# An Interplay of Topology and Quantized Geometric Phase for two Different
Symmetry-Class Hamiltonians

Rahul S  Ranjith Kumar R  Y R Kartik  Poornaprajna Institute of Scientific
Research, 4, Sadashivanagar, Bangalore-560 080, India.  Manipal Academy of
Higher Education, Madhava Nagar, Manipal - 57610 4, India.  Amitava Banerjee
Department of Physics, University of Maryland, College Park, MD 20742, USA.
Sujit Sarkar  Poornaprajna Institute of Scientific Research, 4,
Sadashivanagar, Bangalore-560 080, India.

###### Abstract

Study of symmetry, topology and geometric phase can reveal many new and
interesting results on the topological states of matter. Here we present a
completely new and interesting result of symmetry, topology and quantization
of geometric phase along with the physical explanation for two different
symmetry classes. We present a detailed study of the auxiliary space for two
different symmetry classes of Hamiltonians. We show explicitly that the origin
of the auxiliary space inside the curve is only a necessary condition but it
is not a sufficient condition for the topological state. One of the most
interesting results is that same symmetry-class Hamiltonians show different
behaviour in topology and quantized geometric phase.

Keywords : Geometric phase, Topological quantization, Quantum phase
transition, Auxiliary space.

Introduction :
Symmetry plays a significant role in the study of different physical problems.
It represents a transformation that leaves the physical system invariant. In
quantum mechanics symmetry transformations can be classified as continuous
(rotation, translation) and discrete (parity, lattice translations, time
reversal). Continuous symmetry transformations give rise to conservation of
probabilities and discrete symmetry transformations give rise to the quantum
numbers. The second revolution of quantum mechanics is topological states of
matter [1]. Symmetry also plays an important role in the study of topological
states of matter[2].
Topological insulators and superconductors are the important topological
states of matter which can be distinguished based on the symmetry constraints
[3, 4]. On the basis of presence or absence of non-spatial symmetries like
time-reversal, particle-hole and chiral, one can classify a single-particle
Hamiltonian into different symmetry classes [5]. In each symmetry class one
can distinguish between topological distinct phases using the topological
invariant. There are ten distinct symmetry classes of random matrices, which
can be interpreted as first-quantized Hamiltonians of certain non-interacting
fermionic systems [6]. Among the ten symmetry classes for 1D Hamiltonians,
only the AIII, BDI, D, DIII and CI classes show topological states [6]. We
present basic symmetry classes in table 1. Thus there will be topological
quantum phase transition between two distinct phases within a symmetry class
by closing the gap. This can also be characterized by the quantization of
geometric phase [7, 8].

Geometric phase more commonly known as Berry phase [9], is a phase difference
acquired by the state when subjected to cyclic adiabatic process [10, 11, 12].
The geometric phase in a 1D Bloch band system is called Zak phase [13]. For a
given Bloch wave $\psi_{k}$ with a quasi-momentum $k$, reciprocal lattice
vector $G$, lattice spacing $d$, the Zak phase can be expressed as

$\varphi_{Zak}=i\int_{-\frac{G}{2}}^{\frac{G}{2}}\left\langle
u_{k}|\partial_{k}|u_{k}\right\rangle,$

where $u_{k}(x)=e^{-ikx}\psi_{k}(x)$ and $G=\frac{2\pi}{d}$. The physics of
geometric phase reveals many important aspects of topological state of matter
[14].

There have been many breakthroughs in the field of topological quantum
condensed matter starting with integer quantum Hall effect [15] and fractional
quantum Hall effect [16], and later the idea of topological insulators [17,
18, 19, 20, 21, 22, 23]. One of the classic examples of this kind is 1D
topological superconductor [4]. In this case topologically trivial and non-
trivial phases are distinquished by gap closing. This can be characterized by
the Pfaffian of Majorana representation matrix. In general, the Pfaffian
$Pf[A]$ of a $2n\times 2n$ skew-symmetric matrix A is defined as

$Pf[A]=\frac{1}{2^{n}n!}\sum_{\sigma\in\Pi_{2n}}sign\Pi^{n}_{i=1}A_{\sigma(2i-1),\sigma(2i)}.$
(1)

If $A$ is a $2\times 2$ matrix, $A=\left(\begin{matrix}0&&a\\\
-a&&0\end{matrix}\right),$ then $Pf[A]=a$. In the case of topological states
of matter Pfaffian of Majorana representation matrix is a topological
invariant. On the other hand, one can also write the Hamiltonian in the form
of Bogoliubov de Gennes (BdG) mean-field Hamiltonian using Nambu spinor. Then
the anti-unitary particle-hole constraint of the BdG Hamiltonian gives rise to
the quantization of geometric phase [24], which indicates the topological
quantum phase transition. The topological configuration space of a system
gives rise to the particular value of a topological invariant quantity like
winding number. The closed curves in the configuration space are the auxiliary
space curves which also specify the topological states of the system.
Auxiliary space curves have a unique way of representing the topological
quantum phase transition. When the system is in the topological state, the
auxiliary space curve encircles the origin; for the non-topological state, the
origin lies outside the space curve; and at the point of phase transition, the
origin lies on the space curve in which case the topological invariant number
is ill defined [8]. In the present study we consider two different symmetry
classes of model Hamiltonians. One is BDI and the other is AIII symmetry
class. These two different classes belong to the ten symmetry classes of the
topological state of matter (see table 1).

Table 1: Periodic table of topological insulators and topological
superconductors [6]. Here $\Theta$ are time-reversal, $\Xi$ are particle-hole
and $\Pi$ are chiral symmetry operators.

Topological state of quantum matter is a very rapidly growing field. One may
quantum-simulate different topological properties in different physical
systems [25, 26, 27]. Therefore the different topological symmetry-classes
Hamiltonians we present here may be important in the study of quantum
simulation physics. Simulating quantum systems like topological state of
matter is practically not possible using classical systems. One can also use
quantum devices that mimic the evolution of other quantum systems as quantum
simulators [28]. Therefore the experimental realization of these quantum
devices helps us in understanding many interesting quantum systems.

Motivation and Goals:
1) We consider four model Hamiltonians
($\mathcal{H}^{(1)}(k)$,$\mathcal{H}^{(2)}(k)$,
$\mathcal{H}^{(3)}(k)$,$\mathcal{H}^{(4)}(k)$) in the momentum space and study
explicitly the symmetry classes of these model Hamiltonians [30, 29, 31].
2) We study the quantization of geometric phase and also calculate the
Pfaffian for different model Hamiltonians to characterize the topological
phase transition.
3) We also study the auxiliary space curve and its closseness condition for
different model Hamiltonians to show explicitly the possibility for
topological non-trivial phase to exist. This is also verified by the study of
dispersion plots for different model Hamiltonians.
4) These new and important results based on two different topological symmetry
classes motivate quantum simulation physicists to quantum-simulate these types
of model Hamiltonian systems.

Model Hamiltonians:
First case :We consider the Kitaev’s chain in the matrix form as our model
Hamiltonian,

$\mathcal{H}^{(1)}(k)=\left(\begin{matrix}0&(\epsilon_{k}-\mu)+2i\Delta\sin{k}\\\
(\epsilon_{k}-\mu)-2i\Delta\sin{k}&0\end{matrix}\right),$ (2)

where $\epsilon_{k}=-2t\cos k$.

Second case: Here we add variant term, $\alpha k$ to $\sigma_{y}$ component.
This is also a plausible system to quantum-simulate, since the Hamiltonian is
linear in momentum $k$. With this motivation we consider the Kitaev’s
Hamiltonian with this variant term,

$\mathcal{H}^{(2)}(k)=\left(\begin{matrix}0&(\epsilon_{k}-\mu)+2i\Delta\sin{k}+i\alpha
k\\\ (\epsilon_{k}-\mu)-2i\Delta\sin{k}-i\alpha k&0\end{matrix}\right).$ (3)

We will show that this Hamiltonian has the same symmetry properties of Kitaev
model and both the Hamiltonians fall under BDI symmetry class.

Third case: Here we consider the variant term in the $\sigma_{x}$ component.
Kitaev’s Hamiltonian in presence of variant term becomes

$\mathcal{H}^{(3)}(k)=\left(\begin{matrix}0&(\epsilon_{k}-\mu+\alpha
k)+2i\Delta\sin{k}\\\ (\epsilon_{k}-\mu+\alpha
k)-2i\Delta\sin{k}&0\end{matrix}\right).$ (4)

We will show that this Hamiltonian falls under the AIII symmetry class.

Fourth case: Here we consider the variant term in both $\sigma_{x}$ and
$\sigma_{y}$ components, which gives

$\mathcal{H}^{(4)}(k)=\left(\begin{matrix}0&(\xi_{k}+\alpha_{1}k)+2i\Delta\sin{k}+i\alpha_{2}k\\\
(\xi_{k}+\alpha_{1}k)-2i\Delta\sin{k}-i\alpha_{2}k&0\end{matrix}\right),$ (5)

where $\xi_{k}=\epsilon_{k}-\mu$. We will show that this Hamiltonian has the
same symmetry properties as $\mathcal{H}^{(3)}(k)$ and falls under the AIII
symmetry class. All these Hamiltonians are hermitian in character.
Results and discussion for the Hamiltonians of BDI symmetry class: In this
section we present the detailed study and results of the two model
Hamiltonians $\mathcal{H}^{(1)}(k)$ and $\mathcal{H}^{(2)}(k)$.
$\mathcal{H}^{(1)}(k)$ in the matrix form can be written as

$\mathcal{H}^{(1)}(k)=\left(\begin{matrix}-2t\cos(k)-\mu&&2i\Delta\sin(k)\\\
-2i\Delta\sin(k)&&2t\cos(k)+\mu\end{matrix}\right).$ (6)

Now we study the basic symmetries, i.e., time-reversal ($\mathcal{T}$),
particle-hole ($\mathcal{C}$) and chiral ($\mathcal{S}$) symmetry operations
for the model Hamiltonian. Among these three basic symmetries time-reversal
operator commutes with the Hamiltonian while particle-hole and chiral
operators anti-commute with the Hamiltonian [5, 12]. The operator forms of
these symmetry operations are $\mathcal{T}=K$ (where $K$ is complex conjugate
operator), $\mathcal{C}=\sigma_{x}K$ (where $\sigma_{x}$ is one of the Pauli’s
matrices) and $\mathcal{S}=\sigma_{x}$. The commutation relations can be
checked as

$\displaystyle\mathcal{T}\mathcal{H}^{(1)}(k)\mathcal{T}^{-1}$
$\displaystyle={K}\mathcal{H}^{(1)}(k){K}^{-1}$ (7)
$\displaystyle=\hat{K}\left(\begin{matrix}-2t\cos(k)-\mu&&2i\Delta\sin(k)\\\
-2i\Delta\sin(k)&&2t\cos(k)+\mu\\\ \end{matrix}\right)\hat{K}$
$\displaystyle=\mathcal{H}^{(1)}(k),$

$\displaystyle\mathcal{C}\mathcal{H}^{(1)}(k)\mathcal{C}^{-1}$
$\displaystyle=(\sigma_{x}{K})\mathcal{H}^{(1)}(k)(\sigma_{x}{K})^{-1}$ (8)
$\displaystyle=\left(\begin{matrix}0&&1\\\ 1&&0\\\
\end{matrix}\right)\left(\begin{matrix}-2t\cos(k)-\mu&&i2\Delta\sin(k)\\\
-i2\Delta\sin(k)&&2t\cos(k)+\mu\\\
\end{matrix}\right)\left(\begin{matrix}0&&1\\\ 1&&0\\\ \end{matrix}\right)$
$\displaystyle=-\mathcal{H}^{(1)}(k),$

$\displaystyle\mathcal{S}\mathcal{H}^{(1)}(k)\mathcal{S}^{-1}$
$\displaystyle=(\sigma_{x})\mathcal{H}^{(1)}(k)(\sigma_{x})^{-1}$ (9)
$\displaystyle=\left(\begin{matrix}0&&1\\\ 1&&0\\\
\end{matrix}\right)\left(\begin{matrix}-2t\cos(k)-\mu&&i2\Delta\sin(k)\\\
-i2\Delta\sin(k)&&2t\cos(k)+\mu\\\
\end{matrix}\right)\left(\begin{matrix}0&&1\\\ 1&&0\\\ \end{matrix}\right)$
$\displaystyle=-\mathcal{H}^{(1)}(k).$

Thus we have

$\mathcal{T}\mathcal{H}^{(1)}(k)\mathcal{T}^{-1}=\mathcal{H}^{(1)}(k),\;\;\;\;\mathcal{C}\mathcal{H}^{(1)}(k)\mathcal{C}^{-1}=-\mathcal{H}^{(1)}(k),$
$\mathcal{S}\mathcal{H}^{(1)}(k)\mathcal{S}^{-1}=-\mathcal{H}(k).$ (10)

The Hamiltonian falls under the symmetry class BDI of the ten symmetry classes
of topological insulators and superconductors [6] with topological invariant
number $\mathbb{Z}$, which takes integer values (see table 1). Each value of
$\mathbb{Z}$ indicates a set of $\mathcal{H}^{(1)}(k)$ which can be
interpolated continuously without breaking the symmetries and without closing
the energy gap. Topological quantum phase transition can be observed by tuning
the parameters of $\mathcal{H}^{(1)}(k)$ to get gapless state. This closing
the gap involves changes in $\mathbb{Z}$ by one unit. To get the condition for
the parameters which distinguish between topological trivial and non-trivial
phases one can calculate the Majorana number ($\mathcal{M}$).
The Kitaev’s chain in lattice is [32]

$H_{0}=[\sum_{n}-{t}({c_{n}}^{\dagger}c_{n+1}+h.c)-{\mu}{c_{n}}^{\dagger}{c_{n}}+{|\Delta|}({c_{n}}{c_{n+1}}+h.c)],$
(11)

where $t$ is the hopping matrix element, $\mu$ is the chemical potential, and
$|\Delta|$ is the magnitude of the superconducting gap. We write the
Hamiltonian in the momentum space as

$H_{1}=\sum_{k>0}(\mu+2t\cos
k)({\psi_{k}}^{\dagger}{\psi_{k}}+{\psi_{-k}}^{\dagger}{\psi_{-k}})+2i\Delta\sum_{k>0}\sin
k({\psi_{k}}^{\dagger}{\psi_{-k}}^{\dagger}+{\psi_{k}}{\psi_{-k}}),$ (12)

where ${\psi^{\dagger}}_{k}(\psi_{k})$ is the creation (annihilation) operator
of the spinless fermion of momentum $k$.
One can rewrite eq. 11 in the Majorana fermion operators by using the relation
$\gamma_{2i-1}=c_{i}^{\dagger}+c_{i}$ and
$\gamma_{2i}=i(c_{i}^{\dagger}-c_{i})$, where $\gamma_{j}$ represent real
fermions with properties $\gamma_{j}^{\dagger}=\gamma_{j}$ and
$\left\\{\gamma_{i},\gamma_{j}\right\\}=2\delta_{ij}$. Thus eq. 11 can be
written in the Majorana representation as [32]

$\mathcal{H}^{(1)}(k)=\frac{i}{2}\sum_{\alpha,\beta}\gamma_{\alpha}A_{\gamma\beta}\gamma_{\beta},$
(13)

where $A$ is a real and anti-symmetric matrix.

One can verify the existence of topological trivial and non-trivial phases by
calculating the Pfaffian of Majorana representation matrix, with a real
orthogonal transformation, $W$, and using the property of Pfaffian, i.e.
$Pf[WAW^{T}]=Pf[A]\det[W]$. One can write the Majorana number
$\mathcal{M}=\det(W(0))\det(W(\pi))$. The property of $W(k)$, i.e.
$W(k)^{*}=W(-k)$, implies quantized geometric phase indicating topological
quantum phase transition [33]. One can write the Hamiltonian in block-diagonal
form by using the real orthogonal transformation,

${\small A(k)=\left(\begin{matrix}0&&-2t\cos k-\mu+i2\Delta\sin k\\\ 2t\cos
k+\mu+i2\Delta\sin k&&0\\\ \end{matrix}\right).}$ (14)

Majorana number can be expressed in terms of Pfaffian of the matrix $A(k)$ as

$\mathcal{M}=sign\left\\{Pf[A(0)]Pf[A(\pi)]\right\\}.$ (15)

Pfaffian of the matrix $A(k)$ at $k=0$ and $k=\pi$ is calculated as

$Pf[A(0)]=-(2t+\mu),\;\;\;\;Pf[A(\pi)]=2t-\mu.$ (16)

For the parametric condition $\mu>2t$, the Majorana number $\mathcal{M}=+1$
indicates that the system is in the non-topological phase; also, for $\mu<2t$,
the Majorana number $\mathcal{M}=-1$ indicates that the system is in the
topological phase. It is clear from this that the topological phase transition
occurs at $\mu=\pm 2t$.

Existence of topological states can also be confirmed from fig. 1.

Figure 1: Parametric plots for $\mathcal{H}^{(1)}(k)$ Hamiltonian for
different values of $\mu$.

This shows the auxiliary space of the Kitaev chain, which is a closed
trajectory with origin inside the curve. The integral along this trajectory
takes quantized values indicating the topological quantum phase transition.
The value of Zak phase is $\pi$ when the origin is inside the curve of the
auxiliary space and 0 when it is outside. Since $\mathcal{H}^{(1)}(k)$ has
anti-unitary particle-hole symmetry it gives the reality condition for
Majorana representaion matrix ($A(k)^{*}=A(-k)$), i.e.

$\mathcal{C}A(k)\mathcal{C}^{-1}=A(-k).$ (17)

This results in the quantized Zak phase which is $0$ for trivial and $\pi$ for
non-trivial phases. We verify this by calculating the geometric phase for the
system. First we write Kitaev’s chain in momentum space as

$H_{k}=(\epsilon_{k}-\mu)\sigma_{z}-2\Delta\sin k\sigma_{y}.$

One can also write the same Hamiltonian in a rotated basis as

$H_{k}=(\epsilon_{k}-\mu)\sigma_{x}-2\Delta\sin k\sigma_{y},$ (18)

where $\epsilon_{k}=-2t\cos k$.

$H_{k}=\begin{bmatrix}0&&(\epsilon_{k}-\mu)\\\ (\epsilon_{k}-\mu)&&0\\\
\end{bmatrix}-2\Delta\sin k\begin{bmatrix}0&&-i\\\ i&&0\\\ \end{bmatrix},$
(19)

$H_{k}=\begin{bmatrix}0&&(\epsilon_{k}-\mu)+2i\Delta\sin k\\\
(\epsilon_{k}-\mu)-2i\Delta\sin k&&0\\\ \end{bmatrix},$ (20)

where $\epsilon_{k}-\mu=r\cos\theta$ and $2\Delta\sin k=r\sin\theta.$ Now the
Hamiltonian is reduced to

$H=\begin{bmatrix}0&&re^{i\theta}\\\ re^{-i\theta}&&0\end{bmatrix},$

where $r=\sqrt{(2t\cos k+\mu)^{2}+4\Delta^{2}\sin^{2}k}$ and
$\theta=-\tan^{-1}(\frac{2\Delta\sin k}{2t\cos k+\mu}).$

Here we are calculating the geometric phase for the lowest-energy
eigenfunction. The basic aspects of the geometric phase are given in the
appendix. The Berry phase is given by
$\gamma_{n}=i\oint\left\langle\Psi_{n}|\nabla_{R}|\Psi_{n}\right\rangle dR,$

where
$\Psi_{n}=\left(\frac{-1}{\sqrt{2}}\right)\left(-e^{i\tan^{-1}(\frac{2\Delta\sin
k}{\mu+2t\cos k})}\right),$ and finally the Berry connection is given by

$A(k)=\left\langle\Psi_{n}|\dfrac{d\Psi_{n}}{dk}\right\rangle=\dfrac{\Delta(2t+\mu\cos
k)}{(\mu+2t\cos k)^{2}+4\Delta^{2}\sin^{2}k}.$

Finally,

$\gamma_{n}=i\oint
Adk=i\oint_{C}\;\;\dfrac{\Delta(2t+\mu\cos(k))}{(\mu+2t\cos(k))^{2}+4\Delta^{2}\sin^{2}(k)}dk.$
(21)

Figure 2: (color online) Variation of $\gamma$ with $\mu$. The red, blue,
green, and black curves are for $t$ = 0.1, 0.2, 0.3 and 0.4 respectively.

In fig. 2, we present the variation of geometric phase ($\gamma$) with $\mu$.
It is clear from this study that there is a topological quantum phase
transition from $\gamma=\pi$ to $\gamma=0$ [34]. We have also observed that
the transition occurs at $\mu=2t$ [35].
The physical explanation of this transition can be understood in the following
way: Fig. 3 describes the energy dispersion spectrum of this model
Hamiltonian, where we observe the gapless state at $k=\pm\pi$. A gapless state
is present if the value of the parameters obey the transition relation
$\mu=\pm 2t$. This is when transition takes place from topological to non-
topological state. If the values of the parameters do not obey the transition
relation, then we observe a gapped state (non-topological state), as shown in
the lower panel of fig. 3.

Here we clearly observe that the gapless states, in other words degenerate
states, appear for discrete values of $\mu$, i.e. $\mu=\pm 2t$ for $k=0$ and
$k=\pm\pi$, not for the different values of $\mu$. Thus we justify the
topological characterization of the Kitaev chain from the perspective of
symmetry Pfaffian number calculation, winding number calculation and also the
gapless state at the topological quantum phase transition point.

Figure 3: Dispersion curve $E_{k}$ vs k for the Hamiltonian
$\mathcal{H}^{(1)}(k)$.

Results and discussion for the Hamiltonian $\mathcal{H}^{(2)}(k)$ : The
Hamiltonian $\mathcal{H}^{(2)}(k)$ in the matrix form can be written as

$\mathcal{H}^{(2)}(k)=\left(\begin{matrix}-2t\cos(k)-\mu&&2i\Delta\sin(k)+i\alpha
k\\\ -2i\Delta\sin(k)-i\alpha k&&2t\cos(k)+\mu\end{matrix}\right).$ (22)

Here we consider three variants of Kitaev chain in momentum space. One can
express the Kitaev chain in the basis of $\sigma_{x}$ and $\sigma_{y}$. The
present study is only for the theoretical exercise over the Kitaev chain. But
it has some interesting physics that all these variants of Kitaev chain
($\mathcal{H}^{(2)}(k),\mathcal{H}^{(3)}(k),\mathcal{H}^{(4)}(k)$) are
hermitian in character. Upon adding of the $\alpha k$ term, either in
$\sigma_{x}$ or $\sigma_{y}$, the two components of the Kitaev chain, we
observe very interesting results from the perspective of different topological
symmetry classes of the auxiliary space and the condition for topological
characterization. The pairing mechanism of the Kitaev chain in real space is
the well-defined p-wave pairing mechanism. But for the other three model
Hamiltonians
($\mathcal{H}^{(2)}(k),\mathcal{H}^{(3)}(k),\mathcal{H}^{(4)}(k)$), we do not
know the nature of the interaction. We consider these model Hamiltonians for
the complete theoretical interest and at the same time they satisfy the
hermiticity property of the Hamiltonian and also one can express these model
Hamiltonians into two different symmetry classes (BDI and AIII). Kitaev’s
model belongs to the BDI symmetry class. Therefore the present study has
relevence from the perspective of symmetry class. We would like to study how
the symmetry and topological properties are modified in the presence of
additional terms. Therefore we present an extensive study from the perspective
of topological symmetry class in momentum space.
Many studies in theoretical physics do not find proper physical justification
at the very beginning. But in due course of time, they are justified. The new
and important results of this study may motivate quantum simulation physicists
to quantum-simulate this system.
This model Hamiltonian satisfies the conditions for time-reversal, particle-
hole and chiral symmetry operations,

$\displaystyle\mathcal{T}\mathcal{H}^{(2)}(k)\mathcal{T}^{-1}$
$\displaystyle={K}\mathcal{H}^{(2)}(k){K}^{-1}$ (23)
$\displaystyle=\hat{K}\left(\begin{matrix}-2t\cos(k)-\mu&&2i\Delta\sin(k)+i\alpha
k\\\ -2i\Delta\sin(k)-i\alpha k&&2t\cos(k)+\mu\\\ \end{matrix}\right)\hat{K}$
$\displaystyle=\mathcal{H}^{(2)}(k),$

$\displaystyle\mathcal{C}\mathcal{H}^{(2)}(k)\mathcal{C}^{-1}$
$\displaystyle=(\sigma_{x}{K})\mathcal{H}^{(2)}(k)(\sigma_{x}{K})^{-1}$ (24)
$\displaystyle=\left(\begin{matrix}0&&1\\\ 1&&0\\\
\end{matrix}\right)\left(\begin{matrix}-2t\cos(k)-\mu&&2i\Delta\sin(k)+i\alpha
k\\\ -2i\Delta\sin(k)-i\alpha k&&2t\cos(k)+\mu\\\
\end{matrix}\right)\left(\begin{matrix}0&&1\\\ 1&&0\\\ \end{matrix}\right)$
$\displaystyle=-\mathcal{H}^{(2)}(k),$

$\displaystyle\mathcal{S}\mathcal{H}^{(2)}(k)\mathcal{S}^{-1}$
$\displaystyle=\sigma_{x}\mathcal{H}^{(2)}(k)\sigma_{x}^{-1}$ (25)
$\displaystyle=\left(\begin{matrix}0&&1\\\ 1&&0\\\
\end{matrix}\right)\left(\begin{matrix}-2t\cos(k)-\mu&&2i\Delta\sin(k)+i\alpha
k\\\ -2i\Delta\sin(k)-i\alpha k&&2t\cos(k)+\mu\\\
\end{matrix}\right)\left(\begin{matrix}0&&1\\\ 1&&0\\\ \end{matrix}\right)$
$\displaystyle=-\mathcal{H}^{(2)}(k).$

Thus we have

$\mathcal{T}\mathcal{H}^{(2)}(k)\mathcal{T}^{-1}=\mathcal{H}^{(2)}(k),\;\;\;\mathcal{C}\mathcal{H}^{(2)}(k)\mathcal{C}^{-1}=-\mathcal{H}^{(2)}(k),$
$\mathcal{S}\mathcal{H}^{(2)}(k)\mathcal{S}^{-1}=-\mathcal{H}^{(2)}(k).$ (26)

As in the case of Kitaev chain ($\mathcal{H}^{(1)}(k)$),
$\mathcal{H}^{(2)}(k)$ also belongs to the symmetry class BDI [6] with the
topological invariant $\mathbb{Z}$ (see table 1). One can also expect the
topological quantum phase transition in $\mathcal{H}^{(2)}(k)$. This system
has anti-unitary particle-hole symmetry, thus one may consider that the system
is in the topological state and Zak phase must be quantized [7, 13]. But we
observe that even though the symmetry class of the Hamiltonian
$\mathcal{H}^{(1)}(k)$ and $\mathcal{H}^{(2)}(k)$ are the same, there is no
topological non-trivial phase for these model Hamiltonians. Addition of the
term $\alpha k$ breaks the anti-symmetric property of the Majorana
representation matrix for $k=\pi$,

${\small A(\pi)=\left(\begin{matrix}0&&2t-\mu+i\alpha\pi\\\
-2t+\mu+i\alpha\pi&&0\\\ \end{matrix}\right).}$ (27)

For this case one cannot calculate the Majorana number since the Pfaffian does
not exist due to lack of the anti-symmetric property. This also indicates that
there is no topological non-trivial phase for this system. This result is also
evident from the nature of auxiliary space of $\mathcal{H}^{(2)}(k)$.

Figure 4: Parametric plots for $\mathcal{H}^{(2)}(k)$ for different values of
$\mu$, $\alpha$ and $t$.

Fig. 4 shows that the trajectory in the auxiliary space is not closed and the
integral along the trajectory will not take quantized values. We observe an
interesting feature from this behaviour in the auxiliary space that, although
the origin of auxiliary space is encircled by the curve, the system is in the
non-topological state. At the same time there is no mirror symmetry with
$\alpha$. This can also be verified from the energy dispersion curve (fig. 5).
It shows there is no gapless state for topological quantum phase transition to
occur. The Hamiltonian $\mathcal{H}^{(2)}(k)$ does not have the mirror-
symmetric auxiliary space curves for positive and negative values of $\alpha$.

It is clear from this section that even though both the Hamiltonians
$\mathcal{H}^{(1)}(k)$ and $\mathcal{H}^{(2)}(k)$ belong to the BDI symmetry
class they have different topological properties. In the case of
$\mathcal{H}^{(2)}(k)$ there is no topological non-trivail phase as expected
for the BDI symmetry class. Majorana representation matrix is anti-symmetric
for $\mathcal{H}^{(1)}(k)$ while it does not satisfy this condition for
$\mathcal{H}^{(2)}(k)$. Thus it is not possible to calculate topological
invariant to characterize the topological state for $\mathcal{H}^{(2)}(k)$.
Energy dispersion for $\mathcal{H}^{(1)}(k)$ is gapless, while for
$\mathcal{H}^{(2)}(k)$ there is no gapless dispersion, indicating there is no
topological phase transition for $\mathcal{H}^{(2)}(k)$. Auxiliary space curve
also gives the same conclusion. It is closed curve for $\mathcal{H}^{(1)}(k)$,
indicating integer topological invariant. In the case of
$\mathcal{H}^{(2)}(k)$ it is open curve, which shows the absence of
topological non-tivial phase. Thus it is clear from this study that the same
symmetry class does not reveal the same topological properties.

Figure 5: Dispersion curve of $E_{k}$ vs k for the $\mathcal{H}^{(2)}(k)$
Hamiltonian.

Results and discussion for the Hamiltonians of AIII symmetry class
Here we present the results for Hamiltonians $\mathcal{H}^{(3)}(k)$ and
$\mathcal{H}^{(4)}(k)$. The matrix form of the Hamiltonian
$\mathcal{H}^{(3)}(k)$ can be written as

$\mathcal{H}^{(3)}(k)=\left(\begin{matrix}-2t\cos(k)-\mu-\alpha
k&&2i\Delta\sin(k)\\\ -2i\Delta\sin(k)&&2t\cos(k)+\mu+\alpha k\\\
\end{matrix}\right).$ (28)

We observe that $\mathcal{H}^{(3)}(k)$ does not satisfy the condition for
time-reversal and particle-hole symmetry operations but satisfies only the
chiral symmetry condition.

$\displaystyle\mathcal{T}\mathcal{H}^{(3)}(k)\mathcal{T}^{-1}$
$\displaystyle={K}\mathcal{H}^{(3)}(k){K}^{-1}$ (29)
$\displaystyle=\hat{K}\left(\begin{matrix}-2t\cos(k)-\mu-\alpha
k&&2i\Delta\sin(k)\\\ -2i\Delta\sin(k)&&2t\cos(k)+\mu+\alpha k\\\
\end{matrix}\right)\hat{K}$ $\displaystyle\neq\mathcal{H}^{(3)}(k),$

$\displaystyle\mathcal{C}\mathcal{H}^{(3)}(k)\mathcal{C}^{-1}$
$\displaystyle=(\sigma_{x}{K})\mathcal{H}^{(3)}(k)(\sigma_{x}{K})^{-1}$ (30)
$\displaystyle=\left(\begin{matrix}0&&1\\\ 1&&0\\\
\end{matrix}\right)\left(\begin{matrix}-2t\cos(k)-\mu+\alpha
k&&2i\Delta\sin(k)\\\ -2i\Delta\sin(k)&&2t\cos(k)+\mu-\alpha k\\\
\end{matrix}\right)\left(\begin{matrix}0&&1\\\ 1&&0\\\ \end{matrix}\right)$
$\displaystyle\neq-\mathcal{H}^{(3)}(k),$

$\displaystyle\mathcal{S}\mathcal{H}^{(3)}(k)\mathcal{S}^{-1}$
$\displaystyle=(\sigma_{x})\mathcal{H}^{(3)}(k)(\sigma_{x})^{-1}$ (31)
$\displaystyle=\left(\begin{matrix}0&&1\\\ 1&&0\\\
\end{matrix}\right)\left(\begin{matrix}-2t\cos(k)-\mu-\alpha
k&&i2\Delta\sin(k)\\\ -i2\Delta\sin(k)&&2t\cos(k)+\mu+\alpha k\\\
\end{matrix}\right)\left(\begin{matrix}0&&1\\\ 1&&0\\\ \end{matrix}\right)$
$\displaystyle=-\mathcal{H}^{(3)}(k).$

Thus we have

$\mathcal{T}\mathcal{H}^{(3)}(k)\mathcal{T}^{-1}\neq\mathcal{H}^{(3)}(k)\;\;\;\mathcal{C}\mathcal{H}^{(3)}(k)\mathcal{C}^{-1}\neq-\mathcal{H}^{(3)}(k),$
$\mathcal{S}\mathcal{\mathcal{missing}}{H}^{(3)}(k)\mathcal{S}^{-1}=-\mathcal{\mathcal{missing}}{H}^{(3)}(k).$
(32)

Thus, from the symmetry analysis, $\mathcal{H}^{(3)}(k)$ falls under AIII
class in the ten symmetry classes with topological invariant number
$\mathbb{Z}$. This indicates that there is a possibility for the topological
quantum phase transition by tuning the parameters to obtain the gapless state
with the change in the topological invariant number $\mathbb{Z}$. But we
observe a strange behaviour of the system, that it does not allow one to
calculate Majorana number $\mathcal{M}$ (eq. 15) to get the condition for the
parameters. The Majorana representation matrix $A_{\alpha\beta}$ for
$\mathcal{H}^{(3)}(k)$ is given by

${\small A(k)=\left(\begin{matrix}0&&\xi_{k}+\alpha k+i2\Delta\sin k\\\
-\xi_{k}+\alpha k+i2\Delta\sin k&&0\\\ \end{matrix}\right),}$ (33)

where $\xi_{k}=-2t\cos k-\mu$. Here $A(k)$ loses the anti-symmetric property
for $k=\pi$, which does not allow one to calculate the Pfaffian of the matrix.
This indicates that there is no Majorana number, which implies that the system
is in non-trivial topological phase. This can also be verifed by the
trajectory of the system in the auxiliary space, given in fig. 6.

Figure 6: Parametric plots for $\mathcal{H}^{(3)}(k)$ for different values of
$\mu$, $\alpha$ and $t$.

A very interesting observation can be made from fig. 6: although the auxiliary
space curve in the upper panel encircles the origin, the curve is not closed.
For the system to be in the topological state, the auxiliary space curve
encirling the origin is a necessary condition, but the closedness of the curve
is the sufficient condition, as we observe in the Kitaev chain (fig. 1). The
left and the right panels represent the auxiliary space curves for positive
and negative values of $\alpha$. They show mirror-symmetric behaviour for
positive and negative values of $\alpha$. Although the auxiliary space curve
is symmetric with respect to positive and negative values of $\alpha$, it is
not symmetric with respect to $k$. Thus we conclude that the topological
properties of the system do not depend on the symmetry of the auxiliary space.

Figure 7: Dispersion curve of $E_{k}$ vs k for the Hamiltonian
$\mathcal{H}^{(3)}(k)$.

We observe that there is no closed trajectory in the auxiliary space. We also
observe that the addition of $\alpha k$ to Hamiltonian $\mathcal{H}^{(3)}(k)$
results in breaking of the periodicity of the Brillouin zone. One can observe
this from energy dispersion curve (fig. 7) that $E(\pi)\neq E(-\pi)$. This
lack of periodicity does not allow one to calculate the geometric/Zak phase
[7, 13]. In other words the integral over the non-periodic Brillouin zone will
not be a Cauchy integral and does not take the quantized value [39], which
again indicates that there is no topological quantum phase transition. We
discuss this in the next section.

Results and discussion for the Hamiltonian $\mathcal{H}^{(4)}(k)$ : The matrix
form of the Hamiltonian $\mathcal{H}^{(4)}(k)$ is given by

$\mathcal{H}^{(4)}(k)=\left(\begin{matrix}-2t\cos(k)-\mu-\alpha_{1}k&&2i\Delta\sin(k)+i\alpha_{2}k\\\
-2i\Delta\sin(k)-i\alpha_{2}k&&2t\cos(k)+\mu+\alpha_{1}k\\\
\end{matrix}\right).$ (34)

We observe that $\mathcal{H}^{(4)}(k)$ belongs to the symmetry class AIII,
i.e. it only obeys the chiral symmetry condition.

$\displaystyle\mathcal{T}\mathcal{H}^{(4)}(k)\mathcal{T}^{-1}$
$\displaystyle={K}\mathcal{H}^{(4)}(k){K}^{-1}$ (35)
$\displaystyle=\hat{K}\left(\begin{matrix}-2t\cos(k)-\mu-\beta_{1}k&&2i\Delta\sin(k)+i\beta_{2}k\\\
-2i\Delta\sin(k)-i\beta_{2}k&&2t\cos(k)+\mu+\beta_{1}k\\\
\end{matrix}\right)\hat{K}$ $\displaystyle\neq\mathcal{H}^{(4)}(k),$

$\displaystyle\mathcal{C}\mathcal{H}^{(4)}(k)\mathcal{C}^{-1}$
$\displaystyle=(\sigma_{x}\hat{K})\mathcal{H}^{(4)}(k)(\sigma_{x}{K})^{-1}$
(36) $\displaystyle=\left(\begin{matrix}0&&1\\\ 1&&0\\\
\end{matrix}\right)\left(\begin{matrix}-2t\cos(k)-\mu+\beta_{1}k&&2i\Delta\sin(k)+i\beta_{2}k\\\
-2i\Delta\sin(k)-i\beta_{2}k&&2t\cos(k)+\mu-\beta_{1}k\\\
\end{matrix}\right)\left(\begin{matrix}0&&1\\\ 1&&0\\\ \end{matrix}\right)$
$\displaystyle\neq-\mathcal{H}^{(4)}(k),$

$\displaystyle\mathcal{S}\mathcal{H}^{(4)}(k)\mathcal{S}^{-1}$
$\displaystyle=(\sigma_{x})\mathcal{H}^{(4)}(k)(\sigma_{x})^{-1}.$ (37)
$\displaystyle=\left(\begin{matrix}0&&1\\\ 1&&0\\\
\end{matrix}\right)\left(\begin{matrix}-2t\cos(k)-\mu-\beta_{1}k&&2i\Delta\sin(k)+i\beta_{2}k\\\
-2i\Delta\sin(k)-i\beta_{2}k&&2t\cos(k)+\mu+\beta_{1}k\\\
\end{matrix}\right)\left(\begin{matrix}0&&1\\\ 1&&0\\\ \end{matrix}\right)$
$\displaystyle=-\mathcal{H}^{(4)}(k).$

Thus we have

$\mathcal{T}\mathcal{H}^{(4)}(k)\mathcal{T}^{-1}\neq\mathcal{H}^{(4)}(k),\;\;\;\mathcal{C}\mathcal{H}^{(4)}(k)\mathcal{C}^{-1}\neq-\mathcal{H}^{(4)}(k),$
$\mathcal{S}\mathcal{H}^{(4)}(k)\mathcal{S}^{-1}=-\mathcal{H}^{(4)}(k).$ (38)

Figure 8: Parametric plots for $\mathcal{H}^{(4)}(k)$ for different values of
$\mu$, $\alpha$ and $t$.

Thus Hamiltonian $\mathcal{H}^{(4)}(k)$ also falls under the same symmetry
class as $\mathcal{H}^{(3)}(k)$ i.e. AIII symmetry class. Here also one can
expect the topological quantum phase transition with change in the value of
topological invariant number $\mathbb{Z}$. But similar to
$\mathcal{H}^{(2)}(k)$ and $\mathcal{H}^{(3)}(k)$, we observe the Majorana
representation matrix breaks its anti-symmetric property as we add the $\alpha
k$ term,

${\small A(\pi)=\left(\begin{matrix}0&&2t-\mu+\alpha_{1}\pi+i\alpha_{2}\pi\\\
-2t+\mu+\alpha_{1}\pi+i\alpha_{2}\pi&&0\\\ \end{matrix}\right).}$ (39)

Thus Pfaffian does not exist for this system, which shows there is no
topological non-trivial phase.

Figure 9: Dispersion curve of $E_{k}$ vs k for the $H^{(4)}(k)$ Hamiltonian.

Here the trajectory in the auxiliary space is not closed and integral along
the trajectory is not quantized. Absence of the origin inside the perimeter of
the trajectory in fig. 8 shows there is no topological state for the system.
The curves in the auxiliary space are neither symmetric with $\alpha$ nor with
$k$. A curve in the auxiliary space encircling the origin is a necessary
condition but the closedness of the curve is the sufficient condition for the
system to be in the topological state. This can also be verified from fig. 9,
which shows the energy dispersion for $\mathcal{H}^{(4)}(k)$. We observe that
there are no gapless states responsible for topological quantum phase
transition.

It is clear from this section that both the Hamiltonians
$\mathcal{H}^{(3)}(k)$ and $\mathcal{H}^{(4)}(k)$ belong to the symmetry class
AIII. They are expected to get the topological non-trivial phase for this
symmetry class but both the Hamiltonians do not show any property indicating
the presence of topological non-trivial phase. Majorana representaion matrix
does not show anti-symmetric property, and auxiliary space curve is not closed
and thus shows there is no possibility of having topological non-trivial
phase.

A Comparision of Results for two different symmetry classes :
We have noticed in the BDI symmetry class that the topological behaviour of
the Hamiltonians $\mathcal{H}^{(1)}(k)$ and $\mathcal{H}^{(2)}(k)$ are
different. The Kitaev Hamiltonian $\mathcal{H}^{(1)}(k)$ shows the topological
phase transition while there is no topological phase transition for
$\mathcal{H}^{(2)}(k)$. It results from the dispersion curve, which for
$\mathcal{H}^{(1)}(k)$ shows a gapless state indicating the transition point.
But for BDI symmetry-class Hamiltonians obey
$\mathcal{H}(k=-\pi)=\mathcal{H}(k=\pi)$, while for the case of AIII symmetry
$\mathcal{H}(k=-\pi)\neq\mathcal{H}(k=\pi)$. We also observe that there is no
closing of the gap at both $k=\pm\pi$, as we find for $\mathcal{H}^{(1)}(k)$
dispersion. The energy dispersion for $\mathcal{H}^{(3)}(k)$ and
$\mathcal{H}^{(4)}(k)$ are different although these two belong to the same
symmetry class. Thus it is clear from this study of two different symmetry
classes that gap closedness is the prime condition to get topological quantum
phase transition of the system.

A Few Relevant Calculations and Discussion for the Topological
Characterization of these two symmetry classes - BDI and AIII.

(A). A derivation for sufficient condition for topological characterization
from the behaviour of curves in auxiliary space.

Here we mathematically prove the sufficient condition for the topological
characterization of the system from the behaviour of auxiliary space. We have

$\mathcal{H}^{(3)}(k)=(-2t\cos
k-\mu+\alpha_{1}k)\sigma_{x}+(\alpha_{2}k-2\Delta\sin k)\sigma_{y}.$ (40)

We plot the parametric plot $(x(k),y(k))$,

$x(k)=-2t\cos k-\mu+\alpha_{1}k=r(k)\cos\theta(k),$ (41)
$y(k)=\alpha_{2}k-2\Delta\sin k=r(k)\sin\theta(k),$ (42)

so that, in the auxiliary plane,

$r^{2}(k)=(-2t\cos k-\mu+\alpha_{1}k)^{2}+(\alpha_{2}k-2\Delta\sin k)^{2},$
(43) $\theta(k)=\tan^{-1}\left[\frac{\alpha_{2}k-2\Delta\sin k}{-2t\cos
k-\mu+\alpha_{1}k}\right].$ (44)

To have a closed curve for $k$ running between $\left[-\pi,\pi\right]$ the
curve must come back to its starting point, i.e.

$r^{2}(k=\pi)=r^{2}(k=-\pi),\;\;\;\;\theta(k=\pi)=\theta(k=-\pi)\;\;mod(2\pi).$
(45)

Putting these two conditions in the expression of $r^{2}(k)$ and $\theta(k)$,

$(-2t\cos(-\pi)-\mu+\alpha_{1}(-\pi))^{2}+(\alpha_{2}(-\pi)-2\Delta\sin(-\pi))^{2}=(-2t\cos(\pi)-\mu+\alpha_{1}(\pi))^{2}+(\alpha_{2}(\pi)-2\Delta\sin(\pi))^{2},$
(46)

and

$\tan^{-1}\left[\frac{\alpha_{2}(-\pi)-2\Delta\sin(-\pi)}{-2t\cos(-\pi)-\mu+\alpha_{1}(-\pi)}\right]=\tan^{-1}\left[\frac{\alpha_{2}(\pi)-2\Delta\sin(\pi)}{-2t\cos(\pi)-\mu+\alpha_{1}(\pi)}\right].$
(47)

The conditions for the curve to be closed, i.e. eq. 46 and eq. 47, can be
simultaneously satisfied if $\alpha_{1}=\alpha_{2}=0$. Thus it is clear from
this study that this equation ($\alpha_{1}=0=\alpha_{2})$ is only satisfied by
the Hamiltonian $\mathcal{H}^{(1)}(k)$, i.e. Kitaev chain.
Therefore it is clear from this study that the addition of $\alpha k$ term
either in $\sigma_{x}$ or $\sigma_{y}$ components or both does not favour the
existence of the topological state of matter. The field of quantum simulation
physics is rich and it has application in different branches of physics [7,
26]. Our strong belief is that this result motivates quantum simulation
physicists to find this behaviour in different physical systems. To the best
of our knowledge this study is the first in the literature to study the
necessary and sufficient conditions for topological characterization from the
behaviour of auxiliary space.

(B). A general physical explanation for the existence of topological state:
from the perspective of Berry connection and geometric phase.

BdG Hamiltonian obeys an anti-unitary particle-hole symmetry,
$\mathcal{C}=\sigma_{x}K$ ($\sigma_{x}$ and $K$ are Pauli spin matrix and
complex conjugate operator respectively.),

$\left\\{\mathcal{H}_{BdG},\mathcal{C}\right\\},$ (48)

with $\mathcal{C}^{2}=1$. This symmetry implies that the bands below and above
the energy gap are conjugates of each other, i.e.

$\mathcal{C}\ket{\Psi^{o}(-k)}=e^{-i\phi(k)}\ket{\Psi^{e}(-k)}.$ (49)

Here $\ket{\Psi^{o}}$ and $\ket{\Psi^{e}}$ are the Bloch states of the
occupied and empty bands respectively. Using this abelian Berry connection
[40, 41] of the occupied bands can be written as

$\mathcal{A}^{o}(k)=-i\sum_{\alpha}\bra{\psi^{o}_{\alpha}(k)}\partial_{k}\ket{\psi^{o}_{\alpha}(k)},$
(50)

where $\alpha=1,...,n$ represent the independent Bloch bands. One can observe
the equivalence of the Berry connection of the occupied bands at $k$ and the
empty bands at $-k$ up to a gauge transformation,

$\mathcal{A}^{o}(-k)=\mathcal{A}^{e}(k)-\sum_{\alpha}\partial_{k}\phi_{\alpha}(k).$
(51)

This constraint implies that the Zak phase over half of the Brillouin zone can
be written as

$\gamma=\int\limits_{-\pi}^{\pi}\mathcal{A}^{o}(k)dk=\int\limits_{0}^{\pi}\left[\mathcal{A}(k)-\partial_{k}\phi(k)\right]dk,$
(52)

with $\mathcal{A}(k)=\mathcal{A}^{o}(k)+\mathcal{A}^{e}(k)$. In the Majorana
representation one can write the $\mathcal{H}_{BdG}$ in diagonal form as

$W(k)\mathcal{H}_{BdG}W^{\dagger}(k)=\sigma_{z}diag(\epsilon_{\lambda}),$ (53)

where from the particle-hole symmetry $W$ safisfies the condition

$\mathcal{C}W(k)\mathcal{C}^{-1}=W(-k).$ (54)

This implies that the phase factor $\phi(k)$ in eq. 52 vanishes. Thus in the
presence of anti-unitary particle-hole symmetry the Zak phase is quantized to
integer multiples of $\pi$, indicating the gapless state with topological
quantum phase transition.

But for the present case, in presence of $\alpha k$ term, one cannot express
the Berry connection $\mathcal{A}^{0}(-k)$ to $\mathcal{A}^{e}(k)$ by eq. 51
and also geometric phase, $\gamma$, as in eq. 52.

C. Topological characterization from the perspective of winding number

In general BdG Hamiltonian in the symmetry class BDI can be written in the
special form as

$\mathcal{H}^{(1)}(k)=\left(\begin{matrix}h_{0}(k)&&i\Delta(k)\\\
-i\Delta(k)&&-h_{0}(k)\end{matrix}\right).$ (55)

This can be written in the diagonal form as

$\mathcal{H}^{(1)}(k)=\left(\begin{matrix}0&&A(k)\\\
A^{T}(-k)&&0\end{matrix}\right),$ (56)

where $A(k)=h_{0}(k)+i\Delta(k)$, satisfying the condition $A^{*}(k)=A(-k)$.
Winding number in this case can be defined from the phase of $A(k)$,

$A(k)=|A(k)|e^{i\theta(k)}.$ (57)

Considering $z(k)=e^{i\theta(k)}$ one can define the winding number as

$w=\frac{-i}{\pi}\int\limits_{0}^{\pi}\frac{dz(k)}{z(k)}=\frac{1}{\pi}(\theta(\pi)-\theta(0)).$
(58)

In the case of Kitaev chain, i.e. $\mathcal{H}^{(1)}(k)$, we have

$|A(k)|=\sqrt{(2t\cos k+\mu)^{2}+(2\Delta\sin k)^{2}},$ (59)
$\theta(k)=\tan^{-1}\left[\frac{2\Delta\sin k}{-2t\cos k-\mu}\right].$ (60)

Thus the winding number for Kitaev chain is given by

$w=\frac{1}{\pi}(\pi-0)=1.$ (61)

Since $\tan^{-1}(0)$ can be either $0$ or $\pi$, winding number can take
quantized values, $w=\pm 1,0$.

In the case of $\mathcal{H}^{(2)}(k)$, $\mathcal{H}^{(3)}(k)$ and
$\mathcal{H}^{(4)}(k)$ one cannot define the integral in eq. 58. Even if we
try to calculate the winding by brute force for $\mathcal{H}^{(4)}(k)$, then
we have

$\theta(k)=\tan^{-1}\left[\frac{2\Delta\sin k+\alpha_{2}k}{-2t\cos
k-\mu+\alpha_{1}k}\right].$ (62)

The winding number is given by

$\displaystyle w$
$\displaystyle=\frac{1}{\pi}\left(\tan^{-1}\left(\frac{\alpha_{2}\pi}{2t-\mu+\alpha_{1}k}\right)-0\right)$
(63)
$\displaystyle=\frac{1}{\pi}\tan^{-1}\left(\frac{\alpha_{2}\pi}{2t-\mu+\alpha_{1}k}\right).$

We observe that the winding number is not quantized to integer values but
takes continuous values, thus the system will be in non-topological state for
all non-zero values of $\alpha_{1}$ and $\alpha_{2}$.

Conclusions:

We have presented the results of symmetry, topology and quantization of
geometric phase along with the physical explanation for two different
topological symmetry classes. The nature of these model Hamiltonians and the
results should motivate quantum simulation physics to simulate these types of
model Hamiltonian in different physical systems. We have shown explicitly that
the symmetry criteria are not sufficient to characterize the topological state
of the systems. We have also presented the results based on auxiliary space to
derive the necessary and sufficient conditions for topological
characterization.

Acknowledgments
The authors would like to acknowledge DST (EMR/2017/000898) for the funding
and Raman Research Institute library for the books and journals. The authors
would like to acknowledge Mr.N.A.Prakash for critical reading of this
manuscript. Finally authors would like to acknowledge International Centre for
Theoretical Sciences Lectures/seminars/workshops /conferences/discussion
meetings on different aspects of physics.

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Appendix

Berry phase in Bloch band

The basic Hamiltonian for the Bloch band can be written as
$H=\frac{k^{2}}{2m}+V(r)$, with the value $V(r+a)=V(r)$, where $a$ is the
distance between two lattice points. The Bloch state satisfies the following
conditions for the edge state: $\psi_{nK}(r+a)=e^{iq.a}\psi_{nk}(r)$, i.e. the
wavefunction at the points $r$ and $r+a$ are related to the phase. One can
also write the model Hamiltonian as $H(q)=\frac{(k+\hbar q)^{2}}{2m}+V(r),$
and the wavefunction

$\psi_{nq}(r)=u_{nq}(r)e^{iq.r},$

where $u_{nq}(r)$ is the free particle wavefunction in the presence of
periodic potential, which also satisfies the following condition:

$u_{nq}(r+a)=u_{nq}(r).$

The most interesting point to be noted is the Brillouin zone in the parameter
space for the transformed Hamiltonian with the eigen basis $|u_{nq}\rangle$.
The states $|\psi_{n}(q)\rangle$ and $|\psi_{n}(q+h)\rangle$ satisfy the same
boundary condition as that of the torus. The crystal momentum q is found to
vary and the Bloch state picks up a Berry phase. This is nothing but the Zak
phase

$\gamma_{n}=\oint dq.\langle u_{n}(q)|i\Delta_{q}|u_{n}(q)\rangle.$

For the Kitaev chain (Hamiltonian $\mathcal{H}^{(1)}(k)$) the physics is all
right, but in the presence of $\alpha k$ term the wavefunction

$|\psi_{n}(q)\rangle\neq|\psi_{n}(q+h)\rangle,$

and the parameter of the Zak phase is not valid.


11institutetext: New York University, New York NY 10012, USA
11email<EMAIL_ADDRESS>
https://www.abdoulayendiaye.com/

# Blockchain Price vs. Quantity Controls

Abdoulaye Ndiaye  0009-0000-7466-6444

###### Abstract

This paper studies the optimal transaction fee mechanisms for blockchains,
focusing on the distinction between price-based ($\mathcal{P}$) and quantity-
based ($\mathcal{Q}$) controls. By analyzing factors such as demand
uncertainty, validator costs, cryptocurrency price fluctuations, price
elasticity of demand, and levels of decentralization, we establish criteria
that determine the selection of transaction fee mechanisms. We present a model
framed around a Nash bargaining game, exploring how blockchain designers and
validators negotiate fee structures to balance network welfare with
profitability. Our findings suggest that the choice between $\mathcal{P}$ and
$\mathcal{Q}$ mechanisms depends critically on the blockchain’s specific
technical and economic features. The study concludes that no single mechanism
suits all contexts and highlights the potential for hybrid approaches that
adaptively combine features of both $\mathcal{P}$ and $\mathcal{Q}$ to meet
varying demands and market conditions.

Keywords: blockchain, transaction fees, EIP-1559, EIP-4844, price elasticity,
decentralization, validator costs, demand uncertainty, cryptocurrency
volatility

##  1 Introduction

Transaction fee mechanisms play a crucial role in maintaining blockchains’
network stability, efficiency, and user satisfaction. These policies can be
broadly categorized into three main types, each with distinct characteristics
and implications for the network’s economic and operational dynamics.

Quantity Controls ($\mathcal{Q}$): This approach involves setting a maximum
limit on resource usage, such as the block size limit. Bitcoin’s fee policies
are a prime example of this strategy, where the block size limit is the
primary mechanism controlling the volume of transactions processed in each
block. By capping the block size, the network effectively manages transaction
throughput through a first-price auction.

Price Controls ($\mathcal{P}$): Under this mechanism, a minimum price per unit
of resource usage is set, often dynamically adjusted to reflect current
network conditions. Ethereum’s EIP-1559 and EIP-4844 illustrate cases where a
base fee is determined algorithmically, rising or falling based on the
network’s congestion levels. This approach allows the quantity of transactions
to adjust in response to demand changes.

Fixed Unit Price ($\emptyset$): Some networks implement a fixed fee structure
without adjusting for market conditions. This straightforward approach, often
adopted at the beginning of the life-cycle of blockchains, can simplify
transactions for users but may lack the flexibility needed to address
fluctuating network demand effectively.

The selection between these transaction fee mechanisms—whether a blockchain
opts for quantity control ($\mathcal{Q}$), price control ($\mathcal{P}$), or a
fixed price approach ($\emptyset$)—is influenced by several factors. Key among
these is the interplay between the objectives of welfare-maximizing blockchain
designers and profit-maximizing validators. Blockchain designers typically aim
to enhance overall network efficiency and user satisfaction, while validators
are incentivized to maximize their earnings from transaction fees and block
rewards.

The central question in this paper is: What specific conditions or
characteristics of a blockchain affect the decision between implementing a
price control or a quantity control mechanism? By analyzing various blockchain
architectures and economic factors, such as uncertainty in user demand,
uncertainty in costs to a validator to process transactions, cryptocurrency
price fluctuations, price elasticity of demand, and levels of
decentralization, we establish criteria that determine the selection of
transaction fee mechanisms.

Table 1 summarizes those criteria. First, in environments with high demand
uncertainty, exemplified by blockchains with various use cases, following a
price control is preferred to adjust the block size to match fluctuations in
demand. Second, a quantity control is favored for blockchains with a consensus
mechanism, such as Proof of Work (PoW), where there is a significant positive
correlation between demand uncertainty and marginal costs (hash rate). The
reason is that if the block size limit was allowed to adjust, validators may
face higher costs when demand—and hence their workload—increases. Third, when
token prices fluctuate widely, implementing quantity controls helps avoid base
fees that are too high and leads to more stable transaction fees denominated
in the native token. Fourth, blockchains characterized by a high elasticity of
demand for block space, such as those with faster blocks or quicker
confirmations, benefit from price controls, which allow more flexible and
responsive fee adjustments. Fifth, quantity controls are adequate in highly
decentralized networks with low validator bargaining power, as they become
easier to enforce.

The optimal choice between an EIP-1559 type policy ($\mathcal{P}$) or a
traditional block size limit ($\mathcal{Q}$) is determined by the relative
balance of these five economic factors that emerge from the blockchain
technical specificities. Both Bitcoin and Ethereum blockchains face
significant uncertainties in user demand. The marginal cost for Ethereum
validators has been essentially constant since the Proof of Stake (PoS)
upgrade. In addition, Ethereum features faster blocks and is arguably less
decentralized than Bitcoin. Therefore, our results help explain Ethereum’s
adoption and planned adoption of price control mechanisms such as EIP-1159 and
EIP-4844. At the same time, they help explain why Bitcoin still uses a sole
block size limit as quantity control.

FACTORS | EXAMPLE | OUTCOME
---|---|---
High demand uncertainty | Different use cases | $\mathcal{P}$
\+ Corr. btw demand uncertainty and MC | PoW | $\mathcal{Q}$
Token price fluctuations | Fees in native token | $\mathcal{Q}$
High elasticity of inclusion in next block | Faster blocks or confirmation | $\mathcal{P}$
Low validator bargaining power | High decentralization | $\mathcal{Q}$
Table 1: Summary of factors leading to a $\mathcal{P}$ or $\mathcal{Q}$
equilibrium between ”welfare-maximizing” blockchain designers and ”profit-
maximizing” validators.

###  1.1 Literature Review

This research builds upon the foundational studies on price versus quantity
controls initiated by [8] The choice of selecting a supply function under
uncertain conditions has been extensively discussed in the work of [4] My
methodology closely resembles the analysis by [6], who investigates the
implications of these choices from a macroeconomic standpoint However, my
approach diverges by focusing on the unique challenges blockchain designers
face, who must balance multiple technical and strategic objectives, such as
setting block size limit The results of this study contribute to the
development of transaction fee mechanisms.

The microeconomic mechanism design perspective on TFMs has seen substantial
growth, particularly with the contributions of [1], who analyze ”credible
mechanisms” that resist manipulation by designers. These are particularly
relevant in the blockchain sphere, where [7] demonstrates that Ethereum’s
EIP-1559 and related models meet these criteria, offering a myopically
credible solution for validators and users. This is further supported by the
findings of [3], ensuring that the TFMs underlying the price controls studied
here are incentive-compatible.

From a broader macroeconomic angle, this paper applies results in [5] to
enhance our understanding of the tradeoffs in choosing the families of
blockchain TFMs.

##### Outline:

The paper is organized as follows: Section 2 provides intuition on the
economics of price and quantity controls with perfect enforcement. Section 3
presents how each factor affects the choice of controls in the blockchain
context where protocol designers cannot fully enforce policies. Section 4
takes stock of the results, discusses the general choice of a supply function
for block space, and concludes the paper.

##  2 Prices vs. Quantity Controls with Perfect Enforcement

In this section, we analyze the welfare implications of price and quantity
controls amid demand uncertainty and social cost uncertainty in a general
setting to illustrate the idea of [8].

Consider a general private benefit of block space $q$, denoted by $B(q)$.
These private benefits may reflect the utility users derive from generating a
block of size $q$ that matches their total demand. Social costs could include
costs beyond those incurred by validators, such as centralization costs.
Uncertainty in private benefits and social costs are introduced through the
$B(q,\Psi)$ and $C(q,\eta)$. From the perspective of the blockchain designer,
the value of setting the quantity $\bar{q}$ in advance is given by:

$\max_{\bar{q}\in\mathbb{R}_{+}}\mathbb{E}_{\Psi,\eta}[B(\bar{q},\Psi)-C(\bar{q},\eta)].$
(1)

Alternatively, if the blockchain designer sets the price $\underline{p}$, in
advance, quantities adjust ex-post to match demand:

$B_{1}(q^{adj}(\underline{p},\Psi),\Psi)=\underline{p}.$ (2)

The value of this price control takes quantity adjustments into account:

$\max_{\underline{p}\in\mathbb{R}_{+}}\mathbb{E}_{\Psi,\eta}[B(q^{adj}(\underline{p},\Psi),\Psi)-C(q^{adj}(\underline{p},\Psi),\eta)],$
(3)

For intuition on the choice of instruments between a minimum price (base fee)
and a maximum quantity (block size limit), consider a setting where there is
no uncertainty in the cost function but with a 50% chance demand is high and a
50% chance demand is low Figure 1 illustrates this example with a quantity
limit (top panel) and price control (bottom panel) under such demand
uncertainty.

marginalsocial costprivate demandBlock spacePricehigh demandlow demandsize
limitDWL Block spacePricebase priceDWL Figure 1: Welfare improvement from
price controls under demand uncertainty. Top panel: deadweight loss of a
quantity limit. Bottom panel: deadweight loss of price control.

Consider, without loss of generality, that the block size limit $q^{\max}$ is
equal to block space demand when demand is low but is binding when demand is
high, as depicted in the top panel of the figure. In this context, the concept
of deadweight loss comes into play. Deadweight loss refers to the loss in
total social surplus due to an inefficient market outcome—it occurs when
supply and demand are not in equilibrium. Here, the deadweight loss is
represented by the shaded area between the high demand curve, the marginal
cost curve, and the block size limit. This area is twice the deadweight loss
that results from the block size limit.

Now, consider a scenario where the blockchain designer introduces a minimum
price that exceeds the low-demand market price, as shown in the bottom panel
of the figure. In this scenario, the dark-shaded area represents twice the
deadweight loss that results from this price control. It can be observed that
the deadweight loss from the price control is lower than the deadweight loss
from the block size limit. This situation arises whenever the uncertainty in
demand exceeds the uncertainty in marginal costs.

Formally, the demand curve can be approximated around a quantity limit
$\bar{q}$ as:

$B_{1}(q,\Psi)\approx
B^{\prime}+\beta(\Psi)+B^{\prime\prime}\cdot(q-\bar{q}),$ (4)

and the marginal cost is approximated by:

$C_{1}(q,\eta)\approx C^{\prime}+\eta(\eta)+C^{\prime\prime}\cdot(q-\bar{q}),$
(5)

where it is assumed that there is uncertainty around a fixed demand curve
$B_{1}(q)\approx B^{\prime}+B^{\prime\prime}\cdot(q-\bar{q})$ and marginal
cost curve $C_{1}(q)\approx C^{\prime}+C^{\prime\prime}\cdot(q-\bar{q})$ such
that $\mathbb{E}[\beta]=\mathbb{E}[\eta]=0$.

The comparative advantage of a price control over a quantity control, denoted
by $\Delta$, can be expressed as:

$\Delta\equiv\mathbb{E}[(B(\tilde{q}(\Psi),\Psi)-C(\tilde{q}(\Psi),\eta))-(B(\bar{q},\Psi)-C(\bar{q},\eta))].$
(6)

This comparative advantage of price control is:

$\Delta\propto\frac{B^{\prime\prime}}{C^{\prime\prime
2}}+\frac{1}{C^{\prime\prime}},$ (7)

and if $|B^{\prime\prime}|>C^{\prime\prime}$, a price control improves welfare
over a quantity control.

This is the main result of [8]. When demand is more uncertain than marginal
cost, price controls can lead to quantity adjustments that better match
demand, while marginal costs do not vary much. This result determines when
price controls lead to welfare improvements over quantity controls.

###  2.1 EIP-1559: The Ethereum $\mathcal{P}$ Transaction Fee Mechanism

Inspired by Weitzman’s work about environmental regulation, [2] introduced a
revised pricing mechanism, EIP-1559, for the Ethereum blockchain. The system
has a target block size, currently set at $q^{target}=15M\text{ gas}$ (the
unit of block size), and a maximum block size of $q^{\max}=2q^{target}$ The
minimum gas price, $p_{t}$, is adjusted based on the formula

$p_{t}=p_{t-1}\cdot(1+d\frac{q_{t-1}-q^{target}}{q^{target}})$ (8)

where the adjustment parameter $d$ is set by default for the minimum price to
double in 8 blocks when blocks are full, i.e., $d=\frac{1}{8}$.

In the Ethereum blockchain, each transaction indexed by $j$ has an associated
gas limit $q_{j}$, computed based on a fixed fee schedule $p_{x}$ Transaction
senders pay an amount in ETH, the native currency of the blockchain, equal to
$q_{j}\cdot\min\\{p_{t}+\delta_{j},c\\}$, where $\delta_{j}$ is the tip and
$c$ is the fee cap, with $c\geq p_{t}$ The base fee revenue,
$\sum_{j=1}^{N}q_{j}p_{t}$, s burned, mainly for reasons related to off-chain
incentives of validators [7], while the effective tips are transferred to the
validator.

###  2.2 Modeling Key Differences between Blockchains Fees and Environment
Regulation

Blockchains, particularly in permissionless systems, present unique challenges
that differentiate them from traditional economic regulation under
uncertainty, as illustrated above. These challenges stem primarily from the
decentralized nature of blockchains and the diverse stakeholders involved,
each with different objectives and influences on market equilibrium.

##### Diverse Stakeholders:

The key actors in a blockchain ecosystem include developers, validators
(proposers or builders), and users. Each group holds varying degrees of power
and influence over the network’s operations and policies.

##### Absence of Central Authority:

Unlike a government, a blockchain designer cannot unilaterally enforce
policies. Validators, who play a critical role in processing transactions and
creating new blocks, must be incentivized to follow proposed changes, which
may not always align with their interests.

##### Uncertainty:

Blockchains face multiple sources of uncertainty that affect their operation
and the feasibility of different transaction fee mechanisms. These include
fluctuations in user demand, the variable costs faced by validators, and the
volatile prices of cryptocurrencies.

To address these challenges, we propose in [5] a model that conceptualizes the
decision-making process regarding transaction fee mechanisms as a Nash
bargaining game between blockchain designers and validators. The model is
structured around the following components:

##### Decisions:

Blockchain designers must commit ex-ante to either a fixed base fee
($\mathcal{P}$-setting) or a fixed blockspace ($\mathcal{Q}$-setting) before
the full extent of uncertainties is realized.

##### Bargaining Model of Decentralization:

The bargaining game is formalized by the following optimization problem:

$\displaystyle\max_{\mathcal{P},\mathcal{Q}}\mathbb{E}[\textrm{Social
Benefit}(\Psi,\eta)]^{1-\beta}\mathbb{E}[\textrm{Validator
Profits}(\Psi,\eta)]^{\beta}$ (9)

where $\beta\in[0,1]$ represents the bargaining power of validators.

##  3 How Different Factors Affects the $\mathcal{P}$ vs. $\mathcal{Q}$ choice

In this section, we discuss the contribution of each factor in the choice of
price or quantity controls.

###  3.1  Uncertainty in User Demand

####  3.1.1 Definition and Examples:

In our model, user demand for blockchain transactions can be defined by the
equation:

$\displaystyle q=\frac{\Psi}{p^{\varepsilon}}$ (10)

Where $q$ represents the resource quantity used by transactions, $\Psi$
captures factors influencing demand such as transaction utility or network
activity, and $p$ is the price per transaction.

Demand uncertainty is quantitatively expressed through the variance of $\Psi$,
denoted as $Var[\Psi]$. High variability in $\Psi$ indicates high uncertainty
in user demand. This variability can be attributed to several causes:

##### Different Use Cases:

Blockchains can serve various applications with distinct demand patterns.

##### Adoption Phase:

As the technology matures and gains wider acceptance, demand for block space
can increase and be less volatile.

##### Cycles:

Economic and speculative cycles can cause significant fluctuations in activity
levels on the blockchain.

####  3.1.2 Economic Implications:

To address the challenges posed by high demand uncertainty, one intuitive
solution is to allow the block size to adjust with a base fee ($\mathcal{P}$)
and better match the fluctuating demand. When demand spikes, increasing the
block size can help accommodate more transactions, alleviating congestion.

###  3.2 Uncertainty in Validator Costs

Validator costs play an important role in the operation of blockchains. These
costs can vary significantly depending on the consensus mechanism.

####  3.2.1 Definition and Examples

The marginal cost of transaction validation, denoted by $C^{\prime}(q)$, is
represented by $\eta$. In practice, the covariance between $\Psi$ (user demand
uncertainty) and $\eta$ (validator marginal costs) is often positive, implying
that the costs to validators also tend to rise as demand increases. This
relationship can be mathematically expressed as:

$\displaystyle Cov[\Psi,\eta]>0$ (11)

For example, in PoW blockchains like Bitcoin, the hash rate—a proxy for
computational effort and energy consumption—typically increases with higher
transaction demand, reflecting a positive and high covariance. Conversely, in
PoS systems, the marginal cost of block production is relatively constant and
less dependent on fluctuating transaction volumes since the cost to validators
is mainly the fixed cost of staking.

####  3.2.2 Economic Implications:

A large positive covariance between validator marginal costs and user demand
favors quantity controls $\mathcal{Q}$ In particular, under a price control
mechanism ($\mathcal{P}$), validators may face higher costs exactly when
demand—and hence their workload—increases. This scenario could lead to
inefficiencies where blocks become more expensive to produce precisely when
they are most needed.

###  3.3 Cryptocurrency Price Fluctuations

Cryptocurrencies are notoriously volatile, which presents unique challenges
for blockchain transaction fee mechanisms.

####  3.3.1 Definition and Examples:

In economic terms, people generally value their wealth in stable currencies
like the dollar or in terms of real goods rather than in the native tokens of
blockchains. One measure of price volatility for Ethereum, for instance, is
the variance of its token $Var[\$ETH]$. This variability means that
transaction fees’ real (USD) cost can fluctuate widely, even if the fee in
native tokens remains constant.

####  3.3.2 Economic Implications:

Large fluctuations in cryptocurrency prices tilt the balance against
$\mathcal{P}$ mechanisms and towards $\mathcal{Q}$ mechanisms During periods
when the value of a cryptocurrency like Ethereum is high, the corresponding
USD value of base fees in a $\mathcal{P}$ mechanism can become prohibitively
expensive. Another interpretation of this result is that if blockchain
designers care about implementing a $\mathcal{P}$ mechanism, they should
consider allowing fees to be paid in USD and stablecoins.

###  3.4 Price Elasticity of Demand

####  3.4.1 Definition and Examples:

The price elasticity of demand for inclusion in the next block refers to the
responsiveness of the number of transactions (i.e., block space used) to
changes in the base fee. Mathematically, the exponent $\varepsilon$ of our
demand function denotes the price elasticity of demand. A high value of
$\varepsilon$ suggests that users are highly sensitive to changes in
transaction costs. For instance, fast blockchains with short block times or
confirmation times, such as Ethereum, exhibit higher price elasticities. The
estimate of $\varepsilon_{\textrm{ethereum}}\approx 12.6$ in [5] suggests that
even minor adjustments in the base fee can lead to significant changes in the
demand for block space.

####  3.4.2 Economic Implications:

A high price elasticity of demand for inclusion in the next block amplifies
gains from $\mathcal{P}$ controls. A higher elasticity facilitates faster
adjustments in block size in response to changes in demand, amplifying the
benefits of dynamically matching block space to user needs.

###  3.5 Decentralization

####  3.5.1 Definition and Examples:

Decentralization is an aspect of blockchains, often measured by the
distribution of power among participants in the network. Validator bargaining
power, denoted by $\beta$, inversely correlates with the level of
decentralization within the network. A lower value of $\beta$ implies higher
decentralization, indicating reduced power concentration among validators.

####  3.5.2 Economic Implications:

In a more decentralized blockchain, the reduced bargaining power of validators
makes it easier for blockchain designers to implement and enforce their
preferred policies, such as adjustments to block size. This fact is
particularly relevant when designing mechanisms to adjust block size in
response to fluctuating demand dynamically, ensuring the network can
efficiently respond to user needs without undue influence from a concentrated
group of validators.

##  4 Conclusion

This paper has examined various factors influencing the decision between
price-based ($\mathcal{P}$) and quantity-based ($\mathcal{Q}$) transaction fee
mechanisms in blockchain systems. Below is a summary table that encapsulates
the main findings:

The choice between implementing a policy akin to Ethereum’s EIP-1559
($\mathcal{P}$) or opting for a traditional block size limit ($\mathcal{Q}$)
hinges on the blockchain’s technical specificities. We found that high demand
uncertainty and faster blockchains should favor price controls, while highly
decentralized and PoW blockchains should favor simple block size limits.
Furthermore, EIP-1559-type mechanisms, when needed, should compute base fees
in USD due to cryptocurrency price volatility. The relative balance between
five factors -demand uncertainty, validator cost uncertainty, token price
volatility, demand elasticity, and decentralization- determines the optimal
transaction fee mechanism, which aligns the objectives of welfare-maximizing
blockchain designers and profit-maximizing validators.

The optimal choice between a price-based and a quantity-based transaction fee
mechanism is not one-size-fits-all but depends on a blockchain’s specific
technical characteristics and the economic environment in which it operates.
Future research could further quantify these choices and explore optimal
supply schedules that incorporate the advantages of both $\mathcal{P}$ and
$\mathcal{Q}$ mechanisms, with the potential to offer more flexible and robust
fee structures as blockchain technology evolves and matures.

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