# The self-adjoint toroidal dipole operator in nanostructures

Mircea Dolineanu, Amanda Teodora Preda and Dragoş-Victor Anghel Institutul
National de Cercetare-Dezvoltare pentru Fizica si Inginerie Nucleara Horia
Hulubei, University of Bucharest, Faculty of Physics,
<EMAIL_ADDRESS>National de Cercetare-Dezvoltare
pentru Fizica si Inginerie Nucleara Horia Hulubei, University of Bucharest,
Faculty of Physics<EMAIL_ADDRESS>National de Cercetare-
Dezvoltare pentru Fizica si Inginerie Nucleara Horia Hulubei,
<EMAIL_ADDRESS>corresponding author

###### Abstract

The parity violation in nuclear reactions led to the discovery of the new
class of toroidal multipoles. Since then, it was observed that toroidal
multipoles are present in the electromagnetic structure of systems at all
scales, from elementary particles, to solid state systems and metamaterials.
The toroidal dipole ${\bf T}$ (the lowest order multipole) is the most common.
In quantum systems, this corresponds to the toroidal dipole operator
$\hat{{\bf T}}$, with the projections $\hat{T}_{i}$ ($i=1,2,3$) on the
coordinate axes. Here we analyze a quantum particle in a system with
cylindrical symmetry, which is a typical system in which toroidal moments
appear. We find the expressions for the Hamiltonian, momenta, and toroidal
dipole operators in adequate curvilinear coordinates, which allow us to find
analytical expressions for the eigenfunctions of the momentum operators. While
the toroidal dipole is hermitian, it is not self-adjoint, but in the new set
of coordinates the operator $\hat{T}_{3}$ splits into two components, one of
which is (only) hermitian, whereas the other one is self-adjoint. The self-
adjoint component is the one which is physically significant and represents an
observable. Furthermore, we numerically diagonalize the Hamiltonian and the
toroidal dipole operator and find their eigenfunctions and eigenvalues. We
write the partition function and calculate the thermodynamic quantities for a
system of ideal particles on a torus. Beside proving that the toroidal dipole
is self-adjoint and therefore an observable (a finding of fundamental
relevance) such systems open up the possibility of making metamaterials which
exploit the quantization and the quantum properties of the toroidal dipoles.

Keywords: toroidal dipole operator; quantum observables; nano-systems;
metamaterials.

##  1 Introduction

In 1957, Zeldovich introduced a new type of electromagnetic interaction, in
order to explain the parity nonconservation in $\beta$-decays [1]:

$\hat{H}_{\beta}\sim{\bf S}\cdot{\bf J}^{ext}={\bf S}\cdot(\nabla\times
H^{ext})$ (1)

(where ${\bf S}$ is the spin of the particle, ${\bf J}^{ext}$ is the external
current, and ${\bf H}^{ext}$ is the external magnetic field). Since neither
electric nor magnetic multipoles lead to an interaction of the type (1), he
introduced for the first time the notion of ”anapole”, a type of distribution
that he intuitively described as a toroidal solenoid–a wire solenoid curved
into a torus, as in Fig. 1 (a). A magnetic field is produced inside this
distribution, due to the static toroidal current, while the electric field
everywhere is zero.

Figure 1: A toroidal dipole ${\bf T}$ generated by the poloidal currents ${\bf
j}$ on a toroidal solenoid.

In the context of classical electrodynamics, Dubovik and Cheshkov proposed in
the late 1960s that a whole new multipole family has to be introduced to
complement the electric and magnetic moments [2, 3]. They reached the
conclusion that a dynamic nonradiating anapole can only be achieved if
toroidal moments are taken into consideration. Fundamental symmetry
considerations also explain the need for these new moments. For example, the
electric dipole moment is odd under spatial inversion and even under time
reversal, while the magnetic dipole moment is odd under time reversal and even
under spatial inversion. For a complete picture, one needs to include dipole
moments which are even or odd under both transformations and these are the
axial and polar toroidal dipole moments [4]. It has also been realized that
the toroidal moments and anapole configurations represent an important tool to
describe the properties of systems at all scales, from particle physics to the
physics of macroscopic systems and this new field expanded rapidly [5, 6]. For
example, it was observed that particles that may be good candidates for dark
matter constituents can only have toroidal moments [7, 8] and the CPT
invariance alone requires that the electromagnetic structure of Majorana
fermions consists only of toroidal moments [9, 10].

In solid state physics, toroidal ordering was first studied theoretically by
Charles Kittel [11] and it indicates the existence of a new type of
magnetoelectric effect. In the framework of condensed matter, toroidal moments
are linked to another kind of order parameter known as toroidization (toroidal
polarization). Media that exhibit macroscopic toroidization are called
ferrotoroids and since they are expected to have promising technological
applications (for example in data storage), it is crucial to investigate
whether ferrotoroidicity is on an equal footing with the other feroic states
(ferroelectric, ferromagnetic) [12, 13, 14, 15, 16, 17, 18, 19].

Recently, anapole states have generated tremendous interest in the field of
nanophotonics, optics, and metamaterials, and have great potential for a wide
range of applications like lasers, sensing, and nonscattering objects that may
be used for their cloaking behavior [20, 21, 22, 23]. It has even been
proposed that since anapole states interact weakly with electomagnetic fields,
they could be used to protect qubits from environmental disturbance [24].

To analyze, manipulate, or engineer systems with toroidal properties at small
scales (e.g., particles, molecules, nano-systems, metamaterials), one needs a
quantum description of the toroidal moments. The toroidal dipole operator was
introduced in Ref. [25], whereas its eigenvalues and eigenfunctions were found
in [26]. It was also shown that the toroidal dipole operator, when defined on
the whole space, is hypermaximal (that is, it has several self-adjoint
extensions) [26], whereas on a finite definition domain, the Dirichlet
boundary conditions impose that the operator is hermitian, but not self-
adjoint.

In this paper, we analyze the energy and the toroidal dipole of a quantum
particle on a quasi-two-dimensional (quasi-2D) domain folded into a shape with
cylindrical symmetry, similar to a torus, but of a general cross-section. We
write the momentum, the Hamiltonian, and the projection $\hat{T}_{3}$ of
toroidal dipole operators in the curved coordinates corresponding to such a
domain. We split $\hat{T}_{3}$ into two components, one of which is hermitian
(but not self-adjoint) and the other one is self-adjoint. By this, we
eventually show for the first time that the toroidal dipole represents a
quantum observable.

We particularize the system to a torus (made of a quasi-2D sheet and of
circular cross-section) and calculate the eigenvalues and eigenfunctions of
the studied operators. Further, we show that while the Hamiltonian eigenstates
of lower energies have zero toroidal dipole, at higher energies there is a
crossover above which the toroidal dipole of the particle deviates from zero
with an absolute value which increases with energy. In the energy range above
the crossover, the momentum, energy, and toroidal dipole eigenstates are
(almost) identical.

Finally, we apply the formalism to calculate the thermodynamics of ideal
bosons and fermions in such quasi-2D domains. While in the high temperature
limit the thermodynamics is similar to the thermodynamics of a 2D system, in
the low temperature limit, the system of ideal fermions exhibit a
“toroidization” property, due to the crossover between the energy domain of
eigenstates with zero toroidal dipole expectation value to the domain with
finite expectation values.

The paper is organized as follows. In Section 2 we introduce the toroidal
dipole operator and its basic properties. In Section 3 we introduce the
curvilinear system of coordinates, we write the momentum, Hamiltonian, and
toroidal dipole operators in terms of them, and we give the general recipe to
calculate the eigenvalues and eigenfunction. In Section 4 we prove the self-
adjointness of one of the components of $\hat{T}_{3}$, showing in this way
that the toroidal dipole is an observable. In Section 5 we particularize the
formalism to a torus and explicitly calculate the eigenvalues and
eigenfunctions for all the relevant operators. In Section 6 we calculate the
thermodynamics of a system of identical particles on a torus and in Section 7
we draw the conclusions. In Appendix A we explicitly prove the hermiticity of
the toroidal dipole operator in the curvilinear coordinates and introduce some
notations necessary in the main body of the article.

##  2 The toroidal diople of the particle

The toroidal dipole is the lowest order toroidal moment. It is a polar vector
and, for a current distribution ${\bf j}({\bf r})$, has the expression [27]

${\bf T}=\frac{1}{10}\int_{V}\Big{[}{\bf r}({\bf r}\cdot{\bf j})-2r^{2}{\bf
j}\Big{]}d^{3}{\bf r},$ (2)

where $r\equiv|{\bf r}|$. For quantum systems, the toroidal dipole (2)
corresponds to the operator $\hat{{\bf T}}$, of components [25, 26]

$\hat{T}_{i}\equiv\frac{1}{10m_{p}}\sum_{j=1}^{3}\left(x_{i}x_{j}-2r^{2}\delta_{ij}\right)\hat{p}_{j}$
(3)

in Cartesian coordinates, where we used the notations
$(x_{1},x_{2},x_{3})\equiv(x,y,z)$ for the components of the position vector
${\bf r}$, $\hat{p}_{j}\equiv-i\hbar\partial/\partial x_{j}$ is the momentum
operator along the $j$ axis, and $m_{p}$ is the mass of the particle. The
projection $\hat{T}_{3}$ has a simpler form in cylindrical coordinates [26],

$\hat{T}_{3}=\frac{-i\hbar}{10m_{p}}\left[z\rho\frac{\partial}{\partial\rho}-\left(2\rho^{2}+z^{2}\right)\frac{\partial}{\partial
z}\right],$ (4)

where $\rho\equiv\sqrt{x^{2}+y^{2}}$ and $r=\sqrt{\rho^{2}+z^{2}}$. One may
define the vector field ${\bf T}_{3}\equiv
z\rho\hat{\bf\rho}-(2\rho^{2}+z^{2})\hat{{\bf z}}$ and the “natural
coordinates” $(k,u,\phi)$ of $\hat{T}_{3}$, such that

$\hat{T}_{3}\equiv-i\hbar\frac{\partial}{\partial u},$ (5)

(notice that $u$ was introduced in [26] by a different definition, namely
$\hat{T}_{3}=-[i\hbar/(10m_{p}c)]\partial/\partial u$). We observe that the
vector field ${\bf T}_{3}$ is tangent in any point to the curves that define
the coordinate $u$. Furthermore, the definition (5) leads to [26]

$k\equiv\left[\rho^{2}\left(z^{2}+\rho^{2}\right)\right]^{1/4}\equiv\sqrt{\rho
r}\quad{\rm and}\quad u\equiv
f_{u}(k,z)=-10m_{p}\int_{0}^{z}\frac{dt}{\sqrt{t^{4}+4k^{4}}}=\pm
10m_{p}\int_{\rho}^{k}\frac{dt}{\sqrt{-t^{4}+k^{4}}}.$ (6)

The variable $u$ takes values in a finite interval $(-a(k),a(k))$, where [26]

$a(k)=\frac{10m_{p}}{4k}\frac{\Gamma\left(\frac{1}{4}\right)\Gamma\left(\frac{1}{2}\right)}{\Gamma\left(\frac{3}{4}\right)}\equiv\frac{10m_{p}}{k}C_{a}$
(7)

and $C_{a}\approx 1.31103$ (see Fig. 2).  Å The end points $u\to-a(k)$ and
$u\to a(k)$ (at fixed $k$) correspond to the points $(z\to\infty,\rho\to 0)$
and $(z\to-\infty,\rho\to 0)$, respectively.

Figure 2: The “natural coordinate” $u$ takes values in the interval
$u\in(-a(k),a(k))$, for any $k$.

##  3 The quantum particle in a system with thin walls and cylindrical
symmetry

We analyze a quantum particle on a quasi-two-dimensional (2D) sheet of
material $\Lambda$ with cylindrical symmetry, similar to the one presented in
Fig. 1, but of arbitrary cross-section, as exemplified in Fig. 3 (a).
Following the procedure outlined in [28, 29], we define the set of curvilinear
orthogonal coordinates $(l,q,\phi)$, where $\phi$ is the azimuthal angle, $q$
is the coordinate perpendicular to the surface in any point (Fig. 3 c), and
$l$ is the coordinate along the cross-section, starting and ending in $A$, as
shown in Fig. 3 (a) and (b). Although the system is considered quasi-2D, the
coordinate $q$ takes finite values and varies from $0$ (the interior surface
of the system) to $q_{max}$ (the exterior surface), where $q_{max}$ is much
smaller than all the other dimensions of the system (see Fig. 3 c). In such a
case, the “point” $A$ from the 2D representation of $\Lambda$ corresponds
actually to $l=0$ and $l=L$ (the maximum value of $l$) for any
$q\in[0,q_{max}]$.

If $(\hat{{\bf x}},\hat{{\bf y}},\hat{{\bf z}})\equiv(\hat{{\bf
x}}_{1},\hat{{\bf x}}_{2},\hat{{\bf x}}_{3})$ is the Cartesian coordinates
system and the position vector is ${\bf r}\equiv\sum_{i}x_{i}\hat{{\bf
x}}_{i}$, then we define the set of reciprocally orthogonal vectors

${\bf h}_{l}\equiv\frac{\partial{\bf r}}{\partial l},\quad{\bf
h}_{q}\equiv\frac{\partial{\bf r}}{\partial q},\quad{\bf
h}_{\phi}\equiv\frac{\partial{\bf r}}{\partial\phi}=-y\hat{{\bf x}}+x\hat{{\bf
y}},$ (8)

and the local basis

${\bf b}_{l}\equiv\frac{{\bf h}_{l}}{h_{l}},\quad{\bf b}_{q}\equiv\frac{{\bf
h}_{q}}{h_{q}},\quad{\bf b}_{\phi}\equiv\frac{{\bf
h}_{\phi}}{h_{\phi}},\quad{\rm where}\quad h_{l}\equiv|{\bf h}_{l}|,\quad
h_{q}\equiv|{\bf h}_{q}|,\quad h_{\phi}\equiv|{\bf
h}_{\phi}|=\sqrt{x^{2}+y^{2}}.$ (9)

The partial derivatives may be written in the local basis as

$\left(\begin{array}[]{c}\frac{\partial}{\partial x}\\\
\frac{\partial}{\partial y}\\\ \frac{\partial}{\partial
z}\end{array}\right)=[J_{l,q,\phi}]\left(\begin{array}[]{c}\frac{\partial}{\partial
l}\\\ \frac{\partial}{\partial q}\\\
\frac{\partial}{\partial\phi}\end{array}\right),\quad\left(\begin{array}[]{c}\frac{\partial}{\partial
l}\\\ \frac{\partial}{\partial q}\\\
\frac{\partial}{\partial\phi}\end{array}\right)=[J_{l,q,\phi}]^{-1}\left(\begin{array}[]{c}\frac{\partial}{\partial
x}\\\ \frac{\partial}{\partial y}\\\ \frac{\partial}{\partial
z}\end{array}\right),\quad{\rm
where}\quad[J_{l,q,\phi}]\equiv\left(\begin{array}[]{ccc}\frac{\partial
l}{\partial x}&\frac{\partial q}{\partial x}&\frac{\partial\phi}{\partial
x}\\\ \frac{\partial l}{\partial y}&\frac{\partial q}{\partial
y}&\frac{\partial\phi}{\partial y}\\\ \frac{\partial l}{\partial
z}&\frac{\partial q}{\partial z}&\frac{\partial\phi}{\partial
z}\end{array}\right).$ (10)

The volume element in the curvilinear coordinates is

$d^{3}{\bf r}\equiv dx\,dy\,dz\equiv|\det[J_{l,q,\phi}]|dl\,dq\,d\phi=\rho
h_{l}h_{q}dl\,dq\,d\phi.$ (11)

We choose the coordinates such that

$h_{q}=1,\quad{\rm so}\quad h_{l}=1+q\frac{\partial h_{l}}{\partial
q}=1+\frac{q}{r(l)},\quad|\det[J_{l,q,\phi}]|=\rho h_{l},$ (12)

and $r(l)$ is the radius of the local curvature, centered in $O^{\prime}(l)$
(see Figs. 3 and 4). We assume $r(l)\neq 0$, for any $l\in[0,L]$ (which
implies also that $r(0)=r(L)\neq 0$).

Figure 3:  The cross-sections of two geometries with cylindrical symmetry: (a)
a general shape and (b) a torus. The cross-section of the torus is centered in
$O^{\prime}$, $|OO^{\prime}|\equiv R$ is the major radius, and $r$ is the
minor radius. In both, (a) and (b), $A$ is the origin of the coordinate $l$.
The local basis is $(\hat{{\bf l}}(l),\hat{{\bf q}}(l))$ and the angle
$\alpha$ is defined such that $\hat{{\bf
l}}\cdot\hat{\mbox{\boldmath$\rho$}}=\cos\alpha,\hat{{\bf l}}\cdot\hat{{\bf
z}}=\sin\alpha,\hat{{\bf
q}}\cdot\hat{\mbox{\boldmath$\rho$}}=\sin\alpha,\hat{{\bf q}}\cdot\hat{{\bf
z}}=-\cos\alpha$. In (a), the local curvature is centered at $O^{\prime}(l)$
and has a radius $r(l)$ In (b), $O^{\prime}$ does not depend on $l$ and
$\theta=\alpha-\pi/2$. The finite thickness of the sheet is shown in (c) and
is measured by the coordinate $q$ ($0\leq q\leq q_{max}$)

Whereas $\hat{L}_{z}\equiv-i\hbar\partial/\partial\phi$ is the projection of
the angular momentum on the $z$ axis and is self-adjoint due to the periodic
boundary conditions in $\phi$, the operators $-i\hbar\partial/\partial l$ and
$-i\hbar\partial/\partial q$ may not be self-adjoint or even hermitian. The
hermitian operators associated to $-i\hbar\partial/\partial l$ and
$-i\hbar\partial/\partial q$ are the momentum operator projection on the local
$l$ and $q$ axes [28]

$\hat{p}^{({\bf l})}\equiv-i\hbar\left(\frac{1}{h_{l}}\frac{\partial}{\partial
l}+\frac{1}{2\rho h_{l}}\frac{\partial\rho}{\partial l}\right)\quad{\rm
and}\quad\hat{p}^{({\bf q})}\equiv-i\hbar\left[\frac{\partial}{\partial
q}+\frac{1}{2\rho h_{l}}\frac{\partial(\rho h_{l})}{\partial q}\right].$ (13)

For the choices (12) we obtain

$\displaystyle\hat{p}^{({\bf l})}$ $\displaystyle=$
$\displaystyle-i\hbar\left(\frac{1}{h_{l}}\frac{\partial}{\partial
l}+\frac{\hat{\mbox{\boldmath$\rho$}}\cdot\hat{{\bf l}}}{2\rho}\right)$ (14a)
$\displaystyle\hat{p}^{({\bf q})}$ $\displaystyle=$
$\displaystyle-i\hbar\left(\frac{\partial}{\partial
q}+\frac{\hat{\mbox{\boldmath$\rho$}}\cdot\hat{{\bf
q}}}{2\rho}+\frac{1}{2h_{l}r}\right).$ (14b)

Since the points $l=0$ and $l=L$ coincide for all $q$ and $\phi$, the
wavefunction satisfies periodic boundary conditions in $l$.

Figure 4:  The method for calculating $\partial\hat{{\bf l}}/\partial l$ and
$\partial h_{l}/\partial q$: the variation $\delta l$ of $l$ leads to the
variation $\delta\hat{{\bf l}}$ of the direction $\hat{{\bf l}}$, whereas the
variation $\delta q$ of $q$ causes a variation $\delta h_{l}$ of $h_{l}$.
Then, $|\partial\hat{{\bf l}}/\partial l|=1/|r(l)|$, $\partial h_{l}/\partial
q=1/r(l)$, where $|r(l)|$ is the radius of the local curvature, centered at
$O(l)$. The sign of $r(l)$ is chosen such that $r(l)\equiv-(\delta\hat{{\bf
l}}\cdot\hat{{\bf q}})|r(l)|$, assuming that $\hat{{\bf q}}$ points towards
the outside of the system for any $l$ (see also Fig. 3).

Similarly, the toroidal dipole operator (4) may be written

$\displaystyle\hat{T}_{3}$ $\displaystyle=$
$\displaystyle\frac{-i\hbar}{10m_{p}}\left[z\rho\,\hat{\mbox{\boldmath$\rho$}}-\left(2\rho^{2}+z^{2}\right)\,\hat{{\bf
z}}\right]\cdot\left(\frac{\hat{{\bf l}}}{h_{l}}\frac{\partial}{\partial
l}+\hat{{\bf q}}\frac{\partial}{\partial q}\right).$ (15)

One can easily check from Eqs. (3) and (4) that the operators $\hat{T}_{i}$,
with periodic or Dirichlet boundary conditions, are hermitian [26]. This
implies that $\hat{T}_{3}$ is hermitian also in the coordinates $(l,q,\phi)$
(Eq. 15), but in order to exemplify the action of the partial derivatives on
different coordinates and to introduce some notations, we explicitly prove the
hermiticity in the coordinates $(l,q,\phi)$ in Appendix A.

The Hamiltonian was calculated in Refs. [28, 29], so (without taking into
account the fields) we use the expression

$\displaystyle\hat{{\cal H}}$ $\displaystyle=$
$\displaystyle\frac{-\hbar^{2}}{2m_{p}}\Delta=\frac{-\hbar^{2}}{2m_{p}}\frac{1}{h_{l}h_{q}h_{\phi}}\left(\frac{\partial}{\partial
l}\frac{h_{q}h_{\phi}}{h_{l}}\frac{\partial}{\partial
l}+\frac{\partial}{\partial
q}\frac{h_{l}h_{\phi}}{h_{q}}\frac{\partial}{\partial
q}+\frac{\partial}{\partial\phi}\frac{h_{l}h_{q}}{h_{\phi}}\frac{\partial}{\partial\phi}\right)$
(16a) $\displaystyle=$
$\displaystyle\frac{-\hbar^{2}}{2m_{p}}\left\\{\frac{1}{\rho
h_{l}}\left[\left(\hat{\mbox{\boldmath$\rho$}}\cdot\hat{{\bf l}}-\frac{\rho
q}{h_{l}^{2}}\frac{d}{dl}\left(\frac{1}{r}\right)\right)\frac{\partial}{\partial
l}+\frac{\rho}{h_{l}}\frac{\partial^{2}}{\partial
l^{2}}+\frac{h_{l}}{\rho}\frac{\partial^{2}}{\partial\phi^{2}}\right]+M^{2}-K\right\\}+\frac{(\hat{p}^{({\bf
q})})^{2}}{2m_{p}}$ where $\displaystyle M^{2}-K$ $\displaystyle\equiv$
$\displaystyle-\frac{1}{2h_{l}h_{\phi}}\frac{\partial^{2}(h_{l}h_{\phi})}{\partial
q^{2}}+\frac{1}{(2h_{l}h_{\phi})^{2}}\left(\frac{\partial(h_{l}h_{\phi})}{\partial
q}\right)^{2}=\frac{1}{4}\left(\frac{\hat{\mbox{\boldmath$\rho$}}\cdot\hat{{\bf
q}}}{\rho}-\frac{1}{h_{l}r}\right)^{2}.$ (16b)

###  3.1 The operators for the quasi-two-dimensional quantum particle

Assuming that the system is tightly confined in the $q$ direction and
$q_{max}\ll r(l)$, one may use the approximations
$\rho(l,q)\approx\rho(l,0),z(l,q)\approx z(l,0)$, $r(l,q)\approx r(l,0)$ and
simplify the expression of the Hamiltonian to

$\displaystyle\hat{{\cal H}}_{0}$ $\displaystyle\approx$
$\displaystyle\frac{-\hbar^{2}}{2m_{p}}\left[\frac{1}{\rho}\left(\rho\frac{\partial^{2}}{\partial
l^{2}}+\hat{\mbox{\boldmath$\rho$}}\cdot\hat{{\bf l}}\frac{\partial}{\partial
l}+\frac{1}{\rho}\frac{\partial^{2}}{\partial\phi^{2}}\right)+\frac{1}{4}\left(\frac{\hat{\mbox{\boldmath$\rho$}}\cdot\hat{{\bf
q}}}{\rho}-\frac{1}{r}\right)^{2}\right]+\frac{(\hat{p}^{({\bf
q})})^{2}}{2m_{p}},$ (17)

which may be split into three components

$\hat{{\cal H}}_{0}\equiv\hat{{\cal H}}^{(l)}+\hat{{\cal
H}}^{(\phi)}+\hat{{\cal H}}_{(q)},\quad\hat{{\cal
H}}^{(l)}\equiv\frac{-\hbar^{2}}{2m_{p}}\left[\frac{1}{\rho}\left(\rho\frac{\partial^{2}}{\partial
l^{2}}+\hat{\mbox{\boldmath$\rho$}}\cdot\hat{{\bf l}}\frac{\partial}{\partial
l}\right)+\frac{1}{4}\left(\frac{\hat{\mbox{\boldmath$\rho$}}\cdot\hat{{\bf
q}}}{\rho}-\frac{1}{r}\right)^{2}\right],\quad\hat{{\cal
H}}^{(\phi)}\equiv\frac{-\hbar^{2}}{2m_{p}}\frac{1}{\rho^{2}}\frac{\partial^{2}}{\partial\phi^{2}},\quad\hat{{\cal
H}}^{(q)}\equiv\frac{(\hat{p}^{({\bf q})})^{2}}{2m_{p}},$ (18a) acting
separately on the variables $l,\phi,q$. So, its eigenfunctions may be written
$\Psi_{k_{l},k_{q},m}(l,q,\phi)\equiv\psi_{k_{l}}(l)\psi_{k_{q}}(q)\psi_{m}(\phi),\quad\psi_{k_{q}}(q)\equiv\frac{\sin(k_{q}q)}{\sqrt{q_{max}}},\quad\psi_{m}(\phi)\equiv\frac{e^{im\phi}}{\sqrt{2\pi}},\quad
k_{q}=\frac{n_{q}\pi}{q_{max}},\ n_{q}\in\mathbb{N}^{+},\ m\in\mathbb{Z},$
(18b)

where $\mathbb{N}^{+}$ is the set of strictly positive integers and
$\mathbb{Z}$ is the set of integers. Since $\hat{{\cal
H}}_{(q)}\psi_{k_{q}}(q)=[n_{q}^{2}(\hbar\pi/q_{max})^{2}/(2m_{p})]\psi_{k_{q}}(q)$,
due to the tight confinement in the $q$ direction, the energies of the excited
states (i.e., $n_{q}>1$) may be too high to be physically relevant and the
system stays in the ground state, with $n_{q}=1$. Then, our problem becomes
(quasi-)2D, in the variables $l$ and $\phi$, with the Hamiltonian

$\hat{{\cal H}}^{(2D)}\equiv\hat{{\cal H}}^{(l)}+\hat{{\cal H}}^{(\phi)}.$
(19)

The confinement affects the action of all operators, including $\hat{T}_{3}$.
We use Eqs. (42) from the Appendix and denote

$\left\langle
t^{(q)}_{1}\right\rangle\equiv\left\langle\Psi\left|\frac{-i\hbar}{10m_{p}}\left[z\rho\,\hat{\mbox{\boldmath$\rho$}}-\left(2\rho^{2}+z^{2}\right)\,\hat{{\bf
z}}\right]\cdot\hat{{\bf q}}\,\frac{\partial}{\partial
q}\right|\Psi\right\rangle\quad{\rm and}\quad\left\langle
t^{(q)}_{2}\right\rangle\equiv\left\langle\Psi\left|\frac{-i\hbar}{10m_{p}}\frac{T_{q}(l,q)}{h_{l}\rho}\right|\Psi\right\rangle.$
(20a) Derivating by parts, with Dirichlet boundary conditions in $q$, we
observe that $\langle t^{(q)}_{1}\rangle=\langle
t^{(q)}_{1}\rangle^{*}+\langle t^{(q)}_{2}\rangle,\quad{\rm so}\quad\langle
t^{(q)}_{2}\rangle=\langle t^{(q)}_{1}\rangle-\langle
t^{(q)}_{1}\rangle^{*}\in\mathbb{I},\quad\langle
t^{(q)}_{1}\rangle-\frac{\langle t^{(q)}_{2}\rangle}{2}=\frac{\langle
t^{(q)}_{1}\rangle+\langle t^{(q)}_{1}\rangle^{*}}{2}\in\mathbb{R}.$ (20b)

Since the expectation value $\langle\Psi|\hat{T}_{3}|\Psi\rangle$ is real
($\hat{T}_{3}$ is hermitian), using Eqs. (20) we define two hermitian
components of $\hat{T}_{3}$,

$\displaystyle\hat{T}^{(l)}_{3}\equiv-\frac{i\hbar}{10m_{p}}\left\\{\left[z\rho\hat{\mbox{\boldmath$\rho$}}-\left(2\rho^{2}+z^{2}\right)\hat{{\bf
z}}\right]\cdot\frac{\hat{{\bf l}}}{h_{l}}\frac{\partial}{\partial
l}+\frac{T_{l}(l,q)}{2\rho h_{l}}\right\\},$ (21a)
$\displaystyle\hat{T}^{(q)}_{3}\equiv-\frac{i\hbar}{10m_{p}}\left\\{\left[z\rho\hat{\mbox{\boldmath$\rho$}}-\left(2\rho^{2}+z^{2}\right)\hat{{\bf
z}}\right]\cdot\hat{{\bf q}}\frac{\partial}{\partial
q}+\frac{T_{q}(l,q)}{2\rho h_{l}}\right\\},$ (21b) such that
$\displaystyle\hat{T}_{3}=\hat{T}^{(l)}_{3}+\hat{T}^{(q)}_{3}$ (21c)

(Eq. 43).

###  3.2 Eigenvalues and eigenvectors for the quasi-two-dimensional operators

The operator $\hat{p}^{({\bf l})}$ has a complete orthonormal set of
eigenfunctions. In the quasi-2D case we ignore the $q$ coordinate and from the
expression (14a) we obtain the eigenfunctions

${\cal
F}_{k_{l}}(l)\equiv\frac{1}{\sqrt{L}}\frac{e^{ilk_{l}}}{\sqrt{\rho(l)}},\quad{\rm
so}\quad\hat{p}^{({\bf l})}{\cal F}_{k_{l}}(l)=\hbar k_{l}{\cal
F}_{k_{l}}(l),\quad{\rm where}\quad k_{l}=\frac{2\pi n}{L}\quad{\rm and}\quad
n={\rm integer}.$ (22a) We can now introduce a basis in the $L^{2}(\Lambda)$
space of functions integrable in modulus square in the volume $\Lambda$,
namely ${\cal F}^{(\Lambda)}_{k_{l},m}\equiv\frac{1}{\sqrt{2\pi
L}}\frac{e^{i(lk_{l}+m\phi)}}{\sqrt{\rho(l)}},\quad k_{l}=\frac{2\pi
n}{L}\quad{\rm and}\quad n,m={\rm integers}.$ (22b) One can easily check that
$\left\langle{\cal F}^{(\Lambda)}_{k_{l},m}|{\cal
F}^{(\Lambda)}_{k_{l}^{\prime},m^{\prime}}\right\rangle=\delta_{k_{l},k_{l}^{\prime}}\delta_{m,m^{\prime}}$
(22c)

and the set $\\{{\cal F}^{(\Lambda)}_{k_{l},m}(l,\phi)\\}$ is complete in the
Hilbert space of functions of coordinates $(l,\phi)$, integrable in modulus
square. For a thin torus $\rho(l)\gg L$ and one may use the approximation

${\cal F}^{(\Lambda)}_{k_{l},m}\approx\frac{e^{i(lk_{l}+m\phi)}}{\sqrt{2\pi
L\rho_{0}}},$ (23)

where $\rho_{0}$ is an approximation for $\rho(l)$ (e.g., the average) and
$\rho_{0},\rho(l)\gg L$ for any $0\leq l<L$.

We calculate the matrix elements

$\displaystyle\langle{\cal
F}^{(\Lambda)}_{k_{1},m_{1}}|\hat{T}_{3}^{(l)}|{\cal
F}^{(\Lambda)}_{k_{2},m_{2}}\rangle$ $\displaystyle=$
$\displaystyle\delta_{m_{1}m_{2}}\langle{\cal
F}_{k_{1}}|\hat{T}_{3}^{(l)}|{\cal F}_{k_{2}}\rangle$ (24a)
$\displaystyle\langle{\cal F}^{(\Lambda)}_{k_{1},m_{1}}|\hat{{\cal
H}}^{(2D)}|{\cal F}^{(\Lambda)}_{k_{2},m_{2}}\rangle$ $\displaystyle=$
$\displaystyle\delta_{m_{1}m_{2}}\langle{\cal F}_{k_{1}}|\hat{{\cal
H}}^{(2D)}|{\cal F}_{k_{2}}\rangle,$ (24b)

and, using them, we calculate the eigenvalues and eigenvectors of the
operators $\hat{T}_{3}^{(l)}$ and $\hat{{\cal H}}^{(2D)}$ from the secular
equation. Notice that because of their hermiticity, the operators $\hat{{\cal
H}}^{(q)}$ and $\hat{T}_{3}^{(q)}$ may contribute only to the diagonal
elements (24), with quantities that do not depend on $k_{l}$ and $m$. For this
reason, their contributions will be neglected, since they only shift the
eigenvalues.

##  4 Self-adjointness of the operator $\hat{T}_{3}^{(l)}$

As we mentioned before, the operator $\hat{T}_{3}$ is hermitian, but not self-
adjoint, due to the Dirichlet boundary conditions on the frontier of
$\Lambda$. Nevertheless, in Section 3 we split $\hat{T}_{3}$ into the
hermitian components $\hat{T}_{3}^{(l)}$ and $\hat{T}_{3}^{(q)}$.
$\hat{T}_{3}^{(q)}$ is hermitian, but not self-adjoint, because the
wavefunctions $\Psi(l,q,\phi)$ satisfy Dirichlet boundary conditions at $q=0$
and $q=q_{max}$. On the other hand, $\Psi(l,q,\phi)$ is periodic in the
variables $l$ and $\phi$ and, based on this property, we show that
$\hat{T}_{3}^{(l)}$ is self-adjoint.

Let’s denote by $L^{(2,l)}$ the Hilbert space of functions in the $l$
variable, which are integrable in modulus square. We further denote by ${\cal
D}(\hat{T}_{3}^{(l)})$ and ${\cal
D}\left[\left({\hat{T}_{3}^{(l)}}\right)^{\dagger}\right]$ the definition
domains of $\hat{T}_{3}^{(l)}$ and
$\left({\hat{T}_{3}^{(l)}}\right)^{\dagger}$, respectively. We define the
derivative $dg/dl$ in a generalized sense by

$\int_{0}^{L}f(l)\frac{dg}{dl}\equiv
f(L)g(L)-f(0)g(0)-\int_{0}^{L}g(l)\frac{df}{dl}$ (25)

where $f,g$ are any functions in $L^{(2,l)}$, but $f$ is of class
$C^{1}\left(L^{(2,l)}\right)$ (that is, it is derivable in any point and the
derivative is continuous and integrable in modulus square). If the derivative
in the expression of $\hat{T}_{3}^{(l)}$ is interpreted in this generalized
sense, then, ${\cal D}\left({\hat{T}_{3}^{(l)}}\right)$ contains all the
functions $f\in L^{(2,l)}$ that satisfy periodic boundary conditions and
$\hat{T}_{3}^{(l)}f\in L^{(2,l)}$. Further, $f\in{\cal
D}\left[\left({\hat{T}_{3}^{(l)}}\right)^{\dagger}\right]$ if and only if
exists a real, positive number $M_{f}$, such that

$|\langle f|{\hat{T}_{3}^{(l)}}|g\rangle|\leq M_{f}||g||$ (26)

for any $g\in{\cal D}(\hat{T}_{3}^{(l)})$. If (26) is true, then (by the
Riesz–Fréchet theorem [30]) exists $F\in L^{(2,l)}$, such that $\langle
f|{\hat{T}_{3}^{(l)}}|g\rangle=\langle F|g\rangle$ and we define the adjoint
operator by the relation $\left({\hat{T}_{3}^{(l)}}\right)^{\dagger}f=F$. It
is obvious that if $f\in{\cal D}(\hat{T}_{3}^{(l)})$, then relation (26) is
satisfied and $F={\hat{T}_{3}^{(l)}}f$. This implies that ${\cal
D}(\hat{T}_{3}^{(l)})\subseteq{\cal
D}\left[\left({\hat{T}_{3}^{(l)}}\right)^{\dagger}\right]$. Therefore, to
prove that ${\hat{T}_{3}^{(l)}}$ is self-adjoint (knowing already that it is
hermitian), we have to show that also ${\cal
D}(\hat{T}_{3}^{(l)})\supseteq{\cal
D}\left[\left({\hat{T}_{3}^{(l)}}\right)^{\dagger}\right]$. For this, we
calculate

$\displaystyle\langle f|{\hat{T}_{3}^{(l)}}|g\rangle$ $\displaystyle\equiv$
$\displaystyle\int_{0}^{L}dl\,\rho
f^{*}(l)\left(-\frac{i\hbar}{10m_{p}}\left\\{\left[z\rho\hat{\mbox{\boldmath$\rho$}}-\left(2\rho^{2}+z^{2}\right)\hat{{\bf
z}}\right]\cdot\hat{{\bf l}}\frac{\partial}{\partial
l}+\frac{T_{l}(l,q)}{2\rho h_{l}}\right\\}\right)g(l)$ (27) $\displaystyle=$
$\displaystyle-\frac{i\hbar}{10m_{p}}\left[f^{*}(L)-f^{*}(0)\right]\left(\rho
g\left\\{\left[z\rho\hat{\mbox{\boldmath$\rho$}}-\left(2\rho^{2}+z^{2}\right)\hat{{\bf
z}}\right]\cdot\hat{{\bf
l}}\right\\}\right)_{l=0}+\langle{\hat{T}_{3}^{(l)}}f|g\rangle$

where we used the properties of ${\hat{T}_{3}^{(l)}}$ derived in Section 3.1.
The property (26) is satisfied if in Eq. (27) $\hat{T}_{3}^{(l)}f\in
L^{(2,l)}$ and $f^{*}(L)-f^{*}(0)$, that is, if and only if $f\in{\cal
D}(\hat{T}_{3}^{(l)})$. This implies that ${\cal
D}(\hat{T}_{3}^{(l)})\supseteq{\cal
D}\left[\left({\hat{T}_{3}^{(l)}}\right)^{\dagger}\right]$, which concludes
the proof.

##  5 The quantum particle on a torus with thin walls

We apply the general formulae of Section 3 to calculate the toroidal dipole
and energy eigenvalues on a torus with thin walls (quasi-2D system). The torus
has the major and minor radii $R$ and $r$, respectively, and its cross-section
is presented in Fig. 3 (b) and (c) (in Fig. 3 c, the thickness of the wall is
exaggerated for clarity). We shall consider only tori with the ratio $a\equiv
R/r>1$. In this case, the coordinate $l$ is along the circle and the curvature
radius $r(l)$ is independent of $l$. We introduce the angle $\theta$, such
that $l=r\theta$ (see Fig. 3). Then, $h_{\theta}=r+q\approx r$ and in the
quasi-2D case the relevant operators are

$\displaystyle\hat{p}^{({\bf l})}$ $\displaystyle=$
$\displaystyle-i\hbar\left\\{\frac{\partial}{\partial
l}-\frac{\sin\left(\frac{l}{r}\right)}{2\left[R+r\cos\left(\frac{l}{r}\right)\right]}\right\\}\equiv-i\hbar\left\\{\frac{1}{r}\frac{\partial}{\partial\theta}-\frac{\sin\theta}{2(R+r\cos\theta)}\right\\}$
(28a) $\displaystyle\hat{{\cal H}}^{(l)}$ $\displaystyle=$
$\displaystyle\frac{-\hbar^{2}}{2m_{p}}\left(\frac{1}{R+r\cos\left(\frac{l}{r}\right)}\left\\{\left[R+r\cos\left(\frac{l}{r}\right)\right]\frac{\partial^{2}}{\partial
l^{2}}-\sin\left(\frac{l}{r}\right)\frac{\partial}{\partial
l}\right\\}+\frac{1}{4}\left[\frac{\cos\left(\frac{l}{r}\right)}{R+r\cos\left(\frac{l}{r}\right)}-\frac{1}{r}\right]^{2}\right)$
(28b) $\displaystyle\equiv$
$\displaystyle\frac{-\hbar^{2}}{2m_{p}}\left(\frac{1}{R+r\cos\theta}\left\\{\frac{R+r\cos\theta}{r^{2}}\frac{\partial^{2}}{\partial\theta^{2}}-\frac{\sin\theta}{r}\frac{\partial}{\partial\theta}\right\\}+\frac{1}{4}\left[\frac{\cos\theta}{R+r\cos\theta}-\frac{1}{r}\right]^{2}\right)$
$\displaystyle\hat{{\cal H}}^{(\phi)}$ $\displaystyle=$
$\displaystyle\frac{-\hbar^{2}}{2m_{p}}\frac{1}{\left[R+r\cos\left(\frac{l}{r}\right)\right]^{2}}\frac{\partial^{2}}{\partial\phi^{2}}\equiv\frac{-\hbar^{2}}{2m_{p}}\frac{1}{\left[R+r\cos\theta\right]^{2}}\frac{\partial^{2}}{\partial\phi^{2}},$
(28c) $\displaystyle\hat{T}^{(l)}_{3}$ $\displaystyle=$
$\displaystyle-\frac{i\hbar}{10m_{p}}\left(-z\left[R+r\cos\left(\frac{l}{r}\right)\right]\sin\left(\frac{l}{r}\right)-\left\\{2\left[R+r\cos\left(\frac{l}{r}\right)\right]^{2}+z^{2}\right\\}\cos\left(\frac{l}{r}\right)\right)\frac{\partial}{\partial
l}-\frac{i\hbar}{20m_{p}\rho}T_{l}(l,0)$ (28d) $\displaystyle=$
$\displaystyle-\frac{i\hbar}{10m_{p}}\frac{-r\sin\theta(R+r\cos\theta)\sin\theta-\left[2(R+r\cos\theta)^{2}+r^{2}\sin^{2}\theta\right]\cos\theta}{r}\frac{\partial}{\partial\theta}-\frac{i\hbar}{20m_{p}\rho}T_{l}(\theta,0),$
$\displaystyle T_{l}(l)$ $\displaystyle=$ $\displaystyle
2z\left[R+r\cos\left(\frac{l}{r}\right)\right]\left[\sin^{2}\left(\frac{l}{r}\right)-\cos^{2}\left(\frac{l}{r}\right)\right]+\left\\{5\left[R+r\cos\left(\frac{l}{r}\right)\right]^{2}+z^{2}\right\\}\sin\left(\frac{l}{r}\right)\cos\left(\frac{l}{r}\right)$
(28e)
$\displaystyle+\frac{1}{r}\left[R+r\cos\left(\frac{l}{r}\right)\right]\left(-z\left[R+r\cos\left(\frac{l}{r}\right)\right]\cos\left(\frac{l}{r}\right)+\left\\{2\left[R+r\cos\left(\frac{l}{r}\right)\right]^{2}+z^{2}\right\\}\cos\left(\frac{l}{r}\right)\right)$
$\displaystyle\equiv$
$\displaystyle-2r\sin\theta(R+r\cos\theta)\cos(2\theta)+\left[5(R+r\cos\theta)^{2}+r^{2}\sin^{2}\theta\right]\frac{\sin(2\theta)}{2}+\frac{R+r\cos\theta}{r}$
$\displaystyle\times\left\\{(2R+r\cos\theta)(R+r\cos\theta)+r^{2}\sin^{2}\theta\right\\}\sin\theta.$

The momenta eigenfunctions on the torus are

${\cal F}^{(\Lambda)}_{k,m}(l,\phi)\equiv\frac{1}{\sqrt{2\pi
L}}\frac{e^{i(lk+m\phi)}}{\sqrt{R+r\cos\left(\frac{l}{r}\right)}}\quad{\rm
or}\quad{\cal
F}^{(\Lambda)}_{n,m}\equiv\frac{1}{2\pi}\frac{e^{i(n\theta+m\phi)}}{\sqrt{R+r\cos\theta}},\quad{\rm
where}\quad k=\frac{2\pi n}{L},\quad n,m={\rm integers}.$ (29)

This simplifies considerably the calculations and allows us to obtain
analytical expressions for matrix elements of $\hat{T}_{3}$ and $\hat{{\cal
H}}^{(2D)}$. For the Hamiltonian we obtain

$\langle{\cal F}^{(\Lambda)}_{n_{1},m_{1}}|\hat{{\cal H}}^{(2D)}|{\cal
F}^{(\Lambda)}_{n_{2},m_{2}}\rangle=\frac{\hbar^{2}}{2m_{p}}\frac{a^{2}}{R^{2}}\left[\left(n_{1}^{2}-\frac{1}{4}\right)\delta_{n_{1}n_{2}}+\frac{\left({|\delta
n|}\sqrt{a^{2}-1}+a\right)\left(\sqrt{a^{2}-1}-a\right)^{{|\delta
n|}}}{\left(a^{2}-1\right)^{\frac{3}{2}}}\left(m_{1}^{2}-\frac{1}{4}\right)\right]\delta_{m_{1}m_{2}}$
(30)

where $\delta n\equiv n_{1}-n_{2}$ ($a\equiv R/r$, as defined before).
Similarly, for the toroidal dipole we have

$\displaystyle\langle{\cal
F}^{(\Lambda)}_{n_{1},m_{1}}|\hat{T}_{3}^{(l)}|{\cal
F}^{(\Lambda)}_{n_{2},m_{2}}\rangle$ $\displaystyle=$
$\displaystyle-\frac{\hbar}{10m_{p}}\frac{a}{R}\frac{n_{1}+n_{2}}{2}\left\\{\frac{5a}{2}\delta_{n_{1}n_{2}}+\left(a^{2}+1\right)\left(\delta_{n_{1},n_{2}+1}+\delta_{n_{1},n_{2}-1}\right)\right.$
(31)
$\displaystyle\left.+\frac{3a}{4}\left(\delta_{n_{1},n_{2}+2}+\delta_{n_{1},n_{2}-2}\right)\right\\}\delta_{m_{1}m_{2}}.$

From Eq. (30) we observe that the diagonal elements are

$\displaystyle\langle{\cal F}^{(\Lambda)}_{n,m}|\hat{{\cal H}}^{(2D)}|{\cal
F}^{(\Lambda)}_{n,m}\rangle$ $\displaystyle=$
$\displaystyle\frac{\hbar^{2}}{2m_{p}}\frac{a^{2}}{R^{2}}\left[\left(n^{2}-\frac{1}{4}\right)+\frac{a}{\left(a^{2}-1\right)^{\frac{3}{2}}}\left(m^{2}-\frac{1}{4}\right)\right]$
(32a) whereas the non-diagonal elements, $n_{1}\neq n_{2}$, $\langle{\cal
F}^{(\Lambda)}_{n_{1},m_{1}}|\hat{{\cal H}}^{(2D)}|{\cal
F}^{(\Lambda)}_{n_{2},m_{2}}\rangle=\frac{\hbar^{2}}{2m_{p}}\frac{a^{2}}{R^{2}}\frac{\left({|\delta
n|}\sqrt{a^{2}-1}+a\right)\left(\sqrt{a^{2}-1}-a\right)^{{|\delta
n|}}}{\left(a^{2}-1\right)^{\frac{3}{2}}}\left(m_{1}^{2}-\frac{1}{4}\right)\delta_{m_{1}m_{2}}$
(32b)

decrease (roughly) in geometrical progression with $|\delta n|$, because of
the factor $\left(\sqrt{a^{2}-1}-a\right)^{{|\delta n|}}$, where
$|\sqrt{a^{2}-1}-a|<1$ and $a>1$. Because of this fast decrease of the non-
diagonal elements, the secular equation can be solved easily and the solution
converges fast with the dimension of the matrix. For example, the maximum
absolute value of the difference between the eigenvalues calculated with
matrices of dimensions $501\times 501$ (shown in Figs. 5 and 6) and with
matrices of dimensions $20001\times 20001$ is $5\times 10^{-6}$.

Figure 5: The energies of the ground state (a), first excited state (b),
second excited state (c), and third excited state (d), as functions of $a$ and
different values of $m$ (as shown in the legend). The legend is shown in (a)
and is the same for all four figures. The energies are in units of
$\hbar^{2}a^{2}/(2m_{p}R^{2})$ and include only the contribution of
$\hat{{\cal H}}^{(2D)}$ (without $\hat{{\cal H}}^{(q)}$).

Figure 6: The energy levels vs. $a$, for different values of $m$: (a) $m=0$,
(b) $m=1$, (c) $m=2$, (d) $m=3$. The legend, shown in (a), is the same for all
figures. The energies are in units of $\hbar^{2}a^{2}/(2m_{p}R^{2})$ and
include only the contribution of $\hat{{\cal H}}^{(2D)}$ (without $\hat{{\cal
H}}^{(q)}$).

In Fig. 5 we show the energies of the ground state, $1^{\rm st}$, $2^{\rm
nd}$, and $3^{\rm rd}$ excited states, as functions of $a$ and for the angular
momentum quantum number $m=0,1,2,3$. We observe that the lines converge, as
$a$ increases. This is due to the decrease with $a$ of the influence of $m$ on
the matrix elements (30). For comparison, in Fig. 6 we show the first five
energy levels as functions of $a$, for different values of $m=0,1,2,3$. We
notice that for any value of $m$, the excited energy levels converge two-by-
two as $a$ increases. We can better understand this result if we think of the
expansion of the energy eigenstates $\Psi^{{\cal H}^{(l)}}$ in the basis
formed by the set (29):

$\Psi^{{\cal
H}^{(2D)}}_{h,m,a}(\theta,\phi)\equiv\sum_{n=-\infty}^{\infty}C_{n,m}^{(h,a)}{\cal
F}^{(\Lambda)}_{n,m}(\theta,\phi),$ (33)

where $h$ represents the energy level and $m$ is the orbital quantum number.
In Fig. 7 we plot the relevant coefficients $C_{n,m}^{(h)}$ for
$h=0,\ldots,5$, $a=2$ (left panel) and $a=3$ (right panel). We see that for
the ground state ($h=0$) only the coefficient $C_{0,m}^{(0)}$ is significantly
different from zero, whereas for the excited states the eigenvectors are
roughly symmetric or antisymmetric superpositions of the functions ${\cal
F}^{(\Lambda)}_{n,m}(\theta,\phi)$. We may approximately write

$\displaystyle\Psi^{{\cal H}^{(2D)}}_{0,0,a}(\theta,\phi)\approx{\cal
F}^{(\Lambda)}_{0,0}(\theta,\phi),\quad\Psi^{{\cal
H}^{(2D)}}_{1,0,a}(\theta,\phi)\approx\dfrac{1}{\sqrt{2}}\left[{\cal
F}^{(\Lambda)}_{-1,0}(\theta,\phi)+{\cal
F}^{(\Lambda)}_{1,0}(\theta,\phi)\right],$ $\displaystyle\Psi^{{\cal
H}^{(2D)}}_{2,0,a}(\theta,\phi)\approx\dfrac{1}{\sqrt{2}}\left[{\cal
F}^{(\Lambda)}_{-1,0}(\theta,\phi)-{\cal
F}^{(\Lambda)}_{1,0}(\theta,\phi)\right],\quad\Psi^{{\cal
H}^{(2D)}}_{3,0,a}(\theta,\phi)\approx\dfrac{1}{\sqrt{2}}\left[{\cal
F}^{(\Lambda)}_{-2,0}(\theta,\phi)+{\cal
F}^{(\Lambda)}_{2,0}(\theta,\phi)\right],$ $\displaystyle\Psi^{{\cal
H}^{(2D)}}_{4,0,a}(\theta,\phi)\approx\dfrac{1}{\sqrt{2}}\left[{\cal
F}^{(\Lambda)}_{-2,0}(\theta,\phi)-{\cal
F}^{(\Lambda)}_{2,0}(\theta,\phi)\right],\quad\Psi^{{\cal
H}^{(2D)}}_{5,0,a}(\theta,\phi)\approx\dfrac{1}{\sqrt{2}}\left[{\cal
F}^{(\Lambda)}_{-3,0}(\theta,\phi)+{\cal
F}^{(\Lambda)}_{3,0}(\theta,\phi)\right],$ (34)

for both, $a=2$ and $a=3$ (the overall sign is not relevant). Equations (34)
explain qualitatively the (approximative) degeneracy of the excited energy
levels.

A similar situation appears also for the higher values of $m$, as it is
exemplified in Fig. 8 for $m=1$. Nevertheless, the quasi-degeneracy appears at
higher and higher values of $a$, as $m$ increases, as we can see in Fig. 6.
Furthermore, we observe that an energy gap is formed for $m=0$, between the
ground state and the first excited state, as $a$ decreases towards 1.

Figure 7: The coefficients of the expansion of the energy eigenvectors in the
basis formed by the functions ${\cal F}^{(\Lambda)}_{n,m}(\theta,\phi)$, for
$m=0$, $a=2$ (left) and $a=3$ (right).

Figure 8: The coefficients of the expansion of the energy eigenvectors in the
basis formed by the functions ${\cal F}^{(\Lambda)}_{n,m}(\theta,\phi)$, for
$m=1$, $a=2$ (left) and $a=3$ (right).

In Fig. 9 we plot the eigenvalues of the toroidal dipole operator for four
different values of $a$. In this case, the numerical calculations converge
much slower than in the case of the Hamiltonian. The results in Fig. 9 are
obtained by diagonalizing matrices of dimension $10001\times 10001$. We
observe that the eigenvalues of $\hat{T}_{3}^{(l)}$ are almost linearly
distributed and are separated by distances which increase with $a$.

Figure 9: The eigenvalues of the toroidal dipole operator for four values of
$a$.

It is interesting to calculate the expectation values of $\hat{T}_{3}^{(l)}$
on the Hamiltonian eigenstates. We plot these in Fig. 10, separately, for
different values of $a$ and observe that up to a certain energy level the
expectation values are almost zero. They start to differ from zero at energy
levels that depend on $a$ and have a weak dependence on $m$. The low energy
behavior may be understood based on Figs. 7 and 8 and the approximations (34).
The coefficients of the expansion of the Hamiltonian eigenstates appear in
pairs of the form $C_{n,m}^{(h,a)}$ and $C_{-n,m}^{(h,a)}$ and since the
matrix elements (31) have the property

$\langle{\cal F}^{(\Lambda)}_{n,m}|\hat{T}_{3}^{(l)}|{\cal
F}^{(\Lambda)}_{n,m}\rangle=-\langle{\cal
F}^{(\Lambda)}_{-n,m}|\hat{T}_{3}^{(l)}|{\cal
F}^{(\Lambda)}_{-n,m}\rangle\quad{\rm and}\quad\langle{\cal
F}^{(\Lambda)}_{n,m}|\hat{T}_{3}^{(l)}|{\cal F}^{(\Lambda)}_{-n,m}\rangle=0,$
(35)

then they add to approximately zero in the expectation value.

To understand the behavior at higher energy levels, we plot in Figs. 11 and 12
the coefficients of the expansion of the Hamiltonian eigenstates corresponding
to higher eigenvalues. We observe that at higher energies only one of the
coefficients is significantly different from zero, meaning that the energy and
the momentum $\hat{p}^{(\theta)}$ have approximately the same eigenfunctions,
that is, they (approximately) commute. From Eq. (30) we understand this
approximate “commutation” relation because while the diagonal elements
increase quadratically with the excitation energy, the non-diagonal elements
are independent of it and, therefore, they may be neglected at higher
energies.

Figure 10: The expectation values of the toroidal dipole operator on the
eigenstates of the Hamiltonian, for $a=1.5,2,2.5,3$.

Figure 11: The coefficients of the expansion of the energy eigenvectors in the
basis formed by the functions ${\cal F}^{(\Lambda)}_{n,m}(\theta,\phi)$, for
$m=0$, $a=2$ (left) and $a=3$ (right).

Figure 12: The coefficients of the expansion of the energy eigenvectors in the
basis formed by the functions ${\cal F}^{(\Lambda)}_{n,m}(\theta,\phi)$, for
$m=1$, $a=2$ (left) and $a=3$ (right).

##  6 Thermodynamics of particles on the torus

Using the results of Section 5, we can calculate the statistics of ideal
particles on the torus. We calculate the partition function as the trace over
multi-particle states of symmetrized (for bosons) or antisymmetrized (for
fermions) products of eigenfunctions ${\cal F}^{(\Lambda)}_{n_{1},m_{1}}$.
Generally, the partition function is

$\displaystyle{\cal Z}$ $\displaystyle\equiv$
$\displaystyle\operatorname{Tr}\left\\{e^{-\beta(\hat{{\cal
H}}^{(2D)}-\mu\hat{{\cal N}})}\right\\},$ (36)

where $\hat{{\cal N}}$ is the particle number operator, $\mu$ is the chemical
potential, $\beta\equiv 1/(k_{B}T)$, $T$ is the inverse temperature, and
$k_{B}$ is Boltzmann’s constant. We ignore $\hat{{\cal H}^{(q)}}$, since this
contributes only by a constant, which may be incorporated into $\mu$. Then,
the logarithm of the partition function may be written in a compact form for
both, bosons ($-$) and fermions ($+$), as

$\displaystyle\ln{\cal Z}^{(\pm)}$ $\displaystyle=$
$\displaystyle\pm\sum_{n,m}\ln\left[1\pm e^{-\beta(\epsilon_{nm}-\mu)}\right]$
(37)

where we used (32a) to introduce the short-hand notation

$\epsilon_{nm}\equiv\frac{\hbar^{2}}{2m_{p}}\frac{a^{2}}{R^{2}}\left[n^{2}+\frac{a}{\left(a^{2}-1\right)^{\frac{3}{2}}}m^{2}\right]$
(38)

and the constant factors from $\langle{\cal F}^{(\Lambda)}_{n,m}|\hat{{\cal
H}}^{(2D)}|{\cal F}^{(\Lambda)}_{n,m}\rangle$ have also been assimilated into
$\mu$.

If $\frac{\hbar^{2}}{2m_{p}}\frac{a^{2}}{R^{2}}\beta\ll 1$, then we can
transform the summations in (37) into integrals and we obtain a 2D quasi-
continuous gas of ideal massive particles of constant density of states. Then,
the expression (37) may be written as

$\displaystyle\ln{\cal Z}^{(\pm)}$ $\displaystyle=$ $\displaystyle\pm
k_{B}T\int_{-\infty}^{\infty}\frac{dk_{n}}{\sqrt{\frac{\hbar^{2}}{2m_{p}}\frac{a^{2}}{R^{2}}}}\int_{-\infty}^{\infty}\frac{dk_{m}}{\sqrt{\frac{\hbar^{2}}{2m_{p}}\frac{a^{2}}{R^{2}}\frac{a}{\left(a^{2}-1\right)^{\frac{3}{2}}}}}\ln\left[1\pm
e^{-\beta(\kappa_{n}^{2}+\kappa_{m}^{2}-\mu)}\right]$ (39) $\displaystyle=$
$\displaystyle\frac{2\pi
m_{p}}{\hbar^{2}}\frac{R^{2}}{a^{2}}\sqrt{\frac{\left(a^{2}-1\right)^{\frac{3}{2}}}{a}}\int_{0}^{\infty}\frac{\epsilon
d\epsilon}{e^{-\beta(\epsilon-\mu)}\pm 1}$

and all the thermodynamics follows.

On the other hand, if we have fermions at very low temperature
($\frac{\hbar^{2}}{2m_{p}}\frac{a^{2}}{R^{2}}\beta\gg 1$), the Hamiltonian
eigenstates are occupied one-by-one, in the order of increasing energy. As
explained in Section 5, at lower energies, the expectation value of the
toroidal dipole operator are (almost) zero, but above a certain energy, the
expectation value deviates from zero quite abruptly, producing an overall
toroidal dipole moment of the system.

##  7 Conclusions

We developed here a quantum formalism for calculating the properties of a
particle on a quasi-2D domain folded into a shape with cylindrical symmetry.
We used a set of curvilinear coordinates adapted to our problem and expressed
the relevant operators (momentum, Hamiltonian, and toroidal dipole) in terms
of these coordinates. We split the projection $\hat{T}_{3}$ of the toroidal
dipole operator into two components and showed that the physically relevant
one is self-adjoing and, therefore, represents an observable. This is
eventually the first time when the toroidal dipole is associated with a
quantum observable. We applied the formalism to a torus and obtained the
eigenvalues and eigenfunctions for these operators (Eqs. 22, 29, Figs. 5, 6,
9). We also calculated the expectation value of the toroidal dipole operator
on the Hamiltonian eigenstates and observed a crossover between states of zero
toroidal dipole to states with non-zero toroidal dipole, as the energy
increases (Fig. 10) and explained this crossover by analyzing the coefficients
of the expansion of the energy eigenstates in terms of momentum eigenstates
(Figs. 7, 8, 11, 12). We also calculated the thermodynamics of a system of
identical particles on a torus. At higher temperatures, the system is
equivalent to a quasi-2D system (as expected), whereas in the low temperature
limit, a Fermi system becomes degenerate and, when the Fermi level exceeds the
crossover energy, the system exhibits toroidal polarizability alternating with
the number of particles.

In Appendix A we explicitly show the hermiticity of the toroidal dipole
operator written in the curvilinear coordinates.

##  8 Acknowledgments

This work has been financially supported by the ELI-RO contract 81-44 /2020.
Travel support from Romania-JINR collaboration projects positions 19, 22,
Order 366/11.05.2021 is gratefully acknowledged.

##  Appendix A The explicit proof of the hermiticity of the $\hat{T}_{3}$
operator in the $(l,q,\phi)$ coordinates

We prove here the hermiticity of the $\hat{T}_{3}$ in the $(l,q,\phi)$
coordinates, to check the formula (15) and introduce some notations. The
expectation value of $\hat{T}_{3}$ on an arbitrary state $\Psi$ is

$\displaystyle\langle\Psi|\hat{T}_{3}|\Psi\rangle$ $\displaystyle=$
$\displaystyle\frac{-i\hbar}{10m_{p}}\int_{0}^{2\pi}d\phi\int_{0}^{L}dl\int_{0}^{q_{max}}dq\rho
h_{l}\Psi^{*}(l,q)\left[z\rho\,\hat{\mbox{\boldmath$\rho$}}-\left(2\rho^{2}+z^{2}\right)\,\hat{{\bf
z}}\right]\cdot\left(\frac{\hat{{\bf l}}}{h_{l}}\frac{\partial}{\partial
l}+\hat{{\bf q}}\frac{\partial}{\partial q}\right)\Psi(l,q),$ (40)

Integrating by parts Eq. (40), we obtain

$\displaystyle\langle\Psi|\hat{T}_{3}|\Psi\rangle$ $\displaystyle=$
$\displaystyle\frac{-i\hbar}{10m_{p}}\int_{0}^{2\pi}d\phi\int_{0}^{L}dl\int_{0}^{q_{max}}dq\frac{\partial}{\partial
l}\left\\{\rho\left[z\rho\,\hat{\mbox{\boldmath$\rho$}}-\left(2\rho^{2}+z^{2}\right)\,\hat{{\bf
z}}\right]\cdot\hat{{\bf l}}\,\Psi^{*}(l,q)\Psi(l,q)\right\\}$
$\displaystyle-\frac{i\hbar}{10m_{p}}\int_{0}^{2\pi}d\phi\int_{0}^{L}dl\int_{0}^{q_{max}}dq\frac{\partial}{\partial
q}\left\\{\rho
h_{l}\left[z\rho\,\hat{\mbox{\boldmath$\rho$}}-\left(2\rho^{2}+z^{2}\right)\,\hat{{\bf
z}}\right]\cdot\hat{{\bf q}}\,\Psi^{*}(l,q)\Psi(l,q)\right\\}$
$\displaystyle+\frac{i\hbar}{10m_{p}}\int_{0}^{2\pi}d\phi\int_{0}^{L}dl\int_{0}^{q_{max}}dq\frac{\partial}{\partial
l}\left\\{\rho\left[z\rho\,\hat{\mbox{\boldmath$\rho$}}-\left(2\rho^{2}+z^{2}\right)\,\hat{{\bf
z}}\right]\cdot\hat{{\bf l}}\right\\}\,\Psi^{*}(l,q)\Psi(l,q)$
$\displaystyle+\frac{i\hbar}{10m_{p}}\int_{0}^{2\pi}d\phi\int_{0}^{L}dl\int_{0}^{q_{max}}dq\frac{\partial}{\partial
q}\left\\{\rho
h_{l}\left[z\rho\,\hat{\mbox{\boldmath$\rho$}}-\left(2\rho^{2}+z^{2}\right)\,\hat{{\bf
z}}\right]\cdot\hat{{\bf q}}\right\\}\,\Psi^{*}(l,q)\Psi(l,q)$
$\displaystyle+\frac{i\hbar}{10m_{p}}\int_{0}^{2\pi}d\phi\int_{0}^{L}dl\int_{0}^{q_{max}}dq\rho
h_{l}\left[z\rho\,\hat{\mbox{\boldmath$\rho$}}-\left(2\rho^{2}+z^{2}\right)\,\hat{{\bf
z}}\right]\cdot\left[\hat{{\bf
l}}\Psi(l,q)\frac{1}{h_{l}}\frac{\partial}{\partial l}\Psi^{*}(l,q)+\hat{{\bf
q}}\Psi(l,q)\frac{\partial}{\partial q}\Psi^{*}(l,q)\right].$

The integrals in the lines (41) and (41) are zero by periodic (in $l$) or
Dirichlet (in $q$) boundary conditions. From the lines (41) and (41) we
calculate

$\displaystyle T_{l}(l,q)$ $\displaystyle\equiv$
$\displaystyle\frac{\partial}{\partial
l}\left\\{\rho\left[z\rho\,\hat{\mbox{\boldmath$\rho$}}\cdot\hat{{\bf
l}}-\left(2\rho^{2}+z^{2}\right)\,\hat{{\bf z}}\cdot\hat{{\bf
l}}\right]\right\\}=\frac{\partial}{\partial
l}\left\\{z\rho^{2}\,\hat{\mbox{\boldmath$\rho$}}\cdot\hat{{\bf
l}}-2\rho^{3}\,\hat{{\bf z}}\cdot\hat{{\bf l}}-z^{2}\rho\,\hat{{\bf
z}}\cdot\hat{{\bf l}}\right\\}$ (42a) $\displaystyle=$ $\displaystyle
h_{l}\left\\{\rho^{2}\,(\hat{\mbox{\boldmath$\rho$}}\cdot\hat{{\bf
l}})\,(\hat{{\bf z}}\cdot\hat{{\bf
l}})+2z\rho\,(\hat{\mbox{\boldmath$\rho$}}\cdot\hat{{\bf
l}})^{2}-6\rho^{2}\,(\hat{\mbox{\boldmath$\rho$}}\cdot\hat{{\bf
l}})\,(\hat{{\bf z}}\cdot\hat{{\bf l}})-2z\rho\,(\hat{{\bf z}}\cdot\hat{{\bf
l}})^{2}-z^{2}\,(\hat{\mbox{\boldmath$\rho$}}\cdot\hat{{\bf l}})\,(\hat{{\bf
z}}\cdot\hat{{\bf l}})\right\\}$ $\displaystyle+\rho
h_{l}\left[z\rho\,\hat{\mbox{\boldmath$\rho$}}-\left(2\rho^{2}+z^{2}\right)\,\hat{{\bf
z}}\right]\cdot\frac{\partial\hat{{\bf l}}}{\partial
l}=h_{l}\left\\{2z\rho\,[(\hat{\mbox{\boldmath$\rho$}}\cdot\hat{{\bf
l}})^{2}-(\hat{{\bf z}}\cdot\hat{{\bf
l}})^{2}]-[5\rho^{2}+z^{2}]\,(\hat{\mbox{\boldmath$\rho$}}\cdot\hat{{\bf
l}})\,(\hat{{\bf z}}\cdot\hat{{\bf l}})\right\\}$ $\displaystyle+\rho
h_{l}\left[z\rho\,\hat{\mbox{\boldmath$\rho$}}-\left(2\rho^{2}+z^{2}\right)\,\hat{{\bf
z}}\right]\cdot\frac{\partial\hat{{\bf l}}}{\partial l},$ $\displaystyle
T_{q}(l,q)$ $\displaystyle\equiv$ $\displaystyle\frac{\partial}{\partial
q}\left\\{\rho h_{l}\left[z\rho\,\hat{\mbox{\boldmath$\rho$}}\cdot\hat{{\bf
q}}-\left(2\rho^{2}+z^{2}\right)\,\hat{{\bf z}}\cdot\hat{{\bf
q}}\right]\right\\}=h_{l}\frac{\partial}{\partial
q}\left\\{z\rho^{2}\,\hat{\mbox{\boldmath$\rho$}}\cdot\hat{{\bf
q}}-2\rho^{3}\,\hat{{\bf z}}\cdot\hat{{\bf q}}-z^{2}\rho\,\hat{{\bf
z}}\cdot\hat{{\bf q}}\right\\}$ (42b)
$\displaystyle+\rho\left[z\rho\,\hat{\mbox{\boldmath$\rho$}}\cdot\hat{{\bf
q}}-\left(2\rho^{2}+z^{2}\right)\,\hat{{\bf z}}\cdot\hat{{\bf
q}}\right]\frac{\partial h_{l}}{\partial
q}=h_{l}\left\\{2z\rho\,[(\hat{\mbox{\boldmath$\rho$}}\cdot\hat{{\bf
q}})^{2}-(\hat{{\bf z}}\cdot\hat{{\bf
q}})^{2}]-[5\rho^{2}+z^{2}]\,(\hat{\mbox{\boldmath$\rho$}}\cdot\hat{{\bf
q}})\,(\hat{{\bf z}}\cdot\hat{{\bf q}})\right\\}$
$\displaystyle+\rho\left[z\rho\,\hat{\mbox{\boldmath$\rho$}}\cdot\hat{{\bf
q}}-\left(2\rho^{2}+z^{2}\right)\,\hat{{\bf z}}\cdot\hat{{\bf
q}}\right]\frac{\partial h_{l}}{\partial q}.$ From Fig. 3 we see that
$\hat{\mbox{\boldmath$\rho$}}\cdot\hat{{\bf l}}\equiv\cos\alpha=-\hat{{\bf
z}}\cdot\hat{{\bf q}}$ and $\hat{{\bf z}}\cdot\hat{{\bf
l}}=\sin\alpha=\hat{\mbox{\boldmath$\rho$}}\cdot\hat{{\bf q}}$ and using these
relations we see that
$T_{l}+T_{q}=\rho\left\\{h_{l}\left[z\rho\,\hat{\mbox{\boldmath$\rho$}}-\left(2\rho^{2}+z^{2}\right)\,\hat{{\bf
z}}\right]\cdot\frac{\partial\hat{{\bf l}}}{\partial
l}+\left[z\rho\,\hat{\mbox{\boldmath$\rho$}}-\left(2\rho^{2}+z^{2}\right)\,\hat{{\bf
z}}\right]\cdot\hat{{\bf q}}\frac{\partial h_{l}}{\partial q}\right\\}.$ (42c)

From Fig. 4 we see that $|\partial\hat{{\bf l}}/\partial l|=1/|r(l)|=|\partial
h_{l}/\partial q|$. Using the signs rules explained in Fig. 4, we observe that

$T_{l}+T_{q}=0,$ (43)

which, together with Eqs. (40) and (41), implies that

$\displaystyle\langle\Psi|\hat{T}_{3}|\Psi\rangle=\frac{-i\hbar}{10m_{p}}\int_{0}^{2\pi}d\phi\int_{0}^{L}dl\int_{0}^{q_{max}}dq\rho
h_{l}\Psi^{*}(l,q)\left[z\rho\,\hat{\mbox{\boldmath$\rho$}}-\left(2\rho^{2}+z^{2}\right)\,\hat{{\bf
z}}\right]\cdot\left(\frac{\hat{{\bf l}}}{h_{l}}\frac{\partial}{\partial
l}+\hat{{\bf q}}\frac{\partial}{\partial q}\right)\Psi(l,q)$
$\displaystyle=\frac{-i\hbar}{10m_{p}}\int_{0}^{2\pi}d\phi\int_{0}^{L}dl\int_{0}^{q_{max}}dq\rho
h_{l}\Psi(l,q)\left[z\rho\,\hat{\mbox{\boldmath$\rho$}}-\left(2\rho^{2}+z^{2}\right)\,\hat{{\bf
z}}\right]\cdot\left(\frac{\hat{{\bf l}}}{h_{l}}\frac{\partial}{\partial
l}+\hat{{\bf q}}\frac{\partial}{\partial
q}\right)\Psi^{*}(l,q)\equiv\langle\Psi|\hat{T}_{3}|\Psi\rangle^{*},$ (44)

so, $\hat{T}_{3}$, expressed in the coordinates $(l,q,\phi)$, is hermitian, as
expected.

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# The Dehn function for Palindromic Sub-Group of $Aut(F_{n})$

KRISHNENDU GONGOPADHYAY  and  LOKENATH KUNDU  Indian Institute of Science
Education and Research (IISER) Mohali, Knowledge City, Sector 81, SAS Nagar,
Punjab 140306, India<EMAIL_ADDRESS>Indian Institute of Science
Education and Research (IISER) Mohali, Knowledge City, Sector 81, SAS Nagar,
Punjab 140306, India<EMAIL_ADDRESS>
###### Abstract.

In this paper, we prove that the Dehn function of the palindromic automorphism
group $\Pi A(F_{n})$ is exponential.

###### Key words and phrases:

Dehn function, palindromic automorphism group.

###### 2020 Mathematics Subject Classification:

Primary 20F65; Secondary 57S25, 53C24, 05E16, 20E07

##  1\. Introduction

Let $G~{}=~{}\langle~{}X~{}|~{}R~{}\rangle$ be a finitely generated group. The
word problem for $G$ is the method of determining whether two words in the
generators represent the same element. The Dehn function is about how hard it
is to solve word problems in a finitely presented group. To gain a deeper
understanding of the Dehn function see [4, 5]. If a group $G$ acts cocompactly
and properly on a simplicial complex $S$, then the Dehn function of $G$ is
asymptotically equivalent to the function that provides the optimal upper
bound on the area of the least area discs in $S.$ Here, the restriction is
stated as a function of the length of the boundary of the disc. In [2] authors
proved that the Dehn function is exponential for $Out(F_{n})$ and $Aut(F_{n})$
for $n\geq 3$.

The subgroup of $Aut(F_{n})$, consisting of all automorphisms that transform
the generator of $F_{n}$ into palindromic words of $F_{n},$ is known as the
palindromic automorphism group. The palindromic group was introduced by
Collins in [13]. Let $a_{1},a_{2},\dots,a_{n}$ be the $n$-generators of
$F_{n}.$ Following the notion by the authors in [6], a reduced word
$a_{1}^{l_{1}}a_{2}^{l_{2}}\dots a_{n}^{l_{n}}$ is called a palindrome if
$a_{1}^{l_{1}}a_{2}^{l_{2}}\dots
a_{n}^{l_{n}}=a_{n}^{l_{n}}a_{n-1}^{l_{n-1}}\dots a_{1}^{l_{1}}.$ An
automorphism $\alpha~{}\in~{}Aut(F_{n})$ is said to be a palindromic if
$\alpha(a_{i})$ is a palindromic for all $i$. The set of all palindromic
automorphisms forms a subgroup of $Aut(F_{n})$, and we denote it by $\Pi
A(F_{n})$. For more details on the palindromic automorphism group, see [6, 13,
14].

In this article, we will determine the Dehn function of $\Pi A(F_{n})$. We
prove the following:

######  Theorem 1.1.

The Dehn function of $\Pi A(F_{n})$ is exponential, for $n\geq 3$.

After discussing preliminary results and constructions, we established two
lemmas in Section 3 that play a crucial role in proving the
$Theorem\ref{palin}$. We prove this theorem in Section 4.

###  1.1. Acknowledgements

The authors acknowledge support from the SERB core research grant
CRG/2022/003680 during the course of this work.

###  1.2. Competing Interest

We declare that there are no competing interests associated with this
manuscript.

##  2\. Preliminaries

In this section, we establish certain notations and revisit essential
background information that will be employed consistently in the subsequent
discussions within this document.

###  2.1. Dehn function of a group

Consider a finitely generated group $G$ with presentation $G=\langle
S~{}|~{}R\rangle$. So $G$ is isomorphic to the quotient group $F(S)/N,$ where
$F(S)$ represents the free group generated by the set $S,$ and $N$ is a normal
subgroup of the free group $F(S)$. The elements in $R$ consist of cyclically
reduced words, and every generator in $N$ is a conjugate of an element in $R.$
If a reduced word $w$ in $F(S)$ represents the identity in $G$, we say that
$w$ is null-homotopic and it is denoted as $w\equiv 1$ in $G$. Furthermore,
this implies that $w$ belongs to $N=\langle\langle R\rangle\rangle.$

######  Definition 2.1.

(Area of null-homotopic words) If $w$ is equivalent to $1$ in the group $G,$
then $w$ can be expressed as the product
$\displaystyle\prod_{i=1}^{d}g_{i}r_{i}^{\pm 1}g_{i}^{-1},$ where each $r_{i}$
belongs to the set $R.$ The area of $w$ is defined by the expression:

(2.1) $Area(w)=\min\\{d~{}|~{}w=\displaystyle\prod_{i=1}^{d}g_{i}r_{i}^{\pm
1}g_{i}^{-1}\\}.$

The Dehn function quantifies the complexity of the word problem for any given
group, and its definition is as follows:

######  Definition 2.2.

The Dehn function for a group $G$, denoted as
$\delta_{(G,S)}:\mathbb{N}\rightarrow\mathbb{N}$, is defined as:

(2.2) $\displaystyle\delta_{(G,S)}(n)=\sup\\{Area(w)~{}|~{}|w|\leq
n,~{}w\equiv 1\\}.$

Here, $S$ is any generating set for $G$, and $|w|$ represents the word length
of an element $w\in F(S)$.

The Dehn function $\delta_{(G,S)}(n)$ is independent of generating sets up to
an equivalence relation for any group $G=\langle S~{}|~{}R\rangle$. Henceforth
we will denote the Dehn function by $\delta(n)$ of any group $G$. To explore
more about the Dehn function see [4, 5].

In this context, we will have the next two outcomes without proof which will
be crucial for demonstrating the Theorem 1.1. To find a detailed proof of the
following results, see [2].

######  Lemma 2.3.

If $A$ and $B$ are 1-connected simplicial complexes, $F:~{}A\rightarrow~{}B$
is a simplicial map, and $l$ is a loop in the 1-skeleton of $A,$ then
$Area_{A}(l)~{}\geq~{}Area_{B}(F\circ l).$

######  Corollary 2.4.

Let $A,B$ and $C$ 1-connected simplicial complexes with simplicial maps

$A~{}\rightarrow~{}B\rightarrow~{}C.$

Let $l_{n}$ be a sequence of simplicial loops in $A$ whose length is bounded
above by a linear function of $n,$ let $\overline{l_{n}}$ be the image loops
in $C$ and let $\alpha(n)=Area_{C}(\overline{l_{n}}).$ Then the Dehn function
of $B$ satisfies $\delta_{B}(n)\succeq\alpha(n).$

###  2.2. Palindromic automorphism group

The palindromic automorphism group $\Pi A(F_{n})$ is generated by the
following automorphisms:

  * •

Maps $A_{ij}=(a_{i}||a_{j}),~{}i\neq j,$ which maps $a_{i}\mapsto
a_{j}a_{i}a_{j}$ and fix all other generators $a_{k},~{}k\neq i$.

  * •

Maps $\sigma_{a_{i}}$ which maps $a_{i}\mapsto a_{i}^{-1}$ and fix all other
generators $a_{k}.$

  * •

Maps corresponding to elements of the symmetric group $S_{n}$ which permute
the $a_{1},a_{2},\\\ \dots,a_{n}$ among themselves.

The subgroup of $\Pi A(F_{n})$ which is generated by $A_{ij}$ only is called
elementary palindromic subgroup $E\Pi A(F_{n}).$ Moreover $\Pi
A(F_{n})=~{}E\Pi A(F_{n})\rtimes(\mathbb{Z}/2~{}\wr~{}S_{n}).$ Now

(2.3) $E\Pi A(F_{n})=\langle
A_{ij}~{}|~{}1\leq~{}i,j\leq~{}n,~{}[A_{ik},A_{jk}],~{}[A_{ik},A_{jl}],~{}A_{ik}A_{jk}A_{ij}=A_{ij}A_{jk}A_{ik}^{-1}\rangle.$

To explore more about the palindromic group see [6].

###  2.3. Simplicial complex associated to the $\Pi A(F_{n})$.

In this section we briefly describe a simplicial complex $L_{\Pi A(F_{n})}$ on
which the group $\Pi A(F_{n})$ acts with finite stabilizer and finite
quotient. The space $L_{\Pi A(F_{n})}$ was introduced by Glover and Jensen in
[6].
Let $X_{n}$ be the spine of auter space [6, 7] which is a contractible
simplicial complex with a cocompact proper action by $Aut(F_{n})$. Let
$\sigma_{n}~{}\in~{}Aut(F_{n})$ which sends all the generators of $F_{n}$ to
its inverse. So, $\langle~{}\sigma_{n}~{}\rangle~{}=\mathbb{Z}/2.$ Let $R_{n}$
be a graph consisting of one vertex and $n$-edges. Hence
$\langle~{}\sigma_{n}~{}\rangle$ acts on $R_{n}$ by inverting each edge. It is
a notable point that $\Pi A(F_{n})~{}=~{}C_{Aut(F_{n})}(\sigma_{n}).$ Let
$X_{n}^{\sigma_{n}}$ be the fixed point subspace of $X_{n}$ corresponding to
$\langle~{}\sigma_{n}~{}\rangle$. From [12] $C_{Aut(F_{n})}(\sigma_{n})$ acts
on the contractible space $X_{n}^{\sigma_{n}}$ with finite stabilizer and
finite quotient. Moreover $X_{n}^{\sigma_{n}}$ is
$\langle~{}\sigma_{n}~{}\rangle$-equivariantly deformation retraction to
$L_{\sigma_{n}}.$ The space $L_{\sigma_{n}}$ constructed from
$X_{n}^{\sigma_{n}}$ by considering the essential marked grpahs only. We will
denote $L_{\sigma_{n}~{}}$ by $L_{\Pi A(F_{n})}.$ The vertices of $L_{\Pi
A(F_{n})}$ can be represented as essential marked graphs with a chosen based
point, and two vertices are adjacent if one of them is obtained from the other
by a forest collapse.

Recall that an edge $e$ of a $G$ graph $\Gamma$ is said to be inessential if
it is in every maximal $G$-invariant subforest of $\Gamma$. A graph is said to
be inessential if it contains at least one inessential edge. A graph is said
to be essential if it is not inessential. For more details about $L_{\Pi
A(F_{n})}$, see [6], pp. 646-655.

Since $\Pi A(F_{n})$ acts on the contractible space $L_{\Pi A(F_{n})}$ with
finite stabilizers and finite quotient so the Dehn function of $\Pi A(F_{n})$
is asymptotically equivalent to the function that provides the optimal upper
bound on the area of the least area discs in $L_{\Pi A(F_{n})}$, where the
bound is expressed as a function of the length of the boundary of the disc.

###  2.4.  Simplicial complexes associated to $Out(F_{n})$.

Let $K_{n}$ denote the spine of the outer space as defined in [3]. The group
$Out(F_{n})$ acts on $K_{n}$, and the action is cocompact and proper. A marked
graph is a finite metric graph $\Gamma$ together with a homotopy equivalence
$g:~{}R_{n}\rightarrow\Gamma$, where $R_{n}$ is a fixed graph with one vertex
and $n$-loops. In a marked graph, two vertices are adjacent if one can be
obtained from the other by a forest collapse. A vertex of $K_{n}$ can be
represented as a marked graph $(g,\Gamma)$ with degree of all vertices at
least three. For details see [3].

###  2.5. Nielsen transformation and Nielsen graph

Let $S=\\{s_{1},s_{2},\dots,s_{n}\\}$ be an ordered set. The following
transformations are known as elementary Nielsen transformations:

  1. N1.

Replace some $s_{i}$ by $s_{i}^{-1}.$

  2. N2.

Replace $s_{i}$ by $s_{i}s_{k}$ for some $i\neq k.$

  3. N3.

Delete $s_{i}$ if $s_{i}=1.$

In the above transformations, all $s_{j}$ will be unchanged if $i\neq j.$ The
product of elementary Nielsen transformations is known as Nielsen
transformation.
Let $G$ be a finitely generated group and the minimal number of generators for
$G$ is at least $k.$ The Nielsen graph $N_{k}(G)$ is defined as follows:

  * •

The vertex set of $N_{k}(G)$ consists of tuple $(g_{1},g_{2},\dots,g_{k})$
such that $G=\langle g_{1},g_{2},\dots,g_{k}\rangle.$

  * •

Two vertices of $N_{k}(G)$ are connected by an edge if one of them is obtained
from the other by an elementary Nielsen move.

To explore more deeply into the Nielsen transformations and Nielsen graphs,
refer to [11, 10].

##  3\. Natural maps between simplicial complexes of $Out(F_{n})$ and $\Pi
A(F_{n})$.

There is a forgetful map $\phi_{n}~{}:~{}L_{\Pi
A(F_{n})}~{}\rightarrow~{}K_{n}$ which simply forgets the base point,
essentiality of marked graphs; this is a simplicial map.
Let $m<n.$ We fix an ordered basis for $F_{n}$, identify $F_{m}$ with the
subgroup generated by the first $m$-elements of the basis, and identify $\Pi
A(F_{m})$ with the subgroup of $\Pi A(F_{n})$ that leaves $F_{m}<F_{n}$
invariant and fixes the $n-m$ basis elements.
First, there is an equivariant map

(3.1) $i:~{}L_{\Pi A(F_{m})}\rightarrow L_{\Pi A(F_{n})}$

which attaches a bouquet of $n-m$ circles essentially to the basepoint of each
essential marked graph and marks them with the last $n-m$ basis elements of
$F_{n}.$ This map is simplicial since a forest collapse has no effect on the
bouquet of circles at the base point.
Secondly, there is a restriction map $\rho:~{}K_{n}\rightarrow K_{m},$ which
can be describe in the following way: The chosen embedding $F_{m}<F_{n}$
corresponds to choosing an $m$-petal subrose $R_{m}\subset R_{n}.$ A vertex in
$K_{n}$ is given by a graph $\Gamma$ marked with a homotopy equivalence
$g:~{}R_{n}~{}\rightarrow~{}\Gamma,$ and the restriction of $g$ to $R_{m}$
lifts to a homotopy equivalence
$\hat{g}:~{}R_{m}~{}\rightarrow~{}\hat{\Gamma},$ where $\hat{\Gamma}$ is the
covering space corresponding to $g_{*}(F_{m}).$ There is a canonical
retraction $r$ of $\hat{\Gamma}$ onto its compact core, i.e. the smallest
connected subgraph containing all nontrivial embedded loops in $\Gamma.$ Let
$\hat{\Gamma_{0}}$ be the graph obtained by erasing all vertices of valence
$2$ from the compact core and define
$\rho(g,\Gamma)~{}=~{}(r~{}\circ~{}\hat{g},\hat{\Gamma_{0}}).$ To gain a
deeper understanding see [2]. We now present the following lemma without
proof, for details of proof see [2],

######  Lemma 3.1.

For $m<n,$ the restriction map $\rho:~{}K_{n}\rightarrow K_{m}$ is simplicial.

######  Lemma 3.2.

For $m<n,$ the following diagram of simplicial maps commutes:

$\begin{CD}L_{\Pi A(F_{m})}@>{i}>{}>L_{\Pi A(F_{m})}\\\
@V{}V{\phi_{m}}V@V{}V{\phi_{n}}V\\\ K_{m}@<{\rho}<{}<K_{n}\end{CD}$

###### Proof.

Let $(g,\Gamma;v)$ be an essential marked graph with a base point in $L_{\Pi
A(F_{m})}.$ Then the essential marked graph $i(g,\Gamma;v)$ with base point at
$v$ is obtained by attaching $n-m$ loops essentially at the base point $v$
labelled by the elements $a_{m+1},\dots,a_{n}$ of our fixed basis for $F_{n}.$
Now $(g_{n},\Gamma_{n}):~{}=~{}\phi_{n}\circ i(g,\Gamma;v)$ is obtained by
forgetting the base point $v$ and the essentiality of $(g,\Gamma)$, and the
cover of $(g_{n},\Gamma_{n})$ corresponding to $F_{m}<F_{n}$ is obtained from
a copy of $(g,\Gamma)$ (with labels) by attaching $2(n-m)$ trees. (These trees
are obtained from the Cayley graph of $F_{n}$ as follows: one cuts at an edge
labeled $a_{i}^{\epsilon},$ with $i=m+1,\dots,n$ and $\epsilon=\pm 1,$ takes
one component of the result, and then attaches the hanging edge to the base
point $v$ of $\Gamma.$) The effect of $\rho$ is to delete these trees. ∎

##  4\. Proof of the theorem 1.1.

###### Proof.

In [7] authors proved an exponential upper bound on the Dehn function of
$Aut(F_{3})$ and $Out(F_{3})$ for all $n\geq 3.$ In [9] authors established an
exponential lower bound by using their general results on quasi-retractions to
reduce to the case $n=3.$ So it is enough to prove our Theorem 1.1 for $n=3.$
In light of the Corollary 2.4 and Lemma 3.1, it suffices to exhibit a sequence
of loops $l_{p}$ in the 1-skeleton of $L_{\Pi A(F_{3})}$ whose lengths are
bounded by a linear function of $p$ and whose filling area when projected to
$K_{3}$ grows exponentially as a function of $p.$ Instead of loops we will
consider words in $\Pi A(F_{3})$, but standard quasi-isometric arguments show
this is equivalent. Consider words of the form:

(4.1)
$W_{p}:=A_{ij}^{-p}A_{kj}^{-p}A_{ij}^{p}A_{ki}A_{ij}^{p}A_{kj}^{-p}A_{ij}^{-p}A_{ki}^{-1}.$

We will prove that the word $W_{p}$ is null-homotopic word in $\Pi A(F_{3}).$

(4.2)
$A_{ij}(a_{i})=a_{j}~{}a_{i}~{}a_{j},~{}A_{ij}^{-1}(a_{i})=a_{j}^{-1}~{}a_{i}~{}a_{j}^{-1},~{}A_{ij}(a_{k})=a_{k}~{}\forall~{}k\neq
i.$ (4.3)
$\begin{split}W_{p}(a_{k})&=A_{ij}^{-p}A_{kj}^{-p}A_{ij}^{p}A_{ki}A_{ij}^{p}A_{kj}^{-p}A_{ij}^{-p}A_{ki}^{-1}(a_{k})\\\
&=A_{ij}^{-p}A_{kj}^{-p}A_{ij}^{p}A_{ki}A_{ij}^{p}A_{kj}^{-p}A_{ij}^{-p}(a_{i}^{-1}a_{k}a_{i}^{-1})\\\
&=A_{ij}^{-p}A_{kj}^{-p}A_{ij}^{p}A_{ki}A_{ij}^{p}A_{kj}^{-p}(a_{i}^{-1}a_{k}a_{i}^{-1})\\\
&=A_{ij}^{-p}A_{kj}^{-p}A_{ij}^{p}A_{ki}A_{ij}^{p}(a_{i}^{-1}a_{j}^{-p}a_{k}a_{j}^{-p}a_{i}^{-1})\\\
&=A_{ij}^{-p}A_{kj}^{-p}A_{ij}^{p}A_{ki}(a_{i}^{-1}a_{j}^{-p}a_{k}a_{j}^{-p}a_{i}^{-1})\\\
&=A_{ij}^{-p}A_{kj}^{-p}A_{ij}^{p}(a_{i}^{-1}a_{j}^{-p}a_{i}a_{k}a_{i}a_{j}^{-p}a_{i}^{-1})\\\
&=A_{ij}^{-p}A_{kj}^{-p}(a_{j}^{p}a_{k}a_{j}^{p})\\\
&=A_{ij}^{-p}(a_{j}^{p}a_{j}^{-p}a_{k}a_{j}^{-p}a_{j}^{p})\\\
&=A_{ij}^{-p}(a_{k})\\\ &=a_{k}\end{split}$

(4.4)
$\begin{split}W_{p}(a_{i})&=A_{ij}^{-p}A_{kj}^{-p}A_{ij}^{p}A_{ki}A_{ij}^{p}A_{kj}^{-p}A_{ij}^{-p}A_{ki}^{-1}(a_{i})\\\
&=A_{ij}^{-p}A_{kj}^{-p}A_{ij}^{p}A_{ki}A_{ij}^{p}A_{kj}^{-p}(a_{j}^{-p}a_{i}a_{j}^{-p})\\\
&=A_{ij}^{-p}A_{kj}^{-p}A_{ij}^{p}A_{ki}(a_{j}^{-p}a_{j}^{p}a_{i}a_{j}^{p}a_{j}^{-p})\\\
&=A_{ij}^{-p}A_{kj}^{-p}(a_{j}^{p}a_{i}a_{j}^{p})\\\
&=A_{ij}^{-p}(a_{j}^{p}a_{i}a_{j}^{p})\\\
&=a_{j}^{p}a_{j}^{-p}a_{i}a_{j}^{-p}a_{j}^{p}\\\ &=a_{i}\end{split}$

Similarly, we can compute $W_{p}(a_{j})~{}=~{}a_{j}$. Therefore,
$W_{p}~{}\equiv~{}1$, indicating $W_{p}$ represents a null-homotopic word in
$\Pi A(F_{3}).$

We will interpret these words as loops. Note that
$A_{ij}=\rho_{ij}\lambda_{ij}$ where $\lambda_{ij}$ and $\rho_{ij}$ are
elementary transvection defined as follows:

(4.5) $\lambda_{ij}(a_{i})=a_{j}a_{i},~{}~{}\rho_{ij}(a_{i})=a_{i}a_{j}.$

So $W_{p}$ is the product of $12p+4$ elementary transvections. There is a
connected subcomplex of the 1-skeleton of $L_{\Pi A(F_{3})}$ spanned by roses
and Nielsen graphs. We say roses are adjacent if they have distance $2$ in
this graph.

Let $I\in L_{\Pi A(F_{3})}$ be the rose marked by the identity map
$R_{3}\rightarrow R_{3}.$ Each elementary transvection $\tau$ moves $I$ to an
adjacent rose $\tau I,$ which is connected to $I$ by a Nielsen graph
$N_{\tau}$ in $L_{\Pi A(F_{3})}$. A composition $\tau_{1}\dots\tau_{s}$ of
elementary transvections gives a path through adjacent roses
$I,\tau_{1}I,\dots,\tau_{1}\dots\tau_{s}I;$ the Nielsen graph connecting
$\sigma I$ to $\sigma\tau I$ is $\sigma N_{\tau}.$ Thus the word $W_{p}$
corresponds to a loop $l_{p}$ in $L_{\Pi A(F_{3})}.$ [1, Theorem A] which
states ”$Aut(F_{3})$ and $Out(F_{3})$ satisfy a strictly exponential
isoperimetric inequality.” Therefore there is an exponential lower bound on
the filling area of $\phi_{3}\circ l_{p}$ in $K_{3}.$ Using Lemma 2.3 and
Corollary 2.4, we will have the exponential lower bound of the filling area of
$l_{p}$ in $L_{\Pi A(F_{3})}$. Hence the Dehn function of the palindromic
automorphism group $\Pi A(F_{n})$ is exponential. ∎

## References

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  * [2] Martin R. Bridson, Karen Vogtmann: The Dehn function of $Out(F_{n})$ and $Aut(F_{n})$. Annales De LÍnstitut Fourier, Gernoble, 62, 5 (2012), 1811-1817.
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Since changing $u$ to $u^{\prime}$ does not change this type, using our
assumption on $\varphi$, and since $\bar{a}_{q}\subseteq A_{\alpha}$ this
truth value remains the same.

As before, we can modify $\bar{t}$ so it is nonempty and all its elements have
the same height.

Our strategy to prove the claim is to fix a set of parameters
$\bar{a},\bar{t}$ defining $D$ as above, and for each possible $p\colon
n\times\bar{a}\to 2$, we find a dense open set which is an intersection of
length $\ell^{p}$ of sets of the form $D^{1}_{t^{p}_{i},b^{p}_{i},k^{p}_{i}}$
such that if $q\in\bigcap_{i<\ell^{p}}D^{1}_{t^{p}_{i},b^{p}_{i},k^{p}_{i}}$
and $q\leq p$ then $q\in D$. Then,

$D\supseteq\bigcap_{k<n,a\in\bar{a}}D^{0}_{k,a}\cap\bigcap_{p\colon
n\times\bar{a}\to 2}\bigcap_{i<\ell^{p}}D^{1}_{t^{p}_{i},b^{p}_{i},k^{p}_{i}}$

Fix $p$ to be a condition with domain $n\times\bar{a}$ and let $q\in D$,
$q\leq p$. We may assume that $\operatorname{dom}q=n\times\bar{a}_{q}$. Let
$\bar{t}^{\prime}$ be an extension of $\bar{t}$ such that for every $a$ such
that $(m,a)\in\operatorname{dom}q$ for some $m<n$, there is some
$r\in\bar{t}^{\prime}$, $r\leq_{T}a$, and the level of $r$ is the same as the
level of the elements of $\bar{t}$. Let $\bar{a}_{q}\in A^{<\omega}$ be an
enumeration of $\\{b\mid\exists m<n(m,b)\in\operatorname{dom}q\\}$ and let
$\psi$ be a formula that defines $q$ from $\bar{a}_{q}$ in a natural way and
also describes the full type of $\bar{a}_{q}\cup\bar{t}^{\prime}\cup\bar{a}$.
One can verify that the type of $\bar{a}_{q}$ over
$\bar{a}^{\smallfrown}\bar{t}$ can be computed from the number of elements in
$\bar{a}_{q}\setminus\bar{a}$ in each cone. Thus, there is a sequence
$\\{\langle t_{i},b_{i},k_{i}\rangle\mid i<\ell^{p}\\}$ such that $q\in\bigcap
D^{1}_{t_{i},b_{i},k_{i}}$ and for every other condition $q^{\prime}\in\bigcap
D^{1}_{t_{i},b_{i},k_{i}}$ such that $q^{\prime}\leq p$, and $q^{\prime}$ is
stronger than a condition $q^{\prime\prime}$ such that
$\psi(q^{\prime\prime},\bar{a}_{q^{\prime\prime}})$ holds and the type of
$\bar{a}_{q^{\prime\prime}}$ over $\bar{a}\cup\bar{t}^{\prime}$ is the same as
the type of $\bar{a}_{q}$. By indiscernibility, $q^{\prime\prime}\in D$, and
so $q^{\prime}\in D$, as wanted.

∎

Thus, we conclude that a collection of $n$ subsets of $A_{\alpha}$ is
$V(A_{\alpha},T_{\alpha})$-generic for $\mathbb{Q}_{n}(A_{\alpha})$ if and
only if their intersection with each cone above $t\in A_{<\alpha}$ is $n$
independent subsets.

Since this happens at $\langle x_{m}^{\alpha}\mid m<n\rangle$ by construction,
we conclude that $G_{n}$ is $M_{\alpha}$-generic.

∎

####  8.2.2. Adding $T_{\alpha+1}$

As a corollary of the indiscernibility result, it is quite easy to find
$\mathbb{T}(A_{\alpha+1},A_{\alpha})$-generics over
$V(A_{\alpha+1},T_{\alpha})$:

######  Lemma 8.9.

Let $f\colon A_{\alpha+1}\to A_{\alpha}$ be onto and with every fiber
infinite. Define $F=\\{p\in\mathbb{T}(A_{\alpha+1},A_{\alpha})\mid p\subseteq
f\\}$. Then $F$ is $\mathbb{T}(A_{\alpha+1},A_{\alpha})$-generic over
$V(A_{\alpha+1},T_{\alpha})$.

###### Proof.

Let $D$ be a dense open set in $V(A_{\alpha+1},T_{\alpha})$. As in the proof
of Claim 8.8, using the indiscernibility we may assume that $D$ is definable
using a formula $\varphi$ and only parameters $\bar{a},\bar{b},\bar{x}$, where
$\bar{a}\in A_{\leq\alpha}^{<\omega}$, $\bar{b}\in A_{\alpha+1}^{<\omega}$,
and $\bar{x}\subseteq\operatorname{trcl}A_{\alpha+1}$, and a parameter from
$V$. We may also assume that $[x_{i}]$ appears in $\bar{b}$ for every
$x_{i}\in\bar{x}$.

Pick $p\in F$ such that $\operatorname{dom}p\supseteq\bar{b}$ and
$\operatorname{range}p\supseteq\bar{a}$. Let $q\leq p$ be a condition in $D$
and let $\bar{b}^{\prime}\supseteq\bar{b}$ be the domain of $q$ and
$\bar{a}^{\prime}\supseteq\bar{a}$ be the range of $q$. Let us look at the
type of $\bar{b}^{\prime}\cup\bar{a}^{\prime}$ over $\bar{a},\bar{b},\bar{x}$.

For each $a\in\bar{a}^{\prime}$, let $n_{a}$ be the number of elements
$b\in\bar{b}^{\prime}\setminus\bar{b}$ such that $q(b)=a$.

######  Claim 8.10.

Let $q^{\prime}\leq p$ be a condition such that
$\operatorname{range}q^{\prime}=\bar{a}^{\prime}$ and for every
$a\in\bar{a}^{\prime}$,
$|\\{b\in\operatorname{dom}q^{\prime}\setminus\bar{b}\mid
q^{\prime}(b)=a\\}|=n_{a}$. Then $q^{\prime}\in D$.

###### Proof.

The condition $q$ is definable using an explicit formula $\psi$ and the
parameters $\bar{b}^{\prime},\bar{a}^{\prime}$, where the formula $\psi$
simply states which element is being sent to which element. By our assumption
on $q^{\prime}$, there is an enumeration of
$\bar{b}^{\prime\prime}=\operatorname{dom}q^{\prime}$ such that $q^{\prime}$
is definable using the same $\psi$ and parameters $\bar{b}^{\prime\prime}$ and
$\bar{a}^{\prime}$. Since the type of $\bar{b}^{\prime}\cup\bar{a}^{\prime}$
over $\bar{a},\bar{b},\bar{x}$. is the same as the type of
$\bar{b}^{\prime\prime}\cup\bar{a}^{\prime}$ over $\bar{a},\bar{b},\bar{x}$.,
we apply indiscernibility and conclude that $q^{\prime}\in D$. ∎

Finally, since $f$ is surjective with infinite fibers, there is
$q^{\prime}\leq p$ as in the claim in $F$. ∎

By construction, $\pi_{\alpha}$ is a surjective function from $A_{\alpha+1}$
to $A_{\alpha}$, with the property that every fiber is infinite, so the filter
$F=\\{p\in\mathbb{T}(A_{\alpha+1},A_{\alpha})\mid p\subseteq\pi_{\alpha}\\}$
is generic over $V(A_{\alpha+1},T_{\alpha})$.

Assume now that
$G\subseteq\mathbb{P}_{\alpha}\ast\mathbb{Q}(\dot{A}_{\alpha})$ is generic
over $V$ which computes the correct $A_{\alpha},T_{\alpha}$ and
$A_{\alpha+1}$. We want to find a filter $G^{\prime}\ast F^{\prime}$ which is
generic over $V$ for
$\mathbb{P}_{\alpha}\ast\mathbb{Q}(\dot{A}_{\alpha})\ast\mathbb{T}(\dot{A}_{\alpha+1},\dot{A_{\alpha}})$.

This is obtained by Corollary 7.12. Indeed, we may assume (by genericity) that
$G$ was obtained by applying a generic permutation on the symmetrically
generic filter for $\mathbb{P}_{\alpha}\ast\mathbb{Q}(\dot{A}_{\alpha})$ and
thus we can force with the quotient forcing for adding the generic permutation
for $\mathbb{T}(\dot{A_{\alpha+1}},\dot{A_{\alpha}})$ and obtain a generic
iteration of the two-step iteration that moves the symmetrically generic
filter to an actual $V$-generic filter.

###  8.3. Limit step

######  Lemma 8.11.

Let $\theta$ be a countable ordinal and let us assume that the inductive
hypothesis holds below $\theta$. Assume further that in $V[c]$ there is a
well-ordering of a large enough initial segment, to apply Claim 7.7. Then the
inductive hypothesis holds for $\theta$.

###### Proof.

By assumption $\theta$ is countable. Let $\langle\alpha_{n}\mid
n<\omega\rangle$ be an increasing cofinal sequence at $\theta$. We may think
of the poset $\mathbb{P}_{0}^{\alpha}$ as an $\omega$-length finite support
iteration, with the $n$-th step being
$\mathbb{P}_{\alpha_{n}}^{\alpha_{n+1}}$. Recall from Section 4.3.2 that for
each $\mathbb{P}_{\alpha_{n}}^{\alpha_{n+1}}$ there is a subgroup $S_{n}$ of
automorphisms of the poset such that

  * •

for any $p,q$ there is $s\in S_{n}$ sending $p$ to be compatible with $q$;

  * •

any $s\in S_{n}$ fixes $\dot{A}_{\alpha},\dot{T}_{\alpha}$ for all $\alpha$.

These properties verify the hypothesis of Claim 7.7. We conclude from the
claim that there is a “shuffled” filter $\tilde{G}$ (in some further generic
extension) so $A_{\alpha}=\dot{A}_{0}^{\alpha}[\tilde{G}]$ and
$T_{\alpha}=\dot{T}_{0}^{\alpha}[\tilde{G}]$ for any $\alpha<\theta$. ∎

######  Lemma 8.12.

Let $\theta$ be arbitrary. If the inductive hypothesis holds below $\theta$
then it holds at $\theta$.

###### Proof.

Forcing with a large enough collapse $G$ over $V[c]$ so that in $V[c][G]$,
$\theta$ is countable and we have enough choice to apply Claim 7.7. Note that
the definition of $A_{\alpha},T_{\alpha}$ is the same in $V[G]$ and $V[G][c]$,
since the recipe for constructing them is absolute. Now we can apply the
previous lemma. ∎

###  8.4. Tree step

Let $\theta$ be a limit ordinal. Suppose
$M_{<\theta}=V(A_{<\theta},T_{<\theta})$ has been constructed. Recall the
definitions of $\mathbb{B}$ from Section 4.1, adding by finite approximations
$\omega$ many branches through $T_{<\theta}$.

To find a generic for $\mathbb{B}$ over $M_{<\theta}$, we will instead use the
following slightly different $\tilde{\mathbb{B}}$, which produces the same
symmetric extensions as $\mathbb{B}$.

######  Definition 8.13.

Let $\tilde{\mathbb{B}}$ be the poset whose conditions are partial finite
functions from $\theta$ to $A_{<\theta}$, ordered by the reverse ordering of
the tree $T_{<\theta}$ on each coordinate. Given a generic
$\tilde{G}\subseteq\tilde{\mathbb{B}}$, which we identify with a
$\theta$-sequence of generic branches, let $A_{\theta}^{\tilde{G}}$ be the
unordered set of these branches.

######  Lemma 8.14.

Let $\tilde{G}\subseteq\tilde{\mathbb{B}}$ be
$V(A_{<\theta},T_{<\theta})$-generic. There is, in a further generic
extension, a $V(A_{<\theta},T_{<\theta})$-generic filter
$G\subseteq\mathbb{B}$ so that $A_{\theta}^{G}=A_{\theta}^{\tilde{G}}$.

###### Proof.

Simply note that after collapsing $\theta$ to be countable $\mathbb{B}$ and
$\tilde{\mathbb{B}}$ are forcing isomorphic. ∎

First we note that in $M_{<\theta}$ there are very few symmetric dense open
subsets of $\tilde{\mathbb{B}}$

######  Lemma 8.15.

Let $\mathcal{S}=\langle\tilde{\mathbb{B}},\mathcal{G},\mathcal{F}\rangle$,
where $\mathcal{G}$ is the group generated by $b_{\sigma}$ and $\mathcal{F}$
is the filter of subgroups generated by
$\langle\\{b_{\sigma}\mid\sigma\restriction a=id_{a}\\}\mid
a\subseteq\omega,\text{ finite}\rangle$.

A filter $\tilde{G}$ is symmetrically generic for $\mathcal{S}$ in
$M_{<\theta}$ if and only if

  1. (1)

Every branch defined by $\tilde{G}$ is cofinal, namely for every
$\zeta,\zeta^{\prime}<\theta$, there is an element in the $\zeta$-th
coordinate of $\tilde{G}$ of level $\zeta^{\prime}$ in the tree.

  2. (2)

Every pair of branches is different.

  3. (3)

For every $t\in A_{<\theta}$ there is a branch in $\tilde{G}$ that contains
$t$.

###### Proof.

Let $D\subseteq\mathbb{B}$ be a symmetrically dense open set in
$V(A_{<\theta},T_{<\theta})$ (with respect to the group of finite permutations
of coordinates). Let $\tilde{G}$ be a filter that meets the dense open sets as
in the statement of the lemma. Fix $a_{0},\dots,a_{m-1}\in A_{<\alpha}$ be
such that $D$ is definable from $a_{0},\dots,a_{m-1},A_{<\alpha},T_{<\alpha}$
and a parameter in $V$.

Let $p\in\tilde{G}$ be a condition such that $\operatorname{dom}p$ is the
range of $q$ and that all its elements are in a level higher than all the
$a_{i}$’s. Let $p^{\prime}\leq p$ be in $D$. Let $t_{0},\dots,t_{n-1}$ be the
elements of $p^{\prime}$. Assume without loss of generality that the level of
each $t_{k}$ is greater than the level of each $a_{i}$.

By the tree indescirnibility, the truth value of $p^{\prime}\in D$ is a
function of the type of $t_{0},\dots,t_{n-1}$ over $a_{0},\dots,a_{m-1}$.
Since the levels of $t_{i}$ are (without loss of generality) much higher than
the parameters in $\vec{a}$ and all are different and in the same level, the
diagram is determined by the level and the values of $a_{j}\leq_{T}t_{i}$ for
all $i,j$. In particular, $p^{\prime}\in D$ if and only if $p\cup r\in D$
where
$r=p^{\prime}\restriction(\operatorname{dom}p^{\prime}\setminus\operatorname{dom}p)$.
By the assumptions of the lemma, there are branches in $\tilde{G}$ which
contain the elements of $r$. Let $p^{\prime\prime}\leq p^{\prime}$ be a
condition in $\tilde{G}$ that contains those elements (note that we may move
the domain, as $D$ is invariant under a large group of permutations of the
domain), then $p^{\prime\prime}$ meets $D$. ∎

Let $\tilde{G}\subseteq\tilde{\mathbb{B}}$ be the filter generated by the set
of branches $A_{\theta}$, enumerated by $\theta$, constructed in Section 8.1.
Note that $\tilde{G}\subseteq\tilde{\mathbb{B}}$ satisfies conditions
(1),(2),(3) above.

Finally, given a generic $G_{<\theta}$ for the iteration
$\mathbb{P}_{<\theta}$, we conclude from Corollary 7.12 that after applying a
generic automorphism of
$\mathbb{P_{\theta}}=\mathbb{P}_{<\theta}\ast\tilde{\mathbb{B}}$ to
$G_{<\theta}\ast\tilde{G}$, we get a generic filter over $V$. This concludes
the proof that the inductive hypothesis can be extended.

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