# Spontaneous dimerization, spin-nematic order, and deconfined quantum
critical point in a spin-$1$ Kitaev chain with tunable single-ion anisotropy

Qiang Luo<EMAIL_ADDRESS>College of Physics, Nanjing University of
Aeronautics and Astronautics, Nanjing, 211106, China  Key Laboratory of
Aerospace Information Materials and Physics (NUAA), MIIT, Nanjing, 211106,
China  Shijie Hu  Beijing Computational Science Research Center, Beijing
100084, China  Jinbin Li  College of Physics, Nanjing University of
Aeronautics and Astronautics, Nanjing, 211106, China  Key Laboratory of
Aerospace Information Materials and Physics (NUAA), MIIT, Nanjing, 211106,
China  Jize Zhao  School of Physical Science and Technology $\&$ Key
Laboratory for Magnetism and Magnetic Materials of the MoE, Lanzhou
University, Lanzhou 730000, China  Lanzhou Center for Theoretical Physics,
Lanzhou University, Lanzhou 730000, China  Hae-Young Kee
<EMAIL_ADDRESS>Department of Physics, University of Toronto,
Toronto, Ontario M5S 1A7, Canada  Canadian Institute for Advanced Research,
Toronto, Ontario, M5G 1Z8, Canada  Xiaoqun Wang<EMAIL_ADDRESS>School
of Physics, Zhejiang University, Hangzhou 310058, China

(August 28, 2024)

###### Abstract

The Kitaev-type spin chains have been demonstrated to be fertile playgrounds
in which exotic phases and unconventional phase transitions are ready to
appear. In this work, we use the density-matrix renormalization group method
to study the quantum phase diagram of a spin-1 Kitaev chain with a tunable
negative single-ion anisotropy (SIA). When the strength of the SIA is small,
the ground state is revealed to be a spin-nematic phase which escapes
conventional magnetic order but is characterized by a finite spin-nematic
correlation because of the breaking spin-rotational symmetry. As the SIA
increases, the spin-nematic phase is taken over by either a dimerized phase or
an antiferromagnetic phase through an Ising-type phase transition, depending
on the direction of the easy axis. For large enough SIA, the dimerized phase
and the antiferromagnetic phase undergo a “Landau-forbidden” continuous phase
transition, suggesting new platform of deconfined quantum critical point in
spin-1 Kitaev chain.

###### pacs:

##  I Introduction

The celebrated Kitaev model on the honeycomb lattice Kitaev2006  and its
multitudinous variants offer unprecedented opportunities for our understanding
of exotic states of matter arising from bond-directional exchange couplings
ZhangWHB2019 ; WangNmdLiu2019 ; RalkoMerino2020 ; LuoNPJ2021 ; LeeKCetal2020 ;
GohlkeCKK2020 ; Luo2022PRB  and unconventional quantum phase transitions
(QPTs) that are beyond the Landau-Ginzburg-Wilson (LGW) paradigm FengZX2007 ;
ShiYYN2009 ; GoJungMoon2019 ; LiKimKee2022 . It is rigorously demonstrated
that the ground state of the Kitaev honeycomb model is a quantum spin liquid
(QSL) with fractionalized excitations consisting of itinerant majorana
fermions and localized $\mathbb{Z}_{2}$ vortices (visons) KnolleKCM2014 . The
quantum fluctuation can be greatly enhanced by including further nearest-
neighbor interactions and off-diagonal exchanges, giving rise to emergent
phases such as the vison crystal ZhangWHB2019 , QSLs of different nature
WangNmdLiu2019 ; RalkoMerino2020 ; LuoNPJ2021 , nematic paramagnet that breaks
lattice rotational symmetry LeeKCetal2020 ; GohlkeCKK2020 , and spin-flop
phase which can be interpreted as superfluid phase Luo2022PRB . At the same
time, smoking-gun signals of the topological QPTs are observed by the change
of Chern number and the onset of the peak in the thermal Hall conductivity
GoJungMoon2019 ; LiKimKee2022 .

While substantial efforts have been devoted to studying extended Kitaev models
in two dimensions, many intriguing phenomena regarding the collective
behaviors of the excitations remain elusive because of the numerical
challenges and limitations of different computational methods. One of the
prominent examples is the antiferromagnetic (AFM) Kitaev model subject to a
[111] magnetic field, which is shown to have an intermediate region between
the low-field non-Abelian QSL and the high-field polarized phase ZhuKSF2018 ;
GohlkeMP2018 ; HickeyTrebst2019 ; PatelaTrv2019 . The plausible perspective
which asserts that the intermediate region is a gapless QSL with spinon Fermi
surface has been challenged by a recent study, where a different scenario of
gapped QSL with a Chern number of 4 is proposed ZhangHalBat2022 . Also, it is
revealed by another work that the intermediate region is composed of two
gapped phases with finite Chern number JiangLCQLWang2020 . To reconcile these
seemingly conflicting results, attempts have been made on the spin-ladder
analogue in which a staggered chiral phase as well as a few possible
incommensurate phases appears SorensenCGK2021  and on the spin-chain limit
where a chiral soliton phase is observed SorensenGRWK2022 . Therefore, the
(quais-) one-dimensional Kitaev-type spin chains serve as fruitful grounds to
offer insights into the enigmatic phases in higher dimensions.

Over the years, the Kitaev-type spin chains have been the focus of intensive
research efforts since they can harbor interesting phases and unconventional
QPTs SenShankar2010 ; AgrBrkNis2018 ; YangKG2020 ; YangJKG2020 ; YangSN2021 ;
YouSunRen2020 ; LuoPRB2021 ; LuoPRR2021 . In the Kitaev-$\Gamma$ chain where
$\Gamma$ interaction is an off-diagonal exchange coupling YangKG2020 , a
magnetically ordered state that displays a spin-nematic correlation occurs in
the neighbor of the dominant AFM Kitaev interaction YangSN2021 ; LuoPRB2021 .
Thus, these studies provide a promising way towards pursuing the spin-nematic
order in the models with bond-directional exchanges. The spin-nematic state is
characterized by a quadrupolar order in which the spin-rotational symmetry is
broken whereas both translational and time-reversal symmetries are retained,
constituting the magnetic analogue of liquid crystal Andreev1984 ; Chandra1991
; Chubukov1991 . Despite an active search for several decades, theoretical
proposals of the spin-nematic order is rare and experimental detection has
been hindered by the fact that the spin-nematic order parameter is not coupled
to the external magnetic field directly ManmanaPRB2011 ; OrlovaPRL2017 . On
the other hand, a continuous QPT between two magnetically ordered states with
different symmetry breaking is reported in the Kitaev spin chain with
multiple-spin interaction Macedo2022 . Such an exotic transition is forbidden
by the conventional LGW paradigm, providing another concrete example of the
deconfined quantum critical point (DQCP) in one dimension Senthil2004Science .

In contrast to the spin-$1/2$ Kitaev-type chains that have gained much
attention, the rich physics of their spin-1 counterparts remains hitherto
largely unexplored. For example, although it is revealed that the spin-1
Kitaev chain can host unusual excitations and display an alluring double-peak
structure in its specific heat LuoPRR2021 , nature of its ground state has not
been understood thoroughly. To this end, in this paper we consider a spin-1
Kitaev chain with a negative single-ion anisotropy (SIA) whose easy axis
varies from [001] direction to [110] direction, passing through the [111]
direction. We propose that the Kitaev phase is a sort of spin-nematic phase
that can further be classified into two kinds, depending on the structures of
their low-lying excited states. In the presence of an overwhelmingly dominant
SIA, we find a continuous QPT between the dimerized phase and the AFM phase
which break different discrete symmetries, showing that a DQCP is likely
realized in the spin-1 Kitaev-type chain.

The remainder of the paper is constructed as follows. In Sec. II we construct
the theoretical model, introduce the numerical methods, and show the resultant
quantum phase diagram. Section III is devoted to presenting the nature of
spin-nematic phase and relevant QPTs, which include QPTs from the dimerized
(AFM) phase to the spin-nematic phase for the [001]-type ([111]-type) SIA,
behavior of the four-spin correlation function in the spin-nematic phase, and
emergence of the DQCP in the continuous dimer-AFM transition. Finally, a brief
conclusion is stated in Sec. IV.

Figure 1: Quantum phase diagram of the spin-1 Kitaev chain with a tunable SIA
in which $-1.5\leq D\leq 0$ and $0\leq\vartheta\leq\pi/2$. The spin-nematic
(SN) phase is gapped and can be classified into two types termed SN-I and SN-
II, based on the degeneracy of their first excited states. They undergo a
crossover rather than a phase transition as the lowest excitation gap never
closes. The transitions from the dimerized phase and AFM phase to the spin-
nematic phase belongs to the Ising universality class. In particular, the
quantum critical point is $-0.6551(2)$ for the [001]-type SIA (i.e.,
$\vartheta=0$), while it is $-0.6035(2)$ for the [111]-type SIA (i.e.,
$\vartheta=\tan^{-1}(\sqrt{2})$).

##  II Model and Method

We consider the spin-1 Kitaev chain with a tunable SIA whose Hamiltonian reads

$\displaystyle\mathcal{H}=$ $\displaystyle
K\sum_{i=1}^{L/2}\big{(}S_{2i-1}^{x}S_{2i}^{x}+S_{2i}^{y}S_{2i+1}^{y}\big{)}$
$\displaystyle+D\sum_{i=1}^{L}\left[\frac{\sin\vartheta}{\sqrt{2}}(S_{i}^{x}+S_{i}^{y})+\cos\vartheta
S_{i}^{z}\right]^{2},$ (1)

where $S_{i}^{\gamma}$ ($\gamma=x,y,z$) are the three components of the spin
operator at the $i$th site, and $L$ is the total length of the chain which is
a multiple of 4. The first term is the Kitaev ($K$) interaction with
alternating $x$\- and $y$-type bonds. The second term represents the SIA, in
which $D<0$ is the strength and $\vartheta\in[0,\pi/2]$ determines the
direction of the easy axis. The SIA term is reduced to the simple form
$(S_{i}^{z})^{2}$ and $(S_{i}^{x}+S_{i}^{y})^{2}/2$, respectively, when
$\vartheta=0$ and $\pi/2$, while it exhibits the form $(S_{i}^{c})^{2}$ with
$S_{i}^{c}=(S_{i}^{x}+S_{i}^{y}+S_{i}^{z})/\sqrt{3}$ when
$\vartheta=\tan^{-1}(\sqrt{2})\approx 0.3041\pi$. Although the full $SU(2)$
spin-rotational symmetry is absent, the Hamiltonian in Eq. (II) respects a
time-reversal symmetry $\mathcal{T}$ ($S_{i}^{\gamma}\mapsto-S_{i}^{\gamma}$)
and a link-inversion symmetry $I$ ($S_{i}^{\gamma}\mapsto
S_{L+1-i}^{\gamma}$). In light of a proper basis rotation
$(S_{i}^{x},S_{i}^{y},S_{i}^{z})^{T}=\hat{R}_{z}\hat{R}_{y}\cdot(\tilde{S}_{i}^{x},\tilde{S}_{i}^{y},\tilde{S}_{i}^{z})^{T}$
with

$\hat{R}_{z}=\left[\begin{array}[]{ccc}\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}&0\\\
\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}&0\\\ 0&0&1\\\
\end{array}\right],\hat{R}_{y}=\left[\begin{array}[]{ccc}\cos\vartheta&0&\sin\vartheta\\\
0&1&0\\\ -\sin\vartheta&0&\cos\vartheta\\\ \end{array}\right],$ (2)

it is further revealed to have a
$\mathbb{Z}_{2}^{\tilde{x}}\times\mathbb{Z}_{2}^{\tilde{z}}$ dihedral symmetry
$D_{2}$ where $\mathbb{Z}_{2}^{\tilde{x}/\tilde{z}}$ stands for the spin
inversion in $\tilde{x}/\tilde{z}$ direction. Due to the bond-alternating
nature of the Kitaev interaction, the model possesses a two-site translational
symmetry $T_{2}$ apparently. However, at least in the limit cases where
$\vartheta=0$ and $\pi/2$, $\mathcal{H}$ enjoys an one-site translational
symmetry $T_{1}$. This can be seen by exerting the following unitary
transformation on the even sites:
$(S_{2i}^{x},S_{2i}^{y},S_{2j}^{z})\mapsto(S_{2i}^{y},S_{2i}^{x},-S_{2i}^{z})$
SenShankar2010 . Consequently, the Kitaev term takes the form
$\sum_{i}S_{i}^{x}S_{i+1}^{y}$ while the SIA term remains unchanged, both of
which are translationally invariant.

Figure 2: Bond strength $\big{|}\langle
S_{i}^{\gamma}S_{j}^{\gamma}\rangle\big{|}$ of the $x$-bond (red dash-dot
line) and $y$-bond (blue dashed line) as a function of temperature $T$ in the
Kitaev spin chain with $\vartheta=0.0$. Two different values of $D$ are
chosen, which are $D=-0.4$ (in spin-nematic phase) and $D=-0.8$ (in dimerized
phase).

In fact, the SIA is naturally expected in all high-spin materials under a
slight distortion from their ideal structures Stavropoulos2019PRL , and it has
been identified in various Kitaev materials like CrI3, CrGeTe3 and CrSiTe3
Xu2018npjCM ; Xu2020PRL ; Stav2021PRR ; Zhou2021PRB . Meanwhile, the role
played by the [001]-type and [111]-type SIAs in the spin-1 and spin-$3/2$
Kitaev honeycomb models has been studied extensively Bradley2022PRB ;
Jin2022NC . In the large-$S$ limit, it is revealed that the SIA can stabilize
an interesting triple-meron crystal consisting of three merons, leading to a
finite topological number and a quantized topological Hall conductance
Chen2023NJP . These studies imply that Eq. (II) should also harbour a rich
physics.

In what follows we set $K$ = 1 as the energy unit unless stated otherwise. The
quantum phase diagram is mapped out by the density-matrix renormalization
group (DMRG) method White1992 ; Peschel1999 ; Schollwock2005 . In the DMRG
calculation we adopt both open (OBC) and periodic (PBC) boundary conditions
alternatively, depending on the prominent issue that matters. To improve the
numerical accuracy, 2000 block states are kept in order to maintain a small
truncation error of $\sim 10^{-7}$ or less. The sweep is executed twelve times
basically, with the potential to increase by several times in the vicinity of
the quantum critical point. When necessary, the transfer-matrix
renormalization group (TMRG) method is also employed to study the finite-
temperature evolution of physical quantities BurXiangGeh1996 ; WangXiang1997 .
During the calculation, we set the Trotter-Suzuki step $\tau=0.01$ and the
block states $m=1024$.

Figure 1 illustrates the quantum phase diagram in the region of
$D\in[-1.5,0.0]$ and $\vartheta\in[0,\pi/2]$ in the spin-1 Kitaev chain with
tunable SIA. Firstly, by calculating the four-spin correlation function
pertaining to the spin-nematic order, we find that the small-$D$ region,
including the Kitaev limit whose ground state is previously termed Kitaev
phase LuoPRR2021 , exhibits a nonzero spin-nematic correlation over the
vanishing magnetic moment. This area is thus arguably a spin-nematic phase
that has long been pursued in the past decades Andreev1984 ; Chandra1991 ;
Chubukov1991 . The spin-nematic phase has a unique ground state, above which a
finite excitation gap is acquired. According to the degeneracy of its first
excited state, however, it can be further divided into two parts where a
crossover occurs between them. Secondly, the dimerized phase and the AFM
phase, which break translational symmetry and dihedral and time-reversal
symmetries, respectively, appear as the strength of the SIA increases. When
the strength of the SIA is moderate, the spin-nematic order is intervened
between the two, in accordance with the fact that the spin-nematic order
preserves the translational symmetry and time-reversal symmetry. Last but not
the least, a continuous QPT between the dimerized phase and the AFM phase,
which is advocated by a central charge of 1, is identified if the SIA is
overwhelmingly dominant. Hence, a DQCP is likely realized in the spin-1
Kitaev-type chain.

##  III Results and Discussion

###  III.1 Dimerized phase and AFM phase

The dimerized phase and the AFM phase are two representative symmetry-breaking
phases that have been widely recognized in the field of quantum magnetism. For
concreteness, we consider the Kitaev chain in the [001]-type ([111]-type) SIA
to study the dimerized phase (AFM phase) and its transition to the spin-
nematic phase. The dimerized phase breaks the translational symmetry
spontaneously, leading to a gapped ground state with a two-fold degeneracy. In
the spin-1 Heisenberg chain, the dimerized phase is demonstrated to be
realized by adding competing biquadratic interaction Lauchli2006 ; Hu2014PRL ,
three-spin interaction ChepigaAM2016 , or spatial alternation Kitazawa1996 .
Nevertheless, the SIA itself cannot induce the dimerized phase Chen2003PRB ;
Hu2011PRB .

In the Kitaev chain with a [001]-type SIA, however, the intrinsic bond-
directional interaction opens the possibility of realizing the dimerized
phase. Since there is only one site in each unit cell due to the translational
symmetry $T_{1}$, a natural way to check for the dimerized phase is by
measuring the dimer order parameter defined as $O=\lim_{L\to\infty}O_{L}$ with

$O_{L}=\big{|}\langle S_{L/2-1}^{x}S_{L/2}^{x}\rangle-\langle
S_{L/2}^{y}S_{L/2+1}^{y}\rangle\big{|}.$ (3)

Thus, the dimerized phase occurs as long as the bond strength of
$\big{|}\langle S_{L/2-1}^{x}S_{L/2}^{x}\rangle\big{|}$ and $\big{|}\langle
S_{L/2}^{y}S_{L/2+1}^{y}\rangle\big{|}$ differ. Figure 2 shows the finite-
temperature TMRG calculation of the bond strength $\big{|}\langle
S_{i}^{\gamma}S_{j}^{\gamma}\rangle\big{|}$ ($\gamma=x,y$) with $D=-0.4$ and
$-0.8$. As the temperature $T$ evolves from 10 to 0.0033, the curves of the
bond strength between neighboring $x$ bond (red dot-dashed line) and $y$ bond
(blue dotted line) overlap persistently when $D=-0.4$. By contrast, there is a
sharp differentiation of the bond strength as long as the temperature is lower
than $\sim 0.01$ when $D=-0.8$, indicating a spontaneous dimerization thereof.
In the ultra-low temperature region ($T<0.01$), the bond strength is
insensitive to the temperature and the fact that strength of the weak bond
strength remains finite down to the zero temperature reveals a partially
dimerized phase.

Figure 3: The finite-size scaling of the dimer order parameter $O_{L}$ as a
function of $D$ in the Kitaev spin chain with $\vartheta=0.0$. The chain
length $L$ chosen are 128 (red circle), 192 (green triangle), 256 (blue
square), 320 (cyan pentagram), and 384 (pink diamond).

To study the nature of the QPT, we use the DMRG method to calculate the dimer
order parameter $O_{L}$ for different length $L$. According to the finite-size
scaling ansatz Fisher1972PRL , the dimer order parameter $O_{L}$ satisfies the
formula

$O_{L}(D)\simeq L^{-\beta/\nu}f_{O}\big{(}|D-D_{c}|L^{1/\nu}\big{)},$ (4)

where $\beta$ and $\nu$ are critical exponents of order parameter and
correlation length, and $f_{O}(\cdot)$ is a nonuniversal function that relies
on $O_{L}$. To extract the critical exponents, we adjust parameters
$\mu_{1,2}$ until we see the intersection of $O_{L}L^{\mu_{1}}$ as a function
of $D$ and the collapse of $O_{L}L^{\mu_{1}}$ as a function of
$|D-D_{c}|L^{\mu_{2}}$ for all length $L$. The critical exponents are then
given by $\beta=\mu_{1}/\mu_{2}$ and $\nu=1/\mu_{2}$. Figure 3 shows the
finite-size scaling result of the dimer order parameter $O_{L}$ with $L$ =
128, 192, 256, 320, and 384. By using of the least-square fitting method, we
obtain the quantum critical point $D_{c}=-0.6551(2)$, and the critical
exponents $\beta=0.123(4)$ and $\nu=0.98(3)$. These values are consistent with
the critical exponents of the Ising transition which says that $\beta=1/8$ and
$\nu=1$, suggesting that the transition between the dimerized phase and the
spin-nematic phase belongs to the Ising universality class.

Figure 4: The finite-size scaling of the AFM order parameter $M_{L}$ as a
function of $D$ in the Kitaev spin chain with $\vartheta=\tan^{-1}(\sqrt{2})$.
The chain length $L$ chosen are 128 (red circle), 192 (green triangle), and
256 (blue square).

Before proceeding further, we wish to note that the dimer order parameter in
Eq. (3) is still suitable even though the easy-axis direction of the SIA is
away from the [001] direction. After applying the local transformation
$(S_{2i}^{x},S_{2i}^{y},S_{2j}^{z})\mapsto(S_{2i}^{y},S_{2i}^{x},-S_{2i}^{z})$,
the Kitaev interaction is translationally invariant while the SIA term takes
the form
$\left[\frac{\sin\vartheta}{\sqrt{2}}(S_{i}^{x}+S_{i}^{y})-(-1)^{i}\cos\vartheta
S_{i}^{z}\right]^{2}$. In the dimerized phase, $S_{i}^{z}$ is only weakly
coupled to $S_{i}^{x}$ and $S_{i}^{y}$ when compared to the dominating
$(S_{i}^{z})^{2}$. In addition, although the intensity of $(S_{i}^{x})^{2}$
and $(S_{i}^{y})^{2}$ are different, all the components of
$S_{i}^{\alpha}S_{i}^{\beta}$ ($\alpha,\beta=x,y,z$) are uniformly
distributed, suggesting an effective one-site translational symmetry.

Next, we turn to study the AFM phase which is known to break the dihedral
symmetry and time-reversal symmetry and exhibits a gapped doubly-degenerate
ground state. The magnetic moments along the three spin directions are all
finite except for the case where $\vartheta=\pi/2$. Due to symmetric
structures of the Kitaev interaction and SIA, the $x$ and $y$ components of
magnetic moments are equal but are larger than that of the $z$ component. We
apply an staggered pinning field of value $\mathcal{O}(1)$ at two end sites to
slightly break the degenerate manifold. The nondegenerate ground state thus
displays a well-behaved magnetic pattern, and the magnetic order parameter can
be calculated as $M=\lim_{L\to\infty}M_{L}$ with

$M_{L}=\sqrt{\big{(}\langle S_{L/2}^{x}\rangle\big{)}^{2}+\big{(}\langle
S_{L/2}^{y}\rangle\big{)}^{2}+\big{(}\langle S_{L/2}^{z}\rangle\big{)}^{2}}.$
(5)

Figure 4 shows the finite-size scaling result of the magnetic order parameter
$M_{L}$ ($L$ = 128, 192, and 256) in the Kitaev chain with a [111]-type SIA.
Following a similar procedure mentioned above, we get the quantum critical
point $D_{c}=-0.6035(2)$, and the critical exponents $\beta=0.127(3)$ and
$\nu=0.99(2)$, demonstrating that the transition between the AFM phase and the
spin-nematic phase also falls in the Ising universality class.

Figure 5: The excitation gaps $\Delta_{1}$ (open symbols) and $\Delta_{2}$
(filled symbols) as a function of $D$ in the Kitaev spin chain with
$\vartheta=\tan^{-1}(\sqrt{2})$. In the DMRG simulation, PBC are imposed and
$L=24$ (red circle), 48 (green triangle), and 72 (blue square). Inset: Linear
extrapolation of $\Delta_{2}$ at the quantum critical point, with $L$ changing
from 24 to 144.

To further verify the continuous QPT, we calculate the lowest excitation gaps
$\Delta_{1,2}=E_{1,2}-E_{0}$ in the vicinity of the quantum critical point.
Here, $E_{0,1,2}$ are the three lowest energy levels in the energy spectrum,
with $E_{0}$ being the ground-state energy. In the calculation we use the PBC
to remove the boundary effect, and the ground state of the spin-nematic phase
is unique while it is doubly degenerate in the AFM phase. Behaviors of the
excitation gaps $\Delta_{1}$ (open symbols) and $\Delta_{2}$ (filled symbols)
as a function of $D$ are shown in Fig. 5. Deep in the AFM phase, $\Delta_{1}$
is vanishingly small and $\Delta_{2}$ is robust against the chain length. As
the SIA approaches the quantum critical point, the finite-size effect is
significant since $\Delta_{2}$ decreases apparently with the increase of the
system size. The inset of Fig. 5 shows the evolution of $\Delta_{2}$ as a
function of $1/L$ for a series of chain length $L$ ranging from 24 to 144. The
linear extrapolation gives an estimate of $0.002(5)$ for $\Delta_{2}$,
corroborating a continuous QPT with a closure of the lowest excitation gap.

We wish to comment on the influence of the sign of the Kitaev interaction on
the QPT. For the [001]-type SIA with $\vartheta=0$, the transformation of
$(S_{i}^{x},S_{i}^{y},S_{i}^{z})\mapsto(-S_{i}^{x},-S_{i}^{y},S_{i}^{z})$ on
all even sites implies that $\mathcal{H}(K,D)$ = $\mathcal{H}(-K,D)$, showing
that the sign of the Kitaev interaction does not alter the position of
transition point. By contrast, for the [111]-type SIA with
$\vartheta=\tan^{-1}(\sqrt{2})$, $\mathcal{H}(K,D)$ and $\mathcal{H}(-K,D)$
are no longer equivalent. While the QPT is still of the Ising universality
class when the Kitaev interaction is ferromagnetic, the transition point is
$-0.5531(2)$, which is larger than that of the AFM case.

Figure 6: The (a) chiral-chiral correlation function $K(l_{0},l_{0}+r)$ and
(b) four-spin correlation function $Q_{1}(l_{0},l_{0}+r)$ at selected data
points in an open chain with the length of $L=128$. Here, $l_{0}=L/2$ and $r$
represents the distance from the reference point. In panel (a), the parameters
are $D=-0.2$ and $\vartheta=\tan^{-1}(\sqrt{2})$ for the point I and $D=-0.3$
and $\vartheta/\pi=0.05$ for the point II. In panel (b), the parameters are
equal to these of point I.
Figure 7: Analysis of the spin-nematic correlation $Q_{1}(r)$ [defined in Eq.
(7)] as a function of $D$ in the Kitaev spin chain with
$\vartheta=\tan^{-1}(\sqrt{2})$. In panels (a) and (b), the chain length is
fixed at $L=128$. (a) The real part $\Re(Q_{1}(r))$ (blue symbols) and
imaginary part $\Im(Q_{1}(r))$(cyan symbols) of the spin-nematic correlation,
with $r$ being even ($\square$) or odd ($\Diamond$). (b) The left axis shows
the intensity $\mathcal{O}_{SN}^{2}$ of the spin-nematic correlation with $r$
being even (filled square) or odd (open diamond), while the right axis
represents the nontrivial phase factor $\phi$ of the spin-nematic correlation
when $r$ is odd. (c) The first-order derivative of the phase factor
$\partial\phi/\partial D$ for a chain of length $L=128$ (red circle), 192
(green triangle), and 256 (blue square).

###  III.2 Spin-nematic phase

The spin-nematic order is an intriguing phase which lacks the conventional
magnetic order but breaks the spin-rotational symmetry, giving rise to a
nonzero quadrupolar order and possessing unusual excitations Andreev1984 ;
Chandra1991 ; Chubukov1991 . Therefore, emergence of the spin-nematic order is
often related to the geometrical frustration and competing interactions which
enhance quantum fluctuations Tsunetsugu2006 ; Kohamaa2019PNAS . Hitherto,
several different scenarios have been proposed to theoretically realize the
spin-nematic phase. The spin-$1/2$ ferromagnetic chain with frustrated next-
nearest-neighbor interaction is perhaps the most realistic model since it is
believed to characterize a couple of quasi-one-dimensional magnets like
LiCuVO4 Mourigal2012PRL ; Buttgen2014PRB . According to the proposal by
Zhitomirsky and Tsunetsugu Zhitomirsky2010EPL , just below the saturation
field, the gapped magnon excitations and the attractive interaction between
them enforce the energy of the two-magnon bound state is lower than that of
the single-magnon state, thereby favoring the spin-nematic phase
Hikihara2008PRB ; Sudan2009PRB ; Arlego2011PRB ; Syromyatnikov2012 ;
Parvej2017PRB . Theoretical analysis and numerical calculation suggest that
the spin-nematic phase can be stabilized in spin-1 chains with the biquadratic
interaction ManmanaPRB2011 . In addition, the spin-nematic phase is also
demonstrated to manifest itself in spin-1 chains whose Hamiltonians do not
have $U(1)$ symmetry ZvyaginPRB2019 .

We start by checking for the possible existence of vector spin chirality
$\hat{\kappa}_{i}=(\textbf{S}_{i}\times\textbf{S}_{i+1})_{z}=-\imath(S_{i}^{+}S_{i+1}^{-}-S_{i}^{-}S_{i+1}^{+})/2$,
which is the vector product of two adjacent spins along the chain Chubukov1991
; Hikihara2008PRB ; Sudan2009PRB ; Parvej2017PRB . The chiral order preserves
the time-reversal symmetry but breaks the inversion symmetry. The chiral-
chiral correlation function is defined as

$\displaystyle
K\big{(}i,j\big{)}=\langle\hat{\kappa}_{i}\hat{\kappa}_{j}\rangle,$ (6)

in which $i$ and $j$ are site indices and we assume that
$r\equiv\left|j-i\right|\rightarrow\infty$. For concrete, we set
$(i,j)=(l_{0},l_{0}+r)$ with $l_{0}=L/2$ and calculate the correlator
$K\big{(}l_{0},l_{0}+r\big{)}$ at two representative points, see Fig. 6(a). It
is observed that $K\big{(}l_{0},l_{0}+r\big{)}$ decays rapidly with the
distance $r$ and tends to zero, indicating that the chiral order is not
favored in the ground state. On the other hand, the spin-nematic order can be
confirmed by the spin-nematic order parameter $\mathcal{O}_{SN}$, which is
extracted from the four-spin correlation function Syromyatnikov2012 ;
Sato2013PRL

$\displaystyle Q_{\delta}\big{(}i,j\big{)}=\langle
S_{i}^{+}S_{i+\delta}^{+}S_{j}^{-}S_{j+\delta}^{-}\rangle\simeq\mathcal{O}_{SN}^{2}e^{-\imath\phi}.$
(7)

Here, $\delta$ is fixed as 1 throughout the paper, and $\phi$ is a phase
factor that varies as the interaction strength changes. The real (blue color)
and imaginary (cyan color) parts of
$Q_{1}(r)=Q_{1}\big{(}l_{0},l_{0}+r\big{)}$ at a specific point
$Q_{1}\big{(}l_{0},l_{0}+r\big{)}$ in which $D=-0.2$ and
$\vartheta=\tan^{-1}(\sqrt{2})$ are shown in Fig. 6(b). It is observed that
depending on the odevity of $r$, $Q_{1}(r)$ has a strong even-odd effect. When
$r$ is even, $Q_{1}(r)$ is real as ${\Im}\big{(}Q_{1}(r)\big{)}$ is
vanishingly small. By contrast, both ${\Re}\big{(}Q_{1}(r)\big{)}$ and
${\Im}\big{(}Q_{1}(r)\big{)}$ saturate to finite values for odd $r$. In any
circumstance, the fact that the spin-nematic order parameter
$\mathcal{O}_{SN}$ is nonzero manifests the existence of the spin-nematic
order. Of note is that the spin-rotational symmetry pertaining to the spin-
nematic order is explicitly broken in the Hamiltonian.

We proceed to focus on the Kitaev chain with a [111]-type SIA to study the
behavior of the spin-nematic order parameter. The real (blue color) and
imaginary (cyan color) parts of $Q_{1}(r\gg 1)$ for a chain of length $L=128$
are shown in Fig. 7(a). Irrelevant of the strength of $|D|$,
${\Im}\big{(}Q_{1}(r)\big{)}$ vanishes when $r$ is even, and is finite except
for the limit case where $D=0$ and an accidental point with $D\approx-0.49$
when $r$ is odd. In the former case the phase factor $\phi$ is 0 while in the
latter case it is nontrivial. The left axis of Fig. 7(b) illustrates the
amplitude of $Q_{1}(r)$ when $r$ is even (pink square) and odd (brown
diamond), respectively. The fact that all the data points are overlapped
indicates that $\mathcal{O}_{SN}^{2}$ is uniformly distributed and can be
safely extracted from either case. The right axis of Fig. 7(b), on the other
hand, shows the behavior of the phase factor $\phi$ as $D$ changes. It
decreases from $\pi$ in the pure Kitaev limit where $D$ is zero to $0$ when
$|D|$ is large enough such that the system is in the deep AFM phase. A
nontrivial observation is that the phase factor $\phi$ undergoes a rapid
change near the quantum critical region, indicating that it may serve as a
tool to probe the QPT. To reveal the relation between phase factor $\phi$ and
quantum criticality, we show the derivative of $\phi$ with respect to tuning
parameter $D$ in Fig. 7(c). The quantity $\partial\phi/\partial D$ displays a
singular peak in the vicinity of the quantum critical point $D_{c}$, with the
height of peak growing and the position of peak approaching $D_{c}$ as the
chain length $L$ increases. Thus, $\partial\phi/\partial D$ is predicted to
diverge as $L\to\infty$ and should in principle display a scaling behavior. We
note in passing that derivative of the geometric Berry phase associated with
the many-body ground state has already been demonstrated to exhibit
universality in the neighbor of the quantum critical point ZhuPRL2006 .

Figure 8: The spatial pattern of the bond-parity operator $\langle
W_{i}^{[l]}\rangle$ at site $i$ for the $l$-th energy level with
$l=0,1,2,\cdots$. The symbol $p$ marked in each panel stands for the
periodicity of $\langle W_{i}^{[l]}\rangle$ in a chain of length $L=60$.
Panels (a)-(d) represent the spin-nematic phase of the type-I with $D=-0.3$
and $\vartheta/\pi=0.30$, while Panels (e)-(g) represent the spin-nematic
phase of the type-II with $D=-0.3$ and $\vartheta/\pi=0.05$.

Whereas the spin-nematic phase is characterized by a unique ground state under
PBC, its excited states are quite involved and display distinct patterns. We
find that all the excited states are doubly degenerate except for the first
excited ground state. The first excited ground state is unique in the wide
region, as compared to the twofold case observed in a specific area where
$|D|$ and $\vartheta$ are small. Therefore, we distinguish the spin-nematic
phase as type-I and type-II, respectively, based on its degeneracy of the
first excited state (for illustration, see Fig. 1). However, since the lowest
excitation gap of the spin-nematic phase does not close throughout its whole
region, there is not a QPT but a likely crossover between the two. To
illustrate it, we have calculated the phase factor $\phi$ at fixed SIA, saying
$D=-0.3$. The derivative of $\phi$ with respect to $\vartheta$ shows a broad
hump and suffers from an insignificant finite-size effect, characteristic of
crossover phenomenon.

To further discriminate the two different types of spin-nematic phase, we
resort to the bond-parity operator $\hat{W}_{i}$ defined as SenShankar2010 ;
YouSunRen2020

$\hat{W}_{2i-1}=\Sigma_{2i-1}^{y}\Sigma_{2i}^{y},\quad\hat{W}_{2i}=\Sigma_{2i}^{x}\Sigma_{2i+1}^{x},$
(8)

where $\Sigma_{i}^{\alpha}=e^{\imath\pi S_{i}^{\alpha}}$ is the on-site
operator. For the pure Kitaev chain, $\hat{W}_{i}$ commutes with the
Hamiltonian such that its eigenvalues should only be $\pm 1$ for the ground
state. By switching on the SIA, the relation
$\big{[}\hat{W}_{i},\mathcal{H}\big{]}=0$ does not hold as long as
$\vartheta\neq 0$, indicating that $\langle\hat{W}_{i}\rangle$ will deviate
from 1.

The spatial patterns of $\langle W_{i}^{[l]}\rangle$ in a closed chain of
$L=60$ at different energy levels $l=0,1,3,5$ for the type-I and type-II spin-
nematic phases are shown in Fig. 8, with $D=-0.3$ and $\vartheta/\pi$ = $0.30$
and $0.05$ for the left and right panels, respectively. It can be seen from
Fig. 8(a) and Fig. 8(e) that the ground-state patterns of $\langle
W_{i}^{[0]}\rangle$ for both types are uniformly distributed with a
periodicity $p=1$ along the chain. For the excited-state patterns, they
display a similarity within the twofold degenerate states and thus only three
selected energy levels are shown. For the spin-nematic phase of the type-I,
the first excited state is again unique and $\langle W_{i}^{[1]}\rangle$ is
completely flat. $\langle W_{i}^{[3]}\rangle$ and $\langle W_{i}^{[5]}\rangle$
are smoothly changed within the chain, with periodicity $p=30$ (see Fig. 8(c))
and $p=15$ (see Fig. 8(d)), respectively. By contrast, while $\langle
W_{i}^{[l]}\rangle$ ($l=1,3,5$) exhibits periodicity of $p=10$ (see Fig.
8(f)), $p=3$ (see Fig. 8(g)), or $p=15$ (see Fig. 8(h)), its values are quite
fluctuating and elusive. However, pertaining to the behavior of the first
excited state, the flatness versus oscillation of $\langle W_{i}^{[1]}\rangle$
is the hallmark of the difference between the type-I and type-II spin-nematic
phases. It is in this sense that we can identify the crossover boundary of the
two by the standard deviation of $\langle W_{i}^{[1]}\rangle$, i.e.,
$\sigma_{W}$. In our calculation on three closed chains of length $L$ = 24,
48, and 72, the quantity $\sigma_{W}$ undergoes a sharp jump at
$\vartheta/\pi\approx 0.13$, as depicted in Fig. 1. We note that the
periodicity of $\langle W_{i}^{[l]}\rangle$ in the excited states should be
different as we change the chain length, and such a periodicity can be
discerned by the discrete Fourier transform of $\langle W_{i}^{[l]}\rangle$.
Nevertheless, the most remarkable feature that the curves of $\langle
W_{i}^{[l]}\rangle$ ($l>0$) are smooth and discrete, respectively, in the
type-I and type-II spin-nematic phases remains preserved.

Finally, we comment on the mechanism of the spin-nematic phase. Hitherto, the
two-magnon bound state picture in frustrated spin-$1/2$ systems with the
nearly saturated magnetic field and the description of the on-site quadrupolar
order in spin-1 models with the biquadratic interaction are widespread to
describe the spin-nematic phase. More interestingly, an attempt to unify these
scenarios based on the language of spin-1 dimers has been proposed
Tanaka2020PRB . Physically, the presence of magnetic step of two in
magnetization curve or the Anderson tower of states containing only the even
total spin sectors Lauchli2005PRL  is known as the fingerprint of the spin-
nematic phase. However, it seems to be infeasible to check the picture as the
total spin is not a conserved quantity for the lack of $U(1)$ symmetry.
Nevertheless, one can calculate the one-magnon and two-magnon dynamical
spectra, from which the magnon and magon-pair gaps can be extracted. This may
give some clues on the nature of the spin-nematic phase and deserves future
study.

Figure 9: (a) and (b) show behaviors of order parameters and central charge as
a function of $\vartheta/\pi$ in the Kitaev spin chain with $D=-1$. (a) The
dimerized order parameter $O_{L}$ (left axis, open symbols) and AFM order
parameter $M_{L}$ (right axis, filled symbols) for an open chain of length
$L=128$ (red circle), 192 (green triangle), and 256 (blue square). The
intervening pink region represents the spin-nematic phase. (b) The central
charge for a periodic chain of length $L=48$ (red circle), 72 (green
triangle), and 96 (blue square). (c) and (d) are the same as these of (a) and
(b) but for $D=-100$, in which a direct QPT between the dimerized phase and
AFM phase occurs.

###  III.3 Deconfined quantum critical point

Dating back to 2004, the DQCP is a fascinating proposal which asserts a
continuous QPT between two spontaneous symmetry-breaking phases with
completely unrelated broken symmetries Senthil2004Science . Right at the DQCP,
deconfined fractionalized particles appear, accompanying by an emergent
symmetry to reconcile the two different order parameters nearby. This scenario
is clearly beyond the conventional LGW paradigm as the latter predicts that
this kind of QPT should be of first order. While the transition between the
AFM phase and the valence-bond-solid phase in two dimension is regarded as the
possible realization of the deconfined criticality, decisive evidences are
still lacking as a weakly first-order QPT cannot be ruled out Sandvik2007 .
The one-dimensional analogy was put forward in 2019, providing another
feasible way towards unraveling the enigmatic DQCP JiangMotrunich2019 .
Massive numerical work has been devoted to studying the DQCP in one-
dimensional spin-$1/2$ models during the past few years, including the
ferromagnetic frustrated spin chain Roberts2019PRB ; Huang2019PRB ; Sun2019PRB
; Luo2019PRB , the spin ladder with ring-exchange interaction Ogino2021PRB ,
and the Kitaev spin chain with multiple-spin interaction Macedo2022 .

We will demonstrate that the spin-1 Kitaev chain with tunable SIA is another
promising platform that exhibits the DQCP. To begin with, we focus on the line
of $D=-1$ and calculate the dimer order parameter $O_{L}$ and magnetic order
parameter $M_{L}$, see Fig. 9(a). It can be seen that both order parameters
decrease smoothly as the driving parameter $\vartheta$ approaches their
corresponding quantum critical points. In the intervening region where
$0.0601\lesssim\vartheta/\pi\lesssim 0.1683$, the two order parameters vanish
and the spin-nematic phase of type-I survives, in accordance with the fact
that the spin-nematic phase preserves the translational symmetry and time-
reversal symmetry. Next, we appeal to the central charge to pin down the
nature of QPTs. The central charge $c$ is usually extracted from the
entanglement entropy which is known to obey the conformal field theory
CalCar2004 . Although the OBC is frequently adopted in the DMRG calculation,
it can induce an intrinsic alternating term which decays away from the
boundary with an approximately power-law behavior in the entanglement entropy,
making the fitting formula more intricate Laflorencie2006PRL . Therefore, we
turn to the PBC and the entanglement entropy is well described by the
following expression CalCar2004

$\mathcal{S}_{L}(x)=\frac{c}{3}\ln\left[\frac{L}{\pi}\sin\Big{(}\frac{\pi
x}{L}\Big{)}\right]+c^{\prime},$ (9)

where $x$ is the length of a subsystem and $c^{\prime}$ is a nonuniversal
constant. Results of the fitted central charge for three different lengths $L$
are shown in Fig. 9(b). It is found that at $\vartheta/\pi\approx 0.0601$ and
$\vartheta/\pi\approx 0.1683$ the central charges are slightly decreases with
the increase of the system but saturate to $1/2$ eventually, indicating that
both QPTs belong to the Ising universality class.

As the intensity of the SIA increases, region of the spin-nematic phase
shrinks slightly and does not disappear until $|D|$ is large enough. After a
careful inspection of the quantum criticality, we take $D=-100$ as an example
to illustrate the direct QPT between the dimerized phase and the AFM phase.
The behaviors of order parameters $O_{L}$ and $M_{L}$ in a narrow window of
$0.01\leq\vartheta/\pi\leq 0.02$ are shown in Fig. 9(c). They are smoothly
changed as $\vartheta$ varies and the finite-size scaling [see Eq. (4)]
suggests that there is only a sole quantum critical point at
$\vartheta/\pi\approx 0.0158$. In Fig. 9(d), we also fit the central charge in
the same parameter range as that of Fig. 9(c). Far away from the critical
region, the central charge is vanishingly small and tends to be zero with the
increase of the system size, indicative of the gapped ground states. In the
critical region, the central charge is sizable and its maximal value is
extremely close to 1. Such a finite central charge is also confirmed in
several independent calculations like $D=-200$. Since the nonzero central
charge is crucial to corroborate the continuous QPT, our result thus
demonstrates that the dimer-AFM transition is continuous. Nevertheless,
determining the nature of this QPT is numerically challenging, albeit a
conceivable possibility is the Gaussian transition which has been proposed in
other similar situations Mudry2019PRB . Notably, because the broken
translational symmetry and dihedral symmetry are totally irrelevant, the
continuous QPT is forbidden by the LGW paradigm and thus the quantum critical
point is interpreted as a DQCP.

##  IV Conclusion

We have studied the quantum phase diagram of a spin-1 Kitaev chain with
tunable SIA by the DMRG method, which is identified to host a dimerized phase,
an AFM phase, and two distinct spin-nematic phases. In line with the previous
research effort which reveals that the ground state of the spin-1 Kitaev chain
is a nonmagnetic Kitaev phase LuoPRR2021 , we further clarify that it is a
spin-nematic order which preserves the translational symmetry and time-
reversal symmetry but breaks spin-rotational symmetry, giving rise to a finite
spin-nematic correlation. The four-spin correlation function pertaining to the
spin-nematic order parameter can exhibit a nontrivial phase factor that varies
as the SIA $|D|$ changes, and the derivative of the phase factor is
demonstrated to be a useful probe to capture QPTs. Depending on the degeneracy
of the first excited state, the spin-nematic phase can be classified into two
types and the model undergoes a crossover between the two. Notably, the nature
of the spin-nematic phase is an intriguing topic which deserves future study.

As the strength of SIA increases, the dimerized phase and the AFM phase with
broken translational symmetry and dihedral and time-reversal symmetries set in
when the SIA is aligned along the [001] direction and [111] direction,
respectively. Of particular note is that the spontaneous dimerization is
induced by the SIA only, highlighting the unique role played by the Kitaev
interaction. When the SIA is modest, the spin-nematic phase is intervened
between the two spontaneous symmetry-breaking phases, and both QPTs belong to
the Ising universality class. By contrast, the spin-nematic phase is destroyed
by strong SIA, leading to a continuous QPT between the dimerized phase and the
AFM phase. Thus, our result demonstrates that the Kitaev-type spin chain can
offer a promising playground to study the DQCP. In the future, it is desirable
to study the emergent symmetry Huang2019PRB , the dynamic signatures Xi2020CPL
, the fidelity and entanglement from the quantum information aspect Sun2019PRB
; Yang2021PRE , and the nonequilibrium critical dynamics described by Kibble-
Zurek mechanism Huang2020PRR  in the critical region so as to corroborate this
exotic QPT.

###### Acknowledgements.

This work is supported by the National Program on Key Research Project (Grant
No. MOST2022YFA1402700), the Natural Science Foundation of Jiangsu Province
(Grant No. BK20220876), the National Natural Science Foundation of China
(Grants No. 12247183, No. 12274187, No. 12247101, No. 12174020, No. 11974244,
and No. U1930402), and the NSERC Discovery (Grant No. 2022-04601). Q.L. also
acknowledges the Fundamental Research Funds for the Central Universities
(Grant No. NS2022097) and the startup Fund of Nanjing University of
Aeronautics and Astronautics (Grant No. YAH21129). H.-Y.K. also acknowledges
funding from the Canadian Institute for Advanced Research and the Canada
Research Chairs Program. The computations are partially supported by High
Performance Computing Platform of Nanjing University of Aeronautics and
Astronautics (NUAA) and Tianhe-2JK at the Beijing Computational Science
Research Center (CSRC). Computations are also performed on the Niagara
supercomputer at the SciNet HPC Consortium. SciNet is funded by the Canada
Foundation for Innovation under the auspices of Compute Canada, the Government
of Ontario, Ontario Research Fund, Research Excellence, and the University of
Toronto.

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