# Robustness Evaluation of the Butterfly Optimization Algorithm on a Control
System

Ramadan Abdul-Rashid and Basit Olakunle Alawode
Electrical Engineering Department, King Fahd University of Petroleum and
Minerals, Dhahran, Saudi Arabia
Email<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this paper, the Butterfly Optimization Algorithm (BOA) proposed by [1] is
adopted to optimize the parameters of a designed Lead-Lad Controller so as to
obtain a stabilized control system. Numerical analysis was carried out for BOA
on the control problem and the results are compared to those obtained from the
well known Genetic Algorithm (GA) and Differential Evolution (DE) Algorithm.
BOA performs better in terms of eigenvalue analysis but similar to GA and DE
in terms of optimizing the minimum damping coefficient for the control system.

##  I Introduction

Recent engineering requirements has led to the development of several
metaheuristic algorithms that mimic natural phenomenon in order to search for
an optimal solution. These algorithms adopt different natural phenomena from
the human nervous system used to derive Neural Networks (NN), to the theories
of evolutions used to design evolutionary algorithms like the Genetic
Algorithm (GA) and the swarming nature of birds, and colonial organisms such
as bees used to design the infamous Particle Swarm Optimization algorithm.
Owing to the intuitive nature of these algorithms, and their effectiveness in
reaching optimal solutions even in non-linear, complex problems, more
algorithms continue to be developed based on unexplored natural phenomena and
behavior of living organisms. In this paper, we adopt the novel optimization
paradigm proposed by Arora et al [1] that impersonates the sustenance
searching conduct of butterflies named the Butterfly Optimization Algorithm
(BOA). This algorithm been relatively new presents promising results on on a
set of 30 benchmark test functions and some classical mechanical engineering
problems. Despite it performance, to the best of our knowledge, this new
paradigm of optimization has not been used to optimize a controller design in
a typical control problem. This work seeks to evaluate the robustness of the
BOA procedure on a Lead-Lag Controller (LLC) design in other to achieve a
desirable frequency response.

The rest of the paper is organized as follows, Section II expounds on the
method of operation and relevant system parameters of the BOA. Section III
defines the control system used for the evaluation and formulate the
optimization problem. Section IV presents the simulation evaluation of BOA
against GA and PSO for the design of an optimal LLC controller. The paper is
then concluded in Section V.

##  II Butterfly Optimization Algorithm

To comprehend the BOA, some natural certainties and step by step instructions
to display them in BOA are examined in following subsections.

###  II-A Butterfly

In view of logical perceptions, it is discovered that butterflies have an
extremely precise feeling of finding the wellspring of scent. Moreover, they
can isolate diverse scents and sense their forces [2]. Butterflies are look
specialists of BOA to perform streamlining. A butterfly will create aroma with
some power which is connected with its wellness, i.e., as a butterfly moves
from one area to another, its wellness will fluctuate likewise.

The scent will engender over separation and different butterflies can sense it
and this is the means by which the butterflies can share its own data with
different butterflies and shape an aggregate social information arrange. At
the point when a butterfly can detect aroma from some other butterfly, it will
advance toward it and this stage is named as worldwide hunt in the proposed
calculation. In another situation, when a butterfly can’t sense scent from the
encompassing surrounding, at that point it will move haphazardly and this
stage is named as neighborhood look in the proposed calculation. In this
paper, terms smell and aroma are utilized interchangeably.

###  II-B Fragrance

In BOA, every aroma has its own special fragrance and individual contact. It
is one of the fundamental attributes that recognizes BOA from different
metaheuristics. With the end goal to see how aroma is computed in BOA, first
we have to get how a methodology like smell, sound, light, temperature, and so
on is handled by a stimulus. The entire idea of detecting and handling the
methodology depends on three imperative terms viz. sensory modality ($c$),
stimulus intensity ($I$) and power exponent ($a$).

In sensory modality, sensory intends to gauge the type of vitality and process
it in comparative ways and methodology alludes to the crude info utilized by
the sensors. Presently extraordinary modalities can be smell, sound, light,
temperature and in BOA, methodology is scent. $I$ is the extent of the
physical/genuine improvement. In BOA, $I$ is associated with the wellness of
the butterfly/arrangement. This implies when a butterfly is discharging a more
noteworthy measure of scent, alternate butterflies in that encompassing can
detect it and gets pulled in toward it. Power is the type to which force is
raised. The parameter $a$ takes into consideration customary articulation,
direct reaction what’s more, reaction pressure. Reaction development is the
point at which $I$ builds, the scent ( $f$ ) expands more rapidly than $I$.
Reaction pressure is the point at which $I$ builds, f increments more
gradually than I. Straight reaction is the point at which $I$ builds, $f$
increments relatively [3]. Researchers have led a few investigations on creepy
crawlies, creatures and people for greatness estimation and they have inferred
that occasionally as the upgrade gets more grounded, bugs turn out to be
progressively less touchy to the boost changes [4]. So in BOA, to evaluate the
extent of $I$, reaction pressure is utilized.

The common wonder of butterflies depends on two essential issues: the variety
of $I$ and definition of $f$. For straightforwardness, $I$ of a butterfly is
related with the encoded target work. In any case, $f$ is relative, i.e., it
ought to be detected by different butterflies. As indicated by Steven’s
capacity law [4] with the end goal to separate smell from other modalities,
$c$ is utilized. Presently, as the butterfly with less $I$ moves toward
butterfly with more I, f builds more rapidly than $I$. So we ought to enable f
to fluctuate with a level of ingestion which is accomplished by the power
example parameter $a$. Utilizing these ideas, in BOA, the scent is figured as
a component of the physical force of improvement as follows:

$f=c\times I^{a}$ (1)

where $f$ is the perceived magnitude of the fragrance, i.e., how stronger the
fragrance is perceived by other butterflies, $c$ is the sensory modality, $I$
is the stimulus intensity and a is the power exponent dependent on modality,
which accounts the varying degree of absorption. For most of the cases in our
evaluation, we can take $a$ and $c$ in the range [0, 1]. The parameter $a$ is
the power exponent dependent on modality (fragrance in our case) which means
it characterizes the changes in absorption. In one extreme, $a=1$, this means
there is no absorption of fragrance, i.e., the measure of aroma radiated by a
specific butterfly is detected in a similar limit by alternate butterflies.
This is proportionate to stating that aroma is proliferated in a glorified
domain. Consequently, a butterfly emanating scent can be detected from
anyplace in the space. Subsequently, a solitary (normally worldwide) ideal can
be come to effortlessly. Then again, if $a=0$, it implies that the scent
produced by any butterfly can’t be detected by alternate butterflies by any
stretch of the imagination. Along these lines, the parameter a controls the
conduct of the algorithm.

Another important parameter is $c$ which is also crucial parameter in
determining the speed of convergence and how the BOA algorithm behaves.
Theoretically $c\in[0,\infty]$ but practically it is determined by the
characteristic of the system to be optimized. The values of $a$ and $c$
crucially affect the convergence speed of the algorithm.On account of
expansion issue, the force can be relative to the goal work. Different
structures of power can be characterized comparably to the wellness work in
firefly calculation [5], hereditary calculations or the bacterial scavenging
calculation.

###  II-C Movement of Butterflies

To exhibit above dialogs as far as a hunt calculation, the above attributes of
butterflies are idealized as follows:

  1. 1.

All butterflies are supposed to emit some fragrance which enables the
butterflies to attract each other.

  2. 2.

Every butterfly will move randomly or toward the best butterfly emitting more
fragrance.

  3. 3.

The stimulus intensity of a butterfly is affected or determined by the
landscape of the objective function.

There are three phases in BOA:(1) Initialization phase, (2) Iteration phase
and (3) Final phase. In each run of BOA, first the initialization phase is
executed, then searching is performed in an iterative manner and in the last
phase, the algorithm is terminated finally when the best solution is found. In
the initialization phase, the algorithm defines the objective function and its
solution space. The values for the parameters used in BOA are also assigned.
After setting the values, the algorithm proceeds to create an initial
population of butterflies for optimization. As the total number of butterflies
remains unchanged during the simulation of BOA, a fixed size memory is
allocated to store their information. The positions of butterflies are
randomly generated in the search space, with their fragrance and fitness
values calculated and stored. This finishes the initialization phase and the
algorithm starts the iteration phase, which performs the search with the
artificial butterflies created.

The second phase of the algorithm, i.e., iteration phase, a number of
iterations are performed by the algorithm. In each iteration, all butterflies
in solution space move to new positions and then their fitness values are
evaluated. The algorithm first calculates the fitness values of all the
butterflies on different positions in the solution space. Then these
butterflies will generate fragrance at their positions using Eq. (1). There
are two key steps in the algorithm, i.e., global search phase and local search
phase. In global search phase, the butterfly takes a step toward the fittest
butterfly/solution $g^{\textasteriskcentered}$ which can be represented using
Eq.(2)

$x_{i}^{t+1}=x_{i}^{t}+(r^{2}\times g^{\textasteriskcentered}-x_{i}^{t})\times
f_{i}$ (2)

where $x_{i}^{t}$ is the solution vector $x_{i}$ for $ith$ butterfly in
iteration number $t$. Here, $g^{\textasteriskcentered}$ represents the current
best solution found among all the solutions in current iteration. Fragrance of
$ith$ butterfly is represented by $f_{i}$ and $r$ is a random number in [0,
1]. Local search stage of the algorithm can be given by:

$x_{i}^{t+1}=x_{i}^{t}+(r^{2}\times x_{j}^{t}-x_{k}^{t})\times f_{i}$ (3)

where $x_{i}^{t}$ and $x_{k}^{t}$ are $j$th and $k$th butterflies from the
solution space. If $x_{i}^{t}$ and $x_{k}^{t}$ t belongs to the same swarm and
$r\in[0,1]$ is a random number then Eq. (3) becomes a local random walk.

Search for nourishment and mating accomplice by butterflies can happen at both
nearby and worldwide scale. Considering physical nearness and different
components like rain, wind, and so on., search for nourishment can have a
critical division $p$ in a general mating accomplice or sustenance looking
exercises of butterflies. So a switch likelihood p is utilized in BOA to
switch between normal worldwide hunt to concentrated nearby inquiry. Till the
halting criteria isn’t coordinated, the cycle stage is proceeded. The halting
criteria can be characterized in distinctive ways like greatest CPU time
utilized, most extreme cycle number achieved, the most extreme number of
emphases with no enhancement, a specific estimation of mistake rate is come to
or some other proper criteria. At the point when the emphasis stage is
finished up, the calculation yields the best arrangement found with its best
solution found with its best fitness.

###  II-D Generalized Butterfly Optimization Algorithm

The complete flowchart of BOA is also as shown in figure 1 and the above-
mentioned three steps that make up the complete algorithm of BOA [1] and its
pseudo code is explained in Algorithm 1.

Figure 1: BOA Flowchart

##  III Control Problem Formulation

To test the robustness of the Butterfly Optimization Algorithm (BOA) against
other evolutionary algorithms, we formulate a control problem to be used for
evaluating BOA. Consider a linearized control system modeled in the state-
space form as:

$\dot{x}=A_{o}x+Bu$ (4)

Where $A_{o}\in\mathbb{R}^{n\times n}$, $B\in\mathbb{R}^{n\times 1}$,
$x=[x_{1}(t)\ x_{2}(t)\ \ldots\ x_{n}(t)]^{T}$ and $\dot{x}=[\dot{x}_{1}(t)\
\dot{x}_{2}(t)\ \ldots\ \dot{x}_{n}(t)]^{T}$

Assuming the control system is unstable with undesirable frequency response,
i.e., $|\lambda(A_{o})|>1$. To address this problem, a Lead-Lag compensator
(LLC) is designed to mitigate oscillations in the frequency response and also
optimize the damping coefficient. The control parameters of an LLC are the
controller gain, $K_{c}$, and time constants, $T_{1}$ and $T_{2}$.

The closed loop control system can thus be written as:

$\dot{x}=A_{c}x+Bu$ (5)

and can be transformed into,

$\dot{z}=\bar{A}_{c}z$ (6)

Where:
$\bar{A}_{c}\in\mathbb{R}^{m\times m},\ m=n+2$, $z=[x_{1}(t)\ x_{2}(t)\
\ldots\ x_{n}(t)\ x_{n+1}(t)\ u]^{T}$ and $\dot{z}=[\dot{x}_{1}(t)\
\dot{x}_{2}(t)\ \ldots\ \dot{x}_{n}(t)\ \dot{x}_{n+1}(t)\ \dot{u}]^{T}$

Definition 1: The damping coefficient, $\zeta$ which describes how fast the
system response oscillations decay with time can be related to the complex
eigenvalues [6], $\lambda_{i}(\bar{A}_{c}),\ i=1,2,\ldots,m$ of the closed
loop system matrix, $A_{c}$ as:

$\lambda_{i}=-\sigma+j\omega_{d}=-\zeta\omega_{n}+j\omega_{n}\sqrt{1-\zeta^{2}}$
(7)

Where: $\zeta$ is the damping ratio and $\omega_{n}$ is the natural frequency
assume to be unity.

From Definition 1 and the description of the contro problem, the objective
function value associated with the randomly generated solution is found by
calculating the minimum damping ratio ($\zeta_{min}$) which is computed from
the minimum of the closed eigenvalues associated with the values of the
controller parameters ($K_{c}$, $T_{1}$, and $T_{2}$). $\zeta_{min}$ can be
expressed as:

$\zeta_{min}=\frac{-\sigma}{\sqrt{\sigma^{2}+\omega^{2}}}=cos\bigg{(}arctan\Big{(}\frac{\omega_{d}}{\sigma}\Big{)}\bigg{)}=cos\bigg{(}arctan\Big{(}\frac{Im(\lambda_{min})}{-Re(\lambda_{min})}\Big{)}\bigg{)}$
(8)

We can therefore formulate an optimization problem given as follows:

$max\ \zeta_{min}$ (9)

such that:

$K_{c}^{min}\leq K_{c}\leq K_{c}^{max}$ $T_{1}^{min}\leq T_{1}\leq
T_{1}^{max}$ $T_{2}^{min}\leq T_{2}\leq T_{2}^{max}$

##  IV Simulation Evaluation

To evaluate the performance of BOA on the LLC control system, we develop a
program to design the optimal controller. This section describes the
simulation setup, the system matrices and controller design used. We then
carryout eigenvalue analysis of the system with the optimal controller. The
results obtained is compared with those obtained using Genetic Algorithm (GA)
and Differential Evolution (DE) algorithm. Genetic Algorithm, one of the most
populous evolutionary techniques employs general principles developed in the
theory of evolution and genetics to search for concurrent global solutions to
complex optimization problems. The performance of most newly developed
evolutionary algorithms are tested against GA [7]. Differential Evolution (DE)
is a widely implemented population based evolutionary optimization technique
which is simple, robust, fast and requiring minimal control parameters capable
of solving non-linear and non-differentiable optimization problems [8].

###  IV-A Control Problem Parameters

To conduct our evaluation, we employ an open-loop linearized model system
given by:

$\begin{bmatrix}\dot{x_{1}}\\\ \dot{x_{2}}\\\ \dot{x_{3}}\\\
\dot{x_{4}}\end{bmatrix}=\begin{bmatrix}0&377&0&0\\\ -0.0587&0&-0.1303&0\\\
-0.0899&0&-0.1956&0.1289\\\
95.605&0&-816.0862&-20\end{bmatrix}\begin{bmatrix}x_{1}\\\ x_{2}\\\ x_{3}\\\
x_{4}\par\end{bmatrix}.$

The open loop eigenvalues are the eigenvalues of the matrix A which is:

$A=\begin{bmatrix}0&377&0&0\\\ -0.0587&0&-0.1303&0\\\
-0.0899&0&-0.1956&0.1289\\\ 95.605&0&-816.0862&-20\end{bmatrix}$

So the open loop eigenvalues are:$-10.3932\pm 1$ j3.2910 and $0.2954\pm
1j4.9577$. From these eigenvalues obtained, it can be observed that the system
is unstable since it contains some of them have positive real part.

The LLC controller with a structure given in Figure 2 is employed for
stabilizing the system.

Figure 2: Lead-lag Controller.

This structure is described by the following equations:

$\displaystyle x_{5}=\dfrac{3s}{1+3s}x_{2}$ (10) $\displaystyle
x_{5}+3x_{5}s=3x_{2}s$ (11) $\displaystyle x_{5}+3\dot{x_{5}}=3\dot{x_{2}}$
(12) $\displaystyle\dot{x_{5}}=\dot{x_{2}}-\dfrac{1}{3}x_{5}$ (13)

To derive the closed loop system, two more states are added to form the
transformed model (6) and the system is described by:

$\displaystyle\dot{x_{5}}=-0.0587x_{1}-0.1303_{2}-\dfrac{1}{3}x_{5}$ (14)
$\displaystyle u=\dfrac{K(1+sT_{1})}{(1+sT_{2})}x_{5}$ (15) $\displaystyle
u+suT_{2}=Kx_{5}+Ksx_{5}T_{1}$ (16)
$\displaystyle\dot{u}=\dfrac{KT_{1}}{T_{2}}\dot{x_{5}}+\dfrac{K}{T_{2}}x_{5}-\dfrac{1}{T_{2}}u$
(17)
$\displaystyle\dot{u}=\dfrac{-0.0587KT_{1}}{T_{2}}x_{1}-\dfrac{0.1303KT_{1}}{T_{2}}x_{3}+\left(\dfrac{K}{T_{2}}-\dfrac{KT_{1}}{3T_{2}}\right)x_{5}-\dfrac{1}{T_{2}}u$
(18)

Hence, the new two states will be $(x_{5},u)$, therefore, the closed loop
state vector will be: $z=〖[x_{1},x_{2},x_{3},x_{4},x_{5},u]〗^{T}$
Then, the close loop form of state equations of the system is given by:

$\dot{z}=\bar{A}_{c}z$ (19)

Where:

$\bar{A}_{c}=\begin{bmatrix}0&377&0&0&0&0\\\ -0.0587&0&-0.1303&0&0&0\\\
-0.0899&0&-0.1956&0.1289&0&0\\\ 95.605&0&-816.0862&-20&0&1000\\\
-0.0587&0&-0.1303&0&\dfrac{-1}{3}&0\\\
\dfrac{-0.0587KT_{1}}{T_{2}}&0&-\dfrac{0.1303KT_{1}}{T_{2}}&0&\left(\dfrac{K}{T_{2}}-\dfrac{KT_{1}}{3T_{2}}\right)&-\dfrac{1}{T_{2}}\end{bmatrix}$

###  IV-B Test Parameters and Setup

To evaluate the performance of the BOA algorithm against Genetic Algorithm
(GA) and Differential Evolution (DE) in solving the control problem in Section
IV.A, we develop a programs to design the optimal controller using each
scheme. The parameters used for each algorithm in the simulations is given in
Table I. The limits of the controller parameters adopted for all simulations
are: $1.0\leq K_{c}\leq 50$, $0.1\leq T_{1}\leq 1.0$, and $0.01\leq T_{2}\leq
0.1$

Table I: Simulation Parameters for GA, DE and BOA GA | DE | BOA
---|---|---
Mutation Probability = 0.05 | Mutation Probability = 0.05 | Sensory Modality $c$ = 0.01
Crossover Probability = 0.9 | Crossover Probability = 0.9 | Switching Probability $p$ = 0.8
Crossover coefficient = 0.5 |  | Power Exponent $a$ = 0.1
Population size = 50 | Population size = 50 | Population size = 50
Number of generations = 200 | Number of generations = 200 | Number of generations = 200

###  IV-C Results

The code has been run for the same population size and the optimal objective
function variation with respect to the the number of generations is shown in
Figure 3. It can be noticed that the curves are quite similar to each other
which indicates that all algorithms are robust to the same initial population
of the solution such that they converge quickly to the optimal values.

Table II: Optimal Controller Parameters Algorithm | Kc | T1 | T2
---|---|---|---
GA | 18.3998 | 0.2619 | 0.1
DE | 18.402 | 0.2618 | 0.1
BOA | 18.1352 | 0.2714 | 0.1

It is however observed that the convergence time of BOA is much lower as
compared to GA and DE algorithms. Hence BOA outperforms these evolutionary
algorithms in term of the time taken to reach the optimal values of
$\zeta_{min}$.

Figure 3: Optimal Objective Function vs. Number of Generations Performance
Curves for GA, DE and BOA

Comparing the results of BOA algorithm against GA and DE we observe that
eigenvalues of BOA are further left as compared to GA which in turn has
eigenvalues further left than DE as shown in Table III. We also observe that
BOA achieves a lower constant $K_{c}$ as compared to the DE and GA. However
for the time constant $T_{1}$, BOA has a higher value as compared to GA and DE
but registers the same value of $T2$ as compared to these algorithms. The
results for the optimal controller parameters are summarized in Table II. On a
whole, BOA performs similarly to GA and DE in terms of optimizing the minimum
damping coefficient ($\zeta_{min}$). The values for this parameter for each
algorithm in the design of the controller is given by Table IV .

Table III: Eigenvalues of Optimal Parameters Eigenvalues of Optimal Parameters
---
GA | DE | BOA
-18.2 + 0i | -18.199 + 0i | -18.296 + 0i
-3.032 + 5.5839i | -3.0183 + 5.5576i | -3.2845 + 6.1484i
-3.032 - 5.5839i | -3.0183 - 5.5576i | -3.2845 - 6.1484i
-2.9595 + 5.4499i | -2.9737 + 5.4754i | -2.6591 + 4.9738i
-2.9595 - 5.4499i | -2.9737 - 5.4754i | -2.6591 - 4.9738i
-0.34543 + 0i | -0.34544 + 0i | -0.34519 + 0i
Table IV: Optimal Controller Parameters Algorithm | Objective Value ($\zeta_{min}$)
---|---
GA | 0.4772
DE | 0.4772
BOA | 0.4712

##  V Conclusion

In this work, we adopted the relatively new and novel global/local Butterfly
Optimization algorithm (BOA) in optimizing the control parameters of a Lead-
Lag control system. The general overview and detailed mode of operation of BOA
is discussed. A control system with unstable damping coefficient is used to
formulate an optimization problem for the design of a optimal controller
parameters for a Lead-Lad Controller so as to design a stable control system.
On a whole, BOA performs similarly to GA and DE in terms of optimizing the
minimum damping coefficient for the cotrol system.

## References

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  * [2] T. D. Wyatt, Pheromones and animal behaviour: communication by smell and taste.  Cambridge university press, 2003.
  * [3] J. C. Baird and E. J. Noma, Fundamentals of scaling and psychophysics.  John Wiley & Sons, 1978.
  * [4] J. J. Zwislocki, Sensory neuroscience: Four laws of psychophysics.  Springer Science & Business Media, 2009.
  * [5] X.-S. Yang, “Firefly algorithm, stochastic test functions and design optimisation,” arXiv preprint arXiv:1003.1409, 2010.
  * [6] H. K. Abdulkhader, J. Jacob, and A. T. Mathew, “Fractional-order lead-lag compensator-based multi-band power system stabiliser design using a hybrid dynamic ga-pso algorithm,” IET Generation, Transmission & Distribution, 2018.
  * [7] D. Gong, J. Sun, and Z. Miao, “A set-based genetic algorithm for interval many-objective optimization problems,” IEEE Transactions on Evolutionary Computation, vol. 22, no. 1, pp. 47–60, 2018.
  * [8] P. P. Biswas, P. N. Suganthan, R. Mallipeddi, and G. A. Amaratunga, “Optimal power flow solutions using differential evolution algorithm integrated with effective constraint handling techniques,” Engineering Applications of Artificial Intelligence, vol. 68, pp. 81–100, 2018.


# Exclusive interplay between topological quasiparticles and strongly
correlated fermions

Byungkyun Kang<EMAIL_ADDRESS>College of Arts and Sciences, University of
Delaware, Newark, Delaware 19716, USA  Zachary Brown  Department of Physics
and Astronomy, Texas Tech University, Lubbock, Texas 79409, USA  Myoung-Hwan
Kim  Department of Physics and Astronomy, Texas Tech University, Lubbock,
Texas 79409, USA  Hyunsoo Kim  Department of Physics, Missouri University of
Science and Technology, Rolla, 65409, MO, USA  Chul Hong Park  Quantum Matter
Core-Facility and Research Center of Dielectric and Advanced Matter Physics,
Pusan National University, Busan 46240, Republic of Korea

###### Abstract

The low-energy excitations of topological quantum matter have been
increasingly comprehended. However, the impact of these excitations on
strongly correlated fermions remains rarely explored. Here, we report the
discovery of an incompatible dynamic of 4$f$ quasiparticle and topological
quasiparticle. The former forms through screening processes, while the latter
suppresses the screening channel. By employing ab initio many-body
perturbation theory combined with dynamical mean field theory, we show that
this effect prompts two types of topological semimetals. The type-I semimetal
PrPtBi exhibits a topological quasiparticle away from the Fermi level due to
the formation of a 4$f$ quasiparticle. The type-II semimetals HoPtBi and
PrAlGe feature a robust topological quasiparticle at the Fermi level, which
pushes the 4$f$ quasiparticle away. Our work provides an avenue to harness
semimetals’ anomalous quantum effects from topological quasiparticle to heavy-
fermion behavior by the incorporation of 4$f$ quasiparticles.

In condensed matter, electrons at the Fermi energy preside over the majority
of intriguing physics. In strongly correlated electron systems, the presence
of a moderate Coulomb repulsion can give rise to the emergence of Fermi-liquid
quasiparticles at low-energy scales. These excitations occur through the
creation of screening holes, which effectively reduce the strength of the
Coulomb repulsion (Fig. 1a). In topological semimetals, the emergence of
topological quasiparticles is observed in the immediate vicinity of Weyl
points or quadratic band touchings. Here, we report our discovery that the
presence of topological quasiparticles hinders the formation of 4$f$
quasiparticles due to their interference with the essential hybridization
channel (Fig. 1b). In order to circumvent the prohibited coexistence of 4$f$
and topological quasiparticles at the Fermi level, 4$f$-based topological
semimetals exhibit two distinct modes of dynamic expulsion, effectively
repelling one of the quasiparticles from the Fermi level. In type-I (Fig. 1c),
topological quasiparticles descend from the Fermi level, providing a
hybridization channel through which 4$f$ electrons can overcome the on-site
Coulomb interaction $U_{\textrm{C}}$ and form 4$f$ quasiparticles. In type-II
(Fig. 1d), the 4$f$ quasiparticle is pushed away from the Fermi level due to
the effective Coulomb repulsion $U_{\textrm{eff}}$. This repulsion is
intensified from $U_{\textrm{C}}$ by the diminished screening effect caused by
the highly reduced hybridization with the robust topological quasiparticles at
the Fermi level.

The strongly correlated 4$f$ electrons of rare-earth elements (R-4$f$) are
capable of accommodating a quasiparticle state at the Fermi level. At low
temperatures, the Nd-4$f$ electrons in superconducting nickelate NdNiO2 induce
a quasiparticle with Kondo hybridization, resulting in a three-dimensional
Fermi surface, which is responsible for the inconsistent superconductivity
observed in nickelate superconductors [1]. The band-dependent quasiparticle
dispersion around the Fermi level of the Ce-4$f$ in the heavy-fermion
superconductor CeCu2Si2 has been observed, lending credence to the $d$+$d$
pairing scenario for unconventional superconductors [2]. The hybridization of
Pr-4$f$ electrons with conduction electrons in filled-skutterudite PrRu4P12
has been observed to result in anomalous behaviors, the most notable of which
is a giant negative magnetoresistance [3].

In the lattice model, Dirac fermions can be observed at distinct points within
the first Brillouin zone, exhibiting the presence of massless relativistic
particles in a non-relativistic framework. Three-dimensional Dirac semimetals
are known to feature fourfold degenerate gapless crossing points in their
electronic structures [4]. Additionally, twofold degenerate two-dimensional
Dirac cones, such as those found in graphene, are presented [5]. The Dirac
cones located in proximity to the Fermi level have been observed to result in
remarkable effects such as ultrahigh mobility and giant magnetoresistance in
the Dirac semimetal Cd3As2 [6, 7]. Recently, a Lifshitz transition with
decreasing temperature in the three-dimensional Dirac semimetal Au2Pb has been
discovered, offering a tunable Dirac cone to control electron transport
without external doping [8]. The degeneracy of these materials is protected by
time-reversal symmetry, inversion symmetry, and certain crystalline rotation
symmetries. However, should any of these symmetries be broken, the massless
Dirac fermions can be annihilated, resulting in the emergence of Weyl fermions
[9].

Figure 1:  Schematic diagram of exclusive interplay between 4$f$ quasiparticle
and topological quasiparticle. a, In general, a strongly correlated electron
can be in a Mott phase with an upper Hubbard band (UH) and lower Hubbard band
(LH) at high temperatures due to the Coulomb repulsion ($U_{\textrm{C}}$). The
emergence of 4$f$ quasiparticle ($f$-QP) occurs in the presence of a weak
$U_{\textrm{C}}$ at low temperatures, where the $U_{\textrm{C}}$ is screened
by 4$f$ hybridization ($\Delta$) with neighboring orbitals. b, In 4$f$-based
topological semimetals, the 4$f$ $\Delta$ is primarily influenced by the
density of states (DOS) of the topological bands. The topological
quasiparticle (t-QP) at the Fermi level ($E_{\textrm{F}}$) does not provide
the necessary hybridization environment for the formation of $f$-QP, since the
DOS at the t-QP energy is diminished. As a result, the formation of $f$-QP at
$E_{\textrm{F}}$ is hindered. c, In the type-I PrPtBi, the t-QP shifts from
$E_{\textrm{F}}$ and a hybridization channel is provided for the formation
process of $f$-QP. d, On the other hand, in the type-II HoPtBi and PrAlGe, the
t-QP is formed at $E_{\textrm{F}}$. As a result, the DOS at $E_{\textrm{F}}$
is subtle, and the t-QP does not provide a hybridization environment for the
formation of $f$-QP. The absence of screening leads to a strong effective
Coulomb interaction ($U_{\textrm{eff}}$), which in turn gives rise to the
formation of lower (LH) and upper (UH) Hubbard-like 4$f$ bands.

The rare-earth elements present in the attractive Weyl semimetals RAlGe
(R=rare-earth) have been suggested to be associated with exotic electronic
properties [10, 11, 12, 13, 14, 15, 16, 17, 18]. These include high Fermi
velocities [16], Lorentz-violating type II Weyl fermions [12] in LaAlGe, a
strong anomalous Hall effect in the ferromagnetic Weyl semimetal PrAlGe [13,
17, 18], and a topological magnetic phase in CeAlGe [14]. A recent study
suggested that the formation of defects in LaAlGe can significantly alter the
chemical potential, thereby impacting these unique electronic characteristics
resulting from Weyl physics [19]. The rare-earth elements are of paramount
importance in topological half-Heusler compounds. The tuning of R-4$f$
electron in topological semimetals RPdBi provides a distinct opportunity to
modulate the strength of magnetic interaction, thereby enabling a precise
adjustment of the normal-state band inversion strength, superconducting
pairing, and magnetically ordered ground states [20]. The antiferromagnetic
Weyl semimetal compounds GdPtBi and NdPtBi [21], as well as the topological
semimetal HoPtBi [22], have been observed to exhibit an anomalous Hall effect.
This effect is a result of Weyl points, which are induced by a magnetic field,
and are caused by exchange splitting of the electronic bands at or near the
Fermi energy [21].

The conundrum of rare-earth-based semimetals lies in their possession of both
strongly correlated 4$f$ electrons and topological quasiparticles. The most
intriguing physical phenomenon occurs when the Fermi level is precisely
situated at the topological quasiparticles of Weyl cones and quadratic bands.
At low temperatures, therefore, the emergence of R-4$f$ quasiparticles in
semimetals could result in a dramatic alteration of their physical properties.
We investigated the electronic structure of RAlGe (R=La, Pr) and RPtBi (R=La,
Pr, Ho) and uncovered a strong interplay between R-4$f$ quasiparticles and
topological quasiparticles. As temperature decreases, the prominence of 4$f$
quasiparticles in the vicinity of the Fermi level is augmented by the
influence of $U_{\textrm{C}}$ and screening effect, mainly through
hybridization with topological bands. This precludes the simultaneous
emergence of R-4$f$ quasiparticles and topological quasiparticles at the Fermi
level ($E_{\textrm{F}}$) (Fig. 1b) and gives rise to two types of semimetals
exhibiting distinct interaction behaviors, as depicted in Fig. 1c and d. As
the type-I, for PrPtBi, the topological quasiparticle is shifted away from
$E_{\textrm{F}}$, as the 4$f$-driven quasiparticle emerges at $E_{\textrm{F}}$
at low temperature. Conversely, as the type-II, for HoPtBi and PrAlGe, the
topological quasiparticle remains at $E_{\textrm{F}}$, whereas the 4$f$
quasiparticles are displaced from $E_{\textrm{F}}$ due to the exclusive
interplay.

### Electronic structure of PrPtBi as a type-I semimetal

Figure 2:  Effect of temperature on the electronic structure of PrPtBi and
LaPtBi. a, Spectral functions of LaPtBi. b, Spectral functions of PrPtBi. c,
Projected DOS of Pr-4$f$, the total DOS with the Pr-4$f$ component subtracted,
and the imaginary part of the hybridization function ($\Delta$) of Pr-4$f$. d,
Projected DOS of $f_{2}$, imaginary part of self-energies on the real
frequency axis, and on the imaginary frequency axis. e, Product of temperature
and local total angular momentum susceptibility $T\chi_{loc}^{J_{Z}}$ for
Pr-4$f$. f, The results of the electronic structure using a small static
$U_{\textrm{C}}$ are presented. This includes the projected DOS of Pr-4$f$,
the total DOS with the Pr-4$f$ component subtracted, and the imaginary part of
the hybridization function ($\Delta$) of Pr-4$f$.  Table 1: The electron
occupation of Rare Earth-4$f$ orbitals and the $Z$ factor were calculated. The
$Z$ factor was presented only when it was greater than zero. The 4$f$ orbitals
are labelled for convenience in this work.

|  4$f$ occupancy ($Z$ factor)
---|---
$j$ | 5/2 | 7/2
$j_{z}$ |  | -2.5 | -1.5 | -0.5 | 0.5 | 1.5 | 2.5 | -3.5 | -2.5 | -1.5 | -0.5 | 0.5 | 1.5 | 2.5 | 3.5
label |  | $f_{1}$ | $f_{2}$ | $f_{3}$ | $f_{4}$ | $f_{5}$ | $f_{6}$ | $f_{7}$ | $f_{8}$ | $f_{9}$ | $f_{10}$ | $f_{11}$ | $f_{12}$ | $f_{13}$ | $f_{14}$
PrPtBi |  $T$=100 K | 0.30 | 0.27 | 0.29 | 0.35 | 0.28 | 0.33 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01
| (0.12) | (0.13) | (0.15) | (0.15) | (0.11) | (0.13) |  |  |  |  |  |  |  |
$T$=300 K | 0.34 | 0.24 | 0.35 | 0.35 | 0.24 | 0.32 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01
| (0.12) | (0.09) | (0.12) | (0.12) | (0.09) | (0.12) |  |  |  |  |  |  |  |
HoPtBi | T=100 K | 0.98 | 0.98 | 0.98 | 0.98 | 0.98 | 0.98 | 0.54 | 0.52 | 0.53 | 0.50 | 0.51 | 0.47 | 0.48 | 0.47
| (0.51) | (0.55) | (0.59) | (0.58) | (0.55) | (0.56) | (0.04) | (0.03) | (0.03) | (0.02) | (0.02) | (0.01) | (0.01) | (0.01)
T=300 K | 0.97 | 0.96 | 0.96 | 0.97 | 0.97 | 0.97 | 0.52 | 0.52 | 0.51 | 0.48 | 0.56 | 0.51 | 0.51 | 0.51
| (0.36) | (0.67) | (0.41) | (0.38) | (0.32) | (0.43) | (0.03) | (0.03) | (0.03) | (0.02) | (0.04) | (0.03) | (0.03) | (0.03)
PrAlGe |  $T$=100 K | 0.43 | 0.42 | 0.01 | 0.01 | 0.42 | 0.44 | 0.01 | 0.01 | 0.01 | 0.00 | 0.00 | 0.01 | 0.01 | 0.01
| (0.11) | (0.13) | (0.13) | (0.13) | (0.13) | (0.12) |  |  |  |  |  |  |  |
$T$=300 K | 0.38 | 0.37 | 0.16 | 0.16 | 0.36 | 0.38 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01
| (0.12) | (0.13) |  |  | (0.13) | (0.12) |  |  |  |  |  |  |  |

The ab initio linearized quasiparticle self-consistent GW (LQSGW) method in
combination with dynamical mean field theory (DMFT) [23] was employed to
calculate the electronic structure of correlated quantum materials in their
paramagnetic phase. Fig. 2b shows the spectral functions of PrPtBi. Below 200
K, the topological quasiparticles arising from the quadratic bands touching
are at $\sim$0.15 eV, while the flat 4$f$ bands are closer to
$E_{\textrm{F}}$. In contrast, the topological quasiparticle in LaPtBi remains
at $E_{\textrm{F}}$ (Fig. 2a). Fig. 2c shows the projected density of states
(DOS) of Pr-4$f$ and the total DOS with the Pr-4$f$ component subtracted. As
the temperature decreases, the Pr-4$f$ peak shifts towards $E_{\textrm{F}}$.
Below 100 K, a shoulder-like peak appears at $E_{\textrm{F}}$. The minimum of
the (total $-$ Pr-4$f$) DOS of the topological quasiparticle moves towards
$E_{\textrm{F}}$ as well, but remains at around 0.15 eV. In PrPtBi, the change
in the energy level of topological quasiparticles is a result of their
interaction with 4$f$ electrons. This is further supported by the observation
that there is negligible alteration of topological quasiparticles in LaPtBi,
where 4$f$ electrons are absent.

The driving factor behind this interaction can be elucidated by understanding
the mechanism of 4$f$ quasiparticle formation. The emergence of the Pr-4$f$
quasiparticle is accompanied by a modest static $U_{\textrm{C}}$ of 6.34 eV
for Pr-4$f$ (see Fig. S2), wherein the $U_{\textrm{C}}$ is effectively
screened through hybridization with adjacent orbitals [24] at low
temperatures. As shown in Fig. 2c, the shape of the (total $-$ Pr-4$f$) DOS
and the Pr-4$f$ hybridization function ($\Delta$) are almost identical at all
temperature ranges, indicating that the Pr-4$f$ mainly hybridizes with
adjacent orbitals rather than with other Pr-4$f$ orbitals. Therefore, the
energy level of the topological quasiparticle, where the DOS reaches zero,
plays an important role in blocking the hybridization channel and hindering
the formation of Pr-4$f$ quasiparticles. In PrPtBi, the Pr-4$f$ quasiparticle
is more robust than the topological quasiparticle at the Fermi level. This is
achieved by shifting topological quasiparticles and providing additional
hybridization channels for Pr-4$f$ quasiparticles. To better understand this
interplay, we show Fig. 2f, which is plotted in a similar manner as Fig. 2c
but with a smaller $U_{\textrm{C}}$. This value is obtained by considering
Pr-4$f$ and Pt-5$d$ as correlated orbitals using cRPA to promote the formation
of 4$f$ quasiparticle. This figure clearly illustrates the exclusive interplay
between 4$f$ and topological quasiparticles. At high temperatures, only
topological quasiparticles are present at $E_{\textrm{F}}$, while at low
temperatures, only 4$f$ quasiparticles are present at $E_{\textrm{F}}$. The
absence of experimental evidence of a topological phase in PrPtBi [25] may be
attributed to this exclusive interaction that drives the topological
quasiparticle away from $E_{\textrm{F}}$ at low temperatures.

We calculated the Pr-4$f$ self-energy as shown in Fig. 2d and Fig. S7 (see
Table 1 for labeling of $f$ orbitals). At 700 K, the Pr-4$f$ self-energy on
the imaginary axis exhibits a divergence at a low frequency, indicating a
significant presence of Mott-like physics resulting from the influence of the
$U_{\textrm{C}}$. As the temperature decreases, the self-energy gradually
decreases, exhibiting Fermi-liquid-like characteristics and resulting in the
appearance of a quasiparticle peak in the vicinity of the Fermi level. The
quasiparticle weight $Z$ was calculated and is presented in Table 1. This
value was determined by the gradient of the imaginary component of the self-
energy, as shown in the bottom panel of Fig. 2d,
$Z=1/(1-\frac{\partial\text{Im}\Sigma(i\omega)}{\partial
i\omega}\Bigr{|}_{i\omega\rightarrow 0^{+}})$. The increase in mass caused by
electronic correlations can be calculated as the inverse of the quasiparticle
weight, $\frac{m^{\star}}{m}=\frac{1}{Z}$ [26]. The $Z$ factor is of interest,
as it ranges from 1 for weak electron correlation to 0 for strong electron
correlation [27]. The self-energy of Pr-4$f$ exhibits a prominent shoulder
feature and a small $Z$ factor of $\sim$0.05 (Table 1) at low temperature,
indicating significant correlation in Pr-4$f$ orbitals.

At low temperatures, the emergence of 4$f$ quasiparticles can give rise to
heavy-fermion behavior. The DOS of Pr-4$f$ (Fig. 2c) showed a noticeable
sharpening, and the Pr-4$f$ bands appeared as flattened features in the
vicinity of $E_{\textrm{F}}$ in the spectral functions (Fig. 2b). The
shoulder-like Pr-4$f$ peaks, particularly, appear at 100 and 50 K at
$E_{\textrm{F}}$. The flat Pr-4$f$ bands at $E_{\textrm{F}}$ can hybridize
with the conduction electrons, resulting in the Kondo screening phenomenon
[28]. Table 1 demonstrates a significant occupancy of Pr-4$f$ orbitals in the
$j=5/2$ multiplet. The total occupation of Pr-4$f$ orbitals, calculated for
both temperatures, is 1.9, implying that the substantial local magnetic moment
of Pr-4$f$ is subjected to interaction by conduction electrons at low
temperatures. In order to explore the heavy-fermion behavior, we have computed
the local total angular momentum susceptibility, which is expressed as

$\begin{split}\chi_{\mathrm{loc}}^{J_{Z}}=\int_{0}^{\beta}d\tau\langle
J_{z}(\tau)J_{z}(0)\rangle.\end{split}$ (1)

Fig. 2e shows the product of temperature and $\chi_{\mathrm{loc}}^{J_{Z}}$ of
Pr-4$f$ as a function of temperature. A deviation from Curie-Weiss behavior is
observed above 800 K, with a significant deviation at 200 K indicating the
onset of the strong interaction with conduction electrons. The non-conformity
of the spin susceptibility to the Curie law can be an indication of a heavy-
fermion system [29]. Our findings suggest that the heavy-fermion effect may be
responsible for the non-metallic behavior of the electrical resistivity
($T<300$ K) of PrPtBi [30].

Figure 3:  Effect of temperature on the electronic structure of HoPtBi. a,
Spectral functions of HoPtBi at $T$=100 K (left panel), projected DOS of
Ho-4$f$ and the total DOS with the Ho-4$f$ component subtracted (middle
panel), and the imaginary part of the hybridization function ($\Delta$) of
Ho-4$f$. b, $f_{10}$ projected DOS, imaginary part of self-energies on the
real frequency axis and on the imaginary frequency axis of HoPtBi, at varying
temperature in unit of Kelvin. c, Local angular moment correlation functions
in the imaginary time $\tau$ for HoPtBi, PrPtBi, and PrAlGe. Calculated
instantaneous $J$ are presented with corresponding colors.  Figure 4:  Effect
of temperature on the electronic structure of PrAlGe and LaAlGe. a, Calculated
spectral functions and orbital projected density of states of PrAlGe at
simulation temperatures of 100, 200, 300 K. b, Calculated spectral functions
and orbital projected density of states of LaAlGe at simulation temperatures
of 100 K. c, Fermi surfaces of LaAlGe and PrAlGe. d, $f_{2}$ and e, $f_{3}$
projected density of states, imaginary part of self-energies on the real
frequency axis, and on the imaginary frequency axis. The temperature unit is
Kelvin. In (d), the inset shows magnified views of DOS and self-energy in the
vicinity of the Fermi level.

### Electronic structure of HoPtBi as a type-II semimetal

Simulations of HoPtBi revealed a distinct self-energy form, exhibiting a
pronounced correlation between 4$f$ quasiparticle and topological
quasiparticle. Fig. 3a shows a topological quasiparticle at the $\Gamma$ point
at $E_{\textrm{F}}$, while the Ho-4$f$ orbitals projected spectral weight is
prohibited at $E_{\textrm{F}}$, as shown in Fig. S11. Table 1 and Fig. S10
demonstrate that the six $f$ orbitals in the $f_{5/2}$ multiplet are
completely filled and exhibit similar behavior, while the eight $f$ orbitals
in the $f_{7/2}$ multiplet are partially filled. Fig. 3b illustrates that the
DOS for Ho-$f_{10}$, which is representative of the $f_{7/2}$ multiplet, is
zero at $E_{\textrm{F}}$. As shown in Fig. 3b and Fig. S9, the topological
quasiparticle remains at $E_{\textrm{F}}$ upon cooling. The self-energy in the
real axis remains strong, and the calculated $Z$ factor remains at $\sim$0.03
in the temperature range, indicating that strong correlation persists, pushing
the $f$ DOS away from $E_{\textrm{F}}$ and maintaining the topological
quasiparticle at $E_{\textrm{F}}$ at low temperature. This indicates that the
self-energy of Ho-4$f$ appears to remain relatively unaffected by the Mott-
like physics solely attributed to $U_{\textrm{C}}$, as evident from the modest
static $U_{\textrm{C}}$ of 2.03 eV for Ho-4$f$ (Fig. S2). The 4$f$
quasiparticle is strongly correlated with the topological quasiparticle, which
is responsible for the majority of the self-energy of Ho-4$f$ in HoPtBi.

### Topological quasiparticle-induced null screening

Examining the local angular moment correlation functions
$\chi_{J_{Z}}(\tau)=\langle J_{z}(\tau)J_{z}(0)\rangle$ for HoPtBi, PrPtBi,
and PrAlGe, as depicted in Fig. 3c, provides insight into the degree of
magnetic moment localization [31]. The $\chi_{J_{Z}}(\tau)$ of all three
semimetals exhibit a small reduction as the $\tau$ increases, implying the
presence of substantial local magnetic moments. The almost flat
$\chi_{J_{Z}}(\tau)$ of R-4$f$ is consistent with that of the transition
metal-$d$ in the Mott phase of NiO and FeSe [32], indicating that the behavior
of R-4$f$ electrons in these semimetals can be elucidated through the lens of
local atomic characteristics, where the on-site Coulomb interaction plays a
significant role. Fig. S2 shows the dynamical on-site Coulomb interaction
$U_{\textrm{C}}$. In HoPtBi, a small static $U_{\textrm{C}}$ for Ho-4$f$
exerts a significant local screening effect on the local dynamics, ultimately
enhancing the probability of 4$f$ quasiparticle formation, as schematically
shown in Fig. 1a. The pronounced frequency dependence of $U_{\textrm{C}}$ for
Ho-4$f$ within the low-frequency range suggests the presence of a narrow and
distinct 4$f$ quasiparticle peak at $E_{\textrm{F}}$ [33]. However, as shown
in Fig. 3a, the DOS of neighboring orbitals in the vicinity of the Fermi level
approaches zero as a result of the formation of the topological quasiparticle.
This leads to a substantial reduction in 4$f$ hybridization functions, which
present the extent to which 4$f$ orbitals are mixed with adjacent orbitals,
and consequently, the shielding of the 4$f$ Coulomb repulsion [24]. Hence, the
4$f$ quasiparticle located at $E_{\textrm{F}}$ is exposed to negligible
screening and undergoes a significant effective Coulomb interaction
$U_{\textrm{eff}}$ leading to a Mott-like phase transition (see Fig. 1d). As
shown in Fig. 3b, this exclusive interplay resulted in a decrease in self-
energy at temperatures from 300 to 150 K due to the weakening of Mottness and
an increase in self-energy at temperatures from 150 to 50 K due to the
enhanced null screening effect caused by a higher probability of 4$f$
quasiparticles at lower temperatures. This effect was also reflected in the
half-filling occupancy (Table 1) and the emergence of a Mott-like
characteristic in both the upper and lower Hubbard-like bands (Fig. S10).

### Electronic structure of PrAlGe as a type-II semimetal

We have demonstrated a unique interaction between 4$f$ electrons and
topological quasiparticles arising from quadratic bands touching. In order to
further investigate this interaction with other types of topological
quasiparticles, we have conducted a study on PrAlGe, a semimetal with
topological quasiparticles originating from linear band crossings near the
Fermi level. Fig. 4a presents the spectral functions and DOS of PrAlGe at
simulation temperatures of 100, 200, and 300 K. At 300 K, a flat Pr-4$f$ band
is observed at an energy level of $\sim$-0.3 eV. As the temperature decreases,
this band gradually moves closer to $E_{\textrm{F}}$, centering at $\sim$-0.2
eV at 100 K. At a low temperature, the Pr-4$f$ band exhibits a more coherent
character, which is evidenced by its brighter, narrower, and two-peak
structure, in conjunction with adjacent orbitals at 100 K. The evolution of
the Pr-4$f$ quasiparticle has a profound impact on the topological
quasiparticle. At the $\Sigma$ high-symmetry point, the emergence of Weyl node
W1 has been reported in previous studies [10, 12]. As the temperature
decreases, it is noted that the two intersecting points positioned just above
$E_{\textrm{F}}$ exhibit a closer proximity. The Pr-4$f$ quasiparticle is
responsible for the observed effect, as indicated by the unaltered state of
the corresponding nodes in LaAlGe across the same range of temperatures, as
illustrated in Fig. S5.

As shown in Fig. 4b and Fig. S4, the spectral function of LaAlGe closely
resembles the LQSGW bands, which exhibit a renormalization of the DFT bands.
This suggests that there is not a significant level of strong correlation
present. Nevertheless, when the correlated Pr-4$f$ electrons in PrAlGe are
treated within the DMFT approach, there is a modification to the band
structure, particularly near the $\Gamma$ point, as depicted in Fig. S4,
compared to the results obtained from DFT and LQSGW methods. In Fig. 4c, both
the Fermi surfaces of LaAlGe and PrAlGe are in agreement with the ones that
were measured [12, 18], and there is a significant contrast between the Fermi
surface of LaAlGe and that of PrAlGe. These findings suggest that the Pr-4$f$
electrons are located near $E_{\textrm{F}}$, which has a significant impact on
the electronic configuration of PrAlGe. This results in a distinct electronic
structure compared to LaAlGe, where the 4$f$ electrons are not present.

The Pr-4$f$ orbital-resolved DOS and self-energy are shown in Fig. 4d and e.
As the temperature decreases, there is an increase in the DOS for $f_{2}$, and
its peak position shifts towards $E_{\textrm{F}}$, manifesting a tendency to
converge at $\sim$-0.2 eV. Of greater significance is the fact that the
$f_{2}$ DOS is suppressed at $E_{\textrm{F}}$ across all temperature ranges.
The quasiparticle peak is augmented as the temperature is lowered, due to a
decrease in the self-energy, as demonstrated by the imaginary part of the
self-energy in Fig. 4d. As illustrated in the inset of the imaginary self-
energy on the real axis, similar to the $f_{2}$ DOS, the self-energy shows a
tendency to converge below 150 K. This suggests that the enduring self-energy
at lower temperatures is derived from $U_{\textrm{eff}}$, which is amplified
when the 4$f$ quasiparticle approaches the topological quasiparticle.

Due to the exclusive interaction, a captivating phenomenon is observed in
$f_{3}$, which bypasses $E_{\textrm{F}}$ by lowering the temperature. At 300
K, the imaginary part of the self-energy on the imaginary frequency axis
exhibits a singularity, as shown in Fig. 4e. This implies that the $f_{3}$ at
high temperatures is governed by Mott-like physics, resulting in a divergent
peak of the self-energy on the real frequency axis at $E_{\textrm{F}}$ and the
formation of a pseudogap in the $f_{3}$-driven partial DOS. The $f_{3}$
exhibits a behavior akin to that of Nd-4$f$ at elevated temperatures in NdNiO2
[1]. Nevertheless, a distinguishing feature is that $f_{3}$ in PrAlG at 300 K
still exhibits the suppressed DOS in the vicinity of the Fermi level. At 200
K, the self-energy decreased, resulting in $f_{3}$ DOS peak towards
$E_{\textrm{F}}$, and the occupied DOS decreased while the unoccupied DOS
increased, leaving a still suppressed DOS at $E_{\textrm{F}}$. Below 150 K,
$f_{3}$ remains unoccupied. The weight of $f_{3}$ was redistributed to other
partially occupied $f$ orbitals, as shown in Table 1.

### Conclusion

The topological quasiparticles found at the Fermi level in LaAlGe and LaPtBi
exhibit resilience against alterations in electronic temperature (Fig. S5 and
S8), suggesting that these topological quasiparticles play a dominant role in
the low-energy excitations in the absence of 4$f$ electrons. In the context of
topological and Weyl semimetals featuring rare-earth 4$f$ quasiparticles, the
4$f$ electrons are subject to substantial Coulomb repulsion as a result of the
null screening caused by topological quasiparticles (type-II), or 4$f$
electrons become quasiparticles pushing the topological quasiparticle away to
enhance the screening effect (type-I). It thus predicts an interplay between
4$f$ and topological quasiparticles in the atypical low-energy excitations,
providing an explanation for the inconsistent topological phases observed in
rare-earth-based semimetals. Remarkably, type-I semimetal has the potential to
exhibit a novel synergy between topological and heavy-fermion phenomena within
a single material.

### Methods

LQSGW+DMFT calculation
We adopted experimental lattice constants of space group $I4_{1}md$ (No. 109)
(PrAlGe: $a=$ 4.253, $c=$ 14.641 Å) [34], (LaAlGe: $a=$ 4.336, $c=$ 14.828 Å)
[35], and space group $F\bar{4}3m$ (No. 216) (HoPtBi: $a=$ 6.631 Å), (PrPtBi:
$a=$ 6.78 Å), (LaPtBi: $a=$ 6.829 Å) [36] (see Fig. S1). We employed the
$ab$-$initio$ linearized quasiparticle self-consistent GW (LQSGW) method
combined with dynamical mean field theory (DMFT) [23]. This LQSGW+DMFT is
based on a simplified full GW+DMFT approach [37, 38, 39, 40, 32]. The
electronic structure was calculated using LQSGW approaches [41, 42], and the
local part of the GW self-energy was diagrammatically corrected within DMFT
[43, 44, 45]. We also explicitly calculated the double-counting energy and
dynamically screened Coulomb interaction tensor (see Fig. S2). Local self-
energies for R-4$f$ and R-5$d$ were obtained by solving impurity models, with
spin-orbital coupling included in all calculations. The self-energy is
computed along an imaginary frequency axis and then analytically continued
through a maximum entropy method [46]. Please see the details of this method
in the Supplementary information.

## Acknowledgments

We acknowledge the High Performance Computing Center (HPCC) at Texas Tech
University for providing computational resources that have contributed to the
research results reported within this paper. M.K. and Z.B acknowledge the
support from the National Science Foundation (Grant No. NSF DMR-2317013). C.H.
Park acknowledges the support by the National Research Foundation of Korea
(NRF) grant (Grant No. NRF-2022R1A2C1005548).

Competing Interests The authors declare no competing interests.

Data availability The data that support the ﬁndings of this study are
available from the corresponding authors upon reasonable request.

Author contributions B.K. designed the project. B.K. performed the LQSGW+DMFT
calculations and conducted the data analysis. All authors wrote the
manuscript, discussed the results, and commented on the paper.

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#  Gromov–Hausdorff distances between normed spaces

I. N. Mikhailov

###### Abstract

In the present paper we study the original Gromov–Hausdorff distance between
real normed spaces. In the first part of the paper we prove that two finite-
dimensional real normed spaces on a finite Gromov–Hausdorff distance are
isometric to each other. We then study the properties of finite point sets in
finite-dimensional normed spaces whose cardinalities exceed the equilateral
dimension of an ambient space. By means of the obtained results we prove the
following enhancement of the aforementioned theorem: every finite-dimensional
normed space lies on an infinite Gromov–Hausdorff distance from all other non-
isometric normed spaces.

##  1 Introduction

The Gromov–Hausdorff distance is one of the most beautiful constructions in
metric geometry. It allows to compare how similar two arbitrary metric spaces
are. The concept was first introduced in [4] by D. Edwards. Later it became
famous due to M. Gromov’s paper [5] (see historical details in [12]). The
classical Gromov–Hausdorff distance between metric spaces $X$ and $Y$ is
defined as the infinum of the Hausdorff distances between the images
$X^{\prime}$ and $Y^{\prime}$ of the spaces $X$ и $Y$ under all possible
isometric embeddings $\phi\colon X\to Z$ и $\psi\colon Y\to Z$ into an
arbitrary metric space $Z$.

In the majority of applications the Gromov–Hausdorff distance is used to study
compact metric spaces. In case of unbounded metric spaces the punctured spaces
are considered, i.e., the pairs $(X,\,p)$ where $X$ is an arbitrary metric
space and $p$ is one of its points. Traditionally, in such situation the
topology generated by the Gromov–Hausdorff distance is of more interest than
the distance itself. By $B_{r}(x)$ let us denote a closed ball of radius $r$
centered at the point $x$. Then the sequence of punctured metric spaces
$(X_{n},\,p_{n})$ converges by Gromov–Hausdorff to a punctured metric space
$(Z,\,p)$, iff the following conditions hold (see details in [3], [8]): for
every $r>0$ and $\varepsilon>0$ there exists a natural number $n_{0}$ such
that for every $n>n_{0}$ there exists a mapping $f\colon B_{r}(p_{n})\to X$
with properties

  1. (1)

$f(p_{n})=p$;

  2. (2)

$\operatorname{dis}(f):=\sup\Bigl{\\{}\bigl{|}|xx^{\prime}|-|f(x)f(x^{\prime})|\bigr{|}\colon
x,\,x^{\prime}\in X\Bigr{\\}}<\varepsilon$;

  3. (3)

$\varepsilon$-neighbourhood of the set $f(B_{r}(p_{n}))$ in $X$ contains the
ball $B_{r-\varepsilon}(p)$.

In the present paper we study the Gromov–Hausdorff distance in its original
sense. Restrictions of the Gromov–Hausdorff distance to various classes of
metric spaces possess fascinating and often unexpected geometric properties.
For example, let us consider the class (in the sense of NBG set theory)
$\mathcal{GH}$ of representatives of all isometry classes of metric spaces.
The Gromov–Hausdorff distance defines a generalised pseudometric on this
class, i.e., is a symmetric, non-zero function, that satisfies the triangle
inequality. This space is called _the Gromov-Hausdorff class_. The equivalence
classes in $\mathcal{GH}$ under the relation: $X\sim Y$ iff
$\operatorname{d}_{GH}(X,\,Y)<\infty$ — are called _clouds_. In the monography
[6] M. Gromov announced that each cloud in $\mathcal{GH}$ is contractible and
gave an example of the cloud corresponding to $\mathbb{R}^{n}$. However, it
turned out later that there exist clouds which are not even invariant under
the multiplication of all their spaces by some positive number $\lambda$ (see
details in paper [1]).

In the paper we are studying the Gromov–Hausdorff distance between real normed
spaces. Unlike in the paper [9], we consider the global Gromov–Hausdorff
distance rather than the Gromov–Hausdorff distance between the closed unit
balls of normed spaces. For an arbitrary $\varepsilon>0$ let us call a linear
mapping $T$ between normed spaces $E$ and $F$ an $\varepsilon$-isometry, if
for all $x,\,y\in E$ an inequality holds
$\Bigl{|}\|x-y\|-\|T(x)-T(y)\|\Bigr{|}\leqslant\varepsilon$. In the paper [7]
the following theorem is proved: if $E$ and $F$ are finite-dimensional normed
spaces and $T\colon E\to F$ is a surjective $\varepsilon$-isometry for some
$\varepsilon$, then there exists an isometry $I\colon E\to F$ such that
$\|T(x)-I(x)\|\leqslant 5\varepsilon$ for all $x\in E$. It follows from this
result that any two finite-dimensional normed spaces on a finite
Gromov–Hausdorff distance are isometric. In section $3$ we give a simpler
proof of this theorem, based on Theorem 7, which provides a natural sufficient
condition for isometric embeddability of a given bounded metric space into a
given finite-dimensional normed space.

_Equilateral dimension_ of a metric space is the largest cardinality of such
its subset that all the distances between different points of this subset are
pairwise equal to each other. In the paper [11] it is proved that the
equilateral dimension of an arbitrary $n$-dimensional normed space does not
exceed $2^{n}$. In section $4$ we study some properties of finite sets of
points in normed spaces whose cardinalities exceed the equilateral dimension
of the ambient space. Based on the obtained results, we prove a new lower
estimate on the Gromov–Hausdorff distance between a finite-dimensional normed
space and an arbitrary metric space of a larger equilateral dimension (see
Theorem 11). From this estimate we obtain the following enhancement of the
main theorem of section 3: every finite-dimensional normed space lies on an
infinite Gromov–Hausdorff distance from all other non-isometric normed spaces.

The author expresses deep gratitude to his scientific advisor professor A. A.
Tuzhilin and also professor A. O. Ivanov for posing the problem and constant
attention to the work.

##  2 Main definitions and preliminary results

We start with the introduction of some basic notation. Let $X$ be an arbitrary
metric space. We denote the distance between the points $x$ and $y$ of $X$ by
$|xy|$. Let $U_{r}(a)=\\{x\in X\colon|ax|<r\\}$, $B_{r}(a)=\\{x\in
X\colon|ax|\leqslant r\\}$ be an open and a closed ball of radius $r$ centered
at the point $a$ correspondingly. For an arbitrary subset $A\subset X$ we
define $U_{r}(A)=\cup_{a\in A}U_{r}(a)$ — an open $r$-neighbourhood of $A$.
For non-empty subsets $A\subset X$ and $B\subset X$ by
$\operatorname{d}(A,\,B)$ we denote a simple distance between these subsets,
namely, $\operatorname{d}(A,\,B)=\inf\bigl{\\{}|ab|\colon\,a\in A,\,b\in
B\bigl{\\}}$.

Definition 1. Let $A$ and $B$ be non-empty subsets of a metric space $X$. The
Hausdorff distance between $A$ and $B$ is the value

$\operatorname{d}_{H}(A,\,B)=\inf\\{r>0\colon A\subset U_{r}(B),\,B\subset
U_{r}(A)\\}.$

Definition 2. Let $X$ and $Y$ be metric spaces. If $X^{\prime},\,Y^{\prime}$
are subsets of a metric space $Z$ such that $X^{\prime}$ is isometric to $X$
and $Y^{\prime}$ is isometric to $Y$, then we call the triple
$(X^{\prime},\,Y^{\prime},\,Z)$ a metric realization of a pair $(X,\,Y)$.

Definition 3. The Gromov–Hausdorff distance $\operatorname{d}_{GH}(X,\,Y)$
between two metric spaces $X$, $Y$ is the infinum of positive numbers $r$ such
that there exists a metric realization $(X^{\prime},\,Y^{\prime},\,Z)$ of a
pair $(X,\,Y)$ with $\operatorname{d}_{H}(X^{\prime},\,Y^{\prime})\leqslant
r$.

Let now $X,\,Y$ be non-empty sets.

Definition 4. Any subset $\sigma\subset X\times Y$ is called a relation
between $X$ and $Y$.

Denote the set of all non-empty relations between $X$ and $Y$ by
$\mathcal{P}_{0}(X,\,Y)$.

Define

$\pi_{X}\colon X\times Y\rightarrow X,\;\pi_{X}(x,\,y)=x,$ $\pi_{Y}\colon
X\times Y\rightarrow Y,\;\pi_{Y}(x,\,y)=y.$

Definition 5. Relation $R\subset X\times Y$ is called a correspondence, if
$\pi_{X}|_{R}$ and $\pi_{Y}|_{R}$ are surjective.

Denote the set of all correspondences between $X$ and $Y$ by
$\mathcal{R}(X,\,Y)$.

Definition 6. Let $X,\,Y$ be arbitrary metric spaces. Then for every
$\sigma\in\mathcal{P}_{0}(X,\,Y)$ the distortion of $\sigma$ is defined as

$\operatorname{dis}\sigma=\sup\Bigl{\\{}\bigl{|}|xx^{\prime}|-|yy^{\prime}|\bigr{|}\colon(x,\,y),\,(x^{\prime},\,y^{\prime})\in\sigma\Bigr{\\}}.$

######  Claim 1 ([3]).

For arbitrary metric spaces $X$ and $Y$ the following equality holds

$2\operatorname{d}_{GH}(X,\,Y)=\inf\bigl{\\{}\operatorname{dis}\,R\colon
R\in\mathcal{R}(X,\,Y)\bigr{\\}}.$

Definition 7. Let $X$ be an arbitrary metric space. By $\mathcal{H}(X)$ we
denote the set of all non-empty closed bounded subsets in $X$. The Hausdorff
distance defines a finite metric on $\mathcal{H}(X)$ (see, for example, [3]).
The resulting metric space is called a hyperspace of the space $X$.

######  Theorem 1 ([3]).

Let $X$ be an arbitrary metric space. Then $X$ is compact iff $\mathcal{H}(X)$
is compact.

######  Theorem 2 ([3]).

If $X$ is a compact metric space and $Y$ is a complete metric space such that
$\operatorname{d}_{GH}(X,\,Y)=0$, then $X$ is isometric to $Y$.

Definition 8. The subset of an arbitrary metric space $X$ is called
equilateral if all the distances between its distinct points are pairwise
equal to each other. The equilateral dimension of a metric space $X$ is the
largest cardinality of its equilateral subset. We denote the equilateral
dimension of $X$ by $\operatorname{ed}(X)$.

######  Theorem 3 ([11]).

Let $V$ be an $n$-dimensional normed space. Then
$\operatorname{ed}(V)\leqslant 2^{n}$.

######  Theorem 4 ([2]).

Let $X$ be an arbitrary infinite-dimensional normed space. Then for an
arbitrary $m\in\mathbb{N}$ there exists a finite equilateral subset in $X$ of
$m$ points.

Definition 9. Let $X$ be an arbitrary metric space, and $\varepsilon>0$ an
arbitrary positive number. The subset $P\subset X$ is called
$\varepsilon$-_separated_ , if for any two points $p$ and $q$ in $P$ the
inequality holds $|pq|\geqslant\varepsilon$.

Definition 10. Let $V$ be a normed space, and $m$ be some positive integer. We
denote the infinum of positive numbers $r$ such that there exists a
$1$-separated set of $m$ points in $B_{r}(0)$ by $R_{m}(V)$.

Finally, we need a few classical results from analysis.

######  Theorem 5 ([3], [13]).

Let $K$ be a compact metric space. Then an arbitrary isometric mapping
$f\colon K\to K$ is surjective and, hence, is an isometry.

######  Theorem 6 ([10]).

Two arbitrary normed spaces are isometric iff their closed unit balls are
isometric.

##  3 Finite-dimensional case

In this section we show that any two finite-dimensional normed spaces on a
finite Gromov–Hausdorff distance are isometric to each other.

######  Theorem 7.

Let $V$ be a finite-dimensional normed space and $X$ be an arbitrary bounded
metric space. Suppose there exists a sequence of positive numbers
$(\varepsilon_{n})_{n\in\mathbb{N}}$ converging to $0$ such that for any $n$
there exists a mapping $f_{n}\colon X\to V$ with
$\operatorname{dis}(f_{n})\leqslant\varepsilon_{n}$. Then the completion
$\tilde{X}$ of $X$ is compact and both $X$ and $\tilde{X}$ can be
isometrically embedded into $V$.

###### Proof.

Fix an arbitrary $x\in X$. By $t_{n}$ denote a translation of $V$ by the
vector $-f_{n}(x)$. Let $g_{n}=t_{n}\circ f_{n}$. Since $t_{n}$ is an
isometry, we have $\operatorname{dis}\,g_{n}=\operatorname{dis}\,f_{n}$. For
every $n\in\mathbb{N}$ define $X_{n}=\overline{g_{n}(X)}$ — a closure of
$g_{n}(X)$ in the topology on $V$ generated by the norm. By the definition of
the distortion, for any points $a,\,b\in X$ the inequality holds
$\bigl{|}\|a-b\|-\|g_{n}(a)-g_{n}(b)\|\bigr{|}\leqslant\varepsilon_{n}$.
According to Claim 1, we obtain that
$\operatorname{d}_{GH}\bigl{(}X,\,g_{n}(X)\bigr{)}\leqslant\frac{\varepsilon_{n}}{2}$.
Since $\operatorname{d}_{GH}\bigl{(}g_{n}(X),\,\overline{g_{n}(X)}\bigr{)}=0$,
by applying the triangle inequality we obtain that

$\operatorname{d}_{GH}(X,\,X_{n})\leqslant\operatorname{d}_{GH}\bigl{(}X,\,g_{n}(X)\bigr{)}+\operatorname{d}_{GH}(g_{n}(X),\,X_{n})\leqslant\frac{\varepsilon_{n}}{2}.$

Note that for every $n\in\mathbb{N}$ the point $0$ belongs to the set $X_{n}$.
Also
$\operatorname{diam}\,X_{n}\leqslant\operatorname{diam}\,X+\varepsilon_{n}$.
Hence, all $X_{n}$ belong to the ball $B:=B_{\operatorname{diam}(X)+c}(0)$,
where $c$ is some constant such that $\varepsilon_{n}\leqslant c$ for every
$n$. The space $V$ is finite-dimensional, thus, $B$ is compact. Therefore,
$\mathcal{H}(B)$ is compact by Theorem 1. Sets $X_{n}$ are closed and bounded,
so they form a sequence of points in $\mathcal{H}(B)$. Then it follows from
the compactness of $\mathcal{H}(B)$ that there exists a subsequence
$X_{n_{s}}$ that converges to some non-empty closed and bounded subset $A$ of
$B$. Note that $A$ is compact as the closed subset of a compact ball $B$. Then
for every $s$ the following inequalities hold

$\operatorname{d}_{GH}(X,\,A)\leqslant\operatorname{d}_{GH}(X,\,X_{n_{s}})+\operatorname{d}_{GH}(X_{n_{s}},\,A)\leqslant\operatorname{d}_{GH}(X,\,X_{n_{s}})+\operatorname{d}_{H}(X_{n_{s}},\,A).$

The right side tends to $0$ when $s\to\infty$. Thus,
$\operatorname{d}_{GH}(X,\,A)=0$. Since
$\operatorname{d}_{GH}(X,\,\tilde{X})=0$, the triangle inequality implies,
that
$\operatorname{d}_{GH}(\tilde{X},\,A)\leqslant\operatorname{d}_{GH}(\tilde{X},\,X)+\operatorname{d}_{GH}(X,\,A)=0$,
i.e., $\operatorname{d}_{GH}(\tilde{X},\,A)=0$. Then from Theorem 2 it follows
that $\tilde{X}$ and $A$ are isometric. It implies that $\tilde{X}$ is compact
and can be isometrically embedded into $V$. In particular, $X$ can be
isometrically embedded into $V$. ∎

######  Lemma 1.

If $X,\,Y$ are normed spaces such that $\operatorname{d}_{GH}(X,\,Y)<\infty$,
then $\operatorname{d}_{GH}(X,\,Y)=0$.

###### Proof.

Suppose $\operatorname{d}_{GH}(X,\,Y)=c<\infty$. Then for an arbitrary
$\lambda>0$ the following equalities hold
$c=\operatorname{d}_{GH}(X,\,Y)=\operatorname{d}_{GH}(\lambda X,\,\lambda
Y)=\lambda\operatorname{d}_{GH}(X,\,Y)=\lambda c$. Hence, $c=0$. ∎

######  Theorem 8.

If $V,\,W$ are arbitrary finite-dimensional normed spaces with
$\operatorname{d}_{GH}(V,\,W)<\infty$, then $V$ and $W$ are isometric.

###### Proof.

From Lemma 1 we have $\operatorname{d}_{GH}(V,\,W)=0$. Then, by Claim 1, for
an arbitrary positive number $\varepsilon>0$ there exists a correspondence $R$
between $V$ and $W$ with the distortion
$\operatorname{dis}\,R\leqslant\varepsilon$.

Let us consider the unit balls $B_{1}=B^{V}_{1}(0),\,B_{2}=B^{W}_{1}(0)$ of
the spaces $V,\,W$ correspondingly. It follows from Theorem 7 that there exist
isometric maps $f_{1}\colon B_{1}\to B_{2}$ и $f_{2}\colon B_{2}\to B_{1}$.
Then $f_{2}\circ f_{1}\colon B_{1}\to B_{1}$ and $f_{1}\circ f_{2}\colon
B_{2}\to B_{2}$ are isometric embeddings. From Theorem 5 we conclude that the
constructed mappings $f_{2}\circ f_{1}$, $f_{1}\circ f_{2}$ are isometries. It
follows immediately that $f_{1}$ and $f_{2}$ are bijective, so the balls
$B_{1}$ and $B_{2}$ are isometric to each other. From Theorem 6 we conclude
that the spaces $X$ and $Y$ are isometric. ∎

Example 1. Let us consider any two non-isometric normed spaces $X$ and $Y$.
For example, two copies of $\mathbb{R}^{2}$ one with the euclidean norm and
one with the $\max$-norm. Let us also consider any $\varepsilon_{1}$-network
$\sigma_{1}$ in $X$ and any $\varepsilon_{2}$-network $\sigma_{2}$ in $Y$.
Then it immediately follows from the proven theorem that
$\operatorname{d}_{GH}(\sigma_{1},\,\sigma_{2})=\infty$. In particular, the
Gromov–Hausdorff distance between $\mathbb{Z}^{2}$ with metrics induced from
$X$ and $Y$ respectively is infinite.

##  4 Metric imbalance

We start this section with a few results about point sets in finite-
dimensional normed spaces whose cardinalities exceed the equilateral dimension
of the corresponding space (which is finite due to Theorem 3). Firstly, we
prove an important inequality that characterises such point sets.

######  Theorem 9.

Let $V$ be a finite-dimensional normed space with $\operatorname{ed}(V)=p$.
Then there exists a constant $c>0$ such that for an arbitrary set of $m>p$
distinct points $v_{1},\,\ldots,\,v_{m}$ in $V$ there exist three of them
$\\{v_{i},\,v_{j},\,v_{k}\\}$ such that

$\varphi(v_{i},\,v_{j},\,v_{k}):=\left|\frac{\|v_{i}-v_{k}\|}{\|v_{j}-v_{k}\|}-1\right|\geqslant
c.$

###### Proof.

Arguing by contradiction, suppose that for every
$\varepsilon_{n}=\frac{1}{n},\,n\in\mathbb{N},\,n\geqslant 6$ there exist
distinct points $v_{1}^{n},\,\ldots,\,v_{m}^{n}$ such that for all different
$i,\,j,\,k$ the following inequalities hold
$\varphi(v^{n}_{i},\,v^{n}_{j},\,v^{n}_{k})<\varepsilon_{n}$. For every
$\lambda>0$ we have $\varphi(\lambda v^{n}_{i},\,\lambda v^{n}_{j},\,\lambda
v^{n}_{k})=\varphi(v^{n}_{i},\,v^{n}_{j},\,v^{n}_{k})$, so without loss of
generality we assume that $\|v^{n}_{1}-v^{n}_{2}\|=1$. Then inequalities
$\varphi(v^{n}_{i},\,v^{n}_{2},\,v^{n}_{1})<\varepsilon_{n}$ imply that
$\bigl{|}\|v^{n}_{1}v^{n}_{i}\|-1\bigr{|}\leqslant\varepsilon_{n}$ for each
$i=1,\,\ldots,\,m$. Hence, from
$\varphi(v^{n}_{j},\,v^{n}_{1},\,v^{n}_{i})<\varepsilon_{n}$ it now follows
that

$\bigl{|}\|v^{n}_{i}-v^{n}_{j}\|-\|v^{n}_{1}-v^{n}_{i}\|\bigr{|}\leqslant\varepsilon_{n}\|v_{1}-v_{i}\|\leqslant\varepsilon_{n}(1+\varepsilon_{n})<2\varepsilon_{n}.$

By triangle inequality,

$\bigl{|}\|v^{n}_{i}-v^{n}_{j}\|-1\bigr{|}\leqslant\bigl{|}\|v^{n}_{i}-v^{n}_{j}\|-\|v^{n}_{1}-v^{n}_{i}\|\bigr{|}+\bigl{|}\|v^{n}_{1}-v^{n}_{i}\|-1\bigr{|}<3\varepsilon_{n}.$

Thus, for every distinct $i$ and $j$ the following equality holds
$\|v^{n}_{i}-v^{n}_{j}\|=1+\delta_{n}$ where $|\delta_{n}|<3\varepsilon_{n}$.
In particular, $\|v^{n}_{i}-v^{n}_{j}\|\geqslant\frac{1}{2}$ for $i\neq j$.

Let us put $\xi^{n}=(0,\,v_{2}^{n}-v_{1}^{n}\ldots,\,v_{m}^{n}-v_{1}^{n})\in
V^{m}$ and consider $V^{m}$ with the norm

$\|(v_{1},\,v_{2},\,\ldots,\,v_{m})\|=\max_{q}\|v_{q}\|.$

Note that the sequence $\xi^{n}$ is bounded due to already established
inequalities

$\bigl{|}\|v^{n}_{i}-v^{n}_{1}\|-1\bigr{|}\leqslant\varepsilon_{n},\;i=1,\,\ldots,\,m.$

Consider the subset
$P=\bigl{\\{}(v_{1},\,\ldots,\,v_{m})\colon\|v_{i}-v_{j}\|\geqslant\frac{1}{2}\;\forall\,i\neq
j\bigr{\\}}\subset V^{m}$ with the metric induced from $V^{m}$. Since
$\|v^{n}_{i}-v^{n}_{j}\|\geqslant\frac{1}{2}$ for $i\neq j$, the sequence
$\xi^{n}$ lies in $P$. The space $V^{m}$ is finite-dimensional so there exists
a subsequence $\xi^{n_{s}}$ that converges to some
$\xi=(w_{1},\,\ldots,\,w_{m})$, which belongs to $P$ due to the closeness of
$P$ in $V^{m}$. Define a function $\tilde{\varphi}\colon V^{m}\to\mathbb{R}$
by the formula

$\tilde{\varphi}(v_{1},\,\ldots,\,v_{m})=\max_{i\neq j\neq k\neq
i}\varphi(v_{i},\,v_{j},\,v_{k}).$

Note that $\tilde{\varphi}$ is continuous on $P$. Since
$\varphi(v^{n}_{i},\,v^{n}_{j},\,v^{n}_{k})=\varphi(v^{n}_{i}-v^{n}_{1},\,v^{n}_{j}-v^{n}_{1},\,v^{n}_{k}-v^{n}_{1})$,
it follows from the inequalities
$\varphi(v^{n}_{i},\,v^{n}_{j},\,v^{n}_{k})<\varepsilon_{n}$ that
$\varphi(v^{n}_{i}-v^{n}_{1},\,v^{n}_{j}-v^{n}_{1},\,v^{n}_{k}-v^{n}_{1})<\varepsilon_{n}$.
The latter inequalities are equivalent to
$\tilde{\varphi}(0,\,v^{n}_{2}-v^{n}_{1},\,\ldots,\,v^{n}_{m}-v^{n}_{1})<\varepsilon_{n}$.
Hence, from the continuity of $\tilde{\varphi}$ on $P$ we conclude that
$\tilde{\varphi}(w_{1},\,\ldots,\,w_{m})=0$. Thus, for every different
$i,\,j,\,k$ we have $\varphi(w_{i},\,w_{j},\,w_{k})=0$ which is equivalent to
$\|w_{i}-w_{k}\|=\|w_{j}-w_{k}\|$. Therefore, $(w_{1},\,\ldots,\,w_{m})$ is an
equilateral set in $V$ of cardinality $m>\operatorname{ed}(V)=p$ —
contradiction. ∎

Let $V$ be an arbitrary finite-dimensional normed space. Let us also fix a
positive integer $m$. We define a _metric imbalance_ of $V$ of order $m$ as
follows

$c_{m}(V)=\sup\left\\{c\colon\forall\,(v_{1},\,\ldots,\,v_{m})\in
V^{m}\,\exists\,i\neq j,\,j\neq k,\,k\neq
i\colon\left|\frac{|v_{i}v_{k}|}{|v_{j}v_{k}|}-1\right|\geqslant c\right\\}.$

We conclude immediately from this definition that metric imbalance is a non-
decreasing function of $m$. Besides, Theorem 9 implies that if
$m>\operatorname{ed}(V)$ then $c_{m}(V)>0$.

######  Theorem 10.

Let $V$ be a finite-dimensional normed space. Denote its metric imbalance by
$c_{m}:=c_{m}(V)$ and $R_{m}:=R_{m}(V)$. Then the following inequalities hold

$2R_{m}+1\geqslant c_{m}\geqslant R_{m}-2.$

###### Proof.

Let us put $t_{n}=c_{m}+\frac{1}{n}$, $n\in\mathbb{N}$. By definition of
$c_{m}$ there exists such set of distinct points
$(x^{n}_{1},\,\ldots,\,x^{n}_{m})$ that for all different $i,\,j,\,k$ the
following inequalities hold
$\varphi(x^{n}_{i},\,x^{n}_{j},\,x^{n}_{k})\leqslant t_{n}$.

Note that for every $\lambda>0$ it holds
$\varphi(x^{n}_{i},\,x^{n}_{j},\,x^{n}_{k})=\varphi(\lambda
x^{n}_{i},\,\lambda x^{n}_{j},\,\lambda x^{n}_{k})$. Hence, by putting
$\lambda=\min_{i\neq j}|x^{n}_{i}x^{n}_{j}|$ and
$y^{n}_{i}=\frac{x^{n}_{i}}{\lambda}$, we obtain such set of points
$(y^{n}_{1},\,\ldots,\,y^{n}_{m})$ that for all different $i,\,j,\,k$ we have
$\varphi(y^{n}_{i},\,y^{n}_{j},\,y^{n}_{k})\leqslant t_{n}$, and also
$\min_{i\neq j}|y^{n}_{i}y^{n}_{j}|=1$. By renumerating the points, without
loss of generality we can assume that $\min_{i\neq
j}|y^{n}_{i}y^{n}_{j}|=|y^{n}_{1}y^{n}_{2}|$. Let us put
$z^{n}_{j}=y^{n}_{j}-y^{n}_{1}$. Since
$\varphi(y^{n}_{i},\,y^{n}_{j},\,y^{n}_{k})=\varphi(y^{n}_{i}-y^{n}_{1},\,y^{n}_{j}-y^{n}_{1},\,y^{n}_{k}-y^{n}_{1})=\varphi(z^{n}_{i},\,z^{n}_{j},\,z^{n}_{k})$,
a set of points $(z^{n}_{1},\,\ldots,\,z^{n}_{m})$ has the following
properties: for every different $i,\,j,\,k$ the inequalities hold
$\varphi(z^{n}_{i},\,z^{n}_{j},\,z^{n}_{k})\leqslant t_{n}$; $\min_{i\neq
j}|z^{n}_{i}z^{n}_{j}|=|z^{n}_{1}z^{n}_{2}|=1$, and, finally, $z^{n}_{1}=0$.

For each $k$ distinct from $1$ and $2$ the inequality holds
$\varphi(z^{n}_{k},\,z^{n}_{2},\,z^{n}_{1})\leqslant t_{n}$, therefore,
$\left|\frac{|z^{n}_{k}z^{n}_{1}|}{|z^{n}_{2}z^{n}_{1}|}-1\right|=\Bigl{|}|z^{n}_{1}z^{n}_{k}|-1\Bigr{|}\leqslant
t_{n}$ from which it follows that $|z^{n}_{1}z^{n}_{k}|\leqslant t_{n}+1$.
Hence, for every $n$ all the points $z^{n}_{k}$ belong to the ball
$B_{t_{n}+1}(0)$. Besides, $\min_{i\neq j}|z^{n}_{i}z^{n}_{j}|=1$. Then by
definition of $R_{m}$ for every $n$ the inequality holds $R_{m}\leqslant
c_{m}+1+\frac{1}{n}$. Therefore, $R_{m}\leqslant c_{m}+1$.

Let us prove the second inequality. Consider an arbitrary $1$-separated set
$a_{1},\,\ldots,\,a_{m}$ in $V$. By definition of the metric imbalance there
exist such different indices $i,\,j,\,k$ that
$\Bigl{|}\frac{|a_{i}a_{k}|}{|a_{j}a_{k}|}-1\Bigr{|}\geqslant c_{m}$. Then

$|a_{i}a_{k}|\geqslant(c_{m}-1)|a_{j}a_{k}|\geqslant c_{m}-1,$

which implies that it is impossible for a $1$-separated set of $m$ points to
lie in a ball of a radius less than $\frac{c_{m}-1}{2}$. Hence,
$R_{m}\geqslant\frac{c_{m}-1}{2}$ which is equivalent to $2R_{m}+1\geqslant
c_{m}$.

∎

######  Corollary 1.

If $V$ is a finite-dimensional normed space then
$\lim_{m\to\infty}c_{m}(V)=\infty$.

###### Proof.

Inequalities 10 imply that it suffices to prove that
$\lim_{m\to\infty}R_{m}=\infty$. Suppose that is not true. From the definition
of $R_{m}$ we conclude that it is a non-decreasing function of $m$. Then it
follows from our assumption that there exists a constant $C$ such that an
inequality $C\geqslant R_{m}$ holds for all $m$. But this inequality would
mean that it is possible to find an arbitrarily large $1$-separated set in
$B_{C}(0)$ which contradicts its compactness. ∎

######  Theorem 11.

Let $Y$ be a finite-dimensional normed space with $\operatorname{ed}(Y)=n$.
Also let $X$ be an arbitrary metric space with an equilateral subset
$\\{x_{1},\,\ldots,\,x_{m}\\}$ of diameter $d$, where $m>n$. Denote
$c_{m}:=c_{m}(Y)$. Then

$\operatorname{d}_{GH}(X,\,Y)\geqslant\frac{1}{2}\min\Bigl{\\{}\frac{d}{2},\,\frac{dc_{m}}{2+c_{m}}\Big{\\}}>0.$

###### Proof.

Let $R$ be an arbitrary correspondence between $X,\,Y$ with the distortion
$\operatorname{dis}\,R=t$. At first, we consider the case $t<\frac{d}{2}$.
Choose an arbitrary $y_{i}\in R(x_{i})$. In the considered case all $y_{i}$
are automatically different. By the definition of distortion
$\Bigl{|}|y_{i}y_{j}|-|x_{i}x_{j}|\Bigr{|}\leqslant t$. Hence, the following
inequalities hold

$\frac{-2t}{d+t}=\frac{|x_{i}x_{k}|-t}{|x_{j}x_{k}|+t}-1\leqslant\frac{|y_{i}y_{k}|}{|y_{j}y_{k}|}-1\leqslant\frac{|x_{i}x_{k}|+t}{|x_{j}x_{k}|-t}-1=\frac{2t}{d-t}.$

According to Theorem 9 there exist such indices $i,\,j,\,k$ that
$\displaystyle\left|\frac{|y_{i}y_{k}|}{|y_{j}y_{k}|}-1\right|\geqslant
c_{m}$. Then $\displaystyle\frac{2t}{d-t}\geqslant c_{m}$, which implies
$\displaystyle t\geqslant\frac{dc_{m}}{2+c_{m}}$. Thus, for an arbitrary
correspondence $R\in\mathcal{R}(X,\,Y)$ its distortion is either not less than
$\frac{d}{2}$, or not less than $\frac{dc_{m}}{2+c_{m}}$. It follows that
$\operatorname{dis}\,R\geqslant\min\Bigl{\\{}\frac{d}{2},\,\frac{dc_{m}}{2+c_{m}}\Bigr{\\}}$.
Hence, by Claim 1 we obtain that
$\operatorname{d}_{GH}(X,\,Y)\geqslant\frac{1}{2}\min\Bigl{\\{}\frac{d}{2},\,\frac{dc_{m}}{2+c_{m}}\Big{\\}}$.
∎

Note that Corollary 1 implies that
$\lim\limits_{m\to\infty}\frac{dc_{m}}{2+c_{m}}=d$. So for sufficiently large
$m$ the estimate from Theorem 11 turns into
$\operatorname{d}_{GH}(X,\,Y)\geqslant\frac{d}{4}.$

######  Theorem 12.

Let $X,\,Y$ be normed spaces such as $\dim(Y)<\infty$ and
$\operatorname{d}_{GH}(X,\,Y)<\infty$. Then $X$ is isometric to $Y$.

###### Proof.

According to Theorem 8, it suffices to prove that $X$ is finite-dimensional.
Suppose $\dim(X)=\infty$. Theorem 3 implies that the equilateral dimension of
$Y$ is finite. Denote $\operatorname{ed}(Y)=n$. By Theorem 4 there exists an
equilateral set $x_{1},\,\ldots,\,x_{m}\in X$ of $m>n$ points. Denote its
diameter by $d$. Then for an arbitrary $\lambda>0$ points $\lambda
x_{1},\,\ldots,\,\lambda x_{m}$ form an equilateral set in $X$ of diameter
$\lambda d$. Hence, there are equilateral sets of $m>n$ points and
sufficiently large diameters in $X$. Now Theorem 11 implies that
$\operatorname{d}_{GH}(X,\,Y)=\infty$ — contradiction. ∎

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