaystyle\Delta U\cong 5.8\times 10^{-11}\text{eV}.$ (15)

This energy difference corresponds to a difference between the enantiomeric
quantities L and R in a thermal bath at temperature $T\cong 300$ K given by

$\displaystyle\Delta N=N_{L}-\displaystyle
N_{R}=N_{0}\left[\exp{\left(-\frac{U_{R}}{K_{B}T}\right)}-\exp{\left(-\frac{U_{L}}{K_{B}T}\right)}\right],$
(16)

with $N_{0}=N_{L}+N_{R}$, resulting in an excess of L molecules with respect
to R of

$\displaystyle\displaystyle\Delta N\cong N_{0}\left(\frac{\Delta
U}{K_{B}T}\right)\cong 10^{13}\;\;\text{molecules/mol. }$ (17)

##  IV CONCLUSION

Therefore, within our proposal of considering a $4D$ Chern-Simons theory, we
found, in the terrestrial scenario, a PVED between molecular enantiomers of
$5.8\times 10^{-11}$ eV, implying a difference between the levogyrous and
dextrogyrous quantities of the order of $10^{13}\;\text{molecules/mol}$. In
this sense, from Table 1, the Carrol-Field-Jackiw electrodynamics presents a
quite favorable PVED in that scenario within the range of PVED’s predicted in
other models of fundamental interactions.

Figure 1: PVED based on some models of fundamental interactions.

## REFERENCES

## References

  * (1) J. Gal, The discovery of biological enantioselectivity: Louis pasteur and the fermentation of tartaric acid, 1857—a review and analysis 150 yr later. Chiralitty, v. 20, p. 5–19, 2008.
  * (2) J. Leitner and S. Okubo, Parity, charge conjugation, and time reversal in the gravitational interaction. Physical Review, v. 136, p. 1542–1546, 1964.
  * (3) S. Alexander and N. Yunes, Chern–Simons modified general relativity. Physics Reports, v. 480, p. 1–55, 2009.
  * (4) L. Freidel et al., Quantum gravity, torsion, parity violation, and all that. Physical Review, v. 72, p. 104002, 2005.
  * (5) A. Dorta-Urra and P. Bargueño, Homochirality: A perspective from fundamental physics. Symmetry, v. 11, p. 1–9, 2019.
  * (6) P. Bargueño and I. Gonzalo, Effect of cosmological neutrinos on discrimination between the two enantiomers of a chiral molecule. Origins of Life and Evolution of Biospheres, v. 36, p. 171–176, 2006.
  * (7) P. Bargueño and R. P. de Tudela, The role of supernova neutrinos on molecular homochirality. Origins of Life and Evolution of Biospheres, v. 37, p. 253–257, 2007.
  * (8) S. M. Carrol, G. B. Field, and R. Jackiw, Limits on a Lorentz- and parity-violating modification of electrodynamics. Physical Review D, v.41, p.1231-1240, 1990.
  * (9) P. Bargueño and R. P. de Tudela, Constraining long-range parity violation in gravitation using high resolution spectroscopy of chiral molecules. Physical Review, v. 78, p. 102004, 2008.
  * (10) P. Bargueño, Chirality and gravitational parity violation. Chirality, v. 27, p. 375–381, 2015.


# Magnetic excitations in the one-dimensional Heisenberg-Ising model with
external fields and their experimental realizations

Jiahao Yang  Tsung-Dao Lee Institute, Shanghai Jiao Tong University, Shanghai
200240, China  Xiao Wang  Tsung-Dao Lee Institute, Shanghai Jiao Tong
University, Shanghai 200240, China  Jianda Wu<EMAIL_ADDRESS>Tsung-Dao Lee
Institute, Shanghai Jiao Tong University, Shanghai 200240, China  School of
Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China

###### Abstract

The one dimensional (1D) spin-1/2 Heisenberg-Ising model, a prototype quantum
many-body system, has been intensively studied for many years. In this review,
after a short introduction on some basic concepts of group theory for the
octahedral group, a detailed pedagogical framework is laid down to derive the
low-energy effective Hamiltonian for the Co-based materials. The 1D spin-1/2
Heisenberg-Ising model is obtained when applying the analysis to quasi-1D
antiferromagnetic materials $\rm BaCo_{2}V_{2}O_{8}$ and $\rm
SrCo_{2}V_{2}O_{8}$. After the preparation, we review the theoretical
progresses of a variety of novel magnetic excitations and emergent physics in
the 1D spin-1/2 Heisenberg-Ising model, and further summarize their recent
experimental realizations.

##  I Introduction

The one-dimensional (1D) Heisenberg model and its extension 1D Heisenberg-
XY/Ising model were born many decades ago bethe1931 ; Kasteleijn_1952104 ;
Sutherland_1970 . Due to their quantum integrability and algebraic beauty
Jimbo1995 ; Faddeev1979 ; slavnov_algebraic_2019 ; Baxter_1982_ExactlySM ;
caux_remarks_2011 , close-entwined physical and mathematical progress have
been continuously surging since the birth of the models. The studies not only
significantly deepen and extend our understanding on magnetic excitations and
many-body physics, but also expand frontiers of mathematics and even create
new mathematical directions. From physical side, the studies reveal a series
of charming many-body excitations and rich emergent phenomena. The revealed
excitations include fractional types such as spinon FADDEEV_spinwave_1981 ;
faddeev_spectrum_1984 ; Muller_spinon_1981 ; Karbach_spinon_1997 ;
Bougourzi_spinon_1996 ; Bougourzi_spinon_1998 ; caux2008 , (anti)psinon
Karbach_pp_pap_2002 ; karbach2000III , and exotic ones such as string
excitations bethe1931 ; takahashi_1D_1971 ; Gaudin_XXZ_1971 ;
Taka_suzuki_XXZ_1972 ; Takahashi1999 , where each of the former two
excitations always appears in pairs while the later one always contains
bounded magnons. Depending on the way to tune the anisotropy of the model and
how the magnetic field is applied, a variety of emergences appear: Tomonaga-
Luttinger liquid (without field or with longitudinal field)
tomonaga_remarks_1950 ; Luttinger_1960 ; Luther_TLL_1975 , integrability with
supersymmery (with a fine-tuned anisotropy without field) Fendley_SUSY_2003 ,
transverse-field Ising universality (with transverse field) Pfeuty1970 ;
sachdev_2011 ; Jianda_crossover_2018 , and $E_{8}$ physics (with transverse
and longitudinal fields) a_b_zamolodchikov_integrals_1989 ; jianda_E8_2014 ;
DELFINO_1995 ; xiao_cascade_2021 ; DELFINO199440 . From mathematical side, the
Yang-Baxter equation and algebraic structure discovered in the model
significantly extend our understandings on algebraic structure, and play an
essential role in building up the mathematical framework for quantum group and
quantum algebra Jimbo1995 ; Faddeev1979 ; slavnov_algebraic_2019 ;
Baxter_1982_ExactlySM ; drinfeld_hopf_1990 ; drinfeld_quantum_1988 ;
drinfeld_Yangians_1988 ; drinfeld_QHA_1990 ; drinfeld_YBE_1983 ;
Maillet_Drifeld_twists_2000 .

Along the course, experimental probes of the aforementioned exciting physics
meet great difficulties in early years due to limited techniques and lack of
clean crystals. With the experimental developments, the first progress is made
for the realization of spinon in the $\rm SrCuO_{2}$ kim_spinon_2006 ;
Zaliznyak_spinon_2004  and even higher-order spinon states are observed in
$\rm Sr_{2}CuO_{3}$ mourigal_XXX_2013 . Soon after the first observation of
the spinon, the magnetic excitations with clear Tomonaga-Luttinger liquid
behavior are also observed in the $\rm BaCo_{2}V_{2}O_{8}$ (BCVO)
Kimura_TLL_2007 . However, the demonstrations in material for the string
excitations, transverse field Ising chain (TFIC) universality along with the
exotic $E_{8}$ physics remain as a challenge until recently. The difficulty is
largely due to the challenge to figure out proper parameter regions for the
real material to realize those exotic physics. Substantial progresses are made
with a series of theoretical work Jianda_crossover_2018 ; jianda_E8_2014 ;
xiao_cascade_2021 ; Zou_universality_2019 ; yang_string_one-dimensional_2019
which clearly identify proper parameters to realize those physics in
corresponding materials. Meanwhile, experiments are carried out and
unambiguously realize for the first time the long-desired physics in
corresponding quasi-1D materials kimura_high_2006 ; bera_magnetic_2014 ;
wang_string_experimental_2018 ; bera_string_dispersions_2020 ;
cui_tfic_quantum_2019 ; wang_tfic_quantum_2018 ; zou_e_8_2021 ;
zhang_e8_observation_2020 ; amelin_e8_experimental_2020 ; yang_Local_2022 .

Figure 1:  Phase diagram of the XXZ model Eq. (1).

In this review, we focus on the 1D spin-1/2 Heisenberg-Ising model (XXZ model)
with Ising anisotropy $\Delta>1$,

$H=J\sum_{i=1}^{N}\left(S^{x}_{i}S^{x}_{i+1}+S^{y}_{i}S^{y}_{i+1}+\Delta
S^{z}_{i}S^{z}_{i+1}\right)-h\sum_{i=1}^{N}S^{z}_{i},$ (1)

with the spin component $S_{i}^{\alpha}=\sigma_{i}^{\alpha}/2$ at site $i$
$(\alpha=x,y,z;\;{\rm and}\;\sigma_{i}^{\alpha}\;{\rm
labels\;the\;Pauli\;matrix})$, the coupling $J>0$ between neighbouring spins,
and the external field $h$ along $z$ direction. The phase diagram for the
model is illustrated in Fig. 1 CNYang_XXZ_1966 ; Johnson1991 ; Franchini2017 .
Since total spin along $z$-direction $S^{z}_{T}$ is conserved, the Hilbert
space can be divided into different subspaces according to the quantum numbers
of $S^{z}_{T}$ which is the magnetization along $z$ direction. If $\Delta=1$,
the Hamiltonian recovers the XXX model with an $SU(2)$ symmetry.

Figure 2: From the AFM ground state (a), a pair of spinons illustrated as two
domain walls, created by a single spin-flip (b), propagate along the chain
(c).

At zero field $h=0$, the 1D spin-1/2 XXZ model, has an antiferromagnetic (AFM)
ground state. Its elementary excitations referred to spinons are fractional
spin-1/2 quasiparticles appear in pairs in the spin dynamics. The dynamical
structure factor of two-spinon excitations can be calculated exactly in both
transverse and longitudinal channels caux2008 ; castillo_exact_2020 . The two-
spinon spectrum is directly observed in $\rm SrCo_{2}V_{2}O_{8}$ (SCVO) by
inelastic neutron scattering (INS) and agrees with theoretical prediction
bera_spinon_2017 .

Figure 3: An illustration for the 2 (3)-string excitation [red arrows in top
(bottom) row] in spin configurations.

One appealing excitation in Heisenberg models is the string state which is a
magnetic soliton state bethe1931 ; takahashi_1D_1971 ; Gaudin_XXZ_1971 ;
Taka_suzuki_XXZ_1972 . In the Ising limit ($\Delta\gg 1$) the magnetic soliton
corresponds to a bound object of flipped spins [Fig. 3]. Despite of long-time
theoretical recognition and continuously theoretical studies of the string
excitations and their essentially compelling many-body nature bethe1931 ;
Takahashi1999 , feasible proposals to directly observe those exotic states
remain a challenge ishimura_dynamical_1980 ; Imambekov_1D_2012 ;
kohno_string_dynamically_2009 ; Pereira_edge_2008 ; Pereira_spectral_2009 ;
Caux_comput_2005 ; Caux_computation_2005 ; Shashi_Nonuniversal_2011 ;
Ganahl_Observation_2012 . The difficulty lies in locating a proper parameter
region such that the string contribution to the spin dynamics is non-
negligible. A silver lining appears in 2017 that the Ref. yang_string_one-
dimensional_2019  proposes a promising field-induced quantum critical region
of the 1D spin-1/2 XXZ model Eq. (1), to directly probe the string
excitations, where the string excitations are shown to dominate the dynamic
spectrum in the quantum critical region with small magnetization. Following
the proposal, the string excitations are clearly detected and observed in the
zone center of SCVO for the first time via the high-resolution terahertz (THz)
spectroscopy measurement wang_string_experimental_2018 . Under concrete
theoretical guidance, the exotic 2- and 3-string excitations as well as novel
low energy fractional magnetic excitations are identified in the field-induced
quantum-critical region. In 2020, with INS measurments on the same material,
complete dispersions of the string exciations over the full Brillouin zone are
obtained bera_string_dispersions_2020 . In both experiments, the obtained
excitation spectrum and its magnetic field dependency perfectly agree with
theoretical calculations from low to high energy, demonstrating a rare success
in understanding strongly correlated magnetic systems.

When the longitudinal field is replaced by a transverse field in Eq. (1), a
quantum phase transition arises with field tuning. The corresponding quantum
criticality around the quantum critical point (QCP) falls into the class of
TFIC universality dmitriev_1D_2002 . Near the QCP of the TFIC universality,
the Grüneisen ratio, the ratio of magnetic expansion coefficient to specific
heat Yu_Gruneisen_2020 , is found to exhibit a unique singular behavior: it
can be either divergent or convergent when the system approaches its QCP with
field or temperature tuning, respectively Jianda_crossover_2018 . The exotic
singular behavior can serve as a smoking gun to justify the TFIC universality
in real materials. Meanwhile, a relevant experiment is carried out for BCVO in
the transverse field wang_tfic_quantum_2018 , where a 1D QCP appears at a
strong field. Near the 1D QCP, the obtained quantum critical behaviors for the
Grüneisen ratio exactly follow the aforementioned scaling relation in the
class of TFIC universality, which thus beautifully realizes the TFIC
universality.

More surprisingly, when the quantum critical TFIC is perturbed by a
longitudinal field, an integrable massive relativistic quantum field theory,
dubbed as quantum $E_{8}$ integrable model, further emerges. In the model, it
contains eight types of massive particles whose scattering can be fully
described by the $E_{8}$ exceptional Lie algebra
a_b_zamolodchikov_integrals_1989 . In 2010, an INS experiment on the quasi-1D
ferromagnetic material $\rm CoNb_{2}O_{6}$ (CNO) observes two lowest
excitations whose energy ratio matches the mass ratio of the two lightest
$E_{8}$ particles, providing a preliminary evidence for the existence of the
exotic $E_{8}$ physics in real material coldea_quantum_2010 . However, the
measured DSF in the continuum region is not accurate enough to distinguish
other $E_{8}$ particles, which makes inconclusive a complete realization of
the $E_{8}$ physics in the material CNO. In 2020, a further THz spectroscopy
experiment is performed on the same material, which obtained a detailed
spectrum in the continuum region amelin_e8_experimental_2020 . However, the
obtained spectrum in the continuum region apparently deviates from
theoretically results xiao_cascade_2021 , implying more systematic and careful
studies are needed to conclude the realization of the $E_{8}$ physics in the
CNO morris_duality_2021 ; fava_glide_2020 . In the same year, by taking
advantage of the delicate microscopic structure of BCVO, combined efforts made
from theoretical analysis, numerical simulation, THz spectroscopy, nuclear
magnetic resonance (NMR), and INS experiments unambiguously reveal beautiful
$E_{8}$ physics for the first time in the material, which not only confirms
the eight single $E_{8}$ particles but also the excitations composed by
multi-$E_{8}$ particles. xiao_cascade_2021 ; zou_e_8_2021 ;
zhang_e8_observation_2020 .

Above brief summary clearly shows rich physics in the Co-based materials, such
as BCVO, SCVO, and CNO etc., where the $\rm Co^{2+}$ ion carries on the
magnetism with $3d^{7}$ electrsons. The ground state of the local $\rm
Co^{2+}$ ion possesses total orbital angular momentum $L=3$ and total spin
$S_{tot}=3/2$ Fazekas1999 . Due to the superexchange mechanism the interaction
of the neighbouring spin is isotropic, thus the effective model becomes a
spin-3/2 XXX model Anderson1959 ; Anderson1994 ; SAWATZKY1976 . In those
materials, the local environment and the spin-orbit coupling (SOC) are non-
negligible, which can cause further splitting of the local ground state
manifold. Since time-reversal symmetry is still preserved, the new ground
state in general at least have a degeneracy of Kramers doublet. If we project
the spin-3/2 XXX model to the new ground state, we can obtain an effective 1D
spin-1/2 XXZ model to describe the low-energy physics, as manifested in a
large number of Co-based materials shiba_exchange_2003 ; lines_magnetic_1963 ;
Sarte2018 .

The remainder of the review is organized as follows. Sec. II introduces some
basic properties of octahedral group and atomic basis for the $\rm CoO_{6}$
octahedron. Sec. III is devoted to derive an effective 1D spin-1/2 XXZ model
for the Co-based materials whose ground state is a Kramers doublet, and gives
a detailed discussion for the Zeeman effect in SCVO and BCVO. Sec. IV reviews
recent theoretical progresses on the magnetic excitations and emergent physics
in the 1D spin-1/2 XXZ model without or with the presence various magnetic
fields, and summarizes their experimental realizations. Sec. V is the
conclusions.

##  II The Octahedral Group and Atomic Basis

Figure 4: The local octahedron environment for a $\rm Co^{2+}$ with six oxygen
ions surrounded.  Figure 5: Equivalent axes belonging to $\mathcal{O}$
symmetry of the octahedron and the cube.

For the local octahedron environment the crystal field produced by six oxygen
ions in the octahedron $\rm CoO_{6}$ has the octahedral symmetry described by
$\mathcal{O}$ group, consisting of all the rotations that keep an octahedron
invariant [Fig. 4]. $\mathcal{O}$ group can also characterize the symmetry of
a crystal field induced by eight corner atoms of a cube [Fig. 5]. Thus, the
crystal field of an octahedron is also referred to the “cubic field”.

From Fig. 5, we can enumerate the elements of $\mathcal{O}$ group, and
distribute them to different classes of inequivalent sets of operations
Dresselhaus2008 ; Zee2016 ; Tinkham1964 . Rotations by the same angle about
equivalent axes form a class and two axes are equivalent if they are related
to each other by a symmetry operation Fazekas1999 . Thus, there are six
$\pi/2$\- and $3\pi/2$-rotations in the $\mathcal{C}_{4}$ class. The
$\pi$-rotations about the coordinate axis form the three-element class
$\mathcal{C}_{2}$. And there is another type of $\pi$-rotation
$\mathcal{C}_{2}^{\prime}$ about axes, where the axes pass through the centre
of the cube and are parallel to the face diagonals. Rotations by $\pi/3$ and
$2\pi/3$ about body diagonals of the cube generate the eight-element class
$\mathcal{C}_{3}$. Finally, the identity forms a class in itself. In total,
there are five different classes in $\mathcal{O}$ group with 24 group
elements.

In group theory Dresselhaus2008 ; Zee2016 , the action of a symmetry operation
$\hat{R}$ on a given basis set can be described by a matrix $M({R})$. The
matrices of all symmetry operations form a group which is homomorphic to the
corresponding symmetry group. And the matrix group is referred to a
representation of the corresponding symmetry group. The representation
characterizes the symmetry properties which must be independent of the choice
of the basis. Furthermore, the trace of a matrix representation is invariant
under basis transformation, which is introduced to serve as a character of the
representation,

$\chi(R)=\sum_{i}M_{ii}(R)=\text{Tr}(M(R)),$ (2)

For the identity $E$ of the group, $\chi(E)=d$, the dimensionality of the
representation. For the $\mathcal{O}$ group, Table 1 lists the characters of
all classes and irreducible representations. The 1D representation $A_{1}$ is
known as the identity representation, and other four irreducible
representations have dimensions 1, 2, 3, 3.

|  | basis | $E$ | $8\mathcal{C}_{3}$ | $3\mathcal{C}_{2}$ | $6\mathcal{C}_{2}^{\prime}$ | $6\mathcal{C}_{4}$
---|---|---|---|---|---|---|---
$A_{1}$ | $\Gamma_{1}$ | $\\{x^{2}+y^{2}+z^{2}\\}$ | 1 | 1 | 1 | 1 | 1
$A_{2}$ | $\Gamma_{2}$ | $\\{xyz\\}$ | 1 | 1 | 1 | -1 | -1
$E$ | $\Gamma_{3}$ | $\\{x^{2}-y^{2},3z^{2}-r^{2}\\}$ | 2 | -1 | 2 | 0 | 0
$T_{1}$ | $\Gamma_{4}$ | $\\{x,y,z\\}$ | 3 | 0 | -1 | -1 | 1
$T_{2}$ | $\Gamma_{5}$ | $\\{xy,yz,zx\\}$ | 3 | 0 | -1 | 1 | -1
Table 1:  The character table of the $\mathcal{O}$ group with two standard
notations of the irreducible representations. One commonly used basis set is
also listed.

A representation $\Gamma$ of a group is reducible if we can find a
transformation $U$ which makes all matrices block-diagonal. Then the reducible
representation $\Gamma$ can be decomposed into a direct sum of irreducible
representations $\Gamma_{j}$,

$\Gamma=\bigoplus_{j}a_{j}\Gamma_{j}.$ (3)

where $a_{j}$ is the number of times that the irreducible representations
occur. Using the characters of different classes, $a_{j}$ can be determined as
follows,

$a_{j}=\frac{1}{\mathcal{N}}\sum_{k=1}^{r}\mathcal{N}_{k}\chi_{j}^{*}(\mathcal{C}_{k})\chi(\mathcal{C}_{k}).$
(4)

where the summation is over the classes and $\mathcal{N}_{k}$ is the number of
elements in the $k$th class $\mathcal{C}_{k}$ with
$\sum_{k}\mathcal{N}_{k}=\mathcal{N}$, the total number of the group elements.
And the $\chi(\mathcal{C}_{k})$ and $\chi_{j}(\mathcal{C}_{k})$ are the
characters of the reducible representation $\Gamma$ and irreducible
representation $\Gamma_{j}$ for the class $\mathcal{C}_{k}$, respectively.

| $E$ | $8\mathcal{C}_{3}$ | $3\mathcal{C}_{2}$ | $6\mathcal{C}_{2}^{\prime}$ | $6\mathcal{C}_{4}$
---|---|---|---|---|---
$\Gamma_{l=0}$ | 1 | 1 | 1 | 1 | 1
$\Gamma_{l=1}$ | 3 | 0 | -1 | -1 | 1
$\Gamma_{l=2}$ | 5 | -1 | 1 | 1 | -1
$\Gamma_{l=3}$ | 7 | 1 | -1 | -1 | -1
Table 2: A character table of the representations of the octahedral group
$\mathcal{O}$ with different orbital angular momentum $l=0,1,2,3$.

At the level of the atomic (ionic) Hamiltonian, the angular dependence of the
atomic wave functions is described by the familiar spherical harmonics
$Y_{l}^{m}(\theta,\varphi)$. This comes from the fact that the pure atomic
potential is spherically symmetrical and the symmetry group of the Hamiltonian
is the continuous group $SO(3)$ which contains all rotations in three-
dimensional space. As a result, the (total) angular momentum $l$ is a good
quantum number with eigenstates classified by $l$. For an isotropic system,
all rotation axes are equivalent thus rotations around equivalent axes by a
same angle $\alpha$ belong to a same class. The corresponding character of an
$\alpha$-rotation is $\chi^{l}(\alpha)={\sin[(l+1/2)\alpha]}/{\sin(\alpha/2)}$
where $l$ is the angular momentum quantum number. Since the octahedral group
is a subgroup of the rotation group $\mathcal{O}\in SO(3)$, every irreducible
representation of $SO(3)$ can provide a representation of $\mathcal{O}$, which
usually is reducible. In Table 2, we list the characters of four
representations with different orbital angular momentum $l$. As for the $\rm
Co^{2+}$ with $L=3$, the seven-fold representation $\Gamma_{L=3}$ is reducible
in $\mathcal{O}$ symmetry. Following Eq. (3), $\Gamma_{L=3}$ can be decomposed
into a direct sum of a singlet $A_{2}$ and two triplets $T_{1}$ and $T_{2}$,
i.e., $\Gamma_{L=3}=A_{2}\oplus T_{1}\oplus T_{2}$. Within this seven
dimensional Hilbert space, we are free to choose any seven linearly
independent basis which follow the transformation of $\mathcal{O}$ group as a
basis set. These seven basis functions can be constructed after taking into
account the cubic symmetry AABB1970 ; Bleaney_1953 :

$\small\begin{cases}|T_{1},2\rangle=\sqrt{\frac{3}{8}}Y_{3}^{-1}+\sqrt{\frac{5}{8}}Y_{3}^{3}\\\
|T_{1},0\rangle=Y_{3}^{0}\\\
|T_{1},-2\rangle=\sqrt{\frac{3}{8}}Y_{3}^{1}+\sqrt{\frac{5}{8}}Y_{3}^{-3}\end{cases},\begin{cases}|T_{2},2\rangle=\sqrt{\frac{5}{8}}Y_{3}^{-1}-\sqrt{\frac{3}{8}}Y_{3}^{3}\\\
|T_{2},0\rangle=\frac{1}{\sqrt{2}}(Y_{3}^{2}+Y_{3}^{-2})\\\
|T_{2},-2\rangle=\sqrt{\frac{5}{8}}Y_{3}^{1}-\sqrt{\frac{3}{8}}Y_{3}^{-3}\end{cases},|A_{2}\rangle=\frac{1}{\sqrt{2}}(Y_{3}^{2}-Y_{3}^{-2}).$
(5)

It’s easy to verify that the above basis functions are invariant under
rotation within corresponding subspace and the trace of matrix representation
of every rotation operation is consistent with the character table 1.

Group theory alone can not distinguish which irreducible representation is the
lowest lying one. To figure out this, the details of the Hamiltonian have to
be specified. So next we shall find out the ground state manifold of $\rm
CoO_{6}$ with the help of crystal field Hamiltonian.

##  III Crystal Field and Effective Hamiltonian

In this section, we show details to determine the lowest-lying Kramers doublet
followed by the crystal field Hamiltonian of the octahedron $\rm CoO_{6}$ via
the equivalent operator method AABB1970 ; Bleaney_1953 ; hutchings1964 .
Within the obtained ground-state doublet, an effective XXZ model with
pseudospin $S=1/2$ is built up for SCVO and BCVO.

###  III.1 Crystal field Hamiltonian

We start from the point charge model where the electrostatic potential is
generated from the surrounding anions, like the oxygen ions in Fig. 4. This
potential is referred to crystal field potential (in coordinate
representation):

$\mathcal{V}(\mathbf{r})=\sum_{j}\frac{q_{j}}{\left|\mathbf{R}_{j}-\mathbf{r}\right|}$
(6)

where $q_{j}=q=-Ze$ is the charge at the $j$th anion with a distance
$|\mathbf{R}_{j}|=a$ from the origin (e.g., Co2+) and $\bf r$ is an arbitrary
spatial point from the origin. Since the crystal field potential $\mathcal{V}$
satisfies Laplace’s equation $\Delta\mathcal{V}=0$, it can be conveniently
expanded as a sum of spherical harmonics $Y_{n}^{m}(\theta,\varphi)$:

$\mathcal{V}(r,\theta,\varphi)=\sum_{n=0}^{\infty}\mathcal{V}_{n}=\sum_{n=0}^{\infty}\sum_{m=-n}^{n}A_{n}^{m}Y_{n}^{m}(\theta,\varphi)r^{n}$
(7)

where $A_{n}^{m}$ are coefficients to be determined. Thus, the crystal field
potential energy of a magnetic ion at the origin, e.g., Co2+ in Fig. 4, is
$W=\sum_{i}(-e)\mathcal{V}({\bf r}_{i})$ where ${\bf r}_{i}$ is the position
of an electron of the cobalt ion. We need to calculate the matrix element of
it between two basis functions, i.e. $\int{\rm d}{\bf
r}\phi^{*}\mathcal{V}\psi$. In order to calculate it efficiently, we can
reduce the right hand side of Eq. (7) by symmetry argument, since not all the
terms in the sum have the same symmetry of surrounding anions. For instance,
all terms with $n>2L$ vanish, because both $\psi$ and $\phi$ are $L=3$ basis
functions Eq. (5) for $\rm Co^{2+}$ ion. Thus, the terms with $n>6$ vanish due
to the orthogonality of spherical harmonics. By a similar argument, all terms
with odd $n$ vanish because $\psi$ and $\phi$ have the same parity, i.e.
$\phi^{*}\psi$ has even parity. Finally, due to the $C_{4}$ rotational
symmetry along the $z$ direction of the octahedron, the potential
$\mathcal{V}(r,\theta,\varphi)$ must be the same as
$\mathcal{V}(r,\theta,\varphi+\pi/2)$. And as the only $\varphi$ dependence is
$\exp(im\varphi)$, this means the only non-vanishing $A_{n}^{m}$ are those
terms with $m=0$ and $\pm 4$. After neglecting the $Y_{0}^{0}$ term, which is
a constant, the remaining non-trivial contributions are $\mathcal{V}_{4}$ and
$\mathcal{V}_{6}$ hutchings1964 :

$\mathcal{V}(\mathbf{r})=\mathcal{V}_{4}(\mathbf{r})+\mathcal{V}_{6}(\mathbf{r})\,,$
(8)

where

$\begin{split}\mathcal{V}_{4}(\mathbf{r})&=D_{4}\left[Y_{4}^{0}(\theta,\varphi)+\sqrt{\frac{5}{14}}\left(Y_{4}^{4}(\theta,\varphi)+Y_{4}^{-4}(\theta,\varphi)\right)\right]r^{4}\\\
&=A_{4}\left(x^{4}+y^{4}+z^{4}-\frac{3}{5}r^{4}\right)\end{split}\,,$ (9)

and

$\begin{split}\mathcal{V}_{6}(\mathbf{r})&=D_{6}\left[Y_{6}^{0}(\theta,\varphi)-\sqrt{\frac{7}{2}}\left(Y_{6}^{4}(\theta,\varphi)+Y_{6}^{-4}(\theta,\varphi)\right)\right]r^{6}\\\
&=A_{6}\left[x^{6}+y^{6}+z^{6}+\frac{15}{4}\left(x^{2}y^{4}+x^{2}z^{4}+y^{2}x^{4}+y^{2}z^{4}+z^{2}x^{4}+z^{2}y^{4}\right)-\frac{15}{14}r^{6}\right]\end{split}\,.$
(10)

The prefactors depend on the detailed geometry of the anions. For instance, if
the potential $\mathcal{V}(x,y,z)$ is generated by anions $q$ at the corners
of an octahedron; i.e., at $(a,0,0)$, $(0,a,0)$, $(0,-a,0)$, $(0,0,a)$,
$(0,0,-a)$, one finds [c.f. appendix. A]

$D_{4}=\frac{7\sqrt{\pi}}{3}\frac{q}{a^{5}},\quad
D_{6}=\frac{3}{2}\sqrt{\frac{\pi}{13}}\frac{q}{a^{7}},\quad
A_{4}=\frac{35q}{4a^{5}},\quad A_{6}=\frac{-21q}{2a^{7}}.$ (11)

To further improve the efficiency of calculation, based on the Wigner-Eckart
theorem Edmonds_2016 , Stevens proposed an equivalent operator approach
Stevens_1952 , whose recipe is to replace $\\{x,y,z\\}$ and $(x\pm iy)$ by
$\\{L_{x},\ L_{y},\ L_{z}\\}$ and $L^{\pm}$, respectively. Although $x$ and
$y$ commute, $L_{x}$ and $L_{y}$ do not. Thus, the symmetrical replacement is
needed, for instance, $xy$ is replaced by the $(L_{x}L_{y}+L_{y}L_{x})/2$.
Following the recipe, the crystal field Hamiltonian Eq. (8) can be converted
to,

$H=B_{4}^{0}\left[O_{4}^{0}+5O_{4}^{4}\right]+B_{6}^{0}\left[O_{6}^{0}-21O_{6}^{4}\right],$
(12)

where

$O_{4}^{0}=\left[35L_{z}^{4}-30L(L+1)L_{z}^{2}+3L^{2}(L+1)^{2}+25L_{z}^{2}-6L(L+1)\right]\,,$
(13)
$O_{4}^{4}=\frac{1}{2}\left[\left(L^{+}\right)^{4}+\left(L^{-}\right)^{4}\right]\,,$
(14)
$\begin{split}O_{6}^{0}&=231L_{z}^{6}-(315L(L+1)-735)L_{z}^{4}+(105(L(L+1))^{2}-525L(L+1)+294)L_{z}^{2}\\\
&\quad-5(L(L+1))^{3}+40(L(L+1))^{2}-60L(L+1)\,,\end{split}$ (15)
$\begin{split}O_{6}^{4}&=\frac{1}{4}\left[(11L_{z}^{2}-L(L+1)-38)(L_{+}^{4}+L_{-}^{4})+(L_{+}^{4}+L_{-}^{4})(11L_{z}^{2}-L(L+1)-38)\right]\,.\end{split}$
(16)

And the coefficients

$B_{4}^{0}=\frac{7}{16}\frac{-eq}{a^{5}}\cdot\beta\langle r^{4}\rangle,\quad
B_{6}^{0}=\frac{3}{64}\frac{-eq}{a^{7}}\cdot\gamma\langle r^{6}\rangle\,,$
(17)

where

$\beta=\mp\frac{3(2l+1-4S)[-7(l-2S)(l-2S+1)+3(l-1)(l+2)]}{(2l-3)(2l-1)(2l+3)(2l+5)(L-1)(2L-1)(2L-3)}\,.$
(18)

The minus (plus) sign is used for a shell less (more) than half full. The
values of $\gamma$ in different configurations are listed in TABLE 20 of
AABB1970 . Because $\beta<0$ and $q<0$, $B^{0}_{4}<0$ for $3d^{7}$ electrons
case. And for the $d$-electrons, only $\mathcal{V}_{4}$ needs to be taken into
account since $\mathcal{V}_{6}$ has much weaker effect on $d$-electrons. Thus,
the crystal field Hamiltonian for $d$-electrons follows

$H=B_{4}^{0}\left[O_{4}^{0}+5O_{4}^{4}\right]\,.$ (19)

###  III.2  Effective 1D spin-1/2 XXZ model

For the octahedron $\rm CoO_{6}$, the crystal field Hamiltonian Eq. (19) has
already been diagonalized in the $\Gamma_{L=3}$ manifold Eq. (5). As a result,
the triplet $T_{1}$ has $360B_{4}^{0}$, the triplet $T_{2}$ has
$-120B_{4}^{0}$ and the singlet $A_{2}$ has $-720B_{4}^{0}$. Because
$B_{4}^{0}<0$ for $3d^{7}$ electrons, the ground manifold of $\rm CoO_{6}$ is
triplet $T_{1}$. If the energy gap between the ground state and the excited
state is large, we can use a pseudo angular momentum to describe the ground
manifold Fazekas1999 ; AABB1970 . One can find that the matrix representation
of ${L}$ in the $T_{1}$ manifold can be effectively considered as it in
$p$-basis (i.e. $Y^{1}_{1},\ Y_{1}^{0},\ Y_{1}^{-1}$) with projection
coefficient $-3/2$. Thus, we can replace ${L}$ by $-3/2\tilde{L}$ where
$\tilde{L}=1$. In this sense, $3d^{7}$ electrons in $T_{1}$ manifold behave as
if they were $p$-electrons carrying $\tilde{L}=1$.

Figure 6:  The six Kramers doublets due to crystal field splitting.

Recalling the total spin $S_{tot}=3/2$, the ground state is $3\times
4=12$-fold degenerate, which can be labeled as $|m_{l},m_{s}\rangle$ where
$m_{l}=-1,0,1$ and $m_{s}=-3/2,-1/2,1/2,3/2$ are components of $\tilde{L}$ and
$S_{tot}$, respectively. The SOC effect and the distortion play non-negligible
roles in the materials we discuss, which can lift the 12-fold degeneracy to
certain degree. The perturbation Hamiltonian follows

$H^{\prime}=-\frac{3}{2}\lambda{\bf\tilde{L}}\cdot\mathbf{S}_{tot}-\delta(\tilde{L}_{z}^{2}-2/3)$
(20)

where $\lambda$ and $\delta$ are coupling coefficient and distortion strength,
respectively. The Hamiltonian $H^{\prime}$ has a diagonal form in states
$|m_{j}\rangle$ where $m_{j}=m_{l}+m_{s}$ is the component of the total
angular momentum $J=\tilde{L}+S_{tot}$. Due to the time reversal symmetry of
$H^{\prime}$, $|\pm m_{j}\rangle$ are degenerate and form a Kramers doublet.
In total, there are $2\times 5/2+1=6$ Kramers doublets whose energies as a
function of $\delta/\lambda$ are shown in Fig. 6. The ground state doublet
with energy $E_{1}$ is given by:

$|\pm 1/2\rangle=c_{1}|\mp 1,\pm 3/2\rangle+c_{2}|0,\pm 1/2\rangle+c_{3}|\pm
1,\mp 1/2\rangle$ (21)

where $c_{1}$, $c_{2}$ and $c_{3}$ are coeffecients dependent on
$\delta/\lambda$. When the ground state doublet is well separated from the
second lowest one, we can construct the matrix representation for the total
spin $S_{tot}=3/2$ within the ground state doublet

$S^{x}_{tot}=\frac{\hbar}{2}\begin{pmatrix}0&q\\\ q&0\end{pmatrix},\quad
S^{y}_{tot}=\frac{\hbar}{2}\begin{pmatrix}0&-iq\\\ iq&0\end{pmatrix},\quad
S^{z}_{tot}=\frac{\hbar}{2}\begin{pmatrix}p&0\\\ 0&-p\end{pmatrix}$ (22)

where $q=c_{2}^{2}+\sqrt{3}c_{1}c_{3}$ and
$p=({3}c_{1}^{2}+c_{2}^{2}-c_{3}^{2})/2$. Then, in the lowest-lying Kramers
doublet, the $S_{tot}=3/2$ can be effectively described by the pesudospin
$S=1/2$,

$S_{tot}^{x}=qS^{x},\quad S_{tot}^{y}=qS^{y},\quad S_{tot}^{z}=pS^{z}.$ (23)

In light of this, the pesudospin S=1/2 exhibits an anisotropy in $S^{z}$
component. Similar results for trigonal distortion can also be obtained
lines_magnetic_1963 ; shiba_exchange_2003 ; abragam_1951 ; achiwa_linear_1969
.

Based on superexchange mechanism Anderson1959 ; Anderson1994 ; SAWATZKY1976 ,
the interaction of the neighbouring total spin is isotropic, thus the
effective spin-3/2 Hamiltonian is the XXX model,

$H_{XXX}=J\sum_{i=1}^{N}\left(S^{x}_{i}S^{x}_{i+1}+S^{y}_{i}S^{y}_{i+1}+S^{z}_{i}S^{z}_{i+1}\right).$
(24)

However, for SCVO and BCVO, the low-energy physics can be effectively
described by pseudospin $S=1/2$ instead of the total spin $S_{tot}=3/2$ [c.f.
Eq. (23)]. Thus, the effective spin-1/2 Hamiltonian for SCVO and BCVO follows

$H_{XXZ}=J\sum_{i=1}^{N}\left(S^{x}_{i}S^{x}_{i+1}+S^{y}_{i}S^{y}_{i+1}+\Delta
S^{z}_{i}S^{z}_{i+1}\right),$ (25)

where $J>0$ and $\Delta={p^{2}}/{q^{2}}$ represents the Ising anisotropy. In
the BCVO and SCVO the energy gap between the ground state doublet and second
lowest one is large bera_magnetic_2014 ; bera_string_dispersions_2020 , thus
the effective Hamiltonian Eq. (25) provides a very good theoretical
description for those materials, which is detailedly discussed in Sec. IV.
With similar crystal field environment, Co-based materials usually exhibits an
Ising-anisotropy in the spin-spin interaction, such as the quasi-2D material
$\rm Ba_{3}CoSb_{2}O_{9}$ Kamiya2018 ; Ghioldi2015 . But this is not always
the case Kim_2021 , since the origin of the spin-spin interaction is rather
complicated compared with the simple superexchange mechanism.

###  III.3  the Landé $g$ tensor of $\rm CoO_{6}$ screw chains

Figure 7: Schematic view of (a) $\rm CoO_{6}$ screw chain and (b) the relation
between the $xyz$\- and $\xi\psi\zeta$-coordinate system.

In the following we shall analyze the Landé $g$ factor for the SCVO and BCVO
materials bera_magnetic_2014 ; He_SCVO_2006 ; kimura_collapse_2013 ;
faure_topological_2018 . The SCVO and BCVO are quasi-1D material with four-
fold periodicity along the $\rm CoO_{6}$ screw chain [Fig. 7 (a)]. Due to
local slight inclinations, the local $\xi\psi\zeta$-coordinate system for a
local $\rm CoO_{6}$ octahedron has a tilted angle from the right-handed
$xyz$-coordinate system for the lab frame [Fig. 7 (b)]. As such, the
$\tilde{g}$ factor becomes a tensor from the view of $xyz$ frame where in
experiments the $z$-axis corresponds to the $c$-axis, $x$-axis is set along
the transverse field, and $y$-axis is perpendicular to the $xz$-plane.

The local $\xi\psi\zeta$-coordinate system is given by rotating the
$xyz$-coordinate system around the $y$-axis by $\theta$ and then around the
$z$-axis by $\phi$. Thus, the effective $\tilde{g}$ factor expressed in the
$xyz$-coordinate system follows

$\tilde{g}_{xyz}=\begin{pmatrix}{g_{xx}}&{g_{xy}}&{g_{xz}}\\\
{g_{xy}}&{g_{yy}}&{g_{yz}}\\\ {g_{xz}}&{g_{yz}}&{g_{zz}}\end{pmatrix},$ (26)

where

$\displaystyle g_{xx}$
$\displaystyle=\left(g_{\xi}\cos^{2}\theta+g_{\zeta}\sin^{2}\theta\right)\cos^{2}\phi+g_{\psi}\sin^{2}\phi,$
(27) $\displaystyle g_{yy}$
$\displaystyle=\left(g_{\xi}\cos^{2}\theta+g_{\zeta}\sin^{2}\theta\right)\sin^{2}\phi+g_{\psi}\cos^{2}\phi,$
$\displaystyle g_{zz}$
$\displaystyle=g_{\xi}\sin^{2}\theta+g_{\zeta}\cos^{2}\theta,$ $\displaystyle
g_{xy}$ $\displaystyle=\left(g_{\xi}\cos^{2}\theta-
g_{\psi}+g_{\zeta}\sin^{2}\theta\right)\frac{\sin 2\phi}{2},$ $\displaystyle
g_{yz}$
$\displaystyle=\left(g_{\zeta}-g_{\xi}\right)\sin\theta\cos\theta\sin\phi,$
$\displaystyle g_{xz}$
$\displaystyle=\left(g_{\zeta}-g_{\xi}\right)\sin\theta\cos\theta\cos\phi.$

Here, $g_{\xi}$, $g_{\psi}$ and $g_{\zeta}$ are the values of $g$ factor in
the $\xi\psi\zeta$-coordinate system. In the screw chain, the angle $\phi$
shifts by $\pi/2$ if we change $\rm Co^{2+}$ site to the next one along the
chain, reflecting the four-fold periodicity of the screw structure. We denote
$\phi_{1}$ the angle between $y$-axis and $\psi$-axis for the site 1 in the
screw chain. Then the four-fold periodicity condition is
$\phi=\phi_{1}+(j-1)\pi/2$ where $j=1,2,\ldots,N$ is the site index. Thus,
along the chain, the Zeeman effect of a transverse field $\bf H_{0}$ follows

$\begin{split}{H}_{Zeeman}&=\mu_{B}\sum_{j}\mathbf{S}_{j}\cdot\tilde{g}_{xyz}\cdot\mathbf{H}_{0}\\\
&=\mu_{B}\sum_{j}\left[S_{j}^{x}g_{xx}\text{H}_{0}+S_{j}^{y}g_{xy}\text{H}_{0}+S_{j}^{z}g_{xz}\text{H}_{0}\right],\end{split}$
(28)

where

$\displaystyle g_{xx}$
$\displaystyle=\left(g_{\xi}\cos^{2}\theta+g_{\zeta}\sin^{2}\theta\right)\cos^{2}\left[\phi_{1}+\frac{\pi}{2}(j-1)\right]+g_{\psi}\sin^{2}\left[\phi_{1}+\frac{\pi}{2}(j-1)\right],$
(29) $\displaystyle g_{xy}$
$\displaystyle=\frac{1}{2}\left(g_{\xi}\cos^{2}\theta-
g_{\psi}+g_{\zeta}\sin^{2}\theta\right)\sin\left[2\phi_{1}+\pi(j-1)\right],$
$\displaystyle g_{xz}$
$\displaystyle=\frac{1}{2}\left(g_{\zeta}-g_{\xi}\right)\sin
2\theta\cos\left[\phi_{1}+\frac{\pi}{2}(j-1)\right].$

Figure 8: Two cases with different directions of magnetic field.

Next we present two cases with different directions of transverse field, i.e.
$\phi_{1}=0^{\circ}$ and $\phi_{1}=45^{\circ}$. We should bear in mind that
$\hat{x}//\mathbf{H}_{0}$. In $\phi_{1}=0^{\circ}$ case, namely,
$\mathbf{H}_{0}$ is along $[110]$ direction [Fig. 8 (a)], then the components
of $\tilde{g}$ factor become

$\displaystyle g_{xx}$
$\displaystyle=\left(g_{\xi}\cos^{2}\theta+g_{\zeta}\sin^{2}\theta\right)\cos^{2}\left[0+\frac{\pi}{2}(j-1)\right]+g_{\psi}\sin^{2}\left[0+\frac{\pi}{2}(j-1)\right],$
(30) $\displaystyle g_{xy}$
$\displaystyle=\frac{1}{2}\left(g_{\xi}\cos^{2}\theta-
g_{\psi}+g_{\zeta}\sin^{2}\theta\right)\sin\left[0+\pi(j-1)\right]=0,$
$\displaystyle g_{xz}$
$\displaystyle=\frac{1}{2}\left(g_{\zeta}-g_{\xi}\right)\sin
2\theta\cos\left[0+\frac{\pi}{2}(j-1)\right],$

which indicates that the transverse field along [110] direction can induce a
four-periodic field but without staggered field.

In $\phi_{1}=45^{\circ}$ case, namely, $\mathbf{H}_{0}$ is along $[100]$
direction [Fig. 8 (b)], then the components of $\tilde{g}$ factor become

$\displaystyle g_{xx}$
$\displaystyle=\frac{1}{2}\left[g_{\xi}\cos^{2}\theta+g_{\zeta}\sin^{2}\theta+g_{\psi}\right],$
(31) $\displaystyle g_{xy}$
$\displaystyle=(-1)^{j}\frac{1}{2}\left(g_{\xi}\cos^{2}\theta-
g_{\psi}+g_{\zeta}\sin^{2}\theta\right),$ $\displaystyle g_{xz}$
$\displaystyle=\frac{1}{2}\left(g_{\zeta}-g_{\xi}\right)\sin
2\theta\cos\left[\frac{\pi}{4}(2j-1)\right],$

which indicates that the transverse field along [100] direction can induce
staggered and four-periodic fields. One can obtain a similar result for the
[010] direction. Now we are ready to discuss magnetic excitations in the 1D
spin-1/2 XXZ model with various external fields, as well as their experimental
realization in the BCVO and SCVO materials.

##  IV Magnetic Excitations in the 1D Spin-1/2 XXZ Model with Various External
Fields

Depending on the directions of the applied external field, the magnetic
excitations in the spin-1/2 XXZ chain exhibit rich emergent phenomena and
exotic physics, e.g., Tomonaga-Luttinger liquid tomonaga_remarks_1950 ;
Luttinger_1960 ; Luther_TLL_1975 , spinon FADDEEV_spinwave_1981 ;
faddeev_spectrum_1984 ; Muller_spinon_1981 ; Karbach_spinon_1997 ;
Bougourzi_spinon_1996 ; Bougourzi_spinon_1998 ; caux2008 , string bethe1931 ;
takahashi_1D_1971 ; Gaudin_XXZ_1971 ; Taka_suzuki_XXZ_1972 ;
kohno_string_dynamically_2009 , and $E_{8}$ particles
a_b_zamolodchikov_integrals_1989 ; jianda_E8_2014 ; DELFINO_1995 . To help
measure or probe those exotic magnetic excitations, a study on the spin
dynamical structure factor (SDSF) becomes a must, which can reveal weight
distributions and dispersions of various magnetic excitations on one hand, and
give a direct guidance for their experimental realization on the other hand
Negele1988 ; chaikin_lubensky_1995 ; Zhu_MTCMM_2005 . At zero temperature, the
SDSF follows the Fourier transform of the spin correlation function in space
and time,

$S^{a\bar{a}}(q,\omega)=\frac{1}{N}\sum_{jj^{\prime}}^{N}e^{-iq(j-j^{\prime})}\int^{+\infty}_{-\infty}dte^{i\omega
t}\langle GS|S_{j}^{a}(t)S_{j^{\prime}}^{\bar{a}}(0)|GS\rangle$ (32)

where $a\in\\{-,+,z\\}$ and $|GS\rangle$ is the ground state. Using the
Lehmann representation, it reduces to a single summation over a complete basis
set,

$S^{a\bar{a}}(q,\omega)=2\pi\sum_{\alpha}|\langle
GS|S_{q}^{a}|\alpha\rangle|^{2}\delta(\omega-\omega_{\alpha})$ (33)

where $\omega_{\alpha}=E_{\alpha}-E_{0}$ is the energy difference between
excited state $|\alpha\rangle$ and ground state $|GS\rangle$. In Eq. (33), the
spectral weight $|\langle GS|S_{q}^{a}|\alpha\rangle|^{2}$ describes the
transition probability between $|GS\rangle$ and $|\alpha\rangle$ after an
external perturbation (the incoming photon or neutron) which couples linearly
to the $S^{a}_{q}$ operator Negele1988 ; chaikin_lubensky_1995 ;
Zhu_MTCMM_2005 . Therefore, based on spectral weight distribution $|\langle
GS|S_{q}^{a}|\alpha\rangle|^{2}$ from different excitations $|\alpha\rangle$
in the energy-momentum space, we can carry out detailed comparison between
theoretical calculation and experimental measurement for recognizing possible
realizations of various types of magnetic excitations. In the following, we
will discuss the magnetic excitations in the spin-1/2 XXZ chain with various
external fields and their experimental realizations.

###  IV.1  Zero field

Figure 9:  The density plots for $S^{-+}$ at different anisotropy $\Delta$.
From caux2008 .

At zero field, ground state of the 1D spin-1/2 XXZ model [Eq. (25)]
accommodates a long-range AFM ordering with an energy gap dependent on the
Ising anisotropic parameter $\Delta$. Above the gap, spin-flip excitations
fractionalize into pairs of spinons with fractional quantum number $S=1/2$
FADDEEV_spinwave_1981 . By studying the SDSF of the system, we are able to
explore how the spinons influence the dynamics of the system. For the
$S^{-+}(q,\omega)$ [c.f. Eq. (33)], the two-spinon excitations provide a
gapped continuum spectrum caux2008 , as shown in Fig. 9. The two-spinon
excitations are a small fraction of the total number of states, however, they
carry almost 100% spectral weight in $S^{-+}$ channel if $\Delta\geq 2$. When
$\Delta<2$ multi-spinon states like four-spinons excitations play a more and
more important role with decreasing $\Delta$ toward the isotropic limit
($\Delta=1$) caux2008 ; Caux_2006 . In $S^{zz}(q,\omega)$ channel, two-spinon
excitations have similar results as in $S^{-+}(q,\omega)$ castillo_exact_2020
. The analytical SDSF provides a concrete ground for the experimental
realization. Indeed, INS measurements on the quasi-1D material SCVO directly
observe the expected excitation spectrum. In SCVO, the long-range AFM order
appears below its Néel temperature $T_{N}=5.2$ K. 1D physics is expected to
take over dominancy when $T>T_{N}$, where SCVO can be effectively described by
the 1D spin-1/2 XXZ model Eq. (25). The INS measurements at 6 K reveal a
gapped scattering spectrum [Fig. 10] with excellent agreement between the
theoretical and experimental results bera_spinon_2017 . Additionally, Fig.
10(a) also exhibits resonance peaks caused by thermal fluctuation and referred
to as the Villian modes Villain_1975 . These modes are firstly observed in
$\rm CsCoBr_{3}$ by INS experiments and can be explained by the scattering
between two domain-wall states Nagler_villian_PRL_1982 ;
Nagler_villian_PRB_1983 ; James_XXZ_FiniteT_2009 . Next, we shall discuss the
magnetic excitations in 1D spin-1/2 XXZ model with the presence of various
external magnetic fields.

Figure 10: Dynamical spectrum of spinons. (a) INS measurement in SCVO and (b)
Bethe ansatz calculation. The theoretical dispersion of the Villian mode is
represented by the dashed line and the yellow points are used to highlight
experimental peaks in (a). From bera_spinon_2017 .

###  IV.2 Longitudinal field

With a longitudinal field $h$, the Hamiltonian becomes

$H=H_{XXZ}-h\sum_{i=1}^{N}S^{z}_{i},$ (34)

where the time reversal symmetry is broken but the $U(1)$ symmetry is
preserved. If the longitudinal field is small such that the gap holds, i.e.,
the AFM ground state maintains, there is no magnetization. The magnetization
starts to develop when $h$ is tuned above the field threshold $h_{c}(\Delta)$
where the gap closes, and then the system enters the quantum critical
(gapless) regime. In this region, the spinon can no longer serve as a good
quasi-particle to describe the excitations in the model. Instead, the dynamics
is dominated by fractional excitations known as “psinon-psinon” (PP) and
“psinon-antipsinon” (PAP) Karbach_pp_pap_2002 ; karbach2000III , as well as
exotic string excitations yang_string_one-dimensional_2019 . For observing
those exotic states, an SDSF study needs to be carried out in order to probe
the excitations and quantify their spectra contributions.

In the $S^{-+}(q,\omega)$ channel, fractional excitations PP have a similar
SDSF shape as that of spinon [Fig. 11] at small magnetization and disappears
when the magnetization becomes saturated yang_string_one-dimensional_2019 . In
both zero and nonzero field cases, these fractional excitations dominate
almost the whole spectrum implying negligibly small contribution from the
string excitations in the $S^{-+}(q,\omega)$ channel. In a sharp contrast, for
the $S^{+-}(q,\omega)$ channel [Fig. 11], contributions from low-energy
factional excitations PAP diminish in the zero field limit and recover the
spin-wave excitations in the limit of full polarization. String excitations
take over the dominance at the low-magnetization regions, and tend to vanish
when the system approaches full polarization. At relatively low magnetization,
three well-separated continuums indicate three different contributions from
fractional PAP, 2-string and 3-string excitations. The theoretical progress
provides a concrete guidance to directly probe the string excitations in real
materials.

Figure 11:  The density plots for $S^{-+}$ (top row) and $S^{+-}$ (bottom row)
of the 1D spin-1/2 XXZ model, where the magnetization is 20% (a1, a2), 50%
(b1, b2) and 80% (c1, c2) of full polarization. From yang_string_one-
dimensional_2019 .

In spite of the intriguing many-body nature of the excitations, especially the
string excitations, it is a great challenge to directly detect or realize
these exotic excitations in real materials Imambekov_1D_2012 ;
kohno_string_dynamically_2009 ; Pereira_edge_2008 ; Pereira_spectral_2009 ;
Caux_comput_2005 ; Caux_computation_2005 ; Shashi_Nonuniversal_2011 ;
Ganahl_Observation_2012 . In 2017, a silver lining appears, which suggests
that the field-induced quantum critical region of the 1D spin-1/2 XXZ model
Eq. (34) is a promising region to directly observe the string excitations
yang_string_one-dimensional_2019 . Following the suggestion, a high-resolution
THz spectroscopy measurement observes the string excitations in SCVO at the
zone cneter for the first time wang_string_experimental_2018 . Under the
concrete theoretical guidance, 2- and 3-string excitations as well as low-
energy fractional excitations (PP and PAP) are probed [Fig. 12 (a)]. In 2020,
the dispersion relation of the string excitations over the full Brillouin zone
is obtained via the INS measurement on the same material
bera_string_dispersions_2020 . Fig. 12 (b) shows that the dispersion for the
2- and 3-string excitations appears in the intermediate (2 - 15 meV at 6 T)
and high (4 - 15.5 meV at 9 T) energy regions, respectively. In both
experiments, the obtained excitation spectra and their magnetic field
dependencies are perfectly consistent with theoretical predictions, closing
the long-time open problem to directly observe the string excitations.

Figure 12: (a) THz spectroscopy; (b1, b2) INS measurement of SCVO; and (b3,
b4) Bethe ansatz calculations. From wang_string_experimental_2018 ;
bera_string_dispersions_2020 .

###  IV.3 Transverse field

With presence of a transverse field the XXZ Hamiltonian becomes

$H=H_{XXZ}+H_{x}\sum_{i=1}^{N}S^{x}_{i}\,.$ (35)

With increasing field, a quantum phase transition emerges, which falls in the
class of TFIC universality dmitriev_1D_2002 . For realizing the TFIC
universality in real material, distinctive criticality is desired for
discerning the TFIC universality from others. It is found that the Grüneisen
ratio, directly related to the magnetocaloric effect, exhibits a very unique
quantum critical scaling. The Grüneisen ratio either approaches to a constant
or divergence when the QCP of the TFIC universality is accessed via decreasing
temperature or tuning field, respectively Jianda_crossover_2018 . The critical
behaviors are different from general quantum criticality where the Grüneisen
ratio in general approaches to divergence regardless how to access the QCP
Zhu_Universally_2003 . As such measurements of Grüneisen ratio near a QCP of
the real material can serve as a smoking gun to justify whether the
universality falls in the TFIC universality.

As discussed in Sec. III.3, in the BCVO and SCVO, when the applied field is
along [100] direction, there are two field-induced terms [c.f. Eq. (28)] in
the Hamiltonian,

$\begin{split}H&=H_{XXZ}-H_{x}\sum_{i=1}^{N}\left\\{S_{i}^{x}+h_{y}(-1)^{i}S_{i}^{y}+h_{z}S^{z}_{i}\cos\left[{\pi(2i-1)}/{4}\right]\right\\}\,,\end{split}$
(36)

where $h_{y}\approx 0.4$ and $h_{z}\approx 0.14$ are the internal-induced
staggered and four-periodic fields (reduced by $H_{x}$), respectively. And if
the applied field is along [110] direction, only the four-periodic field with
slightly different form is present [c.f. Eq. (30)]. Although the Hamiltonian
Eq. (36) becomes more involved compared with Eq. (35), it still preserves the
TFIC universality when the system is tuned to its QCP. It is found that the
four-periodic perturbation term only slightly changes the location of the QCP,
while the staggered-field can significantly reduce the field strength to
access the QCP Zou_universality_2019 ; faure_topological_2018 . Thus, from
above theoretical analysis, both BCVO and SCVO effectively described by 1D
spin-1/2 XXZ model, can accommodate the TFIC universality via tuning the
transverse field properly.

Figure 13:  The Grüneisen parameter for BCVO vs. transverse field (a) and
temperature (b). Phase diagram of SCVO vs. transverse field (c). From
cui_tfic_quantum_2019 ; wang_tfic_quantum_2018 .

As expected, by applying a magnetic field along [110] direction, the TFIC
universality is realized in BCVO where the critical field is around 40 T
wang_tfic_quantum_2018 . The measured Grüneisen ratio is divergent when the
transverse field approaches 40 T but converges when the temperature decreases
at 40 T [Fig. 13 (a) and (b)], consistent with the theoretical analysis
Jianda_crossover_2018 ; wang_tfic_quantum_2018 . If the magnetic field is
along [100] direction, much lower critical fields can be obtained in both SCVO
and BCVO Zou_universality_2019 ; cui_tfic_quantum_2019 ; zou_e_8_2021 . For
instance, in SCVO, two QCPs at $H_{C1}=7$ T and $H_{C2}=7.7$ T are determined
by NMR experiment cui_tfic_quantum_2019 . The former one is a (3+1)D Ising
critical point while the later one falls in the class of TFIC universality
[Fig. 13 (c)]. Near the critical field $H_{C2}$, the critical exponent of
transverse field is found to be consistent with the 1D TFIC universality class
sachdev_2011 ; cui_tfic_quantum_2019 .

###  IV.4 More field and the $E_{8}$ physics

Figure 14:  The quantum $E_{8}$ integrable model emerges in the region of blue
solid line. Masses of the eight particles are expressed in units of the
lightest two particles’ masses $m_{1}$ and $m_{2}$. From xiao_cascade_2021 .

In the vicinity of QCPs, the low energy excitations in the infrared (IR) limit
could be described by conformal field theory (CFT) due to the scaling
invariant and large conformal symmetry. Moreover, with perturbation
deformation from relevant primary field of corresponding CFT, the original
physics in the IR limit will be greatly influenced. For instance, it may turn
to massive field theory associated with certain Lie algebra, dubbed as affine
toda field theory (ATFT) BRADEN1990689 ; MUSSARDO1992215 . In the scaling
limit, the TFIC universality can be described by a CFT with central charge
$c=1/2$. With an additional small longitudinal-field perturbation, an
integrable massive quantum field theory (QFT) dubbed as quantum $E_{8}$ model
emerges.

The quantum $E_{8}$ integrable model contains eight different types of massive
particles with their scattering fully described by the $E_{8}$ exceptional Lie
algebra a_b_zamolodchikov_integrals_1989 ; jianda_E8_2014 ; DELFINO_1995 ;
xiao_cascade_2021 . The $E_{8}$ Hamiltonian follows

$H_{E_{8}}=H_{c=1/2}+\lambda\int\sigma(x)dx,$ (37)

with the $c=1/2$ CFT Hamiltonian $H_{c=1/2}$ and the intensity of the small
longitudinal field in the scaling limit $\lambda$. The mass of the lightest
quasiparticle $m_{1}=C\lambda^{8/15}$, $C\approx 4.40490858$ DELFINO_1995 ;
fateev . And $\sigma(x)$ is the spin density operator, a relevant field in the
perturbed CFT a_b_zamolodchikov_integrals_1989 ; DELFINO_1995 . Fig. 14 shows
the phase diagram and massive excitations of the $E_{8}$ model.

The eigenstates of ATFTs satisfy Faddeev-Zamolodchikov algebra and can be
described by exact S-matrix theory ZAMOLODCHIKOV1979253 ; BRADEN1990689 . For
solving the spectrum of ATFTs, a programmable method dubbed as conformal
bootstrap is proposed by Cardy and Mussardo CARDY1990387 , which could
generate expressions of form factor for certain operator. Later, the form
factor theory of the $E_{8}$ model is further developed by Zamolodchikov,
Mussardo and Delfino YUROV1991 ; DELFINO_1995 ; DELFINO1996327 ;
DELFINO1996469 ; Delfino_2004 . A relevant and complete process of the
conformal bootstrap can be found in the appendix of Ref. xiao_cascade_2021 .
With another relevant field energy density (corresponding to $\sigma^{x}$ in
the lattice model), we can use $E_{8}$ form factor theory and bootstrap
approach to determine the SDSFs $S^{xx}(\omega)$, $S^{zz}(\omega)$. And
$S^{yy}(\omega)$ can be obtained via an exact relation
$S^{yy}(\omega)=S^{zz}(\omega)\omega^{2}/(2gJ)^{2}$ jianda_E8_2014 . Fig. 15
shows the analytical SDSF with transfer momentum $q=0$ (corresponding to the
Brillouin zone center in lattice), which includes contributions from single-
and multi-$E_{8}$ particle channels up to total energy $5m_{1}$.

Figure 15: Analytical calculation of DSF with boradening of $0.05m_{1}$ has
been exhibited. (a), (b) and (c) are corresponding to $S^{xx}(\omega)$,
$S^{yy}(\omega)$ and $S^{zz}(\omega)$ respectively. All the contributions of
different peaks have been highlighted inside the figures, from
xiao_cascade_2021 .

The existence of the TFIC universality in the BCVO and SCVO materials brings
in a hope to realize the $E_{8}$ model. In order to realize the model in the
AFM materials, the perturbed longitudinal field needs to be a stagger field
along $z$ direction (the Ising spin direction) from site to site along the
chain. It is extremely difficult to directly apply such a field for condensed-
matter experiments. Different from SCVO where the 1D QCP is always outside its
3D ordering dome, for BCVO when applying a field along [100] (or [010])
direction the field strength for the corresponding 1D QCP is at
$H_{C}^{1D}\approx 4.7$T $<H_{C}^{3D}\approx 10$T, deep inside the BCVO’s 3D
ordering AFM dome [Fig. 16]. Inside the 3D ordering phase, due to the weak
interchain coupling, we can conveniently apply chain mean-field theory to
effectively describe the corresponding physics, whose effective Hamiltonian
now becomes

$\small\begin{split}H&=H_{XXZ}-H_{x}\sum_{i=1}^{N}\left\\{S_{i}^{x}+h_{y}(-1)^{i}S_{i}^{y}+h_{z}S^{z}_{i}\cos\left[{\pi(2i-1)}/{4}\right]\right\\}-H^{\prime}\sum_{i}(-1)^{i}S_{i}^{z},\end{split}$
(38)

where $J=5.8$ meV in $H_{XXZ}$ and the staggered perturbation
$-H^{\prime}\sum_{i}(-1)^{i}S_{i}^{z}$ with $H^{\prime}=0.018J$ comes from the
chain mean-field of the interchain coupling, which can not be neglected in the
3D AFM ordering region. The effective staggered field along $z$ direction now
can serve as the key perturbation field to realize the quantum $E_{8}$ model.
Near $H_{C}^{\rm 1D}$, the INS is carried out to probe the magnetic
excitations in the BCVO, and the long-desired $E_{8}$ spectrum consists of all
the eight single-$E_{8}$ particle peaks and multi-particle continuum are
observed for the first time zou_e_8_2021 , as shown in Fig. 17. This result is
fully consistent with the analytical calculation and the iTEBD calculation
zou_e_8_2021 .

Figure 16:  (a) Phase diagram of the BCVO with a transverse field along [010]
direction; (b) the quantum criticality of TFIC universality verified by NMR
experiment. From zou_e_8_2021 .  Figure 17: The $E_{8}$ spectrum. (a) INS
data, (b) analytical results, and (c) numerical results, where $0.1m_{1}$
energy resolution broadening is applied to the latter two results for
comparison with INS data. (d), (e), (f) and (g) reveal the analytical
contributions of the spectrum from different $E_{8}$ channels. From
zou_e_8_2021 .

It is worth discussing a little more about how the comparison is completed,
since it is a nontrivial task to compare continuous results (of field theory)
with discrete results (of lattice calculations and experiments). In the field
theory the mass of lightest $E_{8}$ particle $m_{1}$ is related to the
longitudinal field as $m_{1}=C\lambda^{8/15}$, which plays the role of IR
cutoff energy scale. When down to lattice we have a simple estimation
$m_{1}^{lattice}=C^{\prime}H^{\prime 8/15}\approx 1.20$ meV where
$C^{\prime}\approx 0.91C$ CASELLE2000667 . The estimated $m_{1}^{lattice}$ is
close to the real experimental observation, therefore, we can safely pick up
the first peak in the experimental data as the corresponding $m_{1}$ in the
$E_{8}$ field theory. Now we can make a detailed comparison of the analytical
and experimental (numerical) zero-transfer-momentum DSF by using $m_{1}$ and
the first peak of experimental (and numerical data) to re-scale their DSF
spectra, respectively. The comparison as shown in Fig. 17, shows excellent
agreement between discrete and analytical results, which allows us to
confidently claim that the $E_{8}$ physics is indeed observed in real
experiments.

In the comparison, we should also note that when taking the space continuum
limit, $J$ for the nearest spin interaction in the lattice model indeed should
serve as a UV energy cutoff for the applicability of the corresponding quantum
field theory. However, since $J$ is the energy scale for local spin
interaction in the lattice, then, after taking the space continuum limit, it
actually appears as a cutoff scale for the energy of local or short-range
fluctuations in the quantum field theory. In the experiment (Fig. 17), the
observed excitations at the zone center are coherent excitations involving
large amounts of spins at long-range scale (zone center implies zero transfer
momentum thus corresponding to long-range scale). The excitation energy at
such long-range scale cannot be simply considered as the same energy for the
local or short-range spin interaction. Although it is highly nontrivial to
accurately determine the energy of local fluctuation (here it is the strength
of nearest spin interaction) from a many-body wavefunction, it should be safe
to claim that even the energy cost to get a many-body coherent excitation at
long-range scale is close to $J$, on average the corresponding energy of local
fluctuation should be still far smaller than $J$. This should provide an
underlying physical reason for the excellent comparison up to one $J$ as shown
in Fig. 17. Actually, even the energy cost is beyond $J$ for the long-range-
scale (corresponding to small (transfer) momentum) coherent excitation, the
field-theory prediction can still agree well with the lattice result, as
demonstrated in Ref. Keselman_Dynamical_2020  which carefully compares field
theory results based on Bosonization with lattice calculation of the
Heisenberg chain, and in Ref. Wybo_Quantum_2022  which carefully compares
sine-Gordon field theory results with lattice calculation of ladder XXZ chain.
Therefore, when considering the applicability of quantum field theory to the
corresponding lattice model, it is not simply about energy scale (or time
scale), the momentum scale (or length scale) also plays a crucial role.

##  V Conclusion

This review details rich physics in the 1D spin-1/2 XXZ model with the
presence of various external fields and their experimental realization.
Because the experimental realization is highly relevant to the quasi-1D AFM
materials, we first provide a systematic and pedagogical analysis for setting
up the effective Hamiltonian for the BCVO and SCVO, which can also be applied
to other Co-based materials with cautions on possible changes of symmetry for
the local environment. Then, we review recent theoretical progress on a
variety of magnetic excitaions of the 1D spin-1/2 XXZ model, such as spinon
(without magnetization), (anti)psinon and string excitations (with finite
magnetization), and their experimental observations. With the presence of
transverse field, we further discuss how the TFIC universality and the exotic
$E_{8}$ physics (with additional small longitudinal field) emerge near the
corresponding 1D QCP, as well as their material realizations.

The 1D spin-1/2 XXZ model with various external fields contains rich magnetic
physics, whose exotic excitations from low to high energy are now successfully
realized in real materials. Those concrete progresses can further inspire and
bridge many research fields such as quantum statistical field theory, cold
atom, and AdS/CFT etc.. Moreover, it is worth to further explore physical
properties of those exotic magnetic excitations, which will be an important
step toward practical control and application of those exotic magnetic
excitations.

Acknowledgements
The work is support by the Innovation Program for Quantum Science and
Technology No. 2021ZD0301900, the Natural Science Foundation of Shanghai with
grant No. 20ZR1428400, and Shanghai Pujiang Program with grant No.
20PJ1408100. J.W. acknowledges additional support from a Shanghai talent
program.

##  Appendix A Cubic field potential in octahedron

We consider the potential $\mathcal{V}(x,y,z)$ provided by the ions with
charge $q$ at the corners of an octahedron; i.e. at $(\pm a,0,0)$, $(0,\pm
a,0)$ and $(0,0,\pm a)$ [Fig. 4 in main text]. Then it reads,

$\begin{split}\mathcal{V}(x,y,z)&=V_{x}+V_{y}+V_{z}\\\
&=\sum_{\alpha=x,y,z}\left[\frac{q}{\left(r^{2}+a^{2}-2a\alpha\right)^{1/2}}+\frac{q}{\left(r^{2}+a^{2}+2a\alpha\right)^{1/2}}\right]\\\
&=\frac{q}{\sqrt{A}}\left[(1+X)^{-1/2}+(1-X)^{-1/2}\right.\\\
&\quad\left.(1+Y)^{-1/2}+(1-Y)^{-1/2}+(1+Z)^{-1/2}+(1-Z)^{-1/2}\right]\end{split}$
(39)

where $r^{2}=x^{2}+y^{2}+z^{2}$ is an arbitrary spatial point near the origin
and

$\begin{split}&A=(r^{2}+a^{2}),\quad B=\frac{r^{2}}{a^{2}},\quad
X=\frac{2ax}{A}=\frac{2x}{a}\frac{1}{1+B},\quad Y=\frac{2ay}{A},\quad
Z=\frac{2az}{A}.\end{split}$ (40)

Next, considering $a>r$, we expand Eq. (39) up to terms with sixth order of
$r/a$. Note that,

$(1+X)^{-1/2}+(1-X)^{-1/2}=2+\frac{3}{4}X^{2}+\frac{35}{64}X^{4}+\frac{231}{512}X^{6}+\mathcal{O}\left(\frac{r^{8}}{a^{8}}\right),$
(41)

Eq. (39) becomes,

$\begin{split}\mathcal{V}(x,y,z)&=\frac{q}{\sqrt{A}}\left[6+\frac{3}{4}(X^{2}+Y^{2}+Z^{2})+\frac{35}{64}(X^{4}+Y^{4}+Z^{4})+\frac{231}{512}(X^{6}+Y^{6}+Z^{6})\right]\\\
&\quad\frac{q}{a}\left\\{6-3\frac{r^{2}}{a^{2}}+\frac{9}{4}\frac{r^{4}}{a^{4}}-\frac{15}{8}\frac{r^{6}}{a^{6}}+\frac{r^{2}}{a^{2}}\left(3-\frac{15}{2}\frac{r^{2}}{a^{2}}+\frac{105}{8}\frac{r^{4}}{a^{4}}\right)\right.\\\
&\quad\left.+\frac{x^{4}+y^{4}+z^{4}}{a^{4}}\left(\frac{35}{4}-\frac{315}{8}\frac{r^{2}}{a^{2}}\right)+\frac{693}{24}\frac{(x^{6}+y^{6}+z^{6})}{a^{6}}\right\\}+\mathcal{O}\left(\frac{r^{8}}{a^{8}}\right).\end{split}$
(42)

After some manipulations, we can obtain the desired expression,

$\begin{split}\mathcal{V}(x,y,z)&=\frac{6q}{a}+\frac{35q}{4a^{5}}\left[(x^{4}+y^{4}+z^{4})-\frac{3}{5}r^{4}\right]+\frac{-21q}{2a^{7}}\\\
&\quad\left[(x^{6}+y^{6}+z^{6})+\frac{15}{4}(x^{2}y^{4}+x^{2}z^{4}+y^{2}x^{4}+y^{2}z^{4}+z^{2}x^{4}+z^{2}y^{4})-\frac{15}{14}r^{6}\right]\end{split}$
(43)

where $\frac{35q}{4a^{5}}$ and $\frac{-21q}{2a^{7}}$ are prefactors $A_{4}$ in
Eq. (9) and $A_{6}$ Eq. (10), respectively.

For an alternative way, we can express the potential
$\mathcal{V}(r,\theta,\phi)$ in terms of spherical harmonics, or tesseral
harmonics. The tesseral harmonics are defined as,

$\begin{split}Z_{n0}&=Y_{n}^{0},\\\
Z_{nm}^{+}&=\frac{1}{\sqrt{2}}[Y_{n}^{-m}+(-1)^{m}Y_{n}^{m}],\quad\quad m>0\\\
Z_{nm}^{-}&=\frac{i}{\sqrt{2}}[Y_{n}^{-m}-(-1)^{m}Y_{n}^{m}],\quad\quad
m>0\end{split}$ (44)

which are always real and can be easily converted to cartesian coordinate
functions. Then, the potential $\mathcal{V}(r,\theta,\phi)$ reads,

$\mathcal{V}(r,\theta,\phi)=\sum_{n}^{\infty}\sum_{\alpha}r^{n}\gamma_{n\alpha}Z_{n\alpha}(\theta,\phi)$
(45)

and

$\gamma_{n\alpha}=\sum_{j=1}^{k}q_{j}\frac{4\pi}{(2n+1)}\frac{Z_{n\alpha}}{R_{j}^{(n+1)}},$
(46)

where the sum runs over all the neighboring anions, $k=6$ in our octahedron
case. Based on the analysis in the main text, only $\gamma_{00}$,
$\gamma_{40}$, $\gamma_{60}$, $\gamma_{44}^{+}$ and $\gamma_{64}^{+}$ have
non-vanishing contributions. And using Eq. (46), they can be obtained

$\begin{split}&\gamma_{00}=\frac{12\sqrt{\pi}}{a}q,\quad\gamma_{40}=\frac{7\sqrt{\pi}}{3}\frac{q}{a^{5}},\quad\gamma_{60}=\frac{3\sqrt{\pi}}{2\sqrt{13}}\frac{q}{a^{7}},\\\
&\gamma_{44}^{+}=\frac{\sqrt{35\pi}}{3}\frac{q}{a^{5}},\quad\gamma_{64}^{+}=-\frac{3\sqrt{7\pi}}{2\sqrt{13}}\frac{q}{a^{7}}.\end{split}$
(47)

Then, if we neglect the $Y_{0}^{0}$, the potential
$\mathcal{V}(r,\theta,\phi)$ becomes the desired form,

$\begin{split}\mathcal{V}(r,\theta,\phi)&=\frac{7\sqrt{\pi}}{3}\frac{q}{a^{5}}\left[Y_{4}^{0}(\theta,\varphi)+\sqrt{\frac{5}{14}}\left(Y_{4}^{4}(\theta,\varphi)+Y_{4}^{-4}(\theta,\varphi)\right)\right]r^{4}\\\
&+\frac{3\sqrt{\pi}}{2\sqrt{13}}\frac{q}{a^{7}}\left[Y_{6}^{0}(\theta,\varphi)-\sqrt{\frac{7}{2}}\left(Y_{6}^{4}(\theta,\varphi)+Y_{6}^{-4}(\theta,\varphi)\right)\right]r^{6}\end{split}$
(48)

where $\frac{7\sqrt{\pi}}{3}\frac{q}{a^{5}}$ and
$\frac{3\sqrt{\pi}}{2\sqrt{13}}\frac{q}{a^{7}}$ are prefactors $D_{4}$ in Eq.
(9) and $D_{6}$ Eq. (10), respectively.

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