# Chiral Majorana Edge Modes and Vortex Majorana Zero Modes in Superconducting
Antiferromagnetic Topological Insulator

Beibing Huang<EMAIL_ADDRESS>Department of Physics, Yancheng Institute
of Technology, Yancheng, 224051, China  Xiaosen Yang  Department of physics,
Jiangsu University, Zhenjiang, 212013, China  Qinfang Zhang  School of
Materials Science and Engineering, Yancheng Institute of Technology, Yancheng
224051, China  Ning Xu<EMAIL_ADDRESS>Department of Physics, Yancheng Institute
of Technology, Yancheng, 224051, China

###### Abstract

The antiferromagnetic topological insulator (AFTI) is topologically protected
by the combined time-reversal and translational symmetry $\mathcal{T}_{c}$. In
this paper we investigate the effects of the $s$-wave superconducting pairings
on the multilayers of AFTI, which breaks $\mathcal{T}_{c}$ symmetry and can
realize quantum anomalous Hall insulator with unit Chern number. For the
weakly coupled pairings, the system corresponds to the topological
superconductor (TSC) with the Chern number $C=\pm 2$. We answer the following
questions whether the local Chern numbers and chiral Majorana edge modes of
such a TSC distribute around the surface layers. By the numerical calculations
based on a theoretic model of AFTI, we find that when the local Chern numbers
are always dominated by the surface layers, the wavefunctions of chiral
Majorana edge modes must not localize on the surface layers and show a smooth
crossover from spatially occupying all layers to only distributing near the
surface layers, similar to the hinge states in a three dimensional second-
order topological phases. The latter phase can be distinguished from the
former phase by the measurements of the local density of state. In addition we
also study the superconducting vortex phase transition in this system and find
that the exchange field in the AFTI not only enlarges the phase space of
topological vortex phase but also enhances its topological stability. These
conclusions will stimulate the investigations on superconducting effects of
AFTI and drive the studies on chiral Majorana edge modes and vortex Majorana
zero modes into a new era.

##  I Introduction

The highly effectiveness of quantum computation is rooted in its parallel
properties ensured by the superposition principles of quantum mechanism.
However the fragileness of quantum systems easily causes random errors in the
computational process, which can be overcome by encoding all information into
nonlocal qubits (topological quantum computation, TQC) kitaevtqc ; nayak .
Majorana fermions majorana , realized as the Majorana zero modes in the
superconducting vortex readgreen ; fukane ; hosur ; jia1 ; jia2 ; xugang ;
ding1 ; ding2 ; feng1 ; feng2  or the chiral Majorana edge modes qi ; wangjin
; jau ; alicea  in the topological superconductors (TSC), can be used to
construct topological qubits. However, the computational processes of TQC
require braiding the Majorana fermions nayak ; ivan , which is a very
difficult task until now for the Majorana zero modes in the superconducting
vortex cores. By contrast, the chiral Majorana edge modes can interchange
themselves in the transport process and are more easily manipulated lianbiao .

In 2010, Qi etc suggested that chiral Majorana edge modes can be realized as
the edge states in the heterostructures of s-wave superconduting proximity to
a quantum anomalous Hall insulator (QAHI) qi . But the experiments in this
direction e1 ; e2 ; e3  arouse much debates about the complicated magnetic
disorder effects in the QAHI obtained by magnetically doping the topological
insulator disorder1 ; disorder2 . The appearance of the antiferromagnetic
topological insulator (AFTI) finds a way out of this dilemma to a large extent
moore . In the AFTI, taking the characteristic Van der Waals layer material
$\text{MnBi}_{2}\text{Te}_{4}$ for example, the magnetic exchanges show
intralayer (interlayer) ferromagnetic (antiferromagnetic) structures with the
moments perpendicular to the layer plane, and the topological properties are
protected by the combined time-reversal and lattice translational symmetry
mnbite0 ; mnbite ; mnbite1 ; mnbite2 ; mnbite3 . For an odd (even) number of
multilayers, the average moment is nonzero (zero) and the system is an QAHI
(axion insulator) with unit (zero) Chern number. The related experiments have
observed quantized Hall plateau without the external magnetic field at the
high temperature qah1 ; qah2 ; qah3 . These characters such as stoichiometric
purity and Van der Waals structure hint that the slab of AFTI can be used as
one of potential candidates to realize the chiral Majorana edge modes.

The multilayers of AFTI have two surfaces and show the gapped surface states.
These surface states originate from gapping out the gapless ones of
topological insulator and correspond to a half QAHI qi2 ; fu2 , described by a
massive Dirac fermion. Moreover the calculation of local Chern number shows
that the contribution to Chern number comes from surface layers, and as one
goes deeper, the local Chern number oscillates around zero lcn1 ; lcn2 ; lcn3
. According to Qi’s suggestion qi , once the weakly coupled $s$-wave pairings
are introduced, the multilayers of AFTI with odd number layers become the TSC
with Chern number $C=2$. In this paper, we ask the following two questions.
The first one is whether the local Chern numbers of such TSC with $C=2$
distribute around the surface layers. If the answer is positive, the next
question is whether the chiral Majorana edge modes also distribute around the
surface layers. By the numerical calculations based on a theoretic model of
AFTI, we find that even if the surface Chern numbers dominate, the
wavefunctions of chiral Majorana edge modes must not localize near the surface
layers and show a smooth crossover from spatially occupying all layers to only
occupying surface layers when some parameters are adjusted. In addition we
also study the superconducting vortex phase transition in this system and find
that the exchange field in the AFTI not only enlarges the phase space of
topological vortex phase but also enhances its topological stability. These
conclusions will stimulate the investigations on the superconducting effects
of AFTI.

In this paper, we are interested in the TSC with Chern number $C=2$ realizable
in superconducting multilayers of AFTI, which are out of spotlight in contrast
to the TSC with $C=1$. The plan of this paper is as follows. In the section 2,
starting from the model of topological insulator, we give the tight-binding
model of superconducting multilayer of AFTI by incorporating the
antiferromagnetic exchange field in $z$ direction and $s$-wave on-site
intraorbital pairings. In the section 3 we map out the phase diagrams of the
system and analyze the distribution of Chern number to exemplify its local
properties. We also find that the chemical potential plays an important role
in determining available topological phases in our suggested model. The
section 4 is devoted to study the wavefunctions of chiral Majorana edge modes.
By tracking the evolutions of wavefunctions, we find that the edge modes can
continuously evolve into the state only occupying surface layers. We
illustrate this crossover from the structure of bulk energy bands. In the
section 5, the effects of the antiferromagnetic exchange field on the
superconducting vortex phase transition are studied. Finally a brief
conclusion is made in the section 6.

##  II Model

We start with a four-band model for topological insulators on a cubic lattice
with two orbitals and spins per cubic site ti

$\displaystyle H_{TI}(k_{x},k_{y},k_{z})$ $\displaystyle=$
$\displaystyle[m+t(\cos{k_{x}}+\cos{k_{y}}+\cos{k_{z}})]\tau_{z}$
$\displaystyle+$ $\displaystyle
v(\sin{k_{x}}\sigma_{x}+\sin{k_{y}}\sigma_{y}+\sin{k_{z}}\sigma_{z})\tau_{x},$

where $\sigma_{x,y,z}$ and $\tau_{x,y,z}$ are two sets of Pauli matrices
acting on spin and orbital spaces respectively. This Hamiltonian is in the
strong topological insulator phase protected by the time-reversal symmetry
with $\mathcal{T}=-i\sigma_{y}\mathcal{K}$ if $|t|<|m|<3|t|$ and $v\neq 0$,
where $\mathcal{K}$ is the complex conjugation operator.

The Hamiltonian of AFTI can be achieved by adding the antiferromagnetic
exchange term along $z$ direction into the Hamiltonian (LABEL:1) moore ;
xuyong . This exchange term breaks the translation symmetry in $z$ direction
and magnifies the unit cell of the topological insulator to include four
orbitals. Choosing the magnetic moment colinear with $z$ direction, AFTI is
described by the Hamiltonian

$\displaystyle H_{AFTI}(k_{x},k_{y},k_{z})$ $\displaystyle=$
$\displaystyle[m+t(\cos{k_{x}}+\cos{k_{y}})]\tau_{z}+\Gamma\varrho_{z}\sigma_{z}$
(2) $\displaystyle+$
$\displaystyle\varrho_{x}[t\cos{(k_{z}/2)}\tau_{z}+v\sin{(k_{z}/2)}\sigma_{z}\tau_{x}]$
$\displaystyle+$ $\displaystyle
v(\sin{k_{x}}\sigma_{x}+\sin{k_{y}}\sigma_{y})\tau_{x},$

where the Pauli matrices $\varrho_{x,y,z}$ act on the bilayer space induced by
antiferromagnetic structure and $\Gamma$ is the antiferromagnetic exchange
field. The model (2) has the combined time-reversal and translational symmetry
$\mathcal{T}_{c}=e^{-ik_{z}/2}\rho_{x}\mathcal{T}$ and its topological
properties are decided by $Z_{2}$ number at $k_{z}=0$ plane. We find that as
long as the gap from topological insulator is not closed when $\Gamma$ is
increased, the model (2) is an AFTI. In Fig.1(a) and (b), we show the surface
states of the model (2), consistent with the our expectations.

Figure 1: The excitation spectra in the surface Brillouin zones $k_{x}k_{y}$
with $k_{y}=0$ (a) and $k_{x}k_{z}$ with $k_{z}=0$ (b) for the AFTI described
by the model (2). In (a) [(b)] the surfaces are ferromagnetic
(antiferromagnetic) and breaks (conserves) the combined time-reversal and
translational symmetry $\mathcal{T}_{c}$, thus the surface states correspond
to a gapped (gapless) Dirac fermion. The parameters are $m/t=-2.5$, $v/t=0.5$,
$\Gamma/t=0.6$.

In this paper, we consider a multilayer from the AFTI, which is finite length
$L_{z}$ along $z$ direction. At the same time, we also assume that the
multilayer is superconducting with $s$-wave on-site intraorbital pairings.
Without loss of generality, we assume all pairings are equal, denoted by
$\Delta$. Further introducing Pauli matrices $s_{x,y,z}$ acting on the
particle-hole space, our Hamiltonian is

$\displaystyle H_{SML}(k_{x},k_{y})$ $\displaystyle=$
$\displaystyle[m+t(\cos{k_{x}}+\cos{k_{y}})]s_{z}\tau_{z}-\mu s_{z}$ (3)
$\displaystyle+$ $\displaystyle
v\sin{k_{x}}\sigma_{x}\tau_{x}+v\sin{k_{y}}s_{z}\sigma_{y}\tau_{x}+\Delta
s_{y}\sigma_{y}$ $\displaystyle+$
$\displaystyle\frac{t}{2}s_{z}\widetilde{\varrho}_{x}\tau_{z}+\frac{v}{2}\widetilde{\varrho}_{y}\sigma_{z}\tau_{x}+\Gamma
s_{z}\widetilde{\varrho}_{z}\sigma_{z},$

where we have added the chemical potential $\mu$. The new $L_{z}\times L_{z}$
matrices $\widetilde{\varrho}_{x,y,z}$ are used to describe the interlayer
coupling, which are defined as follows

$\displaystyle\widetilde{\varrho}_{x,y}=\left(\begin{array}[]{cccccc}0&a&0&...\\\
a^{*}&0&a&...\\\ 0&a^{*}&0&...\\\
...&...&...&...\end{array}\right),\widetilde{\varrho}_{z}=\left(\begin{array}[]{cccccc}1&0&0&...\\\
0&-1&0&...\\\ 0&0&1&...\\\ ...&...&...&...\end{array}\right)$ (12)

with $a=1$ for $\widetilde{\varrho}_{x}$ and $a=-i$ for
$\widetilde{\varrho}_{y}$.

The model (3) is our starting point of this paper. We choose the model
parameters to ensure that the gap closing of the system happens at $k=0$. We
set $m/t=-2.5$, $v/t=0.5$.

Figure 2: The phase diagrams for the multilayer model (3) as the function of
the antiferromagnetic exchange field $\Gamma$ and superconducting gap $\Delta$
with the chemical potential $\mu/t=0$ (a) and $\mu/t=0.1$ (b). The lines with
squares separate the normal superconducting phase from the TSC with Chern
number $C=2$; while the lines with circles differentiate two phases in the
TSC. These two phases are defined depending on whether the chiral Majorana
edge modes only occupy a few layers near the surfaces of the multilayer. The
triangle-like regions correspond to such localized phase. It should be noted
that in (b) there exists a small phase space not shown for TSC with $C=1$
between the phases with $C=2$ and $C=0$.

##  III Phase Diagram

We construct the phase diagram by tracking the gap closings and the Chern
number of the system fukui

$\displaystyle C=\frac{1}{2\pi}\int dk_{x}dk_{y}\text{Tr}\hat{\Omega}_{k,xy}$
(13)

where $\hat{\Omega}_{k,xy}$ are the Berry curvature matrix with

$\displaystyle\hat{\Omega}_{k,xy}^{nn^{\prime}}=\nabla_{k}\times<\psi_{nk}|i\nabla_{k}|\psi_{n^{\prime}k}>$
(14)

and $|\psi_{nk}>$ is the eigenstates of the occupied bands with eigenvalues
$E_{nk}$ in ascending order. We study the Chern number as the function of the
antiferromagnetic exchange field $\Gamma$ and superconducting gap $\Delta$.
The phase diagrams for two different chemical potentials are shown in Fig.2.

For $\mu=0$, the Hamiltonian (3) has an emergent symmetry $\Xi
H_{SML}(k_{x},k_{y})\Xi^{-1}=H_{SML}(k_{x},k_{y})$ with
$\Xi=s_{y}\sigma_{y}\tau_{y}$. This symmetry transforms the particles in an
orbital into holes in the other orbital on the same layer but keeps the
momentum and spin invariant. Therefore, if $\psi_{nk}$ is an eigenstate of
$H_{SML}(k_{x},k_{y})$, $\Xi\psi_{nk}$ is also the eigenstate with the same
energy. We have numerically checked that these two eigenstates are not
linearly correlated. Thus for $\mu=0$, all energy bands are double degenerate.
This result directly influences the available phases in the model (3).
Generally the topological phase transition accompanies the gap closings and
the dispersion around the phase transition point are gapless Dirac type.
Before and after the phase transition, the Dirac mass is inverted and for
every this transition, the Chern number of the system changes $\pm 1$ bissard
. Now for $\mu=0$, the double degeneracy means that the Chern number changes
in unit of $\pm 2$. Since a weakly superconducting coupled QAHI with $C=1$
realizes a TSC with $C=2$, thus our system can only show the even Chern number
states at $\mu=0$.

When $\mu\neq 0$, the symmetry $\Xi$ and double degeneracy are broken, thus
the Chern number changes in unit of $\pm 1$ and we can observe the TSC with
$C=1$. But in terms of parameters we have chosen, this phase space is very
small, so that we do not show this phase region in the phase diagram Fig.2(b).
In Fig.2(b), when $\Gamma$ is small, the system is a normal superconductor. In
this region, the normal state is a metal, not a QAHI. With the increase of the
exchange field $\Gamma$, the gap of the normal state increases. When the gap
of the normal state is beyond the chemical potential, the system enters into
the topological phase. A finite chemical potential reduces the regions of TSC.
This conclusion is consistent with the conventional wisdom.

The Chern number used to differentiate the different phases is defined for the
whole multilayer. To study the distribution of Chern number as the function of
layer index of multilayer, we need find how different layers contribute to
$C$. This local Chern number $C_{l}$ for the $l$th layer can be calculated
from the formula lcn1 ; lcn2 ; lcn3

$\displaystyle C_{l}=\frac{1}{2\pi}\int
dk_{x}dk_{y}\text{Tr}[\hat{\Omega}_{k,xy}\hat{P}_{k,l}]$ (15)

by inserting a projection $\hat{P}_{k,l}$ onto layer $l$

$\displaystyle\hat{P}_{k,l}^{nn^{\prime}}=\sum_{j\in
l}\psi_{nk,j}^{*}\psi_{n^{\prime}k,j},$ (16)

where the summation $j\in l$ is done if the $j$th component of the
eigenfunction $\psi_{nk}$ represents the degree of freedom in layer $l$. Since
$\sum_{l}\hat{P}_{k,l}^{nn^{\prime}}=\delta_{n,n^{\prime}}$, we have
$\sum_{l}C_{l}=C$. The formalism of $C_{l}$ was developed for calculating the
full surface anomalous Hall conductivity in a multilayer of magnetoelectric
insulator.

Figure 3: (a) The local Chern number $C_{l}$. With the increase of the
superconducting gap, the oscillation of the local Chern number shows decayed
amplitude. (b) The evolution of first excitation level $E_{1k}$ at $k=0$ when
the hopping in $z$ direction decreases to zero. The spin-orbit coupling in $z$
direction plays no roles for the gap of the system at the time-reversal
invariant momenta. In (a) the parameters for the red solid (blue dashed) line
with squares (circles) are $\Gamma/t=1.0$ and $\Delta/t=0.02$
($\Delta/t=0.65$); In (b) $\Gamma/t=1.0$, $\Delta/t=0.6$.

Below we directly apply this quantity to our case. Fig.3 shows $C_{l}$ for
some different parameters using a $10\times 10$ sampling in the Brillouin
zone. In all cases, we find that $C_{l}$ on the two surface layers approach
half of the Chern number of the superconducting multilayer, and as one goes
deeper into the interior, $C_{l}$ oscillates around zero with the average of
neighbor layers approaching zero. Therefore the contribution to the whole
Chern number comes from a few layers near the surfaces. These results for
$C_{l}$ can be qualitatively understood as follows. We find that the gap of
the topological phase in Fig.2 do not close when the hopping in $z$ is
adjusted to zero (Fig.3(b)); In addition the spin-orbit coupling in $z$
direction does not influence the gap of the system at the time-reversal
invariant momenta. Therefore from the perspective of topology, it is very
appropriate to consider the model (3) adiabatically connected to the model
without interlayer couplings. In this limit, every independent layer can be
regarded as the TSC with $C=\pm 1$ depending on the sign of the
antiferromagnetic exchange field $\Gamma$. As a result the Chern numbers of
the interior layers average to zero, while surface layers at either end of the
multilayer are not completely canceled, realizing a TSC with $C=\pm 1$ at both
surfaces.

##  IV Majorana Wavefunctions

In this paper, we are interested in the TSC with $C=2$. In Fig.3, we have seen
that the local Chern number $C_{l}$ mainly distribute on the surface layers
with the averages of the interior layers approaching zero. Now we want to
consult that when the Chern number concentrates on the surface layers, whether
the chiral Majorana edge modes also concentrates on the surface layers. Below
we numerically calculate the wavefunctions of chiral Majorana edge modes to
illustrate this question, by assuming an open boundary condition in $y$
direction. Fig.4 shows some characteristic cases for Majorana edge modes.

Figure 4: The evolutions of wavefunctions for the chiral Majorana edge modes
as the superconducting gap $\Delta$ is adjusted. The parameters are
$\Gamma/t=1.0$ and $\Delta/t=0.02$, $0.65$, $0.80$ in (a)-(c).

The normal state of our model is a QAHI and always gapped if we decrease the
exchange fields $\Gamma$ on the interior layers to zero but keep them
invariant on the two surface layers, thus its chiral edge modes are trapped on
the whole side surfaces qi2 . On the other hand the TSC with $C=2$ in the
Fig.2 can be continuously connected to the normal state, we naturally expect
that when the superconducting gap $\Delta$ is small, the chiral Majorana edge
modes will occupy all layers of the multilayer, which is consistent with the
results in Fig.4(a). While for a big superconducting gap $\Delta$, the
wavefunctions are absent from the interior layers and localized near surface
layers (Fig.4(b)). This distribution of wavefunctions is similar to the hinge
states in a three dimensional second-order topological phases hinge1 ; hinge2
; hinge3 ; hinge4 . Between these two limits, there is a smooth crossover of
wavefunction evolution. From this perspective, we can differentiate two kinds
of phases of TSC with $C=2$. In order to discriminate these two phases, we
defined the wavefunctions at intermediate layer of chiral Majorana edge modes
equaling to zero as the phase transition point between these two phases. Such
behaviors as the function of superconducting gap $\Delta$ are shown in
Fig.4(d). We have found some phase transition points shown in Fig.2 as the
circles for different parameters. For small $\Gamma$, the search for the
critical point is costly, so we only make some calculations for large
$\Gamma$. Here we note that such two phases are topologically equivalent with
each other, in spite of the differences on the wavefunctions.

Figure 5: (a) The excitation spectrum of the multilayer for $k_{y}=0$. Three
excitation gaps $E_{g1}=E_{10}$, $E_{g2}=E_{20}-E_{10}$,
$E_{g3}=E_{30}-E_{20}$ at $k=0$ are schematically illustrated; (b)-(c) The
wavefunctions of three lowest energy levels $E_{ik}$ at $k=0$ as the function
of layer index; (d) The evolutions of three excitation gaps as the function of
superconducting pairings. The parameters are $\Delta/t=0.2$ in (a),
$\Delta/t=0.1$ in (b), $\Delta/t=0.4$ in (c), $\Gamma/t=0.6$.

The TSC with $C=2$ will transit into another phase when the superconducting
gap is added further. For $\mu=0$, the system transits into the trivial phase.
In this process of phase transition, the chiral Majorana edge modes have the
same localization length along the $y$ direction. Increasing the
superconducting gap, the localization length gradually increases (Fig.4(c))
and even becomes divergent at the critical point, so that edge modes merge
into the bulk states in $y$ direction, leaving a trivial gapped superconductor
on the other side of the transition. Similarly for $\mu\neq 0$, there is phase
transition to TSC with $C=1$. The nonzero chemical potential leads to
independent evolution of edge modes. At the critical point, one edge mode
becomes bulk state while the other is still invariant. But it is worthwhile to
note that, the chiral Majorana edge modes always localized near the surface
layers in these phase transitions. This property is always invariant.

From the distributions of wavefunctions of chiral Majorana edge modes, we find
that although the Chern number is mainly decided by the surface layers,
generally the wavefunctions of edge modes do not only occupy the surface
layers. In order to illustrate localization to the surface layers, we observe
the evolution of bulk excitation spectrum $E_{nk}$ at $k=0$, since as stated
before we have chosen the model parameters to ensure that the gap closing of
the system happens at this point. The excitation spectrum of the multilayer
consists of two parts. The first (second) part comes from gapless surface
states (gapped bulk states) of topological insulator. When imposed on the
staggered magnetic moments and small superconducting pairings, these states
still occupied surface layers (bulk) (Fig.5(b)), contributing a large local
Chern number to the whole Chern number. Further increasing the superconducting
pairings, although the gaps of the system $E_{g1}=E_{10}$ and
$E_{g3}=E_{30}-E_{20}$ decrease, the gap $E_{g2}=E_{20}-E_{10}$ gradually
increases (Fig.5(d)), not only leading to the mixture of surface energy level
$E_{2k}$ into the bulk ones (Fig.5(c)) but also that the low energy properties
of the system is only decided to a larger extent by the first excitation level
$E_{1k}$ belonging to surface layers. Thus once a boundary is imposed, we
expect the chiral Majorana edges modes will also localize near the surface
layers. This illustrates our observations on the wavefunctions.

Figure 6: The local zero-energy density of state $\rho(i_{y}=0,i_{z},\omega)$
at $\omega=0$. In order for calculation, we have added a small imaginary part
$\delta=10^{-3}$ to $\omega$. The local zero-energy density of state
$\rho(i_{y}=L_{y}-1,i_{z},0)$ shows the same structure. The parameters for the
red solid (blue dashed) line with squares (circles) are $\Gamma/t=1.0$ and
$\Delta/t=0.02$ ($\Delta/t=0.65$).

Two kinds of TSC with $C=2$ have the common bulk properties, however the
occupation of the chiral Majorana edge modes near the surface layers should
leave the fingerprints on the local low-energy density of state, which can be
calculated from the retarded Green function $\hat{G}_{ret}(k_{x},\omega)$ for
the system with edges
$\rho(i_{y},i_{z},\omega)=-\sum_{ik_{x}}\text{Im}[\hat{G}_{ret}^{ii}(k_{x},\omega)]$,
where the summation for $i$ concerns the all degree of freedom on the lattice
site $(i_{y},i_{z})$ and the label $\text{Im}[\cdot\cdot\cdot]$ means to take
the imaginary part. Fig.6 presents the featured local zero-energy density of
state $\rho(i_{y}=0,i_{z},0)$ on the side, which has the similar behaviors
with the wavefuntions of the chiral Majorana edge modes. Physically
$\rho(i_{y}=0,i_{z},0)$ can be measured as the zero-bias conductance peak
using the scanning tunneling spectroscopy and can be used as the smoking gun
to discriminate these two phases experimentally.

##  V Vortex Phase Transition

For the topological insulator, a superconducting vortex line will bind two
Majorana zero modes, which are localized near the intersections of two
surfaces and vortex line when the chemical potential $\mu$ is around the
charge-neutral point; while tuning $\mu$ into the bulk states, a quantum
vortex phase transition occurs and two Majorana zero modes disappear hosur .
The similar vortex phase transitions have been investigated in superconducting
Dirac and Weyl semimetal yan . In our model, the antiferromagnetic exchange
field $\Gamma$ gap out the surface Dirac point and provides a new knob to
manipulate the vortex. Below we will explore the effects of $\Gamma$ on vortex
phase transition.

Figure 7: (a) The phase diagram for one-dimensional vortex line in the
superconducting AFTI. The region with Zak phase $\theta_{Z}=1$
($\theta_{Z}=0$) is nontrivially (trivially) topological. For $\Gamma=0$, two
critical points of the vortex phase transition are located at $\mu_{c}/t=\pm
0.41$. (b) The excitation gap $E_{g1}$ as the function of $\Gamma$ for the
different chemical potential. From top to bottom $\mu/t=0.0$, $0.1$, $0.2$,
$0.3$, $0.4$ respectively. (c) is the same as (b) except $\mu/t=0.45$. In this
case the excitation shows gap closing and reopening, signifying a vortex phase
transition. The change of Zak phase $\theta_{Z}$ is also presented. (d) The
scaling behavior of some low-energy eigenvalues for a cubic sample with open
boundary condition in all three directions for $\mu/t=0.1$ and $\Gamma/t=0.7$.
We only show the results for odd number multilayers. Note that for $\mu=0$,
the emergent symmetry $\Theta$ leads to the double of Majorana zero modes. In
all figures $\Delta_{0}/t=0.4$, $\xi=0.5$.

In order to model a superconducting vortex line in $z$ direction, we assume
the superconducting gap $\Delta$ in the model (3) acquires the spatial
dependence
$\Delta_{i_{x},i_{y}}=\Delta_{0}\tanh{(\sqrt{i_{x}^{2}+i_{y}^{2}}/\xi)}e^{i\theta}$,
where the phase $\theta=\arctan{(i_{y}/i_{x})}$ reflects the vortex structure
and $\xi$ is the superconducting coherence length. We also assume the extreme
type II limit to neglect the effect of the magnetic field used to generate the
vortices. The vortex line conserves the translational symmetry in the $z$
direction, so that $k_{z}$ is still a good quantum number and our system can
be viewed as a one-dimensional superconductor. However, the complex
superconducting gap breaks the combined time-reversal and translational
symmetry $\mathcal{T}_{c}$ and our system is characterized by a well-defined
Zak phase $\theta_{z}$ if the energy spectrum is fully gapped xugang ; jpsj ;
budich . The topological phase diagram from this calculation is shown in
Fig.7(a). For $\Gamma=0$, the system has a nontrivial topological region
around the charge-neutral point $\mu=0$, consistent with others results hosur
. With the increase of $\Gamma$, the topological region is gradually enlarged.
On the other hand, we also find the excitation gap nontrivially increases as
the function of $\Gamma$ for the fixed chemical potential in the topological
region (Fig.7(b)), further leading to the robustness of the ground state.
These two points signify that the extra controllable knob of the
antiferromagnetic exchange field $\Gamma$ is very advantageous to enhance the
topological stability. For the chemical potential beyond the critical points
at $\Gamma=0$ the system shows gap closing, where the sudden jump of Zak phase
$\theta_{Z}$ happens (Fig.7(c)). To further support the results above, we also
diagonalize the Hamiltonian for a cubic geometry with open boundary condition
in all three directions. In Fig. 7(d), we present some eigenenergies near zero
energy for $\mu/t=0.1$ and $\Gamma/t=0.7$, their scalings with the system size
clearly demonstrate the existence of vortex Majorana zero modes.

For the chemical potential $\mu=0$, as stated before, there exists an emergent
symmetry $\Xi$. Thus the Hamiltonian can be partitioned into two subspaces
related to this symmetry. Considering that the commutation between this
symmetry and the particle-hole symmetry, every subspace has also the particle-
hole symmetry, thus in every subspace we can also define Zak phase. These two
Zak phases is nontrivial for $\mu=0$. In other words, the emergent symmetry
$\Theta$ leads to the double of Majorana zero modes at $\mu=0$. This can be
used to illustrate near degeneracy between Majorana zero modes and low energy
excitation in Fig.7(d).

##  VI Conclusions

In this paper all calculations have been done for the odd number of
multilayers. However the conclusions about the chiral Majorana wavefunctions
and vortex phase transition are also applicable to the even number of
multilayers. In conclusion we have investigated the effects of the $s$-wave
superconducting pairings on the multilayers of AFTI, which breaks
$\mathcal{T}_{c}$ symmetry and can realize QAHI with unit Chern number. For
the weakly coupled pairings, the system corresponds to the TSC with the Chern
number $C=\pm 2$. By the numerical calculations based on a theoretic model of
AFTI, we answered the following questions whether the local Chern numbers and
chiral Majorana edge modes of such a TSC distribute around the surface layers.
On one hand we find that the local Chern number $C_{l}$ on the two surface
layers approach half of the Chern number of the superconducting multilayer,
and as one goes deeper into the bulk, $C_{l}$ oscillates around zero with the
average of neighbor layers approaching zero. Therefore the contribution to the
whole Chern number comes from a few layers near the surfaces; On the other
hand the wavefunctions of the chiral Majorana edge modes must not localize on
the surface layers and show a smooth crossover from spatially occupying all
layers to only distributing near the surface layers. These two kinds of
topological phases can be distinguished by the measurements of the local
density of state. We also discuss the vortex phase transition in this system
and find that the antiferromagnetic exchange field not only enlarges the phase
space of topological phase but also enhance its the topological stability. Our
conclusion will stimulate the investigation on superconducting effects of AFTI
and drive the studies on chiral Majorana edge modes into a new era.

## Acknowledgement

We thank Ming Gong for stimulating discussions. B. H. and X. Y. are supported
respectively by National Natural Science Foundation of China under Grant No.
11547047 and No. 11504143.

## References

  * (1) A. Y. Kitaev, Ann. Phys. 303, 2 (2003).
  * (2) C. Nayak, S. H. Simon, A. Stern, M. Freedman and S. Das Sarma, Rev. Mod. Phys. 80, 1083 (2008).
  * (3) E. Majorana, Nuovo Cimento 5, 171 (1937).
  * (4) N. Read and D. Green, Phys. Rev. B 61, 10267 (2000).
  * (5) L. Fu and C. L. Kane, Phys. Rev. Lett. 100, 096407 (2008).
  * (6) P. Hosur, P. Ghaemi, R. S. K. Mong and A. Vishwanath, Phys. Rev. Lett. 107, 097001 (2011).
  * (7) J.-P. Xu, C. Liu, M.-X. Wang, J. Ge, Z.-L. Liu, X. Yang, Y. Chen, Y. Liu, Z.-A. Xu, C.-L. Gao, D. Qian, F.-C. Zhang and J.-F. Jia, Phys. Rev. Lett. 112, 217001 (2014).
  * (8) H.-H. Sun, K.-W. Zhang, L.-H. Hu, C. Li, G.-Y. Wang, H.-Y. Ma, Z.-A. Xu, C.-L. Gao, D.-D. Guan, Y.-Y. Li, C. Liu, D. Qian, Y. Zhou, L. Fu, S.-C. Li, F.-C. Zhang and J.-F. Jia, Phys. Rev. Lett. 116, 257003 (2016).
  * (9) G. Xu, B. Lian, P. Tang, X.-L. Qi and S.-C. Zhang, Phys. Rev. Lett. 117, 047001 (2016).
  * (10) P. Zhang, K. Yaji, T. Hashimoto, Y. Ota, T. Kondo, K. Okazaki, Z. Wang, J. Wen, G. D. Gu, H. Ding and S. Shin, Science 360, 182 (2018).
  * (11) D. Wang, L. Kong, P. Fan, H. Chen, S. Zhu, W. Liu, L. Cao, Y. Sun, S. Du, J. Schneeloch, R. Zhong, G. Gu, L. Fu, H. Ding and H.-J. Gao, Science 362, 333 (2018).
  * (12) Q. Liu, C.Chen, T.Zhang, R. Peng, Y.-J. Yan, C.-H.-P.Wen, X.Lou, Y.-L. Huang, J.-P. Tian, X.-L. Dong, G.-W. Wang, W.-C. Bao, Q.-H. Wang, Z.-P. Yin, Z.-X. Zhao and D.-L. Feng, Phys. Rev. X 8, 041056 (2018).
  * (13) C. Chen, Q. Liu, T.-Z. Zhang, D. Li, P.-P. Shen, X.-L. Dong, Z.-X. Zhao, T. Zhang and D.-L. Feng, Chin. Phys. Lett. 36, 057403 (2019).
  * (14) X.-L. Qi, T. L. Hughes, S.-C. Zhang, Phys. Rev. B 82, 184516 (2010).
  * (15) J. Wang, Q. Zhou, B. Lian and S.-C. Zhang, Phys. Rev. B 92, 064520 (2015).
  * (16) J. D. Sau, R. M. Lutchyn, S. Tewari and S. Das Sarma, Phys. Rev. Lett. 104, 040502 (2010).
  * (17) J. Alicea, Phys. Rev. B 81, 125318 (2010).
  * (18) D. A. Ivanov, Phys. Rev. Lett. 86, 268 (2001).
  * (19) B. Lian, X.-Q. Sun, A. Vaezi, X.-L. Qi and S.-C. Zhang, Proc. Natl. Acad. Sci. 115, 10938 (2018).
  * (20) Q. L. He, L. Pan, A. L. Stern, E. C. Burks, X. Che, G. Yin, J. Wang, B. Lian, Q. Zhou, E. S. Choi, K. Murata, X. Kou, Z. Chen, T. Nie, Q. Shao, Y. Fan, S.-C. Zhang, K. Liu, J. Xia and K. L. Wang, Science 357, 294 (2017).
  * (21) J. Shen, J. Lyu, J. Z. Gao, Y.-M. Xie, C.-Z. Chen, C. Cho, O. Atanov, Z. Chen, K. Liu, Y. J. Hu, K. Y. Yip, S. K. Goh, Q. L. He, L. Pan, K. L. Wang, K. T. Law and R. Lortz, Proc. Natl. Acad. Sci. 117, 238 (2020).
  * (22) M. Kayyalha, D. Xiao, R. Zhang, J. Shin, J. Jiang, F. Wang, Y.-F. Zhao, R. Xiao, L. Zhang, K. M. Fijalkowski, P. Mandal, M. Winnerlein, C. Gould, Q. Li, L. Molenkamp, M. H. W. Chan, N. Samarth and C.-Z. Chang, Science 367, 64 (2020).
  * (23) W. Ji and X.-G. Wen, Phys. Rev. Lett. 120, 107002 (2018).
  * (24) Y. Huang, F. Setiawan and J. D. Sau, Phys. Rev. B 97, 100501 (2018).
  * (25) R. S. K. Mong, A. M. Essin and J. E. Moore, Phys. Rev. B 81, 245209 (2010).
  * (26) M. M. Otrokov et al., Nature (London) 576, 416 (2019).
  * (27) M. M. Otrokov, I. P. Rusinov, M. Blanco-Rey, M. Hoffmann, A. Y. Vyazovskaya, S. V. Eremeev, A. Ernst, P. M. Echenique, A. Arnau and E. V. Chulkov, Phys. Rev. Lett. 122, 107202 (2019).
  * (28) Y. Gong, J. Guo, J. Li, K. Zhu, M. Liao, X. Liu, Q. Zhang, L. Gu, L. Tang, X. Feng, D. Zhang, W. Li, C. Song, L. Wang, P. Yu, X. Chen, Y. Wang, H. Yao, W. Duan, Y. Xu, S.-C. Zhang, X. Ma, Q.-K. Xue and K. He, Chin. Phys. Lett. 36, 076801 (2019).
  * (29) D. Zhang, M. Shi, T. Zhu, D. Xing, H. Zhang and J. Wang, Phys. Rev. Lett. 122, 206401 (2019).
  * (30) J. Li, Y. Li, S. Du, Z. Wang, B.-L. Gu, S.-C. Zhang, K. He, W. Duan and Y. Xu, Sci. Adv. 5, eaaw5685 (2019).
  * (31) Y. Deng, Y. Yu, M. Z. Shi, Z. Guo, Z. Xu, J. Wang, X. H. Chen and Y.Zhang, Science 367, 895 (2020).
  * (32) C. Liu, Y. Wang, H. Li, Y. Wu, Y. Li, J. Li, K. He, Y. Xu, J. Zhang and Y. Wang, Nat. Mater. 19, 522 (2020).
  * (33) J. Ge, Y. Liu, J. Li, H. Li, T. Luo, Y. Wu, Y. Xu and J. Wang, Natl Sci. Rev. 7, 1280 (2020).
  * (34) X.-L. Qi, T. L. Hughes and S.-C. Zhang, Phys. Rev. B 78, 195424 (2008).
  * (35) L. Fu and C. L. Kane, Phys. Rev. B 76, 045302 (2007).
  * (36) A. M. Essin, J. E. Moore and D. Vanderbilt, Phys. Rev. Lett. 102, 146805 (2009).
  * (37) T. Rauch, T. Olsen, D. Vanderbilt and I. Souza, Phys. Rev. B 98, 115108 (2018).
  * (38) N. Varnava1 and D. Vanderbilt, Phys. Rev. B 98, 245117 (2018).
  * (39) P. Hosur, S. Ryu and A. Vishwanath, Phys. Rev. B 81, 045120 (2010).
  * (40) P. Yang and Y. Xu, Phys. Rev. B 99, 195431 (2019).
  * (41) T. Fukui, Y. Hatsugai and H. Suzuki, J. Phys. Soc. Japan 74, 1674 (2005).
  * (42) J. Bellissard, arXiv:cond-mat/9504030.
  * (43) W. A. Benalcazar, B. A. Bernevig and T. L. Hughes, Science 357, 61 (2017).
  * (44) W. A. Benalcazar, B. A. Bernevig and T. L. Hughes, Phys. Rev. B 96, 245115 (2017).
  * (45) F. Schindler, A. M. Cook, M. G. Vergniory, Z. Wang, S. S. P. Parkin, B. A. Bernevig and T. Neupert, Sci. Adv. 4, eaat0346 (2018).
  * (46) Z. Song, Z. Fang and C. Fang, Phys. Rev. Lett. 119, 246402 (2017).
  * (47) Y. Hatsugai, J. Phys. Soc. Jpn. 75, 123601 (2006).
  * (48) J. C. Budich and E. Ardonne, Phys. Rev. B 88, 075419 (2013).
  * (49) Z. Yan, Z. Wu and W. Huang, Phys. Rev. Lett. 124, 257001 (2020).


# Impurity-induced increase in the thermal quantum correlations and
teleportation in an Ising-$XXZ$ diamond chain

Saulo L. L. Silva1, Moises Rojas2 1Centro Federal de Educação Tecnológica de
Minas Gerais , 37250-000, Nepomuceno-MG, Brazil  2Departamento de Física,
Universidade Federal de Lavras, 37200-900, Lavras-MG, Brazil

###### Abstract

In this work we analyze the quantum correlations in a spin-$1/2$ Ising-$XXZ$
diamond chain with one plaquette distorted impurity. We have shown that the
introduction of impurity into the chain can significantly increase
entanglement as well as quantum correlations compared to the original model,
without impurity. Due to the great flexibility in the choice of impurity
parameters, the model presented is very general and this fact can be very
useful for future experimental measurements. In addition to entanglement and
quantum coherence, we studied quantum teleportation through a quantum channel
composed by a coupled of Heisenberg dimers with distorted impurity in an
Ising-$XXZ$ diamond chain, as well as fidelity in teleportation. Our analysis
shows that the appropriate choice of parameters can greatly increase all the
measures analyzed. For comparison purposes, we present all our results
together with the results of the measurements made for the original model,
without impurity, studied in previous works.

##  I Introduction

Quantum correlations not only has interesting properties of quantum mechanics
but also is very important due to the powerful applications in quantum
information process and quantum computing Ben1 ; Ben2 ; lamico ; cirac .

In the last decades, the spin chains with Heisenberg interaction have received
attention due to the fact that they are promising candidates for physical
implementation of quantum information processing loss ; apo-1 ; ame ; divi ;
bose ; kam ; zhou .

On the other hand, quantum coherence that originates from the superposition
principles states is one of the central concepts in quantum mechanical systems
levi ; stre ; winter . Recently various approaches have been put forward to
develop a resource theory of coherence on Heisenberg spin models kar ; wu .
Several measures of coherence have been proposed, and their properties have
been investigated in detail baum ; Hu ; xi ; tan .

Recently, the spin-1/2 Heisenberg diamond chain, as well as an exactly
solvable Ising-Heisenberg have been largely explored. The motivation to
research the Ising-Heisenberg diamond chain model is based in fact that over-
simplification, the generalization version of the spin-1/2 ising-Heisenberg
diamond chain qualitatively reproduces thermodynamics data reported on the
real material $\mathrm{Cu_{3}\left(CO_{3}\right)_{2}\left(OH\right)_{2}}$
known as azurite. Owing to this fact, a lot of attention has been paid to a
rigorous treatment of various versions of the Ising-Heisenberg diamond chain
kiku ; cano ; boh ; val . Furthermore, the thermal entanglement of the Ising-
Heisenberg diamond chain was studied in moi ; moi2 ; moi3 ; moi4 ; gao ; cheng
. Rojas et. al. used the standard teleportation protocol to study an arbitrary
entangled state teleportation through a couple of Heisenberg dimers in an
infinite Ising-$XXZ$ diamond chain tele .

The impurity plays an important role in solid state physics fal ; apo . The
impurities can be understood by a local bound disorder of the interchain
exchange coupling and this changes in the structure affects strongly the
quantum correlations. The impurity effects on quantum entanglement have been
considered in Heisenberg spin chain sale ; osen ; solano ; sun ; fub .
Recently, the quantum discord gon  and dissipative effects ming  based in
Heisenberg chain with impurities has also been studied. More recently, the
research in this field was extended to the undestanding of thermal quantum
entanglement mr  and quantum correlations in Ising-Heisenberg diamond chain
with impurities mro . Besides spin impurity is possible to include a variety
of spin chains with magnetic impurities fu ; gal ; brene .

In this sense, the main goal of this work is the investigate the thermal
entanglement and quantum coherence on exactly solvable spin-1/2 Ising-$XXZ$
diamond chain with one distorted impurity plaquette inserted in the structure.
We will examine in detail the thermal entanglement and quantum coherence,
which exhibits a clear performance improvement when manipulate the impurity
compared to the original model without impurity. On the other hand, our
results show the impurity are helpful not only for improving the quality of
teleportation, but also for enhancing the critical temperature for valid
teleportation.

This paper is organized as follows. In Section II we describe the physical
model and the method of its exact treatment. In Section III, is given a brief
review concerning the definition of concurrence $\mathcal{C}$ as well as the
quantum coherence quantifier $l_{1}$-norm, and next analytical expression is
found for the concurrence and $l_{1}$-norm, respectively. We analyzed the
effects of impurity parameters and several parameter of the model on thermal
entanglement and $l_{1}$-norm. In Section IV, we briefly review the protocol
of the quantum teleportation. Then the average fidelity behavior our distorted
diamond chain model with impurity and of the original model without are
investigated detailed. Finally, in Section V, we summarize our conclusions.

##  II The model and Approach

Our model consists of a infinite spin-1/2 Ising-$XXZ$ diamond chain with one
distorted impurity plaquette under an external magnetic field $B$, which is
schematically depicted in Fig.1. The total hamiltonian is given by

$\mathcal{H}=\sum_{i=1}^{N}\mathcal{H}_{i},$ (1)

where

$\mathcal{H}_{i}=\mathcal{H}_{i}^{host}+\mathcal{H}_{i}^{imp},$

whith

$\begin{array}[]{cl}\mathcal{H}_{i}^{host}=&J\left(\mathbf{S}_{a,i},\mathbf{S}_{b,i}\right)_{\Delta}+J_{0}\left(S_{a,i}^{z}+S_{b,i}^{z}\right)\left(\mu_{i}+\mu_{i+1}\right)\\\
&-B\left(S_{a,i}^{z}+S_{b,i}^{z}\right)-\frac{B}{2}\left(\mu_{i}+\mu_{i+1}\right),\\\
&\mathrm{for}\>i=1,2,\ldots,r-1,r+1,\ldots,N\end{array}$

where
$\left(\mathbf{S}_{a,i},\mathbf{S}_{b,i}\right)_{\Delta}=S_{a,i}^{x}S_{b,i}^{x}+S_{a,i}^{y}S_{b,i}^{y}+\Delta
S_{a,i}^{z}S_{b,i}^{z}$, corresponds to the interstitial anisotropic
Heisenberg spins coupling ($J$ and $\Delta$), while the nodal-interstitial
(dimer-monomer) spins $\mu_{i}=\pm 1/2$ are representing the Ising-type
exchanges ($J_{0}$).
And

Figure 1: Schematic representation of the spin-1/2 Ising-$XXZ$ diamond chain
with distorted impurity. The Ising spins are denoted by $\mu_{i}$ and the
Heisenberg spins are represented by $S_{a(b),i}$

$\begin{array}[]{cl}\mathcal{H}_{i}^{imp}=&\widetilde{J}\left(\mathbf{S}_{a,i},\mathbf{S}_{b,i}\right)_{\widetilde{\Delta}}+\left(J_{1}\mu_{i}+J_{2}\mu_{i+1}\right)S_{a,i}^{z}+\left(J_{1}\mu_{i+1}+J_{2}\mu_{i}\right)S_{b,i}^{z}\\\
&-B\left(S_{a,i}^{z}+S_{a,i}^{z}\right)-\frac{B}{2}\left(\mu_{i}+\mu_{i+1}\right),\\\
&\mathrm{for}\>i=r,\end{array}$

is the Hamiltonian of the impurity. Here $J_{1}=J_{0}(1+\eta)$,
$J_{2}=J_{0}(1.0+\gamma)$, $\widetilde{J}=J(1.0+\alpha)$,
$\widetilde{\Delta}=\Delta(1.0+\Omega)$ represents the impurity parameters.
Throughout the text, the symbol $\widetilde{}$ on the parameter indicates that
this parameter is related to impurity in the chain.

The eigenvalues of $\mathcal{H}_{i}^{host}$ is given by

$\begin{array}[]{cl}\varepsilon_{1,4}=&\frac{J\Delta}{4}\pm\left(J_{0}\mp\frac{B}{2}\right)\left(\mu_{i}+\mu_{i+1}\right)\mp\frac{B}{2},\\\
\varepsilon_{2,3}=&-\frac{J\Delta}{4}\pm\frac{J}{2}-\frac{B}{2}\left(\mu_{i}+\mu_{i+1}\right),\end{array}$

and the corresponding eigenvectors are

$\displaystyle|\varphi_{i1}\rangle$ $\displaystyle=$
$\displaystyle|00\rangle_{i},$ (2) $\displaystyle|\varphi_{i2,i3}\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(|01\rangle_{i}\pm|10\rangle_{i}\right),$
(3) $\displaystyle|\varphi_{i1}\rangle$ $\displaystyle=$
$\displaystyle|11\rangle_{i}.$ (4)

Likewise eigenvalues of $\mathcal{H}_{i}^{imp}$ is given by

$\begin{array}[]{cl}\widetilde{\varepsilon}_{1,4}=&\frac{\widetilde{J}\widetilde{\Delta}}{4}\pm\left(\frac{(J_{1}+J_{2})}{2}\mp\frac{B}{2}\right)\left(\mu_{r}+\mu_{r+1}\right)\mp
B,\\\
\widetilde{\varepsilon}_{2,3}=&-\frac{\widetilde{J}\widetilde{\Delta}}{4}-\frac{B}{2}\left(\mu_{r}+\mu_{r+1}\right)\pm\frac{1}{2}\sqrt{\Sigma^{2}+\widetilde{J}^{2}},\end{array}$

where $\Sigma=\left(J_{1}-J_{2}\right)\left(\mu_{r}-\mu_{r+1}\right)$. The
corresponding eigenvectors are given by

$\displaystyle|\widetilde{\varphi}_{r1}\rangle$ $\displaystyle=$
$\displaystyle|00\rangle_{r},$ (5)
$\displaystyle|\widetilde{\varphi}_{r2,r3}\rangle$ $\displaystyle=$
$\displaystyle M_{\pm}|01\rangle_{r}+N_{\pm}|10\rangle_{r},$ (6)
$\displaystyle|\widetilde{\varphi}_{r4}\rangle$ $\displaystyle=$
$\displaystyle|11\rangle_{r},$ (7)

where

$M_{\pm}=\frac{\widetilde{J}}{\sqrt{2\widetilde{J}^{2}+2\Sigma^{2}\mp
2\Sigma\sqrt{\Sigma^{2}+\widetilde{J}^{2}}}},$

and

$N_{\pm}=\frac{-\Sigma\pm\sqrt{\Sigma^{2}+\widetilde{J}^{2}}}{\sqrt{2\widetilde{J}^{2}+2\Sigma^{2}\mp
2\Sigma\sqrt{\Sigma^{2}+\widetilde{J}^{2}}}}.$

Notice that, in general $J_{1}\neq J_{2}$, therefore $\mathcal{H}_{i}^{imp}$
has diferent eigenvector from $\mathcal{H}_{i}^{host}.$ In addition, the
probabilities of obtaining $|01\rangle_{r}$ and $|10\rangle_{r}$ are different
in eigenstate $|\widetilde{\varphi}_{r2,r3}\rangle$. This is the main
difference between our results and those presented in previous works moi ; mr
; mro .

###  II.1 The partition function

The state at thermal equilibrium can be described by the Gibb’s density
operators $\rho(T)=\frac{\exp(-\beta\mathcal{H})}{Z}$, where $\beta=1/k_{B}T$,
with $k_{B}$ being the Boltzmann’s constant, $T$ is the absolute temperature,
whereas the partition function of the system is defined by
$Z=Tr\left[\exp(-\beta\mathcal{H})\right]$. Similarly to that made by Freitas
$\mathit{et\,al}$ mro  we consider the chain with periodic boundary condition
and use the transfer-matrix notation to obtain the partition function. Thus

$\begin{array}[]{cl}Z=&\sum_{\\{\mu\\}}w(\mu_{1},\mu_{2})...w(\mu_{r-1},\mu{r})\widetilde{w}(\mu_{r},\mu_{r+1})w(\mu_{r+1},\mu_{r+2})...w(\mu_{N},\mu_{1}),\end{array}$

where

$w(\mu_{i},\mu_{i+1})=\sum_{j=1}^{4}e^{-\beta\epsilon_{ij}(\mu_{i},\mu_{i+1})},$
(8)

and

$\widetilde{w}(\mu_{i},\mu_{i+1})=\sum_{j=1}^{4}e^{-\beta\widetilde{\epsilon}_{ij}(\mu_{i},\mu_{i+1})}.$
(9)

The partition function can be simplified by taking
$Z=\mathrm{Tr}(\widetilde{W}W^{N-1})$, in such a way that

$W=\left[\begin{array}[]{cc}w(\frac{1}{2},\frac{1}{2})&w(\frac{1}{2},-\frac{1}{2})\\\
w(-\frac{1}{2},\frac{1}{2})&w(-\frac{1}{2},-\frac{1}{2})\end{array}\right],$

and

$\widetilde{W}=\left[\begin{array}[]{cc}\widetilde{w}(\frac{1}{2},\frac{1}{2})&\widetilde{w}(\frac{1}{2},-\frac{1}{2})\\\
\widetilde{w}(-\frac{1}{2},\frac{1}{2})&\widetilde{w}(-\frac{1}{2},-\frac{1}{2})\end{array}\right].$

In order to simplify the notation we will start to denote $w_{\pm\pm}\equiv
w(\pm\frac{1}{2},\pm\frac{1}{2})$ and $w_{\pm\mp}\equiv
w(\pm\frac{1}{2},\mp\frac{1}{2})$. Through the diagonalization of the
transfer-matrix we obtain

$Z=a\Lambda_{+}^{N-1}+d\Lambda_{-}^{N-1},$ (10)

where

$\Lambda_{\pm}=\frac{w_{++}+w_{--}\pm Q}{2}$

with $Q=\sqrt{(w_{++}-w_{--})^{2}+4w_{+-}^{2}}$ are the eigenvalues associated
with the matrix W. Furthermore

$a=\frac{4w_{+-}\widetilde{w}_{+-}+(w_{++}-w{--})(\widetilde{w}_{++}-\widetilde{w}_{--})+Q(\widetilde{w}_{++}+\widetilde{w}_{--})}{2Q},$
(11)

and

$d=\frac{-4w_{+-}\widetilde{w}_{+-}-(w_{++}-w{--})(\widetilde{w}_{++}-\widetilde{w}_{--})+Q(\widetilde{w}_{++}+\widetilde{w}_{--})}{2Q}.$
(12)

At the thermodynamic limit, $N\rightarrow\infty$, we have

$Z=a\Lambda_{+}^{N-1},$

since $a\Lambda_{+}^{N-1}>d\Lambda_{-}^{N-1}.$

###  II.2 Average reduced density operator

In order to obtain the reduced density matrix let’s take the thermal average
for each two-qubit Heisenbergmr ; mro ,. We start by defining the operator as
a function of spin particles $\mu_{i}$ and $\mu_{i+1}$

$\varrho(\mu_{i},\mu_{i+1})=\sum_{i=1}^{4}e^{-\beta\epsilon_{i,j}}|\phi_{i,j}\rangle\langle\phi_{i,j}|.$
(13)

For impurity we define the operator $\widetilde{\varrho}$ of the two-qubit
Heisenberg operator

$\widetilde{\varrho}(\mu_{r},\mu_{r+1})=\left[\begin{array}[]{cccc}\widetilde{\varrho}_{1,1}&0&0&0\\\
0&\widetilde{\varrho}_{2,2}&\widetilde{\varrho}_{2,3}&0\\\
0&\widetilde{\varrho}_{3,2}&\widetilde{\varrho}_{3,3}&0\\\
0&0&0&\widetilde{\varrho}_{4,4}\end{array}\right],$

where

$\begin{array}[]{cl}\widetilde{\varrho}_{1,1}(\mu_{r},\mu_{r+1})=&\mathrm{e}^{-\beta\widetilde{\varepsilon}_{r1}},\\\
\widetilde{\varrho}_{2,2}(\mu_{r},\mu_{r+1})=&\mathrm{e}^{-\beta\widetilde{\varepsilon}_{r2}}M_{+}^{2}+\mathrm{e}^{-\beta\widetilde{\varepsilon}_{r3}}M_{-}^{2},\\\
\widetilde{\varrho}_{2,3}(\mu_{r},\mu_{r+1})=&\mathrm{e}^{-\beta\widetilde{\varepsilon}_{r2}}M_{+}N_{+}+\mathrm{e}^{-\beta\widetilde{\varepsilon}_{r3}}M_{-}N_{-},\\\
\widetilde{\varrho}_{4,4}(\mu_{r},\mu_{r+1})=&\mathrm{e}^{-\beta\widetilde{\varepsilon}_{r4}}.\end{array}$

Using the transfer-matrix approach we can write the elements of the density
matrix for impurity $\widetilde{\rho}_{k,l}$ in the form mro

$\widetilde{\rho}_{k,l}=\frac{1}{Z}\mathrm{Tr}\left(\widetilde{P}_{k,l}W^{N-1}\right),$
(14)

where

$\widetilde{P}_{k,l}=\left[\begin{array}[]{cc}\widetilde{\rho}_{k,l}(\frac{1}{2},\frac{1}{2})&\widetilde{\rho}_{k,l}(\frac{1}{2},-\frac{1}{2})\\\
\widetilde{\rho}_{k,l}(-\frac{1}{2},\frac{1}{2})&\widetilde{\rho}_{k,l}(-\frac{1}{2},-\frac{1}{2})\end{array}\right],$

consider
$\widetilde{\rho}_{k,l}(\pm\pm)\equiv\widetilde{\rho}_{k,l}(\pm\frac{1}{2},\pm\frac{1}{2})$
and
$\widetilde{\rho}_{k,l}(\pm\mp)\equiv\widetilde{\rho}_{k,l}(\pm\frac{1}{2},\mp\frac{1}{2})$.
Being $U$ the matrix that diagonalizes $W$ we can rewrite Eq. (14) in the form

$\begin{array}[]{ccc}\widetilde{\rho}_{k,l}&=&\frac{\mathrm{Tr}\left(U^{-1}\widetilde{P}_{k,l}U\left[\begin{array}[]{cc}\Lambda_{+}^{N-1}&0\\\
0&\Lambda_{-}^{N-1}\end{array}\right]\right)}{a\Lambda_{+}^{N-1}+d\Lambda_{-}^{N-1}},\end{array}$

that at the thermodynamic limit, $N\rightarrow\infty$, it reduces to

$\begin{array}[]{ccc}\widetilde{\rho}_{k,l}&=&\frac{\mathcal{A}_{k,l}+\mathcal{B}_{k,l}}{\mathcal{M}_{k,l}},\end{array}$

where

$\begin{array}[]{cl}\mathcal{A}_{k,l}=&Q\left[\widetilde{\varrho}_{k,l}(++)+\widetilde{\varrho}_{k,l}(--)\right]+4w_{+-}\widetilde{\varrho}_{k,l}(+-)\\\
\mathcal{B}_{k,l}=&\left[\widetilde{\varrho}_{k,l}(++)-\widetilde{\varrho}_{k,l}(--)\right]\left(w_{++}-w_{--}\right)\\\
\mathcal{M}=&Q\left(\widetilde{w}_{++}+\widetilde{w}_{--}\right)+4w_{+-}\widetilde{w}_{+-}+\left(\widetilde{w}_{++}-\widetilde{w}_{--}\right)\left(w_{++}-w_{--}\right).\end{array}$

Finally, we conclude that the elements of the reduced density matrix take the
form

$\widetilde{\rho}(T)=\left[\begin{array}[]{cccc}\widetilde{\rho}_{11}&0&0&0\\\
0&\widetilde{\rho}_{22}&\widetilde{\rho}_{23}&0\\\
0&\widetilde{\rho}_{23}&\widetilde{\rho}_{33}&0\\\
0&0&0&\widetilde{\rho}_{44}\end{array}\right].$ (15)

The state at thermal equilibrium can be described by the Gibb’s density
operators $\rho(T)$ represents a thermal density operator, the entanglement in
the thermal state is simply called thermal entanglement.

##  III Quantum Correlations

In order to describe the thermal entanglement of two-qubit system, the
concurrence is used as a measure of the entanglement. The concurrence
$\mathcal{C}$ is defined as wootters

$\displaystyle\mathcal{C}={\rm{max}\left\\{0,\sqrt{\lambda_{1}}-\sqrt{\lambda_{2}}-\sqrt{\lambda_{3}}-\sqrt{\lambda_{4}}\right\\},}$

where $\lambda_{i}\>(i=1,2,3,4)$ are the eigenvalues in decreasing order of
the matrix

$\displaystyle
R=\rho\left(\sigma^{y}\otimes\sigma^{y}\right)\rho^{\ast}\left(\sigma^{y}\otimes\sigma^{y}\right),$

with $\sigma^{y}$ being the Pauli matrix. We will explore the thermal
entanglement of the model. To this end, we should obtain the eigenvalues of
$R$. Thus, the concurrence $\mathcal{C}$ can be written as

$\mathcal{C}(\widetilde{\rho})=2\mathrm{max}\\{|\widetilde{\rho}_{2,3}|-\sqrt{\widetilde{\rho}_{1,1}\widetilde{\rho}_{4,4}},0\\}\;.$

In this case, the analytical expression of the thermal concurrence is too
large to be explicitly provided in this paper.

In addition to entanglement, the system may also have other quantum
correlations, such as quantum discord. Quantum coherence is more robust than
entanglement, so it can be a key ingredient in the development of quantum
information. We calculate the quantum coherence by the $l_{1}$-norm defined by
the sum of the absolute values of the elements off the main diagonal of the
reduced density operator of the system baum . So

$\mathcal{C}_{l_{1}}(\rho)=\sum_{i\neq j}|\langle i|\rho|j\rangle|.$

###  III.1 Concurrence

In our model $J_{1}$ and $J_{2}$ they can be taken independently. This
guarantees a much greater generality of this model in relation to those
previously studied moi ; mr ; mro . This greater generality is of great
interest in experimental measurements since it allows greater flexibility in
modeling.

Figure 2: The concurrence $\mathcal{C}$ as a function of temperature $T$ for
$J_{0}/J=1.0$, $\Delta=1.0$, $\alpha=0.0$, solid curve ($\eta=0.0$,
$\gamma=0.0$, $\Omega=0.0$), dashed curve ($\eta=-0.5$, $\gamma=-0.6$,
$\Omega=0.8$).

With an appropriate choice of parameters, we can obtain optimal entanglement
results. In Fig. 2, we show the concurrence $\mathcal{C}$ as a function of
temperature $T$ and for different values of magnetic field. The solid curve
indicates the original model (without impurity) while the dashed curve
indicates the model with impurity described by the Hamiltonian $\mathcal{H}$.
In addition, the red curve refers to a magnetic field $B=1.0$ and the blue
curve refers to $B=2.0$. For the impurity model, the adopted parameters were
($\eta=-0.5$, $\gamma=-0.6$, $\Omega=0.8$). In relation to the original model
moi , we can clearly see the advantage of the model with impurity. The
impurity model presents greater entanglement and also a greater critical
entanglement temperature. For higher fields we can observe a different
behavior for the two models. In this case, the entanglement in the model with
impurity is zero for $T=0.0$ and different from zero for higher temperatures,
until it becomes null at the critical entanglement temperature. In this case,
the triplet splits and $|\widetilde{\varphi}_{r_{1},r_{2}}\rangle$ becames the
ground states. In this case there is no entanglement at $T=0.0$. But the
increase in temperature increases entanglement by bringing in some singlet
component into the mixture.

Figure 3: The concurrence $\mathcal{C}$ as a function of the temperature $T$,
$J_{0}/J=1.0$ , $\Delta=1.0$, solid curve(original model), dashed curve
($\alpha=0.5$, $\eta=-0.5$, $\gamma=-0.6$, $\Omega=0.8$).  Figure 4: Schematic
representation of the form taken by the impurity when we take $\eta=-1.0$
($J_{1}=0.0$). Note that instead of the diamond shape, the impurity structure
now takes on a horizontal shape.

In Figure 3 we analyze how a change in the interaction between the spins of
the impurity affects the entanglement. This interaction is represented by
$\widetilde{J}$ and was obtained by taking $\alpha=0.5$. In other words, the
interaction between the Heisenberg spins in the impurity was greater than
between the Heisenberg spins in the rest of the chain. The advantage of this
small modification is evident in relation to the situation analyzed in Figure
2. In this case we can see that the entanglement is different from zero even
for high field. In addition, the increase in the critical entanglement
temperature was very significant compared to the original model (solid curve).

Figure 5: The concurrence $\mathcal{C}$ as a function of the temperature $T$,
we set $J_{0}/J=1.0$, $\Delta=1.3$, $\eta=-1.0$, (a) dashed curve
($\alpha=0.0$, $\gamma=-0.8$, $\Omega=0.8$), (b) dashed curve ($\alpha=0.5$,
$\gamma=-0.8$, $\Omega=0.8$)

We observe an interesting behavior in the system when we take $J_{1}=0.0$ or
$\eta=-1.0$. This condition is equivalent to a change in the structure of the
impurity. Instead of the diamond shape shown in Figure 1, the impurity term
have a horizontal interaction as shown in Figure 4. This situation is analyzed
in Figure 5 with the following parameters $J_{0}/J=1.0$, $\Delta=1.3$,
$\eta=-1.0$. In Figure 5(a) we considered $\alpha=0.0$ ($J=\widetilde{J}$) and
we observed entanglement and critical entanglement temperature higher than the
original model (solid curve). In Figure 3 we see an increase in entanglement
with $\alpha=0.5$, the same we observed in this case. In Figure 5 (b) is
possible observe the entanglement gain in relation to the original model
(solid curve), both for low and high fields.

Figure 6: The concurrence $\mathcal{C}$ as a function of the magnetic field
$B$, we set $J_{0}/J=1.0$, $\alpha=0.0$, $\eta=-0.5$, $\gamma=-0.6$,
$\Omega=0.8$. (a) dashed curve ($\Delta=1.0$, $T=0.002$, $T=0.400$), (b)
dashed curve ($\Delta=1.3$, $T=0.200$, $T=0.400$)

Our results also show that with an appropriate choice of parameters our model
presents more robust entanglement for application of magnetic field than the
original model. In figure 6 we present some curves of the concurrence as a
function of the magnetic field. As before, here solid curves refer to the
original model and the dashed curves refer to the current model. Although the
critical magnetic field (above which the entanglement is zero) is the same for
both models, the entanglement is significantly greater in the current model.

###  III.2 $l_{1}$-norm

Here we will discuss the results regarding the calculation of quantum
coherence using the $l_{1}$-norm. In Figure 7 we adopt the same parameters
adopted in Figure 2. The comparison between the two Figures is instructive.
The behavior of the curves is similar in the two Figures. Again there is a
clear advantage of the model with impurity (dashed curve) compared to the
original model(solid curve). The main difference is in the scale of magnitude.
Quantum coherence $\mathcal{C}_{l_{1}}$ is significantly greater than
concurrence. This result is not surprising since coherence measures, in
addition to entanglement, other quantum correlations. Coherence is a basis-
dependent measure and states without any quantum correlations can be fully
coherent on an orthogonal basis. In addition, coherence is present also in
single pure quantum systems which does not share any correlation with other
systems.

Figure 7: The quantum coherence $\mathcal{C}_{l_{1}}$ as a function of
temperature $T$ for $J_{0}/J=1.0$, $\Delta=1.0$, $\alpha=0.0$, solid curve
(original model), dashed curve ($\eta=-0.5$, $\gamma=-0.6$, $\Omega=0.8$).
Figure 8: The quantum coherence $\mathcal{C}_{l_{1}}$ as a function of
temperature $T$ for $J_{0}/J=1.0$, $\Delta=1.0$, solid curve (original model),
dashed curve ($\alpha=0.5$, $\eta=-0.5$, $\gamma=-0.6$, $\Omega=0.8$).

Analogous to what was done in Figure 3, in Figure 8 we analyze how a change in
the interaction between the spins of the impurity affects the coherence
$\mathcal{C}_{l_{1}}$. As with concurrence, the increase in coherence with
this small change was very significant. The coherence of the impurity is
greater than $0.2$ up to the temperature of $2.5$ for all fields presented.
The change in $\widetilde{J}$ provides considerable gain in both concurrence
and coherence.

Figure 9: The concurrence $\mathcal{C}_{l_{1}}$ as a function of the
temperature $T$, we set $J_{0}/J=1.0$, $\Delta=1.3$, $\eta=-1.0$, (a) dashed
curve ($\alpha=0.0$, $\gamma=-0.8$, $\Omega=0.8$), (b) dashed curve
($\alpha=0.5$, $\gamma=-0.8$, $\Omega=0.8$)

In Figure 9 we will analyze the coherence by taking the parameter $J_{1}=0.0$.
It is interesting to note that in this case the coherence of the original
model and the model with impurity are close when we take the parameter
$\alpha=0.0$ (Figure 9(a)). From Figure 5 it is evident that the same does not
occur with concurrence. This allows us to conclude that in this case the
coherence is proportionally greater than the concurrence in the original model
than in the model with impurity. In Figure 9(b) we can see the gain in
coherence in relation to the original model increasing the interaction between
Heisenberg spins in impurity ($\alpha=0.5$).

##  IV Quantum teleportation

In this section, we study the quantum teleportation via a distorted
Ising-$XXZ$ diamond chain using the standard teleportation protocol. By means
of an entangled mixed state as resource, the standard teleportation can be
regarded as a general depolarizing channel peres  with probabilities given by
the maximally entangled components of the resource. An unknown two-qubit pure
state to be teleported can be written as
$|\psi_{in}\rangle=\cos\left(\frac{\theta}{2}\right)|10\rangle+e^{i\phi}\sin\left(\frac{\theta}{2}\right)|01\rangle\;$,
where $0\leq\theta\leq\pi$ and $0\leq\phi\leq 2\pi$. In the density operator
formalism, the concurrence $\mathcal{C}_{in}$ of the input state, can be
written as

$\mathcal{C}_{in}=2|e^{i\phi}\sin\left(\frac{\theta}{2}\right)\cos\left(\frac{\theta}{2}\right)|=|\sin(\theta)|\;.$

When a two-qubit state $\rho_{in}$ is teleported via the mixed channel
$\widetilde{\rho}_{ch}$, then the output state $\widetilde{\rho}_{out}$ is
given by peres

$\widetilde{\rho}_{out}=\sum_{i,j=\left\\{0,x,y,z\right\\}}p_{i}p_{j}\left(\sigma_{i}\otimes\sigma_{j}\right)\rho_{in}\left(\sigma_{i}\otimes\sigma_{j}\right)\;,$

in which $p_{i}=tr\left[E^{i}\widetilde{\rho}_{ch}\right]$,
$E^{0}=|\Psi^{-}\rangle\langle\Psi^{-}|$,
$E^{1}=|\Phi^{-}\rangle\langle\Phi^{-}|$,
$E^{2}=|\Phi^{+}\rangle\langle\Phi^{+}|$ and
$E^{3}=|\Psi^{+}\rangle\langle\Psi^{+}|$, where
$|\Phi^{\pm}\rangle=\frac{1}{\sqrt{2}}\left(|00\rangle\pm|11\rangle\right)$
and
$|\Psi^{\pm}\rangle=\frac{1}{\sqrt{2}}\left(|01\rangle\pm|10\rangle\right)$
are Bell states. Here, we consider the density operator channel as
$\widetilde{\rho}_{ch}\equiv\widetilde{\rho}(T)$. Then the output density
operator $\widetilde{\rho}_{out}$ is given by

$\widetilde{\rho}_{out}=\left[\begin{array}[]{cccc}c&0&0&0\\\ 0&f&\chi&0\\\
0&\chi&g&0\\\ 0&0&0&c\end{array}\right]\;.$ (16)

The elements of the operators can be expressed as

$\begin{array}[]{cl}c=&\left(\widetilde{\rho}_{2,2}+\widetilde{\rho}_{3,3}\right)\left(\widetilde{\rho}_{1,1}+\widetilde{\rho}_{4,4}\right),\\\
f=&\left(\widetilde{\rho}_{1,1}+\widetilde{\rho}_{4,4}\right)^{2}\cos^{2}\left(\frac{\theta}{2}\right)+\left(\widetilde{\rho}_{2,2}+\widetilde{\rho}_{3,3}\right)^{2}\sin^{2}\left(\frac{\theta}{2}\right),\\\
g=&\left(\widetilde{\rho}_{2,2}+\widetilde{\rho}_{3,3}\right)^{2}\cos^{2}\left(\frac{\theta}{2}\right)+\left(\widetilde{\rho}_{1,1}+\widetilde{\rho}_{4,4}\right)^{2}\sin^{2}\left(\frac{\theta}{2}\right),\\\
\chi=&2e^{i\phi}\widetilde{\rho}_{2,3}^{\,2}\sin\theta.\end{array}$

According to the definition, the concurrence of the input state is given as
$\mathcal{C}_{in}=\sin(\frac{\theta}{2})$, and the output one follows by

$\mathcal{C}_{out}(\widetilde{\rho})=2max\left\\{2\widetilde{\rho}_{2,3}^{\,2}\mathcal{C}_{in}-2|\widetilde{\rho}_{2,2}||\widetilde{\rho}_{1,1}-\widetilde{\rho}_{4,4}|,0\right\\}.$

More recently, the teleportation of the same entangled state was studies in
this model without impurities tele  and with impurities mro .

##  V Average fidelity of teleportation

Figure 10: The average fidelity $F_{A}$ as a function of temperature $T$ for
$J_{0}/J=1.0$, $\Delta=1.0$, solid curve (original model), dashed curve
($\alpha=0.0$, $\eta=-0.5$, $\gamma=-0.6$, $\Omega=0.8$).

In the following discussion, we turn our attention to the quality of the
entanglement teleportation. The quality of the teleportation is measurement by
the fidelity between the input state $\rho_{in}$ and the output state
$\rho_{out}$. If the input state is pure, the fidelity can be written as

$F=\langle\psi_{in}|\rho_{out}|\psi_{in}\rangle\;.$

For this model $F$ can be written as

$\begin{array}[]{cl}F=&\frac{\sin^{2}\theta}{2}\left[\left(\widetilde{\rho}_{1,1}+\widetilde{\rho}_{4,4}\right)^{2}+4\widetilde{\rho}_{2,3}^{\,2}-\left(\widetilde{\rho}_{2,2}+\widetilde{\rho}_{3,3}\right)^{\,2}\right]+\left(\widetilde{\rho}_{2,2}+\widetilde{\rho}_{3,3}\right)^{\,2}.\end{array}$

In general the state to be teleported is unknown, it is more useful to
calculate the average fidelity. Thus the average fidelity of teleportation can
be formulated as

$F_{A}=\frac{1}{4\pi}\intop_{0}^{2\pi}d\phi\intop_{0}^{\pi}F\sin\theta
d\theta\;.$

The average fidelity of teleportation can be written as

$\begin{array}[]{cl}F_{A}=&\frac{1}{3}\left[\left(\widetilde{\rho}_{1,1}+\widetilde{\rho}_{4,4}\right)^{2}+4\widetilde{\rho}_{2,3}^{\,2}-\left(\widetilde{\rho}_{2,2}+\widetilde{\rho}_{3,3}\right)^{\,2}\right]+\left(\widetilde{\rho}_{2,2}+\widetilde{\rho}_{3,3}\right)^{\,2}.\end{array}$

Figure 11:  The average fidelity ${F}_{A}$ as a function of temperature $T$
for $J_{0}/J=1.0$, $\Delta=1.0$, solid curve (original model), dashed curve
($\alpha=0.5$, $\eta=-0.5$, $\gamma=-0.6$, $\Omega=0.8$).

To transmit a quantum state $|\psi_{in}\rangle\;$ better than any classical
communication protocol, $F_{A}$ must be great than $\frac{2}{3}$ which is the
best fidelity in the classical world joz . To demonstrate the effects of the
impurity on the average fidelity, we plot the behavior of average fidelity. In
Fig. 10, we show the average teleportation $F_{A}$ as a function of
temperature $T$ for $J_{0}/J=1.0$, $\Delta=1.0$ and several values of magnetic
field. For the model with impurity, we set $\alpha=0.5$, $\eta=-0.5$,
$\gamma=-0.6$, $\Omega=0.8$. The horizontal dashed lines at
$F_{A}=\frac{2}{3}$ denote the limit of quantum fidelities. We can see that in
the original model the average fidelity is below $\frac{2}{3}$, signaling that
it is not possible to teleport information. Meanwhile, when consider the
impurity, we have a considerable improvement in quantum teleportation. For
weak magnetic field, the average fidelity reaches maximum value at low
temperatures as the temperature increases, the average fidelity decays up to
the critical temperature, beyond which the teleportation of information is no
longer valid. On the other hand, for strong magnetic field, the effect of the
impurity on the quantum teleportation does not occurs. It means that, average
fidelity remains below $\frac{2}{3}$ regardless of the temperature rise.

In Fig. 11, the behavior of the average fidelity as a function of temperature
$T$ with fixed parameters values, $J_{0}/J=1.0$, $\Delta=1.0$ $\eta=-0.5$,
$\gamma=-0.6$, $\Omega=0.8$ and different values of magnetic field $B$ is
depicted. In this figure, we also consider the impurity in the parameter
$\alpha$ of Heisenberg exchange interaction, we select $\alpha=0.5$, under
this condition a considerable increase in average fidelity is observed,
reaching its maximum value $F_{A}=1$ for several values of magnetic field,
including strong magnetic fields (dashed curves). After that, it gradually
decreases with increase the temperature. However, the average fidelity is
always less than $F_{A}<\frac{2}{3}$ for original model (solid curves), that
is, in this case the quantum teleportation fails. All these show again that
considerable enhancement of teleportation can be achieved by tuning the
strenght of impurity parameters.

Figure 12: The average fidelity $F_{A}$ as a function of the temperature $T$,
we set $J_{0}/J=1.0$, $\Delta=1.3$, $\eta=-1.0$, (a) dashed curve
($\alpha=0.0$, $\gamma=-0.8$, $\Omega=0.8$), (b) dashed curve ($\alpha=0.5$,
$\gamma=-0.8$, $\Omega=0.8$)

Finally, in Fig. 12, let us study the behavior of the system in extreme cases.
We analyze the case when $J_{1}=0.0$ or $\eta=-1.0$ which corresponds to
collapse of one of the impurities of the Ising-like interaction, as depicted
in Fig. 4. We set $J_{0}/J=1.0$, $\Delta=1.3$. In Fig. 12(a), we illustrate
the average fidelity as a function of temperature $T$ for different values of
the magnetic field and the fixed values of impurities parameters $\alpha=0.0$,
$\gamma=-0.8$, $\Omega=0.8$. As can be seen, the results of $F_{A}$ indicate
that, for low temperatures, the average fidelity is maximum, that is,
$F_{A}=1$, for both models. However, for the higher temperature, average
fidelity with impurity is more robust in comparison to that without it.

On the other hand, Fig. 12(b) shows the average fidelity for two values of
magnetic field and we fixed $\alpha=0.5$, $\gamma=-0.8$, $\Omega=0.8$. This
figure shows clearly that the behavior the average fidelity is even more
robust when we consider the impurity in the Heisenberg spin exchange parameter
$\alpha=0.5$, allowing the teleportation of information at higher temperature.
Altogether, it could be concluded that impurity generated by a local bound
disorder of the interchain exchange coupling affects strongly the quantum
correlations, which generates a considerable increase in the teleportation of
information.

##  VI Conclusions

In summary, we have investigated the effects due to the inclusion of an
distorted impurity plaquette on the spin-1/2 Ising-$XXZ$ diamond chain. Next,
the quantum correlations and teleportation of the model under investigation
have been exactly calculated using the transfer-matrix technique. In
particular, our attention has been paid to a rigorous analysis of the quantum
correlations of the interstitial distorted impurity Heisenberg dimers through
the concurrence and norm-$l_{1}$. We observed that the impurity strongly
influence at the behavior of the entanglement and quantum coherence. More
specifically, the results show the existence of the thermal entanglement and
quantum thermal coherence in regions beyond the reach of the original model.
One of our notable results is that quantum correlations can be controlled and
tuned by Ising and Heisenberg parameters into model.

In addition, we have also discussed the teleportation of the two-qubits in an
arbitrary state through a quantum channel composed by a couple of Heisenberg
dimers with distorted impurity in an Ising-$XXZ$ diamond chain structure. We
have found that, the quantum teleportation increases when increases some
suitable impurity parameters. The influence of impurity is more evident in the
average fidelity when we consider the impurity in the Heisenberg parameter.

Finally, our results show that the impurities can be manipulated to locally
control the thermal quantum correlations, unlike the original model where it
is done globally.

## Acknowledgments

This work was partially supported by CNPq, CAPES and Fapemig. M. Rojas would
like to thank CNPq grant 432878/2018-1.

## References

  * (1) C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, W. K. Wootters, Phys. Rev. Lett. 70, 1895 (1993).
  * (2) C. H. Bennett, H. J. Bernstein $\mathit{et\,al}$, Phys. Rev. A 53, 2046 (1996).
  * (3) L. Amico, R. Fazio, A. Osterloh and V. Vedral, Rev. Mod. Phys. 80, 517 (2008).
  * (4) J. I. Cirac, P. Zoller, Nature, 404, 579 (2000).
  * (5) D. Loss, D. P. DiVincenzo, Phys. Rev. A 57, 120 (1998).
  * (6) T. J. G. Apollaro, G. M. A. Almeida, S. Lorenzo, A. Ferraro, S. Piganelli, Phys. Rev A 100, 052308 (2019).
  * (7) M. C. Amesen, S. Bose, V. Vedral, Phys. Rev. Lett. 87, 017901 (2001).
  * (8) D. P. DiVincenzo, D. Bacon, J. Kempe, G. Burkard and K. B. Whaley, Nature, 408, 339 (2000).
  * (9) S. Bose, Phys. Rev. Lett. 91, 207901 (2003).
  * (10) G. L. Kamta, A. F. Starace, Phys. Rev. Lett. 88, 107901 (2002).
  * (11) L. Zhou, H. S. Song, Y. Q. Guo, C. Li, Phys. Rev. Lett. 68, 024301 (2003).
  * (12) F. Levi, F. Mintert, New J. Phys. 16, 033007 (2014).
  * (13) A. Streltsov, G. Adesso, M. B. Plenio, Rev. Mod. Phys. 89, 041003 (2017).
  * (14) A. Winter, D. Yang, Phys. Rev. Lett. 116, 120404 (2016).
  * (15) G. Karpat, B. Çakmak, F. Fanchini, Phys. Rev. B 90, 104431 (2014).
  * (16) W. Wu, J. Xu, Phys. Lett. A 381, 239 (2017).
  * (17) T. Baumgratz, M. Cramer, M. B. Plenio, Phys. Rev. Lett. 113, 140401 (2014).
  * (18) M. L. Hu, X. Hu, J. C. Wang, Y. Peng, Y. R. Zhang, H. Fan, Phys. Rep. 762, 1 (2018).
  * (19) Z. Xi, Y. Li, H. Fan, Sci. Rep. 5, 10922 (2015).
  * (20) K. C. Tan, H. Kwon, C. -Y. Park, H. Jeong, Phys. Rev. A 94, 022329 (2016).
  * (21) H. Kikuchi, Y. Fujii, M. Chiba, S. Mitsudo, T. Idehara, Phys. B 329, 967 (2003).
  * (22) L. Canová, J. Strecka, M. Jascur, J. Phys.: Condens. Matter 18, 4967 (2006).
  * (23) B. Lisnyi, J. Strecka, Phys. Status Solidi B 251, 1083 (2014).
  * (24) J. S. Valverde, O. Rojas, S. M. de Souza, J. Phys.: Condens. Matter 20, 345208 (2008).
  * (25) O. Rojas, M. Rojas, N. S. Ananikian, S. M. de Souza, Phys. Rev. A 86, 042330 (2012).
  * (26) J. Torrico, M. Rojas, S. M. de Souza, Onofre Rojas, N. S. Ananikian, Europhys. Lett. 108, 50007 (2014).
  * (27) J. Torrico, M. Rojas, M. S. S. Pereira, J. Strecka, M. L. Lyra, Phys. Rev. B 93, 014428 (2016).
  * (28) I. M. Carvalho, J. Torrico, S. M. de Souza, M. Rojas, O. Rojas, J. Magn. Magn. Matter. 465, 323 (2018).
  * (29) K. Gao, Y.-L. Xu, X.-M. Kong, Z.-Q. Liu, Physica A 429, 10 (2015).
  * (30) W. W. Cheng, X. Y. Wang, Y. B. Sheng, L. Y. Gong, S. M. Yhao, J. M. Liu, Sci. Rep. 7, 42360 (2017).
  * (31) M. Rojas, S. M. de Souza, O. Rojas, Ann. Phys. 377, 506 (2017).
  * (32) H. Falk, Phys. Rev. 151, 304 (1966).
  * (33) T. J. G. Apollaro, F. Plastina, L. Banchi, A. Cuccoli, R. Vaia, P. Verrucchi, M. Paternostro, Phys. Rev. A 88, 052336 (2013).
  * (34) H. Saleur, P. Schmitteckert, R. Vasseur, Phys. Rev. B 88, 085413 (2013).
  * (35) O. Osenda, Z. Huang, S. Kais, Phys. Rev. A 67, 062321 (2003).
  * (36) E. Solano-Carrillo, R. Franco, J. Silva-Valencia, Physica A 390, 2208 (2011).
  * (37) Y. Sun, X. Huang, G. Min, Phys. Lett. A 381, 387 (2017).
  * (38) T. J. Apollaro, A. Cuccoli, A. Fubini, F. Plastina, P. Verrucchi, Phys. Rev. A 77, 062314 (2008).
  * (39) J. -M. Gong, Z. -Q. Hui, Physica B 444, 40 (2014).
  * (40) M. -L. Hu, X. -Q. Xi, Opt. Commun. 282, 4819 (2009).
  * (41) I. M. Carvalho, O. Rojas, S. M. de Souza, M. Rojas, Quantum Inf. Process. 18, 134 (2019).
  * (42) M. Freitas, C. Filgueiras, M. Rojas, Ann. Phys. (Berlin) 531, 1900261 (2019).
  * (43) H. Fu, A. I. Salomon, X. Wang, J. Phys. A 35, 4293 (2002).
  * (44) A. Galindo, M. A. Martin-Delgado, Rev. Mod. Phys. 74, 347 (2002).
  * (45) M. Brenes, E. Mascarenhas, M. Rigol, J. Goold, Phys. Rev. B 98, 235128 (2018).
  * (46) G. F. Zhang, S. S. Li, Phys. Rev. A 72, 034302 (2005); M. C. Amesen, S. Bose, V. Vedral, Phys. Rev. Lett. 87, 017901 (2001).
  * (47) W. K. Wootters. Phys. Rev. Lett. 80, 2245 (1998).
  * (48) A. Peres. Phys. Rev. Lett. 77, 1413 (1996).
  * (49) R. Jozsa. J. Mod. Opt. 41, 2315 (1994).