# Phase transitions and magnetization of the mixed-spin Ising-Heisenberg
double sawtooth frustrated ladder

Hamid Arian Zad  Young Researchers and Elite Club, Mashhad Branch, Islamic
Azad University, Mashhad, Iran<EMAIL_ADDRESS>Nerses Ananikian
Alikhanyan National Science Laboratory, Alikhanian Br. 2, 0036 Yerevan,
Armenia<EMAIL_ADDRESS>
###### Abstract

The mixed spin-(1,1/2) Ising-Heisenberg double sawtooth ladder containing
mixture of both spin-1 and spin-1/2 nodal atoms, and the spin-1/2 interstitial
dimers is approximately solved by the transfer-matrix method. Here, we study
in detail the ground-state phase diagrams, also influences of the bilinear
exchange coupling on the rungs and cyclic four-spin exchange interaction in
square plaquette of each block on the magnetization and magnetic
susceptibility of the suggested ladder at low temperature. Such a double
sawtooth ladder may be found in a Shastry-Sutherland Lattice-type. In spite of
odd and even blocks spin ordering are different from each other, but due to
the commutation relation between all different block Hamiltonians, phase
diagrams, magnetization behavior and thermodynamic properties of the model are
the same for odd and even blocks. We show that at low temperature, both
exchange couplings can change the quality and quantity of the magnetization
plateaus versus the magnetic field changes. Specially, we find a new
magnetization plateau $\mathcal{M}/\mathcal{M}_{s}=5/6$ for this model.
Besides, we examine the magnetic susceptibility and specific heat of the model
in detail. It is proven that behaviors of the magnetization and the magnetic
susceptibility coincide at low temperature. The specific heat displays diverse
temperature dependencies, which include a Schottky-type peak at a special
temperature interval. We observe that with increase of the bilinear exchange
coupling on the rungs, second peak temperature dependence grows.

###### pacs:

03.67.Bg, 03.65.Ud, 32.80.Qk
Keywords: Double sawtooth, Phase diagrams, Ring exchange, Magnetization
plateau

##  1 Introduction

Various versions of the spin chains has been encountered with a lot of
attentions to display a wide variety of $T=0$ quantum phase transitions caused
by changing the tunable parameters of the system,s Hamiltonian such as
external magnetic field and exchange couplings [5, 1, 2, 3, 4, 6, 7, 8, 9, 10,
11, 12, 13]. A typical example is a ladder of spins for which spins on the
vertices have interaction along the legs by exchange coupling
$\mathcal{J}_{\parallel}$ and along the rungs via exchange coupling
$\mathcal{J}_{\perp}$ [16, 14, 15, 17, 18, 19]. Moreover, sawtooth chains as
another interesting solvable models have been studied from theoretical [20,
21, 22, 23, 24] and experimental [25, 26] point of view.

Geometrically frustrated Ising-like lattices in tetragonal shape have been
theoretically and experimentally investigated. For instance, in Ref. [27]
authors have synthesized single crystals of $Yb_{2}Pt_{2}Pb$ as a quasi-two
dimensional system, where strong magnetic frustration may arise from the
geometry of the underlying Shastry-Sutherland lattice-type (SSLT) and they
reported experimental results of the magnetization, the specific heat, and
magnetic susceptibility. They pointed out to existence of the Schottky-type
peak in the specific heat and the deference in magnetic susceptibility
behavior with respect to the temperature. In Ref. [28] Y. Dublenych determine
a complete and exact solution of the ground-state problem for the Ising model
on the SSLT. He found a magnetization plateau at the one-third of the
saturation value and it was offered as the only possible fractional plateau in
this model. Recently, Linda Ye et al. studied on the Ising-type rare earth
tetraborides $RB_{4}$ ($R=Er,Tm$), which is a tetragonal lattice topologically
equivalent to the SSLT with visible magnetization plateaus [29]. Furthermore,
numerical calculations and related simulations for the magnetic susceptibility
have been carried out.

In the last two decades, mixed spin models are one of the areas in the
framework of the intensive research on the magnetism [30]. We can refer to
[31] as one of the beneficial references in which Ivanov presented a brief
survey of the theoretical results in this hot area, also he investigated some
basic quantum spin models of quasi one-dimensional quantum ferrimagnets with
competing interactions, and compared them with each other.

On the theoretical side, mixed spin models have been constructed and discussed
by several authors [10, 31, 32, 33]. On the experimental side, besides the
exactly solvable spin-1/2 models such as $CuPzN$ [34], several real magnetic
materials such as azurite $Cu_{3}(CO_{3})_{2}(OH)_{2}$ and
$Cu_{3}(TeO_{3})_{2}Br_{2}$ can be described by Heisenberg spin models. Hence,
motivated by the experimental observations, several authors have studied the
case of antiferromagnetic mixed spin-(1,1/2) chain [31, 35]. T. Sakai and S.
Yamamoto numerically showed a quantized magnetization plateau as a function of
the field appearing at the one-third of the saturated magnetization for the
mixed spin-(1,1/2) Heisenberg ferrimagnet with anisotropic exchange coupling
in [36]. Plateaus in magnetization curves as functions of the magnetic field
and anisotropy parameter, also thermodynamic properties for mixed spin models
have been investigated both numerically and analytically [37]. In the most of
studies, beside of the magnetization, some thermodynamic parameters of the
mixed spin models have been investigated in detail. For example, specific heat
which may exhibit Schottky-type peak at special temperature interval. Under
the thermodynamic conditions, this peak might tend to a double-peak
temperature dependence [31, 33, 36, 38].

The diamond chain with Ising interaction have been exactly solved within the
classical transfer-matrix technique in Ref. [39]. The magnetization plateaus
for the mixed spin-(1,1/2) Ising diamond chain have been argued in [40]. In
Ref. [41] O. Rojas et al. presented the analysis of the zero temperature
ground-state phase diagrams for the mixed spin-(1,1/2) Ising-Heisenberg
diamond chain and investigated the magnetization processes at low temperature,
they obtained magnetization plateaus at $\mathcal{M}=1/5$ and
$\mathcal{M}=3/5$ in the units of saturation magnetization.

The approach we are offering in the present work is based on the idea of using
the mixed spin-(1,1/2) Ising-XXZ double sawtooth ladder which can be found in
a mixed-spin SSLT (figure 1) as an approximately solvable spin model, and
developing the transfer matrix formalism. The main goal of the present paper
is to verify whether or not more intermediate magnetization plateaus can be
detected at low temperature in a magnetization process of the mixed
spin-(1,1/2) Ising-XXZ double sawtooth ladder. Also, comparing the
thermodynamic parameters of the model with each other is another significant
intention of the paper.

Figure 1: Spin configuration for antiferromagnetic mixed-spin Ising-XXZ
Heisenberg SSLT. Exchange couplings $\mathcal{J}_{\perp}$,
$\mathcal{J}_{\parallel}$, $\mathcal{J}_{Is}$ and cyclic four-spin exchange
coupling $K$ in square plaquette of the unit blocks (shaded areas) are shown.
Note that each spin-1/2 has Heisenberg XXZ interaction with its fermionic
next-next nearest neighbors. Shaded regions can be considered as a mixed-spin
double sawtooth ladder in the mixed-spin SSLT with periodic boundary
conditions when the number of blocks $\longrightarrow\infty$.

This paper is organized as follow: in the next section we define the mixed
spin double sawtooth ladder. Section 3 deals with the most interesting results
obtained for the ground-state phase diagrams at zero temperature. Further in
section 4, we present the thermodynamic solution of the model using the
enhanced transfer-matrix formalism. Indeed we show that, how one can
investigate this spin model by using an approximate procedure. In section 5,
we have numerically discussed the magnetization, the magnetic susceptibility
and the specific heat of the model at low temperature. Finally in section 6,
we summarize our results and draw the conclusions.

##  2 Model

Recently, we investigated the Ising-XXZ Heisenberg double sawtooth ladder
consist of half-spins in Ref. [20]. Phase transition and some thermodynamic
parameters such as the heat capacity, the magnetization and magnetic
susceptibility have been numerically investigated in detail. In this work, we
study the Ising-XXZ double sawtooth ladder with mixed nodal Ising spins
including both spin-1/2 and spin-1 on the legs and interstitial dimer
Heisenberg half-spins, in the presence of an external magnetic field. The
disordered spin ladder of 16-spins with periodic boundary conditions is
schematically illustrated in figure 2(a). The number of spins in the ladder
are selected even and our method is used for $N\geq 8$. To introduce a chain
with $N>8$, the number of spins will grow as $N+4$, i.e.,
$N\in\\{8,12,16,20,24\cdots\\}$ (figures 2(b) and 2(c) are related to $N=20$
and $N=24$, respectively).

Figure 2: Mixed-spin Ising-XXZ Heisenberg double sawtooth ladder with
geometric frustration for (a) $N=16$, (b) $N=20$ and (c) $N=24$.

The Hamiltonian of the antiferromagnetic mixed spin double sawtooth ladder
under the periodic boundary conditions is given by the following formula

$\begin{array}[]{lcl}H_{SL}=\\\
\sum\limits_{i=1}^{M}\mathcal{J}_{H}{\bm{\sigma}}_{i,2}\cdot{\bm{\sigma}}_{i,5}+\sum\limits_{i=1}^{M}\mathcal{J}_{\parallel}\big{(}{J}_{i,1}^{z}{\sigma}_{i,3}^{z}+{J}_{i,4}^{z}{\sigma}_{i,6}^{z}\big{)}+\\\
\sum\limits_{i=1}^{M}\mathcal{J}_{Is}\big{(}{J}_{i,1}^{z}{\sigma}_{i,2}^{z}+{\sigma}_{i,2}^{z}{\sigma}_{i,3}^{z}+{J}_{i,4}^{z}{\sigma}_{i,5}^{z}+{\sigma}_{i,5}{\sigma}_{i,6}^{z}\big{)}+\\\
\sum\limits_{i=1}^{M}\frac{\mathcal{J}_{\perp}}{2}\big{(}{J}_{i,1}^{z}{\sigma}_{i,6}^{z}+{\sigma}_{i,3}{J}_{i,4}^{z}\big{)}+K\sum\limits_{\langle
1346\rangle i}^{even}P_{1346}^{i\circlearrowleft}-\\\
\sum\limits_{i=1}^{4M}\frac{{B}^{\prime
z}}{2}\big{(}{J}_{i,1}^{z}+{\sigma}_{i,3}^{z}+{J}_{i,4}^{z}+{\sigma}_{i,6}^{z}\big{)}-\sum\limits_{i=1}^{4M}{B}^{\prime\prime
z}\big{(}{\sigma}_{i,2}^{z}+{\sigma}_{i,5}^{z}\big{)},\end{array}$ (1)

where $M$ is the number of blocks (shaded regions in figure 2), and
$\mathcal{J}_{H}$ is the exchange coupling between half-spins of the
interstitial Heisenberg dimer on each block and

$P^{\circlearrowleft}\left(\begin{array}[]{cc}{S}_{i,1}^{z}&{s}_{i,3}^{z}\\\
{s}_{i,6}^{z}&{S}_{i,4}^{z}\\\
\end{array}\right)=\left(\begin{array}[]{cc}{s}_{i,3}^{z}&{S}_{i,4}^{z}\\\
{S}_{i,1}^{z}&{s}_{i,6}^{z}\\\ \end{array}\right).$ (2)

Here, $2{\bf s}={\bm{\sigma}}=\\{{\sigma}^{x},{\sigma}^{y},{\sigma}^{z}\\}$
are the spin-1/2 operators (with $\hbar=1$) and

$\begin{array}[]{lcl}\sqrt{2}{S^{x}}={J^{x}}=\left(\begin{array}[]{ccc}0&1&0\\\
1&0&1\\\ 0&1&0\\\
\end{array}\right),\sqrt{2}{S^{y}}={J^{y}}=\left(\begin{array}[]{ccc}0&-i&0\\\
i&0&-i\\\ 0&i&0\\\ \end{array}\right),\\\
{S^{z}}={J^{z}}=\left(\begin{array}[]{ccc}1&0&0\\\ 0&0&0\\\ 0&0&-1\\\
\end{array}\right),\end{array}$ (3)

where ${S}^{x,y,z}$ are spin-1 operators, ${B}^{\prime z}$ and
${B}^{\prime\prime z}$ are applied homogeneous magnetic fields considered in
the $z$-direction. $\mathcal{J}_{\perp}$ and $\mathcal{J}_{\parallel}$ are the
bilinear exchange couplings on the rungs and along the legs of the block,s
plaquette, respectively. $K$ is the coupling of the cyclic four-spin
permutation operator per plaquette, and $\mathcal{J}_{Is}$ is the Ising
coupling between the spins on the legs of the block,s plaquette and two spins
of the interstitial Heisenberg dimer. Note that here, all of introduced
parameters are considered dimensionless.

The Heisenberg part of the Hamiltonian $H_{SL}$ is introduced as

$\begin{array}[]{lcl}{\bm{\sigma}}_{i,2}\cdot{\bm{\sigma}}_{i,5}=\mathcal{J}_{x}\big{(}{\sigma}^{x}_{i,2}{\sigma}^{x}_{i,5}+{\sigma}^{y}_{i,2}{\sigma}^{y}_{i,5}\big{)}+\Delta{\sigma}^{z}_{i,2}{\sigma}^{z}_{i,5}.\\\
\end{array}$ (4)

The Ising-type cyclic four-spin permutation operator can be written as the
product of three transposition operators
$P_{1346}^{\circlearrowleft}=P_{13}^{\circlearrowleft}P_{14}^{\circlearrowleft}P_{16}^{\circlearrowleft}$
where $P_{13}^{\circlearrowleft}=1/2(1+{J}^{z}_{1}\sigma^{z}_{3})$, then we
can obtain the following result which contains both bilinear and biquadratic
terms of the spin-1/2 and spin-1 operators as

$\begin{array}[]{lcl}P_{1346}^{i\circlearrowleft}=\frac{1}{8}\big{[}1+{J}^{z}_{1}\sigma^{z}_{3}+{J}^{z}_{1}{J}^{z}_{4}+{J}^{z}_{1}\sigma^{z}_{6}+\\\
({J}^{z}_{1}\sigma^{z}_{3})\cdot({J}^{z}_{1}{J}^{z}_{4})+({J}^{z}_{1}\sigma^{z}_{3})\cdot({J}^{z}_{1}\sigma^{z}_{6})+({J}^{z}_{1}{J}^{z}_{4})\cdot({J}^{z}_{1}\sigma^{z}_{6})+\\\
({J}^{z}_{1}\sigma^{z}_{3})\cdot({J}^{z}_{1}{J}^{z}_{4})\cdot({J}^{z}_{1}\sigma^{z}_{6})\big{]}.\end{array}$
(5)

The $i$-th block Hamiltonian ${h}_{i}$ can be written as

$\begin{array}[]{lcl}{h}_{i}=\big{[}\mathcal{J}_{x}\big{(}{\sigma}^{x}_{i,2}{\sigma}^{x}_{i,5}+{\sigma}^{y}_{i,2}{\sigma}^{y}_{i,5}\big{)}+\Delta{\sigma}^{z}_{i,2}{\sigma}^{z}_{i,5}\big{]}\\\
+\mathcal{J}_{\parallel}\big{(}{J}^{z}_{i,1}{\sigma}^{z}_{i,3}+{J}^{z}_{i,4}{\sigma}^{z}_{i,6}\big{)}+\frac{\mathcal{J}_{\perp}}{2}\big{(}{J}^{z}_{i,1}{\sigma}^{z}_{i,6}+{\sigma}^{z}_{i,3}{J}^{z}_{i,4}\big{)}\\\
+\mathcal{J}_{Is}\big{[}{J}^{z}_{i,1}{\sigma}^{z}_{i,2}+{\sigma}^{z}_{i,2}{\sigma}^{z}_{i,3}+{J}^{z}_{i,4}{\sigma}^{z}_{i,5}+{\sigma}_{i,5}{\sigma}^{z}_{i,6}\big{]}+KP_{1346}^{i\circlearrowleft}\\\
-\frac{{B}^{\prime}_{z}}{2}\big{(}{J}^{z}_{i,1}+{\sigma}^{z}_{i,3}+{J}^{z}_{i,4}+{\sigma}^{z}_{i,6}\big{)}-{B}^{\prime\prime}_{z}\big{(}{\sigma}^{z}_{i,2}+{\sigma}^{z}_{i,5}\big{)}.\end{array}$
(6)

In this case, $\mathcal{J}>0$ is denoted antiferromagnetic exchange
interactions and $\mathcal{J}<0$ ferromagnetic ones. The magnetic field
$B^{\prime\prime}$ acts on the Heisenberg dimer spins and $B^{\prime}$ acts on
the spins of the blocks plaquette with Ising-type interaction. Here, the case
$B^{\prime}_{z}=2B^{\prime\prime}_{z}=2B$ is considered.

##  3 Zero temperature phase diagrams

We here explain the ground-state phase diagrams of the introduced ladder. In
figure 3(a), we display the zero temperature phase diagram in the
($B/\mathcal{J}_{Is}-\Delta/\mathcal{J}_{Is}$) plane by assuming fixed values
of the another parameters applied in the Hamiltonian. Classical ferromagnetic
(CFM), quantum ferrimagnetic phase (QFI), frustrated state (S) and quantum
frustrated state (QS) are obtained. Spin arrangements of the relevant phases
can be presented by using the following tensor product of the eigenvectors

$\begin{array}[]{lcl}|CFM\rangle=\displaystyle\prod_{i=1}^{M}|1\uparrow
1\uparrow\rangle_{i}\otimes|\varphi_{1}\rangle_{i},\\\
|QFI\rangle=\displaystyle\prod_{i=1}^{M}|1\downarrow
1\downarrow\rangle_{i}\otimes|\varphi_{2}\rangle_{i},\\\
|S\rangle=\displaystyle\prod_{i=1}^{M}|0\downarrow
0\downarrow\rangle_{i}\otimes|\varphi_{1}\rangle_{i},\\\
|QS\rangle=\displaystyle\prod_{i=1}^{M}|-1\uparrow-1\uparrow\rangle_{i}\otimes|\varphi_{3}\rangle_{i},\\\
\end{array}$ (7)

where

$\begin{array}[]{lcl}|\varphi_{1}\rangle_{i}=|\uparrow\uparrow\rangle_{i},|\varphi_{2}\rangle_{i}=\frac{1}{2}(|\uparrow\downarrow\rangle+|\downarrow\uparrow\rangle)_{i},\\\
|\varphi_{3}\rangle_{i}=\frac{1}{2}(|\uparrow\downarrow\rangle-|\downarrow\uparrow\rangle)_{i},|\varphi_{4}\rangle_{i}=|\downarrow\downarrow\rangle_{i},\\\
\end{array}$ (8)

are the eigenstates of the interstitial XXZ Heisenberg dimer such that
$\sigma^{z}|\uparrow\rangle=|\uparrow\rangle$ and
$\sigma^{z}|\downarrow\rangle=-|\downarrow\rangle$. States $|1\rangle$,
$|0\rangle$ and $|-1\rangle$ are set up in the ${J}^{z}$ basis states.

Figure 3: The ground-state phase diagrams of the mixed spin-(1,1/2) Ising-XXZ
Heisenberg double sawtooth ladder (a) in the
($B/\mathcal{J}_{Is}-\Delta/\mathcal{J}_{Is}$) plane at fixed values of
$\mathcal{J}_{x}=\mathcal{J}_{\parallel}=K=\mathcal{J}_{Is}$ and
$\mathcal{J}_{\perp}=4\mathcal{J}_{Is}$, and (b) in the
($\mathcal{J}_{\perp}/\mathcal{J}_{Is}-\mathcal{J}_{x}/\mathcal{J}_{Is}$)
plane at fixed values of $\Delta=-\mathcal{J}_{Is}$,
$\mathcal{J}_{\parallel}=K=\mathcal{J}_{Is}$ and $B=4\mathcal{J}_{Is}$.

The ground-state energies for fixed values of
$\mathcal{J}_{x}=\mathcal{J}_{\parallel}=K=\mathcal{J}_{Is}$ and
$\mathcal{J}_{\perp}=4\mathcal{J}_{Is}$ are given by

$\begin{array}[]{lcl}\mathcal{E}_{CFM}=\frac{\Delta}{\mathcal{J}_{Is}}+11-\frac{6B}{\mathcal{J}_{Is}},\mathcal{E}_{QFI}=-{\frac{\Delta}{\mathcal{J}_{Is}}-8},\\\
\mathcal{E}_{S}=\frac{\Delta}{\mathcal{J}_{Is}}-\frac{15}{8},\mathcal{E}_{QS}=-{\frac{\Delta}{\mathcal{J}_{Is}}-4}.\end{array}$
(9)

The blue line in figure 3(a) separates regions of CFM and QFI phases with the
curve
$\frac{B}{\mathcal{J}_{Is}}=\frac{1}{3}\frac{\Delta}{\mathcal{J}_{Is}}+\frac{19}{6}$.
The phase boundary between the QS and QFI states is limited by the line
$\Delta\approx-1.1\mathcal{J}_{Is}$, and the boundary between the S and QS
states is limited by the line $\Delta\approx-3.1\mathcal{J}_{Is}$. Line
$B\approx 2.15\mathcal{J}_{Is}$ depicts the phase boundary between the S and
CFM states.

Figure 3(b) illustrates the ground-state phase diagram in the
($\mathcal{J}_{\perp}/\mathcal{J}_{Is}-\mathcal{J}_{x}/\mathcal{J}_{Is}$)
plane where from figure 3(a) we consider fixed values
$\Delta=-\mathcal{J}_{Is}$ and $\mathcal{J}_{\parallel}=K=\mathcal{J}_{Is}$
near the boundary between QS and QFI phases and a formal fixed value of
magnetic field $B=4\mathcal{J}_{Is}$ due to continue our calculations to
achieve second peak in the specific heat diagram. We prove that for other
fixed values of the magnetic field the second peak occurs under spacial
conditions at which the boundary of CFM phase may change. Shaded region shows
CFM phase which plays an important role to describe the behavior of
thermodynamic parameters of the favorite model surrounding critical points at
which quantum phase transitions occur.

##  4 Solution within the transfer-matrix technique

The mixed spin-(1,1/2) Ising-XXZ double sawtooth ladder can be solved through
the transfer matrix technique. The interstitial Heisenberg dimer coupling can
be expressed as

$\big{(}{\bm{\sigma}}_{i,2}\cdot{\bm{\sigma}}_{i,5}\big{)}_{\Delta,\mathcal{J}_{x}}=\left(\begin{array}[]{cccc}\frac{\Delta}{4}&0&0&0\\\
0&-\frac{\Delta}{4}&\frac{\mathcal{J}_{x}}{2}&0\\\
0&\frac{\mathcal{J}_{x}}{2}&-\frac{\Delta}{4}&0\\\
0&0&0&\frac{\Delta}{4}\end{array}\right).$ (10)

Due to the commutation relation between different block Hamiltonians,
$[{h}_{i},{h}_{j}]=0$, the partition function of the ladder can be written in
the form

$\mathcal{Z}=Tr\Big{[}\displaystyle\prod_{i=1}^{M}\exp(-\beta h_{i})\Big{]},$
(11)

where $\beta=\frac{1}{k_{B}T}$, $k_{B}$ is the Boltzmann’s constant and $T$ is
the absolute temperature. We can consider the following matrix representation
for $\exp(-\beta h_{i})$ in the qubit-qutrit standard basis of the eigenstates
of the composite spin operators
$\\{J_{i,1}^{z},\sigma_{i,6}^{z},\sigma_{i,3}^{z},J_{i,4}^{z}\\}$ on the two
consecutive rungs of the plaquette in block $i$. The partition function
$\mathcal{Z}$ can be introduced as

$\begin{array}[]{lcl}\mathcal{Z}=Tr\big{[}\langle
J_{1,1}^{z}\sigma_{1,6}^{z}|\mathcal{T}|\sigma_{1,3}^{z}J_{1,4}^{z}\rangle\langle\sigma_{2,1}^{z}J_{2,6}^{z}|\mathcal{T}|J_{2,3}^{z}\sigma_{2,4}^{z}\rangle\\\
\cdots\langle\sigma_{M,1}^{z}J_{M,6}^{z}|\mathcal{T}|J_{M,3}^{z}\sigma_{M,4}^{z}\rangle\big{]},\end{array}$
(12)

where $\sigma_{i,j}^{z}=\pm 1$ and $J_{i,j}^{z}=\pm 1,0$ and

$\begin{array}[]{lcl}\mathcal{T}(i)=\langle
J_{i,1}^{z}\sigma_{i,6}^{z}|\exp(-\beta
h_{i})|\sigma_{i,3}^{z}J_{i,4}^{z}\rangle=\\\
\sum\limits_{k=1}^{4}\exp\big{[}-\beta\mathcal{E}_{k}(J_{i,1}^{z}\sigma_{i,6}^{z},\sigma_{i,3}^{z}J_{i,4}^{z})\big{]}.\end{array}$
(13)

Four eigenvalues of the $i-$th block with Hamiltonian $h_{i}$ are

$\begin{array}[]{lcl}\mathcal{E}_{1}(i)={\Delta}+\mathcal{J}_{Is}\big{(}J_{i,1}^{z}+\sigma_{i,3}^{z}+J_{i,4}^{z}+\sigma_{i,6}^{z}\big{)}+\Xi-2B,\\\
\mathcal{E}_{2}(i)=\\\
-{\Delta}+\Xi+\sqrt{\mathcal{J}_{Is}^{2}\big{(}J_{i,1}^{z}+\sigma_{i,3}^{z}-J_{i,4}^{z}-\sigma_{i,6}^{z}\big{)}^{2}+4\mathcal{J}_{x}^{2}},\\\
\mathcal{E}_{3}(i)=\\\
-{\Delta}+\Xi-\sqrt{\mathcal{J}_{Is}^{2}\big{(}J_{i,1}^{z}+\sigma_{i,3}^{z}-J_{i,4}^{z}-\sigma_{i,6}^{z}\big{)}^{2}+4\mathcal{J}_{x}^{2}},\\\
\mathcal{E}_{4}(i)={\Delta}-\mathcal{J}_{Is}\big{(}J_{i,1}^{z}+\sigma_{i,3}^{z}+J_{i,4}^{z}+\sigma_{i,6}^{z}\big{)}+\Xi+2B,\end{array}$
(14)

where

$\begin{array}[]{lcl}\Xi=\mathcal{J}_{\parallel}\big{(}J_{i,1}^{z}\sigma_{i,3}^{z}+J_{i,4}^{z}\sigma_{i,6}^{z}\big{)}+\frac{\mathcal{J}_{\perp}}{2}\big{(}J_{i,1}^{z}\sigma_{i,6}^{z}+\sigma_{i,3}^{z}J_{i,4}^{z}\big{)}+\\\
\frac{K}{8}\big{[}J_{i,1}^{z3}\sigma_{i,3}^{z}J_{i,4}^{z}\sigma_{i,6}^{z}+J_{i,1}^{z2}\sigma_{i,3}^{z}J_{i,4}^{z}+J_{i,1}^{z2}J_{i,4}^{z}\sigma_{i,6}^{z}+\\\
J_{i,1}^{z2}\sigma_{i,3}^{z}\sigma_{i,6}^{z}+J_{i,1}^{z}J_{i,4}^{z}+J_{i,1}^{z}\sigma_{i,3}^{z}+J_{i,1}^{z}\sigma_{i,6}^{z}+1\\\
-B\big{(}J_{i,1}^{z}+\sigma_{i,3}^{z}+J_{i,4}^{z}+\sigma_{i,6}^{z}\big{)}-2B\big{]}.\end{array}$
(15)

We can figure out the transfer matrix $\mathcal{T}(i)$ as

$\mathcal{T}(i)=\left(\begin{array}[]{cccccc}\mathcal{T}_{11}&\mathcal{T}_{12}&\mathcal{T}_{13}&\mathcal{T}_{14}&\mathcal{T}_{15}&\mathcal{T}_{16}\\\
\mathcal{T}_{21}&\mathcal{T}_{22}&\mathcal{T}_{23}&\mathcal{T}_{24}&\mathcal{T}_{25}&\mathcal{T}_{26}\\\
\mathcal{T}_{31}&\mathcal{T}_{32}&\mathcal{T}_{33}&\mathcal{T}_{34}&\mathcal{T}_{35}&\mathcal{T}_{36}\\\
\mathcal{T}_{41}&\mathcal{T}_{42}&\mathcal{T}_{43}&\mathcal{T}_{44}&\mathcal{T}_{45}&\mathcal{T}_{46}\\\
\mathcal{T}_{51}&\mathcal{T}_{52}&\mathcal{T}_{53}&\mathcal{T}_{54}&\mathcal{T}_{55}&\mathcal{T}_{56}\\\
\mathcal{T}_{61}&\mathcal{T}_{62}&\mathcal{T}_{63}&\mathcal{T}_{64}&\mathcal{T}_{65}&\mathcal{T}_{66}\\\
\end{array}\right).$ (16)

After straightforward calculations the elements of the transfer matrix are
defined through eigenvalues (14).

Since the mixed spin double sawtooth ladder is translational invariant and all
of $h_{i}$ are independent of the site $i$, equation (12) can be expressed as

$\mathcal{Z}=Tr\big{[}\mathcal{T}^{M}\big{]}.$ (17)

The total partition function can also be expressed in terms of six eigenvalues
of the transfer matrix $\mathcal{T}$

$\mathcal{Z}=\Lambda_{1}^{M}+\Lambda_{2}^{M}+\Lambda_{3}^{M}+\Lambda_{4}^{M}+\Lambda_{5}^{M}+\Lambda_{6}^{M}.$
(18)

In the thermodynamic limit, it is sufficient to consider only the largest
eigenvalue $\Lambda_{max}$ to calculate the partition function. In Ref. [15]
authors investigated an Ising-Heisenberg three-leg tube and obtained a
$8\times 8$ transfer matrix including short components for which four
eigenvalues are zero, while other four eigenvalues can be easily gained. But
here, the situation (obtaining the largest eigenvalue of the transfer matrix)
is much more complex than previous cases.

Now, we are going to explain our approximate solution to extract the largest
eigenvalue of the transfer matrix in detail. Since, we solely look for the
largest eigenvalue of the transfer matrix, it is noteworthy that despite of
the transfer matrix is $6\times 6$, but its elements are so long that
obtaining the eigenvalues of such a matrix is a big challenge for the
accessible processors. We numerically investigated our favorite double
sawtooth ladder and found that for the considered ranges of the Hamiltonian
parameters, several components of the transfer matrix such as
$\mathcal{T}_{66}$ are almost effectless and we can withdraw all of them. On
the other hand, some components like $\mathcal{T}_{11}$, $\mathcal{T}_{13}$,
$\mathcal{T}_{22}$ and other components out of yellow regions shown in figure
4 are the most effective components to determine the largest eigenvalue, which
should be taken into account. Consequently, we reconstruct a new transfer
matrix with less non-zero components. For another fixed values of the applied
parameters in Hamiltonian, the elements of the transfer matrix are changed but
according to our numerical investigations (randomly calculating over than
100-times) at low temperature, we realized that the most effective components
of the transfer matrix deal with its largest eigenvalue are the same.

Figure 4: Numerical matrix plots of the transfer matrix for fixed values of
$\mathcal{J}_{\perp}/\mathcal{J}_{Is}=rand(1..8)$,
$\mathcal{J}_{\parallel}/\mathcal{J}_{Is}=1$,$\mathcal{J}_{x}/\mathcal{J}_{Is}=rand(-2..2)$,
$\Delta/\mathcal{J}_{Is}=rand(-1..6)$, $\beta\mathcal{J}_{Is}=rand(0..3)$ and
$B/\mathcal{J}_{Is}=rand(0..8)$ (a) for 50-times randomly calculating and (b)
over than 100-times. All minimum components (yellow regions) and the most
effective components are unique for each-time randomly calculating.

Accordingly, the new transfer matrix can be characterized as

$\mathcal{T}^{\prime}(i)=\left(\begin{array}[]{cccccc}\mathcal{T}_{11}&\mathcal{T}_{12}&\mathcal{T}_{13}&0&\mathcal{T}_{15}&0\\\
\mathcal{T}_{21}&\mathcal{T}_{22}&0&0&0&0\\\
\mathcal{T}_{31}&0&\mathcal{T}_{33}&0&\mathcal{T}_{35}&0\\\ 0&0&0&0&0&0\\\
\mathcal{T}_{51}&0&\mathcal{T}_{53}&0&\mathcal{T}_{55}&0\\\ 0&0&0&0&0&0\\\
\end{array}\right).$ (19)

Thus, we can consider the largest eigenvalue of the new transfer matrix
$\mathcal{T}^{\prime}(i)$ equivalent to the largest eigenvalue of the primary
transfer matrix $\mathcal{T}(i)$ with high accuracy.

Free energy Gibbs per block for infinitely long chain, when lonely the maximal
eigenvalue survives is given by [42]

$\begin{array}[]{lcl}f=f_{0}+f^{\prime}=2\mathcal{J}_{x}+\Delta-\frac{1}{\beta}\lim\limits_{M\rightarrow\infty}\ln\frac{1}{M}\mathcal{Z}=\\\
2\mathcal{J}_{x}+\Delta-\frac{1}{\beta}\ln\Lambda_{max},\end{array}$ (20)

##  5 Thermodaynamic parameters

Let us examine thermodynamic parameters of the model as functions of the
magnetic field, the bilinear exchange coupling on the rungs
$\mathcal{J}_{\perp}$ and the cyclic four-spin exchange interaction $K$ at low
temperature. Note that the quantum spin ladders are the simplest spin systems
for which the cyclic four-spin exchange appears from the interaction between
nodal spins of the square plaquettes. This specific spin interaction has been
theoretically and experimentally analyzed in the various spin models and some
of its valuable applications (such as explaining the neutron-scattering) have
been demonstrated [5, 43, 44, 45]. The magnetization, the magnetic
susceptibility and the specific heat can be obtained using the maximal
eigenvalue of the transfer matrix $\mathcal{T}^{\prime}(i)$ through the
following formulae

$\begin{array}[]{lcl}\mathcal{M}=-\frac{\partial f}{\partial
B},\chi=\frac{\partial\mathcal{M}}{\partial
B},\mathcal{C}=-k_{B}\beta\frac{\partial}{\partial\beta}\big{(}\beta^{2}\frac{\partial
f}{\partial\beta}\big{)}.\end{array}$ (21)

It was shown that firstly, magnetization plateaus appear for a great number of
spin models, and can be theoretically analyzed by the transfer matrix
techniques as well as the exact diagonalization. Secondly, the spin gap
existence in the spectrum of magnetic excitations creates plateaus at
$\mathcal{M}/\mathcal{M}_{s}\neq 0$ in the external magnetic field. On top of
that, a number of spin-1/2 chains exhibit magnetization plateau at
$\mathcal{M}/\mathcal{M}_{s}=1/3$ and $\mathcal{M}/\mathcal{M}_{s}=2/3$ which
represents a massive phase [46, 47], while mixed spin-(1,1/2) chain [36]
exhibits a plateau at $\mathcal{M}/\mathcal{M}_{s}=1/2$, where $\mathcal{M}$
is the magnetization and $\mathcal{M}_{s}$ is called saturation magnetization.
The zero magnetization plateau $\mathcal{M}/\mathcal{M}_{s}=0$ for the
Heisenberg spin ladders were obtained [20, 47]. Here, we investigate the
magnetization plateaus for the considered mixed spin double sawtooth ladder.
Among the variety of magnetization curves obtained for different sets of
$B/\mathcal{J}_{Is}$ and $\mathcal{J}_{\perp}/\mathcal{J}_{Is}$, the most
remarkable plateau is $\mathcal{M}/\mathcal{M}_{s}=1/3$ in the units of block.

The difference in magnetization behavior with respect to the magnetic field is
shown in figures 5(a) and 5(b). Figure 5(a) demonstrates the magnetization
versus the magnetic field at zero temperature, while figure 5(b) plots this
function versus the magnetic field at low temperature. It is quite obvious
that the magnetization curve has a plateau at
$\mathcal{M}/\mathcal{M}_{s}=1/3$ for all fixed values of
$\mathcal{J}_{\perp}$ and $\Delta<0$ whether the system is considered at zero
temperature or low temperature, on the other hand, this plateau gets wider as
the ratio $\mathcal{J}_{\perp}/\mathcal{J}_{Is}$ strengthens. Interestingly at
zero temperature, the magnetization exhibits two narrow plateaus at
$\mathcal{M}/\mathcal{M}_{s}=2/3$ and $\mathcal{M}/\mathcal{M}_{s}=5/6$ which
are not clear for the case where the system is considered at low temperature.
A stronger evidence of two quantum phase transitions of the mixed spin-(1,1/2)
Ising-XXZ double sawtooth ladder is provided by the magnetic susceptibility.
By inspecting figure 5(c), one can see that the magnetic susceptibility
decreases upon increasing of the magnetic field from zero, until it vanishes
at a critical magnetic field. Indeed, there is an identical rather steep
decrease for the magnetic susceptibility function for all fixed values of
$\mathcal{J}_{\perp}$ at weak magnetic field. When the magnetic field further
strengthens, the magnetic susceptibility displays a double-peak for which
second peak is much more smooth than first. The steep is taken in the interval
$0<B/\mathcal{J}_{Is}\lesssim B_{1}/\mathcal{J}_{Is}$ and the double-peak
appears in the interval
$\frac{(B_{1}/\mathcal{J}_{Is}+\mathcal{J}_{\perp}/\mathcal{J}_{Is})}{2}+1<B/\mathcal{J}_{Is}\lesssim\frac{(B_{1}/\mathcal{J}_{Is}+\mathcal{J}_{\perp}/\mathcal{J}_{Is})}{2}+3$
(in this case $B_{1}\approx\mathcal{J}_{Is}$).

Figure 5(c) also shows the dynamics of the magnetic susceptibility with
respect to the exchange coupling $\mathcal{J}_{\perp}$. Namely, when
$\mathcal{J}_{\perp}$ increases the double-peak moves to the stronger magnetic
fields. According to the phase diagram illustrated in figure 3(b), the steep
shows S phase. The gap between the steep and the double-peak, where the
magnetic susceptibility disappears, denotes CFM phase region. The double-peak
coincides with QS phase and finally, the plateau next to the double-peak is in
accordance with QF phase. Consequently, the CFM boundary gets wider upon
increasing of $\mathcal{J}_{\perp}$.

Figure 5:  The magnetization with respect to its saturation value as a
function of $B/\mathcal{J}_{Is}$ for various fixed values of
$\mathcal{J}_{\perp}/\mathcal{J}_{Is}$, where other parameters are taken as
$\mathcal{J}_{\parallel}=K=\mathcal{J}_{x}=\mathcal{J}_{Is}$ and
$\Delta=-\mathcal{J}_{Is}$ (a) at zero temperature, and (b) at low temperature
$\beta=2\mathcal{J}_{Is}^{-1}$. (c) The magnetic susceptibility of the model
per unit block as function of $B/\mathcal{J}_{Is}$ for various fixed values of
$\mathcal{J}_{\perp}/\mathcal{J}_{Is}$, where other parameters are taken as
$\mathcal{J}_{\parallel}=K=\mathcal{J}_{x}=\mathcal{J}_{Is}$ and
$\Delta=-\mathcal{J}_{Is}$, at low temperature $\beta=2\mathcal{J}_{Is}^{-1}$.
The inset shows the critical points (solid circles) in the
($B/\mathcal{J}_{Is}$-$\mathcal{J}_{\perp}/\mathcal{J}_{Is}$) plane at which
second double-peak of the magnetic susceptibility arises and vanishes,
respectively, at low temperature.

Last but not least, let us examine the bilinear exchange couplings on the
rungs $\mathcal{J}_{\perp}$ and the magnetic field variations of the specific
heat against the temperature. For this purpose, figure 6 illustrates
temperature dependence of the specific heat for the model under consideration
with respect to the various fixed values of $\mathcal{J}_{\perp}$. As one can
see, the specific heat exhibits a steep increase when the temperature
gradually decreases. With further decreasing the temperature, the specific
heat decreases and gradually tends to zero (black circle-line). There is
solely one unique peak for the considered fixed magnetic field
$B=4\mathcal{J}_{Is}$ and $\mathcal{J}_{\perp}=\mathcal{J}_{Is}$. When
$\mathcal{J}_{\perp}/\mathcal{J}_{Is}$ is lifted from 1, the single peak
decreases and a second peak appears, which can be related to an abrupt
$\mathcal{J}_{\perp}$-induced steep increase of the magnetization from plateau
$\mathcal{M}/\mathcal{M}_{s}=1/3$ toward saturation value. The second peak
denotes CFM phase, where with further increase of $\mathcal{J}_{\perp}$ this
peak disappears and simultaneously first peak becomes larger until reaches its
maximum value ($\mathcal{C}\approx 1.8$). The highest value of the first peak
may denote QFI phase.

It is worthwhile to remark that by tuning the magnetic field the double-peak
occurs for all fixed values of the coupling constant $\mathcal{J}_{\perp}$. In
this regard, it could be quite interesting to answer the profound question; in
which intervals of the magnetic field the double-peak may occurs? According to
our investigations we found that the double-peak appears at the critical
magnetic field
$B_{i}\approx\frac{(B_{1}/\mathcal{J}_{Is}+\mathcal{J}_{\perp}/\mathcal{J}_{Is})}{2}+1$,
and will remain until
$B_{f}\approx\frac{(B_{1}/\mathcal{J}_{Is}+\mathcal{J}_{\perp}/\mathcal{J}_{Is})}{2}+3$
(inset of figure 6). It is quite obvious from figure 5(b) and its inset that
in these intervals of the magnetic field, the double-peak occurs for the
magnetic susceptibility function. These intervals determine range of
applicability of the mixed spin-(1,1/2) Ising-XXZ double sawtooth ladder for
the purposes associated with ultra-cold atoms. The peaks in the magnetic
susceptibility and the specific heat can be thus regarded as faithful
indicators of the quantum critical points of the mixed spin-(1,1/2) double
sawtooth ladder.

Finally, let us turn back to the most spectacular magnetic field and exchange
coupling $\mathcal{J}_{\perp}$ dependencies of the magnetization and magnetic
susceptibility, which play an important role to describe quantum critical
points of the investigated model. We examine the magnetization as function of
the magnetic field for various fixed values of the cyclic four-spin exchange
interaction $K$ at low temperature and arbitrary fixed
$\mathcal{J}_{\perp}=3\mathcal{J}_{Is}$. We conceptually understand that $K$
has a substantial role to determine the number of magnetization plateaus. To
support this statement, we have depicted in figure 7(a) typical cyclic four-
spin exchange interaction variations of the magnetization against the magnetic
field. One finds by inspection three separated plateaus
$\mathcal{M}/\mathcal{M}_{s}=1/3$, $\mathcal{M}/\mathcal{M}_{s}=2/3$ and
$\mathcal{M}/\mathcal{M}_{s}=5/6$ (a rare $K$-dependent plateau), which the
last two plateaus gradually appear upon increasing the cyclic four-spin
exchange interaction $K$.

Figure 6: Plot shows the specific heat of the mixed spin-(1,1/2) double
sawtooth ladder as a function of the inverse temperature
$\beta\mathcal{J}_{Is}$ for various fixed values of $\mathcal{J}_{\perp}$,
where other parameters are taken as
$\mathcal{J}_{\parallel}=K=\mathcal{J}_{x}=\mathcal{J}_{Is}$,
$\Delta=-\mathcal{J}_{Is}$ and $B=4\mathcal{J}_{Is}$. The inset shows the
temperature dependence of the specific heat for several fixed values of the
coupling constant $\mathcal{J}_{\perp}$ at different magnetic fields. Brown
dashed line depicts the specific heat changes at fixed
$\mathcal{J}_{\perp}=3\mathcal{J}_{Is}$ and $B=3\mathcal{J}_{Is}$, green
dashed-dot line at fixed $\mathcal{J}_{\perp}=5\mathcal{J}_{Is}$ and
$B=6\mathcal{J}_{Is}$, and blue solid line at fixed
$\mathcal{J}_{\perp}=8\mathcal{J}_{Is}$ and $B=5.5\mathcal{J}_{Is}$ (these
fixed values have been taken up from the intervals containing critical
magnetic fields).

Figure 7: (a) The magnetization and (b) the magnetic susceptibility as a
function of $B/\mathcal{J}_{Is}$ for various fixed values of
$K/\mathcal{J}_{Is}$ where other parameters are taken as
$\mathcal{J}_{\parallel}=\mathcal{J}_{x}=\mathcal{J}_{Is}$,
$\Delta=-\mathcal{J}_{Is}$ and $\mathcal{J}_{\perp}=3\mathcal{J}_{Is}$, at low
temperature $\beta=2\mathcal{J}_{Is}^{-1}$.

Figure 7(b) plots the magnetic susceptibility versus magnetic field for
several fixed values of $K$ at low temperature. We find that with increase of
$K$, peaks of the mentioned double-peak are gradually separated from each
other, which denotes second magnetization plateau
$\mathcal{M}/\mathcal{M}_{s}=2/3$ appearance. With further increase of $K$
another new peak grows and gets away from separated peaks, namely, magnetic
field gaps will arise between all peaks. At low temperature, these gaps and
all available magnetization plateaus coincide exactly.

##  6 Conclusions

In the present article, we have examined the ground state and thermodynamics
of the mixed spin-(1,1/2) Ising-XXZ double sawtooth ladder including
interstitial Heisenberg dimer half-spins connected to the leg sites on each
block by taking advantage of the transfer-matrix technique. First of all, we
have obtained exact results for the ground-state phase diagrams. Then, we have
numerically investigated the transfer matrix and realized that the most of
components are almost effectless to derive its largest eigenvalue, so they
were eliminated and a new transfer matrix with less components was
generalized. The new transfer matrix technique reported in this paper offers
an approximate procedure to solve the model. This procedure provides the
possibility of obtaining analytical expressions for the thermodynamic
quantities such as the magnetization, the magnetic susceptibility and the
specific heat with high accuracy.

The magnetization have been examined in detail with respect to the magnetic
field and the bilinear exchange couplings on the rungs at low temperature.
Under this circumstances, we just observed the plateau
$\mathcal{M}/\mathcal{M}_{s}=1/3$ for various fixed values of the bilinear
exchange couplings on the rungs. On the other hand, we observed a plateau at
$\mathcal{M}/\mathcal{M}_{s}=2/3$ upon increasing the cyclic four-spin
exchange interaction. The most interesting finding stemming from our study
certainly represents a striking plateau $\mathcal{M}/\mathcal{M}_{s}=5/6$ upon
further increase of the cyclic four-spin exchange interaction. This plateau
appears rarely for spin models. Furthermore, we have investigated the magnetic
susceptibility and compared it with the magnetization. We found that the
magnetization behavior is in accordance with the magnetic susceptibility
changes, where by increasing the cyclic four-spin exchange interaction, peaks
of the field-dependent double-peak are gradually separated from each other and
a magnetic field gap grows between them. As the cyclic four-spin exchange
interaction further increases, an extra peak appears and get away from other
peaks, namely, between any two consecutive peaks a distinguishable magnetic
field gap arises. Consequently, magnetization plateaus and magnetic
susceptibility gaps coincide at low temperature.

Finally, we have investigated the specific heat with respect to the inverse
temperature for several values of the bilinear exchange couplings on the
rungs, where other parameters applied in the Hamiltonian were taken as fixed
values. It is quite surprising that at low temperature we observed a Schottky-
type peak. When the bilinear exchange couplings on the rungs increases second
peak arises, so under this condition, the specific heat has a double-peak. The
temperature dependence of this function is almost similar to the specific heat
depicted in Ref. [31]. Namely, when the temperature increases, the specific
heat gradually increases and reaches its maximum at a special temperature
interval. Here, this thermodynamic parameter have been more investigated
numerically and remarkable outcomes have been obtained. For instance, we
obtained critical intervals of the magnetic field at which the specific heat
shows a double-peak for various fixed values of the bilinear exchange
couplings on the rungs. Indeed, we found that the range of these critical
intervals are dependent on the fixed value of the bilinear exchange couplings
on the rungs. Interestingly, in these intervals the field-dependent double-
peak of the magnetic susceptibility occurs for the corresponding fixed values
of the bilinear exchange couplings on the rungs.

Major applications of the considered mixed-spin double sawtooth ladder in the
real world are saving and transferring quantum information (both qbits and
qtrits) through the model which surely can be useful in quantum information
processing, optical lattices, spintronics and applied condensed matter
physics.

We note that the precise method elaborated in the present paper is valid at
low temperature and can be straightforwardly adapted to account for the
another spin models with more complicate transfer matrix and different spin
arrangement.

##  7 Acknowledgments

NA acknowledge by partly financial support of the MC-IRSES no. 612707
(DIONICOS) under FP7-PEOPLE-2013 and ICTP NT-04 grants.

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