Specific Features of Phase States of a Diluted 2D Magnet with Frustration  D.
N. Yasinskaya∗1, V. A. Ulitko1, Yu. D. Panov1 1Ural Federal University 620002,
19 Mira Street, Ekaterinburg, Russia<EMAIL_ADDRESS>
The properties of a dilute Ising magnet are studied using a two-dimensional
spin–pseudospin model with charged impurities and a frustration caused by the
competition of the charge and magnetic orderings. Based on the classical Monte
Carlo method, the ground state phase diagram has been obtained and also
unusual phase states appeared at finite temperatures have been studied. The
regions in which order–order phase transitions and also reentrant phase
transition are observed have been found.

Keywords: dilute Ising magnet, classical Monte Carlo method, frustration,
phase transitions

##  1 Introduction

At present, the study of the properties of disordered and frustrated magnets
and also diluted systems is of significant interest from both fundamental and
practical points of view. Such systems have a rich ground state phase diagram
and exhibit high sensitivity to external actions, demonstrating various types
of phases and phase transitions at finite temperatures [1, 2, 3]. The interest
in unusual phase states of such systems is particularly topical taking into
account their close connection with spin liquids [4].

The spin–pseudospin model considered in this work belongs to a wide class of
the pseudospin Blume–Emery–Griffiths-type models [5] which are widely used for
description of the properties of quantum and classical liquids, binary and
ternary alloys, metamagnets, dilute magnets, cold atoms, and many other
physical systems [6]. The spin–pseudospin model was proposed in [7] for the
description of the competition between magnetic and charge orderings in HTSC
cuprates in normal state. In this model for the cuprate CuO2 plane along with
[CuO4]6- centers with spin 1/2, we consider the interacting [CuO4]5- and
[CuO4]7- centers with spin 0 in the ground state. The more detailed
justification of the model and the possibility of its using for describing the
physical properties of cuprates and also the results obtained in the mean-
field approximation are presented in [8, 9]. The model Hamiltonian includes
the on-site ($\Delta$) and intersite ($V$) density–density correlations for
nonmagnetic centers which have the form of single-ion anisotropy and the Ising
exchange coupling in the language of pseudospin operators, and also the
conventional spin exchange interaction in the Ising form ($J$):

$\mathcal{H}=\Delta\sum_{i}^{\phantom{N}}S_{iz}^{2}+V\sum_{\left\langle
ij\right\rangle}S_{iz}S_{jz}+\tilde{J}\sum_{\left\langle
ij\right\rangle}\sigma_{iz}\sigma_{jz}-\mu\sum_{i}^{\phantom{N}}S_{iz}.$ (1)

Here, $\sigma_{iz}=(1-S^{2}_{iz})s_{iz}$, $\tilde{J}=Js^{2}$, and $\mu$ is the
chemical potential that is necessary to take into account the constant charge
constrain

$n=\frac{1}{N}\sum_{i}\hat{S}_{iz}=const,$ (2)

where $n$ is the charge density. The states corresponding to two pseudospin
projections $S_{z}=\pm 1$ belong to two nonmagnetic [CuO4]5-,7- states with
charges $\pm 1$, which are counted from the charge of the magnetic [CuO4]6-
states with $S_{z}=0$. The magnetic state with $S_{z}=0$ is the spin doublet
$s=1/2$. The summation is performed over the sites of a two-dimensional square
lattice, ${\left\langle ij\right\rangle}$ denotes the nearest neighbors).

The term ‘‘frustration’’ can have various meanings; thus, it should be refined
in this work. The systems with nonzero entropy of the ground state can be
called frustrated systems [10]. In the system considered in this work, nonzero
entropy of the ground state is observed over a wide range of parameters due to
the existence of strongly interacting impurities [11]. In this work, we call
the frustration point the values of the model parameters at which the ordering
type is changed from the charge ordering to the magnetic ordering in the
ground state. It is the classical analog of the quantum critical point.

A feature of our model is the existence of both the disorder (annealed charged
impurities) and also the frustration related to the competition of various-
type interactions. In [12, 13], it was shown that the introduction of
impurities and various structural defects in the system substantially
influences the phase states and the critical behavior, and also extends the
possibilities of applying this model for the description of real physical
systems. Thus, it is interesting to study the influence of charged impurities
on the phase states of our system near the frustration point.

##  2 Methods

The numerical simulation was performed using the classical Monte Carlo (MC)
method. The charge constraint (2) is provided by the modification of the
Metropolis algorithm [14]. The calculations were carried out on a square
lattice with periodic boundary conditions, linear sizes La, and the number of
sites $N=L\times L$, where $a$ is the lattice constant taken to be 1. All the
calculations were carried out for the lattice $L=64$ with annealing of $1\cdot
10^{6}$ MC steps per site, and the data were averaged over 100 copies of the
system. All the effects and results discussed in this report were checked for
$L=256$.

The temperature dependences of the specific heat and the susceptibility are
determined using fluctuation relations

$C=\frac{1}{N}\frac{\langle E^{2}\rangle-\langle
E\rangle^{2}}{k_{B}T^{2}};\qquad\chi=\frac{1}{N}\frac{\langle\mathcal{O}^{2}\rangle-\langle\mathcal{O}\rangle^{2}}{k_{B}T}$
(3)

where $k_{B}$ is the Boltzmann constant and $E$ is the energy of the system
with Hamiltonian (1). Order parameters $\mathcal{O}$ for the checkerboard
antiferromagnetic and charge-ordered phases were determined as follows:

$\mathcal{O}=\left\\{\begin{array}[]{l}m_{1}-m_{2},\\\ M_{1}-M_{2}.\\\
\end{array}\right.\\\ $ (4)

Here, $m_{\lambda}=\sum\limits_{i\in\lambda}s_{iz}$ is the magnetization of a
sublattice $\lambda=1,2$, and $M_{\lambda}=\sum\limits_{i\in\lambda}S_{iz}$ –
is the summary charge of a sublattice $\lambda$ (pseudo-magnetization).

The charge and spin structure factors were calculated using the following
relations:

$\displaystyle
S(\vec{q})=\frac{1}{N^{2}}\sum_{lm}e^{i\vec{q}(\vec{r}_{l}-\vec{r}_{m})}\langle
S_{lz}S_{mz}\rangle,$ (5) $\displaystyle
s(\vec{q})=\frac{1}{N^{2}}\sum_{lm}e^{i\vec{q}(\vec{r}_{l}-\vec{r}_{m})}\langle
s_{lz}s_{mz}\rangle,$

The critical phase transition temperatures were determined by the maxima of
the specific heat and the susceptibilities. In this case, the error of
determination of the critical temperatures was no higher than 1comparison with
the critical temperatures found by the Binder cumulant method. The existence
of the charge and antiferromagnetic orders (both long-range and short-range
orders) was determined using the values of the structure factors in point
$a\vec{q}=(\pi,\pi)$.

##  3 Results and discussion

In [15], it was shown that, in the mean-field approximation (MFA), a change in
the relation of parameters $V$ and $J$ leads to two qualitatively different
ground state phase diagrams. In this work we have restricted ourselves to the
case of a weak spin exchange ($V=4\tilde{J}$). In this case, in MFA, four
phases with the checkerboard-type charge orderings (CO) form in the ground
state (Fig. 1,a). COI phase corresponds to the charge order without spin
centers; COII and COIII are the phases diluted with spin centers distributed
only over one sublattice. In COII phase at $|n|\geqslant 0.5$, one sublattice
is completely filled with charge centers of one type. In MFA, both phases have
a magnetic ordering that evidently is not observed during numerical MC
calculations. MFA predicts the ferromagnetic ordering for FIM phase; however,
the numerical calculation showed the existence of a dilute antiferromagnetic
(AFM) ordering at small $n$ and the shortrange AFM ordering at $0.33\lesssim
n<0.5$. In addition, the ground state phase diagram obtained in terms of the
MC method (Fig. 1,b) differs in the existence of a large region of the short-
range charge ordering (SRO) of one or other type. The asterisks denote the
phases obtained in terms of MFA.

Figure 1: Ground state phase diagrams for the weak spin exchange limit
obtained (a) in MFA and (b) using the MC method.

Studying the phase states at finite temperatures is of specific interest. The
total diagram of possible phase states for $V=4\tilde{J}$ is shown in Fig. 2.
Near the frustration point $\Delta^{*}/J=1.5$, there are three regions labeled
by numerals 1, 2, and 3 in frames in which the ordering type is changed as the
temperature decreases.

Figure 2: Diagram of possible phase states. Near the frustration point
$\Delta^{*}/J=1.5$, there are regions 1, 2, and 3, in which the types of
ordering are changed as the temperature decreases. The horizontal dash–dot
lines correspond to the temperature phase diagrams shown in Figs. 3–5. The
vertical dash–dot line corresponds to the dependences of the structure factors
on the charge density n in Fig. 6

Figures 3–5 show the temperature phase diagrams for $n\,~{}{=}\,~{}0.1$,
$n\,~{}{=}\,~{}0.15$, and $n\,~{}{=}\,~{}0.225$ corresponding to the
horizontal lines in regions 1, 2, and 3 in Fig. 2, respectively. At the right
of the phase diagrams, there are the snapshots of lattice fragments of
$16\times 16$ sites. Different projections of pseudospin $S_{z}=\pm 1$ (two
charge states) and spin $s_{z}=\pm 1/2$ (two magnetic states) are indicated by
different gradations of the grey color. In region 1 for $n=0.1$ (Fig. 3), a
decrease in temperature leads to the phase transition from the nonordered
state (NO) to the AFM phase and then, at lower temperatures, another
order–order phase transition with a change of the AFM ordering to the COIII
phase takes place. The change in the ordering type is observed up to the value
$\Delta/J=1.5$.

Figure 3: Temperature phase diagram for $n=0.1$ near the frustration point. As
temperature decreases, the second phase transition accompanied by changing the
AFM ordering to the COIII phase, takes place. Figure 4: Temperature phase
diagram for $n=0.15$ near the frustration point. A double change in the
ordering type is observed with an increase in temperature; i.e. the reentrant
phase transition to AFM takes place. Figure 5: Temperature phase diagram for
$n=0.225$ near the frustration point. As temperature decreases, another phase
transition from COIII to AFM is observed.

In region 2 (‘‘bridge’’), such a change in the orderings takes place twice as
temperature decreases. This situation is shown in Fig. 4 for $n=0.15$. Thus,
there are three sequential phase transitions: from the high-temperature NO
phase to the ordered AFM state; then, the change to the COIII charge ordering,
and the reentrant phase transition to the AFM phase. In the line for $n=0.225$
(Fig. 5), we also observe the change of one type of ordering to another,
namely of COIII to AFM at $\Delta/J>\Delta^{*}/J=1.5$.

Figure 6: Dependence of the maxima of the charge and spin structure factors in
point ($\pi,\pi$) on charge concentration $n$ for $\Delta/J=1.52$.

Figure 6 shows the dependence of the maximum values of the charge and spin
structure factors in point ($\pi,\pi$) on charge concentration $n$. This
picture corresponds to vertical line $\Delta/J=1.52$ in Fig. 2. The value of
the maximum of the spin (charge) structure factor in point ($\pi,\pi$) allows
us to judge the existence of the AFM (CO) ordering and to determine the
boundary between the long-range and short-range orders. In this case, we
determined this boundary from the structure factor value equal to 0.1. It is
important to note that the certain structure factor can reach its maximum
value not in the ground state but at finite temperatures due to the change in
the ordering types. As shown in Fig. 6, at small $n$, a long-range order forms
in the system. As $n$ increases (regions 2 and 3), CO and AFM ordering
‘‘coexist’’ and change to one another. The condition of conservation of the
charge (2) formally corresponds to the existence of an external field acting
to the pseudospin subsystem. In the case of a weak exchange at large $n$, the
charge ordering minimizes the energy of the density–density correlations. In
this sense, the charge ordering is induced by the charge constraint. In the
FIM phase, one sublattice is completely filled with various magnetic centers
and another sublattice is filled with charge states of one type diluted with
small number of magnetic centers. Thus, the FIM phase appears as the COII-type
order diluted with the shortrange AFM order. It conserves up to $n=0.5$; in
this case, at $n\gtrsim 0.42$, the peak height of the spin structure factor in
point ($\pi,\pi$) is no higher than 0.002.

##  4 Conclusions

Using the classical MC method, we studied the two-dimensional spin–pseudospin
model for the Ising magnet diluted with charged impurities and frustrated by
the competition of the charge and magnetic orderings. The particular attention
was focused on the influence of annealed charged impurities on the phase
states of the system near to the frustration point in the case of a weak spin
exchange.

It is shown that the competition between the charge and magnetic ordering
leads to the formation of unusual phase states at finite temperatures. The MC-
method simulation enabled us to refine the ground state phase diagram obtained
in MFA before and to determine the boundary between the long-range and short-
range orders. We also built the diagram of possible phase states that takes
into account their evolution with an increase in temperature. Near the
frustration point $\Delta^{*}/J=1.5$, we found three regions, in which the
changes in the types of orderings and also reentrant phase transitions take
place at finite temperatures. This effect is caused by the combination of the
following factors. In the case of a weak spin exchange, the substantial
concentration of charged impurities leads to the charge ordering, unlike the
case of strong spin exchange [8], at which we observe the phase separation
into macroscopic regions consisting of charge and magnetic centers. In
addition, in the case of a weak exchange in the frustration point, the ground
state is degenerate in energy for two various ordering types, namely, the
charge and antiferromagnetic orderings. As a result, at finite temperatures
near the frustration point, we can observe orderings that do not correspond to
the minimum energy at $T=0$.

##  5 Acknowledgement

This work was supported by the program of enhancing the competitiveness of the
Ural Federal University (Act 211 of the Government of the Russian Federation,
Agreement no. $02.A03.21.0006$), the Ministry of Education and Science of the
Russian Federation (project FEUZ-2020-0054, and also the Russian Foundation
for Basic Research (project no. 18-32-00837$/18$).

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