# Experimental demonstration of the dynamics of quantum coherence evolving
under a PT-symmetric Hamiltonian on an NMR quantum processor

Akanksha Gautam<EMAIL_ADDRESS>Department of Physical
Sciences, Indian Institute of Science Education & Research Mohali, Sector 81
SAS Nagar, Manauli PO 140306 Punjab India.  Kavita Dorai
<EMAIL_ADDRESS>Department of Physical Sciences, Indian Institute of
Science Education & Research Mohali, Sector 81 SAS Nagar, Manauli PO 140306
Punjab India.  Arvind<EMAIL_ADDRESS>Department of Physical
Sciences, Indian Institute of Science Education & Research Mohali, Sector 81
SAS Nagar, Manauli PO 140306 Punjab India.  Vice Chancellor, Punjabi
University Patiala, 147002, Punjab, India

###### Abstract

In this work, we study the dynamics of quantum coherence (total coherence,
global coherence and local coherence) evolving under a local PT-symmetric
Hamiltonian in maximally entangled bipartite and tripartite states. Our
results indicate that quantum coherence in the bipartite state oscillates in
the unbroken phase regime of the PT-symmetric Hamiltonian. Interestingly, in
the broken phase regime, while the global coherence decays exponentially, the
local and total coherences enter a “freezing” regime where they attain a
stable value over time. A similar pattern is observed for the dynamics of
total and local coherences in the maximally entangled tripartite state, while
the dynamics of global coherence in this state differs from that of the
bipartite state. These results were experimentally validated for a maximally
entangled bipartite state on a three-qubit nuclear magnetic resonance (NMR)
quantum processor, with one of the qubits acting as an ancilla. The
experimental results match well with the theoretical predictions, upto
experimental errors.

##  I Introduction

The hermiticity of physical observables in standard quantum mechanics ensures
both real energy spectra of quantum systems and unitary evolution of closed
systems. In 1998, Bender and Boettcher discovered certain parity-time (PT)
symmetric Hamiltonians that despite being non-hermitian, can still have real
energy spectra, thereby demonstrating that hermiticity is a sufficient
condition but not necessary [1, 2]. It was observed that that the eigenvalues
of the PT-symmetric Hamiltonian are always real and PT-symmetry is unbroken if
the eigenstates of a PT-symmetric Hamiltonian are also eigenstates of the PT
operator; otherwise, the eigenvalues are complex and PT-symmetry is broken.
The point where the eigenvalues and eigenvectors coalesce, causing a
transition from unbroken to broken PT-symmetry is called the exceptional
point[3, 4].

PT-symmetric Hamiltonians have been implemented on various experimental
platforms including ultracold atoms [5], superconducting circuits [6, 7],
nitrogen-vacancy centers in diamonds [8, 9], optical waveguides [10, 11] and
NMR spins [12]. The anti-PT-symmetric Hamiltonian was implemented using NMR
spins where information flow between the system and environment was
experimentally explored [13].

PT-symmetric Hamiltonians have attracted wide attention due to their
interesting features, and numerous studies have been performed including
increase or restoration of entanglement by a local PT-symmetric Hamiltonian
[12, 14], protecting quantum correlations via non-hermitian operations [15],
no-signaling principle violation [16, 17], observing fast evolution in a PT-
symmetric system [18, 19] and observing critical phenomena in a PT-symmetric
system [20, 21]. Recently, it has been shown that quantum coherence is
increased in a single-qubit system under PT-symmetric Hamiltonian and maximal
quantum coherence is observed at the exceptional point[22, 23]. The flow of
quantum coherence of a single qubit under PT-symmetric and anti-PT-symmetric
Hamiltonians has been demonstrated on an optics system [24].

Different measures of basis dependent quantum coherence have been proposed
such as $l_{1}$-norm and relative entropy [33, 36, 37, 38]. Based on these
measures, various types of quantum coherences have been defined in
multipartite systems such as global coherence, local coherence and total
coherence [39, 40]. Recently, bipartite and tripartite entanglement has been
generated in a PT-symmetric system that involves three interacting cavities
[34]. The evolution process of entropy and entanglement in a three-qubit
system has been investigated using nuclear spins [35].

In this work, we use the relative entropy of coherence [33] to quantify
quantum coherence of various kinds present in a multipartite system, and
explicitly study their dynamics in a bipartite and a tripartite system
evolving under a local PT-symmetric Hamiltonian. In the maximally entangled
bipartite (Bell) and tripartite (GHZ) states, only global coherence exists
while local coherence remains zero. Our theoretical results show that local
coherence, which previously did not exist in such maximally entangled states,
gets created under the local PT-symmetric Hamiltonian. Global, local and total
coherences in the bipartite entangled state show an oscillatory behavior in
the unbroken PT-symmetry regime. On the other hand, in the broken PT-symmetry
regime, global coherence first increases then decays exponentially; total
coherence also increases initially but attains a stable value at later times,
while local coherence is first created, then increases and stabilizes at later
times. Similar patterns of coherence dynamics is observed in the tripartite
maximally entangled state, in the unbroken regime of PT-symmetry. In the
broken phase of PT-symmetry, the pattern of total and local coherences are
similar to the bipartite state but the global coherence exhibits a different
dynamics. Further, the dynamics of quantum coherence present in a two-qubit
reduced state of a tripartite state was investigated under the PT-symmetric
Hamiltonian in both unbroken and broken phase regimes. We have also
experimentally demonstrated the dynamics of quantum coherence of a maximally
entangled bipartite state using three NMR qubits. Two of the qubits were
utilized to prepare two-qubit states and the third qubit was used as an
ancillary qubit to simulate a PT-symmetric Hamiltonian acting on one of the
qubits. Both broken as well as unbroken phase regimes of the PT-symmetric
Hamiltonian were simulated experimentally and the dynamics of various types of
of quantum coherences in the maximally entangled bipartite state was verified.

This article is organized as follows: Section II describes the procedure to
simulate a general PT-symmetric Hamiltonian. Section III contains details of
the simulation of the dynamics of quantum coherence (global coherence, local
coherence and total coherence) in a maximally entangled bipartite (Bell) state
and a tripartite (GHZ) state evolving under the local PT-symmetric
Hamiltonian. Section IV contains the experimental results of the
implementation of the local PT-symmetric Hamiltonian on a maximally entangled
bipartite state. Section V contains some concluding remarks.

##  II Simulation of the PT-Symmetric Hamiltonian

Figure 1: (a) Quantum circuit to simulate a PT-symmetric Hamiltonian. The
gates in the first box create a maximally entangled Bell state
($\rho_{{}_{BS}}\otimes\rho_{0}$) on the first two qubits, with the third
qubit acting as an ancillary qubit. The action of the $V$, $U_{1}$ and $U_{2}$
gates is described in the main text. $H$ is a Hadamard gate and $R$ acts as a
NOT gate when $\phi$ (given in Eq.7) is negative, otherwise $R$ is an $I_{2}$
identity operation. (b) The corresponding NMR pulse sequence, where the
unfilled rectangles are $\frac{\pi}{2}$ pulses, the black rectangles denote
$\pi$ pulses, the grey rectangle is a $\theta$ pulse and the green rectangle
denotes a $\phi$ pulse. Pulse phases are given above each pulse and a bar over
a phase represents negative phase. The free evolution time intervals
$\tau_{12}$ and $\tau_{31}$ are given by 1/2$J_{12}$ and 1/2$J_{31}$
respectively, where $J_{12}$ and $J_{31}$ are scalar coupling constants.

While all hermitian Hamiltonians can be realized by unitary operations, the
PT-symmetric Hamiltonian (being non-hermitian) can cause non-unitary evolution
and is hence nontrivial to simulate on a quantum processor. The PT-symmetric
Hamiltonian can be simulated by enlarging the system into a higher-dimensional
Hilbert space by using an ancillary qubit. The PT-symmetric Hamiltonian for a
single qubit can be written as [2]:

$H_{PT}=\sigma_{x}+ir\sigma_{z}$ (1)

where $r>0$ is the amount of non-hermiticity and
$\sigma_{x},\sigma_{y},\sigma_{z}$ are Pauli matrices. The PT-symmetric
Hamiltonian satisfies the condition $(PT)H^{\dagger}(PT)^{-1}=H$, where
$P=\sigma_{x}$ is the parity operator and $T$ denotes the complex conjugation
operator. The energy gap $g$ of the PT-symmetric Hamiltonian is
$g=2\sqrt{1-r^{2}}$ and its eigenvalues are $\pm\sqrt{1-r^{2}}$; for $\mid
r\mid<1$ the eigenvalues are positive, which means that the PT-symmetry is
unbroken and for $\mid r\mid>1$ the eigenvalues become complex, which leads to
a broken PT-symmetry. The Hamiltonian has an exceptional point at $\mid
r\mid=1$, where both the eigenvalues as well as the eigenvectors coalesce.

The evolution of $H_{PT}$ can be realized by introducing an ancillary qubit to
simulate the non-unitary operator $U_{PT}=e^{-iH_{PT}t}$, where $t$ represents
evolution time and the matrix form of the local operation
$U_{PT}=e^{-iH_{PT}t}$ is represented by

$U_{PT}=\begin{bmatrix}\cos\omega t-\frac{ir\sin\omega
t}{\omega}&-\frac{i\sin\omega t}{\omega}\\\ -\frac{i\sin\omega
t}{\omega}&\cos\omega t+\frac{ir\sin\omega t}{\omega}\end{bmatrix}$ (2)

where $\omega=\sqrt{1+r^{2}}$. Thus, this non-unitary operator can be
implemented by a linear combination of unitary operators using an ancillary
qubit [41, 42, 12] and the quantum circuit to implement it is shown in
Fig.1(a), where the initial state $\ket{\psi}\ket{0}$ is first created. Then
the unitary operator $V$ is performed on ancillary qubit which is represented
as [12]:

$V=\begin{bmatrix}\cos{\theta}&-\sin{\theta}\\\ \sin{\theta}&\cos{\theta}\\\
\end{bmatrix}$ (3)

where

$\cos\theta=\sqrt{\frac{1-r^{2}\cos^{2}{gt}/2}{1-r^{2}\cos{gt}}}\,,\sin\theta=\frac{r\sin{gt/2}}{\sqrt{1-r^{2}\cos{gt}}}$
(4)

The two controlled unitary operations $C_{U_{1}}$ and $C_{U_{2}}$ are then
performed on the first qubit of the two-qubit work state using the ancillary
qubit and the matrix form of these operations are represented by [12]:

$C_{U_{1}}=\begin{bmatrix}U_{1}&0\\\ 0&I\\\ \end{bmatrix}\quad\quad
C_{U_{2}}=\begin{bmatrix}I&0\\\ 0&U_{2}\\\ \end{bmatrix}$ (5)

where

$U_{1}=\begin{bmatrix}\cos{\phi}&i\sin{\phi}\\\ i\sin{\phi}&\cos{\phi}\\\
\end{bmatrix}\quad\quad U_{2}=\sigma_{z}$ (6)

and

$\cos\phi=\frac{g\cos{gt/2}}{2\sqrt{1-r^{2}\cos^{2}{gt/2}}},\sin\phi=\frac{-r\sin{gt/2}}{\sqrt{1-r^{2}\cos^{2}{gt/2}}}$
(7)

and the last operation is a Hadamard gate on the ancillary qubit. The state is
then measured in the subspace spanned by $|0\rangle$ of the ancillary qubit,
which leads to the final state where the local PT-symmetric Hamiltonian has
been implemented.

Figure 2: Plots of the evolution of quantum coherence present in the maximally
entangled bipartite (Bell) state under different phases of PT-symmetry. (a),
(b) and (c) represent the dynamics of total coherence ($C_{T}$), global
coherence ($C_{G}$) and local coherence ($C_{L}$) respectively, in the
unbroken phase (at $r=0.6$ and $r=0.8$). (d), (e) and (f) represent the
dynamics of total, global and local coherence respectively, in the broken
phase (at $r=1.4$ and at exceptional point $r=1$).

##  III Dynamics of quantum coherence under the PT-symmetric Hamiltonian

###  III.1 Measures of quantum coherence

Figure 3: Plots of the evolution of quantum coherence present in a maximally
entangled tripartite (GHZ) state under different phases of PT-symmetry. (a),
(b) and (c) represent the dynamics of total coherence ($C_{T}$), global
coherence ($C_{G}$) and local coherence ($C_{L}$) respectively, in the
unbroken phase (at $r=0.6$ and $r=0.8$). (d), (e) and (f) represent the
dynamics of total, global and local coherence respectively, in the broken
phase (at $r=1.4$ and at exceptional point $r=1$).

Several measures have been proposed to quantify quantum coherence that follow
certain properties such as relative entropy and $l_{1}$ norm [33]. These
measures have further been utilized to quantify different types of quantum
coherences present in multipartite systems such as total coherence, local
coherence and global coherence which are described below [39, 40].

Total quantum coherence- The total quantum coherence [33] of the system is
defined as the measure of its distance to the closest incoherent state, where
the incoherent state has the form $\sigma=\sum_{i}p_{i}|i\rangle\langle i|$
and the fixed local basis $\\{|i\rangle\\}$ for each qubit is
$\\{|0\rangle,|1\rangle\\}$. The relative entropy of coherence is defined as
[33]:

$C_{T}(\rho)=min_{\sigma\epsilon
I}S(\rho\parallel\sigma)=S(\rho_{{}_{d}})-S(\rho)$ (8)

where $S(\rho)=-tr(\rho log_{{}_{2}}\rho)$ is the Von-Neumann entropy of
$\rho$, and $\rho_{{}_{d}}$ is the matrix of $\rho$ with all off-diagonal
terms set to zero in the basis $|i\rangle$.

Local coherence- Quantum coherence which is localized on each qubit of the
entire system is called local coherence [39, 40]. For example, in a separable
two-qubit state $|+-\rangle=(|0\rangle+|1\rangle)(|0\rangle-|1\rangle)/2$,
quantum coherence exists in each qubit. Local coherence can also be defined in
terms of the relative entropy [39, 40]:

$C_{L}(\rho)=min_{\sigma\epsilon
I}S(\delta(\rho)\parallel\sigma)=S(\delta_{d}(\rho))-S(\delta(\rho))$ (9)

where $\delta(\rho)=\rho_{{}_{1}}\otimes\rho_{{}_{2}}$
($\delta(\rho)=\rho_{{}_{1}}\otimes\rho_{{}_{2}}\otimes\rho_{{}_{3}}$) for a
two-qubit state (three-qubit state) and
$\rho_{{}_{1}}=Tr_{{}_{2}}\rho_{{}_{12}}$
($\rho_{{}_{1}}=Tr_{{}_{23}}\rho_{{}_{123}}$) is the single-qubit reduced
density matrix; $\delta_{d}(\rho)$ is the matrix produced by eliminating all
off-diagonal elements of $\delta(\rho)$ in the basis $|i\rangle$.

Global Coherence- Quantum coherence that originates due to the collective
nature of the whole system is called global coherence [39, 40].
Mathematically, it is the difference between total quantum coherence and local
coherence [39, 40]:

$C_{G}(\rho)=C_{T}(\rho)-C_{L}(\rho)$ (10)

###  III.2 Two-qubit maximally entangled (Bell) state

Consider a two-qubit Bell state defined as

$\displaystyle\rho_{{}_{BS}}=|\psi_{{}_{BS}}\rangle\langle\psi_{{}_{BS}}|\quad{\rm
with}$
$\displaystyle|\psi_{{}_{BS}}\rangle=\frac{1}{\sqrt{2}}\left(|00\rangle+|11\rangle\right)$
(11)

Our aim is to study the dynamics of various types of quantum coherences,
namely the total, global and local coherence as the state evolves from
$\rho_{BS}$ under the local operation generated by the PT-symmetric non-
hermitian Hamiltonian [2]( Eq.(1)), acting on one of the qubits. The final
density matrix after the action of $U_{PT}^{2}=U_{PT}\otimes I$ on the initial
state $\rho_{BS}$ is given by:

$\rho_{{}_{BS}}(t)=\frac{U_{PT}^{2}\rho_{{}_{BS}}(0)U_{PT}^{2}}{Tr[U_{PT}^{2}\rho_{{}_{BS}}(0)U_{PT}^{2}]}$
(12)

From this state, we calculate the total, global and local coherence by using
the definitions given in Eqs. 8,9 and 10, respectively. The regions of
interest are the broken and unbroken phase regimes of the PT-symmetric
Hamiltonian and the exceptional point [2].

The results of these calculations for the unbroken and broken PT-symmetry
regimes are plotted in Fig. 2. Figs. 2(a)-(c) contain the plots of the total,
global and local coherence as a function of time for for the unbroken PT-
symmetry regime, while Figs. 2(d)-(f) contain the same for the broken PT-
symmetry regime. It can be observed from Fig.2(a)-(c) that all the different
types of coherences (total, global and local coherence) show an oscillatory
behavior for the unbroken PT-symmetry regime. The amplitude of the total and
local coherence increases as the parameter $r$ approaches to $r=1$
(exceptional point), while the amplitude of global coherence decreases as the
parameter $r$ approaches the exceptional point. The transition in the dynamics
of quantum coherence once the exceptional point is reached, can be observed in
Figs.2(d)-(f), where the amplitude of total quantum coherence initially
increases and then freezes at later times. A similar behavior is observed in
the dynamics of local coherence in the broken phase regime (which is initially
absent and is created when acted upon by the PT-symmetric Hamiltonian) where
its amplitude first increases and then attains a stable value. The freezing
behavior of quantum coherence depicted in Fig. 2 is different from the
dynamics of previously observed quantum correlations (both entanglement and
quantum discord) in the maximally entangled state in the broken phase. The
global coherence in the broken phase of PT-symmetry first increases with time
and then decays exponentially to zero. The discussion of the dynamics of
various coherences at the exceptional point ($r=1$) will be taken up later in
Section III.4.

Figure 4: Plots of the evolution of quantum coherence present in the two-qubit
reduced GHZ state under different phases of PT-symmetry. (a), (b) and (c)
represent the dynamics of total coherence ($C_{T}$), global coherence
($C_{G}$) and local coherence ($C_{L}$) respectively, present in
$\rho_{12}(\rho_{13})$ in unbroken PT-symmetry phase ($r=0.6$ and $r=0.8$).
(e),(f) and (g) represent the dynamics of total, global and local coherence
respectively, present in $\rho_{12}(\rho_{13})$ in a broken phase $r=1.4$ and
at the exceptional point $r=1$. (d) represents the dynamics of global
coherence present in $\rho_{23}$ in an unbroken and (h) represents the
dynamics of global coherence present in $\rho_{23}$ in the broken phase of PT-
symmetry. Figure 5:  Contour plots of the dynamics of (a) total coherence, (b)
global coherence, and (c) local coherence, present in the state
$|\psi_{{}_{BS}}^{\alpha}\rangle$ at the exceptional point. Contour plots of
the dynamics of (d) total coherence, (e) global coherence, and (f) local
coherence, present in the state $|\psi_{{}_{GHZ}}^{\beta}\rangle$ at the
exceptional point; the angles $\alpha$ and $\beta$ vary from $0$ to $2\pi$
(6.28) radians.

###  III.3 Three-qubit maximally entangled state (GHZ state)

Next we consider three-qubit states and analyze the evolution of their
coherence properties when one of the qubits is subjected to time evolution
generated by the PT-symmetric Hamiltonian (Eq. (1)). We first consider the
maximally entangled Greenberger-Horne-Zeilinger (GHZ) state:

$\displaystyle\rho_{{}_{GHZ}}=|\psi_{{}_{GHZ}}\rangle\langle\psi_{{}_{GHZ}}|\quad{\rm
where}$
$\displaystyle|\psi_{{}_{GHZ}}\rangle=\frac{1}{\sqrt{2}}(|000\rangle+|111\rangle)$
(13)

Taking the GHZ state as the initial state, we consider the action of the PT-
symmetric Hamiltonian (Eq. (1)) on the first qubit. This amounts to evaluating
the action of the operator $U_{PT}$ defined in Eq. (2) on the first qubit,
leading to the overall operation on the three-qubit system to be implemented
as:

$U_{PT}^{3}=U_{PT}\otimes I\otimes I.$ (14)

The final normalized density operator corresponding to the local action of the
PT-symmetric Hamiltonian on the GHZ state then can be written as:

$\rho_{{}_{GHZ}}^{\prime}(t)=\frac{U_{PT}^{3}\rho_{{}_{GHZ}}(0)U_{PT}^{3}}{Tr[U_{PT}^{3}\rho_{{}_{GHZ}}(0)U_{PT}^{3}]}$
(15)

From this final state, the three coherences under consideration can be
calculated as per the definitions given in Eqs. 8,9 and 10, and their dynamics
can be studied as was done for the case of the two-qubit Bell state in the
previous section.

The results of these calculations for unbroken and broken PT-symmetry regimes
are presented in Figure 3. As can be seen from Figs. 3(a)-(c), in the unbroken
PT-symmetry phase, all quantum coherences in tripartite state show an
oscillatory behavior similar to the bipartite state. As the parameter $r$
increases and reaches the exceptional point, the transition in dynamics of
total and local coherence in tripartite state can be observed, wherein the
quantum coherence freezes and attains a stable value at later times. On the
other hand, the dynamics of global coherence in the GHZ state is different
from the Bell state, as can be observed from Fig.3(c), where the global
coherence initially decays, then increases and finally freezes at later times.

Next, the dynamics of quantum coherences in three possible two-qubit reduced
states ($\rho_{ij}=tr_{k}\left(\rho_{ijk}\right)$, where $i,j,k\in(1,2,3)$) of
the GHZ state are studied. The results of the calculations of dynamics of
various coherences for the two-qubit reduced states associated with the GHZ
state are presented in Figure 4. It is observed that various quantum coherence
(total, local and global) are generated in two-qubit reduced states,
$\rho_{12}$ and $\rho_{13}$ due to the local operation of the PT-symmetric
Hamiltonian on the first qubit in both unbroken and broken phase. The pattern
of quantum coherence evolution in the two-qubit reduced states ($\rho_{12}$
and $\rho_{13}$) are similar to the bipartite maximally entangled Bell state
as shown in Fig.4(a)-(c) and Fig.4(e)-(g). However in the two-qubit reduced
state $\rho_{23}$, only global coherence is generated, which contributes to
overall coherence. From Figs.4(d) and (h) it can be seen that the dynamics of
global coherence present in $\rho_{23}$ shows an oscillatory behavior in
unbroken phase while in broken phase it initially increases and then freezes
at later times.

###  III.4 Dynamics of coherence at the exceptional point

The results of the calculations at the exceptional point ($r=1$) for the two-
qubit Bell state and the three-qubit GHZ state are depicted as contour plots
in Fig. 5. The dynamics of various quantum coherences present in
$|\psi_{{}_{BS}}^{\alpha}\rangle=\cos\alpha|00\rangle+\sin\alpha|11\rangle$
were explored at the exceptional point where $\alpha$ is varied from $0$ to
$2\pi$ i.e.the state evolves from the ground state to an excited state,
passing through the maximally entangled state. From the contour plots shown in
Fig.5(a)-(c), it can be observed that the total coherence stabilizes with time
but the amplitude of total coherence varies with $\alpha$. Similar dynamics of
local coherence can be observed from the contour plot, where it can be seen
that the local coherence decays with time and does not depend on the $\alpha$
value. The amplitude of the global coherence at the exceptional point first
increases with time and then decays exponentially to zero. The dynamics of
different quantum coherences in the broken phase are similar to the dynamics
of quantum coherence observed at the exceptional point except that their
amplitude decreases as the parameter $r$ increases.

The dynamics of various types of quantum coherences present in
$|\psi_{{}_{GHZ}}^{\beta}\rangle=\cos\beta|000\rangle+\sin\beta|111\rangle$
were investigated at the exceptional point. The parameter $\beta$ is varied
from $0$ to $2\pi$, i.e.the state goes from the ground state $|000\rangle$ to
the excited state $|111\rangle$, through the maximally entangled state. The
contour plots of the dynamics of the quantum coherences are shown in
Figs.5(d)-(f), from where it can be observed that the total coherence freezes
with time and the amplitude of the frozen coherence varies with varying
$\beta$ values of the state. Similar dynamics of global coherence can be
observed from the contour plot, which shows that the global coherence freezes
with time and its amplitude depends on the $\beta$ value. The contour plot of
the local coherence depicts that the local coherence stabilizes to a value $1$
with time and does not depend on the $\beta$ value. The dynamics of all
quantum coherences in the broken phase are similar to the dynamics of the
quantum coherences observed at the exceptional point but their amplitude
decreases as the parameter $r$ increases as displayed in Figure 3(d)-(f) where
panel (d) shows evolution of total coherence, panel (e) shows evolution of the
global coherence and evolution of the local coherence is shown in panel (f).

##  IV Experimental demonstration of the dynamics of quantum coherence under a
PT-symmetric Hamiltonian

As discussed in Sec.II, the PT-symmetric Hamiltonian can be simulated by the
dilation method using an ancillary qubit. The experimental implementation of
PT-symmetric Hamiltonian on an NMR quantum processor is realized using a
three-qubit system where the three spin-1/2 nuclei (1H, 19F and 13C) of
13C-labeled diethylfluoromalonate dissolved in acetone-D6, are encoded as the
first, second and third qubit respectively; the 1H and 19F are used as a two-
qubit system while the 13C spin is utilized as the ancillary qubit.

Figure 6: (a) Molecular structure of 13C-labeled diethylfluoromalonate with
the three qubits encoded as 1H, 19F and 13C and the system parameters given
alongside. (b) NMR spectrum of thermal equilibrium state obtained after a
$\pi/2$ readout pulse. (c) NMR spectrum of the pseudopure state. Each
resonance peak is labeled in the logical basis and the horizontal scale of
each spectrum denotes the chemical shift in ppm.

The molecular structure of the three-qubit system along with tabulated system
parameters and the NMR spectrum of each qubit at thermal equilibrium are shown
in Fig.6(a) and Fig.6(b), respectively. The NMR Hamiltonian for a three-qubit
system in a rotating frame under the weak approximation [43] is given by

$H=-\sum_{i=1}^{3}\nu_{i}I_{z}^{i}+\sum_{i>j,i=1}^{3}J_{ij}I_{z}^{i}I_{z}^{j}$
(16)

where $i,j=$1, 2 and 3 label the nuclear spin, $v_{i}$ is the chemical shift
of the $i$th spin, $J_{ij}$ represents the scalar spin-spin coupling strength
and $I_{z}^{i}$ is the $z$ component of the spin angular momentum operator for
the $i^{th}$ spin. The three-qubit system was initialized into a pseudopure
state (PPS) $|000\rangle$ using the spatial averaging technique [44, 45], with
the density $\rho_{000}$ given by

$\rho_{000}=\frac{(1-\epsilon)}{8}I+\epsilon|000\rangle\langle 000|$ (17)

where $\epsilon\sim 10^{-5}$ represents the spin polarization at room
temperature and $I$ is the 8$\times$8 identity operator. The NMR spectrum of
the three-qubit PPS is shown in Fig.6(c). The experimentally prepared PPS was
tomographed and its fidelity was computed to be $0.9857\pm 0.0001$ using the
Uhlmann-Jozsa fidelity measure [46, 47].

The density matrix of the initial state was reconstructed using least square
constrained convex optimization method [48] by performing full quantum state
tomography using a set of seven preparatory pulses $\\{III$, $XXX$, $IIY$,
$XYX$, $YII$, $XXY$, $IYY\\}$ where $I$ represents ’no-operation’ and $X(Y)$
is the local $\pi/2$ unitary operator with phases $x(y)$ and local operations
are achieved using spin-selective $\pi/2$ radiofrequency (rf) pulses. All
experiments were performed at room temperature 293K on a Bruker Avance-III 600
MHz FT-NMR spectrometer equipped with a QXI probe.

Figure 7: Real (left) and imaginary (right) parts of the theoretically
expected and experimentally reconstructed density matrix of the Bell State
after implementation of the PT-symmetric Hamiltonian at time $t=4s$ under (a)
a broken phase at non-hermiticity strength r=1.4 and (b) an unbroken phase at
non-hermiticity strength r=0.6.

The quantum circuit to implement PT-symmetric Hamiltonian is shown in
Fig.1(a). The first block creates the maximally entangled bipartite state
(Bell State, $\rho_{{}_{BS}}\otimes\rho_{0}$) on two qubits ($\rho_{{}_{BS}}$)
and the third qubit ($\rho_{0}$) is utilized as an ancillary qubit. The next
block of the circuit implements a PT-symmetric Hamiltonian on the first qubit
of the two-qubit state using an ancillary qubit where the parameter $r$, and
the time period ($t$) are varied by $\theta$ and $\phi$ as given in Eq. [4]
and Eq. [7] respectively; $\theta$ is controlled by the operator $V$ and
$\phi$ by the operator controlled-$U_{1}$. All quantum gates were implemented
using rf-pulses and free evolution under the system Hamiltonian. Local
rotations were implemented by spin-selective pulses with different phases; the
durations of $\pi/2$ pulses for 1H, 19F and 13C nuclei were 9.38 $\mu$s, 21.8
$\mu$ s and 15.1 $\mu$ s at power levels of 18.14 W, 42.27 W and 179.47 W,
respectively.

Figure 8: Dynamics of quantum coherence present in the two-qubit maximally
entangled Bell State under a $PT$-Symmetric Hamiltonian. Plots in (a), (b) and
(c) represent the dynamics of the total coherence ($C_{T}$), global coherence
($C_{G}$) and local coherence ($C_{L}$) respectively, in the unbroken phase of
PT-Symmetry ($r=0.6$). (d), (e) and (f) represent the dynamics of the total
coherence ($C_{T}$), global coherence ($C_{G}$) and local coherence ($C_{L}$)
respectively, in the broken phase of PT-Symmetry ($r=1.4$). The black lines
are the theoretical curves (Theor.) and the black filled circles are the
experimental points (Exp.).

The NMR pulse sequence used to experimentally prepare a maximally entangled
bipartite state is shown in Fig.1(b). The fidelity of the initial state
$\left(\rho_{{}_{BS}}\otimes\rho_{0}\right)$ created was found to be
$0.9745\pm 0.0102$. The method to simulate the PT-symmetric Hamiltonian is
described in Sec.II, where $\theta$ and $\phi$ control the degree of non-
hermiticity $r$ and the time point of evolution $t$. We experimentally
simulated the PT-symmetric Hamiltonian by choosing parameters $r=0.6$ and
$r=1.4$ corresponding to unbroken and broken phase respectively, which was
implemented on the first qubit of the bipartite state, using the third qubit
as an ancillary qubit. The parameter $r$ was fixed depending on different
phases of PT-symmetry, and the state was allowed to evolve by increasing the
time period $t$ and accordingly varying $\theta$ and $\phi$. Each state was
tomographed at different time points after implementing the PT-symmetric
Hamiltonian in both the unbroken and broken phase. The final density matrices
of the two qubit-Bell states were reconstructed by measuring in the
$|0\rangle$ subspace of the ancillary qubit, where only three tomographic
pulses ($III$, $IIY$, $YII$) were required to reconstruct the entire density
matrix. A representative tomograph of the experimentally reconstructed density
matrix of the Bell state after the implementation of the PT-symmetric
Hamiltonian in two different phases (unbroken phase at $r=0.6$ and broken
phase at $r=1.4$) of PT-symmetry at time $t=4$ s is shown in Fig.7.

A single qubit coherence flow in PT-symmetric and anti-PT-symmetric systems
was demonstrated experimentally using an optical setup and it was observed
that coherence flow oscillates back and forth in the unbroken PT-symmetric and
anti-PT-symmetric regimes while in broken phase, coherence flow attains a
stable value in both PT-symmetric and anti-symmetric systems [24]. We
investigated the dynamics of quantum coherence (total, local and global
coherence) in the maximally entangled (Bell) state, when one qubit of the Bell
state is acted upon by the PT-symmetric Hamiltonian at $r=0.6$ (unbroken
phase) and at $r=1.4$ (broken phase). The results are plotted in Fig.8 where
(a), (b) and (c) depict the dynamics of various types of quantum coherences in
the unbroken phase and (d), (e) and (f) depict the dynamics of various types
of quantum coherence in the broken phase. Similar patterns of coherence
dynamics are observed as in the case of the optics setup [24]. Our results in
Fig.8 (a)-(c) show that in the Bell state, the total, local and global
coherences oscillate with time in the unbroken phase ($r=0.6$) while the
amplitudes are different for various quantum coherences. Another interesting
feature of local coherence generation is observed that it oscillates with
time, and is not present in the initial Bell state. In the broken phase
($r=1.4$), the total coherence initially increases and then freezes at later
times. Similarly, the results of Fig.8(c) indicate that the local coherence,
which is not present initially in the Bell state, is created and increases
under the broken phase of PT-symmetry and freezes at later times. Fig.8 (b)
shows that the the global coherence initially increases and then exponentially
decays with time. Our experimental results match well with the theoretically
predicted behavior of quantum coherence, within experimental errors.

Similar patterns of information flow have been observed in NMR in a single-
qubit system where information flow oscillates in the broken phase while it
decays with time in the unbroken phase of anti-PT-symmetry [12]. A similar
freezing pattern of quantum coherence has also been observed in a two-qubit
subsystem of a three-qubit GHZ state under the PT-symmetric Hamiltonian using
nuclear spins, where it was observed that entropy and entanglement stabilize
to a certain value [13].

##  V Conclusion

We have theoretically investigated the dynamics of quantum coherence (total,
global and local coherence) present in maximally entangled bipartite and
tripartite states in both the unbroken and the broken phase of PT-symmetry and
at an exceptional point when one qubit is acted upon by the local PT-symmetric
Hamiltonian. Our results show that the different types of quantum coherence
oscillate in the unbroken phase of PT-symmetry. In the broken phase of PT-
symmetry and at the exceptional point, the total and local coherence first
increase and then attain a stable value over time in both the maximally
entangled bipartite and tripartite states. The global coherence decays
exponentially in the bipartite state but freezes over time in the tripartite
state. Our results indicate that quantum coherence behaves differently under
PT-symmetry, when the dimensionality of the quantum system is increased. The
dynamics of quantum coherence present in the maximally entangled bipartite
state was experimentally verified by implementing the PT-symmetric Hamiltonian
on an NMR quantum processor. Our work sheds some light on the effect of the
PT-symmetric Hamiltonian on quantum coherences in a multipartite system. Our
results can help in gaining an understanding of the effects of the PT-
symmetric Hamiltonian in quantum thermodynamics and in quantum information
processing[49, 50].

###### Acknowledgements.

All the experiments were performed on a Bruker Avance-III 600 MHz FT-NMR
spectrometer at the NMR Research Facility of IISER Mohali. A. acknowledges
financial support from DST/ICPS/QuST/Theme-1/2019/Q-68. K .D. acknowledges
financial support from DST/ICPS/QuST/Theme-2/2019/Q-74.

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# Linear mass density of vertically suspended heavy springs.

Cyril Belardinelli<EMAIL_ADDRESS>Lucerne University of Applied
Sciences and Arts, Technikumstrasse 21, CH-6048 Horw, Switzerland

###### Abstract

In the present article the linear mass density $\lambda$ for vertically
suspended heavy springs is calculated for two different cases. First for a
spring of invariable length suspended at the top and fixed at the bottom of
the spring, then for a hanging heavy spring with an additional load. Both
cases are solved by minimizing the total energy $E[\lambda]$ which is a sum of
potential energy and energy due to the deformation.

Hooke’s law, Gravitational field, Euler-Lagrange-equation, Variational
Calculus

##  I Introduction

Hooke’s law for ideal springs is a recurring subject in elementary physics
courses. In most textbook problems the mass of the spring itself is neglected.
An interesting question arises when one asks for the static deformation of the
spring when it is suspended vertically with or without a mass hanging at the
free end. Due to its own weight and the loaded mass the spring deforms in a
non-uniform way giving rise to a non-constant mass density function
$\lambda(x)$. The issue of heavy springs has already been treated rather
intensely by various authors.cushing:1984 ; mak:1987 ; ruby:2000 ; essen:2010
; galloni:1979 ; serna:2010  However, an explicit formula for the linear mass
density of suspended heavy springs does not seem to occur in the literature.
The issues presented in this article are nicely solved by the standard methods
of variational calculus.courant:43 ; goldstein:1980

##  II Suspended spring with fixed ends

In this section we consider an ideal spring of mass m, natural length $l_{0}$
and stiffness k. The spring is vertically suspended where both ends are fixed
at a distance $\textit{l}>\textit{l}_{0}$.

Figure 1: vertically suspended ideal spring of stiffness $k$, natural length
$l_{0}$ and mass $m$ with fixed ends. The spring has in its stretched form a
length of $l$. The spring’s linear mass density in its undeformed state is
$\lambda_{0}=\frac{m}{l_{0}}$. In its suspended position, the spring’s linear
mass density $\lambda(x)$ increases from top to bottom due to its weight.

To find the linear mass density $\lambda(x)$ one minimizes the energy
functional $E_{\text{tot}}[\lambda]$ which is given by the sum of the
potential energy $E_{pot}$ and deformation energy $E_{d}$:

$E_{\text{tot}}[\lambda]=E_{\text{pot}}[\lambda]+E_{d}[\lambda]$ (1)

The potential energy formulated as a functional of $\lambda(x)$ is given by:

$E_{\text{pot}}[\lambda]=-g\int_{0}^{\textit{l}}x\lambda(x)dx$ (2)

The energy functional $E_{d}[\lambda]$ due to deformation is given by:

$E_{d}[\lambda]=\frac{kl_{0}}{2}\int_{0}^{\textit{l}}\left(\frac{\lambda_{0}}{\lambda(x)}-2+\frac{\lambda(x)}{\lambda_{0}}\right)dx$
(3)

One has to minimize Eq. (1) under the constraints of invariable mass and non-
negativity of $\lambda(x)$:

$m=\int_{0}^{\textit{l}}\lambda(x)dx\quad\text{and}\quad\lambda(x)>0\quad\forall
x\in[0,l]$ (4)

In order to take account of the constraints one introduces a Lagrange-
multiplier $\mu$ with which the functional reads as follows:

$F[\lambda]=E_{\text{tot}}[\lambda]-\mu\left[m-\int_{0}^{\textit{l}}\lambda(x)dx\right]$
(5)

Variation of the functional Eq. (5) leads to the following Euler-Lagrange-
equation.courant:43 ; goldstein:1980 :

$\frac{kl_{0}\lambda_{0}}{2}\left(\frac{1}{\lambda_{0}^{2}}-\frac{1}{\lambda^{2}(x)}\right)-gx+\mu=0$
(6)

From the latter equation one deduces ($\lambda_{0}l_{0}=m$):

$\lambda(x)=\frac{\lambda_{0}}{\sqrt{1+\frac{2\mu
m}{kl^{2}_{0}}-\frac{2mg}{kl^{2}_{0}}x}}$ (7)

The Lagrange-multiplier $\mu$ in Eq. (5) is fixed by the constraint Eq. (4). A
short calculation yields:

$\mu=\frac{k}{2m}(l^{2}-l^{2}_{0})+\frac{mg^{2}}{8k}+\frac{gl}{2}$ (8)

By inserting $\mu$ into Eq. (7) one gets the final form for the linear mass
density $\lambda(x)$. For further calculations an expression in terms of
dimensionless variables is convenient. The reduced mass density
$\tilde{\lambda}:=\frac{\lambda}{\lambda_{0}}$ reads then:

$\tilde{\lambda}(\tilde{x})=\frac{1}{\sqrt{(\frac{\alpha}{2}+\tilde{l})^{2}-2\alpha\tilde{x}}},\quad
0<\tilde{x}<\tilde{l}$ (9)

where the dimensionless parameters read:

$\alpha:=\frac{mg}{kl_{0}}\quad\tilde{x}:=\frac{x}{l_{0}}\quad\tilde{l}:=\frac{l}{l_{0}}$
(10)

Figure 2: reduced mass density $\tilde{\lambda}(\tilde{x})$ of Eq. (9) for
$\alpha=0.8$ (dotted curve) and $\alpha=1.5$ (solid). For both plots:
$\tilde{l}=1.3$

##  III heavy spring suspended at the top end with load

Figure 3: free hanging ideal spring of stiffness $k$, natural length $l_{0}$
and mass $m$. The elongation $\Delta x$ is due to the spring’s own weight and
additional load $M$.

In this section we calculate the linear mass density $\lambda(x)$ for a static
spring of mass $m$ and stiffness $k$ which is suspended only at the top end.
The spring is loaded with an additional mass $M$ at the bottom end. The
calculation can be done in almost the same manner as in the previous section.
The calculation starts again with the same functional Eq. (7) which has to be
minimized. To fix the Lagrange-multiplier $\mu$ one can use the fact that at
the bottom end the spring is only deformed by the additional load $M$. It is a
simple task to derive the spring’s mass density $\lambda(l)$ at the bottom
end, as shown in the following.
$dx$ denotes an infinitesimal intervall of the suspended spring at the bottom
end. The corresponding natural (undeformed) length of $dx$ is denoted by
$dx_{0}$. By Hooke’s law one has:

$k^{\prime}(dx-dx_{0})=Mg$ (11)

where $k^{\prime}$ denotes the stiffness of the spring of infinitesimal length
$dx_{0}$ which is given by:

$k^{\prime}=k\frac{l_{0}}{dx_{0}}$ (12)

Inserting the latter expression into Eq. (11) one gets:

$kl_{0}\left(\frac{dx}{dx_{0}}-1\right)=Mg$ (13)

By using the relation: $\frac{\lambda_{0}}{\lambda(l)}=\frac{dx}{dx_{0}}$ one
arrives at:

$kl_{0}\left(\frac{\lambda_{0}}{\lambda(l)}-1\right)=Mg$ (14)

Solving for $\lambda(l)$ yields:

$\lambda(l)=\frac{\lambda_{0}}{1+\frac{Mg}{kl_{0}}}$ (15)

For an unloaded spring (i. e. $M=0$) the density at the bottom is, as
expected, identical to $\lambda_{0}$ since the spring does not experience any
weight at the bottom. Then by Eq. (7) one gets the Lagrange-multiplier $\mu$:

$\mu=gl+\frac{M}{m}gl_{0}+\frac{M^{2}g^{2}}{2mk}$ (16)

At this stage one can determine the total length $l$ of the spring by imposing
once more the constraint of invariable mass:

$\int_{0}^{l}\lambda(x)dx=m$ (17)

Rewriting the latter integral in terms of the usual dimensionless variables
(by adding the dimensionless parameter $\beta:=\frac{Mg}{kl_{0}}$) one gets:

$\int_{0}^{\tilde{l}}{\frac{d\tilde{x}}{\sqrt{(1+\beta)^{2}+2\alpha(\tilde{l}-\tilde{x})}}}=1$
(18)

Evaluation of the integral and solving for the total relative length
$\tilde{l}$ gives:

$\tilde{l}=1+\frac{\alpha}{2}+\beta$ (19)

or respectively:

$l=l_{0}+\frac{mg}{2k}+\frac{Mg}{k}$ (20)

For an unloaded spring (i. e. $M=0$) the elongation $\Delta x$ due to its own
weight is given by $\Delta x=\frac{mg}{2k}$ which is a well known
result.mak:1987 ; galloni:1979  By Eq. (19) one gets immediately an expression
for the reduced mass density:

$\tilde{\lambda}(x)=\frac{1}{\sqrt{(1+\alpha+\beta)^{2}-2\alpha\tilde{x}}},\quad
0<\tilde{x}<\tilde{l}$ (21)

Figure 4: Typical plot of the reduced mass density
$\tilde{\lambda}(\tilde{x})$ given by Eq. (21) for no load ($M=0$) and
$\alpha=1.5$. At the bottom end of the spring (i. e. $x=l_{0}+\frac{mg}{2k}$)
one has a reduced mass density which assumes the value
$\tilde{\lambda}=\frac{\lambda(l)}{\lambda_{0}}=1$ as shown by the plot.

##  IV relative change of $\Delta E_{\text{pot}}$ vs. $\Delta E_{\text{d}}$

The distribution of mass along the suspended spring is fixed by the condition
of minimizing total energy. It is interesting to determine the ratio
$\frac{\Delta E_{pot}}{\Delta E_{d}}$ of relative change in potential energy
and deformation energy induced by the spring’s deformation.

###  IV.1  spring with fixed ends

The relative change in potential energy is given by:

$\Delta
E_{\text{pot}}=-\int_{0}^{l}x\lambda(x)dx+\frac{1}{2}mgl_{0}=-mgl_{0}\int_{0}^{\tilde{l}}\frac{\tilde{x}d\tilde{x}}{\sqrt{(\frac{\alpha}{2}+\tilde{l})^{2}-2\alpha\tilde{x}}}+\frac{1}{2}mgl_{0}$
(22)

The latter integral can be evaluated easily, one gets then

$\Delta E_{\text{pot}}=-mgl_{0}\frac{\alpha}{12}$ (23)

From the latter result one concludes that the lowering of the center of mass
is given by:

$\Delta x=\frac{\alpha}{12}l_{0}=\frac{mg}{12k}$ (24)

On the other hand, the relative change in the deformation energy is given by
the following expression:

$\Delta
E_{d}=\frac{1}{2}kl_{0}\int_{0}^{l}\left(\frac{\lambda_{0}}{\lambda(x)}-2+\frac{\lambda(x)}{\lambda_{0}}\right)dx-\frac{1}{2}kl_{0}^{2}(\tilde{l}-1)^{2}$
(25)

In dimensionless form the later expression becomes:

$\Delta
E_{d}=\frac{1}{2}kl_{0}^{2}\int_{0}^{\tilde{l}}\left[\sqrt{\left(\frac{\alpha}{2}+\tilde{l}\right)^{2}-2\alpha\tilde{x}}-2+\left(\sqrt{\left(\frac{\alpha}{2}+\tilde{l}\right)^{2}-2\alpha\tilde{x}}\right)^{-1}\right]dx-\frac{1}{2}kl_{0}^{2}(\tilde{l}-1)^{2}$
(26)

Evaluation of the latter expression yields:

$\Delta E_{d}=\frac{1}{24}kl_{0}^{2}\alpha^{2}$ (27)

We conclude therefore:

$\frac{\Delta E_{\text{pot}}}{\Delta E_{d}}=2$ (28)

###  IV.2  spring suspended at top end

To calculate the change in potential energy $\Delta E_{\text{pot}}$ one
determines the center of mass (CM) of the suspended spring. It is given by the
integral:

$CM=\frac{1}{l_{0}}\int_{0}^{1+\frac{mg}{2k}}\frac{xdx}{\sqrt{(1+\frac{mg}{l_{0}k})^{2}-2\frac{mg}{kl_{0}}x}}$
(29)

Once again, in dimensionless form one has:

$\begin{split}CM&=l_{0}\int_{0}^{1+\frac{\alpha}{2}}\frac{\tilde{x}d\tilde{x}}{\sqrt{(1+\alpha)^{2}-2\alpha\tilde{x}}}=\left(\frac{1}{2}+\frac{\alpha}{3}\right)l_{0}\\\
&=\frac{l_{0}}{2}+\frac{mg}{3k}\end{split}$ (30)

Due to the self weight of the spring the center of mass lowers by an amount of
$\frac{mg}{3k}$ which is a known result.serna:2010  With this result one
determines immediately $\Delta E_{\text{pot}}$:

$\Delta E_{\text{pot}}=-\frac{1}{3}k\left(\frac{mg}{k}\right)^{2}$ (31)

The change in deformation energy is readily evaluated. One has:

$\Delta
E_{d}=\frac{1}{2}kl_{0}^{2}\int_{0}^{1+\frac{\alpha}{2}}\left[\sqrt{\left(1+\alpha\right)^{2}-2\alpha\tilde{x}}-2+\left(\sqrt{\left(1+\alpha\right)^{2}-2\alpha\tilde{x}}\right)^{-1}\right]dx-\frac{1}{2}kl_{0}^{2}(\tilde{l}-1)^{2}$
(32)

Evaluation gives:

$\Delta
E_{\text{d}}=\frac{1}{6}kl_{0}^{2}\alpha^{2}=\frac{2}{3}k\left(\frac{mg}{2k}\right)^{2}$
(33)

Interestingly, one gets the same ratio:

$\frac{{\Delta E_{\text{pot}}}}{\Delta E_{d}}=2$ (34)

*

## References

  * (1) R. Courant and D. Hilbert.  Methoden der mathematischen Physik, 2nd ed., volume 1.  Interscience Publishers, Inc. New York, 1943.
  * (2) J. T. Cushing.  The spring-mass system revisited.  Am. J. Phys., 52:925–933, 1984.
  * (3) H. Essen and A.B. Nordmark.  Static deformation of heavy spring due to gravity and centrifugal force.  Eur. J. Phys., 31:603–609, 2010.
  * (4) E. E. Galloni and M. Kohen.  Influence of the mass of the spring on its static and dynamic effects.  Am. J. Phys., 47:1076–1078, 1979.
  * (5) H. Goldstein.  Classical Mechanics.  Addison-Wesley, Massachusetts, 2nd edition, 1980.
  * (6) S. Y. Mak.  The static effectiveness mass of a slinky.  Am. J. Phys., 55:994–997, 1987.
  * (7) L. Ruby.  Equivalent mass of a coil spring.  The Physics Teacher, 38:140–141, 2000.
  * (8) J. D. Serna and A. Joshi.  Finding the center of mass of a soft spring.  College Mathematics Journal, 42, 2010.


11institutetext:  Dipartimento di Fisica dell’Università di Pisa and INFN \-
Sezione di Pisa, Largo Pontecorvo 3, I-56127 Pisa, Italy 22institutetext:
SISSA, Via Bonomea 265, I-34136, Trieste, Italy 33institutetext: ELTE Eötvös
Loránd University, Institute for Theoretical Physics, Pázmány Péter sétány
1/A, H-1117, Budapest, Hungary

# Localisation of Dirac eigenmodes and confinement in gauge theories: the
Roberge-Weiss transition

Marco Cardinali 11 Massimo D’Elia 11 Francesco Garosi 22 Matteo Giordano 33
<EMAIL_ADDRESS>
###### Abstract

Ample numerical evidence from lattice calculations shows a strong connection
between the confining properties of gauge theories at finite temperature and
the localisation properties of the low-lying Dirac eigenmodes. In this
contribution we discuss recent progress on this topic, focussing on results
for QCD at imaginary chemical potential ${\mu}_{I}/T=\pi$ at temperatures
above the Roberge-Weiss transition temperature. These confirm the general
picture of low modes turning from delocalised to localised at the
deconfinement transition, in a previously unexplored setup with a genuine,
physical transition in the presence of dynamical fermions. This further
supports the use of Dirac eigenmodes as a tool to investigate the mechanisms
behind confinement and the deconfinement transition.

##  1 Introduction

While the crossover nature and the location of the QCD transition at finite
temperature have been firmly established, the mechanism that drives it is
still a matter of active research. In particular, one would like to understand
why both confining and chiral properties change dramatically in the same
relatively narrow temperature range, where quarks and gluons deconfine and
chiral symmetry gets restored. In fact, linear confinement of quarks and
spontaneous breaking of chiral symmetry apply to QCD only in an approximate
sense, originating from opposite quark mass limits, and it is not yet fully
understood how they affect each other.

Quarks are confined by a linear potential only in the “quenched” limit of
infinite quark mass, where QCD reduces to pure SU(3) gauge theory. In this
setting, deconfinement is signalled by the spontaneous breaking of the exact
centre symmetry of the theory by the ordering of the local Polyakov loops,
resulting in a vanishing string tension. The Polyakov loop,

$P(\vec{x}\mkern
2.0mu)=\textrm{Pexp}\left\\{ig\int_{0}^{\frac{1}{T}}dt\,A_{4}(t,\vec{x}\mkern
2.0mu)\right\\}\,,$ (1)

is the holonomy of the gauge field around the temporal direction, which in the
imaginary-time formulation of finite-temperature gauge theories is
compactified to a circle of size equal to the inverse of the temperature, $T$.
In pure gauge theory at high temperature the local Polyakov loops tend to
align to one of the elements of the gauge group centre, $\mathbb{Z}_{3}$, thus
breaking centre symmetry, with the trivial element being the one selected when
pure gauge theory is approached as the quenched limit of QCD. Deconfinement is
then associated with the ordering of local Polyakov loops, and while to a
small extent this is present in QCD already at low temperatures, it becomes
considerably stronger at the crossover.

On the other hand, chiral symmetry is a symmetry only in the limit of massless
quarks, and only there one can speak of its spontaneous breaking in a strict
sense. In this limit, chiral symmetry is spontaneously broken to its vector
part by a quark condensate, resulting from the accumulation of eigenvalues of
the Dirac operator $\not{D}$ near the origin. This is made clear by the Banks-
Casher formula Banks:1979yr ,

$\Sigma=|\langle\bar{\psi}\psi\rangle|=\int_{0}^{\infty}d\lambda\,\rho(\lambda)\frac{2m}{\lambda^{2}+m^{2}}\,,\qquad\rho(\lambda)\equiv\lim_{V\to\infty}\frac{T}{V}\Big{\langle}\sum_{n}\delta(\lambda-\lambda_{n})\Big{\rangle}\,,$
(2)

where $\rho(\lambda)$ is the spectral density of $\not{D}$. Here angular
brackets denote the expectation value in the sense of the Euclidean functional
integral in the imaginary-time formulation of finite-temperature gauge
theories, $i\lambda_{n}$ are the eigenvalues of $\not{D}$ in a given gauge
configuration, and $V$ is the spatial volume of the system. In the chiral
limit one finds precisely $\Sigma\to\pi\rho(0^{+})$. Even for physical quark
masses, the crossover is characterised by a dramatic reduction of the density
of near-zero modes, and so the approximate restoration of chiral symmetry is
signalled by the depletion of the near-zero spectral region.

Despite their very different origins, deconfinement and chiral restoration
show a close relationship not only in QCD, but also in other QCD-like gauge
theories. In theories with a single, genuine deconfining phase transition
(e.g., pure gauge theory, or lattice QCD with $N_{f}=3$ staggered fermions on
coarse lattices Karsch:2001nf ), this is associated with a general improvement
of the chiral properties, in particular with a decrease of the chiral
condensate. In theories with two genuine transitions (e.g., QCD with $N_{f}=2$
adjoint fermions Karsch:1998qj ), deconfinement preceeds chiral restoration.
This suggests that deconfinement strongly affects the chiral properties of the
theory, although it is not yet clear by means of what mechanism.

What could play a role in this mechanism is the localisation of the low-lying
Dirac modes taking place at the finite temperature transition in QCD
GarciaGarcia:2006gr ; Kovacs:2012zq ; Cossu:2016scb ; Holicki:2018sms ;
Giordano:2013taa ; Nishigaki:2013uya ; Ujfalusi:2015nha  and other gauge
theories (see Ref. Giordano:2021qav  for a recent review). While delocalised
at low $T$, low Dirac modes become localised above the transition, up to a
critical point in the spectrum known as mobility edge. Ample numerical
evidence shows that the connection between deconfinement and low-mode
localisation is a general feature of gauge theories. In particular, localised
low modes appear precisely at deconfinement when this takes place through a
genuine phase transition. The general connection between deconfinement and
low-mode localisation is understood qualitatively in terms of the
“sea/islands” picture of localisation Bruckmann:2011cc ; Giordano:2015vla ;
Giordano:2016cjs ; Giordano:2016vhx . In this picture, the ordering of local
Polyakov loops into a “sea”, that signals deconfinement, is responsible for
the opening of a gap in the spectrum of $\not{D}$, that can be filled by
eigenmodes only at the price of localising on “islands” of Polyakov-loop
fluctuations. This leads one to expect the localisation of low Dirac modes in
the deconfined phase of a generic gauge theory, an expectation so far
supported by all the available numerical investigations Giordano:2021qav
.111The connection between localisation and chiral symmetry breaking has
received less attention in recent years. For the possible consequences of the
presence of localised modes for chiral symmetry breaking, see Refs.
Giordano:2020twm ; Giordano:2022ghy .

Until recently, the numerical evidence for a strong deconfinement/localisation
connection, with localised modes appearing right at the critical temperature,
came mostly from pure gauge theories with a sharp transition. The only
exception was the toy model for QCD provided by $N_{f}=3$ staggered fermions
on coarse lattices Giordano:2016nuu , where the transition is only a lattice
artefact and does not survive the continuum limit. The first study in a theory
with a genuine and physical phase transition in the presence of dynamical
fermions was done by us in Ref. Cardinali:2021fpu , looking at the Roberge-
Weiss transition in QCD with an imaginary chemical potential $\mu_{I}$ at
$\hat{\mu}_{I}\equiv\mu_{I}/T=\pi$. Our results confirm the general
expectation, and further support the strong connection between deconfinement
and localisation.

In this contribution, after a brief review of localisation in Sec. 2, of the
sea/islands picture in Sec. 3, and of QCD at finite imaginary chemical
potential in Sec. 4, we discuss our results obtained in QCD at
$\hat{\mu}_{I}=\pi$ above the Roberge-Weiss temperature in Sec. 5.

##  2 Localisation

Eigenmode localisation is a concept originating in the study of random
Hamiltonians mimicking disordered systems lee1985disordered . Qualitatively
speaking, the localised modes of a random Hamiltonian $H$ are those supported
essentially only in a finite region of space, where their amplitude squared
$\|\psi(x)\|^{2}=\sum_{\alpha}\psi_{\alpha}(x)^{*}\psi_{\alpha}(x)$ is not
negligible, $\|\psi(x)\|^{2}\sim 1/L_{0}^{d}$ with $d$ the spatial dimension
and $L_{0}$ the linear size of the localisation region. Here $\sum_{\alpha}$
denotes the sum over any discrete spacetime or internal index. Delocalised
modes instead extend throughout the system, with $\|\psi(x)\|^{2}\sim 1/L^{d}$
in a finite spatial box of side $L$.222Modes with $\|\psi(x)\|^{2}\sim
1/L^{\alpha}$, $0<\alpha<d$, are known as “critical” in the condensed matter
literature. A more accurate characterisation is in terms of how the average
spatial size of the modes scales with the system size. This is obtained from
their inverse partecipation ratio (IPR), averaged over modes in the spectral
region of interest and over realisations of $H$,

$\overline{\textrm{IPR}}(\lambda,L)=\frac{\langle\sum_{n}\delta(\lambda-\lambda_{n})\textrm{IPR}_{n}\rangle}{\langle\sum_{n}\delta(\lambda-\lambda_{n})\rangle}\,,\qquad\textrm{IPR}_{n}=\int
d^{d}x\,\|\psi_{n}(x)\|^{4}\,,$ (3)

where $\psi_{n}$ and $\lambda_{n}$ are the eigenmodes and eigenvalues of $H$
for some disorder realisation, $H\psi_{n}=\lambda_{n}\psi_{n}$, and angular
brackets indicate the average over disorder. The inverse of the IPR measures
the spatial size of the mode: for localised modes
$\overline{\textrm{IPR}}(\lambda,L)$ remains finite as $L\to\infty$, while for
delocalised modes $\overline{\textrm{IPR}}(\lambda,L)\to 0$.

Figure 1: Left: $I_{s_{0}}$ [see Eq. (5)] along the Dirac spectrum (in units
of the light-quark mass $m_{\textrm{ud}}$) for 2+1 flavour lattice QCD at
$T=394\leavevmode\nobreak\ \textrm{MeV}$, for several spatial sizes $L$ (here
the lattice spacing is $a=0.125\leavevmode\nobreak\ \textrm{fm}$, and
$s_{0}\simeq 0.508$). The mobility edge, the critical value of the statistics,
as well as the Poisson and RMT expectations are also shown. (Data from Ref.
Giordano:2013taa .) Right: Sketch of the spectral density in high-temperature
QCD ($\hat{\mu}_{I}=0$). The solid line is the spectral density in the case of
vanishing gauge fields, starting at the gap $\omega=\pi T$, while the dashed
line qualitatively shows the spectral density after the inclusion of localised
modes penetrating the gap, and $\lambda_{c}$ indicates the position of the
mobility edge.

A convenient way to detect localisation is through the study of the
statistical properties of the eigenvalues altshuler1986repulsion . Since
delocalised modes are easily mixed by disorder fluctuations, the corresponding
eigenvalues are expected to obey the statistics of the appropriate ensemble of
Random Matrix Theory (RMT) mehta2004random , depending on the symmetry class
of $H$. Localised modes, on the other hand, are sensitive only to disorder
fluctuations in their localisation region, and so the corresponding
eigenvalues should fluctuate independently for uncorrelated disorder, or more
generally for disorder with short-range correlations, therefore obeying
Poisson statistics. A simple way to analyse the statistical properties of the
spectrum is in terms of the distribution $p_{\lambda}(s)$ of the unfolded
level spacings, computed locally in the spectrum. Unfolded level spacings are
defined as

$s_{i}=\frac{\lambda_{i+1}-\lambda_{i}}{\langle\lambda_{n+1}-\lambda_{n}\rangle_{\lambda}}\,,$
(4)

where ${\langle\lambda_{n+1}-\lambda_{n}\rangle_{\lambda}}$ is the average
level spacing in the spectral region of interest. The behaviour of
$p_{\lambda}(s)$ is universal, depending only on the type of statistics and
not on the details of the model, and is explicitly known both for Poisson and
RMT statistics. Localisation can then be detected by comparing features of
$p_{\lambda}(s)$ to the known expectations. To this end, a particularly
convenient observable is the integrated unfolded level spacing distribution
Shklovskii:1993zz

$I_{s_{0}}=\int_{0}^{s_{0}}ds\,p_{\lambda}(s)\,,$ (5)

where $s_{0}$ is chosen to maximise the difference between the Poisson and RMT
expectations.

Disordered systems typically display disjoint spectral regions of localised
and delocalised modes separated by mobility edges, where the localisation
length of the localised modes diverges and the system undergoes a second-order
phase transition known as Anderson transition Evers:2008zz . As the system
size increases, observables like $I_{s_{0}}$ tend to their Poisson or RMT
expectation depending on the localisation properties of the modes in the
spectral region of interest, except at a mobility edge, where they are scale-
invariant and take the critical value appropriate for the symmetry class of
the system. This allows for an accurate determination of the position of the
mobility edge via a finite-size scaling analysis or, if the critical value is
known, by looking for the point where the observable takes it.

The discussion above carries over unchanged to the study of the eigenmodes of
the Dirac operator, which is precisely ($-i$ times) a random Hamiltonian with
disorder provided by the fluctuations of the gauge fields, over which one
integrates in the functional-integral formulation of the theory. An example of
how to detect localisation of Dirac eigenmodes using spectral statistics is
shown in Fig. 1 (left), in the case of high-temperature lattice QCD. The
volume tendency of $I_{s_{0}}$ is clear, and the scale-invariant point, i.e.,
the mobility edge, is easily spotted.

The presence of localised modes in the spectrum of $\not{D}$ in QCD is by
itself not surprising given the formal analogy with random Hamiltonians.
Similarly, it is not surprising that critical properties at the mobility edge
match those of the 3D unitary Anderson model Hamiltonian Giordano:2013taa ;
Nishigaki:2013uya ; Ujfalusi:2015nha , which has the same dimensionality and
symmetry class as $\not{D}$ in QCD. What is surprising though, and calls for
an explanation, is the sudden appearance of localised modes near the origin at
sufficiently high temperatures.

##  3 Sea/islands picture

A qualitative explanation of localisation in QCD was proposed in Ref.
Bruckmann:2011cc  and refined in Refs. Giordano:2015vla ; Giordano:2016cjs ;
Giordano:2016vhx . This is based on the observation that the local Polyakov
loop acts in practice as a space-dependent effective temporal boundary
condition for the Dirac eigenmodes, modifying the “twist” on the wave
functions coming from the antiperiodic boundary conditions. At high
temperature in the deconfined phase the Polyakov loops get ordered, and so
doing they induce strong correlations in the temporal direction. In a first
approximation one can then imagine the typical gauge configurations as
fluctuating around a strongly ordered configuration with identical Polyakov
loops $P(\vec{x}\mkern 2.0mu)=\textrm{diag}(e^{i\phi_{a}})$,
$\phi_{a}\in(-\pi,\pi]$, and trivial spatial gauge fields. For this kind of
configuration the eigenmodes and the spectrum can be computed exactly:
eigenmodes are delocalised plane waves, and the spectrum has a gap
$\omega(\phi_{\textrm{PL}})=(\pi-|\phi_{\textrm{PL}}|)T$ determined by the
Polyakov-loop phase $\phi_{\textrm{PL}}$ the farthest away from zero - the
larger the distance, the smaller the gap, as well as the effective temporal
twist on the wave functions. Dynamical fermions then select the trivial
Polyakov loop, for which the twist, the gap, and so the fermion determinant
are maximal. On typical configurations, the “sea” of ordered Polyakov loops
contains “islands” of fluctuations, where the Polyakov-loop phases are
nontrivial and so reduce the effective temporal twist on the fermions. Modes
supported on these islands are expected to have eigenvalues inside the gap, at
the price of being localised there. This leads to a relatively low density of
localised modes filling the gap, with a mobility edge developing to separate
them from the delocalised bulk modes [Fig. 1 (right)].

The most attractive feature of this “sea/islands” picture is that it requires
only the ordering of the Polyakov loop to be applicable. This leads to predict
localisation of the low Dirac modes in the deconfined phase of a generic gauge
theory. Moreover, one expects localised modes to appear right at the critical
temperature when the transition is genuine. Such predictions have been
confirmed so far by all the numerical studies carried out on the lattice.
These include, besides QCD, also pure gauge theories with group SU(2) in 3+1D
Kovacs:2010wx , SU(3) in 3+1D Kovacs:2017uiz ; Vig:2020pgq  and 2+1D
Giordano:2019pvc , $\mathbb{Z}_{2}$ Baranka:2021san  and $\mathbb{Z}_{3}$
Baranka:2022dib  in 2+1D, SU(3) with trace deformation in 3+1D Bonati:2020lal
, and a toy model for QCD with $N_{f}=3$ fermions on coarse lattices
Giordano:2016nuu . Localised low modes were not found in the confined phase in
any of these cases.

The sea/islands picture as presented above turns out to be too simplistic, and
to require a refinement involving a more prominent role for spatial gauge
fields Baranka:2022dib . Nonetheless, Polyakov-loop fluctuations away from
order still retain their role as favourable localisation centres also in the
refined version of the picture. Considerations based on Polyakov loops are
then expected to correctly guide our expectations concerning the localisation
of low Dirac modes in the deconfined phase of a gauge theory in most of the
cases.

##  4 QCD at finite imaginary chemical potential

The introduction of an imaginary chemical potential $\mu_{I}$ for the quarks
boils down in practice to a modification of the temporal boundary conditions
from antiperiodic to $\psi(1/T,\vec{x}\mkern
2.0mu)=e^{i(\pi+\hat{\mu}_{I})}\psi(0,\vec{x}\mkern 2.0mu)$. This is the same
effect caused by a centre transformation, that modifies all Polyakov loops to
$P(\vec{x}\mkern 2.0mu)\to zP(\vec{x}\mkern 2.0mu)$, where $z=e^{i\zeta}$ is
an element of the gauge group centre, and leads to $\psi(1/T,\vec{x}\mkern
2.0mu)=e^{i(\pi+\zeta)}\psi(0,\vec{x}\mkern 2.0mu)$. Since a centre
transformation does not change the gauge action, this leads to the well-known
Roberge–Weiss symmetry of the partition function $Z(\hat{\mu}_{I})$, i.e., the
invariance of $Z(\hat{\mu}_{I})$ under $\hat{\mu}_{I}\to\hat{\mu}_{I}+\zeta$
Roberge:1986mm . For QCD where the gauge group is $\textrm{SU}(3)$, with
centre $\mathbb{Z}_{3}$, this means that $Z(\hat{\mu}_{I})$ is periodic with
period $\frac{2\pi}{3}$. This periodicity, however, is realised differently at
low and high temperature: at low $T$ in the confined phase, $Z(\hat{\mu}_{I})$
is a smooth periodic function, while at high $T$ in the deconfined phase it
displays lines of first-order phase transitions at
$\hat{\mu}_{I}=\frac{\pi}{3}+\frac{2\pi}{3}n$, $n\in\mathbb{Z}$. At these
values of $\hat{\mu}_{I}$ two of the centre sectors are equally favoured by
fermions and the system has an exact $\mathbb{Z}_{2}$ centre symmetry, which
at high temperature is spontaneously broken. The first-order lines terminate
at second-order points in the Ising class at the Roberge-Weiss temperature
$T_{\textrm{RW}}=208(5)\leavevmode\nobreak\ \textrm{MeV}$ Bonati:2016pwz .

Figure 2: Left: gap function $\omega(\phi_{\textrm{PL}},\hat{\mu}_{I})$ at
finite imaginary chemical potential, for $\phi_{\textrm{PL}}$ corresponding to
the three centre elements of $\textrm{SU}(3)$. Solid lines indicate the vacuum
selected by fermions. Right: sketch of the spectral density at
$\hat{\mu}_{I}=\pi$. The solid line is the spectral density in the case of
vanishing gauge fields, starting at the gap $\omega=\frac{2\pi}{3}T$, while
the dashed line qualitatively shows the spectral density after the inclusion
of localised modes penetrating the gap, and $\lambda_{c}$ indicates the
position of the mobility edge.

The sea/islands picture discussed in Sec. 3 applies also at finite $\mu_{I}$
after suitably modifying the gap function to
$\omega=\omega(\phi_{\textrm{PL}},\hat{\mu}_{I})$. Its behaviour as a function
of $\hat{\mu}_{I}$ is shown in Fig. 2 (left) in the case of Polyakov loops
ordered along one of the centre elements of $\textrm{SU}(3)$. In the case of
$\hat{\mu}_{I}=\pi$ (corresponding to effectively periodic boundary
conditions) it reads $\omega(\phi_{\textrm{PL}},\pi)=|\phi_{\textrm{PL}}|T$.
This shows that the complex sectors $z=e^{\pm i\frac{2\pi}{3}}$ are equally
favoured over the real sector by dynamical fermions, in agreement with the
residual $\mathbb{Z}_{2}$ centre symmetry at $\hat{\mu}_{I}=\pi$.
Independently of what sector is chosen when the symmetry breaks spontaneously,
Polyakov-loop fluctuations to the real sector are favourable localisation
centres for the Dirac eigenmodes. In fact, the gap function is locally
vanishing there, $\omega(0,\pi)=0$ [see Fig. 2 (left)], and so they can
support localised modes with eigenvalues below the gap
$\omega(\pm\frac{2\pi}{3},\pi)=\frac{2\pi}{3}T$. The resulting spectral
density is sketched in Fig. 2 (right). Moving along the temperature axis at
$\hat{\mu}_{I}=\pi$, one then expects localised modes to appear right at
$T_{\textrm{RW}}$.

##  5 Localisation above the Roberge-Weiss temperature

Studying localisation of low Dirac modes along the $\hat{\mu}_{I}=\pi$ line
provides the rare opportunity of checking the expectations of the sea/islands
picture across a genuine deconfinement phase transition in the presence of
dynamical fermions. This was done by us in Ref. Cardinali:2021fpu  by means of
a numerical study on the lattice. We simulated $N_{f}=2+1$ QCD at physical
values of the quark masses using 2-stout improved rooted staggered fermions
and a tree-level improved Symanzik gauge action, and studied the statistical
properties of the spectrum of the staggered discretisation of the Dirac
operator, in order to infer the localisation properties of its eigenmodes.
Since the staggered operator is anti-Hermitean, the discussion of Sec. 2
applies without modifications. In particular, we computed $I_{s_{0}}(\lambda)$
along the spectrum and identified the mobility edge $\lambda_{c}$ as the point
where it takes its critical value,
$I_{s_{0}}(\lambda_{c})=I_{s_{0}}^{\textrm{crit}}$. To check for
discretisation effects we used lattices of temporal extension $N_{t}=4,6,8$ in
lattice units, and spatial extension $N_{s}$ fixed by setting the aspect ratio
to $N_{s}/N_{t}=6$ (as well as 8 for $N_{t}=4$).

Figure 3: Left: $I_{s_{0}}$ ($s_{0}\simeq 0.508$) in QCD at
$\hat{\mu}_{I}=\pi$ and $T=394\leavevmode\nobreak\ \textrm{MeV}$ for
$N_{s}^{3}\times N_{t}$ lattices of different spatial size. The Poisson and
RMT expectation and the critical value $I_{s_{0}}^{\textrm{crit}}$
Giordano:2013taa  are also shown. Right: temperature dependence of the
mobility edge in units of $m_{\textrm{ud}}$. Figures from Ref.
Cardinali:2021fpu . Figure 4: Localisation temperature
$T_{\textrm{loc}}(N_{t})$ (black dots) and Roberge-Weiss temperature
$T_{\textrm{RW}}(N_{t})$ (blue squares; data from Ref. Bonati:2016pwz ) for
the various temporal extensions $N_{t}$. The error band of the continuum
extrapolation of $T_{\textrm{RW}}$ is also shown (red dashed lines). Figure
from Ref. Cardinali:2021fpu .

An example of our results for $I_{s_{0}}$ is shown in Fig. 3 (left) for
$T=394\leavevmode\nobreak\ \textrm{MeV}$. At this temperature we also checked
for finite-volume effects, obtaining consistent results for $\lambda_{c}$ for
different $N_{s}$. The results for $\lambda_{c}$ in the whole range of
temperatures that we investigated is shown in Fig. 3 (right). There we plot
the ratio $\lambda_{c}/m_{\textrm{ud}}$, with $m_{\textrm{ud}}$ the mass of
the light quarks, against $T=1/(aN_{t})$, for the different choices of $N_{t}$
(and corresponding lattice spacing $a$). The ratio
$\lambda_{c}/m_{\textrm{ud}}$ is a renormalisation-group-invariant quantity
Kovacs:2012zq ; Giordano:2022ghy , expected to depend mildly on $a$ and to
have a finite continuum limit. This is supported by our results.

For each set of data at fixed $N_{t}$ we fitted the temperature dependence of
the mobility edge with a power law,
$\lambda_{c}/m_{\textrm{ud}}=A(N_{t})[T-T_{\textrm{loc}}(N_{t})]^{B(N_{T})}$,
with $T_{\textrm{loc}}(N_{t})$ the localisation temperature where the mobility
edge vanishes.333This is assumed to happen in a continuous fashion, as one
expects at a second-order phase transition where the ordering of Polyakov
loops majorly responsible for localisation disappears gradually. A jump in the
mobility edge has been observed at first-order transitions Bonati:2020lal ;
Kovacs:2021fwq . Results at the lowest temperature for $N_{t}=6,8$ were
excluded from the fits, since they are still affected by large finite-size
effects for the available volumes. Results for $T_{\textrm{loc}}(N_{t})$ are
shown in Fig. 4, together with the corresponding lattice determinations
$T_{\textrm{RW}}(N_{t})$ of the Roberge-Weiss temperature and the resulting
continuum extrapolation $T_{\textrm{RW}}$ Bonati:2016pwz . For each $N_{t}$
the results for $T_{\textrm{loc}}(N_{t})$ and $T_{\textrm{RW}}(N_{t})$ are
compatible within errors, which strongly supports that the continuum
extrapolation $T_{\rm loc}$ of the localisation temperature satisfies
$T_{\textrm{loc}}=T_{\textrm{RW}}$.

Our results provide further evidence of the strong connection between
deconfinement and the localisation of the low Dirac modes, that happens right
at the critical temperature, also in the case of a genuine deconfinement
transition in the presence of dynamical fermions.

Acknowledgements MG was partially supported by the NKFIH grant KKP-126769.

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$\displaystyle\big{\langle}(\mathcal{T},\mathcal{P},\Theta;\mathbf{c}),(\epsilon,\rho,\zeta;\mathbf{z})\big{\rangle}=\frac{1}{\beta}\int\limits_{0}^{\beta}\mathrm{d}\tau\big{(}\mathcal{T}\epsilon+\mathcal{P}\rho+\Theta\zeta\big{)}+c_{i}z_{i}~{}.$
(266)

The quantities $(\mathcal{T},\mathcal{P},\Theta)$ are the components of
appropriate densities with weights $(2,1,0)$, respectively. The dual vector
$\mathbf{c}$ contains the central charges. The coadjoint action of the group
is defined by

$\displaystyle\big{\langle}\text{Ad}^{\ast}_{(f,g,h)}(\mathcal{T},\mathcal{P},\Theta;\mathbf{c}),(\epsilon,\rho,\zeta;\mathbf{z})\big{\rangle}=\big{\langle}(\mathcal{T},\mathcal{P},\Theta;\mathbf{c}),\text{Ad}_{(f,g,h)^{-1}}(\epsilon,\rho,\zeta;\mathbf{z})\big{\rangle}$
(267)

such that it leaves the central terms invariant. The coadjoint action is then
determined as

$\displaystyle\tilde{\mathcal{T}}(\tau)$ $\displaystyle=(f^{-1})^{\prime
2}\big{(}\mathcal{T}\circ f^{-1}-(\mathcal{P}\circ f^{-1})(g\circ
f)^{\prime}\circ f^{-1}+(\Theta^{\prime}f^{\prime})\circ f^{-1}h\big{)}$ (268)
$\displaystyle\quad-
c_{1}\Big{(}h^{\prime}-h\frac{(f^{-1})^{\prime\prime}}{(f^{-1})^{\prime}}\Big{)}-c_{2}g^{\prime}h+c_{3}(g^{\prime})^{2}$
$\displaystyle\tilde{\mathcal{P}}(\tau)$
$\displaystyle=(f^{-1})^{\prime}\,\mathcal{P}\circ
f^{-1}+c_{2}h-2c_{3}g^{\prime}$ (269) $\displaystyle\tilde{\Theta}(\tau)$
$\displaystyle=\Theta\circ f^{-1}+c_{1}\log(f^{-1})^{\prime}-c_{2}g~{}.$ (270)

It is sometimes convenient to rewrite this evaluated on $f(\tau)$,

$\displaystyle\tilde{\mathcal{T}}(f(\tau))$
$\displaystyle=\frac{1}{(f^{\prime})^{2}}\Big{(}\mathcal{T}-\mathcal{P}(g\circ
f)^{\prime}+\Theta^{\prime}f^{\prime}(h\circ f)$ (271)
$\displaystyle\qquad\qquad-c_{1}\Big{(}(h\circ
f)f^{\prime}\Big{)}^{\prime}-c_{2}f^{\prime}(g\circ f)^{\prime}(h\circ
f)+c_{3}(g\circ f)^{\prime 2}\Big{)}$
$\displaystyle\tilde{\mathcal{P}}(f(\tau))$
$\displaystyle=\frac{1}{f^{\prime}}\Big{(}\mathcal{P}+c_{2}(h\circ
f)f^{\prime}-2c_{3}(g\circ f)^{\prime}\Big{)}$ (272)
$\displaystyle\tilde{\Theta}(f(\tau))$ $\displaystyle=\Theta-c_{1}\ln
f^{\prime}-c_{2}g\circ f~{}.$ (273)

Infinitesimally this is

$\displaystyle\text{ad}^{\ast}_{(\epsilon,\rho,\zeta;\mathbf{z})}\mathcal{T}$
$\displaystyle=\epsilon\mathcal{T}^{\prime}+2\epsilon^{\prime}\mathcal{T}+\mathcal{P}\rho^{\prime}-\Theta^{\prime}\zeta+c_{1}\zeta^{\prime}$
(274)
$\displaystyle\text{ad}^{\ast}_{(\epsilon,\rho,\zeta;\mathbf{z})}\mathcal{P}$
$\displaystyle=\epsilon\mathcal{P}^{\prime}+\epsilon^{\prime}\mathcal{P}-c_{2}\zeta+2c_{3}\rho^{\prime}$
(275) $\displaystyle\text{ad}^{\ast}_{(\epsilon,\rho,\zeta;\mathbf{z})}\Theta$
$\displaystyle=\epsilon\Theta^{\prime}+c_{1}\epsilon^{\prime}+c_{2}\rho$ (276)

which matches with the gravitational transformations (170)-(172) if

$\displaystyle c_{1}=1$ $\displaystyle c_{2}=1$ $\displaystyle
c_{3}=-\frac{1}{2}~{}.$ (277)

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