# Magnetic phase diagram of a spin-1/2 XXZ chain with modulated
Dzyaloshinskii-Moriya interaction

G. I. Japaridze  Ilia State University, Center for Condened Matter Theory and
Quantum Computations  Andronikashvili Institute of Physics, Tbilisi, Georgia
Hadi Cheraghi and Saeed Mahdavifar  1Department of Physics, University of
Guilan, 41335-1914, Rasht, Iran

###### Abstract

We consider the ground-state phase diagram of a one-dimensional spin-1/2 XXZ
chain with spatially modulated Dzyaloshinskii-Moriya interaction in the
presence of applied along with the $\hat{z}$ axis alternating magnetic field.
The model is studied using the continuum-limit bosonization approach and the
finite system exact numerical technique. In the absence of the magnetic field,
the ground-state phase diagram of the model includes besides the ferromagnetic
and gapless Luttiger-Liquid (LL) phases two gapped phases, the composite (C1)
phase characterized by the coexistence of the long-range-ordered (LRO)
alternating dimerization and the spin chirality patterns, and the composite
(C2) phase characterized in addition to the coexisting spin dimerization and
alternating chirality patterns, by the presence of LRO antiferromagnetic
order. In the case of two-letter gapped phases, in the case of a uniform
magnetic field, the commensurate-incommensurate type quantum phase transitions
(QPT) from a gapful phase into the gapless phase have been identified and
described using the bosonization treatment and finite chain exact
diagonalizations studies. The upper critical magnetic field corresponding to
the transition into a fully polarized state has been also determined. It has
been shown that the very presence of the staggered component of the magnetic
field vapes out the composite (C1) in favour of the composite gapped (C2)
phase.

###### pacs:

03.65.-w; 03.65.Vf; 03.67.Bg;75.10.Pq

##  I Introduction

The study of the quantum phase transitions (QPTs) [1, ] in low-dimensional
systems has attracted much of current interest particularly in the context of
applications in modern quantum information technologies [Wen_19, ]. The one-
dimensional quantum spin systems, [GNT_book_98, ; Giamarchi_book_04, ;
Takahashi_Book_99, ; Cabra_Pujol_04, ; Vasiliev_Volkova_18, ] because of rich
quantum features and complex ground state phase diagrams, serve as a
fascinating playground to investigate QPTs. Although from the rigid point of
view the phase transitions occur only in the thermodynamic limit i.e. at an
infinite number of system constituent particles, extreme miniaturization of
quantum devices makes it important to search for tools, which allow tracing
the very presence of such a transition in the case of finite clusters.
Achieved during the last years substantial progress in ability to control
matter at the quantum level, realized on various experimental platforms,
including optically trapped ultra-cold atoms [ExPlat_UCA, ], photonics
[ExPlat_7, ], superconducting quantum circuits [ExPlat_8, ] and spin chains
[ExPlat_9, ], has triggered wide activity in the use of finite quantum
clusters as simulators of system behavior in the thermodynamic limit.

Already the seminal studies unambiguously confirm that finite quantum spin,
fermion or boson system may exhibit all the hallmarks of the QPTs [Ent_97, ;
Ent_02, ; QS_Chains_03, ; QS_Chains_03a, ; QS_Chains_03b, ]. The exact
calculation of various quantum parameters, such as entanglement [Ent_Wang_01,
; Ent_Solyom_Noack, ; Ent_SCh_4, ; Ent_SCh_5, ; Ent_SCh_6, ; Ent_SCh_7, ;
Ent_SCh_8, ], Loschmidt echo [Loschmidt_Echo_1, ; Loschmidt_Echo_2, ],
fidelity [Fidelity, ] quantum discord (QD) [Quant_Discord, ], quantum
coherence [Quant_Coherence, ], time-dependent quantum behaviour of a complex
few spin cluster [Grimaudo_etal_19, ; Grimaudo_etal_20, ] and quantum
correlations after sudden quenches [Quenches_18a, ; Quenches_18b, ;
Quenches_20, ] has been successfully used to identify various QPTs using
finite system studies. However, each tool has its limitations for different
types of QPTs, and only a proper combination of used markers allows to get a
correct description.

The one-dimensional spin systems with Dzyaloshinskii-Moriya interaction (DMI)
[DMI, ], have become the subject of intensive studies in the last decades, due
to their rich quantum nature and high potential of applicability in modern
information technologies [Perk_76, ; Zviagin_89, ; Alkaraz_90, ; Aharony_92, ;
Affleck_etal, ; Aristov_00, ; Tsvelik_01, ; YuLu_2003, ; Derzhko_06, ;
Derzhko_07, ; Starykh_etal, ; Mila_etal_11, ; Fazio_14, ; Mahdavifar_08, ;
Mahdavifar_10, ; Amiri_Langari_13, ; Mahdavifar_etal_14, ; Hadi_Saeed_18, ;
Soltani_Saeed_19, ; Japaridze_etal_19a, ; Japaridze_etal_19b, ;
Habibi_etal_19, ; Lima_19, ; Thakur_20, ; Japaridze_etal_20, ].

The Dzyaloshinskii-Moriya interaction, given by the term

${\cal H}_{DM}\sim{\bf D}\cdot\left[{\bf S}_{n}\times{\bf S}_{m}\right],$

where ${\bf D}$ is an axial vector describes an antisymmetric magnetic
exchange between spins located at sites $n$ and $m$. Such an exchange
interaction typically arises in non-centrosymmetric bulk magnets
[Bulk_Systems, ] and at interfaces between a ferromagnet and an adjacent layer
with strong spin-orbit coupling [Interfaces, ]. The DMI is responsible for the
generation of skyrmions in two-dimensional magnetic structures [Skyrmions, ],
composite electric and magnetic order in multiferroic materials
[Multi_Ferroics, ] and for the formation of helical and other unconventional
magnetic ordered states in quasi 1D magnetic materials [1D_Chains_w_DMI_Exp,
].

In real solid-state materials, due to the symmetry restrictions, vector ${\bf
D}$ is either uniform or staggered. Respectively, theoretical studies of the
spin $S=1/2$ Heisenberg chain and ladder systems with DMI [Perk_76, ;
Zviagin_89, ; Alkaraz_90, ; Aharony_92, ; Affleck_etal, ; Aristov_00, ;
Tsvelik_01, ; YuLu_2003, ; Derzhko_06, ; Derzhko_07, ; Starykh_etal, ;
Mila_etal_11, ; Fazio_14, ; Mahdavifar_08, ; Mahdavifar_10, ; Hadi_Saeed_18, ;
Soltani_Saeed_19, ], exception is only the Ref. [Derzhko_07, ] where the $XY$
spin chain with random changes in the sign of DM interactions were studied.

Recently it has been demonstrated that the DMI can be efficiently tailored
with a substantial efficiency factor by an external electric field
[E-Tailored_DMI_2, ; E-Tailored_DMI_3, ; E-Tailored_DMI_4, ]. These
achievements open a possibility to manipulate with magnetic properties of the
spin system by spatially modulated by the applied external electric potential
DMI. In the recent publication, the ground-state phase diagram of a spin-1/2
XXZ Heisenberg chain in the presence of alternating DMI has been studied using
continuum-limit bosonization approach [Japaridze_etal_19a, ]. It has been
shown that the joint effect of the uniform and the staggered components of the
DMI leads to the formation of a gap in the excitation spectrum and, in
addition to the standard ferromagnet (FM) and gapless Luttinger-liquid (LL)
phase, to the formation of two new unconventional gapped phases in the ground-
state: the gapped composite ($C1$) phase characterized by coexistence of the
long-range-ordered (LRO) dimerization and the LRO alternating spin chirality
patterns and, in the limit of strong exchange anisotropy, of the gapped
composite ($C2$) phase characterized in addition to the coexisting spin
dimerization and alternating chirality patterns, by the presence of LRO
antiferromagnetic order. It has been shown, that the transition from the LL to
the $C1$ phase belongs to the Berezinskii-Kosterlitz-Thouless (BKT)
universality class, while the transition from $C1$ to $C2$ phase is of the
Ising type [Japaridze_etal_19a, ]. In the subsequent studies, these QPTs have
been also imprinted using quantum correlations studies in finite clusters
[Japaridze_etal_20, ].

In this paper, we extended our studies of the QPTs in the ground state of the
spin-1/2 XXZ chain with alternating DMI and study magnetic phases of the model
in the presence of, orientated along with the DM D vector, alternating
magnetic field. The Hamiltonian under consideration is

$\displaystyle{\cal H}$ $\displaystyle=$
$\displaystyle\sum\limits_{n=1}^{N}\left[J\left(S_{n}^{x}S_{n+1}^{x}+S_{n}^{y}S_{n+1}^{y}\right)+J_{z}S_{n}^{z}S_{n+1}^{z}\right]$
(1) $\displaystyle+$
$\displaystyle\sum\limits_{n=1}^{N}\left(D_{0}+(-1)^{n}D_{1}\right)\left(S_{n}^{x}S_{n+1}^{y}-S_{n}^{y}S_{n+1}^{x}\right)$
$\displaystyle-$
$\displaystyle\sum\limits_{n=1}^{N}\left(H+(-1)^{n}H_{1}\right)S^{z}_{n},$

where $S_{n}$ is the spin-1/2 operator on the $n$-th site, $J>0$ is the
exchange coupling, while $D_{0}$ and $D_{1}$ and $H_{0}$ and $H_{1}$ are
uniform and staggered parts of the DMI vector and of the magnetic field,
respectively. Studies of the ground-state phase diagram of the XXZ chain in
the presence of uniform and staggered magnetic field counts decades and have
been revived in many excellent books and papers [GNT_book_98, ;
Giamarchi_book_04, ; Takahashi_Book_99, ]. Therefore below we focus on the
study of effects caused by the interplay and competition between the
alternating DMI and magnetic field and hallmarks of the corresponding QPTs
within the finite chain exact quantum calculations.

The paper is organized as follows: In the forthcoming Sec. II, we consider the
exactly solvable limit of the model corresponding to the case $J_{z}=0$. In
the Section III the continuum-limit bosonization analysis of the magnetic
phase diagram of the model (1) is presented. In the Section IV results of the
finite chain, exact diagonalizations studies are presents. Finally, in Section
V the summary is put.

##  II The exactly solvable case

In this section we consider the exactly solvable case of the model (1) at
$J_{z}=0$. It is instructive to rewrite the Hamiltonian in the following form

$\displaystyle{\cal H}$ $\displaystyle=$
$\displaystyle\sum_{n}\Big{[}\,\frac{J}{2}\left(S^{+}_{n}S^{-}_{n+1}+S^{-}_{n}S^{+}_{n+1}\right)$
(2) $\displaystyle+$
$\displaystyle\frac{i}{2}(D_{0}+(-1)^{n}D_{1})\left(S^{+}_{n}S^{-}_{n+1}-S^{-}_{n}S^{+}_{n+1}\right)$
$\displaystyle-$
$\displaystyle\left(H+(-1)^{n}H_{1}\right)S^{z}_{n}\,\Big{]}\,,$

where $S^{+}_{n}=S^{+}_{x}\pm iS^{+}_{y}$. Using the Jordan-Wigner
transformations [JW_1928, ]

$\displaystyle S_{n}^{+}$ $\displaystyle=$ $\displaystyle
a^{\dagger}_{n}\exp\left(i\pi\sum_{m<n}a^{\dagger}_{m}a^{\phantom{\dagger}}_{m}\right)\,,$
$\displaystyle\quad S_{n}^{-}$ $\displaystyle=$
$\displaystyle\exp\left(-i\pi\sum_{m<n}a^{\dagger}_{m}a^{\phantom{\dagger}}_{m}\right)a_{n}\,,$
$\displaystyle S^{z}_{n}$ $\displaystyle=$ $\displaystyle
a^{\dagger}_{n}a^{\phantom{\dagger}}_{n}-1/2,$ (3)

where $a^{\dagger}_{n}$ ($a^{\phantom{\dagger}}_{n}$) is a spinless fermion
creation (annihilation) operator on site $n$, we rewrite the initial lattice
spin Hamiltonian (2) in terms of interacting spinless fermions in the
following way:

$\displaystyle{\cal H}$ $\displaystyle=$
$\displaystyle\sum_{n}\Big{[}\frac{J}{2}\left(a^{\dagger}_{n}a^{\phantom{\dagger}}_{n+1}+a^{\dagger}_{n+1}a^{\phantom{\dagger}}_{n}\right)$
(4)
$\displaystyle+\frac{i}{2}\left(D_{0}+(-1)^{n}D_{1}\right)\left(a^{\dagger}_{n}a^{\phantom{\dagger}}_{n+1}-a^{\dagger}_{n+1}a^{\phantom{\dagger}}_{n}\right)$
$\displaystyle-$
$\displaystyle\left(H+(-1)^{n}H_{1}\right)(a^{\dagger}_{n}a^{\phantom{\dagger}}_{n}-1/2)\Big{]}\,.$

###  II.1 The spectrum

The Hamiltonian (4) can be easily diagonalized in the momentum space. Indeed,
performing the Fourier transform
$a_{n}=\frac{1}{\sqrt{N}}\sum_{k}a_{k}e^{ikn}$ we obtain

$\displaystyle{\cal H}$ $\displaystyle=$
$\displaystyle\sum_{k}\left[\left(\epsilon(k)-H\right)\,a^{\dagger}_{k}a^{\phantom{\dagger}}_{k}+\Delta(k)a^{\dagger}_{k}a^{\phantom{\dagger}}_{k+\pi}\right]\,,$
(5)

where

$\displaystyle\epsilon(k)$ $\displaystyle=$ $\displaystyle J\cos k-D_{0}\sin
k,$ (6) $\displaystyle\Delta(k)$ $\displaystyle=$ $\displaystyle iD_{1}\cos
k+H_{1}\,.$ (7)

Diagonalization of the Hamiltonian (5) is straightforward. It is convenient to
restrict momenta within the reduced Brillouin zone $-\pi/2<k\leq\pi/2$ and to
introduce a new notation $a_{k+\pi}=b_{k}$. In these terms the Hamiltonian
reads

$\displaystyle{\cal H}$ $\displaystyle=$
$\displaystyle\sum_{k}{}^{\prime}\,\Big{[}\left(\epsilon(k)-H\right)a_{k}^{{\dagger}}a^{\phantom{\dagger}}_{k}-\left(\epsilon(k)+H\right)b_{k}^{{\dagger}}b^{\phantom{\dagger}}_{k}$
(8) $\displaystyle\hskip
28.45274pt+\Delta(k)\,a_{k}^{{\dagger}}b^{\phantom{\dagger}}_{k}+\Delta^{\ast}\,(k)b_{k}^{{\dagger}}a^{\phantom{\dagger}}_{k}\Big{]}\,,$

where prime in the sum means that the summation is taken over the reduced
Brillouin zone $-\pi/2<k\leq\pi/2$. Using the unitary transformation

$\displaystyle a_{k}$ $\displaystyle=$
$\displaystyle\cos\phi_{k}\,\alpha_{k}+\sin\phi_{k}e^{i\theta_{k}}\,\beta_{k}\,b,$
$\displaystyle b_{k}$ $\displaystyle=$
$\displaystyle-\sin\phi_{k}e^{-i\theta_{k}}\,\alpha_{k}+\cos\phi_{k}\,\beta_{k}\,.$
(9)

and choosing

$\displaystyle\tan\,\theta_{k}$ $\displaystyle=$ $\displaystyle
D_{1}\cos(k)/\epsilon(k)\,,$ $\displaystyle\tan\,2\phi_{k}$ $\displaystyle=$
$\displaystyle\sqrt{\epsilon^{2}(k)+D_{1}^{2}\cos^{2}k}/H_{1}\,,$ (10)

we finally obtain

$\displaystyle{\cal H}$ $\displaystyle=$
$\displaystyle\sum_{k}{}^{\prime}\,\Big{[}\,E_{+}(k)\alpha_{k}^{{\dagger}}\alpha^{\phantom{\dagger}}_{k}+E_{-}(k)\beta_{k}^{{\dagger}}\beta^{\phantom{\dagger}}_{k}\,\Big{]}\,,$
(11)

where

$\displaystyle E_{\pm}(k)$ $\displaystyle=$
$\displaystyle-H\pm\sqrt{\epsilon^{2}(k)+D_{1}^{2}\cos^{2}k+H_{1}^{2}}\,.$
(12)

The minimum of the upper, ”$\alpha$” band, and the maximum of the lower,
”$\beta$” band, is reached at

$\displaystyle
k^{\ast}=\pi/2+k_{0}=\frac{\pi}{2}-\frac{1}{2}\tan^{-1}\left(\frac{2JD_{0}}{J^{2}-D_{0}^{2}+D_{1}^{2}}\right)\,.$
(13)

At this point the dispersion relation (12) shows a gap

$\displaystyle E_{+}(k^{\ast})-E_{-}(k^{\ast})=2\Delta_{0}\,,$ (14)

where

$\displaystyle\Delta_{0}=\sqrt{H^{2}_{1}+\frac{1}{2}\left(J^{*2}-\sqrt{J^{*4}-(2D_{0}D_{1})^{2}}\right)}$
$\displaystyle\simeq\sqrt{H^{2}_{1}+\frac{D^{2}_{0}D^{2}_{1}}{J^{*2}}}\,,\quad\,{\mbox{a}t}\quad\,\,D_{0}D_{1}\ll
J^{*2}$ (15)

and $J^{*}=\sqrt{J^{2}+D^{2}_{0}+D_{1}^{2}}$.

Respectively the maximum of the upper $\alpha$ band and the minimum of the
lower ”$\beta$” band is reached at $k_{0}$ and are equal to

$\displaystyle E_{\pm}(k_{0})$ $\displaystyle=$
$\displaystyle-H\pm\sqrt{\Delta_{0}^{2}+\sqrt{J^{*4}-(2D_{0}D_{1})^{2}}}\,,$
(16)

As it follows from (16), in the absence of the magnetic field, only synergic
action of the uniform nor staggered components of the DMI provide the
emergence of a gap in the excitation spectrum [Japaridze_etal_19a, ]. In
marked contrast, the effect of uniform and staggered components of the
magnetic field are mutually exclusive – if the staggered component contributes
additively to DMI and enlarges the gap, the uniform component serves as a
chemical potential and at $H>H_{c_{1}}=\Delta_{0}$ leads to the gapless
excitation spectrum (see Fig. 1).

Figure 1: (color online) Single particle dispersion relations for a DMI as
$D_{0}=0.7$ and $D_{1}=0.3$ for the uniform magnetic fields as (a) $H=0.0$,
(b) $H_{c_{1}}=0.17$, (c) $H_{c_{2}}=1.25$, and also for the case of the
staggered magnetic field as (d) $H_{1}=0.5$. The horizontal and vertical black
dashed lines hint $E(k)=0.0$ and $k^{*}$, respectively.

###  II.2 The ground-state structure, order parameters, and critical points

Obtained in the considered exactly solvable case single-particle dispersion
relations (12) allow to identify key points of the ground state magnetic phase
diagram, the character of the low-energy excitation spectrum, and critical
points. These results will serve as a milestone for the general, nonintegrable
case, considered in the following sections.

In the ground state all states with negative energy are filled and with
positive energy are empty. Since $E_{-}(k)<0$ for arbitrary $H$ all states in
the lower band are completely filled and $n_{\beta}(k)=\langle
0|\beta^{{\dagger}}_{k}\beta_{k}|0\rangle=1$ for $k\in[-\pi/2,\pi/2]$.

At $H<H_{c_{1}}$ the upper band is empty $n_{\alpha}(k)=\langle
0|\alpha^{{\dagger}}_{k}\alpha_{k}|0\rangle=0$. However, when the uniform
component of the magnetic field exceeds the critical value, at $H>H_{c_{1}}$
states of the upper band between two Fermi points

$\displaystyle
k_{F}^{\pm}=k_{0}\pm\arcsin\sqrt{\frac{H^{2}-\Delta_{0}^{2}}{\sqrt{J^{*4}-(2D_{0}D_{1})^{2}}}},$
(17)

become occupied, and at

$\displaystyle H\geq
H_{c_{2}}=\sqrt{\Delta_{0}^{2}+\sqrt{J^{*4}-(2D_{0}D_{1})^{2}}}\,$ (18)

all states in the upper band are also occupied. Thus

$\displaystyle
n_{\alpha}(k)=\left\\{{\begin{array}[]{*{15}{c}}{\begin{array}[]{*{20}{c}}0\quad{\mbox{f}or}\quad
k\in[-\pi/2,\pi/2]\quad{\mbox{a}t}\quad{H<{H_{c_{1}}}}\end{array}}\\\
{\begin{array}[]{*{20}{c}}1\quad{\mbox{f}or}\quad
k\in\Lambda(k)\quad{\mbox{a}t}\quad{H>H_{c_{1}}}\end{array}}\end{array}}\right.$
(23)

where

$\displaystyle\Lambda(k)=\left\\{{\begin{array}[]{*{20}{c}}{\begin{array}[]{*{20}{c}}{[k_{F}^{-},k_{F}^{+}]}\\\
{[-\pi/2,\pi/2]}\end{array}}&{\begin{array}[]{*{20}{c}}{H_{c_{1}}<H<H_{c_{2}}}\\\
{H>H_{c_{2}}}\end{array}}\end{array}}\right.$ (29)

Figure 2: (color online) (a) Magnetization and (b) staggered magnetization
versus the magnetic fields for a given DMI as $D_{0}=0.7$ and $D_{1}=0.3$. The
inset in (a) expresses a square-root behavior of the magnetization near to
$H_{c_{1}}$. The vertical black dashed lines in (a) point out the critical
uniform ones.

Using (23) one easily obtains that at $H<H_{c_{1}}$ net magnetization of the
system is zero

$\displaystyle M=\frac{1}{N}\sum_{n}\langle
0|S_{n}^{z}|0\rangle=\frac{1}{2\pi}\int_{-\pi/2}^{\pi/2}dk\,n_{\beta}(k)-\frac{1}{2}=0\,,$

while, for $H>H_{c_{1}}$ the system acquires a finite magnetization per site

$\displaystyle
M=\frac{1}{2\pi}\int_{-k_{F}^{-}}^{k_{F}^{+}}\,dk\,n_{\alpha}(k)=\frac{1}{2\pi}\left(k_{F}^{+}-k_{F}^{-}\right)$
$\displaystyle\hskip
14.22636pt=\frac{1}{\pi}\arcsin\sqrt{\frac{H^{2}-\Delta_{0}^{2}}{\sqrt{J^{*4}-(2D_{0}D_{1})^{2}}}}\,.$
(30)

In the close proximity to the transition point, at $H>H_{c_{1}}=\Delta_{0}$
but $H-H_{c_{1}}\ll H_{c_{1}}$ the magnetization shows a square-root behavior

$\displaystyle M\simeq{\cal C}\left(H-H_{c_{1}}\right)^{1/2}\,,$ (31)

while the magnetic susceptibility is divergent

$\displaystyle\chi\simeq\frac{1}{2}{\cal C}\left(H-H_{c_{1}}\right)^{-1/2}\,,$
(32)

where

${\cal
C}=\frac{1}{\pi}\sqrt{\frac{2\Delta_{0}}{\sqrt{J^{*4}-(2D_{0}D_{1})^{2})}}}\,.$

At $H>H_{c_{1}}$ the excitation spectrum is gapless. As displayed in Fig.
2(a), as soon as the uniform magnetic field exceeds the first critical field,
the magnetization starts to boost and exhibits a monotonic behavior up to the
second critical field $H_{c_{2}}$ where the magnetization reaches its
saturation value $M=1/2$. It is explicit at a certain magnetic field as
$H_{c_{1}}<H_{T}<H_{c_{2}}$, the concavity of the graph of the magnetization
changes. The insert in Fig. 2(a) clearly shows the square-root behavior in the
vicinity of $H_{c_{1}}$. On the other side, Fig. 2(b) exhibits that right
after the exert of the staggered case, the Neel order induces. In such a
situation enhancing the staggered field leads to the increment of the
staggered magnetization, thereby at $H_{1}\to\infty$ it tends to the value of
1/2 albeit the saturation is never achieved for any finite staggered field.

Figure 3: (color online) The alternating chiral order parameter for the DMI
values $D_{0}=0.7$ and $D_{1}=0.3$ versus (a) the uniform and (b) the
staggered magnetic fields.

Order parameters always serve as an excellent tools to identify QPT-s [1, ]
even in the case of nonequilibrium criticality [48, ]. At $H<H_{c_{1}}$ the
system is in the gapped phase, with a fixed LRO structure of correlations in
the ground state. To identify this structure consider the following set of
order parameters:

a) the staggered magnetization

$\displaystyle M_{st}$ $\displaystyle=$
$\displaystyle\frac{1}{N}\sum_{n}(-1)^{n}\langle 0|S_{n}^{z}|0\rangle$ (33)
$\displaystyle=$
$\displaystyle\frac{1}{2N}\sum\limits_{k=-\pi/2}^{\pi/2}\cos(2{\phi_{k}}),\,$

b) the staggered transverse spin dimerization on the plane vertical over the
$z$-axis

$\displaystyle{\cal D}^{\perp}$ $\displaystyle=$
$\displaystyle\frac{1}{N}\sum\limits_{n=1}^{N}(-1)^{n}{\left\langle
S_{n}^{x}S_{n+1}^{x}+S_{n}^{y}S_{n+1}^{y}\right\rangle}$ (34) $\displaystyle=$
$\displaystyle\frac{1}{2N}\sum\limits_{k^{\prime}{}}\sin(2\phi_{k^{\prime}{}})\omega(\theta_{k^{\prime}{}},k^{\prime}{})\,$

where
$\omega(\theta_{k^{\prime}{}},k^{\prime}{})=\cos(\theta_{k^{\prime}{}})-\cos(k^{\prime}{}-\theta_{k^{\prime}{}})$
and

c) the staggered transverse spin chirality order parameter [Chubukov_91, ]

$\displaystyle{\cal K}^{z}$ $\displaystyle=$
$\displaystyle\frac{1}{N}\sum\limits_{n=1}^{N}(-1)^{n}{\left\langle
S_{n}^{x}S_{n+1}^{y}-S_{n}^{y}S_{n+1}^{x}\right\rangle}$ (35) $\displaystyle=$
$\displaystyle\frac{1}{2N}\sum\limits_{k^{\prime}{}}\sin(2\phi_{k^{\prime}{}})\varpi(\theta_{k^{\prime}{}},k^{\prime}{})\,.$

where
$\varpi(\theta_{k^{\prime}{}},k^{\prime}{})=\sin(\theta_{k^{\prime}{}})-\sin(k^{\prime}{}-\theta_{k^{\prime}{}})$.
Note that $k^{\prime}{}$ refers to $[-\pi/2,\pi/2]$ for $H<H_{c_{1}}$ and
$\Lambda(k)$ for $H>H_{c_{1}}$. The computing of the order parameters is put
in detail in Appendix.

In the absence of a magnetic field, the ground state of the model (2) is
characterized by the coexistence of the LRO alternating transverse
dimerization and alternating chirality order [Japaridze_etal_19a, ]. In Fig. 3
and Fig. 4 we have plotted the corresponding order parameters as a function of
the applied uniform or staggered magnetic field for a chain with DMI values
$D_{0}=0.7,~{}D_{1}=0.3$. The very presence of plateau at $H<H_{c_{1}}$ in
Fig. 3(a) and Fig. 4(a) demonstrate that the system endures in front of the
uniform magnetic field. Immediately after the uniform component of the
magnetic field rises from $H_{c_{1}}$ the chirality and dimerization
parameters, which still marks the response of the system on the explicitly
broken by the alternating DM translation symmetry, already in the given
gapless phase, decreases monotonically so as to provide the zero values
exactly at the saturation critical field $H_{c_{2}}$. At $H>H_{c_{2}}$ the
system is fully polarized, without any trace of chirality of dimer order. The
interesting is at $H_{T}$, the value of the dimer is zero. On the contrary, in
this certain field, the chiral order has a nonzero value. As a consequence,
$H_{T}$ is the field that can distinguish the coexistence of these two orders.
Conversely, in the Fig. 3(b) and Fig. 4(b) the alternating chiral and dimer
order parameter are plotted as a function of staggered magnetic field. As it
follows from these figures both parameters monotonically decay with increasing
$H_{1}$ asymptotically approaching zero at $H_{1}\rightarrow\infty$.

Figure 4: (color online) (a) The dimer order parameter for DMI interaction
values as $D_{0}=0.7$ and $D_{1}=0.3$ as function of (a) the uniform and (b)
the staggered magnetic fields.

To conclude this section let us review the obtained results. As was extracted,
the order parameters accurately respond to change of the magnetic fields; in
contrary to the staggered case which just adds Neel order in the system, the
uniform field unveils two critical fields, $H_{c_{1}}$ and $H_{c_{2}}$. In a
range of magnetic fields, we see the conditions where the orders coexist.
Moreover, albeit the staggered field tends the system to have a saturated
staggered magnetization but annihilates the other orders.

##  III The continuum-limit bosonization treatment

To continue our analysis of the model (1) in the whole area
$\gamma=J_{z}/J>-1$, in this Section we present results obtained by the
continuum-limit bosonization treatment. Below we follow the route developed in
[Japaridze_etal_19a, ] to study the ground state magnetic phase diagram of a
spin $S=1/2$ $XXZ$ chain with alternating DMI.

###  III.1 Gauging away the DM interaction

We start from the Hamiltonian (1). In analogy with the case of a spin chain
with uniform DMI it is instructive to gauge away the alternating DM term by a
position-dependent rotation of spins about the ${\hat{z}}$ axis [Perk_76, ;
Japaridze_etal_19a, ]. We introduce new spin variables ${\bf\tau}_{2n}$ and
${\bf\tau}_{2n+1}$ by performing a site-dependent rotation of spins along the
chain around the ${\hat{z}}$ axis with relative angle $\vartheta_{-}$ for
spins at consecutive odd-even sites ($2n-1,2n$) and $\vartheta_{+}$ for spins
at consecutive even-odd sites ($2n,2n+1$), as

$\displaystyle S^{+}_{2n-1}$ $\displaystyle=$ $\displaystyle
e^{i(n-1)(\vartheta_{-}+\vartheta_{+})}\tau^{+}_{2n-1},$ $\displaystyle
S^{+}_{2n}$ $\displaystyle=$ $\displaystyle
e^{in\vartheta_{-}+i(n-1)\vartheta_{+}}\tau^{+}_{2n},$ (36) $\displaystyle
S^{+}_{2n+1}$ $\displaystyle=$ $\displaystyle
e^{in(\vartheta_{-}+\vartheta_{+})}\tau^{+}_{2n+1},$ $\displaystyle
S^{z}_{2n\pm 1}$ $\displaystyle=$ $\displaystyle\tau^{z}_{2n\pm
1}~{}~{};~{}~{}S^{z}_{2n}=\tau^{z}_{2n}\,.$

Inserting (III.1) in (1) and choosing angles $\vartheta_{\pm}$ such that

$\tan\vartheta_{\pm}=D_{\pm}/J,$

one cancels the DM coupling and obtains, in terms of new $\tau$ spin
variables, the Hamiltonian of $XX$ chain with an alternating exchange in the
presence of alternating transverse magnetic field

$\displaystyle{\cal H}$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{N}\Big{[}\frac{\tilde{J}}{2}(1-(-1)^{n}\delta)\left(\tau^{+}_{n}\tau^{-}_{n+1}+\tau^{-}_{n}\tau^{+}_{n+1}\right)$
(37) $\displaystyle+$ $\displaystyle
J_{z}\tau^{z}_{n}\tau^{z}_{n+1}-(H+(-1)^{n}H_{1})\,\tau^{z}_{n}\Big{]}\,.$

Here

$\displaystyle\tilde{J}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(J_{+}+J_{-}\right)\simeq J^{\ast}+{\cal
O}\left(D_{i}/J\right)^{4},$ (38) $\displaystyle\delta\tilde{J}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left(J_{+}-J_{-}\right)\simeq\frac{D_{0}D_{1}}{J^{\ast
2}}+{\cal O}\left(D_{i}/J\right)^{4},$ (39)

at $D_{\pm}\ll J$ ( $i=\pm$) and $J_{\pm}=\sqrt{J^{2}+D_{\pm}^{2}}$.

Thus, after the gauge transformation, we obtain the Hamiltonian of the
spin-$1/2$ $XXZ$ Heisenberg chain with alternating transverse exchange
[Derzhko_07, ] in the presence of an alternating magnetic field. The
analytical and numerical studies of the spin-$1/2$ Heisenberg chain with
dimerized exchange count decades. The gapped excitation spectrum of bond
alternating spin-$1/2$ AFM Heisenberg chain was first predicted by Bulaevskii
in 1963 [Bulaevskii_63, ]. The analytical and numerical studies of the
spin-$1/2$ Heisenberg chain with dimerized exchange include studies of the
ordered phases and of the QPT-s in the ground state[Cross_Fisher_79, ;
Hida_92, ; Yamamoto_97, ; Takayoshi_Sato_10, ; Qiang_et_al_13, ;
Wang_et_al_13, ; Giamarchi_etal_18a, ; Ueda_Onoda_20, ], of the excitation
spectrum [Harris_73, ; Bonner_Bloete_82, ; BKJ_98, ; Affleck_Lect_Notes_07, ]
as well as of the magnetic and thermal properties [Totsuka_97, ;
Chitra_Giamarchi_97, ; Giamarchi_etal_18b, ].

The very powerful and accurate analytical tool to study spin chains and, in
particular, the spin-$1/2$ dimerized Heisenberg chain is the continuum-limit
bosonization approach. A method is well known and discussed in detail in many
excellent reviews and books. Therefore, below we briefly sketch the most
relevant steps and bosonization conventions, while for technical details we
refer the reader to the corresponding references [GNT_book_98, ;
Giamarchi_book_04, ; Cabra_Pujol_04, ]. To obtain the continuum version of the
Hamiltonian (37), we use the standard bosonization expression of the spin
operators [GNT_book_98, ]

$\displaystyle\tau_{n}^{z}$ $\displaystyle\simeq$
$\displaystyle\sqrt{\frac{K}{\pi}}\partial_{x}\phi(x)+(-1)^{n}\frac{{\it
a}}{\pi\alpha}\sin\sqrt{4\pi K}\phi(x)\,,$ (40) $\displaystyle\tau^{\pm}_{n}$
$\displaystyle\simeq$ $\displaystyle\frac{{\it b}}{\pi\alpha}\cos(\sqrt{4\pi
K}\phi)\,e^{\pm i\sqrt{\pi/K}\theta}$ (41) $\displaystyle-$
$\displaystyle(-1)^{n}\frac{{\it c}}{\pi\alpha}\,e^{\pm
i\sqrt{\pi/K}\theta}\,.$

Here $\phi(x)$ and $\theta(x)$ are dual bosonic fields,
$\partial_{t}\phi=u\partial_{x}\theta$, and satisfy the following
commutational relation

$\displaystyle[\phi(x),\theta(y)]=i\Theta(y-x)\,,$
$\displaystyle[\phi(x),\theta(x)]=i/2\,.$ (42)

Here the non-universal real constants a, b and c depend smoothly on the
parameter $\gamma^{\ast}=J_{z}/J^{\ast}$, are of the order of unity at
$\gamma^{\ast}=0$ [Hikihara_Furusaki_98, ] and are expected to be nonzero
everywhere at $|\gamma^{\ast}|<1$. The Luttinger liquid parameter is known
within the critical line $-1<\gamma^{\ast}<1$ to be [Luther_Peschel_75, ]

$\displaystyle K$ $\displaystyle=$
$\displaystyle\frac{\pi}{2\arccos\left(-\gamma^{\ast}\right)}\,.$ (43)

Thus the parameter $K$ decreases monotonically from its maximal value
$K\to\infty$ at $\gamma^{\ast}\to-1$ (ferromagnetic instability point), is
equal to unity at $\gamma^{\ast}=0$ ($J_{z}=0$) and reaches the value $K=1/2$
at $\gamma^{\ast}=1$ (isotropic antiferromagnetic chain). In the case of
dominating Ising type anisotropy, at $\gamma^{\ast}>1$, $K<1/2$.

Using (40)-(41) we finally obtain for the initial lattice Hamiltonian (37):

$\displaystyle{\cal H}$ $\displaystyle=$ $\displaystyle\int
dx\,\Big{[}\frac{u}{2}(\partial_{x}\phi)^{2}+\frac{u}{2}(\partial_{x}\theta)^{2}-H\sqrt{\frac{K}{\pi}}\partial_{x}\phi$
(44) $\displaystyle-\frac{\delta J^{*}}{\pi\alpha}\cos\sqrt{4\pi
K}\phi-\frac{H_{1}}{\pi\alpha}\sin\sqrt{4\pi K}\phi$
$\displaystyle+\frac{J_{z}}{2\pi^{2}\alpha^{2}}\cos\sqrt{16\pi
K}\phi\,\Big{]}\,,$

where $u\simeq J^{\ast}/K$ is the velocity of spin excitation. It is
instructive to unify the first two nonlinear terms with equal arguments into
one and rewrite the Hamiltonian in the standard form of the double-frequency
sine-Gordon model

$\displaystyle{\cal H}=\int
dx\Big{[}\frac{u}{2}(\partial_{x}\phi)^{2}+\frac{u}{2}(\partial_{x}\theta)^{2}-H\,\frac{\beta}{2\pi}\partial_{x}\phi$
$\displaystyle\hskip
14.22636pt-\frac{\Delta_{0}}{\pi\alpha^{2}}\cos\beta(\phi-\phi_{0})+\frac{M_{0}}{\pi\alpha^{2}}\cos
2\beta\phi\,\Big{]}\,.$ (45)

where $\beta=\sqrt{4\pi K}$, $M_{0}=J_{z}/2\pi$,

$\displaystyle\Delta_{0}$ $\displaystyle=$
$\displaystyle\sqrt{H_{1}^{2}+(D_{0}D_{1}/J^{\ast})^{2}},$ (46)

and

$\displaystyle\phi_{0}=\arctan\left(H_{1}/\delta J^{\ast}\right)\,.$ (47)

The scaling dimensions of the first two cosine terms
$d=dim[\cos(\beta\phi)]=\beta^{2}/4\pi=K$, while the scaling dimension of the
third cosine term is $d^{\ast}=dim[\cos(2\beta\phi)]=4K$. Each of two cosine
terms in (III.1) become relevant in the parameter range where the
corresponding scaling dimensionality $d\leq 2$ or $d^{\ast}\leq 2$. Using (43)
we find that $d\leq 2$, i.e. the first cosine term in (III.1) is relevant, at

$\displaystyle\gamma^{\ast}>\gamma^{\ast}_{c_{1}}=-\sqrt{2}/2\,,$ (48)

while the second cosine term in (III.1), for $\gamma^{\ast}>1$. Therefore, at
$H=0$ the antiferromagnetic sector of the phase diagram at $\gamma^{\ast}>-1$
can be divided in two segments of the model parameter range: the gapless LL
sector at $-1<\gamma^{\ast}\leq\gamma^{\ast}_{c_{1}}$and the gapful sector at
$\gamma^{\ast}>\gamma^{\ast}_{c_{1}}$.

###  III.2 The LL sector $-1<\gamma^{\ast}<\gamma^{\ast}_{c_{1}}$

At $-1<\gamma^{\ast}<\gamma^{\ast}_{c_{1}}$, both cosine terms in (III.1) are
irrelevant and can be neglected. The gapless long-wavelength excitations of
the anisotropic spin chain are described by the standard Gaussian theory with
the Hamiltonian

$\displaystyle{\cal H}_{0}$ $\displaystyle=$ $\displaystyle u\int
dx\,\Big{[}\,\frac{1}{2}(\partial_{x}\phi)^{2}+\frac{1}{2}(\partial_{x}\theta)^{2}\,\Big{]}.$
(49)

In this critical LL phase, all correlations show a power-law decay, with
indices, smoothly depending on the parameter $K$ [GNT_book_98, ]. Because the
LL parameter $K$ depends only on the anisotropy parameter $\gamma^{\ast}$, the
very presence of magnetic field does not change the values of the parameter
$\gamma^{\ast}$ where the conine term becomes relevant, however, as we show
below, magnetic field substantially influence properties of the model in the
gapped phase and at leads to the extension of the LL sector.

###  III.3 The gapped sector $\gamma^{\ast}_{c_{1}}<\gamma^{\ast}\leq 1$

At $H=0$ and $\gamma^{\ast}_{c_{1}}<\gamma^{\ast}\leq 1$ the first cosine in
(III.1) is relevant perturbation, while the double-frequency cosine term
remains irrelevant and can be neglected. In this case infrared properties of
the system are described by the standard sine-Gordon (SG) model

$\displaystyle{\cal H}$ $\displaystyle=$ $\displaystyle\int
dx\,\Big{[}\frac{u}{2}(\partial_{x}\phi)^{2}+\frac{u}{2}(\partial_{x}\theta)^{2}$
(50) $\displaystyle-$
$\displaystyle\frac{\Delta_{0}}{\pi\alpha^{2}}\cos\beta(\phi-\phi_{0})\Big{]}.$

At $\gamma^{\ast}=\gamma^{\ast}_{c_{1}}\simeq-0.7$ the Berezinskii-Kosterlitz-
Thouless (BKT) [BKT_tansition, ] quantum phase transition takes place in the
ground state of the system, the excitation gap opens and remains finite in the
whole region $\gamma^{\ast}>\gamma^{\ast}_{c_{1}}$. From the exact solution of
the quantum sine-Gordon model [SG_exact_Sol, ; Zamolodchikov_95, ] it is known
that for arbitrary finite $\Delta_{0}$ the gapped excitation spectrum of the
Hamiltonian Eq. (50) at $-0.7<\gamma^{\ast}\leq 0$ consists of solitons and
antisolitons with masses

${\cal M}_{sol}\sim\left(\Delta_{0}/J^{\ast}\right)^{1/(2-K)},$ (51)

while at $0<\gamma^{\ast}\leq 1$ in addition, also of soliton-antisoliton
bound states (”breathers”) with the lowest breather mass

${\cal M}_{br}=2{\cal M}_{sol}\sin\left(\frac{\pi K}{4-2K}\right)\,.$ (52)

Thus, in the whole parameter range $0<\gamma^{\ast}\leq 1$ the soliton mass
${\cal M}_{sol}$ is the energy scale that determines the size of the spin
excitation gap.

At the BKT phase transition point, the excitation gap is exponentially small

$\Delta_{exc}\sim
J^{\ast}\exp\left(-1/(\gamma^{\ast}-\gamma^{\ast}_{c_{1}})\right)\,,$ (53)

it smoothly increases with increasing $\gamma^{\ast}$, and at
$\gamma^{\ast}=0$

$\Delta_{exc}=2J^{\ast}{\cal M}_{sol}=2\Delta_{0}\,,$ (54)

in a perfect agreement with results obtained in the Sec. II (see Eq.(16)).

Generation of a gap in the excitation spectrum leads to suppression of
fluctuations in the system and the $\phi$ field is condensed in one of its
vacua corresponding to the minimum of the dominating potential energy term
$V(\phi)=-\Delta_{0}\cos\sqrt{4\pi K}(\phi-\phi_{0})$ [Mutalib_Emery_86, ]

$\displaystyle\langle 0|\,\phi\,|0\rangle$ $\displaystyle=$
$\displaystyle\phi_{0}+2n\pi,\hskip 28.45274pt\textrm{ at
\quad$\Delta_{0}>0$}\,.$ (55)

Using the vacuum expectation value of the $\phi$ field (55) one easily obtains
that in the ground state, the system is characterized by the LRO pattern of
the on-site staggered magnetization with amplitude

$\displaystyle M_{st}=(-1)^{n}\langle
0|\tau^{z}_{n}|0\rangle\sim\sin\sqrt{4\pi K}\phi_{0}\,.$ (56)

Moreover, if we consider the link-located degrees of freedom, using (40)-(41)
one obtains that in the ground state of the system the staggered component of
the $\tau$-spin chirality operator given by

$\displaystyle\kappa_{n}^{(\tau)}=-i(-1)^{n}\langle
0|\left(\tau^{+}_{n}\tau^{-}_{n+1}-h.c.\right)|0\rangle\rightarrow$
$\displaystyle\simeq\frac{{\it b}}{\pi\alpha}\sin(\sqrt{4\pi K}\phi_{0})\,$
(57)

together with staggered parts of the $\tau$-spin longitudinal and transverse
nearest-neighbour spin-exchange operators

$\displaystyle{\cal D}_{\perp}^{(\tau)}(n)=\frac{(-1)^{n}}{2}\langle
0|\left(\tau^{+}_{n}\tau^{-}_{n+1}+h.c.\right)|0\rangle$ $\displaystyle\hskip
42.67912pt\sim\frac{{\it a}}{\pi\alpha}\cos(\sqrt{4\pi K}\phi_{0})$ (58)
$\displaystyle{\cal D}_{z}^{(\tau)}(n)=(-1)^{n}\langle
0|\tau^{z}_{n}\tau^{z}_{n+1}|0\rangle$ $\displaystyle\hskip
42.67912pt\sim\frac{b}{\pi\alpha}\cos(\sqrt{4\pi K}\phi_{0})$ (59)

show the long-range ordered patterns in the ground state of the effective
$\tau$-spin model.

Using (III.1), from (56)-(59) we obtain, that in the gapped phase the initial
spin chain shows a long-range dimerization order

$\displaystyle\frac{1}{N}\sum_{n}(-1)^{n}\langle\,\textbf{S}_{n}\cdot\textbf{S}_{n+1}\,\rangle\sim(\cos\vartheta_{+}-\cos\vartheta_{-})\,\epsilon\,,$
(60)

which coexists with the LRO pattern of the alternating spin chirality vector

$\displaystyle\frac{1}{N}\sum_{n}(-1)^{n}\langle\,\kappa^{z}_{n}\rangle\sim(\sin\vartheta_{+}-\sin\vartheta_{-})\,\,\kappa^{z}_{0}.$
(61)

and of the staggered magnetization

$\displaystyle\frac{1}{N}\sum_{n}(-1)^{n}\langle S^{z}_{n}\rangle=\,\,m.$ (62)

###  III.4 The gapped sector $\gamma^{\ast}>1$

At $\gamma^{\ast}>1$ the second cosine term in (III.1) becomes relevant and
the effective continuum-limit version of the initial lattice spin model (37)
is given by the double-frequency sine-Gordon (DSG) model [DSG_Book_80, ]

$\displaystyle{\cal H}=\int
dx\,\Big{[}\frac{u}{2}(\partial_{x}\phi)^{2}+\frac{u}{2}(\partial_{x}\theta)^{2}$
$\displaystyle-\frac{\Delta_{0}}{\pi\alpha^{2}}\cos\beta(\phi-\phi_{0})+\frac{M_{0}}{\pi\alpha^{2}}\cos(2\beta\phi)\,\Big{]}\,.$
(63)

The DSG model describes an interplay between two relevant perturbations to the
Gaussian conformal field theory with the ratio of their scaling dimensions
equal to 4. Infrared properties of the DSG model are determined by an
interplay between these two relevant sources of perturbations to the Gaussian
conformal field theory. Acting separately, each leads to the pinning of the
field $\phi$ in corresponding minima, and depending on sign and amplitudes of
the model parameters, $\Delta_{0}$ and $J_{z}$, these minima either exclude
each other or support synergic ordering [DSG_1, ; DSG_2a, ].

In absence of the staggered component of the magnetic field $\phi_{0}=0$ the
spin dimerization, provided by the alternating DMI interaction and the
staggered magnetization, supported by the Ising part of the exchange
interaction have different parity symmetries, the field configurations which
minimize one perturbation do not minimize the other. This destructive
competition between possible sets of vacuum configurations of the two cosine
terms is resolved via the Ising type QPT in the ground state at
$J^{\ast}_{z}\sim 1+D_{0}D_{1}/J^{\ast}$ from a dimerized phase at
$J_{z}<J^{\ast}_{z}$ into the composite phase with coexisting dimer and
antiferromagnetic order at $J_{z}>J^{\ast}_{z}$ [Japaridze_etal_19a, ]. The
very presence of the corresponding QPT can already be traced performing
minimization of the potential

$\displaystyle V(\phi)$ $\displaystyle=$
$\displaystyle-\Delta_{0}\cos\beta\phi+M_{0}\cos 2\beta\phi\,.$ (64)

The Ising transition corresponds to the crossover from a double well to a
single well profile of the potential (64). indeed, one can easily obtain, that
at $4M_{0}<\Delta_{0}$ the vacuum expectation value of $\phi$ field which
minimizes $V(\phi)$ is $\langle\phi\rangle=0$ and therefore in this case only
the dimerized phase is realized in the ground state. However, at
$4M_{0}>\Delta_{0}$ the $\phi$ field is condensed in the minima

$\displaystyle\beta\langle\phi\,\rangle=\pm\,\arccos\left(\Delta_{0}/4M_{0}\right)+2\pi
n\,$ (65)

and, as the result, the ground state of the $\tau$-spin system, in addition to
the dimerization pattern the long-range antiferromagnetic order is present.

Figure 5: The DSG potential $V(x)=-\Delta_{0}\cos x+M_{0}\cos 2x$ for
$4M_{0}<\Delta_{0}$ (red) and for $4M_{0}>\Delta_{0}$ (blue). We have taken a
DMI as $D_{0}=0.7$ and $D_{1}=0.3$ for $\gamma=0.1$ (red) and $\gamma=1.0$
(blue). It is evident that at $4M_{0}>\Delta_{0}$ the structure of minima is
changed. Appear new transitions between new set of minima corresponding to
solitons capturing topological charge (spin) equal to $\phi_{0}/2\pi$ and
$1-\phi_{0}/2\pi$.

The Ising transition at $J=(J_{z})_{c_{2}}$ is also displayed in the character
of excitations. In absence, as well as for the weak double-frequency cosine
term ($4M_{0}<\Delta_{0}$) the vacuum expectation value of the field is
determined by the set of minima of the $-\Delta_{0}\cos\beta\phi$ term given
by (55). Therefore the spin of the elementary excitation i.e the quantum
number captured by the soliton kink between nearest minima

$\displaystyle
S=\frac{\beta}{2\pi}\int_{-\infty}^{+\infty}dx\partial_{x}\phi(x)=1,$ (66)

and corresponds to the magnon type excitation.

In the case of doubled potential there appear solitons with different masses
(see Fig. 5), one corresponds to the kink between two split minima, and the
second, of the larger mass, corresponds to the kink over the high potential.
The first soliton captures the spin

$\displaystyle
s_{1}=\frac{\beta}{2\pi}\int_{-\infty}^{+\infty}dx\,\partial_{x}\phi(x)=\frac{1}{\pi}\arccos\left(\Delta_{0}/4M_{0}\right),$
(67)

while the second,

$\displaystyle
s_{2}=\frac{\beta}{2\pi}\int_{-\infty}^{+\infty}dx\partial_{x}\phi(x)=1-s_{1}\,.$
(68)

In absence of the dimerization ($\delta=0$), masses and spins of both soliton
$s_{1}=s_{2}=1/2$, which corresponds to the spinon type excitations in the
Heisenberg chain in the gapful sector.

In the presence of finite staggered magnetic field the $\phi\rightarrow-\phi$
symmetry of the model is broken for arbitrary $H_{1}\neq 0$. Already in the
absence of the double frequency cosine term, the minimum of the potential is
reached at $\phi=\phi_{0}+2l\pi$. Presence, at $\gamma^{\ast}>1$, of the
second relevant cosine term in the Hamiltonian results to shift off the minima
positions, make minima deeper, but does not change the structure of the vacua
(see Fig. 6 ), the distance between minima remain unchanged and equal to
$2\pi$ and therefore the ground state and character of excitations is not
influenced.

Thus, in constant with the $H_{1}\neq 0$ case Japaridze_etal_19a , at the
$\gamma^{\ast}>1$ point the Ising transition is absent and in the whole
parameter range $\gamma^{\ast}>\gamma^{\ast}_{c_{1}}$ the ground state is
characterized by the composite order with coexisting dimer, alternating
chirality and antiferromagnetic order, massive excitations have the similar
nature, are magnons and capture spin $S=1$ and the excitation gap smoothly
increases in the whole area $\gamma^{\ast}>\gamma^{\ast}_{c_{1}}$ with weak
anomaly at the crossover point at $\gamma^{\ast}>1$.

To summarize this section, the staggered magnetic field

  * •

leads to the presence of the composite ordered phase in the whole parameter
range $\gamma^{\ast}>\gamma^{\ast}_{c_{1}}$;

  * •

increases the bare value of the soliton mass $\Delta_{0}$ i.e leads to
enlargement of the excitation gap in the whole parameter range
$\gamma^{\ast}>\gamma^{\ast}_{c_{1}}$ .

Figure 6: (color online) The DSG potential $V(x)=-\Delta_{0}\cos x+H_{1}\sin
x+M_{0}\cos 2x$ for values of the parameters $\Delta_{0}=M_{0}=1$ and for
different values of the parameter $M_{0}=0$ (red), $M_{0}=1$ (green) and
$M_{0}=1.5$ (blue). It is clear that the distance between absolute minima of
the potential remain unchanged.

Figure 7: (color online) Magnetization versus the uniform magnetic field for
sizes $N=20,24$ provided for a DMI as $D_{0}=0.7$ and $D_{1}=0.3$ for (a)
$\gamma=-1.0$, (b) $\gamma=0.5$, and (c) $\gamma=2.0$. The vertical black
dashed lines hint the critical uniform fields.

##  IV The C-IC transition

At $H\neq 0$ the Hamiltonian (III.1) is easily recognized as the Hamiltonian
for the commensurate-incommensurate (C-IC) phase transition
[C_IC_transition_1, ; C_IC_transition_2, ] which has been intensively studied
in the past using bosonization approach [JN_79, ] and the Bethe ansatz
technique [JNW_84, ]. Below we use the results obtained in these studies to
give brief sketch of the phase diagram.

At $H\neq 0$, the presence of the gradient term in the Hamiltonian (III.1)
makes it necessary to consider the ground state of the sine-Gordon model in
sectors with nonzero topological charge. The effective chemical potential

$\displaystyle\sim-H\sqrt{\frac{\beta}{4\pi^{2}}}\partial_{x}\phi\,,$ (69)

tends to change the number of particles in the ground state i.e. to create a
finite and uniform density of solitons. It is clear that the gradient term in
(III.1) can be eliminated by a gauge transformation

$\displaystyle\phi\rightarrow\phi+\frac{H\beta}{2\pi}\,x\,,$ (70)

however, this immediately implies that the vacuum distribution of the field
$\phi$ will be shifted with respect to the corresponding minima. Competition
between the uniform magnetic field, which supports the ground state
characterized by the finite value of the gradient of the $\phi(x)$ field and
of the nonlinear cosine terms, which prefer pinned in one of the potential
minima the constant value of the $\phi(x)$ field, is resolved as a continuous
C-IC phase transition from a gapped state at $H<H_{c}=M$ to a gapless
(paramagnetic) phase at $H>H_{c}$, where $M$ is the mass of the soliton of the
corresponding SG or DSG theory [C_IC_transition_1, ].

As usual in the case of C-IC transition, in the close proximity of the
transition point, $H>H_{c}$ $(H-H_{c})\ll H_{c}$, the magnetization shows a
square-root behavior

$\displaystyle M(H)\sim(H-H_{c})^{1/2}\ $ (71)

and the magnetic susceptibility a square-root divergence

$\displaystyle\chi(H)\sim(H-H_{c})^{-1/2}\,.$ (72)

At $H>H_{c}$ the excitation spectrum becomes gapless, all correlations show a
power-law decay with exponents dependent on $\gamma^{\ast}$ and $H$ and the
system exhibits properties of the magnetized spin LL.

Thus in the presence of alternating magnetic field the ground-state phase
diagram depending on the model parameters consist of the following two
sectors:

  * •

The gapless LL phase with finite magnetization at $H>H_{c}$;

  * •

The gapped phase with composite order, characterized by the coexistence of the
LRO dimerization, alternating spin chirality (spin current), and
antiferromagnetic patterns $H<H_{c}$.

  * •

The value of the critical field $H_{c}=M$ is determined by the mass of the
soliton of the corresponding SG or DSG theory.

  * •

The soliton mass is determined by its bare value $\Delta_{0}$ and the
effective anisotropy parameter $\gamma^{\ast}$.

  * •

The effective anisotropy parameter $\gamma^{\ast}$ depends on both symmetric
($J$) and asymmetric ($D_{0},D_{1}$.) components of spin exchange;

  * •

The bare value of the soliton mass $\Delta_{0}$ is additively contributed by
the staggered component of the magnetic field and alternating DMI.

##  V Numerical results for an arbitrary value of $\gamma$.

In this section, we present the results of our numerical studies for finite
chains. We use the numerical Lanczos technique as one of the most frequently
used numerical algorithms in investigating the ground-state phase diagram of
low-dimensional spin-1/2 systems. Applying the Lanczos algorithm to the
transformed Hamiltonian, we diagonalize numerically finite chains up to $N=28$
spin-1/2 particles and calculate the uniform and the staggered magnetizations
for arbitrary values of $\gamma=J_{z}/J$.

In Fig. 7 we have illustrated the magnetization as a function of the uniform
magnetic field for a specified value of DMI as $D_{0}=0.7$ and $D_{1}=0.3$ in
the absence of the staggered magnetic field. As depicted in Fig. 7(a), when
the system is put in the gapless LL phase, as soon as the uniform field is
exerted the magnetization process starts and the system remains in the LL
phase up to a critical uniform field $H_{c}\simeq 0.26\pm 0.01$ where the
system goes into the paramagnetic phase. Observed oscillations of the
magnetization result from the level crossing between the ground and the
excited states of this model in the gapless LL phase. The magnetization curves
started from the gapped composite $C1$ and $C2$ phases are manifested in Figs.
7(b) and 7(c). As is observed, the magnetization remains zero up to when the
value of the uniform field reaches to the first critical value as
$H_{c_{1}}(\gamma=0.5)\simeq 1.76\pm 0.01$ and $H_{c_{1}}(\gamma=2.0)\simeq
3.26\pm 0.01$, that are equal to the spin gaps. In complete agreement with our
analytical calculations, more increment of the uniform field entails the
enhance of the magnetization in such a way that makes it to be saturated in
the second critical field as $H_{c_{2}}(\gamma=0.5)\simeq 0.28\pm 0.01$ and
$H_{c_{2}}(\gamma=2.0)\simeq 0.62\pm 0.01$.

Figure 8: (color online) Staggered magnetization versus the staggered magnetic
field for different sizes $N=20,24,28$ for a DMI as $D_{0}=0.7$ and
$D_{1}=0.3$ supplied for (a) $\gamma=-2.0$, (b) $\gamma=-1.0$, (c)
$\gamma=0.5$, and (d) $\gamma=2.0$. As is viewed in (a), the FM phase resists
the transition up to when the staggered field touches a critical value (the
vertical black dashed line).

We also have studied the effect of the staggered magnetic field on the ground-
state phase diagram of the model. Numerical Lanczos results are presented in
Fig. 8 for values of DMI as $D_{0}=0.7$ and $D_{1}=0.3$. Since the staggered
field has no effect on the FM state a zero-plateau is observed in the curve of
staggered magnetization for $\gamma=-2.0$ [see Fig. 8(a)]. In the gapless LL
region, as is seen in Fig. 8(b), the staggered magnetization process starts
from zero instantly after imposing the staggered field and boosts
monotonically up to attain a saturated state. It is explicit that almost the
same growing behaviour also takes place for when the system is located in the
gapped $C1$ and $C2$ phases [Figs. 8(c) and 8(d)].

##  VI Summary

Quantum phase transition originated from quantum fluctuations at absolute zero
temperature deals with dramatic changes of the ground-state and low-excitation
properties. Despite the many-body systems, it has been found that it may also
outstretch in a few-body system. Understanding emergent quantum phenomena is a
major challenge of physics and a requirement for future technologies that aim
to manipulate in a controlled way the quantum properties of many-spin systems.

In this paper, we have studied the ground-state magnetic phase diagram of a
novel quantum spin chain, a 1D spin-1/2 XXZ model in the presence of the
modulated period of two lattice units Dzyaloshinskii-Moriya interaction (DMI)
and alternating with the same period longitudinal magnetic field. In this
respect, we first have focused on the exactly solvable case of the Hamiltonian
where $J_{z}=0.0$. In the absence of the magnetic fields, the ground state is
constructed from a gapped composite phase made of the coexistence of a long-
range dimer and an alternating spin chirality. The driven outcomes have
indicated that the uniform magnetic field generates two QPTs. The gapped
mentioned composite phase endures in the presence of a uniform field up to a
critical field. Above endurance is emerged as a zero-plateau in the
magnetization curve. More increasing the uniform field, a gapless phase is
appeared up to the second critical field, where the magnetization is
saturated. On the other hand, instantly a staggered magnetic field is exerted,
the Neel LRO creates and coexists with dimer and alternating spin chirality
orders.

In order to consider the non-integrable Hamiltonian ($J_{z}\neq 0$), we have
employed two apparatuses; the continuum-limit bosonization approach as an
analytical technique and the Lanczos algorithm as a numerical method to
confirm the analytical results. Our results in this condition have unveiled
that the uniform field behaves the same as what has come out for the case
$J_{z}=0.0$ with this difference that the critical fields in addition to DMI,
now, are dependent on the value of $\gamma$. The staggered magnetic field
manifests three different behaviors in the range of $\gamma$. When the system
is settled in the FM phase, it does not change the phase up to a critical
staggered field where after that, it induces the Neel phase in the system.
When the system is put in the LL phase, as soon as it is exerted, the Neel
phase is created. Eventually, in the case where the system is placed in the
composite $C1$ ($C2$) phase, the staggered field just creates (amplifies) the
Neel phase.

###### Acknowledgements.

G.I.J. acknowledges support from the Shota Rustaveli Georgian National Science
Foundation through the grant N FR-19-11872. In addition, S. Mahdavifar and H.
Cheraghi wish to acknowledge the support of the Iran National Science
Foundation (INSF) under grant number 98018317.

## Appendix: Calculation of the order parameters

The calculations of the alternating chirality and dimer order parameters are
straightforward. Applying (II) to original definition of these parameters and
afterwards using the Bogoliobov transformation, and also (23) and (29), lead
to

$\displaystyle{\cal D}^{\perp}$ $\displaystyle=$
$\displaystyle\frac{1}{N}\sum\limits_{n=1}^{N}(-1)^{n}{\left\langle
S_{n}^{x}S_{n+1}^{x}+S_{n}^{y}S_{n+1}^{y}\right\rangle}$ (73) $\displaystyle=$
$\displaystyle\frac{1}{2N}\sum\limits_{n=1}^{N/2}{\left\langle{-a_{n}^{\dagger}{b_{n}}-b_{n}^{\dagger}{a_{n}}+a_{n+1}^{\dagger}{b_{n}}+b_{n}^{\dagger}{a_{n+1}}}\right\rangle}$
$\displaystyle=$
$\displaystyle-\frac{1}{2N}\sum\limits_{k}{\sin(2{\Phi_{k}})\omega({\theta_{k}},k)}\left\langle{\alpha_{k}^{\dagger}{\alpha_{k}}-\beta_{k}^{\dagger}{\beta_{k}}}\right\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{2N}\sum\limits_{k^{\prime}{}}\sin(2\phi_{k^{\prime}{}})\omega(\theta_{k^{\prime}{}},k^{\prime}{})\,$

and

$\displaystyle{\cal K}^{z}$ $\displaystyle=$
$\displaystyle\frac{1}{N}\sum\limits_{n=1}^{N}(-1)^{n}{\left\langle
S_{n}^{x}S_{n+1}^{y}-S_{n}^{y}S_{n+1}^{x}\right\rangle}$ (74) $\displaystyle=$
$\displaystyle\frac{{-i}}{{2N}}\sum\limits_{n=1}^{N/2}{\left\langle{a_{n}^{\dagger}{b_{n}}-b_{n}^{\dagger}{a_{n}}-b_{n}^{\dagger}{a_{n+1}}+a_{n+1}^{\dagger}{b_{n}}}\right\rangle}$
$\displaystyle=$
$\displaystyle\frac{1}{{2N}}\sum\limits_{k}{\sin(2{\Phi_{k}})\varpi({\theta_{k}},k)}\left\langle{\alpha_{k}^{\dagger}{\alpha_{k}}-\beta_{k}^{\dagger}{\beta_{k}}}\right\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{2N}\sum\limits_{k^{\prime}{}}\sin(2\phi_{k^{\prime}{}})\varpi(\theta_{k^{\prime}{}},k^{\prime}{})\,.$

where as was mentioned before, $k^{\prime}{}$ refers to $[-\pi/2,\pi/2]$ for
$H<H_{c_{1}}$ and $\Lambda(k)$ for $H>H_{c_{1}}$.

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