# Scaling and renormalization in the modern theory of polarization:
application to disordered systems

Balázs Hetényi1,2,3, Selçuk Parlak3, and Mohammad Yahyavi3 1MTA-BME Topology
and Correlations Research Group, Department of Physics, Budapest University of
Technology and Economics, H-1521 Budapest, Hungary
2Department of Theoretical Physics, Budapest University of Technology and
Economics, Budafoki út 8, H-1111, Budapest, Hungary
3 Department of Physics, Bilkent University, TR-06800 Bilkent, Ankara, Turkey

###### Abstract

We develop a scaling theory and a renormalization technique in the context of
the modern theory of polarization. The central idea is to use the
characteristic function (also known as the polarization amplitude) in place of
the free energy in the scaling theory and in place of the Boltzmann
probability in a position-space renormalization scheme. We derive a scaling
relation between critical exponents which we test in a variety of models in
one and two dimensions. We then apply the renormalization to disordered
systems. In one dimension the renormalized disorder strength tends to infinity
indicating the entire absence of extended states. Zero(infinite) disorder is a
repulsive(attractive) fixed point. In two and three dimensions, at small
system sizes, two additional fixed points appear, both at finite disorder,
$W_{a}$($W_{r}$) is attractive(repulsive) such that $W_{a}<W_{r}$. In three
dimensions $W_{a}$ tends to zero, $W_{r}$ remains finite, indicating metal-
insulator transition at finite disorder. In two dimensions we are limited by
system size, but we find that both $W_{a}$ and $W_{r}$ decrease significantly
as system size is increased.

###### pacs:

††preprint: APS/123-QED

##  I Introduction

Scaling theory of localization in disordered systems Abrahams79 ; Lee85 ;
Evers08 ; Langedijk09  has a long history. A milestone work by Abrahams et al.
Abrahams79 , often referred to as the “gang of four” paper (G4), put forth
this theory to explain the dimensional dependence of criticality. The central
results for systems of no symmetry are that all states are localized in one
dimension (1D), even in two dimensions (2D) there are no extended states, but
here a crossover occurs, and only in three dimensions (3D) does a true metal-
insulator transition occur in the form of an unstable fixed point.

Experimental evidence shows unambiguous support for the G4 conclusions in 1D
Billy08 ; Roati08  and 3D Kondov11 ; Jendrzejewski11 ; Semeghini15 . The 2D
results were more difficult to establish White20  experimentally, due to the
possibility of weak localization RobertdeSaintVincent10 ; Jendrzejewski12 ;
Muller15 . Theoretically a debate Kuzovkov02 ; Markos04 ; Suslov05  about a
possible metal-insulator transition in 2D, rather than the entire absence of
extended states, arose.

An important development in the understanding of crystalline systems in
general (with or without disorder) was the development of the modern theory of
polarization King-Smith93 ; Resta94 ; Resta00  (MTP). This theory provides the
tools Resta98 ; Resta99  to measure localization. Although some studies
Bendazzoli10 ; Varma15 ; Olsen17  have used these tools to assess localization
in disordered systems, how the G4 results concur with the MTP is still an open
question.

In this paper we seek to fill this gap by developing thermodynamic scaling and
renormalization methods Reichl16 ; Kadanoff66 ; Wilson71  within the MTP
context. The idea is to use the MTP characteristic function (also known as the
polarization amplitude Kobayashi18 ) in place of the partition function as a
starting point for both scaling (similar to Widom scaling of critical
exponents) and our position space renormalization scheme.

In 1D our flow lines tend to a high disorder attractive fixed point, meaning
that there are no extended states. In 2D and 3D, for finite system sizes, we
find an attractive ($W_{a}$) and a repulsive ($W_{r}$) fixed point
($W_{a}<W_{r}$). The key question becomes how these fixed points evolve with
system size. In 3D $W_{a}$ tends to zero, while $W_{r}$ tends to a finite
number, indicating a metal-insulator transition. In 2D both the $W_{a}$ and
$W_{r}$ decrease, but due to finite size limitations it is difficult to draw a
definite conclusion valid for the thermodynamic limit. We discuss the possible
scenarios.

Other renormalization approaches to Anderson localized systems, apart from the
G4 scaling theory, also have been used Domany79 ; Sarker81 ; Shapiro82 ;
Foster09 . Traditional real-space renormalization schemes, based on blocking
sites of the lattice (Migdal-Kadanoff procedure) concur with G4. In the
context of MTP renormalization has been applied by Voit and Nakamura
Nakamura02 , but this technique relies on bosonization and is only applicable
in 1D.

There are many investigations Marzari12 ; Yahyavi17 ; Kobayashi18 ; Hetenyi19
; Patankar18 ; Furuya19  which focus on the distribution function and the
scaling of the polarization amplitude in band theoretic and correlated quantum
models. MTP also serves as the starting point for deriving topological
invariants Bernevig13 , for example, the time-reversal polarization, the
topological invariant in the Fu-Kane spin-pump Fu06  is the difference of the
Zak phases of different members of a Kramers pair.

In section II the scaling theory of localization and the MTP are outlined, and
the motivation of this work is stated, also placing our work in contemporary
context. In section III the Hamiltonians of the models studied in this work
are given. In section IV we derive a relation between critical exponents based
on Widom’s thermodynamic scaling and test it for some model systems. In
section V we develop our renormalization approach and apply it to disordered
systems in different dimensions. In section VI we conclude our work.

##  II Background and Motivation

The starting point of the G4 scaling theory Abrahams79  is the specification
of the Thouless number Edwards72  as the relevant quantity to analyze. The
Thouless number is a dimensionless conductance defined as

$g(L)=\frac{G}{e^{2}/2\hbar},$ (1)

where

$G=\frac{\Delta E}{dE/dN},$ (2)

where $\Delta E$ is the difference between energy levels calculated using
periodic boundary conditions and anti-periodic boundary conditions, $dE/dN$ is
the average level spacing. The argument in support of $G$ as a conductance is
that it is localization that determines whether a state is insulating or not.
A delocalized state should be sensitive to changing the boundary conditions,
whereas a localized one should not. G4 then argues that the $g(L)$ depends
only on the system size. The scaling theory then analyzes the scaling function

$\beta(g)=\frac{d\ln g}{d\ln L}.$ (3)

In this function $L$ appears explicitly, since in disordered systems the
system size dependence is more pronounced than in clean systems. Asymptotic
analysis can be applied. When the conductance is small (states are localized),
it is expected that

$g=\exp(-L/\xi),$ (4)

where $\xi$ denotes a correlation length. When extended states dominate, the
conductance is expected to behave as

$g=\sigma_{0}L^{d-2},$ (5)

where $\sigma_{0}$ denotes the conductivity, and $d$ is the dimensionality of
the system. From this information, by plotting $\beta(g)$ as a function of
$\ln g$ the dimensional dependence of the critical behavior can be surmised.
In 1D, all states are localized, since the function $\beta(g)$ is always
negative, even as $\ln g$ goes to infinity. In 2D the curve is negative when
$\ln g$ is negative, and it approaches zero as $\ln g$ goes to infinity,
meaning that even in 2D the states are extended, however, in this case due to
the crossover between exponential and logarithmic behavior, G4 predicts that
experiments may detect a sharp mobility edge. In 3D, since the curve crosses
$\beta(g)=0$, corresponding to an unstable fixed point, a metal-insulator
transition is predicted.

Overall, in assessing a metal-insulator transition (in ordered or disordered)
many-body quantum systems, the 1964 work of Kohn Kohn64  was a starting point.
On the one hand, Kohn argued Kohn64  that assessing whether a system is
metallic or insulating can be done by investigating the response of the system
to twisting the boundary conditions. On the other hand, Kohn also pointed out
that the localization of the center of mass of the charge distribution is the
ultimate measure of whether a system is conducting or insulating. The two
approaches are equivalent.

The central difficulty addressed by MTP was the ill-defined nature of the
position operator in systems with periodic boundary conditions. This
difficulty hindered the application of the hypothesis of Kohn Kohn64  in
calculations for crystalline systems. The problem was overcome by casting the
polarization in terms of a geometric phase Berry84  of the Zak Zak89  variety,
which arises upon integrating across the Brillouin zone. Other relevant
properties, such as the many-body generalization Resta98  of the polarization,
and the variance thereof Resta99  were also derived. The program of Kohn
Kohn64  was later realized through the MTP King-Smith93 ; Resta94 ; Resta00 ;
Resta99 .

In the formalism of Resta and Sorella, the variance of the total position is
cast Resta99  as

$\chi^{(2)}=\sigma^{2}=-\frac{L^{2}}{2\pi^{2}}\mbox{Re}\ln Z_{1}$ (6)

where

$Z_{q}=\langle\Psi_{0}|\exp\left(i\frac{2\pi
q}{L}\hat{X}\right)|\Psi_{0}\rangle,$ (7)

and where $|\Psi_{0}\rangle$ denotes a quantum ground state, $q$ is an
integer, and the total position operator is defined as

$\hat{X}=\sum_{j=1}^{L}\hat{n}_{j}j,$ (8)

where $\hat{n}_{j}$ is the density operator at site $j$. If $\sigma$ tends to
infinity with system size, the system is metallic.

In this work we perform calculations for disordered systems in different
dimensions based on MTP. Since the Thouless number is a measure of
localization, we replace it with the the quantity from the MTP which can be
taken as its analog,

$g(L)\rightarrow 1-|Z_{1}|.$ (9)

This is not an exact correspondence by any means. We justify it by first
stating that the variance can be cast according to an approximation Hetenyi19
different from the one of Resta and Sorella,

$\sigma^{2}=\frac{L^{2}}{2\pi^{2}}(1-|Z_{1}|).$ (10)

This approximation has the advantage that in the limit of the Fermi sea
($Z_{1}=0$) $\sigma$ scales with system size linearly, which is expected on
robust physical grounds. Furthermore, in Fig. 2(a), in the inset, we show the
quantity $1-|Z_{1}|$ as a function of the Thouless number (defined as the sum
over the absolute value of the difference in energy between periodic and anti-
periodic boundary conditions, divided by the total energy difference), for a
1D system of $L=160$, and averaged over one hundred disorder realizations
(replicas) (Hamiltonian in Eq. (11)). The function is monotonic, which also
justifies our replacing of the Thouless number with $|Z_{1}|$ in our
renormalization scheme (section V).

To perform the asymptotic analysis, the G4 scaling theory uses two pieces of
information. In the small conductivity (large disorder) limit it is assumed
that the conductivity localizes exponentially (Eq. (4)). In the opposite limit
the conductance is related to the conductivity via the relation (Eq. (5)). In
the latter, there appears to be no direct prescription to calculate the
conductance based on a microscopic Hamiltonian. To phrase the question
differently: given a disordered Hamiltonian for which we can calculate the
eigenstates, how do we calculate the conductance? Which states do we consider?
Should we consider a distribution of states? In our calculations below we will
average the relevant quantities over all states. This amounts to a high-
temperature approximation, since all states have the same contribution. The
position operator we use Selloni87 ; Fois88 ; Ancilotto92  is a single-body
one.

##  III Models

Most of this paper is devoted to disordered systems. The one-dimensional
version of the disordered Hamiltonian we study can be written

$\hat{H}=\sum_{i=1}^{L}\left[-t(\hat{c}_{i}^{\dagger}\hat{c}_{i+1}+\mbox{H.c.})+W\xi_{i}\hat{n}_{i}\right],$
(11)

where $t$ denotes the hopping parameter (which will be taken as the unit of
energy), $W$ indicates the disorder strength, and $\xi_{i}$ is a normal
distributed random number. On the other hand, in the next section we also test
the result of our Widom scaling theory for the Su-Schrieffer-Heeger Su79
(SSH) and Rice-Mele Rice82  (RM) models. The latter can be written,

$\hat{H}=\sum_{i=1}^{L}\left[-(t+(-1)^{i}\delta
t)(c_{i}^{\dagger}c_{i+1}+\mbox{H.c.})+\Delta(-1)^{i}\hat{n}_{i}\right],$ (12)

where $\delta t$ denotes the alternation in hopping, and $\Delta$ denotes the
strength of an alternating on-site potential. The SSH Hamiltonian is obtained
by setting $\Delta=0$ in Eq. (12).

##  IV Widom scaling in the modern theory of polarization

We consider an $N$-electron system, one-dimensional for convenience, periodic
in $L$. The discrete analog of the characteristic function is given in Eqs.
(7) and (8). The expression for the average position in a many-body
crystalline system given by Resta is

$\langle X\rangle=\frac{L}{2\pi}\mbox{Im}\ln Z_{1},$ (13)

which reduces to the Zak phase if $\Psi_{0}$ is a Slater determinant. The
average position Resta98  (Eq. (13)) and the variance Resta99  (Eq. (6)) are a
finite difference derivatives of $\ln Z_{q}$ at $q=0$, first and second
derivatives, respectively.

To our purposes in this section it is more suitable to take the thermodynamic
limit in Eq. (7) and write

$Z(K)=\langle\Psi_{0}|\exp\left(iK\hat{X}\right)|\Psi_{0}\rangle,$ (14)

and express the variance as

$\chi^{(2)}=-\left.\frac{\partial^{2}\ln Z(K)}{\partial K^{2}}\right|_{K=0}.$
(15)

The quantity $\chi^{(2)}$ can be interpreted as a semi-classical approximation
to the dielectric susceptibility. It is to be expected that if a phase
transition point is approached from the insulating side $\chi^{(2)}$ diverges.

To keep the discussion general we introduce a variable $D$ which characterizes
the approach to the critical point (we assume that $D=0$ is a quantum phase
transition, or gap closure point). We first define

$\chi^{(n)}(D,K)=\frac{1}{i^{n}}\frac{\partial^{n}\ln Z(D,K)}{\partial
K^{n}},$ (16)

and also define three critical exponents characterizing the approach to the
phase transition:

$\displaystyle\chi^{(2)}(D,0)$ $\displaystyle\propto$
$\displaystyle\frac{1}{D^{\beta}},$ (17) $\displaystyle\chi^{(4)}(D,0)$
$\displaystyle\propto$ $\displaystyle\frac{1}{D^{\gamma}},$
$\displaystyle\chi^{(2)}(0,K)$ $\displaystyle\propto$
$\displaystyle\frac{1}{K^{\delta}}.$

We also define the singular quantity,

$\Phi(D,K)=-\ln Z(D,K),$ (18)

which serves as the analog of the free energy. Applying the scaling relation
$\Phi(\lambda^{a}D,\lambda^{b}K)=\lambda\Phi(D,K)$, we obtain the following
relation between the scaling exponents:

$\gamma\delta=\beta(\delta+2).$ (19)

Figure 1: Log-log plot of $\chi^{(2)}(0,K)$ as a function of $K$ for different
order finite difference approximations to the second derivative (see Eq. (16))
for two system sizes. $n$ refers to the order of the approximation,
$\mathcal{O}(K^{-2n})$. The upper panel shows results for $L=100000$, the
lower panel for $L=1000$. The solid line indicates a curve of the form
$f(K)\propto K^{-2}$. The legend is for the upper and lower panels, not the
inset of the lower panel. The inset of the lower panel shows the estimated
ratio of the exponents $\gamma$ and $\beta$ as a function of system size for a
two-dimensional disordered system.

We tested this relation on a number of different models. For the exponent
$\delta$, ($D=0$ limit corresponds to the Fermi sea) our calculations are
shown in Fig. 1. The different panels show different system sizes, the
different curves within each panel are finite difference approximations to the
second derivative (Eq. (17)). From these calculations we can conclude that
$\delta=2$, which also means that $\gamma=2\beta$. We then calculated the
other exponents of the SSH and Rice-Mele (Eq. (12)) models, in 1D and 2D, and
found that $\beta=2$ and $\gamma=4$.

In addition, we also calculated the exponents for a 1D disordered system with
the Hamiltonian given by Eq. (11). When calculating the cumulants (Eq. (17)) a
difficulty faced was that a smooth curve only results for large systems. In
particular, at small $W$, large system sizes are needed for converged results.
We calculated the critical exponents for an $L=1280$ system, averaged over one
hundred realizations of the disorder and found $\beta=3.61(5)$ and
$\gamma=7.16(27)$, consistent with Eq. (19). We also made calculations for
three different realizations of the disorder (three different calculations no
disorder average is taken, only $W$ was swept) for a system of size $L=2560$.
The results for the exponents turned out to be $\beta=3.53(4),3.45(4),3.52(5)$
and $\gamma=7.12(9),6.90(12),7.07(11)$, respectively, again consistent with
Eq. (19) within error bars.

We also performed a calculation for a 2D disordered system. In this
calculation we encountered system size related difficulties, however we
provide an estimate of the ratios of critical exponents (inset of the lower
panel of Fig. 1). As shown below, at the system sizes accessible, there is a
transition in two-dimensional disordered systems (see Fig. 2), but the
position of this transition decreases as system size is increased (see Fig. 4,
upper panel). The relevant range of $\chi^{(2)}$ and $\chi^{(4)}$ as a
function of $W$ is above the transition (which itself changes with system
size), but before the disorder becomes too large, because then the errors are
larger than the values of $\chi^{(2)}$ and $\chi^{(4)}$. We estimated the
critical exponents $\gamma$ and $\beta$ by looking for the maxima of the
derivatives of the functions $\log\chi^{(2)}$ and $\log\chi^{(4)}$ as a
function of the logarithm of disorder strength. This estimate is shown in the
inset of the lower panel of Fig. 1. The estimated ratio is increasing of the
range of system sizes studied, and it is $\gamma/\beta\approx 1.8$ for the
largest size, $L=120$.

Figure 2: Renormalization flow lines for disordered systems of different
dimensions. (a) One-dimensional $L=160$, 100 replicas averaged. The inset
shows the quantity $1-Z_{1}$ vs. the Thouless number. (b) Two-dimensional,
$L=24$, 100 replicas averaged. The repulsive ($W_{r}$) and attractive
($W_{a}$) fixed points are also indicated. The inset shows the size scaling
exponent calculated using system sizes of $L=12,14,16,18$ (small) and
$L=72,80,96$ (large). (c) Three-dimensional, $L=8$, 100 replicas averaged. The
repulsive ($W_{r}$) and attractive ($W_{a}$) fixed points are also indicated.
The inset shows the size scaling exponent calculated using system sizes of
$L=6,8,10$ (small) and $L=20,22,24$ (large).

##  V Real-space renormalization in the modern theory of polarization

In real-space renormalization Reichl16 ; Kadanoff66 ; Wilson71  one starts
with the blocking of sites of the lattice. To each block a single block
variable is assigned. The blocked system is assumed to have the same
Hamiltonian as the original Hamiltonian (although, this can mean an extended
set of couplings). The parameters of the blocked Hamiltonian are tuned to
produce the same Boltzmann probability as the original Hamiltonian, provided
that the configurations of the starting Hamiltonian consistent with a given
configuration of the block variables are summed over (traced out).

A common way to define a blocked system is by the procedure of “decimation”,
in which some of the variables of the original system are traced out. In most
cases the set of equations obtained this way are overdetermined, so either new
parameters have to be introduced (for example, by extending the couplings
included in the Hamiltonian) or via introducing further approximations, for
example equating cumulants Niemeijer73 , between the original and the
renormalized system, rather than the full Boltzmann distribution.

We now apply this set of steps to the quantity

$\tilde{P}_{j}(x_{1},...,x_{L})=|\Psi_{0}(x_{1},...,x_{L})|^{2}\exp\left(i\frac{2\pi}{L}\hat{X}\right).$
(20)

Here, the index $j$ indicates the step in the renormalization. As a first step
we integrate out all the odd sites, resulting in

$\tilde{P}_{j+1}(x_{2},...,x_{L})=\int.\int
dx_{1}...dx_{L-1}\tilde{P}_{j}(x_{1},...,x_{L}).$ (21)

We now require that the $\tilde{P}_{j}(x_{2},...,x_{L})$ of the remaining
variables equals $\tilde{P}_{j+1}(x_{2},...,x_{L})$. It is easy to see, based
on the definition of $\hat{X}$ (Eq. (8)), that
$\tilde{P}_{j}(x_{2},...,x_{L})$ corresponds to the distribution of a system
with half the size of the original one, $L/2$. As in other real-space
renormalization techniques, the requirement is too stringent, so to arrive at
a practical scheme, we use the relaxed requirement,

$Z_{1}(W_{j+1},L/2)=Z_{1}(W_{j},L).$ (22)

In other words, in each renormalization step, with a given disorder strength
and system size, we find the disorder strength at half the system size which
generates the same value of $Z_{1}(W_{j},L)$. It is not the entire probability
distribution that is kept fixed in the course of a renormalization step, but
only one Fourier mode of the distribution of the many-body position. In this
sense, this renormalization is taylored to MTP.

In Fig. 2 the flowlines of the renormalization scheme are shown for systems of
different dimensions. Fig. 3 shows the size scaling exponent ($\gamma$) of the
variance of the polarization. We define $\gamma$ as

$\chi^{(2)}=aL^{\gamma}.$ (23)

In previous studies Hetenyi19  it was shown that metal-insulator transitions
can be accurately determined by investigating $\gamma$. In clean systems a
gapless system will exhibit $\gamma=2$, while in gapped insulators
$\gamma\rightarrow 1$.

Figure 3: Size scaling exponent of the variance of the polarization for a (a)
one-dimensional, (b) two-dimensional, and (c) three-dimensional system.

Fig. 2(a) shows results for a 1D disordered system of size $L=160$, and one
hundred realizations of the disorder averaged. All flow lines which start at a
finite value tend towards infinity indicating that all states are localized.
We found no significant size dependence or dependence on the number of
replicas. Also, the upper panel of Fig. 3(a) is consistent with the
renormalization flows. The clean conducting system ($W=0$) exhibits
$\gamma=2$, and finite disorder strength leads to a rapid decrease in
$\gamma$. These results concur with the G4 Abrahams79 .

Fig. 2(b) shows the flowlines in a 2D sample calculation. The linear dimension
of the system is $L=24$, the calculation was done on a square lattice. The
number of disorder configurations averaged was one hundred. The flow lines
here show qualitatively different behavior from the 1D case. We find two fixed
points on the $W$ axis, one repulsive, $W_{r}\approx 1$, above which the flow
lines tend to infinity, corresponding to a fully localized state. Below
$W_{r}$, the flow lines which start at finite disorder strength tend to a
finite disorder strength of $W_{a}$. $W_{r}$($W_{a}$) is a
repulsive(attractive) fixed point. We have done a number of calculations and
this qualitative behavior is maintained, however, we found variation in the
values of $W_{r}$ and $W_{a}$. Fig. 3(b) shows the size scaling exponent in
2D. In these calculations system sizes up to $L=18$ ($L$ is the linear
dimension) were used, and one hundred replicas were averaged. Note that until
$W\approx 1$, the size scaling exponent is approximately two, above that value
it decreases. The flow lines in 2(b) are consistent with the behavior of the
scaling exponent.

Similar behavior is found in 3D, (see Figs. 2(c) and 3(c)). A repulsive fixed
point $W_{r}$ and an attractive fixed point $W_{a}$ and $W_{r}>W_{a}$. The
flow lines shown in Fig. 2(c) are for an $8\times 8\times 8$ system with 100
replicas averaged.

Figure 4: The behavior of fixed points as a function of system size in 2D and
3D. In 2D both the repulsive and the attractive fixed points are shown. The
asterisks correspond to the raw data (three calculations) also shown in Tables
1 and 2. The red diamonds show the average of the three calculations for each
system size. The blue dashed curve for 3D shows a fit function (see the text
for a detailed discussion).

In the G4 scaling theory the system size itself appears as a relevant variable
(Eq. (3)). For this reason, we further investigate the system size dependence
of the attractive and repulsive fixed points. Our results are shown in Tables
1 and 2 and Fig. 4). To keep the CPU time manageable, we reduced the number of
disorder configurations averaged, proportionately to system size. We made
three calculations for each data point, and the raw data is shown in Tables 1
and 2. A plot is shown in Fig. 4, with the raw results and their average.

In 3D $W_{a}$ tends to zero, and $W_{r}$ approaches a finite value with large
system size. The function fit in the lower panel of Fig. 4 is
$f(L)=a\exp(bL)+c$, where $a=31.8259$, $b=-0.489526$ and $c=4.75641$. A metal-
insulator transition takes place at finite $W_{r}$, in agreement with the G4
Abrahams79 . In 2D, due to finite size limitations, a definite conclusion is
difficult to reach. Several scenarios are possible, depending on what happens
to $W_{r}$ and $W_{a}$ as $L$ becomes large. Our results (upper panel, Fig. 4)
show a sizeable decrease for both $W_{r}$ and $W_{a}$, $W_{r}\approx 1$ for
the smallest size, and $W_{r}\approx 0.53$ for the largest one. Also, while
$W_{a}$ does decrease, we can not say that it reaches zero (as it happens in
3D). There are then several possible scenarios. If $W_{a}$ and $W_{r}$ both go
to zero, then extended states are absent, concurring with the G4. If only
$W_{a}$ goes to zero, that would correspond to a metal-insulator transition,
which would appear to be in accordance with transfer matrix and Lyapunov
exponent based calculations Kuzovkov02 ; Suslov05 . A third possibility is
that $W_{a}$ and $W_{r}$ both tend to finite values in the thermodynamic limit
such that $W_{r}>W_{a}$. This would mean that there is a transition, but the
small disorder states would be localized, rather than extended, even though,
they would scale with system size as $L$ (substitute a finite size dependent
$Z_{1}$ in Eq. (10)). We note that localized states of a qualitatively
different nature, exhibiting power-law decay rather than exponential, have
also been suggested Mott85  in 2D.

We also calculated $\gamma$ using different sets of systems, the results shown
as insets in Fig. 2 middle and lower panel. In 2D we used systems with linear
extension $L=12,16,18$($L=72,80,96$) for the data points designated
“small”(“large”). For the former, $\gamma$ remains two up to $W\approx 1.0$,
while for the latter, the decrease from two starts earlier at $W\approx 0.6$.
In 3D increasing the system size does not lead to a similar decrease. (Here
the linear extensions were $L=6,8,10$($L=20,22,24$) for “small”(“large”).)
Although, these results are also limited by system size limitations, the
suggestion is that when the system size is increased the curve in Fig. 3b
eventually becomes like Fig. 3a, in other words, no sign of extended states
remains, and our results likely concur with G4.

Size(No. of replicas) | $W_{r}(1)$ | $W_{r}(2)$ | $W_{r}(3)$ | $W_{a}(1)$ | $W_{a}(2)$ | $W_{a}(3)$
---|---|---|---|---|---|---
$24\times 24$($100$) | $1.00$ | $0.86$ | $0.95$ | $0.38$ | $0.40$ | $0.43$
$32\times 32$($50$) | $0.78$ | $0.85$ | $0.84$ | $0.30$ | $0.29$ | $0.29$
$48\times 48$($25$) | $0.69$ | $0.75$ | $0.69$ | $0.19$ | $0.17$ | $0.16$
$64\times 64$($15$) | $0.65$ | $0.66$ | $0.65$ | $0.15$ | $0.14$ | $0.14$
$80\times 80$($10$) | $0.60$ | $0.62$ | $0.62$ | $0.11$ | $0.10$ | $0.12$
$96\times 96$($5$) | $0.57$ | $0.55$ | $0.55$ | $0.06$ | $0.09$ | $0.09$
$112\times 112$($4$) | $0.51$ | $0.53$ | $0.56$ | $0.08$ | $0.07$ | $0.06$
$128\times 128$($3$) | $0.48$ | $0.55$ | $0.57$ | $0.06$ | $0.07$ | $0.06$
Table 1: The repulsive ($W_{r}$) and attractive ($W_{a}$) fixed points for three sample calculations as a function of system size for a two-dimensional disordered system. The number of replicas is indicated in parentheses. Size(No. of replicas) | $W_{r}(1)$ | $W_{r}(2)$ | $W_{r}(3)$ | $W_{a}(1)$ | $W_{a}(2)$ | $W_{a}(3)$
---|---|---|---|---|---|---
$8\times 8\times 8$($100$) | $5.49$ | $5.46$ | $5.23$ | $0.42$ | $0.40$ | $0.49$
$12\times 12\times 12$($25$) | $4.96$ | $4.87$ | $4.73$ | $0.30$ | $0.28$ | $0.25$
$16\times 16\times 16$($10$) | $4.60$ | $4.96$ | $4.68$ | $0.08$ | $<0.01$ | $0.23$
$20\times 20\times 20$($5$) | $5.05$ | $4.70$ | $4.58$ | $<0.01$ | $<0.01$ | $0.05$
$24\times 24\times 24$($3$) | $4.62$ | $4.97$ | $4.66$ | $<0.01$ | $<0.01$ | $<0.01$
Table 2: The repulsive ($W_{r}$) and attractive ($W_{a}$) fixed points for
three sample calculations as a function of system size for a three-dimensional
disordered system. The number of replicas is indicated in parentheses.

##  VI Conclusions

Wegner Wegner76  is credited Lee85  with introducing concepts from statistical
mechanics into the study of disordered systems. In this work we applied
statistical mechanical ideas using the characteristic function of the modern
theory of polarization as a starting point. In particular we derived a scaling
relation according to the steps followed by Widom to relate critical
exponents, and we applied a renormalization procedure to the problem of
disorder. In 1D and 3D our method is in full agreement with the common wisdom
Abrahams79  on Anderson localized systems, however, in 2D we encountered
system size limitations.

We note that the case of two dimensions has always been the most difficult
one, both experimentally White20 ; RobertdeSaintVincent10 ; Jendrzejewski12 ;
Muller15  and theoretically Kuzovkov02 ; Markos04 ; Suslov05 ; Mott85 ;
Srivastava90 . Although, it is considered common knowledge that there are no
extended states in two dimensions, the original work of Abrahams et al.
Abrahams79  states that inspite of the absence of extended states, due to the
crossover between exponential and logarithmic behavior, experiments may still
detect a mobility edge.

## Acknowledgements

This research was supported by the Ministry of Innovation and Technology and
the National Research, Development and Innovation Office (NKFIH), within the
Quantum Information National Laboratory of Hungary and the Quantum Technology
National Excellence Program (Project No. 2017-1.2.1-NKP-2017-00001). BH would
like to thank Jan Wehr for a very enlightening discussion.

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††2020 _Mathematics Subject Classification_ : Primary 11F03; Secondary 11F11,
11R37.†† _Key words and phrases_ : Ramanujan continued fraction; modular
equations; ray class fields.

# Analogue of Ramanujan’s function $k(\tau)$ for the continued fraction of
order six

Russelle Guadalupe, Victor Manuel Aricheta  Institute of Mathematics,
University of the Philippines-Diliman
Quezon City 1101, Philippines<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

The Ramanujan’s $k(\tau)$ function is defined as
$k(\tau)=r(\tau)r(2\tau)^{2}$, where $r(\tau)$ is the Rogers-Ramanujan
continued fraction. Inspired by the recent work of Park (2023) about the
analogue of function $k(\tau)$ for the Ramanujan cubic continued fraction, we
study certain modular and arithmetic properties of the function
$w(\tau)=X(\tau)X(3\tau)$, where $X(\tau)$ is the continued fraction of order
six introduced by Vasuki, Bhaskar and Sharath (2010). We consider $w(\tau)$ to
be an analogue of $k(\tau)$ for the continued fraction $X(\tau)$.

##  1\. Introduction

For an element $\tau$ in the complex upper half plane $\mathbb{H}$, define
$q:=e^{2\pi i\tau}$. The Rogers-Ramanujan continued fraction $r(\tau)$ is
defined by

$\displaystyle
r(\tau)=\dfrac{q^{1/5}}{1+\dfrac{q}{1+\dfrac{q^{2}}{1+\dfrac{q^{3}}{1+\ddots}}}}=q^{1/5}\prod_{n=1}^{\infty}\dfrac{(1-q^{5n-1})(1-q^{5n-4})}{(1-q^{5n-2})(1-q^{5n-3})}.$

Ramanujan [17, p. 326] showed that the function
$k:=k(\tau)=r(\tau)r(2\tau)^{2}$ satisfies the relations

(1) $\displaystyle r(\tau)^{5}=k\left(\dfrac{1-k}{1+k}\right)^{5},\qquad
r(2\tau)^{5}=k^{2}\left(\dfrac{1+k}{1-k}\right).$

Lee and Park [9] studied the modularity of $k(\tau)$, where they proved that
it generates the field of all modular functions on $\Gamma_{1}(10)$. They also
proved that there is an algebraic relation between $k(\tau)$ and $k(n\tau)$
for all positive integers $n$, and that $k(\tau)$ generates the ray class
field $K_{(10)}$ over an imaginary quadratic field $K$ modulo $10$ for
$\tau\in K\cap\mathbb{H}$. One may refer to [5, 9, 20] for more details about
the modular and arithmetic properties of $k(\tau)$.

More recently, Park [13] studied the modularity of the function
$u(\tau)=C(\tau)C(2\tau)$, which is an analogue of $k(\tau)$ for Ramanujan’s
cubic continued fraction defined by

$\displaystyle
C(\tau)=\dfrac{q^{1/3}}{1+\dfrac{q+q^{2}}{1+\dfrac{q^{2}+q^{4}}{1+\dfrac{q^{3}+q^{6}}{1+\ddots}}}}=q^{1/3}\prod_{n=1}^{\infty}\dfrac{(1-q^{6n-1})(1-q^{6n-5})}{(1-q^{6n-3})^{2}}.$

He established the following relations for $u:=u(\tau)$ given by

(2) $\displaystyle C(\tau)^{3}=u(1-2u),\qquad
C(2\tau)^{3}=\dfrac{u^{2}}{1-2u},$

and proved that $u(\tau)$ generates the field of all modular functions on
$\Gamma_{0}(12)$. He also showed that there is a polynomial relation between
$u(\tau)$ and $u(n\tau)$ for all positive integers $n$, and that
$u(\frac{\tau}{2})$ generates the ray class field $K_{(2)}$ over $K$ modulo
$2$ for $\tau\in K\cap\mathbb{H}$.

In this paper, we study the modularity of the function
$w(\tau):=X(\tau)X(3\tau)$, which may be considered as an analogue of
$k(\tau)$ for the continued fraction $X(\tau)$ of order six defined by

$\displaystyle X(\tau)$
$\displaystyle=\dfrac{q^{7/4}}{q^{2}-1}\left(1-q^{-3/2}+\dfrac{(q^{1/4}-q^{-1/4})(q^{-7/4}-q^{7/4})}{(1-q^{-3/2})(1+q^{3})+\dfrac{(q^{1/4}-q^{11/4})(q^{-7/4}-q^{19/4})}{(1-q^{-3/2})(1+q^{6})+\ddots}}\right).$
$\displaystyle=q^{1/4}\prod_{n=1}^{\infty}\dfrac{(1-q^{6n-1})(1-q^{6n-5})}{(1-q^{6n-2})(1-q^{6n-4})}$

The function $X(\tau)$ was introduced by Vasuki, Bhaskar and Sharath [19], and
arises as a quotient of mock theta functions. Lee and Park [10] proved that
$X(2\tau)^{2}$ generates the field of all modular functions on
$\Gamma_{0}(12)$. Our first result is that $w(\tau)$ is modular.

######  Theorem 1.1.

The function $w(\tau)$ generates the field of all modular functions on
$\Gamma_{0}(18)$.

Theorem 1.1 follows from standard results on the modularity of
$\eta$-quotients, and the observation that $w(\tau)$ can be written as an
$\eta$-quotient.

For each positive integer $n$, the modular equation for $w(\tau)$ of level $n$
is defined by $F_{n}(w(\tau),w(n\tau))=0$ for some irreducible polynomial
$F_{n}(X,Y)\in\mathbb{C}[X,Y]$. The modular equation of any level can be
explicitly computed, as the next result shows.

######  Theorem 1.2.

One can explicitly obtain modular equations for $w(\tau)$ of level $n$ for all
positive integers $n$.

For example, the modular equations for $w(\tau)$ of level $2$ and level $3$
are, respectively:

$\displaystyle
w(\tau)^{2}-w(2\tau)+2w(\tau)w(2\tau)-3w(\tau)^{2}w(2\tau)+w(2\tau)^{2}=0,$
$\displaystyle w(\tau)^{3}-w(3\tau)$
$\displaystyle+3w(\tau)w(3\tau)-3w(\tau)^{2}w(3\tau)+3w(3\tau)^{2}-9w(\tau)w(3\tau)^{2}$
$\displaystyle+9w(\tau)^{2}w(3\tau)^{2}-3w(3\tau)^{3}+9w(\tau)w(3\tau)^{3}-9w(\tau)^{2}w(3\tau)^{3}=0.$

We provide in the Appendix the modular equations for $w(\tau)$ of prime levels
$p\leq 13$. The modular equation for $X(\tau)$ of level $3$ can be used to
obtain the following identities, which are the analogues of the identities (1)
and (2).

######  Theorem 1.3.

We have the identities

(3) $\displaystyle X^{4}(\tau)=w(\tau)(1-3w(\tau)+3w(\tau)^{2}),\qquad
X^{4}(3\tau)=\dfrac{w(\tau)^{3}}{1-3w(\tau)+3w(\tau)^{2}}.$

Finally, we show the following result that certain values of $w(\tau)$
generate ray class fields over imaginary quadratic fields.

######  Theorem 1.4.

Let $K$ be an imaginary quadratic field and $\tau\in K\cap\mathbb{H}$ be a
root of a primitive equation $0=aX^{2}+bX+c$, with $a,b,c\in\mathbb{Z}$, such
that the discriminant of $K$ is $b^{2}-4ac$. Suppose that $\gcd(a,6)=1$. Then
$K(w(\frac{\tau}{3}))$ is the ray class field modulo $6$ over $K$.

We organize the paper as follows. In Section 2, we review some facts about
modular functions, particularly $\eta$-quotients, on the congruence subgroup
$\Gamma_{0}(N)$. In Section 3, we prove Theorem 1.1, and using an affine model
on the modular curve $X_{0}(18)$, we prove Theorem 1.2. We then prove Theorem
1.3, and establish modular equations for $w(\tau)$ of level $p$ for primes
$p\leq 13$. We also describe some properties of the modular equations for
$w(\tau)$ of level $p$ for primes $p\geq 5$, which satisfy the Kronecker
congruence relation. In Section 4, we describe certain arithmetic properties
of $w(\tau)$ and prove Theorem 1.4. We use Mathematica for our computations.

##  2\. Preliminaries

In this section, we recall some facts about modular functions on the
congruence subgroup

$\Gamma_{0}(N):=\left\\{\begin{bmatrix}a&b\\\
c&d\end{bmatrix}\in\mbox{SL}_{2}(\mathbb{Z}):c\equiv 0\pmod{N}\right\\}.$

Any element $\gamma\in\Gamma_{0}(N)$ acts on the extended upper half plane
$\mathbb{H}^{\ast}:=\mathbb{H}\cup\mathbb{Q}\cup\\{\infty\\}$ by linear
fractional transformations. We define the cusps of $\Gamma_{0}(N)$ to be the
equivalence classes of $r\in\mathbb{Q}\cup\\{\infty\\}$ under this action,
which are known to be finite. We define a modular function on $\Gamma_{0}(N)$
as a complex-valued function $f(\tau)$ on $\mathbb{H}$ such that $f(\tau)$ is
meromorphic on $\mathbb{H}$, $f(\gamma\tau)=f(\tau)$ for all
$\gamma\in\Gamma_{0}(N)$, and $f(\tau)$ is meromorphic at all cusps of
$\Gamma_{0}(N)$. The last condition implies that for every cusp $r$ of
$\Gamma_{0}(N)$ and an element $\gamma\in\mbox{SL}_{2}(\mathbb{Z})$ with
$\gamma(r)=\infty$, $f\circ\gamma^{-1}$ has a $q$-expansion given by
$(f\circ\gamma^{-1})(\tau)=\sum_{n\geq n_{0}}a_{n}q^{n/h}$ for some integers
$h$ and $n_{0}$ with $a_{n_{0}}\neq 0$. We call $n_{0}$ the order of $f(\tau)$
at $r$, denoted by $\mbox{ord}_{r}f(\tau)$, and we say that $f(\tau)$ has a
zero (resp., a pole) at $r$ if $\mbox{ord}_{r}f(\tau)$ is positive (resp.,
negative).

We denote by $A_{0}(\Gamma_{0}(N))$ the field of all modular functions on
$\Gamma_{0}(N)$ and $A_{0}(\Gamma_{0}(N))_{\mathbb{Q}}$ its subfield
consisting of all modular functions on $\Gamma_{0}(N)$ with rational Fourier
coefficients. We identify $A_{0}(\Gamma_{0}(N))$ with the field of all
meromorphic functions on the modular curve
$X_{0}(N):=\Gamma_{0}(N)\backslash\mathbb{H}^{\ast}$. If $f(\tau)\in
A_{0}(\Gamma_{0}(N))$ has no zeros nor poles on $\mathbb{H}$, then the degree
$[A_{0}(\Gamma_{0}(N)):\mathbb{C}(f(\tau))]$ is the total degree of poles of
$f(\tau)$ given by $-\sum_{r}\mbox{ord}_{r}f(\tau)$, where the sum ranges over
the inequivalent cusps $r$ of $\Gamma_{0}(N)$ for which $f(\tau)$ has a pole
at $r$ (see [14, Chap. 6]). The following lemma gives a complete set of
inequivalent cusps of $\Gamma_{0}(N)$.

######  Lemma 2.1 ([4]).

Let $a,a^{\prime},c$ and $c^{\prime}$ be integers with
$\gcd(a,c)=\gcd(a^{\prime},c^{\prime})=1$. Let $S_{\Gamma_{0}(N)}$ be the set
of all inequivalent cusps of $\Gamma_{0}(N)$. We denote $\pm 1/0$ as $\infty$.
Then

  1. (1)

$a/c$ and $a^{\prime}/c^{\prime}$ are equivalent over $\Gamma_{0}(N)$ if and
only if there exist an integer $n$ and an element
$\overline{s}\in(\mathbb{Z}/N\mathbb{Z})^{\times}$ such that
$(a^{\prime},c^{\prime})\equiv(\overline{s}^{-1}a+nc,\overline{s}c)\pmod{N}$,
and

  2. (2)

we have

$\displaystyle S_{\Gamma_{0}(N)}=\\{a_{c,j}/c\in\mathbb{Q}:0<c\mid
N,0<a_{c,j}\leq N,\gcd(a_{c,j},N)=1,$ $\displaystyle
a_{c,j}=a_{c,j^{\prime}}\stackrel{{\scriptstyle\text{def}}}{{\iff}}a_{c,j}\equiv
a_{c,j^{\prime}}\pmod{\gcd(c,N/c)}\\}.$

An $\eta$-quotient is a function of the form

$f(\tau)=\prod_{\delta\mid N}\eta(\delta\tau)^{r_{\delta}}$

for some index set $\\{r_{\delta}\in\mathbb{Z}:\delta\mid N\\}$, where
$\eta(\tau):=q^{1/24}\prod_{n=1}^{\infty}(1-q^{n})$ is the Dedekind eta
function.The next two lemmas give sufficient conditions for an $\eta$-quotient
to be modular.

######  Lemma 2.2 ([6, 15, 16]).

Let $f(\tau)=\prod_{\delta\mid N}\eta(\delta\tau)^{r_{\delta}}$ be an
$\eta$-quotient with $k=\frac{1}{2}\sum_{\delta\mid N}r_{\delta}\in\mathbb{Z}$
such that

$\sum_{\delta\mid N}\delta r_{\delta}\equiv 0\pmod{24}\quad\text{ and
}\quad\sum_{\delta\mid N}\dfrac{N}{\delta}r_{\delta}\equiv 0\pmod{24}.$

Then for all $[\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}]\in\Gamma_{0}(N)$,

$f\left(\dfrac{a\tau+b}{c\tau+d}\right)=\left(\dfrac{(-1)^{k}\prod_{\delta\mid
N}\delta^{r_{\delta}}}{d}\right)(c\tau+d)^{k}f(\tau).$

######  Lemma 2.3 ([1, 11, 12]).

Let $c,d$ and $N$ be positive integers with $d\mid N$ and $\gcd(c,d)=1$ and
let $f(\tau)=\prod_{\delta\mid N}\eta(\delta\tau)^{r_{\delta}}$ be an
$\eta$-quotient satisfying the conditions of Lemma 2.2. Then the order of
vanishing of $f(\tau)$ at the cusp $c/d$ is

$\displaystyle\dfrac{N}{24d\gcd(d,\frac{N}{d})}\sum_{\delta\mid
N}\gcd(d,\delta)^{2}\cdot\dfrac{r_{\delta}}{\delta}.$

##  3\. Modularity of $w(\tau)$

In this section, we establish the modularity of $w(\tau)=X(\tau)X(3\tau)$ and
deduce that the modular equations for $w(\tau)$ exist at any level. Observe
that $w(\tau)$ is an $\eta$-quotient since

$\displaystyle w(\tau)$
$\displaystyle=q\prod_{n=1}^{\infty}\dfrac{(1-q^{6n-1})(1-q^{6n-5})}{(1-q^{6n-2})(1-q^{6n-4})}\dfrac{(1-q^{18n-3})(1-q^{18n-15})}{(1-q^{18n-6})(1-q^{18n-12})}$
$\displaystyle=q\prod_{n=1}^{\infty}\dfrac{(1-q^{2n-1})(1-q^{6n-3})^{-1}}{(1-q^{2n})(1-q^{6n})^{-1}}\dfrac{(1-q^{3n})(1-q^{18n-9})^{-1}}{(1-q^{18n-6})^{2}(1-q^{18n-12})^{2}}$
$\displaystyle=q\prod_{n=1}^{\infty}\dfrac{(1-q^{n})(1-q^{2n})^{-1}}{1-q^{2n}}\dfrac{(1-q^{18n})(1-q^{9n})^{-1}}{(1-q^{18n})^{-1}}$
$\displaystyle=\dfrac{\eta(\tau)\eta^{2}(18\tau)}{\eta^{2}(2\tau)\eta(9\tau)}.$

###### Proof of Theorem 1.1.

By Lemma 2.2, $w(\tau)$ is a modular function on $\Gamma_{0}(18)$. We now take

$S_{\Gamma_{0}(18)}=\\{\infty,0,1/2,1/3,2/3,1/6,5/6,1/9\\}$

as the set of inequivalent cusps of $\Gamma_{0}(18)$ by Lemma 2.1. We compute
the order of $w(\tau)$ at each element of $S_{\Gamma_{0}(18)}$ using Lemma
2.3, as shown in Table 1. Thus, $w(\tau)$ has a simple pole at $1/2$ and a
simple zero at $\infty$. Hence, $A_{0}(\Gamma_{0}(18))=\mathbb{C}(w(\tau))$.

Table 1. The orders of $w(\tau)$ at the cusps of $\Gamma_{0}(18)$ cusp $r$ | $\infty$ | $0$ | $1/2$ | $1/3$ | $2/3$ | $1/6$ | $5/6$ | $1/9$
---|---|---|---|---|---|---|---|---
$\mbox{ord}_{r}s(\tau)$ | $1$ | $0$ | $-1$ | $0$ | $0$ | $0$ | $0$ | $0$

∎

As the modular curve $X_{0}(18)$ has genus zero, the next lemma shows that
there is an affine plane model for $X_{0}(18)$ over $\mathbb{Q}$ defined by
$w(\tau)$.

######  Lemma 3.1.

For all positive integers $n$, we have
$A_{0}(\Gamma_{0}(18n))_{\mathbb{Q}}=\mathbb{Q}(w(\tau),w(n\tau))$.

###### Proof.

Since $w(\tau)$ has rational Fourier coefficients, by Theorem 1.1 we have
$\mathbb{Q}(w(\tau))=A_{0}(\Gamma_{0}(18))_{\mathbb{Q}}$. Since
$\Gamma_{0}(18n)\subseteq\Gamma_{0}(18)$, we have $w(\tau)\in
A_{0}(\Gamma_{0}(18n))_{\mathbb{Q}}$. Observe that
$\Gamma_{0}(18)\cap[\begin{smallmatrix}n&0\\\
0&1\end{smallmatrix}]^{-1}\Gamma_{0}(18)[\begin{smallmatrix}n&0\\\
0&1\end{smallmatrix}]=\Gamma_{0}(18n)$, which implies that
$w(n\tau)=(w\circ[\begin{smallmatrix}n&0\\\ 0&1\end{smallmatrix}])(\tau)\in
A_{0}(\Gamma_{0}(18n))_{\mathbb{Q}}$. Thus, it suffices to show that
$A_{0}(\Gamma_{0}(18n))_{\mathbb{Q}}\subseteq\mathbb{Q}(w(\tau),w(n\tau))$. We
choose matrices $M_{i}\in\Gamma_{0}(18)$ such that

(4) $\Gamma_{0}(18)=\sqcup_{i}(\Gamma_{0}(18n)\cdot M_{i})$

as a disjoint union. We claim that the functions
$w\circ[\begin{smallmatrix}n&0\\\ 0&1\end{smallmatrix}]\circ M_{i}$ are
distinct. Suppose otherwise; that is, $w\circ[\begin{smallmatrix}n&0\\\
0&1\end{smallmatrix}]\circ M_{i}=w\circ[\begin{smallmatrix}n&0\\\
0&1\end{smallmatrix}]\circ M_{j}$ for some distinct $i$ and $j$. Then
$w\circ[\begin{smallmatrix}n&0\\\
0&1\end{smallmatrix}]M_{j}M_{i}^{-1}[\begin{smallmatrix}n&0\\\
0&1\end{smallmatrix}]^{-1}=w$ and $[\begin{smallmatrix}n&0\\\
0&1\end{smallmatrix}]M_{j}M_{i}^{-1}[\begin{smallmatrix}n&0\\\
0&1\end{smallmatrix}]^{-1}\in\mathbb{Q}^{\times}\cdot\Gamma_{0}(18)$. As
$M_{j}M_{i}^{-1}\in[\begin{smallmatrix}n&0\\\
0&1\end{smallmatrix}]^{-1}\Gamma_{0}(18)[\begin{smallmatrix}n&0\\\
0&1\end{smallmatrix}]$ and $M_{j}M_{i}^{-1}\in\Gamma_{0}(18)$, we get
$M_{j}M_{i}^{-1}\in\Gamma_{0}(18n)$, contradicting (4). Hence, the functions
$w(\tau)$ and $w(n\tau)$ generate $A_{0}(\Gamma_{0}(18))_{\mathbb{Q}}$. ∎

We next observe the behavior of $w(\tau)$ at all cusps
$r\in\mathbb{Q}\cup\\{\infty\\}$.

######  Lemma 3.2.

Let $a,c,a^{\prime},c^{\prime}$ and $n$ be integers with $n$ positive. Then:

  1. (1)

The modular function $w(\tau)$ has a pole at
$a/c\in\mathbb{Q}\cup\\{\infty\\}$ if and only if $\gcd(a,c)=1$ and
$c\equiv\pm 2\pmod{6}$.

  2. (2)

The modular function $w(n\tau)$ has a pole at
$a^{\prime}/c^{\prime}\in\mathbb{Q}\cup\\{\infty\\}$ if and only if there
exist integers $a$ and $c$ such that $a/c=na^{\prime}/c^{\prime},\gcd(a,c)=1$
and $c\equiv\pm 2\pmod{6}$.

  3. (3)

The modular function $w(\tau)$ has a zero at
$a/c\in\mathbb{Q}\cup\\{\infty\\}$ if and only if $\gcd(a,c)=1$ and $18\mid
c$.

  4. (4)

The modular function $w(n\tau)$ has a zero at
$a^{\prime}/c^{\prime}\in\mathbb{Q}\cup\\{\infty\\}$ if and only if there
exist integers $a$ and $c$ such that $a/c=na^{\prime}/c^{\prime},\gcd(a,c)=1$
and $18\mid c$.

###### Proof.

Using Table 1, we know that $w(\tau)$ has a simple pole (resp., zero) at
$a/c\in\mathbb{Q}\cup\\{\infty\\}$ which is equivalent to $1/2$ (resp.,
$\infty$) on $\Gamma_{0}(18)$. By Lemma 2.1, $a/c$ is equivalent to $1/2$ on
$\Gamma_{0}(18)$ if and only if $(a,c)\equiv(s^{-1}+2n,2s)\pmod{18}$ for some
integer $n$ and $s\in(\mathbb{Z}/18\mathbb{Z})^{\times}$. This implies that
$\gcd(a,c)=1$ and $c\equiv\pm 2\pmod{6}$, proving (1). On the other hand,
$a/c$ is equivalent to $\infty$ on $\Gamma_{0}(18)$ if and only if
$(a,c)\equiv(s^{-1},0)\pmod{18}$ for some integer $n$ and
$s\in(\mathbb{Z}/18\mathbb{Z})^{\times}$. This implies that $\gcd(a,c)=1$ and
$18\mid c$, proving (3). Statements (2) and (4) immediately follow from (1)
and (3). ∎

Ishida and Ishii [7] proved the following result, which can be used to
describe the coefficients of the modular equation between two modular
functions that are generators of the field $A_{0}(\Gamma^{\prime})$, where
$\Gamma^{\prime}$ is a congruence subgroup of $\mbox{SL}_{2}(\mathbb{Z})$.

######  Proposition 3.3.

Let $\Gamma^{\prime}$ be a congruence subgroup of $\mbox{SL}_{2}(\mathbb{Z})$,
and $f_{1}(\tau)$ and $f_{2}(\tau)$ be nonconstant functions such that
$A_{0}(\Gamma^{\prime})=\mathbb{C}(f_{1}(\tau),f_{2}(\tau))$. For
$k\in\\{1,2\\}$, let $d_{k}$ be the total degree of poles of $f_{k}(\tau)$.
Let

$F(X,Y)=\sum_{\begin{subarray}{c}0\leq i\leq d_{2}\\\ 0\leq j\leq
d_{1}\end{subarray}}C_{i,j}X^{i}Y^{j}\in\mathbb{C}[X,Y]$

satisfy $F(f_{1}(\tau),f_{2}(\tau))=0$. Let $S_{\Gamma^{\prime}}$ be the set
of all the inequivalent cusps of $\Gamma^{\prime}$, and for $k\in\\{1,2\\}$,
define

$S_{k,0}=\\{r\in S_{\Gamma^{\prime}}:f_{k}(\tau)\text{ has a zero at
}r\\}\text{ and }S_{k,\infty}=\\{r\in S_{\Gamma^{\prime}}:f_{k}(\tau)\text{
has a pole at }r\\}.$

Further let

$a=-\sum_{r\in S_{1,\infty}\cap S_{2,0}}\mbox{ord}_{r}f_{1}(\tau)\quad\text{
and }\quad b=\sum_{r\in S_{1,0}\cap S_{2,0}}\mbox{ord}_{r}f_{1}(\tau)$

with the convention that $a=0$ (resp. $b=0$) whenever $S_{1,\infty}\cap
S_{2,0}$ (resp. $S_{1,0}\cap S_{2,0}$) is empty. Then

  1. (1)

$C_{d_{2},a}\neq 0$ and if, in addition, $S_{1,\infty}\subseteq
S_{2,\infty}\cup S_{2,0}$, then $C_{d_{2},j}=0$ for all $j\neq a$;

  2. (2)

$C_{0,b}\neq 0$ and if, in addition, $S_{1,0}\subseteq S_{2,\infty}\cup
S_{2,0}$, then $C_{0,j}=0$ for all $j\neq b$.

By interchanging the roles of $f_{1}(\tau)$ and $f_{2}(\tau)$, one may obtain
properties analogous to (1) - (2).

###### Proof.

See [7, Lem. 3 and Lem. 6] (or [2, Thm. 3]). ∎

We can now apply Lemma 3.1 and Proposition 3.3 to prove Theorem 1.2.

###### Proof of Theorem 1.2.

We follow the argument used to prove [8, Thm. 1.2]. Let
$\mathbb{C}(f_{1}(\tau),f_{2}(\tau))$ be the field of all modular functions on
some congruence subgroup, where $f_{1}(\tau)$ and $f_{2}(\tau)$ are
nonconstant functions. For $j\in\\{1,2\\}$, the degree
$[\mathbb{C}(f_{1}(\tau),f_{2}(\tau)):\mathbb{C}(f_{j}(\tau))]$ is equal to
the total degree $d_{j}$ of poles of $f_{j}(\tau)$. Thus, by [2, Lem. 3.1],
there is a unique polynomial $\Phi(X,Y)\in\mathbb{C}[X,Y]$ such that
$\Phi(X,f_{2}(\tau))$ (resp. $\Phi(f_{1}(\tau),Y)$) is the minimal polynomial
of $f_{1}(\tau)$ (resp. $f_{2}(\tau)$) over $\mathbb{C}(f_{2}(\tau))$ (resp.
$\mathbb{C}(f_{1}(\tau))$) of degree $d_{2}$ (resp. $d_{1}$). As $w(\tau)$ has
rational Fourier coefficients, setting $f_{1}(\tau)=w(\tau)$ and
$f_{2}(\tau)=w(n\tau)$ and using Proposition 3.3, we see that there is a
polynomial $F_{n}(X,Y)\in\mathbb{Q}[X,Y]$ such that
$F_{n}(w(\tau),w(n\tau))=0$ for all positive integers $n$ with
$\deg_{X}F_{n}(X,Y)=d_{2}$ and $\deg_{Y}F_{n}(X,Y)=d_{1}$. The equation
$F_{n}(w(\tau),w(n\tau))=0$ is the modular equation for $w(\tau)$ of level $n$
for all positive integers $n$. ∎

We apply Theorem 1.2 to find explicitly the modular equations for $w(\tau)$ of
levels two and three, and obtain information about the coefficients of the
modular equations for $w(\tau)$ of level $p$ for primes $p\geq 5$.

######  Theorem 3.4 (Modular equation of level two).

We have

$w(\tau)^{2}-w(2\tau)+2w(\tau)w(2\tau)-3w(\tau)^{2}w(2\tau)+w(2\tau)^{2}=0.$

###### Proof.

By Lemma 3.1, we have
$\mathbb{Q}(w(\tau),w(2\tau))=A_{0}(\Gamma_{0}(36))_{\mathbb{Q}}$. From Lemma
2.1, we take

$S_{\Gamma_{0}(36)}=\\{\infty,0,1/2,1/3,2/3,1/4,1/6,5/6,1/9,1/12,5/12,1/18\\}$

as the set of inequivalent cusps of $\Gamma_{0}(36)$. From Lemmata 2.3 and
3.2, we see that $w(\tau)$ has two simple poles at $1/2$ and $1/4$ and two
simple zeros at $\infty$ and $1/18$. Similarly, $w(2\tau)$ has a double pole
at $1/4$ and a double zero at $\infty$. Thus, the total degrees of poles of
both $w(\tau)$ and $w(2\tau)$ are $2$, so by Proposition 3.3, there is a
polynomial

$F_{2}(X,Y)=\sum_{0\leq i,j\leq 2}C_{i,j}X^{i}Y^{j}\in\mathbb{C}[X,Y]$

such that $F_{2}(w(\tau),w(2\tau))=0$. In addition, we get $C_{2,0}\neq 0$ and
$C_{0,1}\neq 0$. Interchanging the roles of $w(\tau)$ and $w(2\tau)$, we
obtain $C_{0,2}\neq 0$ and $C_{1,2}=C_{2,2}=C_{1,0}=C_{0,0}=0$. Using the
$q$-expansion of $w(\tau)$ given by

(5) $w(\tau)=q-q^{2}+q^{3}-2q^{4}+3q^{5}-4q^{6}+5q^{7}+O(q^{8}),$

we can take $C_{0,2}=1$. Hence, we obtain
$F_{2}(X,Y)=X^{2}-Y+2XY-3X^{2}Y+Y^{2}$ and the desired identity follows. ∎

######  Theorem 3.5 (Modular equation of level three).

We have

$\displaystyle w(\tau)^{3}-w(3\tau)$
$\displaystyle+3w(\tau)w(3\tau)-3w(\tau)^{2}w(3\tau)+3w(3\tau)^{2}-9w(\tau)w(3\tau)^{2}$
$\displaystyle+9w(\tau)^{2}w(3\tau)^{2}-3w(3\tau)^{3}+9w(\tau)w(3\tau)^{3}-9w(\tau)^{2}w(3\tau)^{3}=0.$

###### Proof.

By Lemma 3.1, we have
$\mathbb{Q}(w(\tau),w(3\tau))=A_{0}(\Gamma_{0}(54))_{\mathbb{Q}}$. From Lemma
2.1, we take

$S_{\Gamma_{0}(54)}=\\{\infty,0,1/2,1/3,2/3,1/6,5/6,1/9,5/9,1/18,5/18,1/27\\}$

as the set of inequivalent cusps of $\Gamma_{0}(54)$. From Lemmata 2.3 and
3.2, we see that $w(\tau)$ has a triple pole at $1/2$ and three simple zeros
at $\infty,1/18$ and $5/18$. Similarly, $w(3\tau)$ has three simple poles at
$1/2,1/6$ and $5/6$ and a triple zero at $\infty$. Thus, the total degrees of
poles of both $s(\tau)$ and $s(3\tau)$ are $3$, so by Proposition 3.3, there
is a polynomial

$F_{3}(X,Y)=\sum_{0\leq i,j\leq 3}C_{i,j}X^{i}Y^{j}\in\mathbb{C}[X,Y]$

such that $F_{3}(w(\tau),w(3\tau))=0$. We also have $C_{3,0}\neq 0,C_{0,1}\neq
0$ and $C_{3,1}=C_{3,2}=C_{3,3}=0$. Interchanging the roles of $w(\tau)$ and
$w(3\tau)$, we get $C_{0,3}\neq 0$ and $C_{0,0}=C_{1,0}=C_{2,0}=0$. Using the
$q$-expansion (5) of $w(\tau)$, we can take $C_{0,1}=-1$. Hence, we arrive at
the polynomial

$\displaystyle
X^{3}-Y+3XY-3X^{2}Y+3Y^{2}-9XY^{2}+9X^{2}Y^{2}-3Y^{3}+9XY^{3}-9X^{2}Y^{3}$

as $F_{3}(X,Y)$ and the desired identity follows. ∎

######  Theorem 3.6.

Let $p\geq 5$ be a prime and $0=F_{p}(X,Y)=\sum_{0\leq i,j\leq
p+1}C_{i,j}X^{i}Y^{j}\in\mathbb{Q}[X,Y]$ be the modular equation of $w(\tau)$
of level $p$. Then

  1. (1)

$C_{p+1,0}\neq 0$ and $C_{0,p+1}\neq 0$,

  2. (2)

$C_{p+1,j}=C_{j,p+1}=0$ for $j\in\\{1,2,\ldots,p+1\\}$, and

  3. (3)

$C_{0,j}=C_{j,0}=0$ for $j\in\\{0,1,\ldots,p\\}$.

###### Proof.

As $\mathbb{Q}(w(\tau),w(p\tau))=A_{0}(\Gamma_{0}(18p))_{\mathbb{Q}}$ by Lemma
3.1, we choose $S_{\Gamma_{0}(18p)}$ to be the set

$\\{\infty,0,1/2,1/3,2/3,1/6,5/6,1/9,1/18,1/p,1/2p,1/3p,2/3p,1/6p,a_{p}/6p,1/9p\\},$

where $a_{p}=11$ if $p=5$ and $a_{p}=5$ otherwise, as the set of inequivalent
cusps of $\Gamma_{0}(18p)$. From Lemma 3.2, both $w(\tau)$ and $w(p\tau)$ have
poles at $1/2$ and $1/2p$ and a zero at $\infty$ and $1/18$. Using Lemma 2.3,
we get

$\mbox{ord}_{1/2}w(\tau)=\mbox{ord}_{1/2p}w(p\tau)=-p\quad\text{ and
}\quad\mbox{ord}_{1/2p}w(\tau)=\mbox{ord}_{1/2}w(p\tau)=-1.$

Thus, the total degrees of poles of both $w(\tau)$ and $w(p\tau)$ are $p+1$,
so by Proposition 3.3, there is a polynomial

$F_{p}(X,Y)=\sum_{0\leq i,j\leq p+1}C_{i,j}X^{i}Y^{j}\in\mathbb{Q}[X,Y]$

such that $F_{p}(w(\tau),w(p\tau))=0$. We also have $C_{p+1,0}\neq 0$ and
$C_{p+1,j}=0$ for $j\in\\{1,2,\ldots,p+1\\}$. Since
$\mbox{ord}_{1/18}w(\tau)=p$ and $\mbox{ord}_{\infty}w(\tau)=1$ from Lemma
2.3, we get $C_{0,p+1}\neq 0$ and $C_{0,j}=0$ for $j\in\\{0,2,\ldots,p\\}$.
Interchanging the roles of $w(\tau)$ and $w(p\tau)$, we obtain $C_{j,p+1}=0$
for $j\in\\{1,2,\ldots,p+1\\}$ and $C_{j,0}=0$ for $j\in\\{0,1,2,\ldots,p\\}$.
∎

The modular equations $F_{p}(X,Y)=0$ for $w(\tau)$ of prime level $p\leq 13$
are shown at Tables 2 and 3 in the Appendix. We observe that for these primes
$p$, the polynomial $F_{p}(X,Y)$ satisfy the Kronecker congruence relation

$F_{p}(X,Y)\equiv(X^{p}-Y)(X-Y^{p})\pmod{p}.$

We are in a position to describe the properties of the modular equation for
$w(\tau)$ of levels coprime to $6$. For an integer $a$ coprime to $6$, we
define matrices $\sigma_{a}\in\mbox{SL}_{2}(\mathbb{Z})$ such that
$\sigma_{a}\equiv[\begin{smallmatrix}a^{-1}&0\\\
0&a\end{smallmatrix}]\pmod{18}$. Then $\sigma_{a}\in\Gamma_{0}(18)$, and we
may take

$\sigma_{\pm 1}=\pm\begin{bmatrix}1&0\\\ 0&1\end{bmatrix},\qquad\sigma_{\pm
5}=\pm\begin{bmatrix}-7&-72\\\ 18&185\end{bmatrix}\quad\text{ and
}\quad\sigma_{\pm 7}=\pm\begin{bmatrix}-5&-72\\\ 18&259\end{bmatrix}.$

For any integer $n$ coprime to $6$, we have the disjoint union

$\Gamma_{0}(18)\begin{bmatrix}1&0\\\
0&n\end{bmatrix}\Gamma_{0}(18)=\bigsqcup_{0<a\mid
n}\bigsqcup_{\begin{subarray}{c}0\leq b<n/a\\\
\gcd(a,b,n/a)=1\end{subarray}}\Gamma_{0}(18)\sigma_{a}\begin{bmatrix}a&b\\\
0&\frac{n}{a}\end{bmatrix}$

and $\Gamma_{0}(18)\backslash\Gamma_{0}(18)[\begin{smallmatrix}1&0\\\
0&n\end{smallmatrix}]\Gamma_{0}(18)$ has $\psi(n):=n\prod_{p\mid n}(1+1/p)$
elements (see [18, Prop. 3.36]). We now define the polynomial

$\Phi_{n}(X,\tau):=\prod_{0<a\mid n}\prod_{\begin{subarray}{c}0\leq b<n/a\\\
\gcd(a,b,n/a)=1\end{subarray}}(X-(w\circ\alpha_{a,b})(\tau))$

where $\alpha_{a,b}:=\sigma_{a}[\begin{smallmatrix}a&b\\\
0&n/a\end{smallmatrix}]$. Note that the coefficients of $\Phi_{n}(X,\tau)$ are
elementary symmetric functions of $w\circ\alpha_{a,b}$, so these are invariant
under the action of $\Gamma_{0}(18)$ and thus are in
$A_{0}(\Gamma_{0}(18))=\mathbb{C}(w(\tau))$. We may then write
$\Phi_{n}(X,\tau)$ as $\Phi_{n}(X,w(\tau))\in\mathbb{C}(w(\tau))[X]$. Since
$\alpha_{1,0}=\sigma_{1}[\begin{smallmatrix}1&0\\\ 0&n\end{smallmatrix}]$ and
$(w\circ\alpha_{1,0})(\tau)=w(\tau/n)$ is a zero of $\Phi_{n}(X,w(\tau))$, we
have $\Phi_{n}(w(\tau/n),w(\tau))=0$.

The following theorem shows several properties of the polynomial
$\Phi_{n}(X,Y)$.

######  Theorem 3.7.

Let $n$ be a positive integer coprime to $6$ and let $\Phi_{n}(X,Y)$ be the
polynomial defined as above. Then

  1. (1)

$\Phi_{n}(X,Y)\in\mathbb{Z}[X,Y]$ and
$\deg_{X}\Phi_{n}(X,Y)=\deg_{Y}\Phi_{n}(X,Y)=\psi(n)$.

  2. (2)

$\Phi_{n}(X,Y)$ is irreducible both as a polynomial in $X$ over
$\mathbb{C}(Y)$ and as a polynomial in $Y$ over $\mathbb{C}(X)$.

  3. (3)

$\Phi_{n}(X,Y)=\Phi_{n}(Y,X)$.

  4. (4)

(Kronecker congruence relation) If $p\geq 5$ is a prime, then

$\Phi_{p}(X,Y)\equiv(X^{p}-Y)(X-Y^{p})\pmod{p\mathbb{Z}[X,Y]}.$

###### Proof.

We follow the argument used to prove [8, Thm. 2.9]. For a positive integer $k$
coprime to $n$, let $\psi_{k}\in\mbox{Gal}(\mathbb{Q}(\zeta_{n})/\mathbb{Q})$
with $\psi_{k}(\zeta_{n})=\zeta_{n}^{k}$ (here, $\zeta_{n}$ is the primitive
$n$th root of unity). By writing $w(\tau)=\sum_{m\geq 1}c_{m}q^{m}$ for some
integers $a_{n}$, we compute the action of $[\begin{smallmatrix}a&b\\\
0&n/a\end{smallmatrix}]$ on $w$:

$\displaystyle w\circ\begin{bmatrix}a&b\\\
0&\frac{n}{a}\end{bmatrix}(\tau)=w\left(\dfrac{a^{2}\tau+ab}{n}\right)=\sum_{m\geq
1}c_{m}\zeta_{n}^{abm}q^{a^{2}m/n}.$

Thus, $\psi_{k}$ induces an automorphism of $\mathbb{Q}(\zeta_{n})((q^{1/n}))$
over $\mathbb{Q}(\zeta_{n})$ (which is again denoted by $\psi_{k}$) defined by

$\displaystyle\psi_{k}\left(w\circ\begin{bmatrix}a&b\\\
0&\frac{n}{a}\end{bmatrix}(\tau)\right)=\sum_{m\geq
1}c_{m}\zeta_{n}^{abkm}q^{a^{2}m/n}.$

Choose an integer $b^{\prime}$ with $0\leq b^{\prime}<a$ such that $abk\equiv
ab^{\prime}\pmod{n}$. Then the above equations imply that
$\psi_{k}(w\circ\alpha_{a,b})=w\circ\alpha_{a,b^{\prime}}$. Thus, $\psi_{k}$
fixes $\Phi_{n}(X,w(\tau))$ and
$\Phi_{n}(X,w(\tau))\in\mathbb{Q}(w(\tau))[X]$. As
$w\circ\alpha_{1,0}(\tau)=w(\tau/n)$ is a root of $\Phi_{n}(X,w(\tau))$, we
have $[\mathbb{C}(w(\tau/n),w(\tau)):\mathbb{C}(w(\tau))]\leq\psi(n)$.

We now fix integers $a$ and $b$ and choose
$\gamma,\gamma^{\prime},\gamma_{a,b}\in\Gamma_{0}(18)$ such that
$\gamma[\begin{smallmatrix}1&0\\\
0&n\end{smallmatrix}]\gamma_{a,b}=\gamma^{\prime}\alpha_{a,b}$, so that
$[\begin{smallmatrix}1&0\\\
0&n\end{smallmatrix}]\gamma_{a,b}\alpha_{a,b}^{-1}\in\Gamma_{0}(18)$. Define
the embedding $\xi_{a,b}$ of $\mathbb{C}(w(\tau/n),w(\tau))$ into a field of
all meromorphic functions on $\mathbb{H}$ containing
$\mathbb{C}(w(\tau/n),w(\tau))$ over $\mathbb{C}(w(\tau))$ by
$\xi_{a,b}(f)=f\circ\gamma_{a,b}$. Then $\xi_{a,b}$ fixes $w$ and

$\xi_{a,b}\left(w\left(\frac{\tau}{n}\right)\right)=w\circ\begin{bmatrix}1&0\\\
0&n\end{bmatrix}\gamma_{a,b}=w\circ\alpha_{a,b}.$

We see that the functions $w\circ\alpha_{a,b}$ are distinct and there are
exactly $\psi(n)$ embeddings $\xi_{a,b}$ of $\mathbb{C}(w(\tau/n),w(\tau))$
over $\mathbb{C}(w(\tau))$. Hence, $\Phi_{n}(X,w(\tau))$ is irreducible over
$\mathbb{C}(w(\tau))$.

Let $S_{n,\infty}$ (resp., $S_{n,0}$) be the set of cusps of $\Gamma_{0}(18n)$
which are poles (resp., zeros) of $w(n\tau)$. Using the notation of
Proposition 3.3, let $-a_{n}$ be the sum of the orders $\mbox{ord}_{r}w(\tau)$
at the cusps $r\in S_{1,\infty}\cap S_{n,0}$. In view of the proof of Theorem
1.2, let $F_{n}(X,Y)\in\mathbb{Q}[X,Y]$ be a polynomial such that
$F_{n}(w(\tau),w(n\tau))=0$. We have

$\displaystyle
F_{n}(X,Y)=C_{d_{n},a_{n}}X^{d_{n}}Y^{a_{n}}+\sum_{\begin{subarray}{c}0\leq
i<d_{n}\\\ 0\leq j\leq d_{1},\,j\neq a_{n}\end{subarray}}C_{i,j}X^{i}Y^{j},$

where $d_{n}$ is the total number of poles of $w(n\tau)$. Note that
$F_{n}(X,Y)$ is irreducible both as a polynomial of $X$ over $\mathbb{C}(Y)$
and as a polynomial of $Y$ over $\mathbb{C}(X)$. Since $F_{n}(X,w(\tau))$ is
the minimal polynomial of $w(\tau/n)$ over $\mathbb{C}(w(\tau))$ and
$F_{n}(w(\tau/n),Y)$ is the minimal polynomial of $w(\tau)$ over
$\mathbb{C}(w(\tau/n))$, we have

$\displaystyle
w(\tau)^{a_{n}}\Phi_{n}(X,w(\tau))=\dfrac{F_{n}(X,w(\tau))}{C_{d_{n},a_{n}}}\in\mathbb{Q}[X,w(\tau)].$

As the Fourier coefficients in the $q$-expansion of
$w(\tau)^{a_{n}}\Phi_{n}(X,w(\tau))$ are algebraic integers, we see that
$w(\tau)^{a_{n}}\Phi_{n}(X,w(\tau))\in\mathbb{Z}[X,w(\tau)]$. By Lemma 3.2, we
see that $a_{n}=0$ for all $n$, so that
$\Phi_{n}(X,w(\tau))\in\mathbb{Z}[X,w(\tau)]$ and
$\Phi_{n}(X,Y)\in\mathbb{Z}[X,Y]$. This proves (1) and (2).

We next show (3). Note that $w\circ\alpha_{n,0}=w(n\tau)$ is a root of
$\Phi_{n}(X,w(\tau))\in\mathbb{Z}[X,w(\tau)]$, so
$\Phi_{n}(w(\tau),w(\tau/n))=0$. Since $\Phi_{n}(X,w(\tau))$ is irreducible
and $w(\tau/n)$ is a root of $\Phi_{n}(X,w(\tau))$, we have

$\Phi_{n}(w(\tau),X)=G(X,w(\tau))\Phi_{n}(X,w(\tau))$

for some polynomial $G(X,Y)$. Interchanging $X$ and $w(\tau)$ yields

$\Phi_{n}(w(\tau),X)=G(X,w(\tau))G(w(\tau),X)\Phi_{n}(w(\tau),X)$

so that $G(X,Y)=\pm 1$. Suppose $G(X,Y)=-1$. Then
$\Phi_{n}(w(\tau),w(\tau))=-\Phi_{n}(w(\tau),w(\tau))$ and $X-w(\tau)$ is a
factor of $\Phi_{n}(X,w(\tau))$ with multiplicity at least $2$, a
contradiction. Hence, we get $\Phi_{n}(w(\tau),X)=\Phi_{n}(X,w(\tau))$.

We now let $p\geq 5$ be a prime. Given
$f_{1}(\tau),f_{2}(\tau)\in\mathbb{Z}[\zeta_{p}]((q^{1/p}))$, we say that
$f_{1}(\tau)\equiv f_{2}(\tau)\pmod{\alpha}$ to mean
$f_{1}(\tau)-f_{2}(\tau)\in\alpha\mathbb{Z}[\zeta_{p}]((q^{1/p}))$. We compute

$\displaystyle w\circ\alpha_{1,b}(\tau)=\sum_{m\geq
1}c_{m}\zeta_{p}^{bm}q^{m/p}\equiv\sum_{m\geq 1}c_{m}q^{m/p}\equiv
w\circ\alpha_{1,0}(\tau)\pmod{1-\zeta_{p}}$

and

$\displaystyle w\circ\alpha_{p,0}(\tau)=\sum_{m\geq
1}c_{m}q^{pm}\equiv\sum_{m\geq 1}c_{m}^{p}q^{pm}\equiv w(\tau)^{p}\pmod{p}.$

We also have

$\displaystyle(w\circ\alpha_{1,0}(\tau))^{p}=\left(\sum_{m\geq
1}c_{m}q^{m/p}\right)^{p}\equiv\sum_{m\geq 1}c_{m}q^{m}\equiv
w(\tau)\pmod{1-\zeta_{p}}.$

Thus, we see that

$\displaystyle\Phi_{p}(X,w(\tau))$
$\displaystyle=(X-(w\circ\alpha_{p,0})(\tau))\cdot\prod_{b=0}^{p-1}(X-(w\circ\alpha_{1,b})(\tau))$
$\displaystyle\equiv(X-w(\tau)^{p})(X-w\circ\alpha_{1,0}(\tau))^{p}\pmod{1-\zeta_{p}}$
$\displaystyle\equiv(X-w(\tau)^{p})(X^{p}-(w\circ\alpha_{1,0}(\tau))^{p})\pmod{1-\zeta_{p}}$
$\displaystyle\equiv(X-w(\tau)^{p})(X^{p}-w(\tau))\pmod{1-\zeta_{p}},$

so that
$\Phi_{p}(X,w(\tau))-(X-w(\tau)^{p})(X^{p}-w(\tau))\in(1-\zeta_{p})\mathbb{Z}[\zeta_{p}]((q^{1/p}))$.
As the coefficients of $\Phi_{p}(X,w(\tau))-(X-w(\tau)^{p})(X^{p}-w(\tau))$ as
a polynomial in $X$ lie in $\mathbb{Z}[w(\tau)]$, they lie in
$(1-\zeta_{p})\mathbb{Z}[w(\tau)]$, hence in $p\mathbb{Z}[w(\tau)]$. Hence,
(4) follows. ∎

We close this section by showing that $X(\tau)$ and $X(3\tau)$ can be
expressed in terms of $w(\tau)$.

###### Proof of Theorem 1.3.

From [10, Thm. 3.7(2)], the modular equation for $X(\tau)$ of level three is

(6) $\displaystyle
X^{3}(\tau)-X(3\tau)+3X(\tau)X^{2}(3\tau)-3X^{2}(\tau)X^{3}(3\tau)=0.$

Plugging in $X(3\tau)=w(\tau)/X(\tau)$ and $X(\tau)=w(\tau)/X(3\tau)$ in
equation (6) leads, respectively, to

$\displaystyle
X^{3}(\tau)-\dfrac{w(\tau)}{X(\tau)}+\dfrac{3w(\tau)^{2}}{X(\tau)}-\dfrac{3w(\tau)^{3}}{X(\tau)}$
$\displaystyle=0,$
$\displaystyle\dfrac{w(\tau)^{3}}{X(3\tau)^{3}}-X(3\tau)+3w(\tau)X(3\tau)-3w(\tau)^{2}X(3\tau)$
$\displaystyle=0,$

which are equivalent to the identities in (3). ∎

##  4\. The ray class field $K_{(6)}$

In this section, we provide some arithmetic properties of $w(\tau)$. We start
with the following lemma, which asserts that ray class fields over an
imaginary quadratic field can be generated by singular values of a suitable
modular function on some congruence subgroup between $\Gamma(N)$ and
$\Gamma_{1}(N)$. For $N\geq 1$, we denote $\Gamma(N)$ and $\Gamma_{1}(N)$ the
set of all matrices $[\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}]\in\mbox{SL}_{2}(\mathbb{Z})$ congruent to
$[\begin{smallmatrix}1&0\\\ 0&1\end{smallmatrix}]$ and
$[\begin{smallmatrix}1&\ast\\\ 0&1\end{smallmatrix}]$, respectively, modulo
$N$.

######  Lemma 4.1.

Let $K$ be an imaginary quadratic field with discriminant $d_{K}$ and $\tau\in
K\cap\mathbb{H}$ be a root of a primitive equation $aX^{2}+bX+c=0$ such that
$b^{2}-4ac=d_{K}$ and $a,b,c\in\mathbb{Z}$. Let $\Gamma^{\prime}$ be a
congruence subgroup such that
$\Gamma(N)\subset\Gamma^{\prime}\subset\Gamma_{1}(N)$. Suppose that
$\gcd(a,N)=1$. Then the field generated over $K$ by all values of $h(\tau)$,
where $h\in A_{0}(\Gamma^{\prime})_{\mathbb{Q}}$ is defined and finite at
$\tau$, is the ray class field $K_{(N)}$ modulo $N$ over $K$.

###### Proof.

See [3, Cor. 5.2]. ∎

We now use Lemma 4.1 to prove Theorem 1.4.

###### Proof of Theorem 1.4.

Since $w(\tau)$ has rational Fourier coefficients, we have
$A_{0}(\Gamma_{0}(18))_{\mathbb{Q}}=\mathbb{Q}(w(\tau))$. For a subgroup
$\Gamma^{\prime}$ of $\mbox{SL}_{2}(\mathbb{Z})$, define
$\overline{\Gamma^{\prime}}:=\Gamma^{\prime}/\langle-I\rangle$. Since

$\begin{bmatrix}1&0\\\
0&3\end{bmatrix}^{-1}\overline{\Gamma_{0}(18)}\begin{bmatrix}1&0\\\
0&3\end{bmatrix}=\overline{\Gamma_{0}(6)}\cap\overline{\Gamma(3)}=\overline{\Gamma_{1}(6)}\cap\overline{\Gamma(3)},$

we see that
$\mathbb{Q}(w(\frac{\tau}{3}))=A_{0}(\Gamma_{1}(6)\cap\Gamma(3))_{\mathbb{Q}}$.
Let $K$ be an imaginary quadratic field and $\tau\in K\cap\mathbb{H}$ be a
root of a primitive equation $0=aX^{2}+bX+c\in\mathbb{Z}[X]$ such that the
discriminant of $K$ is $b^{2}-4ac$. Since $\gcd(6,a)=1$,
$\Gamma(6)\subset\Gamma_{1}(6)\cap\Gamma(3)\subset\Gamma_{1}(6)$ and
$w(\frac{\tau}{3})$ is well-defined and finite, by Lemma 4.1 with $N=6$, we
see that the ray class field $K_{(6)}$ modulo $6$ over $K$ is
$K(w(\frac{\tau}{3}))$. ∎

######  Remark 4.2.

We note that $f:=f(\tau)=1/w(\tau)$ is an algebraic integer by observing that
$f$ satisfies the identity

$j(\tau)=\dfrac{(f^{3}+3f^{2}-9f+9)^{3}P(f)^{3}}{(f-1)^{2}f^{9}(f-3)^{18}(f^{2}-3f+3)(f^{2}+3)^{2}},$

where $j(\tau)$ is the modular $j$-function with $q$-expansion

$j(\tau):=\frac{1}{q}+744+196884q+21493760q^{2}+O(q^{3})$

and $P(X)\in\mathbb{Z}[X]$ is the polynomial

$\displaystyle X^{9}+225X^{8}-1080X^{7}$ $\displaystyle+3348X^{6}-8262X^{5}$
$\displaystyle+16038X^{4}-23328X^{3}+26244X^{2}-19683X+6561.$

##  Appendix A Modular equations for $w(\tau)$ of prime levels $p\leq 13$

Table 2. Modular equations $F_{p}(X,Y)=0$ for $w(\tau)$ of level $p$ for primes $p\leq 11$ $p$ | $F_{p}(X,Y)$
---|---
$2$ | $X^{2}-Y+2XY-3X^{2}Y+Y^{2}$
$3$ | $X^{3}-Y+3XY-3X^{2}Y+3Y^{2}-9XY^{2}+9X^{2}Y^{2}-3Y^{3}+9XY^{3}-9X^{2}Y^{3}$
$5$ | $\begin{aligned} &(X^{5}-Y)(X-Y^{5})-5XY(-X+2X^{2}-3X^{3}+3X^{4}-Y+4XY-6X^{2}Y+9X^{3}Y-9X^{4}Y\\\ &+2Y^{2}-6XY^{2}+8X^{2}Y^{2}-18X^{3}Y^{2}+18X^{4}Y^{2}-3Y^{3}+9XY^{3}-18X^{2}Y^{3}+36X^{3}Y^{3}\\\ &-27X^{4}Y^{3}+3Y^{4}-9XY^{4}+18X^{2}Y^{4}-27X^{3}Y^{4}+16X^{4}Y^{4})\end{aligned}$
$7$ | $\begin{aligned} &(X^{7}-Y)(X-Y^{7})-7XY(-X+3X^{2}-6X^{3}+9X^{4}-9X^{5}+5X^{6}-Y+7XY-21X^{2}Y\\\ &+42X^{3}Y-63X^{4}Y+59X^{5}Y-27X^{6}Y+3Y^{2}-21XY^{2}+63X^{2}Y^{2}-126X^{3}Y^{2}+197X^{4}Y^{2}\\\ &-189X^{5}Y^{2}+81X^{6}Y^{2}-6Y^{3}+42XY^{3}-126X^{2}Y^{3}+242X^{3}Y^{3}-378X^{4}Y^{3}+378X^{5}Y^{3}\\\ &-162X^{6}Y^{3}+9Y^{4}-63XY^{4}+197X^{2}Y^{4}-378X^{3}Y^{4}+567X^{4}Y^{4}-567X^{5}Y^{4}+243X^{6}Y^{4}\\\ &-9Y^{5}+59XY^{5}-189X^{2}Y^{5}+378X^{3}Y^{5}-567X^{4}Y^{5}+567X^{5}Y^{5}-243X^{6}Y^{5}+5Y^{6}\\\ &-27XY^{6}+81X^{2}Y^{6}-162X^{3}Y^{6}+243X^{4}Y^{6}-243X^{5}Y^{6}+104X^{6}Y^{6})\end{aligned}$
$11$ | $\begin{aligned} &(X^{11}-Y)(X-Y^{11})-11XY(-X+5X^{2}-16X^{3}+38X^{4}-70X^{5}+100X^{6}-110X^{7}+91X^{8}\\\ &-51X^{9}+15X^{10}-Y+12XY-65X^{2}Y+221X^{3}Y-544X^{4}Y+1022X^{5}Y-1478X^{6}Y\\\ &+1613X^{7}Y-1263X^{8}Y+630X^{9}Y-153X^{10}Y+5Y^{2}-65XY^{2}+377X^{2}Y^{2}-1348X^{3}Y^{2}\\\ &+3422X^{4}Y^{2}-6552X^{5}Y^{2}+9616X^{6}Y^{2}-10542X^{7}Y^{2}+8077X^{8}Y^{2}-3789X^{9}Y^{2}+819X^{10}Y^{2}\\\ &-16Y^{3}+221XY^{3}-1348X^{2}Y^{3}+5000X^{3}Y^{3}-12982X^{4}Y^{3}+25214X^{5}Y^{3}-37458X^{6}Y^{3}\\\ &+41403X^{7}Y^{3}-31626X^{8}Y^{3}+14517X^{9}Y^{3}-2970X^{10}Y^{3}+38Y^{4}-544XY^{4}+3422X^{2}Y^{4}\\\ &-12982X^{3}Y^{4}+34186X^{4}Y^{4}-67074X^{5}Y^{4}+100662X^{6}Y^{4}-112374X^{7}Y^{4}+86544X^{8}Y^{4}\\\ &-39906X^{9}Y^{4}+8100X^{10}Y^{4}-70Y^{5}+1022XY^{5}-6552X^{2}Y^{5}+25214X^{3}Y^{5}-67074X^{4}Y^{5}\\\ &+132804X^{5}Y^{5}-201222X^{6}Y^{5}+226926X^{7}Y^{5}-176904X^{8}Y^{5}+82782X^{9}Y^{5}-17010X^{10}Y^{5}\\\ &+100Y^{6}-1478XY^{6}+9616X^{2}Y^{6}-37458X^{3}Y^{6}+100662X^{4}Y^{6}-201222X^{5}Y^{6}+307674X^{6}Y^{6}\\\ &-350514X^{7}Y^{6}+277182X^{8}Y^{6}-132192X^{9}Y^{6}+27702X^{10}Y^{6}-110Y^{7}+1613XY^{7}-10542X^{2}Y^{7}\\\ &+41403X^{3}Y^{7}-112374X^{4}Y^{7}+226926X^{5}Y^{7}-350514X^{6}Y^{7}+405000X^{7}Y^{7}-327564X^{8}Y^{7}\\\ &+161109X^{9}Y^{7}-34992X^{10}Y^{7}+91Y^{8}-1263XY^{8}+8077X^{2}Y^{8}-31626X^{3}Y^{8}+86544X^{4}Y^{8}\\\ &-176904X^{5}Y^{8}+277182X^{6}Y^{8}-327564X^{7}Y^{8}+274833X^{8}Y^{8}-142155X^{9}Y^{8}+32805X^{10}Y^{8}\\\ &-51Y^{9}+630XY^{9}-3789X^{2}Y^{9}+14517X^{3}Y^{9}-39906X^{4}Y^{9}+82782X^{5}Y^{9}-132192X^{6}Y^{9}\\\ &+161109X^{7}Y^{9}-142155X^{8}Y^{9}+78732X^{9}Y^{9}-19683X^{10}Y^{9}+15Y^{10}-153XY^{10}+819X^{2}Y^{10}\\\ &-2970X^{3}Y^{10}+8100X^{4}Y^{10}-17010X^{5}Y^{10}+27702X^{6}Y^{10}-34992X^{7}Y^{10}+32805X^{8}Y^{10}\\\ &-19683X^{9}Y^{10}+5368X^{10}Y^{10})\end{aligned}$
Table 3. Modular equation $F_{p}(X,Y)=0$ for $w(\tau)$ of level $p=13$ $p$ | $F_{p}(X,Y)$
---|---
$13$ | $\begin{aligned} &(X^{13}-Y)(X-Y^{13})-13XY(-X+6X^{2}-23X^{3}+65X^{4}-144X^{5}+255X^{6}-363X^{7}+414X^{8}\\\ &-369X^{9}+243X^{10}-108X^{11}+26X^{12}-Y+13XY-76X^{2}Y+277X^{3}Y-735X^{4}Y+1530X^{5}Y\\\ &-2559X^{6}Y+3459X^{7}Y-3816X^{8}Y+3447X^{9}Y-2457X^{10}Y+1235X^{11}Y-324X^{12}Y+6Y^{2}\\\ &-76XY^{2}+420X^{2}Y^{2}-1398X^{3}Y^{2}+3307X^{4}Y^{2}-6111X^{5}Y^{2}+9126X^{6}Y^{2}-11187X^{7}Y^{2}\\\ &+12096X^{8}Y^{2}-12609X^{9}Y^{2}+11638X^{10}Y^{2}-7371X^{11}Y^{2}+2187X^{12}Y^{2}-23Y^{3}+277XY^{3}\\\ &-1398X^{2}Y^{3}+3973X^{3}Y^{3}-7302X^{4}Y^{3}+9297X^{5}Y^{3}-7731X^{6}Y^{3}+3258X^{7}Y^{3}-4860X^{8}Y^{3}\\\ &+21199X^{9}Y^{3}-37827X^{10}Y^{3}+31023X^{11}Y^{3}-9963X^{12}Y^{3}+65Y^{4}-735XY^{4}+3307X^{2}Y^{4}\\\ &-7302X^{3}Y^{4}+6195X^{4}Y^{4}+9981X^{5}Y^{4}-44883X^{6}Y^{4}+82404X^{7}Y^{4}-73664X^{8}Y^{4}\\\ &-14580X^{9}Y^{4}+108864X^{10}Y^{4}-103032X^{11}Y^{4}+33534X^{12}Y^{4}-144Y^{5}+1530XY^{5}\\\ &-6111X^{2}Y^{5}+9297X^{3}Y^{5}+9981X^{4}Y^{5}-77760X^{5}Y^{5}+193752X^{6}Y^{5}-297231X^{7}Y^{5}\\\ &+247212X^{8}Y^{5}+29322X^{9}Y^{5}-302049X^{10}Y^{5}+280179X^{11}Y^{5}-88209X^{12}Y^{5}+255Y^{6}\\\ &-2559XY^{6}+9126X^{2}Y^{6}-7731X^{3}Y^{6}-44883X^{4}Y^{6}+193752X^{5}Y^{6}-419532X^{6}Y^{6}\\\ &+581256X^{7}Y^{6}-403947X^{8}Y^{6}-208737X^{9}Y^{6}+739206X^{10}Y^{6}-621837X^{11}Y^{6}+185895X^{12}Y^{6}\\\ &-363Y^{7}+3459XY^{7}-11187X^{2}Y^{7}+3258X^{3}Y^{7}+82404X^{4}Y^{7}-297231X^{5}Y^{7}+581256X^{6}Y^{7}\\\ &-699840X^{7}Y^{7}+269487X^{8}Y^{7}+753057X^{9}Y^{7}-1484973X^{10}Y^{7}+1115370X^{11}Y^{7}\\\ &-314928X^{12}Y^{7}+414Y^{8}-3816XY^{8}+12096X^{2}Y^{8}-4860X^{3}Y^{8}-73664X^{4}Y^{8}+247212X^{5}Y^{8}\\\ &-403947X^{6}Y^{8}+269487X^{7}Y^{8}+501795X^{8}Y^{8}-1774386X^{9}Y^{8}+2410803X^{10}Y^{8}-1607445X^{11}Y^{8}\\\ &+426465X^{12}Y^{8}-369Y^{9}+3447XY^{9}-12609X^{2}Y^{9}+21199X^{3}Y^{9}-14580X^{4}Y^{9}+29322X^{5}Y^{9}\\\ &-208737X^{6}Y^{9}+753057X^{7}Y^{9}-1774386X^{8}Y^{9}+2896317X^{9}Y^{9}-3057426X^{10}Y^{9}\\\ &+1817397X^{11}Y^{9}-452709X^{12}Y^{9}+243Y^{10}-2457XY^{10}+11638X^{2}Y^{10}-37827X^{3}Y^{10}\\\ &+108864X^{4}Y^{10}-302049X^{5}Y^{10}+739206X^{6}Y^{10}-1484973X^{7}Y^{10}+2410803X^{8}Y^{10}\\\ &-3057426X^{9}Y^{10}+2755620X^{10}Y^{10}-1495908X^{11}Y^{10}+354294X^{12}Y^{10}-108Y^{11}+1235XY^{11}\\\ &-7371X^{2}Y^{11}+31023X^{3}Y^{11}-103032X^{4}Y^{11}+280179X^{5}Y^{11}-621837X^{6}Y^{11}+1115370X^{7}Y^{11}\\\ &-1607445X^{8}Y^{11}+1817397X^{9}Y^{11}-1495908X^{10}Y^{11}+767637X^{11}Y^{11}-177147X^{12}Y^{11}\\\ &+26Y^{12}-324XY^{12}+2187X^{2}Y^{12}-9963X^{3}Y^{12}+33534X^{4}Y^{12}-88209X^{5}Y^{12}+185895X^{6}Y^{12}\\\ &-314928X^{7}Y^{12}+426465X^{8}Y^{12}-452709X^{9}Y^{12}+354294X^{10}Y^{12}-177147X^{11}Y^{12}+40880X^{12}Y^{12})\end{aligned}$

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# Slow dynamics of phospholipid monolayers
at the air/water interface

Siyoung Q. Choi and Todd M. Squires

Department of Chemical Engineering ,
University of California, Santa Barbara, CA 93106, USA

###### Abstract

Phospholipid monolayers at the air-water interface serve as model systems for
various biological interfaces, e.g. lung surfactant layers and outer leaflets
of cell membranes. Although the dynamical (viscoelastic) properties of these
interfaces may play a key role in stability, dynamics and function, the
relatively weak rheological properties of most such monolayers have rendered
their study difficult or impossible. A novel technique to measure the
dynamical properties of fluid-fluid interfaces have developed accordingly. We
microfabricate micron-scale ferromagnetic disks, place them on fluid-fluid
interfaces, and use external electromagnets to exert torques upon them. By
measuring the rotation that results from a known external torque, we compute
the rotational drag, from which we deduce the rheological properties of the
interface. Notably, our apparatus enable direct interfacial visualization
while the probes are torqued.

In this fluid dynamics video, we directly visualize
dipalmitoylphosphatidylcholine(DPPC) monolayers at the air-water interface
while shearing. At about 9 mN/m, DPPC exhibits a liquid condensed(LC) phase
where liquid crystalline domains are compressed each other, and separated by
grain boundaries. Under weak oscillatory torque, the grain boundaries slip
past each other while larger shear strain forms a yield surface by deforming
and fracturing the domains. Shear banding, which is a clear evidence of yield
stress, is visualized during steady rotation. Remarkably slow relaxation time
was also found due to slow unwinding of the stretched domains.

Two videos are (low quality, high quaility)