# Localization and integrability breaking in weakly interacting Floquet
circuits

Dominik Hahn<EMAIL_ADDRESS>Max Planck Institute for the Physics of Complex
Systems, Nöthnitzer Straße 38, 01187-Dresden, Germany  Luis Colmenarez
<EMAIL_ADDRESS>Max Planck Institute for the Physics of
Complex Systems, Nöthnitzer Straße 38, 01187-Dresden, Germany  Institute for
Quantum Information, RWTH Aachen University, 52056 Aachen, Germany  Institute
for Theoretical Nanoelectronics (PGI-2), Forschungszentrum Jülich, 52428
Jülich, Germany

###### Abstract

We present a family of Floquet circuits that can interpolate between non-
interacting qubits, free propagation, generic interacting, and dual-unitary
dynamics. We identify the operator entanglement entropy of the two-qubit gate
as a good quantitative measure of the interaction strength. We test the
persistence of localization in the vicinity of the non-interacting point by
probing spectral statistics, decay of autocorrelators, and measuring
entanglement growth. The finite-size analysis suggests that the many-body
localized regime does not persist in the thermodynamic limit. Instead, our
results are compatible with an integrability-breaking phenomenon.

##  I Introduction

Thermalization of quantum many-body systems has been an active topic of
research in recent years Nandkishore and Huse (2015); Abanin  _et al._ (2019);
Ueda (2020). It is now known that generic interacting systems reach an
equilibrium state that can be described by a statistical mechanics ensemble
Deutsch (1991); Srednicki (1994); Rigol  _et al._ (2008). Such systems show
quantum chaotic dynamics, which encompasses several dynamical and static
traits Heller (2001); D’Alessio  _et al._ (2016). There are two non-generic
cases where the dynamics is not expected to be chaotic: (i) non-interacting
systems can show free propagation or get localized in the presence of
disorder, the latest being called Anderson localization Anderson (1958). (ii)
Interacting integrable systems whose extensive number of conserved quantities
constrained their dynamics, giving rise to a special type of thermalization
Vidmar and Rigol (2016). When integrability is perturbed in an extensive
manner, quantum chaos is expected to set in the long run and the system
thermalizes Brenes  _et al._ (2020); Rigol (2016); Essler and Fagotti (2016);
D’Alessio  _et al._ (2016); Rigol  _et al._ (2007). However, in recent years,
it was shown that strong disorder potentials may cause the break down of
thermalization, the so-called many-body localization (MBL) Basko  _et al._
(2006); Gornyi  _et al._ (2005); Oganesyan and Huse (2007). This is described
as an ergodicity-breaking transition at finite interaction strength and
disorder. In this regime, quasi-local conserved quantities Huse  _et al._
(2014); Serbyn  _et al._ (2013) are stabilized, leading to _emergent_
integrability out of an initially quantum chaotic system. There are lots of
evidence of MBL in systems of moderate size Schreiber  _et al._ (2015); Luitz
_et al._ (2015), but recently, the fate of MBL in the thermodynamic limit has
been put into question Šuntajs  _et al._ (2020); Sels and Polkovnikov (2023,
2021); Kiefer-Emmanouilidis  _et al._ (2021) and spark debates on the scaling
of the critical disorder and possible de-localization mechanisms Morningstar
_et al._ (2022); Sels (2022); Crowley and Chandran (2020); Sierant and
Zakrzewski (2022); Sierant  _et al._ (2020); Panda  _et al._ (2020); Crowley
and Chandran (2022).

An interesting question arises at the intersection of the dynamical regimes
exposed above: how stable are Anderson localized systems to small
interactions? On the one hand, any small interactions are expected to drive
the system towards thermal equilibrium, on the other hand, disorder may
overcome the interactions and the system becomes MBL. In Ref. Laflorencie  _et
al._ (2022) the authors studied a model where the MBL persists at small enough
interactions. The critical value of the localization length is the one
expected from the avalanche mechanism De Roeck and Imbrie (2017); Thiery  _et
al._ (2018). In Ref. Krajewski  _et al._ (2022) it is shown that small
interactions are not effectively perturbing the Anderson localized orbitals
when the disorder strength is large. In this work, we study the onset of chaos
in a maximally localized system subjected to small and medium-strength
interactions.

Most of the studies that address the strong disorder and weak interaction
limit focus on Hamiltonian systems so far. Here we present a family of Floquet
circuits that can interpolate between non-interacting qubits and strongly
interacting systems. In this work, we test the onset of quantum chaos close to
the trivial non-interacting point and study the fate of the MBL regime with
increasing system size in our model. Our results suggest that there exists no
MBL phase for our system in the thermodynamic limit. Instead, our numerics
suggest a finite-size crossover similar to integrability-breaking phenomena
Bulchandani  _et al._ (2022).

The structure of the paper is as follows: In Sec. II, we introduce our Floquet
model and the quantities we use to study the onset of quantum chaos. In Sec.
III, we present our results for various commonly studied quantities in the
field of thermalization, together with a finite-size scaling analysis. Finally
in Sec. IV we discuss the implications and perspectives of our work.

##  II Model and observables

Before we present our results, we provide technical details about our model
and different quantities to detect signatures of the MBL-regime.

###  II.1 Classification of two-qubit gates

Figure 1: Parametrization of different equivalence classes for two-qubit
gates: Each two-qubit gate is uniquely determined up to single-qubit unitaries
by three parameters ${0\leq c_{3}\leq c_{2}\leq c_{1}\leq 0.5}$ (purple
tetrahedron) that form the Weyl chamber. Colored points indicate common gates.
the dashed lines show the corresponding families of two-qubit gates defined in
Sec. II.3.

For any two-qubit gate $U_{2}$, there exist general single-qubit gates
$u_{1},\,u_{2},v_{1}$ and $v_{2}$ and a two-qubit gate of the form (with
$\bm{c}=(c_{1},c_{2},c_{3})$)

$G(\bm{c})=\exp\left[-i\dfrac{\pi}{2}\left(c_{1}\sigma^{x}\otimes\sigma^{x}+c_{2}\sigma^{y}\otimes\sigma^{y}+c_{3}\sigma^{z}\otimes\sigma^{z}\right)\right],$
(1)

such that it can be decomposed as Kraus and Cirac (2001)

$U_{2}=(u_{1}\otimes u_{2})G(v_{1}\otimes v_{2}),$ (2)

Here $\sigma^{\alpha}$ with $\alpha=x,y,z$ are the Pauli operators acting on
each qubit.

Restricting to the subset ${0.5\geq c_{1}\geq c_{2}\geq c_{3}\geq 0}$, this
allows for a classification of all two-qubit gates in terms of the vector
$\bm{c}$: Two gates which are described by the same vector $\bm{c}$ are
equivalent to each other apart from the multiplication with single-qubit gates
Balakrishnan and Sankaranarayanan (2011a, 2010).

The space enclosed by the independent set of $\mathbf{c}$ is known as the Weyl
chamber Balakrishnan and Sankaranarayanan (2009); Zhang  _et al._ (2003);
Makhlin (2002) and indicated by the purple tetrahedron in Fig. 1. There are a
few special points and lines Balakrishnan and Sankaranarayanan (2011a)(see
Fig. 1):

  1. (i)

$c_{1}=c_{2}=c_{3}=0$ corresponds to independent single-qubit rotations.

  2. (ii)

$c_{1}=c_{2}=c_{3}=0.5$ is the equivalence class of the SWAP gate up to single
qubit rotations.

  3. (iii)

$c_{1}=c_{2}=0.5$ and $c_{3}\geq 0$ denote the family of dual-unitary gates
(shown as an orange line in Fig. 1).

  4. (iv)

$c_{1}=0.5$ and $c_{2}=c_{3}=0$ is the CNOT gate up to single qubit rotations.

Since any two-qubit gate corresponds to a specific point of the Weyl chamber,
the entanglement properties of each gate Eq. (2) are uniquely determined by
the vector ${\bm{c}=(c_{1},c_{2},c_{3})}$ Balakrishnan and Sankaranarayanan
(2010, 2011b, 2011a).

###  II.2 Two-qubit gate entanglement

One way to characterize equivalence classes of two-qubit gates defined in Eq.
(2) is by their operator Schmidt decomposition Balakrishnan and
Sankaranarayanan (2011b):

$\displaystyle U_{2}=\sum_{l=1}^{4}\lambda_{l}(A_{l}\otimes B_{l}),$ (3)

where $A_{l}$ and $B_{l}$ are orthonormal operators for the corresponding
single-qubit spaces. This is analogous to the Schmidt decomposition of states.
The Schmidt coefficients $\lambda_{l}$ are normalized, i.e.
$\sum_{l=1}^{4}\lambda_{l}=1$. It turns out that two-qubit gates Eq. (2) that
are different only in single-qubit rotations will have the same set of Schmidt
coefficients Balakrishnan and Sankaranarayanan (2011b). Henceforth, the set
$(c_{1},c_{2},c_{3})$ uniquely determines any function of the Schmidt
coefficients. In the following, We use the second Renyi entropy Balakrishnan
and Sankaranarayanan (2011b, a) which allows us to define the operator
entanglement per gate as:

$\displaystyle s(\bm{c})=-\log\left(\sum_{l=1}^{4}\lambda^{2}_{l}\right).$ (4)

There are a few remarks with respect to the operator entanglement for some
regions in the Weyl chamber. First, the line $c_{1}=c_{2}=0.5$, $c_{3}<0.5$
has the largest possible $s(\bm{c})=2\log 2$ Balakrishnan and Sankaranarayanan
(2011b). They are the building block of so-called dual-unitary circuits
Bertini  _et al._ (2019a); Bertini and Piroli (2020); Akila  _et al._ (2016),
which exhibit the fastest scrambling in the family of random circuits Bensa
and Žnidarič (2021); Zhou and Harrow (2022). Recently, it was shown that after
perturbing the dual unitary point the gate operator entanglement plays a
crucial role in recovering the more generic quantum chaotic behavior Rampp
_et al._ (2023). This motivates our choice to study long-time many-body
properties - i.e., scrambling, chaos and localization - of quantum circuits as
a function of the operator entanglement of the two-qubit gate they consist of.

The operator entanglement at any point of the Weyl chamber is given by
Balakrishnan and Sankaranarayanan (2011b):

$\displaystyle s(\bm{c})=-\log(P(\bm{c})/32),$ (5)

with

$\displaystyle P(\bm{c})=14+4\cos(2\pi c_{1})+4\cos(2\pi c_{2})+4\cos(2\pi
c_{3})+$
$\displaystyle\cos(2\pi(c_{1}-c_{2}))+\cos(2\pi(c_{1}+c_{2}))+\cos(2\pi(c_{1}-c_{3}))+$
$\displaystyle\cos(2\pi(c_{1}+c_{3}))+\cos(2\pi(c_{2}+c_{3}))+\cos(2\pi(c_{2}-c_{3})).$
(6)

###  II.3 Floquet circuit model

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{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\hskip
8.5359pt{\left\uparrow\vbox to39.83368pt{}\right.\kern-1.2pt}t$ Figure 2:
Diagrammatic notation of the unitary defined in Eq. (7). The single-qubit
gates $u_{i}$ and $w_{i}$ introduce spatial disorder to the system, while the
interaction between neighboring qubit is mediated by the two-qubit gate
$G(\mathbf{c})$.

The characterization of two-qubit unitaries by means of the gate
$G(\mathbf{c})$ allows us to introduce the following family of Floquet
circuits:

$\displaystyle
U(\bm{c})=\prod_{i=0}^{L-1}w_{i}\prod_{k=0}^{L/2-2}G(\bm{c})_{2k+1,2k+2}\prod_{i=0}^{L-1}u_{i}\prod_{k=0}^{L/2-1}G(\bm{c})_{2k,2k+1}$
(7)

A diagrammatic representation of this circuit is shown in Fig. 2: The gates
$u_{i}$ and $w_{i}$ are single qubit rotations drawn from the Haar measure.
They act as a spatial disorder. The gate $G(\bm{c})$ is defined in equation
Eq. (1). All bonds have the same set of parameters
${\bm{c}=(c_{1},c_{2},c_{3})}$. The Floquet dynamics is given by repeatedly
applying $U$ Eq. (7), thus the Floquet circuit ensemble is determined by the
vector $\bm{c}$.

A few special cases of this model have been already studied in the literature:
The case of space-time dual unitary circuits $c_{1}=c_{2}=0.5$, $c_{3}\geq 0$
(see orange dashed line in Fig. 1) has been extensively studied Bertini  _et
al._ (2019b); Fisher  _et al._ (2023). These circuits are quantum chaotic
despite being exactly solvable, which makes them special in the study of
thermalization Piroli  _et al._ (2020); Bertini  _et al._ (2020a, b); Claeys
and Lamacraft (2021); Aravinda  _et al._ (2021); Bertini  _et al._ (2021,
2018); Borsi and Pozsgay (2022). For instance, they are shown to saturate
bounds on information scrambling Aravinda  _et al._ (2021); Bertini  _et al._
(2019c); Bertini and Piroli (2020); Zhou and Harrow (2022). A random version
of this circuit - different single qubit gates at each time step - was studied
in Ref. Bensa and Žnidarič (2021), the authors found that the fastest
scrambler quantum circuit, i.e., highest entanglement rate production, are
random circuits with $c_{1}=c_{2}=0.5$ and $c_{3}\geq 0$.

The vicinity of the non-interacting point $c_{1}=c_{2}=c_{3}=0$ is less
explored. A natural question is whether the system gets many-body localized
Abanin  _et al._ (2019); Alet and Laflorencie (2018) for small finite values
of the coefficients $(c_{1},c_{2},c_{3})$. In principle, there are multiple
possible choices for $\bm{c}$ that can be studied starting from the origin. We
focus on three lines:

  1. (i)

SWAP line: it is determined by $c_{1}=c_{2}=c_{3}$ with $0<c_{1}<0.5$ (see Fig
1). It interpolates between single qubit rotations ($c_{1}=0$) and the SWAP
gate ($c_{1}=0.5$).

  2. (ii)

CNOT line: denoted by $c_{2}=c_{3}=0$ and $0<c_{1}<0.5$, this line ends on a
CNOT gate-based circuit.

  3. (iii)

The imaginary SWAP (iSWAP) Foxen  _et al._ (2020) line given by $c_{3}=0$,
$c_{1}=c_{2}$ and $0<c_{1}<0.5$ that ends on the imaginary-SWAP gate
$c_{1}=c_{2}=0.5$.

In Ref. Garratt and Chalker (2021) the authors report MBL for small
coefficients on the SWAP line. Besides the three lines mentioned above, we
study a subset $\mathbf{c}=(c_{1},c_{2},c_{3})$ with
$c_{1},c_{2}\in(0.02,0.28)$ and $c_{3}\in(0.0,0.18)$ in order to test the
generality of the chosen lines and possible links between long-time dynamics
and two-qubit gate invariants.

###  II.4 Level statistics and eigenstate entanglement entropy

Figure 3: Gap ratio (upper panel) and eigenstate entanglement entropy (lower
panel) for system sizes $L=8,10,12,14,16$ and combination of values:
$c_{1},c_{2}=0.02,0.04,0.06,...,0.2,0.24,0.28$ and
$c_{3}=0,0.02,0.04,...,0.18$. SWAP line (black dashed) and CNOT line (grey
continuous) are also shown for comparison. Error bars are $68\%$ confidence
interval. Figure 4: Upper row (a,b,c): Gap ratio as function of the two-qubit-
gate operator entanglement entropy for system sizes $L=8,10,12,14,16$ (in
addition CNOT line has $L=18$ and SWAP line has $L=18,20$). Lower row (d,e,f):
Average entanglement entropy normalized by the Page value
$S_{\text{Page}}=0.5(L\log 2-1)$ for the same system sizes. Right column: CNOT
line denoted by $c_{1}\in[0.02,0.40]$, $c_{2}=c_{3}=0$. Middle column: SWAP
line denoted as $c_{1}=c_{2}=c_{3}$ with $c_{1}\in[0.02,0.40]$. Left column:
DU line denoted by $c_{2}=c_{1}$, $c_{3}=0$ and $c_{1}\in[0.02,0.28]$. Error
bars (too small for this scale) are $68\%$ confidence interval. Grey vertical
lines are the $s(\bm{c})$ for which dynamics is shown in Sec. III.2. In all
three cases, the operator entanglement entropy $s(\bm{c})$ indicates an MBL-
to-thermal crossover for both gap ratio and eigenstate entanglement entropy.

The long-time dynamics of Floquet circuits can be probed by the spectral
properties of the Floquet operator D’Alessio  _et al._ (2016). In particular,
we are interested in the eigenphases and eigenstates
$U|n\rangle=e^{i\theta_{n}}|n\rangle$. The gaps between consecutive
eigenphases are defined as $\delta_{i}=\theta_{i+1}-\theta_{i}$, the ratio
between two consecutive gaps is denoted as:

$\displaystyle
r_{i}=\min\left(\delta_{i+1},\delta_{i}\right)/\max\left(\delta_{i+1},\delta_{i}\right).$
(8)

The average gap ratio $r$ is known to serve as an order parameter for
ergodicity breaking transitions Pal and Huse (2010); Luitz  _et al._ (2015):
When the Floquet dynamics leads to thermalization, the mean gap ratio is
described by GUE random matrix ensemble $\overline{r}\approx 0.60$ Atas  _et
al._ (2013). In contrast, when the Floquet dynamics gets localized, i.e. MBL,
the gap ratio statistics is Poissonian such that $\overline{r}=2\log
2-1\approx 0.386$ Oganesyan and Huse (2007); D’Alessio  _et al._ (2016). If
our Floquet model undergoes a MBL transition it should show up in the behavior
of $r$ as a function of the Schmidt coefficients.

A second diagnostic for the transition is the structure of eigenstates of the
time evolution operator: We introduce the reduced density matrix over half of
the spin chain as $\rho_{A}=\operatorname{Tr}_{L/2}\left(|n\rangle\langle
n|\right)$. We probe the transition at the level of eigenstates using the
half-chain entanglement entropy

$S=-\operatorname{Tr}\rho_{A}\log\rho_{A}.$ (9)

In the thermal phase, the eigenstates are expected to be essentially random
vectors in Hilbert space; thus their entanglement entropy is proportional to
the chain length $S_{\text{Page}}=(L\log 2-1)/2$, the so-called Page value
Page (1993). On the other hand, in the localized phase the eigenstates only
exhibit short-range entanglement, resulting in an area law for the
entanglement entropy $S\approx const$ Nandkishore and Huse (2015). Hence the
average entanglement entropy $\overline{S}$ signals the MBL transition Khemani
_et al._ (2017); Luitz  _et al._ (2015); Yu  _et al._ (2016).

These quantities are obtained using exact diagonalization. For small system
sizes $L=8,10,12$ the whole eigenspectrum can be computed, while for larger
system sizes $L\geq 14$ polynomial filtered diagonalization Luitz (2021) is
used for extracting 100 eigenpairs of the Floquet unitary. All quantities are
averaged over 3000-6000 disorder realizations (except for $L=20$ where only
500-1000 realizations are used) and all available eigenstates.

###  II.5 Quench dynamics

Another direct way to detect localization in our system is using transport
properties. A common tool is the autocorrelator Long  _et al._ (2023);
Schreiber  _et al._ (2015); Luitz  _et al._ (2016); Lezama  _et al._ (2019,
2021):

$\displaystyle
C(t)=\dfrac{1}{2^{L}}\operatorname{Tr}\left[\hat{O}(t)\hat{O}\right]=\dfrac{1}{2^{L}}\operatorname{Tr}\left[(\hat{U}^{\dagger}(t)\hat{O}\hat{U}(t)\hat{O}\right].$
(10)

Here $O$ is a normalized observable with vanishing mean
($\operatorname{Tr}{\hat{O}}=0$, $\operatorname{Tr}{\hat{O}^{2}}=2^{L}$), and
the time evolution is generated by the circuit introduced in Eq. (7), i.e.
$\hat{U}(t)=U(\bm{c})^{t}$.

Since our goal is to probe scrambling caused by the entangling gates, we
choose an operator $\hat{O}$ such that $C(t)=1$ for $c_{1}=c_{2}=c_{3}=0$. To
achieve this, consider the product of the single-site operators at $i=\lfloor
L/2\rfloor$:

$\displaystyle R=u_{i}w_{i}$ (11)

Since $R$ is unitary, there exists a diagonal matrix $D$ and an unitary $V$
such that

$\displaystyle R=V^{\dagger}DV.$ (12)

By choosing

$\displaystyle\hat{O}=V^{\dagger}\sigma^{z}_{i}V$ (13)

we obtain an autocorrelator $C(t)$ with the desired properties. It is
important to note that the choice of $V$ depends on the specific disorder
configuration. When the system thermalizes, the disorder-averaged
autocorrelator vanishes in the long-time limit Nandkishore and Huse (2015). In
contrast, $C(t)$ is expected to converge to a non-zero value in the MBL regime
Nandkishore and Huse (2015). In summary, we obtain

$\displaystyle\lim_{t\rightarrow\infty}C(t)=\begin{cases}0,&\text{thermalization}\\\
c>0,&\text{localization}.\\\ \end{cases}$ (14)

Another way to probe transport properties in the study of MBL is the
entanglement entropy production starting from a product state
$|\psi(0)\rangle=|\psi_{0}\rangle$ Žnidarič  _et al._ (2008); Kjäll  _et al._
(2014). Analog to Sec. II.4, we compute the half-chain entanglement entropy
$S(t)$ [cf. Eq.(9)], but now for the time-evolved state $|\psi(t)\rangle$
instead. The entanglement growth rate depends on the overall dynamics: quantum
chaotic systems show linear growth in time, while MBL systems exhibit
logarithmic growth of entanglement Žnidarič  _et al._ (2008); Bardarson  _et
al._ (2012). The latest signals the existence of quasi-local integrals of
motion Serbyn  _et al._ (2013); Huse  _et al._ (2014).

##  III Results

###  III.1 Gap ratio and eigenstate entanglement entropy

Figure 5: Crossings of the curves in Fig. 4 between system size $L$ and $L+2$
for the gap ratio (a) and eigenstate entanglement entropy (b) along the CNOT
line (orange), SWAP line (blue), iSWAP line (red). The black dashed lines show
the scaling $1/L$. For accessible system sizes the trend is compatible within
error bars with $s(\bm{c})_{c}\propto 1/L$, suggesting restoring of ergodicity
in the thermodynamic limit for any finite interaction strength. Figure 6: (a)
Gap ratio and (b) entanglement entropy along the SWAP line for system size
$L=8-20$. The two-qubit gate operator entanglement $s(\bm{c})$ is re-scaled by
$L\log L$. The data set is the same as shown in Fig. 4. These visible
crossings suggest a drifting critical value $s(\bm{c})_{crit}\propto 1/L$,
compatible with an integrability-breaking phenomenon (see main text).

As a first check, we probe the operator entanglement per gate $s(\bm{c})$ as a
unifying parameter for the interaction strength. To do so, we show the gap
ratio and the eigenstate entanglement entropy as a function of the gate
entanglement entropy $s(\bm{c})$ in Fig. 3 (a) and (b), respectively. We focus
on parameters $\bm{c}(c_{1},c_{2},c_{3})$ within the range
$c_{1},c_{2}=[0.02,0.28]$ and $c_{3}=[0.0,0.18]$. For the system size and
$s(\bm{c})$ fixed, the gap ratio and half-chain entanglement almost collapse
on top of the corresponding SWAP line value. This supports our motivation to
choose $s(\bm{c})$ as an indicator for the interaction strength.

However, the results for the CNOT and the SWAP line lie not directly on top of
each other. This difference may originate from the specific choice of the
vector $\bm{c}$. On the CNOT line, two Schmidt coefficients $\lambda_{l}$ [cf.
Eq. (8)] are zero Balakrishnan and Sankaranarayanan (2011b), which is not the
case for any other choice of $\bm{c}$ in the Weyl chamber. Therefore the
finite size behavior visible in $r$ and $S/S_{\text{Page}}$ may be affected by
this choice of $\bm{c}$ (see Appendix A for a more quantitative comparison).

For our further studies, we choose three parameter lines that differ in the
number of non-vanishing coefficients $c_{i}$: the SWAP line, the iSWAP line,
and the CNOT line (cf Sec. II.3). The gap ratio and the half-chain
entanglement entropy on these lines of the Weyl chamber and systems sizes are
shown in Fig. 4. In all three cases, we see a crossover between an MBL regime
(indicated by a small eigenstate entanglement entropy and an $r$-value close
to the Poissonian case) for small interactions towards a thermal regime at
large interactions. Besides, an analysis of the entanglement entropy
fluctuations is shown in Appendix Sec. B.

Furthermore, the curves of different system sizes intersect. We then compare
the different crossings between consecutive system sizes in order to check the
stability of the phase in the thermodynamic limit Pal and Huse (2010). In Fig.
5, the crossings of gap ratio and entanglement entropy for consecutive sizes
$L$ and $L+2$ are shown. The trend of the crossing suggests a scaling $\propto
1/L$, at least for the accessible system sizes. Remarkably, finite size
effects for the half-chain entanglement entropy are less pronounced in
comparison to the r-value. The trend of the data suggests that the crossings
are shifting towards zero in the limit $L\rightarrow\infty$, thus ergodicity
is restored at any finite interaction. However, given the smallness of the
accessible system sizes, we can not rule out a change in the trend at larger
sizes.

It has been recently shown that non-interacting spin systems with small
perturbative interaction undergo a Fock space type delocalization “transition”
that marks the onset of quantum chaos Bulchandani  _et al._ (2022). Taking
$\epsilon$ as the integrability breaking parameter, the value $\epsilon_{c}$
denotes the onset of chaos scaling as $\epsilon_{c}\sim(L\log L)^{-1}$ with
increasing system size. In Fig. 6 we test such scaling for $s(\bm{c})$ for
both gap ratio and eigenstate entanglement entropy. Although the available
system sizes do not allow us to discern the $\log L$ component, the linear
scaling is clearly visible. From this perspective, the MBL-thermalization
crossover appears as an ordinary integrability-breaking phenomenon for finite
system sizes.

###  III.2 Single spin autocorrelation and entanglement entropy after a quench

Figure 7: Single spin autocorrelation $\overline{C}(t)$ as a function of time
$t$, gates $G(\bm{c})$ are chosen on the SWAP line for
$c=(0.05,0.06,0.07,0.08)$ (the corresponding interaction strength $s(\bm{c})$
is shown in each panel). Disorder average is taken over 1000-4000 disorder
realizations. Errors bars are $68\%$ confidence interval. Figure 8: Half chain
entanglement entropy after a quench on the SWAP line for
$c=(0.05,0.06,0.07,0.08)$ (same values but translated to $opEE$ are shown in
each panel) and system sizes $L=18,20,22$. Disorder average is taken over
1000-4000 disorder realizations, error bars denote $68\%$ confidence interval.
The black dashed lines are $\log t$ with different offsets. For large
interactions $S(t)$ the logarithmic growth is proceeded by a faster
entanglement entropy growth. Figure 9: Numerical derivative of the half chain
entanglement entropy with respect to $\log t$ after a quench on the SWAP line
for $c=(0.05,0.06,0.07,0.08)$ (same data is shown in Fig. 8) and system sizes
$L=18,20,22$. The entropy growth is faster than pure logarithmic for all
interaction strengths.

From the previous section, we can conclude that the critical operator
entanglement per gate $s(\bm{c})$ is scaling roughly as $1/L$. The largest
system size for which we could extract eigenvalues and eigenvectors is $L=20$.
From Fig. 5, we estimate the crossover region for current system sizes to be
around $s(\bm{c})\sim 1/20=0.05$. In this section, we explore signatures for
this crossover in quench dynamics for up to $10^{5}$ cycles in the regime
$s(\bm{c})\lesssim 0.1$ and system sizes $L\geq 18$.

In Fig. 7, we show the autocorrelation function introduced in Sec. II.5 on the
SWAP line of the model. The circuit dynamics is simulated using Cirq
Developers (2023) that allows to reach $10^{5}$ cycles and up to $L=22$
qubits.

For $s(\bm{c})\approx 0.09,0.07$ close to the crossover region,
$\overline{C}(t)$ decays with a scale either logarithmically or stretched
exponentially, in line with previous work on autocorrelation decay in
prethermal systems Long  _et al._ (2023). The long-time limit
$\lim_{t\rightarrow\infty}C(t)$ decreases with system size, suggesting a trend
towards thermalization. For smaller interactions $s(\bm{c})\approx 0.05,0.03$,
our accessible time-scales are too short to draw conclusions about a drift in
the long-time dynamics: we do not reach a steady state in our numerics.

Finally, we study the entanglement entropy growth for initial product states
along the SWAP line. As discussed in Sec. II.5, a signature of localization is
logarithmic entanglement growth. As is visible in Fig. 8, the entanglement
entropy $S(t)$ is growing faster than logarithmically (black dashed lines) at
time scales $t\gtrsim 10^{3}$ and interactions $s(\bm{c})\gtrsim 0.07$. In
order to confirm this observation, we compute the derivative of $S(t)$ with
respect to $\log t$ (see Fig. 9). A logarithmic curve would be visible as a
constant value. Instead, we see that the derivative keeps growing with system
size even in the regime when the level statistics is Poissonian $s(c)=0.0369$.
We conclude that the entropy growth is faster than pure logarithmic growth
even for the smallest interactions. The latest is in odds with the steady
logarithmic growth in MBL regimes Bardarson  _et al._ (2012).

##  IV Discussion

In this work, we have introduced a generic Floquet circuit model that allows
us to parametrically tune the interaction and keep the disorder maximal. We
have identified the gate entanglement entropy $s(\bm{c})$ as a quantitative
measure for the interaction. In the limit $s(\bm{c})=0$, our model reduces to
a non-interacting system. Our results for various quantities suggest that the
observed MBL regime for small interactions does not persist in the
thermodynamic limit. Moreover, by re-scaling our results with the gate
entanglement $s(\bm{c})$, we conclude that this crossover at finite system
sizes shows a striking similarity to an integrability-breaking transition
Bulchandani  _et al._ (2022).

Our reachable system sizes and our investigated model are not sufficient to
draw conclusions about the general fate of the MBL transition in the
thermodynamic limit. Nevertheless, they suggest analyzing whether the results
for other commonly studied models in the field of many-body localization are
in alignment with an integrability-breaking transition Krajewski  _et al._
(2022, 2023). Apart from that, our model contains dual unitary circuits as
another special case for a specific choice of parameters $\bm{c}$. This model
is thus a good starting point to study the effects of breaking dual-unitarity
in more detail Rampp  _et al._ (2023).

##  V Acknowledgments

We thank Anushya Chandran, Pieter Claeys, David Long, David Luitz and Michael
Rampp for inspiring discussions. L.C. gratefully acknowledges funding by the
U.S. ARO Grant No. W911NF-21-1-0007. All statements of fact, opinion or
conclusions contained herein are those of the authors and should not be
construed as representing the official views or policies of the US Government.

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##  Appendix A Gate operator entanglement, SWAP and CNOT lines

Figure 10: Distance of (a) the gap ratio $r(\bm{c},L)$ and (b) entanglement
entropy $S(\bm{c},L)$ to the reference values $r(\bm{c}_{S},L)$ and
$S(\bm{c}_{S},L)$ on the SWAP line. Dashed lines are error bars of
$r(\bm{c}_{S},L)$ and $S(\bm{c}_{S},L)$. The SWAP reference points are
computed by using spline interpolation of the curves shown in Fig. 4. The
points are the same data as shown in Fig. 3.

Throughout this work, we assume that the results for the SWAP, iSWAP and CNOT
line can be extrapolated to any other set of $\bm{c}=(c_{1},c_{2},c_{3})$ via
the equivalence through the gate operator entanglement $s(\bm{c})$. Inspired
by Ref. Rampp  _et al._ (2023), where the authors found that $s(\bm{c})$ plays
a leading role in generic thermalization, we aim to apply these ideas to the
weakly interacting regime. In this section, we provide quantitative tests for
this assumption by computing the gap ratio and eigenstate operator
entanglement at any point $(c_{1},c_{2},c_{3})$, namely $r(\bm{c},L)$ and
$S(\bm{c},L)$, and the difference

$\displaystyle d_{r}(\bm{c})$ $\displaystyle=|r(\bm{c},L)-r(\bm{c}_{S},L)|$
(15) $\displaystyle d_{S}(\bm{c})$
$\displaystyle=|S(\bm{c},L)-S(\bm{c}_{S},L)|/S_{Page},$ (16)

where $r(\bm{c}_{S},L)$ and $S(\bm{c}_{S},L)$ are the gap ratio and eigenstate
entanglement entropy for the reference choice
${\bm{c}_{S}=(c_{x},c_{x},c_{x})}$ on the SWAP line and the same gate operator
entanglement entropy $s(\bm{c})=s(\bm{c}_{S})$. Our results are shown in Fig.
10. As a reference, we compare these results with the effects of the
statistical error of $r(\bm{c}_{S},L)$ and $S(\bm{c}_{S},L)$ due to the
disorder average, indicated by dashed lines. When a point $d_{r}(\bm{c})$ or
$d_{S}(\bm{c})$ is below the corresponding error bar line, then the difference
of the results for the parameter $\bm{c}$ and the reference point $\bm{c}_{S}$
are the same within statistical errors. Visually, there is a large fraction of
$d_{r}(\bm{c})$ or $d_{S}(\bm{c})$ above the error bar line for all system
sizes. However, the difference in absolute value is small enough, such that
extrapolating the results from the SWAP, iSWAP and CNOT line to other values
of $\bm{c}$ is a fair assumption for the current setup and system sizes.

##  Appendix B Entanglement entropy fluctuations

Figure 11: Entanglement entropy fluctuations
$\sigma_{S}^{2}=\overline{s^{2}}-\overline{s}^{2}$ with $s=S/S_{\text{Page}}$
as function of the two-qubit gate operator entanglement. Right column: CNOT
line denoted by $c_{1}\in[0.02,0.40]$. Middle column: SWAP line with
$c_{1}\in[0.02,0.40]$. Left column: iSWAP line with $c_{1}\in[0.02,0.28]$.
Error bars (too small for this scale) are $68\%$ confidence interval. The data
set is the same shown in Fig.4 for $S/S_{\text{Page}}$.

It is known that fluctuations around the mean eigenstate entanglement entropy
$\overline{S}$, probed by $\sigma_{S}^{2}=\overline{S^{2}}-\overline{S}^{2}$,
peak at the transition point Luitz  _et al._ (2015). Therefore tracking
fluctuation’s peaks also reveals the possible location of the transition. Such
fluctuations are shown in Fig. 11. We see that the peak of the fluctuations is
moving towards smaller $s(\bm{c})$, as has been reported in other models where
MBL might be stable Yu  _et al._ (2016). Importantly, the entanglement entropy
fluctuations also seem to be sensitive only to the operator entanglement
$s(\bm{c})$ of the gate rather than the specific choice $(c_{1},c_{2},c_{3})$
in the Weyl chamber.


# From Kepler’s Laws to Newtonian Motion and the Direction Angle of Hamilton’s
Hodograph

Dr. Klaus Huber
Huber Consult
Sesenheimer Str. 21
10627 Berlin
Germany
email<EMAIL_ADDRESS>
###### Abstract

In this contribution it is shown that the path from Kepler’s results to
Newtonian motion can be remarkably short and simple. Following this path we
also give a straight forward computation of the direction angle of Hamilton’s
Hodograph. Then we show how the speed as function of the direction angle can
be expressed and inverted elegantly using elliptic functions.

Index Terms —  Kepler’s Laws, Newtonian Motion, Inverse Square Law, Hamilton’s
Hodograph, Gauss transform.

##  1 Introduction

Usually Newtonian motion of planets under an inverse square law force is
established using Kepler’s second law. Asuming at the outset that the inverse
square law rules elliptic motion results (or hyperbolic and parabolic motion).

Here we use Kepler’s first and second law to find the inverse square law. This
procedure also gives the opportunity to find a simple formula for the
direction angle of Hamilton’s Hodograph.

##  2 Kepler’s laws

We start ab ovo with Kepler’s three laws:

  * i.

A planet moves on an ellipse around the sun with the sun placed in a focus of
the ellipse.

  * ii.

The straight line from the sun to a planet sweeps out equal areas in equal
times.

  * iii.

The time period $T$ for a revolution and the major semiaxis $a$ of the ellipse
behave as $T^{2}=\mbox{const}\cdot a^{3}$.

##  3 The shortest path from Kepler’s laws to Newtonian motion

All motion happens in a plane. Without loss of generality we consider an
ellipse with reference to perihel (i.e. the position where the distance of the
planet is closest to the sun). The radius is then given by

$r=\frac{a(1-e^{2})}{1+e\cos\theta}\mbox{ . }$

The above parameters are: the major semiaxis $a$, the excentricity of the
ellipse $e$, and the angle $\theta$ called true anomaly in astronomy. It is
convenient to use complex numbers with $\imath=\sqrt{-1}$. The position in the
complex plane is given by

$Z=\frac{a(1-e^{2})}{1+e\cos\theta}\,\exp(\imath\theta)\mbox{ . }$

The derivative with respect to time $t$, denoted by a dot, then immediately
gives the velocity

$\dot{Z}=\frac{a(1-e^{2})\dot{\theta}}{(1+e\cos\theta)^{2}}\left\\{\imath
e+\imath\exp(\imath\theta)\right\\}\mbox{ . }$

From Kepler’s second law we find $\frac{1}{2}r^{2}\dot{\theta}=\frac{\pi
ab}{T}$ where $\pi ab$ is the area of an ellipse with major- and minor
semiaxes $a$ and $b$ and $T$ is the time for a complete revolution. As
$b=a\sqrt{1-e^{2}}$ we get
$\dot{\theta}=\frac{2\pi}{T(1-e^{2})^{\frac{3}{2}}}(1+e\cos\theta)^{2}$ and

$\dot{Z}=\frac{2\pi a}{T}\frac{1}{\sqrt{1-e^{2}}}\left\\{\imath
e+\imath\exp(\imath\theta)\right\\}\mbox{ . }$ (1)

We easily see that $\dot{Z}$ decribes a circle in the complex plane. Taking
the second derivative yields the acceleration

$\ddot{Z}=-\frac{4\pi^{2}a^{3}}{T^{2}}\frac{(1+e\cos\theta)^{2}}{a^{2}(1-e^{2})^{2}}\,\exp(\imath\theta)=-\frac{4\pi^{2}a^{3}}{T^{2}}\frac{1}{r^{2}}\,\exp(\imath\theta)\mbox{
, }$

which, setting $\frac{4\pi^{2}a^{3}}{T^{2}}=G$, leads to the inverse square
law force for an object of mass $m$:

$m\ddot{Z}=-\frac{mG}{r^{2}}\,\exp(\imath\theta)\mbox{ . }$

Note that Kepler’s third law $\frac{4\pi^{2}a^{3}}{T^{2}}=G$ is obtained
automatically.

##  4 Hamilton’s Hodograph

Let $\bf{r}$ be the location vector for the orbit of a planet (see figure 1).
Then $\bf{v}=$ $\frac{d}{dt}\bf{r}$ gives the velocity vector, i.e. the vector
whose length $v=|\bf{v}|$ gives the speed. If we consider all vectors $\bf{v}$
on the curve which $\bf{r}$ reaches and if we translate all vectors to the
same point (e.g. to the focus for Newtonian motion) the endpoints of the
velocity vectors give rise to a curve which is called the Hodograph.

Figure 1:  Elliptic orbit having $e=0.6$$\bf{r}$$\theta$$\bf{v}$

Hamilton introduced the Hodograph in [2] and found out that for Newtonian
motion according to an inverse square law force the hodograph is a circle. A
look at the equation for $\dot{Z}$ immediately shows that it describes the
hodograph. The only task left is to express $\dot{Z}$ using the angle
$\vartheta_{H}$ which goes from the perihel to the vector $\bf{v}$, the
subscript $H$ stands for Hamilton. This is displayed in fig. 2.

Figure 2:  Hodograph $\dot{Z}=v\cdot\exp(\imath\,\vartheta_{H})$ for
$e=0.6$$\theta$$\vartheta_{H}$$v$

For convenience we use the angle $\vartheta=\vartheta_{H}-\frac{\pi}{2}$
instead of $\vartheta_{H}$ and find 111Here and at subsequent places the
reader should have no problems to resolve any ambiguities when applying
inverse functions like the $\arctan$-function.

$\tan\vartheta=\frac{\sin\theta}{e+\cos\theta}\mbox{ . }$ (2)

To find $\theta$ as function of $\vartheta$ we express $\dot{Z}$ using
$\vartheta$ which gives

$\dot{Z}=\frac{2\pi
a}{T}\frac{1}{\sqrt{1-e^{2}}}\left(\sqrt{1-e^{2}\sin^{2}(\vartheta)}+e\cos\vartheta\right)\cdot\imath\,\exp(\imath\vartheta)\mbox{
. }$ (3)

Equating with (1) and separating real- and imaginary parts gives

$\displaystyle\sin\theta$ $\displaystyle=$
$\displaystyle(\sqrt{1-e^{2}\sin^{2}(\vartheta)}+e\cos\vartheta)\sin\vartheta$
$\displaystyle e+\cos\theta$ $\displaystyle=$
$\displaystyle(\sqrt{1-e^{2}\sin^{2}(\vartheta)}+e\cos\vartheta)\cos\vartheta$

which yields

$\tan\theta=\frac{\sin\vartheta\,(\sqrt{1-e^{2}\sin^{2}\vartheta}+e\cos\vartheta)}{\cos\vartheta\,(\sqrt{1-e^{2}\sin^{2}\vartheta}+e\cos\vartheta)-e}\mbox{
. }$ (4)

Clearly the speed is given by

$v=\frac{2\pi
a}{T}\frac{1}{\sqrt{1-e^{2}}}\left(\sqrt{1-e^{2}\sin^{2}(\vartheta)}+e\cos\vartheta\right)\mbox{
. }$ (5)

##  5 Computing $\vartheta$ from $v$

The speed $v$ as function of $\vartheta$ is given in (5). We now invert this
function, i.e. we solve for $\vartheta$. This can be done directly using
elliptic functions 222Some basic knowledge of Elliptic functions is assumed,
see e.g. the book of Whittacker and Watson [3]. For the definition of the
functions sn, cn, and dn see appendix A.. We set $\vartheta=\rm{am}(u,k)$,
where $\rm{am}(u,k)$ is Jacobi’s amplitude function, which has as input $u$
and the modulus $k$ with $k=e$ and get

$v=\frac{2\pi
a}{T}\frac{1}{\sqrt{1-e^{2}}}\,\left(\rm{dn}(u,e)+e\cdot\rm{cn}(u,e)\right)\mbox{
. }$

In appendix A it is shown that this can be expressed using the
$\rm{dn}$-function alone to give

$v=\frac{2\pi
a}{T}\sqrt{\frac{1+e}{1-e}}\cdot\rm{dn}\left(\frac{1+e}{2}u,\frac{2\sqrt{e}}{1+e}\right)\mbox{
. }$ (6)

Hence we immediately can express $\vartheta$ as function of $v$.

$\vartheta=\rm{am}\left(\frac{\rm{dn}^{-1}(\frac{v}{2\pi
a/T}\sqrt{\frac{1-e}{1+e}},\frac{2\sqrt{e}}{1+e})}{(1+e)/2},e\right)\mbox{ .
}$ (7)

Note that all involved functions and inverse functions can be easily and
efficiently computed (in addition they are included in all relevant computer
algebra programs).

##  6 The angle between $\bf{r}$ and $\bf{v}$

Having determined $\vartheta_{H}$ (or $\vartheta$) we immediately get the
angle $\varphi$ between $\bf{r}$ and $\bf{v}$ as

$\varphi=\vartheta_{H}-\theta=\frac{\pi}{2}+\vartheta-\theta\hskip
8.53581pt\Rightarrow\hskip
8.53581pt\varphi=\frac{\pi}{2}+\arctan\left(\frac{\sin\theta}{e+\cos\theta}\right)-\theta\mbox{
. }$ (8)

The angle $\varphi$ is displayed as function of $\theta$ in figure 3 for
$e=0.6$. Clearly, at perihel (at $\theta=0$) and aphel (at $\theta=\pi$) this
angle equals $\frac{\pi}{2}$. For $0<\theta<\pi$ the angle $\varphi$ is
smaller than $\pi/2$ and in the interval $\pi<\theta<2\pi$ the angle $\varphi$
is greater than $\pi/2$. It is a simple matter to compute the angle $\theta$
at which $\varphi$ reaches its extrema. We get the minimum of $\varphi$ at
$\theta=\arccos(-e)$ and the maximum at $\theta=2\pi-\arccos(-e)$. Thus the
smallest $\varphi$ equals $\varphi_{\rm{min}}=\pi-\arccos(-e)=\arccos(e)$ and
the highest $\varphi$ equals $\varphi_{\rm{max}}=\arccos(-e)$. From the
analysis of (8), for the interval $0\leq\theta<2\pi$ of figure 3, the
ambiguity of the $\arctan$-function in equation (8) is resolved as follows.

$\varphi=\left\\{\begin{array}[]{ll}\frac{\pi}{2}+\arctan\left(\frac{\sin\theta}{e+\cos\theta}\right)-\theta&\mbox{
for $0\leq\theta<\arccos(-e)$}\\\
\frac{3\pi}{2}+\arctan\left(\frac{\sin\theta}{e+\cos\theta}\right)-\theta&\mbox{
for $\arccos(-e)\leq\theta<2\pi-\arccos(-e)$}\\\
\frac{5\pi}{2}+\arctan\left(\frac{\sin\theta}{e+\cos\theta}\right)-\theta&\mbox{
for $2\pi-\arccos(-e)\leq\theta<2\pi$.}\end{array}\right.$

Figure 3:  Angle $\varphi$ between $\bf{r}$ and $\bf{v}$ as function of
$\theta$ (for
$e=0.6$)$2\pi$$\arccos(-e)$$2\pi\\!\\!-\\!\\!\arccos(-e)$$\arccos(e)$$\arccos(-e)$$\frac{\pi}{2}$$\theta$$\varphi$

##  7 Conclusion

Starting with Kepler’s first and second laws it is essentially a three line
derivation to obtain Newton’s gravitational law. In addition, from the
computation one essentially gets Hamilton’s hodograph as side effect. The
computation of the speed as function of the angle $\vartheta$ is a straight-
forward task as well as the dependency of $\vartheta$ from the true anomaly
$\theta$ (and its inversion). Using elliptic functions one can also express
elegantly the speed and invert it, i.e. express $\vartheta$ as function of the
speed. The new expression of the speed (eqn.(6)) is essentially an application
of the Gauss transform 333The Gauss transform is the inverse of the Landen
transform. to the elliptic function $\rm{dn}(u,k)+k\cdot\rm{cn}(u,k)$. The
Gauss transform changes the modulus $k$ of the elliptic function to
$2\sqrt{k}/(1+k)$.

##  Appendix A Elliptic Functions used and Derivation of an Identity

We use the standard notation of Gudermann (see [3], p.494):

$\displaystyle\sin\rm{am}(u,k)$ $\displaystyle=$ $\displaystyle\rm{sn}(u,k)$
$\displaystyle\cos\rm{am}(u,k)$ $\displaystyle=$ $\displaystyle\rm{cn}(u,k)$
$\displaystyle\sqrt{1-k^{2}\,\rm{sn}^{2}(u,k)}$ $\displaystyle=$
$\displaystyle\rm{dn}(u,k)\mbox{ . }$

We now derive a formula which leads to equation (6). Using the transformation
formulas 8.152 from [1], p.915 (the 7-th row) – where
$k^{\prime}=\sqrt{1-k^{2}}$, $k_{1}=\frac{1-k^{\prime}}{1+k^{\prime}}$, and
$u_{1}=(1+k^{\prime})u$ – we find

$\rm{dn}(u_{1},k_{1})+k_{1}\,\rm{cn(u_{1},k_{1})}=\frac{1-(1-k^{\prime})\rm{sn}^{2}(u,k)}{dn(u,k)}+\frac{1-k^{\prime}}{1+k^{\prime}}\,\frac{1-(1+k^{\prime})\rm{sn}^{2}(u,k)}{dn(u,k)}$

which leads to

$\rm{dn}(u_{1},k_{1})+k_{1}\,\rm{cn(u_{1},k_{1})}=\frac{2}{1+k^{\prime}}\,\rm{dn}(u,k)$

If we set $k_{1}=e$ it follows that $k^{\prime}=\frac{1-e}{1+e}$,
$k=\frac{2\sqrt{e}}{1+e}$, and $\frac{2}{1+k^{\prime}}=1+e$ hence

$\rm{dn}(u_{1},e)+e\,\rm{cn(u_{1},e)}=(1+e)\rm{dn}\left(\frac{1+e}{2}u_{1},\frac{2\sqrt{e}}{1+e}\right)$

To avoid collision with the different use of $u$ above we replace $u_{1}$ by
$z$ and get

$\frac{\rm{dn}(z,e)+e\,\rm{cn(z,e)}}{\sqrt{1-e^{2}}}=\sqrt{\frac{1+e}{1-e}}\,\rm{dn}\left(\frac{1+e}{2}z,\frac{2\sqrt{e}}{1+e}\right)\mbox{
. }$

## References

  * [1] I.S.Gradsteyn, I.M.Ryzhik, ”Tables of Integrals, Series, and Products”, Academic Press, Sixth Printing, 1979.
  * [2] W.R.Hamilton, ”New mode of geometrically conceiving, and of expressing in symbolical language, the Newtonian law of attraction, and the mathematical problem of determining the orbits and perturbations of bodies which are governed in their motions by that law”, Proceedings of the Royal Irish Academy, Vol. III, 1847, pp. 344-353.
  * [3] E.T.Whittaker, G.N.Watson, ”A course of modern analysis”, fourth edition, Cambridge, reprinted 1969.