# Dynamics of order-disorder and complexity for interacting bosons in optical
lattice

Rhombik Roy1, Barnali Chakrabarti1, N. D. Chavda2, M. L. Lekala3 1Department
of Physics, Presidency University, 86/1 College Street, Kolkata 700073, India.
2Department of Applied Physics, Faculty of Technology and Engineering, The
Maharaja Sayajirao University of Baroda, Vadodara-390001, India.
3Department of Physics, University of South Africa, P.O. Box 392, Pretoria
0003, South Africa

###### Abstract

The present work reports on the dynamical measures of order, disorder and
complexity for the interacting bosons in optical lattice. We report results
both for the relaxed state as well as quench dynamics. Our key observations
are: (1) Lattice depth can be taken as order-disorder parameter. (2) The
superfluid to Mott insulator transition can be treated as ‘order-disorder’
transition. Our main motivation is to find how the system organize by itself
during quench and how it optimizes the complexity. We find dynamical measures
of order and disorder are more sensitive tool than entropy measures. We
specifically calculate the time scale of entry and exit of different phases
during time evolution. Initially the system exhibits collapse revival trend,
however gradually looses its ability to turn back to superfluid phase and
finally Settle to Mott insulator phase.

bosons in optical lattice, cold atom

††preprint: AIP/123-QED

##  I Introduction

In the recent years, different measures of complexity have been proposed in
several scientific disciplines [1, 2, 3, 4, 5, 6, 7, 8]. Out of these,
information theoretical measures and entropic uncertainty relations in quantum
mechanical systems become the trademark [9, 10, 11, 12, 13]. In this context,
the concept of complexity has also attracted considerable interest [14, 15,
16, 17, 18, 19, 20, 21]. Complexity means how a physical or biological system
will organize themselves in response to some change in the external
parameters. It is usually believed that with increase in number of particles
$N$, the complexity would also increase. Although there are several
definitions of complexity in the literature [22, 23, 24, 25], the usual
definition of complexity $\Gamma_{\alpha\beta}$ was introduced by Shiner,
Davison and Landsberg [26] known as SDL complexity. The other statistical
measures of complexity $\bar{C}$ was defined by López- Ruiz [27] and known as
LMC measure. In the context of measuring Shannon information entropy,
complexity is taken as the most efficient measure of order and disorder exist
in a system [28, 29, 30, 31, 32, 33, 34]. The most common case is the ‘convex’
type complexity where it is minimum both for completely ordered and disordered
system. Some system also exhibit complexity which are either increasing or
decreasing function of disorder [26]. The concept of statistical complexity
was first successfully applied in atomic system by the group of Panos [35].
Both SDL and LMC were studied as a function of atomic number $Z$ and main
interest was to explore the connection of the periodicity of shell structure
with the complexity measures.
In the present work, we are interested in the SDL complexity measures for
interacting bosons in the optical lattice. The realization of fully controlled
quantum many-body system is an outstanding challenge since past years.
Interacting bosons in an external trap at ultracold temperature allow an
unprecedented experimental control. Interacting bosons in optical lattice
feature a variety of quantum phases— superfluid phase (SF), Mott insulator
phase (MI), fragmented Mott insulator (FMI) [36]. In the pioneering experiment
of Greiner et. al. [37], a quantum phase transition is observed in a Bose
Einstein condensate (BEC) kept in an optical lattice potential with repulsive
inter atomic interaction. It is observed that weakly interacting bosons in
shallow optical lattice exhibits a superfluid phase. In the SF phase, the
atoms exhibit long range phase coherence across the lattice. Increasing the
depth of the lattice, SF phase makes a transition to Mott insulator phase. In
MI phase, atoms are localized in the individual lattice sites and phase
coherence across the lattice is lost. The quantum phases are studied by Bose-
Hubbard model [38, 39], ab-inito many-body technique—multiconfigurational time
dependent Hartree for bosons (MCTDHB) [40, 41, 42, 36]. The many-body features
are characterised by distinct measures of many-body correlation, collapse and
revival dynamics in lattice depth quench [43]. The collapse-revival dynamics
in the measure of correlation shows the exact behaviour of Fig.2 of ref [44].
Our present work is focused to characterize the quantum phases of the
interacting bosons in optical lattice through the measures of order, disorder
and complexity. Although the measure of complexity uses the defined notations
of order and disorder, but it exhibits considerable degree of flexibility. In
the SF to MI transition, the lattice depth is experimentally controlled and
theoretically corresponding dynamics is recently studied [40, 41, 43, 36]. Our
present calculation exhibits how the system optimizes the complexity. Of
course we need to resolve whether the lattice depth parameter can be taken as
a parameter of disorder. If so, it leads another open question whether SF to
MI transition can be considered as an order-disorder transition. The relevant
issue is to study how the system becomes self organized during the sudden
quench.
We consider a system of $N=3$ bosons interacting with a contact interaction
$W(x_{i}-x_{j})=\lambda\sum_{i<j}^{N}\delta(x_{i}-x_{j})$ in one dimensional
three well optical lattice, $\lambda$ is the strength of interaction and the
bosons are trapped in a lattice of form $V_{OL}(x)=V_{0}sin^{2}(kx)$, where
$V_{0}$ is the depth of the optical lattice and $k$ is the periodicity of the
lattice. Lattice depth is experimentally tunable. In the present work, lattice
depth is tuned to achieve SF phase to MI phase transition. The Bose Hubbard
model is widely successful to describe the phase transition from SF to MI
state. However the most challenging many-body physics is expected in the
regime of strongly interacting boons in shallow optical lattice —- it is
beyond the scope of Bose Hubbard physics. We solve the many-body Schrödinger
equation at a high level of accuracy by MCTDHB method (presented in Sec. II).
Following the definition of information entropy measures, order($\Omega$),
disorder($\Delta$) as given in Sec.III, we calculate complexity
$\Gamma_{\alpha\beta}=\Delta^{\alpha}\Omega^{\beta}$. We observe that lattice
depth parameter acts as a measure of order-disorder and complexity shows all
the three types behaviour as defined in literature [26] depending on the
choice of $\alpha$ and $\beta$. We also note that SF phase is characterised
with maximum order and MI phase exhibits maximum disorder. The ‘SF-MI’
transition can be termed as ‘order-disorder’ transition in the language of
statistical measures. For lattice depth quench, we prepare an initial state
which is a pure SF phase. Instantaneous increase in the lattice depth triggers
the system to go to MI phase. From the time evolution of entropy measures, we
further calculate the time dynamics of order, disorder and complexity. We
observe collapse revival dynamics in all measures in short time dynamics as
discussed in details in Sec.IV. We again observe that maximum order is
associated with SF phase and MI phase is associated with maximum disorder. Of
course, our calculation for time dynamics reconfirm that SF to MI phase can be
termed as ‘order-disorder’ transition. However, the intriguing observation is
that we are able to focus on the time scale of entry and exit of different
phases over several cycles. We also able to find out the holding time of Mott
phase which is exposed as a plateau region in the time dynamics of order and
disorder. We clearly demonstrate that long time dynamics of order-disorder is
more sensitive tool as we are able to calculate the time scale of dynamical
evolution. To showcase how the system is able or disable to organize the
impact of the quench, we measure several different kinds of complexity.
$\Gamma_{1,1}$ is the most fundamental which keep equal weight in order and
disorder. $\Gamma_{0,4}$ is the same with zero disorder and $\Gamma_{1/4,0}$
is with zero order. The gross observation in the time evolution of all the
three measures in complexity, is same —- collapse and revival in the short
time scale. The system of bosons can adjust the effect of quench and is able
to make a transition to MI phase with a subsequent revival to SF phase.
However in the long time dynamics, the system gradually looses its ability to
turn back to SF phase and finally settle to MI phase.

The paper is structured as follows. In Sec.II, we introduce the setup and the
necessary theory. Sec.III deals with the basic equation to measure the
different quantities. Sec.IV explains our numerical results and explanations.
Sec.V draws our conclusion.

Figure 1: Population of the first three natural orbitals as a function of
lattice depth ($V_{0}$). As $V_{0}$ increases, the occupation in the first
orbital starts to decrease while another two orbitals start to contribute. At
$V_{0}=10.0$, the state becomes threefold fragmented ($n_{1}\simeq n_{2}\simeq
n_{3}\simeq\frac{1}{3}$).

##  II Setup and methodology

In the present work, we consider $N=3$ bosons confined in one-dimensional
optical lattice and interacting with contact inter-particle interaction. This
one dimensional regime is easily achieved by tight transverse confinement.
Quantum many-body effect is also important in such reduced dimension as the
quantum fluctuation plays an important role. The Hamiltonian for $N$
interacting bosons in $1D$ optical lattice is given by

$H=\sum_{i=1}^{N}\left(-\frac{1}{2}\frac{\partial^{2}}{\partial
x^{2}}+V_{OL}(x_{i})\right)+\sum_{i<j}\hat{W}(x_{i}-x_{j})$ (1)

Where $V_{OL}$ represents the external lattice potential and
$\hat{W}(x_{i}-x_{j})$ is the two-body interaction. We make the Hamiltonian
dimensionless by dividing it by the factor $\frac{\hbar^{2}}{mL^{2}}$, where
$m$ is the mass of the bosons and $L$ is some arbitrary length scale. The
Hamiltonian is scaled in terms of recoil energy,
$E_{R}=\frac{\hbar^{2}k^{2}}{2m}$. Thus the time is expressed in unit of
$\frac{\hbar}{E_{R}}$ and unit of distance becomes $k^{-1}$. We use natural
units ,i.e., $\hbar=m=k=1$. We fix up the grid at $x_{min}=-\frac{3\pi}{2}$ to
$x_{max}=\frac{3\pi}{2}$ such that we consider three wells. For $N=3$ bosons,
we choose $V_{0}$ and $\lambda$ such that it allowes superfluidity in the
initial state. We find the stationary solution of the many-body Schrödinger
equation by MCTDHB implemented in the MCTDH-X software[45, 46, 47] with
periodic boundary condition. In MCTDHB, the wave-function of the interacting
bosons is expanded over a set of permanents which are the symmetrized bosonic
states of $N$ bosons distributed over $M$ single particle states.

$|\Psi(t)\rangle=\sum_{\bar{n}}C_{\bar{n}}(t)|\bar{n};t\rangle,$ (2)

The vector $\vec{n}=(n_{1},n_{2},\dots,n_{M})$ represent the occupation of the
orbitals and $n_{1}+n_{2}+\dots n_{M}=N$ preserve the total number of
particle.

$|\bar{n};t\rangle=\prod_{i=1}^{M}\left(\frac{\left(b_{i}^{\dagger}(t)\right)^{n_{i}}}{\sqrt{n_{i}!}}\right)|vac\rangle.$
(3)

$b_{k}^{\dagger}(t)$ creates a boson occupying the time-dependent orbital
$\phi_{k}(x,t)$. The number of possible configuration is
$\left(\begin{array}[]{c}N+M-1\\\ N\end{array}\right)$. It is important to
note that both the expansion coefficients ($C_{\vec{n}}(t)$) and time
dependent orbitals ($\phi_{i}(x,t)$) that build the permanents
$|\vec{n},t\rangle$ are time dependent and fully variationally optimized
quantities. Thus MCTDHB has been established as the most efficient way to
solve the time dependent many-body Schrödinger equation [48, 49, 50]. The
efficiency of MCTDHB is to make the sampled Hilbert space dynamically follow
the time evolution of the many-body system. MCTDHB has been widely used in
different theoretical calculations [36, 51, 52, 53] and results are very close
to experimental predictions [54, 55]. For $M\rightarrow\infty$ limit, as the
set of permanents $|\vec{n;t}\rangle$ span the complete Hilbert space, the
expansion is exact. But during computation, we limit the size of the Hilbert
space. As the permanents are now time dependent, a given degree of accuracy is
achieved with the truncated basis as compared to time-independent basis. It is
also proved that significant computational advantage is achieved over exact
diagonalization [56]. To solve the time dependent many-body Schrödinger
equation $\hat{H}|\psi\rangle=i\frac{\partial|\psi\rangle}{\partial t}$ for
the wave function $|\psi\rangle$, we calculate the time evolution of the
coefficients $C_{\vec{n}}(t)$ and the orbitals $\phi_{i}(x,t)$. We utilize
variational principle [57, 58, 59, 60] to obtain the equation of motion of the
time dependent coefficient and orbital [49, 61, 48, 62, 63]. Finally the
coupled nonlinear integrodifferential equations are solved by MCTDHB package
[47]. For the calculation of the eigenstates of the Hamiltonian, we prepare
the MCTDHB equations in imaginary time — called improved relaxation method.
For quench dynamics, we consider the total Hamiltonian

$\hat{H}(x_{1},x_{2},\dots
x_{N})=\sum_{i=1}^{N}\hat{h}(x_{i})+\Theta(t)\sum_{i<j=1}^{N}\hat{W}(x_{i}-x_{j})$
(4)

where $\hat{h}(x)$ is the one-body part includes the external trap and kinetic
energy. $\Theta(t)$ is the Heaviside step function of time $t$ which trigger
the quench at $t=0$.

Figure 2: First-order normalized correlation function
$g^{(1)}(x_{1}^{\prime},x_{1},t)$ as a function of lattice depth ($V_{0}$).
(a) $V_{0}=0.7$ is in superfluid phase. inter-well coherence as well as intra-
well coherence is maintained throughout the lattice. (b)$V_{0}=2.8$; It is the
initiation point of MI state. Diagonal correlation starts to build up which
means the loss of inter-well coherence. (c) $V_{0}=5.0$; MI is the dominating
phase. Intra-well coherence is prominent whereas inter-well coherence faded
away. (d) $V_{0}=10.0$ corresponds to pure MI phase where we find absolute
loss of inter-well coherence and $100\%$ build up of diagonal correlations.

##  III Measures of order-disorder and complexity

Complexity is measured in terms of order and disorder. Often entropy was taken
as an appropriate measure of disorder. However, with increase in the number of
available states, the disorder of the system increases, as well as entropy
increases. Later Landsberg’s definition of the disorder parameter ($\Delta$)
is well accepted which circumvent the previous problem. Disorder is defined
as,

$\Delta=\frac{S}{S_{max}},$ (5)

where $S$ is the actual information entropy of the system. $S_{max}$ is the
maximum entropy which is accessible to the system. Thus, in the Landsberg
definition, entropy and disorder are decoupled. Order is defined as,

$\Omega=(1-\Delta).$ (6)

$\Omega=1$ corresponds to perfect ordered and predictable system. $\Omega=0$
corresponds to complete disorder and randomness. Both order and disorder are
size independent and lies between $0$ and $1$. The measure of complexity is
further defined appropriately in terms of order and disorder. In the
literature, we find three categories of complexity measure as mentioned in the
Introduction. To take into account all three categories, we utilize the most
generic form of complexity defined as,

$\Gamma_{\alpha\beta}=\Delta^{\alpha}\Omega^{\beta}=\Delta^{\alpha}(1-\Delta)^{\beta}=(1-\Omega)^{\alpha}\Omega^{\beta}$
(7)

It defines complexity of disorder strength $\alpha$ and order strength
$\beta$. Thus three categories are subsumed here. With $\beta=0$ and
$\alpha>0$, complexity is an increasing function of disorder; with $\alpha=0$
and $\beta>0$, complexity is an increasing function of order. When $\alpha\neq
0$, $\beta\neq 0$, one finds the most common case of ‘convex’ type of
complexity. Complexity

Figure 3: (a) Plot of Shannon information entropy (S) as a function of lattice
depth ($V_{0}$). Entropy is minimum for shallow lattice depth. As lattice
depth increases, system transit from SF to MI phase followed by saturation in
the entropy at maximum value. (b) Order and disorder of the system of $N=3$
bosons in three well optical lattice. In superfluid to Mott insulator
transition, order gradually decreases to zero and disorder gradually increases
to one. The intersection point refers as the initiation of SF to MI state. SF
phase is characterized as ordered and MI phase is characterized as disordered
state. (c) Plot of Complexity measures ($\Gamma_{\alpha\beta}$) as a function
of $V_{0}$. $\Gamma_{0,4}$, $\Gamma_{1,1}$, $\Gamma_{1/4,0}$ exhibit type I,
type II and type III complexity as discussed in the literature [26]

vanishes at zero disorder and zero order and exhibits a maximum in between.
However, the key quantity for the calculation of order and disorder are the
entropy $S$. Shannon information entropy both in coordinate space and momentum
space are usually taken as the key measure of information entropy. They are
defined as $S_{x}(t)=-\int dx\rho^{(1)}(x,t)ln[\rho^{(1)}(x,t)]$ and
$S_{k}(t)=-\int dk\rho^{(1)}(k,t)ln[\rho^{(1)}(k,t)]$. Where $\rho^{(1)}(x,t)$
is the one-body density in position space and $\rho^{(1)}(k,t)$ is the same in
momentum space. The reduced one-body density in coordinate space defined as

$\displaystyle\begin{split}\rho^{(1)}(x^{\prime}|x;t)=N\int
dx_{2}dx_{3}...dx_{N}\psi^{*}(x^{\prime},x_{2},\dots,x_{N};t)\\\
\psi(x,x_{2},\dots,x_{N};t).\end{split}$ (8)

Its diagonal gives the one-body density $\rho(x,t)$ defined as

$\displaystyle\begin{split}\rho(x;t)=N\int
dx_{2}dx_{3}...dx_{N}\psi^{*}(x,x_{2},\dots,x_{N};t)\\\
\psi(x,x_{2},\dots,x_{N};t).\end{split}$ (9)

Density distributions are normalized to unity. However, the one-body density
are insensetive to address correlations present in the many-body wave
function. So we define an alternative measure of many-body information entropy
as

$S=-\sum_{i}\bar{n}_{i}(t)\hskip 3.44444ptln\hskip 0.86108pt\bar{n}_{i}(t)$
(10)

This can be called as occupational information entropy as it is defined by the
eigenvalues of the reduced one-body density matrix or occupation numbers. For
the mean-field theory as there is only one natural occupation, occupational
entropy is always zero. In our earlier calculation [43], we have also
discussed how time evolution of occupational entropy can be chosen as a good
measure for the description of fragmentation. It is also a key quantity in the
study of non-equilibrium quench dynamics to establish whether thermalization
and relaxation are ubiquitous in nature. Thus the use of occupational entropy
in the measure of order, disorder and complexity is justified. As pointed out
that SF phase exhibits global correlation across the lattice whereas Mott
phase exhibits on-site correlations. To explore the link between order-
disorder to complexity, we further make an analysis of first order correlation
function $g^{(1)}(x_{1}^{\prime},x_{1},t)$ defined as,

$g^{(1)}(x_{1}^{\prime},x_{1},t)=\frac{\rho^{(1)}(x_{1}^{\prime}|x_{1};t)}{\sqrt{\rho(x,t)\rho(x^{\prime},t)}}$
(11)

where $\rho$ is the diagonal part of the one-body density matrix given in Eq.
8. $g^{(1)}(x_{1}^{\prime},x_{1},t)$ quantify how correlated the particles are
in the specific system. In recent experiments, it is also possible to
calculate higher order correlations experimentally to characterize a many-body
system [64, 65, 66, 67].

Figure 4: Time evolution of natural occupation, order and disorder of $N=3$
bosons in optical lattice as a function of time. (a) Plot of first three
natural orbitals exhibit oscillation. At the crossing point of the first three
orbitals, system is in MI phase with $n_{1}\simeq n_{2}\simeq
n_{3}\simeq\frac{1}{3}$. See the text for further discussion. (b) order and
disorder plot where order shows maximum value at SF phase and disorder shows
maximum value at MI state. The plateau region at the maximum point of the
disorder curve tells MI phase holds at that time interval.

##  IV Results

###  IV.1 Relaxed state

The present calculation is performed in the one dimension with $N=3$
repulsively interacting bosons in the optical lattice. The dimensionless
strength parameter $\lambda$ of the repulsive interaction is kept fixed to
$\lambda=0.3$ and the lattice depth potential is varied. For the entire
manuscript, we consider commensurate filling factor $\frac{N}{W}=1$, i.e.,
three repulsively interacting bosons in three well. However, the following
results are also valid for five repulsively interacting bosons in five wells.
As pointed out earlier, fragmentation is the hallmark of MCTDHB, where several
natural orbitals exhibit significant population. Thus convergence is an
important issue and to capture the correct physics, we need adequate number of
orbitals. Before assessing that how much complexity the system is developing
during SF to MI transition, we first need to evaluate the many-body
information entropy for different lattice depth potential. For the stationary
state solution, we follow the improve relaxation method —- we propagate the
MCTDHB equation in imaginary time. For relaxation, we keep $M=6$ orbitals and
find that with further increase in number of orbitals, there is no change in
the computed quantities. In Fig. 1, we plot the occupation of the first,
second and third natural orbitals as a function of lattice depth potential.
For very small lattice depth, the first orbital has close to $100\%$
occupation and second and third orbitals have insignificant occupation. The
many-body wave function can be well approximated by the single orbital mean-
field state $|N=3,0,0,0,0,0\rangle$. To identify the state, we plot the
absolute value of first order correlation function
$g^{(1)}(x_{1}^{\prime},x_{1},t)$ for $V_{0}=0.7$ in Fig. 2(a). We observe
that coherence within and between the sites is maintained. It confirms the
state as superfluid. With increase in lattice depth $V_{0}$, the occupation in
the first natural orbital gradually decreases and contribution from second and
third orbitals increase. From the numerical data, we observe that at
$V_{0}=10.0$, the occupation in first three natural orbitals saturates at
$33.33\%$. We call that the initial SF state is now three fold fragmented.
We further identify that the fragmented many-body state with the configuration
$|1,1,1,0,0,0\rangle$ as Mott state. In Fig. 2(d), we plot the one-body
correlation for $V_{0}=10.0$. It exhibits three separated regimes with non-
zero values along the diagonal and zero contribution from the off-diagonal
terms. Thus the coherence within the well is maintained and inter-well
coherence is completely lost. It is clearly a Mott phase. Thus Fig. 1 and Fig.
2 demonstrate the transition from SF phase to MI phase with gradual change in
the lattice depth potential.
The main aim of our present work is to characterize the superfluid to Mott
transition in terms of order, disorder and complexity. The most important
question whether the lattice depth parameter can be chosen as a measure of
disorder. In Fig. 3(a), we plot the many-body Shannon information entropy as a
function of lattice depth. For SF phase, entropy is minimum but not zero. With
increase in lattice depth, entropy gradually increases and at $V_{0}=10.0$, it
saturates. With further increase in lattice depth parameter, entropy remains
at the saturation value. Thus once the MI phase is reached, system retains at
that phase even with further increase in lattice depth.
In Fig. 3(b), We plot the order and disorder for varying lattice depth. We
observe for SF state, order is maximum and disorder is minimum. As neither the
order is one nor the disorder is zero, the SF state is not perfectly ordered
state. With increase in lattice depth, order gradually decreases and disorder
increases. At $V_{0}=2.8$, order and disorder plot intersects each other which
we considered as the initiation of SF to MI state. At $V_{0}=10.0$, order
become exactly zero and the disorder becomes exactly $1.0$, which confirm that
Mott phase is a random phase or a perfectly disordered phase. Thus our
numerical calculations exhibit that the ‘SF-MI’ transition can be renamed as
‘order-disorder’ transition. The corresponding complexity
$\Gamma_{\alpha\beta}$ is plotted in Fig. 3(c). $\Gamma_{1,1}$ shows convex-
type, i.e., type II complexity. For SF phase, complexity is minimum. With
increase in lattice depth, complexity increases, reaches a maximum and
smoothly reduced to zero for Mott phase. Whereas for $\Gamma_{0,4}$, the
complexity with zero disorder exhibits type II complexity of the literature
[26]. $\Gamma_{1/4,0}$, the complexity with zero order, exhibits type I
complexity [26]. Thus we find the existence of all three types of complexity
in our calculation. From the results based on the stationary state
calculation, we can conclude that ‘lattice depth’ of the optical lattice can
be taken as a measure of disorder in the system. In the language of
statistical measure, SF phase is an ordered state and MI phase is a disordered
state. From all the related measures in the statistical quantities, we
uniquely conclude that SF to MI transition is truly an ordered-disordered
transition.

Table 1: Superfluid to Mott insulator phase transition time statistics.

Period | SF to MI initiation | Mott state entry | Mott state exit | MI phase holding time | MI to SF initiation | SF state
---|---|---|---|---|---|---
First Cycle | 10.6 | 30.0 | 54.0 | 24.0 | 71.0 | 81.0
Second Cycle | 91.0 | 113.0 | 137.0 | 24.0 | 154.0 | 162.0
Third Cycle | 170.0 | 196.0 | 219.0 | 23.0 | 239.0 | 243.0

Figure 5: Time evolution of complexity in $\Gamma_{\alpha\beta}$ of $N=3$
bosons in optical lattice as a function of time. $\Gamma_{1,1}$ shows collapse
revival dynamics between SF and MI phase. Picks in $\Gamma_{0,4}$ signifies
the time when the system is in SF phase. The height of the picks in
$\Gamma_{0,4}$ give a qualitative idea about the strength of the SF phase.
Picks in $\Gamma_{1/4,0}$ quantify the MI phase.

###  IV.2 Lattice depth quench

In this section, we do the dynamical measures of order, disorder and
complexity. The motivation of the study is to estimate the relative
contribution of order and disorder to complexity during evolution. It will
practically demonstrate how the system would optimize the complexity which
basically will exhibit the ability or inability of the system to self organize
with the external change. In the quench dynamics, we monitor the time
evolution of the natural orbitals and entropy production. In the dynamical
evolution, the convergence is a serious issue and we need $M=12$ orbitals for
quench dynamics. We report long-time dynamics (up to t=500) in all measured
quantities. We initially prepare the system in SF phase with weak interaction
strength $\lambda=0.1$ in shallow lattice depth of $V_{0}=3.0$. We quench the
system to MI phase by sudden increase in lattice depth to $V_{0}=10.0$ at
$t=0$. In Fig. 4(a), we plot the natural occupation in first three orbitals as
a function of time. At $t=0.0$, only the first natural orbital contributes,
which corresponds to SF phase. With increase in time, fragmentation occurs. We
observe that at $t=30$, the system is a fully fragmented MI state, the lowest
three natural orbitals $n_{1},n_{2},n_{3}$ have close to $33\%$ population.
Between $t=30$ to $t=54$, $n_{1},n_{2}$ overlap, $n_{3}$ becomes down. From
the one-body correlation plot (not shown here), between $t=30$ and $t=54$, we
find that the system exhibits only the diagonal correlation, which signifies
the system is in MI phase. So, for this choice of parameter, MI state is
retained in this interval. Then at $t=81$, it enters to SF phase with maximum
occupation only in the first orbital, at $t=113$ it again enters the second MI
phase and retained the MI phase till $t=137$. The above scenario repeats with
entry and exit in SF and MI phase. This basically simulates collapse revival
dynamics as observed in Greiner’s experiment [44]. The time evolution of the
many-body information entropy exhibits the same collapse revival picture in
the same time scale (not shown here).
From the time evaluation of many-body information entropy, we further
calculate the time dependence in order, disorder and complexity.
Qualitatively, all the measures exhibit collapse revival scenario. However the
observations based on the time scale need further explanation. From Fig. 4
(b), initially when the many-body state is in SF phase, only one natural
orbital contributes, order is maximum and disorder is minimum. With increase
in time, order decreases and disorder in the system builds up. At $t=10.6$,
order-disorder cross each other. From the natural occupation, we find the
significance of the point of crossing. $t=10.6$ is the initiation of Mott
phase when fragmentation starts —- order-disorder exhibit the same weights
($\simeq 50\%$). Further at $t=30$, disorder becomes close to one and order is
close to zero which signify the pure Mott phase. Between $t=30$ and $t=54$,
both the order and disorder have a plateau region which signify the holding
time of Mott phase. Then at $t=71$, order-disorder again cross each other
which is the initiation of MI to SF phase. At $t=81$, the system enters in SF
phase exhibiting maximum order and minimum disorder. Then again order
decreases and disorder increases further. At $t=113$, system enters in second
MI phase. The time interval $t=113$ to $t=137$ is the holding time of second
MI phase. The time scale for different phases during the time evolution is
presented in Table.1 for the first three cycles. All the measured time scales
are in mutual agreement. Although the time dynamics exhibit the collapse
revival picture on an average, however close scrutiny reveals some additional
information. In the time evolution of order, the peak which corresponds to SF
phase gradually decreases. Initially the peak value was close to one, finally
it settle down to 0.4. It exhibits that the system initially has full ability
to turn back to SF phase, however with complex time evolution, it looses its
ability gradually. We understand that in the long-time dynamics, many excited
states contribute in a complex manner. So it is very hard for the system to
retain its perfect order and coherence. The study of the evolution of the
natural occupation and entropy evolution was not so sensitive tool to detect
the above observation.
Fig. 5 shows the time evolution of complexity ($\Gamma_{1,1}$, $\Gamma_{0,4}$
and $\Gamma_{1/4,0}$). $\Gamma_{1,1}$ repeats the common-type (type II)
behaviour in the dynamics and exhibits same features in same time scale as
observed in the time evolution of order and disorder. We do not find any steep
building in complexity. By sudden increase in lattice depth means we are
pumping energy to the system externally and it will be distributed through
one-body term in the Hamiltonian. The time dynamics of $\Gamma_{1,1}$ exhibits
that the system is able to distribute the extra energy between the interacting
bosons and thus able to self organize the external perturbation.
For further investigation we calculate complexity $\Gamma_{0,4}$ of disorder
weight factor zero and order of weight factor four to quantify the absolute
contribution by order only. Peaks in $\Gamma_{0,4}$ tell about the strength of
order present in the system. The first peak in $\Gamma_{0,4}$ arises at
$t=81$, which is SF phase. Second and third peak positions are at $t=162$ and
$t=343$ respectively. All the peaks indicates the formation of SF phase which
is an ordered state. In the long time dynamics, $\Gamma_{0,4}$ dies out to
zero gradually which indicates the gradual disappearance of SF phase. It also
indicates the gradual loss of coherence in the time evolution.
We next calculate $\Gamma_{1/4,0}$, which defines complexity with order factor
of zero and disorder factor of small finite quantity of 0.25. It will
facilitate to quantify the contribution of disorder to complexity. Initially
dynamics of $\Gamma_{1/4,0}$ is reverse to that of dynamics in $\Gamma_{0,4}$
qualitatively. The peaks correspond to Mott state with maximum disorder. The
peaks are flat as the Mott state is hold for some time (as discussed earlier)
before switching to SF phase. In the long time dynamics it gradually settle to
one.
Thus we conclude that the above analysis nicely exhibit how the system tends
to self organize on the sudden quench in optical lattice. Although initially
all the complexity measure point out the same physics, collapse and revival
dynamics, however in the long time, the system will gradually loose its
ability of revival and finally will settle to MI phase only.

##  V Conclusion

‘Statistical complexity’ is one of the most circulating word in scientific
research of Physics, biology, mathematics, computer science etc. Although
there is no strictly followed definition of ‘what is complexity’, it is
defined in many ways in the literature. From its vast application, it is found
that SDL measure of complexity is well understood.

SDL measure has been extensively applied in different systems including atoms
and molecules. Here we consider ultracold trapped atoms in the optical lattice
which has been proved as a most challenging platform to study the many-body
physics. Interacting bosons in optical lattice exhibit different quantum
phases like superfluid and Mott insulator phase. The quantum phase transition
has been experimentally studied as well as there are numerous theoretical
calculation on the dynamical evolution. Although most of the calculation are
based on mean-field level or utilizing Bose Hubbard model, the strongly
interacting bosons in shallow lattice deserve quantum many-body calculation.
We report our results on small ensemble of few particle system utilizing
MCTDHB method which is exact by its construction and retain many-body
correlation. We obtain many-body states which are the few-body analogy of
different thermodynamic phases.
The main motivation of our work is how to consult the various quantum phases
with the measure of order-disorder and complexity. As the lattice depth is an
easily controllable parameter experimentally, our theoretical analysis both
for the relaxed state as well as dynamically evolve state exhibit that lattice
depth can act as order-disorder parameter. SF phase is characterized as an
ordered but not properly ordered state, whereas Mott phase is characterized as
proper disordered state. Complexity exhibits type II features. SF to MI phase
transition can be renamed as ‘order-disorder’ transition. In the quench
dynamics, we study the entire physics behind the SF to MI transition in the
light of dynamical measures of order, disorder. We find that initially the
system is able to demonstrate the collapse to Mott phase and revival to
superfluid phase. However, the system gradually looses its ability and finally
looses global correlation across the lattice and settles to Mott insulator
phase in long-time dynamics. We also able to present the time scale of
different phases through several cycles.

###### Acknowledgements.

R. Roy acknowledges the University Grant Commission (UGC) India for the
financial support as a senior research fellow.

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Ekaterina Kompantseva, Askar Tuganbaev

Absolute Ideals
of Almost Completely Decomposable Abelian Groups

Abstract. We consider the class $\mathcal{A}_{0}$ of Abelian block-rigid
$CRQ$-groups of ring type. A subgroup $A$ of an Abelian group $G$ is called an
absolute ideal of the group $G$ if $A$ is an ideal in any ring on $G$. We
describe principal absolute ideals of groups in $\mathcal{A}_{0}$. This allows
to prove that any group in $\mathcal{A}_{0}$ is an $afi$-group, i.e., a group
$G$ such that any absolute ideal of $G$ is a fully invariant subgroup.

Key words. Abelian group, almost completely decomposable Abelian group, ring
on an Abelian group

MSC2020 datebase: 16B99, 20K30, 20K99

##  1 Introduction

Any homomorphism $\mu\colon G\otimes G\rightarrow G$ is called a
multiplication on an Abelian group $G$. An Abelian group $G$ with
multiplication defined on $G$ is called a ring on $G$. When studying rings on
an abelian group $G$, one of the most natural questions is the question on
subgroups of $G$ that are ideals in any ring on $G$. Such subgroups are called
absolute ideals of the group $G$.

Absolute ideals of Abelian groups were studied in [10], [5], [7], [8], [9],
[20]. In [7], general properties of absolute ideals are studied. To do this,
it is defined the group $\text{M}(G)$ generated by all homomorphic images of
the group $G$ in its endomorphism group $\text{End }G$ and it is proved that a
subgroup $A$ of an Abelian group $G$ is an absolute ideal of $G$ if and only
if $A$ is $\text{M}(G)$-admissible, i.e., $\text{M}(G)(A)\subseteq A$. The
results of this paper were generalized to [8] for modules over a commutative
unital ring. In [19], it is proved that any absolute ideal of an Abelian
$p$-group $G$ is of the form $G=H\cap\\{g\in G\,|\,p^{n}g\in S\\}$, where $H$
is a fully invariant subgroup of the group $G$, $S$ is a subgroup of the group
$G^{1}$, and $n$ is an integer. In [5], Abelian groups, in which all subgroups
are absolute ideals, are described. It is proved that the class of all such
groups consists of torsion groups, each $p$-component of which is a cyclic or
divisible group, torsion-free nil-groups, as well as cyclic groups of infinite
order.

We remark that for an Abelian group $G$, the subgroup $\text{M}(G)$ of $G$ is
an ideal in the endomorphism ring $\text{E}(G)$. Therefore, there is a close
connection between the completely invariant subgroups of an Abelian group and
its absolute ideals. It is easy to see that any fully invariant subgroup of an
Abelian group is its absolute ideal [10]. However, the converse is not true.
In [7], the problem of describing Abelian groups in which any absolute ideal
is a completely invariant subgroup is formulated. Such groups are called
$afi$-groups. In [7], it is proved that an Abelian group $G$ is an $afi$-group
if and only if for each element $g\in G$, the subgroups
$A(g)=\\{\eta\in\text{End}G\,|\,\eta(g)=0\\}$ and $\text{M}(G)$ together with
the identity endomorphism generate the entire group $\text{End}G$. However, in
reality, this criterion does not bring us closer to solving the problem, but
is only a variant of the definition of an $afi$-group. In [20], it is proved
that a fully transitive $p$-group $G$ is an $afi$-group if and only if its
first Ulm subgroup $G^{1}$ is a cocyclic group. Since any decomposition of a
torsion group $G$ into the direct sum of its $p$-components also is a
decomposition of any ring on $G$ into a direct sum of ideals, it is easy to
obtain a description of fully transitive torsion $afi$-groups. The same result
was obtained in [21], and the methods used in the proof are essentially
different from the methods in [20]. Mixed $afi$-groups are studied in [22] and
[13] and torsion-free $afi$-groups are studied in [14].

In the study of absolute ideals, principal absolute ideals introduced in [21]
play an important role. A principal absolute ideal of the Abelian group $G$
generated by an element $g\in G$ is the least absolute ideal $\langle
g\rangle_{AI}$ of the group $G$ containing $g$. Since any absolute ideal $K$
is a sum of principal absolute ideals, $K=\sum\limits_{g\in K}\langle
g\rangle_{AI}$, the study of the properties of arbitrary absolute ideals is
often reduced to the case of principal absolute ideals.

This paper is devoted to questions related to absolute ideals of almost
completely decomposable Abelian groups. The paper is a continuation of the
papers [15] and [16], where the properties of the group $\text{Mult}\,G$ of
all multiplications of an almost completely decomposable group $G$ were
studied.

The paper considers only additively written Abelian groups and the word
<<group>> everywhere means <<abelian group>>.

A torsion-free group $G$ of finite rank is called an almost completely
decomposable group (an $ACD$-group) if $G$ contains a completely decomposable
subgroup of finite index. $ACD$-groups are studied in [1], [2], [3], [4], [6],
[12], [15], [16], [18] and other papers. Every $ACD$-group $G$ contains a
completely decomposable fully invariant subgroup $\text{Reg}\,G$ of finite
index which is called the regulator of the group $G$. The factor group
$G/\text{Reg}\,G$ is called the regulator quotient and the index $n(G)$ of the
subgroup $\text{Reg}\,G$ in the group $G$ is called the regulator index of the
group $G$. $ACD$-groups with cyclic regulator quotient are called
$CRQ$-groups.

Let $G$ be an almost completely decomposable group. Then the group
$\text{Reg}\,G$ is uniquely, up to isomorphism, represented as a direct sum of
torsion-free groups of rank $1$ [10, Proposition 86.1]. For every type $\tau$,
we denote by $\text{Reg}_{\tau}\,G$ the sum of direct summands of rank 1 and
of type $\tau$ in this direct decomposition of the group $\text{Reg}\,G$. This
sum is called the $\tau$-homogeneous component of the group $G$.

The set of types

$T(G)=T(\text{Reg}\,G)=\\{\tau\,|\,\text{Reg}_{\tau}\,G\neq 0\\}$

is called the set of critical types of the groups $G$ and $\text{Reg}\,G$. If
$T(G)$ consists of pairwise incomparable types, then the groups $G$ and
$\text{Reg}\,G$ are called block-rigid groups. In this case, if for any
$\tau\in T(G)$, the group $\text{Reg}_{\tau}G$ is of rank $1$, then $G$ and
$\text{Reg}\,G$ are called rigid groups. If all types in $T(G)$ are idempotent
types, then $G$ is called a group of ring type.

We denote by $\mathcal{A}_{0}$ the class of all reduced block-rigid
$CRQ$-groups of ring type. In Section 2, we describe principal absolute ideals
of groups in $\mathcal{A}_{0}$ (Theorem 2.4). This allows us to prove in
Section 3 that any group in $\mathcal{A}_{0}$ is an $afi$-group (Theorem 3.2).

A multiplication $\mu\colon G\otimes G\rightarrow G$ is often denoted by
$\times$, i.e., $\mu(g_{1}\otimes g_{2})=g_{1}\times g_{2}$ for all
$g_{1},g_{2}\in G$. The ring on the group $G$, defined by this multiplication,
is denoted by $(G,\times)$. Let $G$ be a group, $g\in G$, and let $(G,\times)$
be a ring on $G$. The rank of the group $G$ is denoted by $r(G)$, the
divisible hull of $G$ is denoted by $\tilde{G}$. The endomorphism group and
the endomorphism ring of the group $G$ are denoted by $\text{End}\,G$ and
$E(G)$, respectively. We denote by $(g)_{\times}$ the principal ideal of the
ring $(G,\times)$ generated by the element $g$. If $G$ is a completely
decomposable group and $\tau$ is a type, then $G_{\tau}$ is the
$\tau$-homogeneous component of the group $G$ and $\tau(p)$ is the component
of the type $\tau$ corresponding to the integer $p$. If $S\subseteq G$, then
$\langle S\rangle$ and $\langle S\rangle_{*}$ are the subgroup and the pure
subgroup of the group $G$ generated by the set $S$, respectively. As usual,
$\mathbb{N}$ and $P$ are the set of all positive integers and the set of all
prime integers, respectively, $\mathbb{Z}$ is the ring of integers and
$\mathbb{Q}$ is the field of rational numbers. If $S$ is a finite subset in
$\mathbb{Z}$, then $\text{gcd}(S)$ is the greatest common divisor of all
integers in $S$ and $\text{lcm}(A)$ is the the least common multiple of all
integers in $S$.

For any type $\tau$, we set

$P_{\infty}(\tau)=\\{p\in P\,|\,\tau(p)=\infty\\},\quad P_{0}(\tau)=P\setminus
P_{\infty}(\tau).$

If $P_{1}\subseteq P$, then a $P_{1}$-number is a non-zero integer such that
any its prime factor (if it exists) is contained in $P_{1}$. If $R$ is a
unital ring, then $Re$ is the cyclic $R$-module generated by the element $e$.
If $\tau$ is an idempotent type, then $R_{\tau}$ is a unital subring of the
field $\mathbb{Q}$ such that the additive group of $R_{\tau}$ is of type
$\tau$. For notation and definitions, unless otherwise specified, we refer to
[11] and [17].

##  2 Principal Absolute Ideals of $CRQ$-Groups

In this section, we describe principal absolute ideals of groups in the class
$\mathcal{A}_{0}$.

In [6], positive integers $m_{\tau}=m_{\tau}(G)$ are defined for a group
$G\in\mathcal{A}_{0}$, and it is proved that the set $\\{m_{\tau}\,|\,\tau\in
T(G)\\}$ is an invariant system of almost isomorphism of the group $G$. In
addition, $n(G)=\text{lcm}\\{m_{\tau}\,|\,\tau\in T(G)\\}$. In [4, Theorem
3.5], it is proved that for any group $G\in\mathcal{A}_{0}$, there exists a
direct decomposition

$G=G_{1}\oplus C,$ $None$

where $C$ is a block-rigid completely decomposable group and $G^{\prime}$ is a
rigid $CRQ$-group satisfying the following conditions:

$\tau\in T(G_{1})\text{ if and only if }m_{\tau}(G)>1,$ $None$
$m_{\tau}(G_{1})=m_{\tau}(G)\text{ for all }\tau\in T(G^{\prime}).$ $None$

Decomposition $(2.1)$, satisfying conditions $(2.1^{\prime})$ and
$(2.1^{\prime\prime})$, is called a principal decomposition of the group $G$.
In a principal decomposition of $G$, the group $G^{\prime}$ does not contain
completely decomposable direct summands; such groups are said to be clipped.
We remark that a principal decomposition of a $CRQ$-group is not uniquely
defined [4]. Let $\text{Reg}\,G_{1}=B$. Then $T(G_{1})=T(B)=\\{\tau\in
T(G)\,|\,m_{\tau}>1\\}$ and $\tilde{G}_{1}=\tilde{B}$.

Let $d$ be an element of $G_{1}$ such that $G/\text{Reg}\,G=\langle
d+\text{Reg}\,G\rangle$. Then there exists a system
$E_{0}=\\{e_{0}^{(\tau)}\in B_{\tau}\,|\,\tau\in T(B)\\}$ such that

$B=\oplus_{\tau\in T(B)}R_{\tau}e_{0}^{(\tau)},$ $None$

the element $d$ of the group $\tilde{B}$ can be represented in the form

$d=\sum_{\tau\in T(B)}\dfrac{s_{\tau}}{m_{\tau}}e_{0}^{(\tau)},$ $None$

where $m_{\tau}=m_{\tau}(G)$, $s_{\tau}\in\mathbb{Z}$, and the following
conditions hold:

$\text{gcd}(s_{\tau},m_{\tau})=1\text{ for all }\tau\in T(B).$ $None$
$s_{\tau}\text{ and }m_{\tau}\text{ are }P_{0}(\tau)\text{-numbers for every
}\tau\in T(B).$ $None$

A system $E_{0}=\\{e_{0}^{(\tau)}\in B_{\tau}\,|\,\tau\in T(B)\\}$, satisfying
conditions $(2.2)$ and $(2.3)$, is called an $rc$-basis of the group $G$
defined by the element $d$. We remark that the pair $(d,E_{0})$ uniquely
determines the numbers $s_{\tau}$ ($\tau\in T(B)$). Equality $(2.3)$ is called
a
standard representation of a block-rigid $CRQ$-group $G$ related to the pair
$(d,E_{0})$.

In its turn, the group $C$ is of the form $C=\oplus_{\tau\in T(C)}C_{\tau}$,
where $C_{\tau}$ can be represented in the form $C_{\tau}=\oplus_{i\in
I_{\tau}(C)}R_{\tau}e_{i}^{(\tau)}$ and $I_{\tau}(C)$ is a non-empty finite
subset in $\mathbb{N}$ for $\tau\in T(C)$. We set $I_{\tau}(B)=\\{0\\}$ for
$\tau\in T(B)$. We also set $I_{\tau}(B)=\varnothing$ for $\tau\notin T(B)$
and $I_{\tau}(C)=\varnothing$ for $\tau\notin T(C)$. For any $\tau\in T(G)$,
we set $I_{\tau}=I_{\tau}(B)\cup I_{\tau}(C)$. A system
$E=\\{e_{i}^{(\tau)}\,|\,\tau\in T(G),\;i\in I_{\tau}\\}$ is called an
$r$-basis of the group $G$ if its subsystem
$E_{0}=\\{e_{0}^{(\tau)}\,|\,\tau\in T(B)\\}$ is an $rc$-basis of the group
$G$.

We set $\text{Reg}\,G=A$. Then $A_{\tau}=B_{\tau}\oplus C_{\tau}$ and
$A=\oplus_{\tau\in T(G)}A_{\tau}$. According to [18, Proposition 2.4.11], such
a decomposition of a completely decomposable group is unique if and only if
$A$ is a block-rigid group. Let $\tilde{G}$, $\tilde{A}$ and
$\tilde{A}_{\tau}$ be divisible hulls of the groups $G$, $A$ and $A_{\tau}$,
respectively. Then we have

$\tilde{G}=\tilde{A}=\oplus_{\tau\in T(G)}\tilde{A}_{\tau}.$

For $\tau\in T(G)$, we denote by $\pi_{\tau}$ the natural projection of the
group $\tilde{G}$ on $\tilde{A}_{\tau}$.

In what follows, we fix a principal decomposition and an $r$-basis of the
group $G$.

Let $G\in\mathcal{A}_{0}$, $\text{Reg}\,G=A$, $E=\\{e_{i}^{(\tau)}\,|\,\tau\in
T(G),\;i\in I_{\tau}\\}$ be an $r$-basis of the group $G$, and let $G$ have
standard representation $(2.3)$. Since $G=\langle d,A\rangle$, any element of
$G$ is of the form

$g=\sum_{\tau\in T(B)}\dfrac{r_{\tau}}{m_{\tau}}e_{0}^{(\tau)}+\sum_{\tau\in
T(C)}\sum_{i\in I_{\tau}(C)}a_{i}^{(\tau)}e_{i}^{(\tau)}\in G,$

where $r_{\tau},a_{i}^{(\tau)}\in R_{\tau}$ for $\tau\in T(G)$, $i\in
I_{\tau}(C)$.

Remark 2.1. Any set of elements $\\{u_{ij}^{(\tau)}\in A_{\tau}\,|\,\tau\in
T(G),\;i,j\in I_{\tau}\\}$ determines a multiplication $\times$ on $A$ such
that $e_{i}^{(\tau)}\times e_{j}^{(\tau)}=u_{ij}^{(\tau)}$ for any $\tau\in
T(G)$ and each $i,j\in I_{\tau}$; in addition, $e_{i}^{(\tau)}\times
e_{j}^{(\sigma)}=0$ for $\tau\neq\sigma$ and any $i\in I_{\tau}$, $j\in
I_{\sigma}$. According to [16, Theorem 2.4], this multiplication can be
extended to a multiplication on $G$ if and only if there exists
$\alpha\in\mathbb{Z}$ such that

$u_{i0}^{(\tau)}=m_{\tau}v_{i0}^{(\tau)},\;u_{0i}^{(\tau)}=m_{\tau}v_{0i}^{(\tau)},\;v_{00}^{(\tau)}=\alpha
s_{\tau}^{-1}e_{0}+m_{\tau}a_{00}^{(\tau)}$ $None$

for some $v_{i0}^{(\tau)},v_{0i}^{(\tau)},a_{00}^{(\tau)}\in A_{\tau}$. Here
$s_{\tau}^{-1}$ is an integer which is converse to $s_{\tau}$ modulo
$m_{\tau}$; such an integer exists by $(2.3^{\prime})$.

If a ring $(G,\times)$ with multiplication $\times$ satisfies condition
$(2.5)$, then we say that the ring corresponds to the integer $\alpha$ with
respect to the pair $(d,E_{0})$. The multiplication, corresponding to the
integer $\alpha$, can be described if we consider it as an element of the
group $\text{Mult }G$ of all multiplications of the group $G$. Let
$M=\text{Mult }G$. In [16], it is proved that the group $M$ also belongs to
the class $\mathcal{A}_{0}$ and it can be represented in the form $M=\langle
X,\text{Reg }M\rangle$, where $X=X(d,E_{0})$ is an element of the group $M$
depending on the pair $(d,E_{0})$. The multiplication $\times$ on the group
$G$ corresponds to $\alpha\in\mathbb{Z}$ with respect to the pair $(d,E_{0})$
if and only if $\times$ belongs to the class $\alpha X+\text{Reg }M$.

The principal absolute ideal of the group $G$ generated by an element $g$ is
the least absolute ideal $\langle g\rangle_{AI}$ of the group $G$ containing
$g$. In [8], it is defined the subgroup
$M(G)=\langle\text{Im}\,\Psi\,|\,\Psi\in\text{Hom}\,(G,\text{End}\,G)\rangle$
of the group $\text{End}\,G$ and it is proved that $M(G)$ is an ideal of the
endomorphism ring $E(G)$. We set $M(G)(g)=\langle\varphi(g)\,|\,\varphi\in
M(G)\rangle$ for $g\in G$.

To describe principal absolute ideals of groups, we need the following lemma
in [22].

Lemma 2.2 [22]. Let $G$ be a group and $g\in G$. Then

1) $M(G)(g)=\langle g\times x\,|\,\times\in\text{Mult}\,(G),\,x\in G\rangle$,

2) $\langle g\rangle_{AI}=\langle g\rangle+M(G)(g)$. $\blacktriangleright$

Remark 2.3. If $\tau$ is a type and $a\in R_{\tau}\setminus\\{0\\}$, then $a$
can be uniquely written in the form $a=\overline{a}y$, where $\overline{a}$ is
a positive $P_{0}(\tau)$-number and $y$ is a $P_{\infty}(\tau)$-fraction. We
define the number $\overline{a}$ for every $a\in R_{\tau}$, by setting
$\overline{0}=0$.

Theorem 2.4. Let $G$ be a block-rigid $CRQ$-group of ring type with $r$-basis
$E=\\{e_{i}^{(\tau)}\,|\,\tau\in T(G),\;i\in I_{\tau}\\}$, standard
representation $(2.3)$ and $\text{Reg }G=A$. Let

$g=\sum_{\tau\in T(B)}\dfrac{r_{\tau}}{m_{\tau}}e_{0}^{(\tau)}+\sum_{\tau\in
T(C)}\sum_{i\in I_{\tau}(C)}a_{i}^{(\tau)}e_{i}^{(\tau)}\in G,$

where $r_{\tau},a_{i}^{(\tau)}\in R_{\tau}$ for $\tau\in T(G)$, $i\in
I_{\tau}(C)$. By setting $r_{\tau}=0$ for $\tau\notin T(B)$ and
$a_{i}^{(\tau)}=0$ for $\tau\notin T(C)$ (in this case, we have
$I_{\tau}(C)=\varnothing$), we set

$\ell_{\tau}(g)=\text{gcd}\,(\overline{r_{\tau}},\\{\overline{a_{i}^{(\tau)}}\,|\,i\in
I_{\tau}(C)\\})\quad\text{ for }\,\tau\in T(G).$

Then

$\langle g\rangle_{AI}=\langle g\rangle+\oplus_{\tau\in
T(B)}r_{\tau}A_{\tau}+\oplus_{\tau\in T(C)}\oplus_{i\in
I_{\tau}(C)}a_{i}^{(\tau)}A_{\tau}=$ $=\langle g\rangle+\oplus_{\tau\in
T(G)}\ell_{\tau}(g)A_{\tau}.$

Remark. We can prove that for any $g\in G$ and every $\tau\in T(G)$, the
number $\ell_{\tau}(g)$ does not depend on the choice of an $r$-basis $E$ and
a standard representation of the group $G$.

$\blacktriangleleft$ Let $\tau\in T(G)$. We set

$L_{\tau}=r_{\tau}A_{\tau}+\oplus_{i\in
I_{\tau}(C)}a_{i}^{(\tau)}A_{\tau},\qquad L=\oplus_{\tau\in T(G)}L_{\tau},$

We prove that $\langle g\rangle_{AI}=\langle g\rangle+L$.

First, we prove that $M(G)(g)\subseteq\langle g\rangle+L$. Let $(G,\times)$ be
a ring. Then there exists a ring $(\tilde{G},\times)$ on the divisible hull
$\tilde{G}=\tilde{A}$ of the group $G$ and $G$ is a subring in
$(\tilde{G},\times)$, [11, Chapter 18, Theorem 1.3]. In the ring $(G,\times)$,
we set $e_{i}^{(\tau)}\times e_{j}^{(\tau)}=u_{ij}^{(\tau)}\in A_{\tau}$ for
all $\tau\in T(G)$ and $i,j\in I_{\tau}$. It follows from Remark 2.1 that
there exists an integer $\alpha$ such that for any $\tau\in T(B)$ and each
$i\in I_{\tau}$, we have

$u_{i0}^{(\tau)}=m_{\tau}v_{i0}^{(\tau)},\;u_{0i}^{(\tau)}=m_{\tau}v_{0i}^{(\tau)},\;v_{00}^{(\tau)}=\alpha
s_{\tau}^{-1}e_{0}+m_{\tau}a_{00}^{(\tau)}$

for some $v_{i0}^{(\tau)},v_{0i}^{(\tau)},a_{00}^{(\tau)}\in A_{\tau}$. Here
$s_{\tau}^{-1}$ is an integer such that the relation
$s_{\tau}s_{\tau}^{-1}=1+m_{\tau}y_{\tau}$ holds for some
$y_{\tau}\in\mathbb{Z}$.

Let $\tau\in T(G)$ and $k\in I_{\tau}$. If $\tau\notin T(B)$, then

$g\times e_{k}^{(\tau)}=\pi_{\tau}(g)\times e_{k}^{(\tau)}=\left(\sum_{i\in
I_{\tau}}a_{i}^{(\tau)}e_{i}^{(\tau)}\right)\times e_{k}^{(\tau)}=\sum_{i\in
I_{\tau}(C)}a_{i}^{(\tau)}u_{ik}^{(\tau)}\in L_{\tau}.$

In addition, $\pi_{\tau}(g\times d)=0$.

Let $\tau\in T(B)$. In the ring $(\tilde{G},\times)$, we have

$g\times e_{k}^{(\tau)}=\pi_{\tau}(g)\times
e_{k}^{(\tau)}=\left(\dfrac{r_{\tau}}{m_{\tau}}e_{0}^{(\tau)}+\sum_{i\in
I_{\tau}(C)}a_{i}^{(\tau)}e_{i}^{(\tau)}\right)\times e_{k}^{(\tau)}=$
$=\dfrac{r_{\tau}}{m_{\tau}}u_{0k}^{(\tau)}+\sum_{i\in
I_{\tau}(C)}a_{i}^{(\tau)}u_{ik}^{(\tau)}=r_{\tau}v_{0k}^{(\tau)}+\sum_{i\in
I_{\tau}(C)}a_{i}^{(\tau)}u_{ik}^{(\tau)}\in L_{\tau}.$

In addition,

$\pi_{\tau}(g\times
d)=\pi_{\tau}(g)\times\pi_{\tau}(d)=\left(\dfrac{r_{\tau}}{m_{\tau}}e_{0}^{(\tau)}+\sum_{i\in
I_{\tau}(C)}a_{i}^{(\tau)}e_{i}^{(\tau)}\right)\times\dfrac{s_{\tau}}{m_{\tau}}e_{0}^{(\tau)}=$
$=\dfrac{r_{\tau}s_{\tau}}{m_{\tau}^{2}}u_{00}^{(\tau)}+\sum_{i\in
I_{\tau}(C)}\dfrac{a_{i}^{(\tau)}s_{\tau}}{m_{\tau}}u_{i0}^{(\tau)}=\dfrac{r_{\tau}s_{\tau}\alpha
s_{\tau}^{-1}}{m_{\tau}}e_{0}^{(\tau)}+r_{\tau}s_{\tau}a_{00}^{(\tau)}+\sum_{i\in
I_{\tau}(C)}a_{i}^{(\tau)}s_{\tau}v_{i0}^{(\tau)}=$
$=\alpha\dfrac{r_{\tau}}{m_{\tau}}(1+m_{\tau}y_{\tau})e_{0}^{(\tau)}+r_{\tau}s_{\tau}a_{00}^{(\tau)}+\sum_{i\in
I_{\tau}(C)}a_{i}^{(\tau)}s_{\tau}v_{i0}^{(\tau)}=$
$=\alpha\dfrac{r_{\tau}}{m_{\tau}}e_{0}^{(\tau)}+r_{\tau}(\alpha
y_{\tau}e_{0}^{(\tau)}+s_{\tau}a_{00}^{(\tau)})+\sum_{i\in
I_{\tau}(C)}a_{i}^{(\tau)}s_{\tau}v_{i0}^{(\tau)}=$
$=\alpha\left(\dfrac{r_{\tau}}{m_{\tau}}e_{0}+\sum_{i\in
I_{\tau}(C)}a_{i}^{(\tau)}e_{i}^{(\tau)}\right)-\alpha\sum_{i\in
I_{\tau}(C)}a_{i}^{(\tau)}e_{i}^{(\tau)}+r_{\tau}b_{\tau}+\sum_{i\in
I_{\tau}(C)}a_{i}^{(\tau)}s_{\tau}v_{i0}^{(\tau)}=$ $=\pi_{\tau}(\alpha
g)+r_{\tau}b_{\tau}+\sum_{i\in
I_{\tau}(C)}a_{i}(s_{\tau}v_{i0}^{(\tau)}-\alpha e_{i}^{(\tau)})=$
$=\pi_{\tau}(\alpha g)+r_{\tau}b_{\tau}+\sum_{i\in
I_{\tau}(C)}a_{i}c_{i}^{(\tau)},$

where $b_{\tau}=\alpha y_{\tau}e_{0}^{(\tau)}+s_{\tau}a_{00}^{(\tau)}\in
A_{\tau}$, $c_{i}^{(\tau)}=s_{\tau}v_{i0}^{(\tau)}-\alpha e_{i}^{(\tau)}\in
A_{\tau}$ for $i\in I_{\tau}(C)$. Therefore, $\pi_{\tau}(g\times
d)\in\pi_{\tau}(\alpha g)+L_{\tau}$, whence $g\times d\in\langle g\rangle+L$.

Since any element $x\in G$ can be represented in the form

$x=\beta d+\sum_{\tau\in T(G)}\sum_{k\in
I_{\tau}}x_{k}^{(\tau)}e_{k}^{(\tau)},$

where $\beta\in\mathbb{Z}$, $x_{k}^{(\tau)}\in R_{\tau}$, we have that
$g\times x\in\langle g\rangle+L$ for any $x\in G$. Since the multiplication
$\times$ is arbitrary, $M(G)(g)\subseteq\langle g\rangle+L$, whence

$\langle g\rangle_{AI}=\langle g\rangle+M(G)(g)\subseteq\langle g\rangle+L,$

by Lemma 2.2.

Now we prove that $L\subseteq\langle g\rangle_{AI}$. Let $\tau\in T(G)$. We
prove that $L_{\tau}\subseteq\langle g\rangle_{AI}$. Let $k\in I_{\tau}$,
$b\in R_{\tau}$.

First, we assume that $\tau\in T(B)$ and prove that
$r_{\tau}be_{k}^{(\tau)}\in\langle g\rangle_{AI}$.

Case 1. $\tau\in T(B)\cap T(C)$.

We fix $t\in I_{\tau}(C)$. According to Remark 2.1, relations

$e_{0}^{(\tau)}\times e_{t}^{(\tau)}=e_{t}^{(\tau)}\times
e_{0}^{(\tau)}=m_{\tau}e_{k}^{(\tau)},$

$e_{i}^{(\tau)}\times e_{j}^{(\sigma)}=0$ if $\sigma\neq\tau$ or
$(i,j)\neq(0,t)$, $(i,j)\neq(t,0)$,

define a multiplication $\times$ on $G$. We consider the ring $(G,\times)$ as
a subring of the ring $(\tilde{G},\times)$ [11, Chapter 18, Theorem 1.3] and
obtain

$g\times be_{t}^{(\tau)}=\dfrac{r_{\tau}}{m_{\tau}}e_{0}^{(\tau)}\times
be_{t}^{(\tau)}=r_{\tau}be_{k}^{(\tau)}\in\langle
g\rangle_{\times}\subseteq\langle g\rangle_{AI}.$

Case 2. $\tau\in T(B)\setminus T(C)$.

In this case, we have $e_{k}^{(\tau)}=e_{0}^{(\tau)}$. For any $\sigma\in
T(B)$, there exist $s_{\sigma}^{-1}$ and $y_{\sigma}\in\mathbb{Z}$ such that
$s_{\sigma}s_{\sigma}^{-1}+m_{\sigma}y_{\sigma}=1$. We set

$e_{0}^{(\sigma)}\times
e_{0}^{(\sigma)}=(s_{\tau}s_{\sigma}^{-1}m_{\tau}+m_{\sigma}^{2}y_{\sigma})e_{0}^{(\sigma)}\,\text{if}\,\sigma\in
T(B),$ $e_{i}^{(\sigma_{1})}\times
e_{j}^{(\sigma_{2})}=0\,\text{if}\,\sigma_{1}\neq\sigma_{2}\,\text{or}\,(i,j)\neq(0,0).$

It follows from Remark 2.1 that these relations define a multiplication
$\times$ on $G$. Then we consider the ring $(G,\times)$ as a subring of the
ring $(\tilde{G},\times)$ and obtain

$g\times be_{0}^{(\tau)}=\dfrac{r_{\tau}}{m_{\tau}}e_{0}^{(\tau)}\times
be_{0}^{(\tau)}=r_{\tau}b(s_{\tau}s_{\tau}^{-1}+m_{\tau}v_{\tau})e_{0}^{(\tau)}=r_{\tau}be_{0}^{(\tau)}\in\langle
g\rangle_{AI}.$

Consequently,

$r_{\tau}A_{\tau}\subseteq\langle g\rangle_{AI}\;\text{for any }\,\tau\in
T(B).$ $None$

Now let $\tau\in T(C)$, $i\in I_{\tau}(C)$. We prove that
$a_{i}^{(\tau)}be_{k}^{(\tau)}\in\langle g\rangle_{AI}$. It follows from
Remark 2.1 that the relations

$\begin{cases}e_{i}^{(\tau)}\times e_{i}^{(\tau)}=e_{k}^{(\tau)},\\\
e_{j}^{(\sigma_{1})}\times
e_{s}^{(\sigma_{2})}=0\qquad\text{if}\,(\sigma_{1},\sigma_{2})\neq(\tau,\tau)\,\text{or}\,(j,s)\neq(i,i),\end{cases}$

define the ring $({G},\times)$. In this ring, we have

$g\times be_{i}^{(\tau)}=a_{i}^{(\tau)}e_{i}^{(\tau)}\times
be_{i}^{(\tau)}=a_{i}^{(\tau)}be_{k}^{(\tau)}\in\langle g\rangle_{AI}.$

Consequently,

$a_{i}^{(\tau)}A_{\tau}\subseteq\langle g\rangle_{AI}\,\text{ for all }\tau\in
T(C),\,i\in I_{\tau}(C).$ $None$

It follows from $(2.5)$ and $(2.6)$ that $L_{\tau}\subseteq\langle
g\rangle_{AI}$ for all $\tau\in T(G)$. Therefore, $L=\oplus_{\tau\in
T(G)}L_{\tau}\subseteq\langle g\rangle_{AI}$, whence $\langle
g\rangle+L\subseteq\langle g\rangle_{AI}$.

Thus, we have proved that

$\langle g\rangle_{AI}=\langle g\rangle+L.$ $None$

Now we prove that if $\tau\in T(G),\,\\{b_{i},\,|\,i\in I_{\tau}\\}\subseteq
R_{\tau}$, $\ell=\text{gcd}\,(\overline{b_{i}}\,|\,i\in I_{\tau})$, then

$\sum_{i\in I_{\tau}}b_{i}A_{\tau}=\ell A_{\tau}.$ $None$

It follows from Remark 2.3 that we can represent the elements $b_{i}$ in the
form $b_{i}=\overline{b_{i}}c_{i}$, where $c_{i}$ is an invertible element in
$R_{\tau}$, $\overline{b_{i}}=\ell b_{i}^{\prime}$ and
$b_{i}^{\prime}\in\mathbb{Z}$.

Let $x=\sum_{i\in I_{\tau}}b_{i}x_{i}\in\sum_{i\in I_{\tau}}b_{i}A_{\tau}$,
where $x_{i}\in A_{\tau}$. Then

$x=\sum_{i\in I_{\tau}}b_{i}x_{i}=\sum_{i\in
I_{\tau}}\overline{b_{i}}(c_{i}x_{i})=\ell\sum_{i\in
I_{\tau}}b_{i}^{\prime}\left(c_{i}x_{i}\right)\in\ell A_{\tau}.$

Conversely, let $\ell x\in\ell A_{\tau}$, where $x\in A_{\tau}$. Since
$\ell=\text{gcd}\,(\overline{b_{i}}\,|\,i\in I_{\tau})$, we can represent
$\ell$ in the form $\ell=\sum_{i\in I_{\tau}}y_{i}\overline{b_{i}}$ for some
$y_{i}\in\mathbb{Z}$. Therefore,

$\ell x=\left(\sum_{i\in
I_{\tau}}y_{i}\overline{b_{i}}\right)x=\left(\sum_{i\in
I_{\tau}}y_{i}b_{i}c_{i}^{-1}\right)x=\sum_{i\in
I_{\tau}}b_{i}(y_{i}c_{i}^{-1}x)\in\sum_{i\in I_{\tau}}b_{i}A_{\tau},$

where $c_{i}^{-1}$ is the element which is converse to $c_{i}$ in the ring
$R_{\tau}$. Consequently, $\sum_{i\in I_{\tau}}b_{i}A_{\tau}=\ell A_{\tau}$.

It follows from $(2.7)$ and $(2.8)$ that $L_{\tau}=\ell_{\tau}(g)A_{\tau}$ for
all $\tau\in T(G)$, whence

$\langle g\rangle_{AI}=\langle g\rangle+L=\langle g\rangle+\oplus_{\tau\in
T(G)}L_{\tau}=\langle g\rangle+\oplus_{\tau\in
T(G)}\ell_{\tau}(g)A_{\tau}.\quad\blacktriangleright$

##  3 $afi$-Groups in the Class of $CRQ$-Groups

The description of the principal absolute ideals allows us to solve the
problem of describing $afi$-groups in the class $\mathcal{A}_{0}$. We recall
that an $afi$-group is a group which has no absolute ideals other than fully
invariant subgroups.

Lemma 3.1 [21]. A group $G$ is an $afi$-group if and only if $\langle
g\rangle_{AI}$ is a fully invariant subgroup of the group $G$ for any $g\in
G$.

Theorem 3.2. Any block-rigid $CRQ$-group of ring type is an $afi$-group.

$\blacktriangleleft$ Let $G$ be a block-rigid $CRQ$-group of ring type with
standard representation $(2.3)$. Let $g_{i}\in G$,

$g=\sum_{\tau\in T(B)}\dfrac{r_{\tau}}{m_{\tau}}e_{0}^{(\tau)}+\sum_{\tau\in
T(C)}\sum_{i\in I_{\tau}(C)}a_{i}^{(\tau)}e_{i}^{(\tau)}\in
G,\;(r_{\tau},a_{i}^{(\tau)}\in R_{\tau}).$

By Theorem 2.4, we have $\langle g\rangle_{AI}=\langle g\rangle+L$, where

$L=\oplus_{\tau\in T(G)}L_{\tau},\quad
L_{\tau}=\ell_{\tau}(g)A_{\tau}=r_{\tau}A_{\tau}+\sum_{i\in
I_{\tau}(C)}a_{i}^{(\tau)}A_{\tau}.$

It is easy to see that $L$ is a fully invariant subgroup of the group $G$.
Therefore, it is sufficient to prove that $\varphi(g)\in\langle g\rangle+L$
for any $\varphi\in\text{End}\,G$.

Let $\varphi\in\text{End}\,G$. Then

$\varphi(g)=\varphi\left(\sum_{\tau\in
T(B)}\dfrac{r_{\tau}}{m_{\tau}}e_{0}^{(\tau)}+\sum_{\tau\in T(C)}\sum_{i\in
I_{\tau}(C)}a_{i}^{(\tau)}e_{i}^{(\tau)}\right)=$ $=\sum_{\tau\in
T(B)}\dfrac{r_{\tau}}{m_{\tau}}\varphi(e_{0}^{(\tau)})+\sum_{\tau\in
T(C)}\sum_{i\in I_{\tau}(C)}a_{i}^{(\tau)}\varphi(e_{i}^{(\tau)}).$ $None$

Since $\varphi(e_{i}^{(\tau)})\in A_{\tau}$ for all $\tau\in T(C)$ and $i\in
I_{\tau}(C)$, we have that

$\sum_{\tau\in T(C)}\sum_{i\in
I_{\tau}(C)}a_{i}^{(\tau)}\varphi(e_{i}^{(\tau)})\in L.$ $None$

It follows from [3, Theorem 4.5] that there exists an $\alpha\in\mathbb{Z}$
such that for any $\tau\in T(B)$, the element $\varphi(e_{0}^{(\tau)})$ is of
the form

$\varphi(e_{0}^{(\tau)})=(\alpha+m_{\tau}y_{0}^{(\tau)})e_{0}^{(\tau)}+\sum_{i\in
I_{\tau}(C)}m_{\tau}y_{i}^{(\tau)}e_{i}^{(\tau)},\;\text{where}\;y_{i}^{(\tau)}\in
R_{\tau}\;(i\in I_{\tau}).$

Consequently,

$\sum_{\tau\in
T(B)}\dfrac{r_{\tau}}{m_{\tau}}\varphi(e_{0}^{(\tau)})=\sum_{\tau\in
T(B)}\dfrac{r_{\tau}}{m_{\tau}}(\alpha+m_{\tau}y_{0}^{(\tau)})e_{0}^{(\tau)}+\sum_{\tau\in
T(B)}\sum_{i\in I_{\tau}(C)}r_{\tau}y_{i}^{(\tau)}e_{i}^{(\tau)}=$
$=\alpha\left(\sum_{\tau\in
T(B)}\dfrac{r_{\tau}}{m_{\tau}}e_{0}^{(\tau)}+\sum_{\tau\in T(C)}\sum_{i\in
I_{\tau}(C)}a_{i}^{(\tau)}e_{i}^{(\tau)}\right)-\sum_{\tau\in T(C)}\sum_{i\in
I_{\tau}(C)}a_{i}^{(\tau)}\alpha e_{i}^{(\tau)}+$ $+\sum_{\tau\in
T(B)}r_{\tau}y_{0}^{(\tau)}e_{0}^{(\tau)}+\sum_{\tau\in T(B)}\sum_{i\in
I_{\tau}(C)}r_{\tau}y_{i}^{(\tau)}e_{i}^{(\tau)}=$ $=\alpha g-\sum_{\tau\in
TCB)}\sum_{i\in I_{\tau}(C)}a_{i}^{(\tau)}\alpha e_{i}^{(\tau)}+\sum_{\tau\in
T(B)}r_{\tau}\left(\sum_{i\in
I_{\tau}}y_{i}^{(\tau)}e_{i}^{(\tau)}\right)\in\langle g\rangle+L.$

It follows from $(3.1)$ and $(3.2)$ that $\varphi(g)\in\langle g\rangle_{AI}$.
Since $L$ is a fully invariant subgroup of the group $G$, we have that
$\langle g\rangle_{AI}$ is a fully invariant subgroup of the group $G$, as
well. It follows from Lemma 3.1 that $G$ is an $afi$-group.
$\blacktriangleright$

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