# Quantum dynamics of frustrated Josephson junction arrays embedded in a
transmission line: an effective $XX$ spin chain with long-range interaction

Benedikt J.P. Pernack<EMAIL_ADDRESS>Mikhail V. Fistul
Ilya M. Eremin  Theoretische Physik III, Ruhr-Universität Bochum, Bochum
44801, Germany

###### Abstract

We study theoretically a variety of collective quantum phases occurring in
frustrated saw-tooth chains of Josephson junctions embedded in a
dissipationless transmission line. The basic element of a system, i.e., the
triangular superconducting cell, contains two $0$\- and one $\pi$\- Josephson
junctions characterized by $E_{J}$ and $\alpha E_{J}$ Josephson energies,
accordingly. In the frustrated regime the low energy quantum dynamics of a
single cell is determined by anticlockwise or clockwise flowing persistent
currents (vortex/antivortex). The direct embedding of $\pi$-Josephson
junctions in a transmission line allows to establish a short/long-range
interaction between (anti)vortices of well separated cells. By making use of
the variational approach, we map the superconducting circuit Hamiltonian to an
effective $XX$ spin model with an exchange spin-spin interaction decaying with
the distance $x$ as $x^{-\beta}$, and the local $\hat{\sigma}_{x,n}$-terms
corresponding to the coherent quantum beats between vortex and antivortex in a
single cell. We obtain that in long arrays as
$N\gg\ell_{0}\simeq\sqrt{C/C_{0}}$, where $C$ and $C_{0}$ are capacitances of
$0$-Josephson junction and transmission line, accordingly, the amplitude of
quantum beats is strongly suppressed. By means of exact numerical
diagonalization, we study the interplay between the coherent quantum beats and
the exchange spin-spin interaction leading to the appearance of various
collective quantum phases such as the paramagnetic ($P$), compressible
superfluid ($CS$) and weakly compressible superfluid ($w$-$CS$) states.

††preprint: APS/123-QED

##  I Introduction

Different kinds of frustrated systems and their novel physics have motivated
many experiments and theoretical studies in recent times. Initially, the
concept of frustration was introduced in magnetic materials, in which either
geometric frustration or frustration emerging due to competing interactions,
e.g., ferromagnetic/anti-ferromagnetic interactions, occur [1, 2, 3, 4]. The
unique properties of frustrated systems are a highly degenerated ground state,
multiple low-lying metastable states with long relaxation times at low
temperatures [2, 4, 5].

In order to give a short flavour of the richness of the collective phases and
phase transitions realized in frustrated systems, we refer to the discovery of
nematic order in ferromagnetic superconductors [6, 7] and frustrated
ferromagnetic chains [8], non-collinear ferrimagnet behaviour [4], magnetic
order-disorder phase transitions [5], spin-liquids [2, 9, 10, 3] and
topological vortices [2, 11]. Such novel phases and phase transitions are
observed in strongly correlated electronic systems [2, 8, 5, 12], e.g., in
kagome superconductors [13], in antiferromagnets [9, 14, 15, 16], in quantum
magnets [10, 11] and also in quasi-one-dimensional molecular systems
displaying a transition to a chiral spin-liquid state [17].

The frustration has been implemented in various artificial intrinsically
quantum platforms such as trapped ion systems [18, 19, 20], photonic crystals
[21, 22], Rydberg arrays [23, 24, 25] and Josephson junction arrays (JJAs)
[26, 27, 28, 29]. In the latter systems, produced in very different geometries
and sizes, fascinating physical effects, e.g., nonlinear classical dynamics of
magnetic Josephson vortices (fluxons) [30, 31], discrete breathers in
Josephson junction ladders [32, 33], superconductor-insulator quantum phase
transitions [34, 35] have been intensively studied experimentally and
theoretically in last decades. Furthermore, Josephson junction arrays provide
a promising platform for future realizations of different superconducting
Josephson meta-materials in the context of novel quantum devices and
technology [36, 37, 38]. These quantum devices might be suitable for
realization of analog quantum simulations in different fields of quantum
science [39, 40, 41, 42].

There is a particular interest in frustrated Josephson junction arrays
($f$-JJAs) with at least two different ways how the frustration can be
introduced in f-JJAs: (i) application of an external magnetic field [43, 35,
26] and (ii) usage of a combination of $0$ and $\pi$-Josephson junctions
implemented in a single triangular unit cell [44, 45]. Depending on the
frustration parameter, f-JJAs exhibits either frustrated or non-frustrated
regime. In the frustrated regime, for a single triangular unit cell, the
potential energy shows a double well shape with two degenerate minima
corresponding to the anticlockwise or clockwise flowing persistent currents
(vortex/antivortex) [43]. This generalizes to a more complex square and
triangular lattices of f-JJAs, where highly degenerated complex ground states
corresponding to the different configurations of vortices/antivortices, e.g.,
checkerboard, ribbon or stripe types have been observed [46, 47, 48].

Figure 1:  (color online) Schematic of a frustrated saw-tooth chain of
Josephson junctions directly embedded in a dissipationless transmission line
characterized by the capacitance to the ground $C_{0}$. The $0$\- and $\pi$\-
Josephson junctions are shown in black (red). $E_{J}$ and $\alpha E_{J}$ are
the Josephson energies of $0$\- and $\pi$ Josephson junctions, respectively.

The coherent quantum dynamics of a single triangular unit cell biased in the
frustrated regime is determined by macroscopic quantum tunneling between two
degenerate minima leading to the coherent quantum beats between the anti-
clockwise/clockwise flowing persistent currents. Such a lump superconducting
quantum circuit is identical to a single flux qubit biased to the symmetry
point [43].

Single triangular unit cells can be arranged in plenty different vertex-
sharing $f$-JJAs: quasi-one-dimensional systems such as saw-tooth and diamond
chains [26, 35, 49, 44] as well as a two dimensional Kagome lattice [50, 45].
In the latter case the collective anisotropic vortex/antivortex states have
been predicted [50] and studied in detail [45]. In particular, it has been
shown in Ref. [45] that the physical origin of these collective states is the
presence of a huge amount of topological constraints caused by the flux
quantization in closed superconducting loops. This induces a highly
anisotropic interaction between Josephson junctions of different unit cells.
At the same time, in Ref. [44] it has been shown that in quasi-1D f-JJAs,
e.g., saw-tooth and diamond chains of Josephson junctions, the interaction
between Josephson junctions of different cells is absent resulting in random
configurations of vortices/antivortices in the classical frustrated regime.

In the absence of interaction, the quantum dynamics of quasi-1D vertex-sharing
f-JJAs is reduced to a set of independent flux qubits, and there are no
collective quantum phases. Therefore, natural questions arise in this field:
which interaction allows to observe various collective quantum phases in
quasi-1D f-JJAs and how it can be realized?

In this paper, we answer these questions demonstrating that a straightforward
embedding of a frustrated saw-tooth chain of Josephson junctions in a
dissipationless transmission line allows to establish a strong, intrinsically
quantum interaction between Josephson junctions of different cells. Moreover,
depending on the physical parameters this interaction is either long- or
short-ranged. We show how various collective quantum phases and quantum phase
transitions between them can be observed in quasi-1D $f$-JJAs.

The paper is organized as follows: In Section II we introduce the
electrodynamic model of frustrated saw-tooth quasi-1D arrays of small
(quantum) Josephson junctions incorporated in a dissipationless transmission
line and define the most important physical parameters and dynamic variables
of the system. We write down the potential and kinetic energies, the
Lagrangian and the Hamiltonian for two Josephson junction arrays: a frustrated
single triangular cell of Josephson junctions (Sec. II.1) and a long
frustrated saw-tooth chain of Josephson junctions (Sec. II.2). In the latter
case we obtain a long-range charge interaction between Josephson junctions of
well separated cells. In Section III, we address the frustrated regime in
which the potential energy of a single cell has two equivalent minima. In
Section III.1 the variational approach is used to reduce the circuit
Hamiltonian to an effective spin Hamiltonian. In the Section III.2, we
illustrate this method for a frustrated single triangular cell of Josephson
junctions. In Section IV, the quantum dynamics of low-lying eigenstates will
be mapped to an effective $XX$ quantum spin chain with a long/short-range
exchange interaction and a local magnetic field applied in the $x$-direction.
The collective quantum phases and corresponding quantum phase transitions will
be identified in Section V. Section VI provides conclusions.

##  II Electrodynamic Model, Dynamic Variables, Lagrangian and Hamiltonian

Let us consider an exemplary frustrated vertex-sharing saw-tooth array of
Josephson junctions. Such an array contains $N$ periodically arranged basic
cells, i.e., triangular superconducting loops interrupted by two $0$\- and one
$\pi$-Josephson junctions. The $0$\- and $\pi$-Josephson junctions connecting
adjacent superconducting islands, i.e., nodes of the array, are characterized
by the Josephson energies $E_{J}=\hbar I_{c}/(2e)$ and $\alpha E_{J}$, and the
charging energies $E_{C}=e^{2}/(2C)$ and $E_{C}/|\alpha|$, respectively. Here,
$I_{c}$ and $C$ are the critical current and the capacitance of the
$0$-Josephson junction, respectively. The parameter $\alpha$ varying between
$-1$ and $1$ determines the frustration of a system and is related to the
commonly used frustration parameter $f$ as $\alpha=(1-2f)$, where $0<f<1$.
Utilizing the $\pi$-Josephson junction with the parameter
$\alpha<\alpha_{c}=-0.5$ ($f>f_{c}=3/4$), one can realize the frustrated
regime in which each basic cell of the array is in one of two equivalent
classical ground states. In such $f$-JJAs, the interaction between Josephson
junctions of different cells is absent, and the classical ground state of the
entire array is $2^{N}$ degenerate.

A natural way to provide a strong intrinsically quantum interaction in vertex-
sharing quasi-1D JJAs is to embed the $\pi$-Josephson junctions as distributed
inductances in a dissipationless transmission line. The transmission line is
also characterized by the capacitance to the ground, $C_{0}$. Although in most
of the experiments so far, the ratio of $C_{0}/C$ was rather small [34, 51],
the opposite limit $C_{0}\geq C$ was also obtained in specially prepared
Josephson metamaterials, e.g., SQUID transmission lines [38, 52]. The
schematic of a frustrated saw-tooth chain of Josephson junctions incorporated
in the transmission line is presented in Fig. 1.

The classical electrodynamics of a system is completely determined by the
_two_ time-dependent phases of the superconducting order parameter per unit
cell, $\chi_{n}=\\{\chi_{0,n};\chi_{+,n}\\}$. By making use of the Kirchhoff’s
laws and choosing a spanning tree (graph theory) as introduced in Refs. [53,
54], we obtain the Lagrangian $L=K-U$, where the potential $U$ and the kinetic
$K$ energies are written as follows

$\displaystyle U(\\{\chi_{n}\\})$
$\displaystyle=E_{J}\sum_{n=1}^{N}[2+\alpha-\cos(\chi_{+,n}-\chi_{0,n})$
$\displaystyle-$
$\displaystyle\cos(\chi_{0,n+1}-\chi_{+,n})-\alpha\cos(\chi_{0,n}-\chi_{0,n+1})]$
(1)

and

$\displaystyle K(\\{\dot{\chi}_{n}\\})$
$\displaystyle=\frac{\hbar^{2}}{16E_{C}}\sum_{n=1}^{N}[(\dot{\chi}_{+,n}-\dot{\chi}_{0,n})^{2}+(\dot{\chi}_{0,n+1}-\dot{\chi}_{+,n})^{2}$
$\displaystyle+|\alpha|(\dot{\chi}_{0,n}-\dot{\chi}_{0,n+1})^{2}]+\frac{\hbar^{2}}{16E_{C_{0}}}\sum_{n=1}^{N+1}(\dot{\chi}_{0,n})^{2},$
(2)

where the charging energy to the ground $E_{C_{0}}=(C/C_{0})E_{C}$.

###  II.1 Effective Hamiltonian of a single building block

Figure 2:  (color online) The schematic of a single building block of
frustrated saw-tooth chains of Josephson junctions. The phases of the order
parameter of superconducting islands, $\chi_{0}$ ($\chi_{+}$), and
corresponding Josephson phases $\varphi$ are shown.

A frustrated saw-tooth chain of Josephson junctions consists of periodically
arranged building blocks, i.e., triangular superconducting loops interrupted
by three (two -$0$ and one -$\pi$) Josephson junctions. The schematic of a
single building block is presented in Fig. 2. To elaborate the classical and
quantum dynamics of a single building block we introduce the Josephson phases
as $\varphi_{1}=\chi_{+,1}-\chi_{0,1}$, $\varphi_{2}=\chi_{0,2}-\chi_{+,1}$
and $\varphi_{3}=\chi_{0,1}-\chi_{0,2}$. For superconducting loops of a small
area the Josephson phases satisfy the flux quantization condition,
$\varphi_{1}+\varphi_{2}+\varphi_{3}=0$. As we are interested in the low-
energy dynamics of the system, we make a few further simplifications. First,
it is not necessary to take into account the $2\pi$-periodicity of the
potential energy landscape, because the energy barriers $\simeq E_{J}$ are
assumed to be large compared to both $k_{B}T$, where $T$ is the temperature,
and $\hbar\Omega$, where $\Omega$ is a typical frequency of small
oscillations.

Taking into account the flux quantization condition, one finds that the
potential energy is determined by two Josephson phases $\varphi_{1,2}$. It is
convenient to symmetrize these variables by introducing the symmetric and
antisymmetric combinations $\varphi_{s,a}=(\varphi_{1}\pm\varphi_{2})/2$. In
the frustrated regime $-1<\alpha<-0.5$, the potential energy has two
equivalent shallow minima at $\varphi_{s}=\pm u_{0}$ with
$u_{0}=\arccos[1/(2|\alpha|)]$ and $\varphi_{a}=0$ (see Fig. 3). These minima
are separated by the potential energy barrier
$E_{J}(\alpha)=-E_{J}[2(1+\alpha)+1/(2\alpha)]$, which becomes zero at the
critical value of $\alpha=\alpha_{c}=-0.5$. Since the frequency of small
oscillations in the $\varphi_{a}$ direction is much larger than those in the
$\varphi_{s}$ direction, they cannot be excited at low temperatures.
Therefore, the potential energy of a single building block of frustrated saw-
tooth chains of Josephson junctions is effectively determined by a single
dynamic variable, $\varphi_{s}$.

In the classical frustrated regime, the two minima of the potential energy
correspond to non-zero persistent currents flowing in two opposite (clockwise
or anticlockwise) directions. Note that the frustrated regime, as defined
here, is equivalent to the so-called flux qubit biased to the symmetry point
[43], i.e., an externally applied magnetic flux $\Phi=(1/2)\Phi_{0}$, where
$\Phi_{0}$ is the flux quantum. The kinetic energy (II) of a single building
block depends on $\dot{\varphi}_{s}$ and $\dot{\chi}_{0,1}$. The latter
determines the dynamics of the whole system and does not affect the collective
states of the system. Thus, we further disregard this dynamics and set
$\dot{\chi}_{0,1}$ to zero.

With these simplifications, we can now express the circuit Lagrangian for a
single building block of a system as (from here, we will omit the index $s$ in
$\varphi_{s}$),

$\displaystyle\mathcal{L}_{sb}$
$\displaystyle=\frac{\hbar^{2}}{8E_{C}}\left(\gamma+1+2|\alpha|\right)\dot{\varphi}^{2}$
(3) $\displaystyle-
E_{J}\left[2+\alpha-2\cos(\varphi)-\alpha\cos(2\varphi)\right],$

where the parameter $\gamma=C_{0}/C$ is introduced. The characteristic
frequency $\Omega$ of small oscillations of the Josephson phase around the
minima of the potential energy is

$\displaystyle\Omega=[2/(\hbar)]\sqrt{E_{C}E_{J}(|4\alpha+1/|\alpha||)/(\gamma+1+2|\alpha|)}.$
(4)

We also define the effective mass:

$\displaystyle m_{eff}=(\hbar^{2}/4E_{C})(\gamma+1+2|\alpha|).$ (5)

Introducing the generalized momentum
$p=\partial\mathcal{L}_{sb}/\partial\dot{\varphi}$ and representing it in the
operator form $\hat{p}=-i\hbar\partial/\partial\varphi$ we arrive at the
circuit Hamiltonian

$\displaystyle\hat{H}_{sb}$
$\displaystyle=\frac{2E_{C}}{\hbar^{2}}\frac{1}{\gamma+1+2|\alpha|}\hat{p}^{2}$
(6)
$\displaystyle+E_{J}\left[2+\alpha-2\cos(\varphi)-\alpha\cos(2\varphi)\right].$

Figure 3: (color online) Calculated effective potential energy of a single
building block. In the non-frustrated regime there is a single minimum at
$\varphi_{s}=0$ (dashed line), and in the frustrated regime there are two
shallow minima $\varphi_{s}=\pm u_{0}$ (solid line).

###  II.2  Effective Hamiltonian of frustrated saw-tooth chains of Josephson
junctions

Using the assumptions discussed in Sec. II.1 and introducing the Josephson
phases as $\vec{\varphi}=\\{\varphi_{n}\\}$, where
$\varphi_{n}=\chi_{+,n}-\chi_{0,n}$, and $n$ is the cell number, the
Lagrangian of a saw-tooth chain of Josephson junctions embedded in the
transmission line (see Fig. 1) is written as follows

$\displaystyle\mathcal{L}$
$\displaystyle=\frac{\hbar^{2}}{8E_{C}}\left[(1+2|\alpha|)\dot{\vec{\varphi}}^{T}\dot{\vec{\varphi}}+\dot{\vec{\varphi}}^{T}\mathbf{C}_{\gamma}\dot{\vec{\varphi}}\right]$
(7) $\displaystyle-
E_{J}\sum_{i}\left(2+\alpha-2\cos(\varphi_{i})-\alpha\cos(2\varphi_{i})\right),$

where the non-diagonal capacitance matrix $\mathbf{C}_{\gamma}$ determines the
interaction between the Josephson junctions of different cells and is
expressed explicitly as

$\displaystyle\mathbf{C}_{\gamma}=2\gamma\begin{bmatrix}N&N-1&N-2&\ldots&1\\\
N-1&N-1&N-2&\ldots&1\\\ N-2&N-2&N-2&\ldots&1\\\
\vdots&\vdots&\vdots&\ddots&1\\\ 1&1&1&1&1\end{bmatrix}.$ (8)

Similar to Sec. II.1, the generalized momenta are defined as
$\vec{p}=\partial\mathcal{L}/\partial\dot{\vec{\varphi}}$. Applying the
Fourier transform $\vec{\varphi}\rightarrow\vec{\varphi}_{k}$ and utilizing
the special tri-diagonal form of the inverse capacitance matrix
$\mathbf{C}_{\gamma}^{-1}$, we derive the circuit Hamiltonian in the $k$-space
as
$\hat{H}_{k}=\frac{2E_{C}}{\hbar^{2}}\sum_{k}\Gamma(k)\hat{p}_{k}\hat{p}_{-k}+\sum_{k}U(\phi_{k})$,
where the non-local kinetic energy factor $\Gamma(k)$ is

$\displaystyle\Gamma(k)=\frac{\sin^{2}(k/2)}{\gamma/2+(1+2|\alpha|)\sin^{2}(k/2)}.$
(9)

Performing the inverse Fourier transform, the effective circuit Hamiltonian is
then written as

$\displaystyle\hat{H}$
$\displaystyle=\frac{2E_{C}}{\hbar^{2}}\left[\Gamma_{d}\sum_{i}\hat{p}_{i}^{2}-\Gamma_{o}\sum_{i\neq
j}e^{-|i-j|/\ell_{0}}\hat{p}_{i}\hat{p}_{j}\right]$ (10)
$\displaystyle+E_{J}\sum_{i}\left[2+\alpha-2\cos(\varphi_{i})-\alpha\cos(2\varphi_{i})\right]$

where the parameters
$\Gamma_{d}=\frac{1}{2|\alpha|+1}-\frac{\gamma}{(2|\alpha|+1)\sqrt{\gamma(4|\alpha|+\gamma+2)}}$
and
$\Gamma_{o}=\frac{\gamma}{(2|\alpha|+1)\sqrt{\gamma(4|\alpha|+\gamma+2)}}$,
and the interaction length,
$\ell_{0}=\left|\log(\frac{2|\alpha|+1+\gamma-\sqrt{\gamma(4|\alpha|+\gamma+2)}}{2|\alpha|+1})\right|^{-1}$.

The term proportional to $\Gamma_{o}$ determines the interaction between the
cells. Therefore, if $\gamma=0$ the quantum dynamics of a saw-tooth chain of
Josephson junctions is reduced to the dynamics of independent cells. As
$\gamma=C_{0}/C\ll 1$, the interaction strength is small but extends over a
long range, $\ell_{0}\simeq 1/\sqrt{\gamma}$. In the limit of $\gamma\gg 1$,
only the nearest neighbor interactions remain. The typical dependence of the
inverse interaction length $1/\ell_{0}$ on the parameter $\gamma$ is presented
in Fig. 4. To conclude this section, we notice that the interaction length
$\ell_{0}$ has a physical meaning of the charge screening length [34].

Figure 4: (color online) Calculated dependence of the inverse interaction
length $1/\ell_{0}$ on the ratio of the ground $C_{0}$ and the Josephson
junction $C$ capacitances, $\gamma=C_{0}/C$. We choose $\alpha=-0.8$.

##  III Effective quantum spin model

In the frustration regime, $\alpha<\alpha_{c}=-0.5$ ($f>f_{c}=3/4$), the
potential energy of a system exhibits $2^{N}$ equivalent minima separated by
small potential barriers (see the solid line in Fig. 3 for a single building
block). At low temperatures, as $k_{B}T\ll\hbar\Omega$, where $\Omega$ is the
characteristic frequency around each minimum, the coherent quantum regime is
established, and in this regime the quantum dynamics of a system consists of
small oscillations around the positions of the minima and quantum tunneling
between them. The latter corresponds to the coherent quantum beats between
persistent currents of opposite directions. Equivalently, it can be understood
as the coherent quantum beats between vortex/antivortex penetrating each cell.
The amplitude of such macroscopic quantum tunneling is exponentially small if
the parameter $\alpha$ is not too close to the critical value $\alpha_{c}$. To
analyze the coherent quantum frustrated regime, we use the variational
approach allowing to reduce the circuit Hamiltonian (10) to an effective spin
Hamiltonian. In this section, we apply this method to a single building block
of a frustrated saw-tooth Josephson junction array (see Fig. 2).

###  III.1 Variational approach

Here, we present an outline of the procedure. We choose the wave function of a
whole system as the product of the localized wave functions taking a Gaussian
form around each minimum

$\Psi(\vec{\varphi},\vec{\sigma}^{(n)}_{z})=\frac{1}{R}\exp\left[-\left(\vec{\varphi}-u_{0}\vec{\sigma}^{(n)}_{z}\right)\mathbf{A}\left(\vec{\varphi}-u_{0}\vec{\sigma}^{(n)}_{z}\right)\right],$
(11)

where $1/R$ is the normalization factor. The presence of two minima in each
cell is denoted by the classical spin component, $\sigma_{z}=\pm 1$. Thus, the
wave function $\Psi_{n}$ is determined by the vector of Josephson phases,
$\vec{\varphi}=\\{\varphi_{i}\\}$, and the $n$-th configuration of the spin
vector, $\vec{\sigma}^{(n)}_{z}=\\{\sigma_{z,i}\\}$, where $i$ is the cell
number. Using these functions as a basis, the matrix elements of an effective
spin Hamiltonian are calculated as

$\mathbf{H}_{nl}=\int
d\vec{\varphi}\Psi(\vec{\varphi},\vec{\sigma}^{(n)}_{z})\hat{H}\Psi(\vec{\varphi},\vec{\sigma}^{(l)}_{z}).$
(12)

For Josephson arrays with $N$ cells, we have $2^{N}$ basis functions, and
$\mathbf{H}$ is a $2^{N}\times 2^{N}$ matrix.

The $N\times N$ matrix $\mathbf{A}$ is obtained by minimizing the ground state
expectation value, $E_{0}=\mathbf{H}_{nn}=\int
d\vec{\varphi}\Psi(\vec{\varphi},\vec{\sigma}^{(n)}_{z})\hat{H}\Psi(\vec{\varphi},\vec{\sigma}^{(n)}_{z})$.
Other matrix elements of the effective spin Hamiltonian $\mathbf{\hat{H}}$
contain exponentially small factors $G_{n,l}$ obtained as

$\mathbf{H}_{nl}\propto
G_{n,l}=\exp\left[-\frac{1}{2}u_{0}^{2}\left(\vec{\sigma}^{(n)}_{z}-\vec{\sigma}^{(l)}_{z}\right)\mathbf{A}\left(\vec{\sigma}^{(n)}_{z}-\vec{\sigma}^{(l)}_{z}\right)\right].$
(13)

Next, we define
$\vec{Z}_{n,l}=\left(\vec{\sigma}^{(n)}_{z}-\vec{\sigma}^{(l)}_{z}\right)$.
This vector is zero if no quantum tunneling events are present upon going from
the configuration $(n)$ to $(l)$ or vice versa. A single tunneling event in
the $j$th-cell leads to a non-zero element $Z_{j}=\pm 2$, and the term
$\propto\hat{\sigma}_{x,j}$ in the effective spin Hamiltonian
$\mathbf{\hat{H}}$. In this framework, a single tunneling event is equivalent
to a spin-flip of the $j$th spin. Multiple simultaneous tunneling events,
i.e., spin-flips occurring in different cells, are interpreted as an
interaction between different spins.

###  III.2  Effective spin Hamiltonian: a single building block

For the single building block of a frustrated Josephson junction array, an
application of the procedure elaborated above results in the effective spin
Hamiltonian:

$\displaystyle\mathbf{\hat{H}}_{sb}=\Delta_{sb}\hat{\sigma}_{x}.$ (14)

The quantum tunneling amplitude, i.e., a single spin-flip, is obtained as
follows: the matrix $\mathbf{A}$ is reduced to a single parameter $A$, and
utilizing harmonic approximation for the potential $U(\varphi)$ we get the
analytical result, $A=m_{eff}\Omega/(2\hbar)$, or explicitly
$A=\sqrt{(E_{J}/16E_{C})(|4\alpha+1/|\alpha||)(\gamma+1+2|\alpha|)}$.
Calculating the Gaussian integrals in Eq. (12) we obtain the non-diagonal
matrix element of Eq. (14) containing a small exponential factor as

$\displaystyle\Delta_{sb}\propto G_{1}=\exp(-2u_{0}^{2}A).$ (15)

The pre-exponential factor in Eq. (15) was calculated numerically, and was
obtained to be of the order $\hbar\Omega$.

Additionally, one can find the quantum tunneling amplitude by making use of
the quasiclassical ($WKB$) approximation [55]. In a complete analogy with
Refs. [43, 55] the tiny energy gap $2\Delta_{sb}$ between the two lowest
eigenvalues is written as:

$\displaystyle\Delta^{WKB}_{sb}=\frac{\hbar\Omega}{2\pi}\exp\left(-\frac{2\sqrt{2m_{eff}E_{J}}}{\hbar}\left(\sqrt{2|\alpha|-\frac{1}{2|\alpha|}}\right.\right.$
$\displaystyle\left.\left.-\frac{1}{\sqrt{2|\alpha|}}\arccos(\frac{1}{2|\alpha|})\right)\right).~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(16)

In order to compare the results for $\Delta_{sb}$ obtained by two methods, one
can study small exponential factors in the limit
$\alpha\rightarrow\alpha_{c}$. In this limit, the variational approach results
in $\ln
G_{1}=-\sqrt{\frac{E_{J}}{E_{C}}(\gamma+2)}\left(2|\alpha|-1\right)^{3/2}$,
while using the quasiclassical approximation one obtains
$\ln\left(\frac{2\pi\Delta^{WKB}_{sb}}{\hbar\Omega}\right)=-\frac{4}{3}\sqrt{\frac{E_{J}}{E_{C}}(\gamma+2)}\left(2|\alpha|-1\right)^{3/2}$.
Therefore, these two methods demonstrate the same universal dependence of a
small exponential factor in Eqs. (15) and (III.2) on the parameter $\alpha$ as
$\alpha\simeq\alpha_{c}$, with only different numerical factors.

Therefore, one can expect that the variational approach can be effectively
used to describe low-energy quantum dynamics of long frustrated Josephson
junction arrays.

##  IV Effective $XX$ spin Hamiltonian with a long-range interaction

Here, we apply the variational approach, elaborated in the Section III, to
derive an effective spin Hamiltonian for a long frustrated saw-tooth chain of
Josephson junctions. Using the trial wave function, Eq. (11), and calculating
the expectation value of the ground state energy,
$E_{0}(\mathbf{A})=\mathbf{H}_{nn}=\int
d\vec{\varphi}\Psi(\vec{\varphi},\vec{\sigma}^{(n)}_{z})\hat{H}\Psi(\vec{\varphi},\vec{\sigma}^{(n)}_{z})$,
we obtain the matrix $\mathbf{A}$ by minimizing $E_{0}$. Due to the specific
structure of the interaction in the Hamiltonian (Eq. (10)), it is convenient
to rewrite the Hamiltonian in the $k$-space and use the harmonic
approximation:

$\displaystyle\hat{H}_{\textrm{harm}}=\frac{1}{2m_{k}}\hat{p}_{k}\hat{p}_{-k}+\frac{m_{k}\Omega_{k}^{2}}{2}\varphi_{k}\varphi_{-k},$
(17)

where we identify $m_{k}=\frac{\hbar^{2}}{4E_{C}}\frac{1}{\Gamma(k)}$ and
$\Omega_{k}=\sqrt{\frac{E_{J}}{m_{k}}(4|\alpha|-1/|\alpha|)}$. In the
$k$-space, the trial wave function has a simple form:
$\Psi_{k}=1/R_{k}\exp(-\zeta_{k}\phi_{k}^{2})$. It allows us to obtain the
parameter $\zeta_{k}$ as

$\displaystyle\zeta_{k}=\frac{m_{k}\omega_{k}}{2\hbar}=\frac{D}{4}\frac{\sqrt{\gamma/2+(2|\alpha|+1)\sin^{2}(k/2)}}{|\sin(k/2)|},$
(18)

where $D=\sqrt{\frac{E_{J}}{E_{C}}\left(4|\alpha|-\frac{1}{|\alpha|}\right)}$.
Making the inverse Fourier transformation to the real space, we obtain the
dependence of the matrix elements $A_{nm}$ on the distance between $n$-th and
$m$-th cells:

$\displaystyle A(|n-m|)=2\sum_{k}\zeta_{k}\cos[k(n-m)].$ (19)

In this analysis we imply open boundary conditions.

The analysis of the non-diagonal matrix elements of the effective spin
Hamiltonian $\mathbf{\hat{H}}$ (Eq. (13)) allows us to conclude that multiple
simultaneous spin flips involving $n$ cells with $n>2$ are exponentially small
compared to the spin flips with $n=1$ and $n=2$ (see the Appendix for
details). Taking this property into account, we write the effective spin
Hamiltonian in the following form:

$\mathbf{\hat{H}}=\sum_{n=1}^{N}\Delta\hat{\sigma}_{x,n}+\frac{1}{2}\sum_{n\neq
m}J(|n-m|)\left[\hat{\sigma}_{x,n}\hat{\sigma}_{x,m}+\hat{\sigma}_{y,n}\hat{\sigma}_{y,m}\right].$
(20)

Here, the first and second terms in the r.h.s. of Eq. (20) describe the
quantum tunneling between a vortex and an antivortex in a single cell, and
induced by the coupling to a transmission line the long/short-range exchange
spins interaction between different cells, accordingly. The amplitude of
quantum tunneling, $\Delta$, and the exchange interaction strength,
$J(|n-m|)$, are determined by the matrix elements $A_{nm}$ as

$\displaystyle\Delta=\Delta_{0}\exp[-2u_{0}^{2}A(0)]$ (21)

and

$J(|n-m|)=J_{0}\exp\\{-4u_{0}^{2}[A(0)-A(|n-m|)]\\},$ (22)

where both pre-exponential factors, $\Delta_{0}$ and $J_{0}$, are of the order
of $\hbar\Omega$.

###  IV.1  The parameters $\Delta$ and $J(|n-m|)$: general properties

The parameters $\Delta$ and $J(|n-m|)$ of the effective spin Hamiltonian
$\mathbf{\hat{H}}$ are determined by two matrix elements $A(0)$ and $A(|n-m|)$
having the following properties: in the absence of the coupling to the
transmission line, i.e., as $C_{0}=0$, the matrix $A(|n-m|)$ has a diagonal
form, $A(|n-m|)=(m_{eff}\Omega/\hbar)\delta_{nm}$. Therefore, the exchange
interaction term in the Hamiltonian (20) is absent. In this case the value of
$\Delta$ coincides with $\Delta_{sb}$ (Eqs. (15) and (III.2) for $C_{0}=0$).

In a general case of $C_{0}\neq 0$, the matrix elements $A(0)$ and $A(|n-m|)$
strongly depend on the ratio of the total length of the array $N$ to the
interaction length $\ell_{0}$. For short arrays, as $N<\ell_{0}$, we obtain
that the exchange interaction term in the Hamiltonian (20) is small and weakly
decays with the distance $|n-m|$. The parameter $\Delta\simeq\Delta_{sb}$.

However, the most interesting case is realized for long arrays as
$N\gg\ell_{0}$. Here, the parameter $A(0)$ increases logarithmically with $N$,
i.e., $A(0)=(D/2)\sqrt{\gamma/2}\ln(N/\ell_{0})$, and therefore, the amplitude
of the quantum tunneling between vortex and antivortex in a single cell is
strongly suppressed. The typical dependence of $\Delta(N)$ is shown in Fig. 5
(green solid line). On the other hand, the parameter $[A(0)-A(1)]$,
determining the strength of the exchange interaction $J(1)$, weakly depends on
$N$ (see, red line in Fig. 5). This means that varying the length of the array
$N$ or the interaction length $\ell_{0}(\gamma)$ enables one to switch between
strong and weak exchange interactions.

At the distance $|n-m|>\ell_{0}$, the parameter $A(|n-m|)$ increases with
$|n-m|$ as $A(|n-m|)=(D/2)\sqrt{\gamma/2}\times\ln(N/|n-m|)$. Therefore, the
exchange interaction term can be accurately approximated by a power-law
$J(x)\approx J(1)(|x|)^{-\beta}$ with $\beta=(Du_{0}^{2})\sqrt{\gamma/2}$. The
typical dependencies of $J(|x|)$, together with the power-law approximations
(shown by dashed lines) for different values of the parameter $\gamma$, are
presented in Fig. 6a. The obtained dependence of the power exponent $\beta$ on
the parameter $\gamma$ is presented in Fig. 6b.

Thus, varying the parameters $E_{J}$, $E_{c}$, $\alpha$ and $\gamma$, one can
realize different regimes of the effective spin Hamiltonian (20): long- or
short-range exchange spin interactions, as well as strong/weak local quantum
tunneling amplitude.

Figure 5: (color online) Calculated dependencies of the amplitude of quantum
tunneling, $\Delta$ (green solid line), and the exchange interaction strength,
$J(1)$ (red solid line), on the total length $N$. The blue dashed line
indicates the logarithmic behavior of $\ln(\Delta/\Delta_{0})$ as
$N>\ell_{0}$. The chosen parameters are $\alpha=-0.8$, $\gamma=1.0$, and
$E_{J}/E_{C}=80$.

(a)

(b)

Figure 6: (color online) (a): Calculated typical dependencies of $J(|x|)/J(1)$
(solid lines) for different values of $\gamma$: $\gamma=1.5$ (green line),
$\gamma=0.5$ (red line) and $\gamma=0.1$ (blue line). The approximations with
the power-law dependence are shown by colored dashed lines. Other parameters
were chosen as $\alpha=-0.8$, $N=1000$, $E_{J}/E_{C}=80$. (b): The calculated
dependence of the power exponent $\beta$ on the parameter $\gamma$. The
colored dashed lines correspond to the $J(|x|)$ curves presented in Fig. 6a.
Other parameters were chosen as $\alpha=-0.8$, $N=1000$, $E_{J}/E_{C}=80$.

##  V  Collective phases of the effective spin Hamiltonian

Here, we present a numerical analysis of the effective $XX$ spin Hamiltonian
in the situation when the exchange spin-spin interaction decays with distance
according to a power law. We rewrite Eq. (20) in the dimensionless form:

$\frac{\mathbf{\hat{H}}}{J(1)}=\sum_{n=1}^{N}\tilde{\Delta}\hat{\sigma}_{x,n}+\frac{1}{2}\sum_{n\neq
m}\frac{1}{|n-m|^{\beta}}\left[\hat{\sigma}_{x,n}\hat{\sigma}_{x,m}+\hat{\sigma}_{y,n}\hat{\sigma}_{y,m}\right],$
(23)

where we introduce the dimensionless parameter $\tilde{\Delta}=\Delta/J(1)$.

This Hamiltonian is not integrable (except the case of $\Delta=0$) [56, 57],
and therefore, our further analysis is based on the direct numerical
diagonalization of Eq. (23) for spin chains of a moderate size, up to $N=16$.
Exploiting open boundary conditions and fixing the parameters
$\tilde{\Delta}$, $\beta$, we obtain the eigenvalues $E_{\alpha}$ and
eigenvectors $\ket{\psi_{\alpha}}$, where $\alpha$ is the label of the
different energy levels. Instead of performing the complete numerical
diagonalization [58], we use Arnoldi method [59] to find the eigenvalues and
eigenvectors of the ground and first excited state $E_{0}$, $E_{1}$,
$\ket{\psi_{0}}$ and $\ket{\psi_{1}}$. Next, we vary the parameters
$|\tilde{\Delta}|<15$ and $0<\beta<5$, to explore the different collective
phases.

We characterize the collective quantum phases by the dimensionless minimum
energy gap, i.e., $G=(E_{1}-E_{0})/J(1)$. In Fig. 7a, the dependencies of the
minimum energy gap $G$ on the relative amplitude of the $\sigma_{x}$ term,
$\tilde{\Delta}$, for different values of $\beta$ are presented. One can see
that for the large values of $|\tilde{\Delta}|$, the interaction term becomes
relatively small, and extensive local spin-flips lead to the paramagnetic
($P$) state. The $P$-state is characterized by a finite value of the minimum
energy gap $G\propto|\tilde{\Delta}|$, a low entanglement, and the quantum-
mechanical average of the total $x$-component of the magnetization,
$M_{x}=1/N<\sum_{i}\sigma_{x,i}>$ is one.

As the parameter $|\tilde{\Delta}|$ decreases below the critical value, the
exchange spin interaction plays a dominant role, and the so-called
compressible superfluid ($CS$) state forms. From a study of the analytically
tractable $XX$ models with a nearest-neighbors interaction [60], it is well
known that at $\Delta=0$, $CS$-state demonstrates zero minimum energy gap in
the limit of infinite number of spins $N$. From Fig. 7a one sees that the
$CS$-state occurs also for the $XX$ model with a short/long-range exchange
interaction and $x$-component of the magnetic field. The small wiggles
observed in the dependence of $G(\tilde{\Delta})$ are attributed to the
finite-size effects (see the discussion below) [61].

We also obtain the well-defined crossover between the $P$\- and $CS$-states,
and the critical value of $|\tilde{\Delta}_{c}(\beta)|$ separating these
phases increases as the parameter $\beta$ decreases, which means that the
range of the parameter $\Delta/J(1)$ for which the $CS$-state is observed,
increases for a long-range interaction as $\beta\leq 2$. Our observations are
summarized in the color plot of Fig. 7b, where the minimum energy gap $G$ as a
function of $\tilde{\Delta}$ and $\beta$ is presented. Notice here, that the
effective spin model (23) with the nearest-neighbor interactions has been
studied using the semi-classical product of states (no entanglement)
approximation in [62], and the lower bound of the critical value
$|\tilde{\Delta}_{c}|>|\tilde{\Delta}_{n}|=2\sqrt{2}$ for the quantum phase
transition between the $P$\- and $CS$-states, was obtained. Using the Jordan-
Wigner transformation and following mean-field analysis the quantum
corrections resulting in a slight increase of the critical value
$|\tilde{\Delta}_{c}|\simeq 3$ have been obtained in Refs. [56, 57]. As
$\beta\gg 1$, i.e., for short-range spin-spin exchange interaction, our
numerical results obtained for the minimum energy gap $G$ dependence on the
$\tilde{\Delta}$ in both $P$ and $CS$ collective quantum states are consistent
with such analysis [56, 57].

The spatial properties of the collective states can be further characterized
through the spatial correlation function of the $y$-component of local spins:

$C_{y}(|i-j|)=<\sigma_{y,i}\sigma_{y,j}>.$ (24)

The obtained dependencies of $C_{y}(N/2-1)$ on the parameter $\tilde{\Delta}$
for different values of $\beta$ are presented in Fig. 8a. In the $P$-state,
spatial correlations between spins located at a large distance are absent, and
in the $CS$-state the negative spatial correlations oscillating with
$\tilde{\Delta}$ were found. However, as the parameter $\beta\leq 2$ (a long-
range exchange spin interaction), the spatial correlations in $y$-direction
are strongly diminished. Thus, the collective phase obtained in the presence
of a long-range exchange spin interaction can be called a weakly compressible
superfluid ($w$-$CS$) state. The complete dependence of $C_{y}(N/2-1)$ on the
parameters $\tilde{\Delta}$ and $\beta$, is presented in Fig. 8b.

The $CS$-state realized in a short-range interaction regime, as $\beta\gg 1$,
also shows pronounced antiferromagnetic oscillations in the $C_{y}(n)$
dependence (see, the blue line in Fig. 9). In the $w$-$CS$-state, obtained in
a long-range interaction regime ($\beta\leq 2$), the amplitude of the
antiferromagnetic oscillations of $C_{y}(n)$ decreases (see, the green and red
lines in Fig. 9). For a non-zero $\tilde{\Delta}$, the oscillatory behaviour
of the correlation function with a dependence of the period and amplitude on
$\tilde{\Delta}$ (similar to the observations in [57]) and $\beta$ (see Fig.
10) persists. We also calculated the correlation function in $x$-direction

$C_{x}(|i-j|)=<\sigma_{x,i}\sigma_{x,j}>-M_{x}^{2},$ (25)

where $C_{x}(n)$ becomes zero for large $\tilde{\Delta}$ in P-state. In Fig.
10 we present $C_{x}(n)$ for $\tilde{\Delta}=4$. $C_{x}(n)$ shows
incommensurate behaviour, indicated previously in Ref. [57], and oscillations
with large amplitude.

To conclude this Section, we notice that in the $CS$ ($w$-$CS$)-states, the
amplitude of oscillations of $C_{y}(n)$ exponentially decays with the distance
$n$, and one can introduce the correlation length, $\xi$. Explicitly, we
define the correlation length as (see Refs. [58])

$\xi=\frac{1}{q_{1}}\sqrt{\frac{S(0)}{S(q_{1})}-1},$ (26)

where $q_{1}=2\pi/N$ and

$S(q)=\sum_{j=0}^{N}|C(j)|\cos(qj).$ (27)

The correlation length $\xi$ depends on the parameters $\tilde{\Delta}$,
$\beta$, as well as the total number of spins $N$. In the $CS$ state, we
obtain that the dependencies of the ratio $\xi/N$ on $\tilde{\Delta}$ for
various $N$ demonstrate a standard scaling behavior [58, 63, 64, 65]. It is
presented in Fig. 11a for $\beta=4.5$, where one can see all curves
intersecting in a single point. Thus, these scaling arguments support the
conclusion that $\tilde{\Delta}\approx 3$ determines the phase transition
between $P$ and $CS$ states [56, 57, 62].

On the contrary, in the $w$-$CS$ the dependencies of $\xi/N$ on
$\tilde{\Delta}$ do not show a single intersection point (see, Fig. 11b), and
therefore, to obtain the critical value $\tilde{\Delta}$ determining the phase
transition between $w$-$CS$ and $P$ states, one has to go beyond the direct
diagonalization procedure.

(a)

(b)

Figure 7:  (color online)(a) Calculated dependence of the minimal energy gap,
$G=(E_{1}-E_{0})/J(1)$, on the ratio of the local tunneling amplitude to the
maximum exchange interaction strength, $\tilde{\Delta}=\Delta/J(1)$ for
various values of $\beta$: $\beta=4.5$ (blue line), $\beta=1.0$ (green line),
$\beta=0.1$ (red line); (b) The color plot showing the dependence of the
minimal gap $G$ on $\tilde{\Delta}$ and $\beta$. The color dashed lines
corresponds to the values of $\beta$ presented in Fig. 7a. The other physical
parameters were chosen as $N=12,\alpha=-0.8,E_{J}/E_{C}=80$.

(a)

(b)

Figure 8: (color online) (a) Calculated dependence of the spatial correlations
along $y$-axis characterized by $C_{y}(N/2-1)$ on the dimensionless parameter
$\tilde{\Delta}$ for various values of $\beta$: $\beta=4.5$ (blue line),
$\beta=1.0$ (green line), $\beta=0.1$ (red line); (b) The color plot of the
spatial correlations $C_{y}(N/2-1)$ as the function of $\tilde{\Delta}$ and
$\beta$. The color dashed lines correspond to the values of $\beta$ presented
in Fig. 8a. The parameters were chosen as $N=12,\alpha=-0.8,E_{J}/E_{C}=80$.
Figure 9: (color online) Calculated spatial correlation function $C_{y}(n)$ in
the absence of the local quantum tunneling ($\tilde{\Delta}=0$) for different
values of $\beta$: $\beta=4.5$ (blue line), $\beta=1.0$ (green line),
$\beta=0.1$ (red line). The other parameters were chosen as:
$\alpha=-0.8,N=15$. Figure 10: (color online) Calculated spatial correlation
function $C_{x}(n)$ for non-zero local quantum tunneling $\tilde{\Delta}=4$
for different values of $\beta$: $\beta=4.5$ (blue line), $\beta=1.0$ (green
line), $\beta=0.1$ (red line). The other parameters were chosen as:
$\alpha=-0.8,N=15$.

(a)

(b)

Figure 11: (color online) Claculated dependencies of the relative correlation
length, $\xi/N$, on $\tilde{\Delta}$ for different values of the spin chain
length, $N$: $N=6$ (black lines), $N=8$ (blue lines), $N=10$ (green lines),
$N=12$ (red lines), $N=14$ (yellow lines). For a short-range interaction (a)
($\beta=4.5$) the typical scaling behavior as a single crossing point of
different graphs, $\tilde{\Delta}\approx 3$ determines the phase transition
between $P$\- and $CS$ states. For a long-range interaction ($\beta=1.5$)
shown in (b) the scaling behavior is absent.

##  VI Conclusions

In conclusion, we theoretically study the collective quantum phases occurring
in frustrated saw-tooth chains of Josephson junctions embedded in a
dissipationless transmission line. The frustration was introduced through the
periodic arrangement of $0$\- and $\pi$-Josephson junctions. The frustrated
regime is realized for the parameter $-1<\alpha<-0.5$, where $\alpha E_{J}$ is
the Josephson coupling energy of $\pi$-Josephson junctions. In this regime a
single basic cell of the system, i.e., a superconducting triangle, composed of
two $0$\- and one $\pi$-Josephson junctions embedded in the superconducting
loop, shows two stable states separated by a potential barrier. These two
stable states correspond to the persistent currents (magnetic
vortex/antivortex) flowing in clockwise or anticlockwise directions. In the
quantum frustrated regime, the macroscopic quantum tunnelling yields coherent
quantum beats between these two states. Thus, a single basic element of a
frustrated saw-tooth chain of Josephson junctions is equivalent to a single
flux qubit biased at the symmetry point [43].

The collective macroscopic quantum dynamics of a large frustrated saw-tooth
chain of Josephson junctions crucially depends on the interaction between
Josephson junctions located in different cells. Direct embedding of
$\pi$-Josephson junctions in the transmission line establishes either a long
or short-range interaction. The range of interaction is determined as
$\ell_{0}\simeq\sqrt{C/C_{0}}$ where $C$ and $C_{0}$ are capacitances of
Josephson junctions and the transmission line, accordingly.

By making use of the variational approach, we reduce the quantum
superconducting circuit Hamiltonian of a system to the effective interacting
spin chain $XX$ Hamiltonian (Eq. (20)), in which the local spin-flips and a
long/short exchange spin-spin interaction are taken into account. We obtain
that the amplitude of a local spin-flips $\Delta$ is drastically suppressed in
the limit of long arrays, $N\gg\ell_{0}$, and therefore, the dimensionless
parameter $\tilde{\Delta}=\Delta/J(1)$ determining the appearance of a
specific collective quantum phase can be tuned in a wide region by changing
the physical parameters $E_{J}$, $\alpha$, $C$, $C_{0}$ and $N$. The exchange
spin-spin interaction decays with distance as a power law with exponent
$\beta$, which can be either larger or smaller than $1$.

Using the direct numerical diagonalization of the Hamiltonian (Eq. (23)) for
the spin chains up to $N=16$ sites, we identify several collective quantum
phases: the paramagnetic ($P$)-state, the compressible superfluid ($CS$) and
weakly compressible superfluid ($w-CS$) states, characterized by absence
(presence) of the minimum energy gap (see, Fig. 7a), and substantial spatial
antiferromagnetic correlations (see, Fig. 8a). Varying the parameters
$\tilde{\Delta}$ and $\beta$, we explored the complete phase diagram (see,
Figs. 7b and 8b).

Finally, we notice that an application of a small magnetic field allows us to
include local $\hat{\sigma}_{z,n}$-terms in the spin interacting Hamiltonian
(20). We anticipate that the quasi-1D $f$-JJAs directly embedded in the low
dissipative transmission line can be used as a convenient experimental
platform to establish the quantum simulations of large strongly interacting
spin systems, and to obtain various quantum collective states.

###### Acknowledgements.

We acknowledge the support provided by the Konrad-Adenauer-Stiftung, Germany.

##  Appendix A Higher-order multiple simultaneous spin flips

Here, we verify the statement that the amplitudes of higher-order spin flips
are exponentially small and can be neglected in the effective Hamiltonian
(20). The amplitude of the process involving spin flips in $n$ cells depends
on the distances between the constituent spins and the parity  of each spin-
flip pair. First, the amplitude of an even two cells ($n=2$) spin flips

$J^{+}(|n-m|)=J_{0}\exp\\{-4u_{0}^{2}\left[A(0)+A(|n-m|)\right]\\}.$ (28)

is exponentially smaller than the odd amplitude $J(n-m)$ (see, Eq. (22)):

$\frac{J^{+}(|n-m|)}{J(|n-m|)}=\exp(-8u_{0}^{2}A(|n-m|)).$ (29)

The higher-order terms for $n\geq 2$ can be written as

$\displaystyle\frac{J^{(n)}(\\{\mathcal{P}_{i},d_{i}\\})}{J_{0}^{(n)}}=$
$\displaystyle\exp\\{-4u_{0}^{2}\left[\frac{nA(0)}{2}+\sum_{i=1}^{\binom{n}{2}}\mathcal{P}_{i}A(d_{i})\right]\\},$
(30)

where $\mathcal{P}_{i}=\\{+1,-1\\}$ is the parity and $d_{i}$ the distances
between the two constituents of the $i$-th spin flip. All the
$\\{\mathcal{P}_{i},d_{i}\\}$ are fixed for a given spin flip configuration.
The sum $i$ runs over all possible spin flips (spin pairs).

As $A(d_{i})$ decays strongly with increasing distance $d_{i}$, we consider
only clusters of spin flips ($\\{d_{i}=1,2\\}$). Thus we get the third order
term

$\displaystyle\frac{J^{(3)}}{J^{(3)}_{0}}=$
$\displaystyle\exp\\{-4u_{0}^{2}\left[\frac{3A(0)}{2}\pm A(1)\pm A(1)\pm
A(2)\right]\\}.$ (31)

Such expression can be easily extended to higher orders. Since the exponent in
(A) increases logarithmically with $N$ the all terms $J^{(n)}$ with odd $n$
are strongly suppressed.

The fact that the higher-order terms with $n>2$ are indeed exponentially small
can also be seen in Fig. 12, which shows all spin flip amplitudes up to $4$th
order for a connected cluster of spin flips. From this, we conclude that the
highest amplitude within a certain order occurs when all short ranged
(nearest-neighbor) spin flips have odd parity. Furthermore, if the order is
increased, we find that the amplitude is exponentially small and can be
neglected in the effective model.

Figure 12: (color online) Comparison of the terms up to $4$th order: 1st order
(green), $2$nd order (blue), $3$rd order (red), $4$th order (black). The
amplitude of quantum tunneling $\Delta$ (green solid line), the odd second
order exchange interaction strength $J(1)$ (blue solid line), the even order
term $J^{+}(1)$ (blue dashed line). The third order combinations are given as
follows: odd-odd (red solid line), odd-even (red dashed line) and even-even
(red dotted line). For the 4th order (black), we have the following
combinations: o-o-o (solid), o-o-e (dashed), o-e-e (dotted) and e-e-e
(dashdot).

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# Existence of Circle Packings on Translation Surfaces

Anton Levonian

(December 2023)

###### Abstract

A translation surface is a surface formed by identifying edges of a collection
of polygons in the complex plane that are parallel and of equal length using
only translations. We determined that the same circle packing can be realized
on varying translation surfaces in a certain stratum. We also determined
possible complexities of contacts graphs and provide a bound on this
complexity in some low-genus strata. Finally, we established the possibility
of certain contacts graphs’ complexities in strata with genus greater than
$2$.

##  1 Introduction

Translation surfaces are very common and important types of Riemann surfaces,
and can be defined through gluing together a collection of polygons in the
complex plane without applying any rotations or reflections [9]. For example,
a torus is a translation surface, formed by identifying opposite sides of a
square. Translation surfaces have applications in billiards on rational
polygons ([10]) and geodesic flow ([2]). The introduction of Veech surfaces in
1989 (see [8]) has increased research on translation surfaces, and they have
since been extensively studied (see [1, 3, 4, 5, 9, 10]).

Before we elaborate any further, we will describe a few topological and
geometric properties that will be freely discussed throughout the paper. We
define the _genus_ of a surface as the maximal number of cuts along simple
closed curves that can be made without making the surface disconnected.
Translation surfaces of genus greater than 2 also contain _singular points_ ,
whose geometry is different from that of other points on the surface.

The results presented in this paper mainly concern circle packings on
different translation surfaces with similar properties. We give more
background on these objects in Section 2. We then settle questions regarding
both the existence of packings on distinct surfaces in Section 3 and the
simplest packings that can be realized on certain families of translation
surfaces in Section 4. Finally, we outline directions that future research may
take in Section 5.

##  2 Background

###  2.1 Definition and Basic Properties of Translation Surfaces

We begin by defining translation surfaces, the geometry of which will be the
main subject of the paper.

######  Definition 2.1.

A _translation surface_ is formed by identifying opposite sides of polygons
from $\mathcal{P}$, where $\mathcal{P}$ is a collection of several polygons in
the complex plane such that $\mathcal{P}$ has an even number of sides and
opposite sides of $\mathcal{P}$ are parallel and of equal length.

A particular family of translation surfaces, the square-tiled surfaces, will
be of importance.

######  Definition 2.2.

A _square-tiled surface_ is a translation surface for which $\mathcal{P}$ is
formed by joining together opposite sides of congruent squares.

Multiple polygons may define the same translation surface:

######  Remark 2.3.

Let $\mathcal{P}$ be a collection of polygons in the complex plane. Cut
polygons in $\mathcal{P}$ along straight lines and translate the resulting
polygons so that newly-created edges are identified to create a new collection
of polygons $\mathcal{P}_{1}.$ After identification of opposite edges,
$\mathcal{P}$ and $\mathcal{P}_{1}$ define the same surface (see Wright [9]).

We now introduce singular points:

######  Definition 2.4.

A _singular point_ is a point with at which the angle is greater than $2\pi$
to which multiple vertices of $\mathcal{P}$ get mapped.

######  Definition 2.5.

The _cone angle_ of a singular point is the total angle measure at this point.
It is the sum of the angles of all the vertices of $\mathcal{P}$ which get
mapped to the singular point.

It is a well-known result, outlined by Zorich in [10], that the total angle at
any point on a translation surface will always be a multiple of $2\pi.$

######  Definition 2.6.

The _order_ , represented by $\delta,$ of a singular point $v$ with cone angle
$2k\pi$ is $\delta(v)=k-1.$

######  Example 2.7.

Consider a square in the complex plane, and identify its opposite sides. This
yields a torus, a surface of genus $1.$ All 4 vertices of the square will be
mapped to a single point, with angle $4\cdot\frac{\pi}{2}=2\pi.$ Since this
point’s angle is not greater than $2\pi$, it is not a singular point.

###  2.2 Genus and Stratum of Translation Surfaces

We begin by introducing a well-known theorem that relates the genus of a
translation surface to the order of its singular points (see Massart [2]).

######  Theorem 2.1.

(Gauss-Bonnet Theorem). Let $X$ be a translation surface with $k$ singular
points $v_{i}$ each of order $\delta(v_{i})$, and let $\chi(X)$ be the Euler
characteristic of $X.$ Then the following is true:

$\sum_{i=1}^{k}\delta(v_{i})+\chi(X)=0.$

Theorem 2.1 will be useful for calculating the genus, $g$, of $X$ by recalling
the well-known formula

$\chi(X)=2-2g.$

######  Example 2.8.

Consider the square-tiled surface with $3$ squares shown in Figure 1. All of
the vertices of each of the 3 squares will be mapped to a single singular
point $v$ of angle $12\cdot\frac{\pi}{2}=6\pi$. So, $\delta(v)=2,$ and since
$\chi(X)=-\sum\delta(v_{i}),$ the Euler characteristic in this case is $-2$.
We now see that the genus is $g=2.$

Figure 1: Genus $2$ 3-squared surface.

One more important topic related to the genus of a translation surface is the
surface’s stratum.

######  Definition 2.9.

A _stratum_ , denoted by $\mathcal{H}(\kappa)$ where $\kappa$ is a partition
of $\chi(X)$, is the set of all translation surfaces with singular point
orders equal to the parts of $\kappa.$

######  Example 2.10.

The surface in Example 2.8 has 1 singular point of order $2$ and no other
singular points of nonzero order, so it is a member of $\mathcal{H}(2).$
Meanwhile, the surface in Figure 2 is a member of $\mathcal{H}(1,1),$ having 2
singular points each of order 1.

Figure 2: Translation surface in $\mathcal{H}(1,1).$

Because $(1,1)$ and $(2)$ are the only partitions of $2,$ all genus $2$
translation surfaces are either in $\mathcal{H}(1,1)$ or in $\mathcal{H}(2).$

######  Example 2.11.

All of the regular octagon’s vertices are mapped to a single point, so its
angle is $\pi(8-2)=6\pi.$ Hence, this point is a singular point of order $2$
and the regular octagon is in $\mathcal{H}(2).$

We now make the following important observation, which will be useful in
Section 4, about polygons which are mapped to $\mathcal{H}(2)$ surfaces.

######  Remark 2.12.

It follows from Proposition 1.16 of Wright [9] and Remark 2.3 that every
$\mathcal{H}(2)$ surface results from gluing opposite edges of a rotationally
symmetric octagon in the plane. This is because any polygon forming a cylinder
and a slit torus upon identification of opposite edges can be cut and reglued
into an octagon with rotational symmetry, and a cutting and regluing in the
other direction is also possible for every octagon. This is illustrated in
Figure 3 below:

Figure 3: The 2 polygons above are identified to the same $\mathcal{H}(2)$
surface.

###  2.3 Circle Packings

Of particular interest are circle packings, popularized by Thurston (see [7]),
on translation surfaces.

######  Definition 2.13.

A _circle packing_ is a collection of circles with disjoint interiors with a
_contacts graph_ $G$ so that each vertex of $G$ corresponds to a circle and 2
circles are tangent if and only if their corresponding vertices on $G$ are
connected by an edge.

In Definition 2.13, note that the radii and placement of the circles is not
determined by $G$.

######  Example 2.14.

Figure 4 contains a possible circle-packing on the 3-squared surface from
Figure 1, along with the corresponding contacts graph in Figure 5. We will
denote this particular packing as $C_{3}$ throughout.

Figure 4: A circle packing $C_{3}$ on a surface in the complex plane.

Figure 5: Contacts graph for $C_{3}.$

######  Remark 2.15.

The metric on a translation surface is not a flat smooth metric near singular
points of nonzero order (see Wright [9]). Hence, circles containing singular
points do not correspond with circles in the complex plane.

######  Example 2.16.

Figure 6 depicts a collection of circular sectors on a 3-squared $\mathcal{P}$
in the complex plane which, when opposite sides of $\mathcal{P}$ are
identified, yield a circle with radius less than $\frac{1}{2}$ centered at a
singular point.

Figure 6: Visualizing a circle centered at a singular point.

We now introduce a concept that will be useful in defining circle packing
equivalence.

######  Definition 2.17.

A _tangency pattern_ is the pattern of intersection of segments connecting
tangency points between pairs of circles. Consider 2 circle packings $C_{1}$
and $C_{2}$ with the same contacts graph $G_{C}$ and 2 quadruplets of not
necessarily distinct circles $(t_{1},u_{1},v_{1},w_{1})$ and
$(t_{2},u_{2},v_{2},w_{2})$ in $C_{1}$ and $C_{2}$, respectively so that:

  1. 1.

$(t_{i},u_{i})$ and $(v_{i},w_{i})$ are pairs of tangent circles for $i\leq
2$.

  2. 2.

$t_{1}$ and $t_{2}$ correspond to the same vertex of $G_{C},$ and likewise for
$u_{i},v_{i},$ and $w_{i}$.

Let $t_{1}u_{1}$ represent the segment connecting tangency points of $t_{1}$
and $u_{1},$ and likewise define $v_{1}w_{1},t_{2}u_{2},$ and $v_{2},w_{2}.$
The tangency patterns of $C_{1}$ and $C_{2}$ are the same if and only if, for
all $(t_{i},u_{i},v_{i},w_{i})$ in $C_{i},$ $t_{1}u_{1}$ intersects
$v_{1}w_{1}$ if and only if $t_{2}u_{2}$ intersects $v_{2}w_{2}$.

######  Example 2.18.

In Figure 7, 2 circle packings on 2 different translation surfaces of the same
stratum have the same contact graph and tangency patterns:

(a) 3-squared surface.

(b) 4-squared surface.

Figure 7: Equivalence of trivial circle packings.

The following theorem (see Stephenson [6]) is instrumental in studying circle
packings, and provides motivation for our explorations in Section 3.

######  Theorem 2.2.

(Koebe-Andreev-Thurston Theorem) Given a Riemann sphere with a graph $G,$
there exists a collection of circles on the surface whose contacts graph is
$G$ such that the circles form a circle packing of the surface that is unique
up to a Möbius transformation.

We now establish equivalence conditions for circle packings on translation
surfaces. These conditions are not the same as those defined by the Koebe-
Andreev-Thurston Theorem for spheres:

######  Remark 2.19.

Unlike in the case of a sphere, equality of contacts graphs is not sufficient
for the equality of circle packings up to a Möbius transform on translation
surfaces, as 2 circle packings with different tangency patterns are not
equivalent on surfaces with genus greater than $0.$ We call 2 circle packings
_equivalent_ if they have the same contacts graph and the same tangency
patterns.

######  Example 2.20.

The circle packing in Figure 8 has the same contacts graph as $C_{3}$ on a
surface of the same stratum, but has a different tangency pattern. Thus, these
2 packings are not equivalent.

Figure 8: Illustration of non-intersecting tangency segments.

##  3 Existence of Equivalent Circle Packings on Distinct Surfaces

Consider a circle packing on a translation surface and its contacts graph $G$.
One interesting question is whether, for all $G$, there exists a circle
packing, unique up to a Möbius transformation, on a different translation
surface of the same stratum with the same contacts graph and tangency
patterns. In other words, do questions of existence of equivalent circle
packings answered by Koebe-Andreev-Thurston Theorem on spheres hold for non-
zero genus strata? We answered this question in the negative for specific
surfaces in the $\mathcal{H}(2)$ stratum.

######  Theorem 3.1.

An equivalent packing to $C_{3}$ cannot be realized on any 4-squared
translation surface in $\mathcal{H}(2)$ without applying an affine
transformation.

###### Proof.

We start by noting that having any of the circles contain a singular point
cannot yield an equivalent packing to $C_{3}$ on a 4-squared surface, as shown
in Figure 9. Due to the equivalence outlined in Remark 2.3, we can, without
loss of generality, only consider 4-squared surfaces formed by joining 3
squares in a horizontal strip, and then vertically identifying a fourth square
to the left-most square in the strip, as in Figure 9.

Figure 9: Packing on a 4-squared surface with the same contacts graph as
$C_{3}.$

While the contacts graph in Figure 9 is the same as that of $C_{3},$ the 2
packings have differing tangency patterns, since the segment connecting the
self-tangency point of the red circle does not intersect the segment
connecting the mutual tangency points between the red and orange circles.
Since the corresponding segments do intersect in $C_{3}$, the packing in
Figure 9 and $C_{3}$ are therefore not equivalent. Observe that it was not
necessary to center the orange circle at the singular point as was done above.

It remains to observe the effect of not allowing any circles to contain
singular points in their interiors.

Figure 10: Configuration with no circles containing singular points.

In Figure 10, the red and blue circles must have radius $\frac{1}{2}$, due to
the self-tangency conditions. The orange circle is tangent to both circles,
with the tangency patterns preserved for the red circle. In fact, this is the
only unique configuration that satisfies the tangency patterns with no circles
containing singular points.

However, the blue circle is not tangent to the orange circle a second time, as
it should be in order to have the given contacts graph. In fact, the orange
circle cannot possibly meet this tangency criterion. The largest possible
radius for the orange circle is $\frac{1}{\sqrt{2}}$, depicted as a dashed
line in Figure 10. Even then, the sums of the diameters of the dashed orange
circle and the blue circle are $\sqrt{2}+1<3,$ so there will only be 1
tangency point even with maximal circle radii.

So, it is impossible to realize an equivalent packing to $C_{3}$ on a
4-squared surface of $\mathcal{H}(2)$ if an affine transformation is not
applied to this surface. ∎

######  Remark 3.1.

In fact, stretching out the 4-squared surface in $\mathcal{H}(2)$ vertically
by a factor of $\frac{4}{3}$ allows for an equivalent packing to $C_{3}$,
shown in Figure 11.

Figure 11: Note that the tangency patterns are the same as in $C_{3}$.

##  4 Simplest Achievable Contacts Graphs

Consider the stratum $\mathcal{H}(2)$ and a contacts graph on a translation
surface in that stratum. One question of particular interest is how simple
this contacts graph can be. We will now answer 2 fundamental questions: first,
how many loops can a single vertex of the graph have, and second, how many
edges can possibly connect 2 vertices.

Theorems 4.1 and 4.2 provide strict bounds on the amount of possible edges
between 2 vertices (multi-edges) and loops on a single vertex (multi-loops) in
contacts graphs on $\mathcal{H}(2)$ surfaces:

######  Theorem 4.1.

A maximum of $8$ multi-edges can be realized on any contacts graph in
$\mathcal{H}(2).$

###### Proof.

The maximal configuration with $8$ multi-edges is achieved on Figure 12 below.
The vertices of the contacts graph corresponding to the red and blue circles
of the circle packing have $8$ edges connecting them.

Figure 12: The configuration with maximum multi-edges before gluing of
opposite sides.

We now demonstrate that no more than $8$ multi-edges can be realized. By
Remark 2.12, the only polygons on which we must analyze circle packings are
octagons with at least 1 axis of rotational symmetry. Having 1 of the circles
contain a singular point cannot produce more than $8$ multi-edges, as the
second circle can only be tangent to each of the $8$ circular sectors in the
complex plane once. If neither circle contains a singular point, there can be
a maximum of $5$ multi-edges: 1 in the interior, and 1 along each pair of
opposite sides of the octagon.

Thus, at most $8$ multi-edges can be realized on any contacts graph in
$\mathcal{H}(2)$, as desired.

∎

######  Theorem 4.2.

A maximum of $9$ multi-loops can be realized on any contacts graph in
$\mathcal{H}(2).$

###### Proof.

Figure 13 displays a packing that, when opposite sides are identified, yields
the maximum amount of realizable multi-loops in the $\mathcal{H}(2)$ stratum.

Figure 13: The configuration with maximum multi-loops before gluing of
opposite sides.

To demonstrate that $10$ multi-loops are not realizable, we examine circular
sectors on rotationally symmetric octagons in the plane which are mapped to
circle packings on $\mathcal{H}(2)$ surfaces. As in the proof of Theorem 4.1,
only packings resulting from gluing of such octagons need to be analyzed by
Remark 2.12.

We now perform casework based on the number of vertices of the octagon whose
interior angles are greater than $\pi$, which we call _concavities_ throughout
the proof. There can only be an even number of these, as opposite sides must
be parallel. There can also be no more than $4$ concavities, as the sum of the
interior angles of an octagon with $6$ concavities is greater than $6\pi.$

  1. 1.

If there are no concavities, there are only 4 possible multi-loops
corresponding to $4$ pairs of opposite sides of the octagon.

  2. 2.

If there are 2 concavities, the vertices with angles greater than $\pi$ must
be opposite each other. A circle that does not contain the singular point can
only have at most $4$ multi-loops by the argument in the previous case. If the
circle contains a singular point, there can be at most $1$ self-tangency in
the interior of the octagon and $4$ self-tangencies on the edges, which again
correspond to $4$ pairs of opposite sides, for a maximum of $5$ multi-loops in
this case.

  3. 3.

Finally, if there are 4 concavities, we again maximize the amount of multi-
loops with a circle containing the singular point. There can be a maximum of 4
self-tangencies on the edges of the polygon and 5 in the interior, since it is
well-known that 4 circles with the same radius in the plane can have a maximum
of 5 mutual tangency points. In fact, the only such configuration (unique up
to an affine transformation in the plane) is pictured in Figure 14, and
placing alternating vertices of the octagon at the centers of these circles
yields the configuration shown in Figure 13. This case therefore yields a
maximum of $9$ multi-loops.

Figure 14: Configuration of 4 circles in the plane with 5 mutual tangencies.

To conclude, 9 is the maximum possible number of multi-loops of any circle
packing on any surface in $\mathcal{H}(2)$. ∎

Theorems 4.3 and 4.4 demonstrate that certain numbers of multi-edges and
multi-loops can be realized on contacts graphs of packings on general genus
$g$ translation surfaces with 1 singular point.

######  Theorem 4.3.

It is possible to realize $4g$ multi-loops on a contacts graph on at least 1
surface of the genus $g$ stratum $\mathcal{H}(2g-2).$

###### Proof.

Consider the regular polygon $\mathcal{Q}_{0}=A_{2}A_{4}\cdots A_{4g}$ with
$2g$ vertices in the complex plane. For each side $\overline{A_{2i}A_{2i+2}}$
of $\mathcal{Q}_{0}$, construct the equilateral triangle
$A_{2i}A_{2i+1}A_{2i+2}$ so that $A_{2i+1}$ is not in the interior of
$\mathcal{Q}_{0}.$ Let $\mathcal{Q}=A_{1}A_{2}\cdots A_{4g-1}A_{4g}.$

For each $2i+1,$ center a circular sector $C_{2i+1}$ contained in
$\mathcal{Q}$ with angle measure $\frac{\pi}{3}$ and radius
$\frac{A_{1}A_{2}}{2}$ at $A_{2i+1},$ and for each $2i,$ center a circular
sector $C_{2i}$ contained in $\mathcal{Q}$ with angle measure
$\left(\frac{2g-2}{2g}+\frac{2}{3}\right)\pi$ and radius
$\frac{A_{1}A_{2}}{2}$ at $A_{2i}.$ Observe that members of the set
$\\{C_{n}|1\leq n\leq 4g\\}$ have $2$ mutual tangency points for each of the
$2g$ pairs of opposite sides of $\mathcal{Q}$, as well as $2g$ mutual tangency
points at the midpoints of the $2g$ sides of $\mathcal{Q}_{0}.$

Now, we identify the opposite sides of $\mathcal{Q}$ to form a surface
$X_{\mathcal{Q}}$ in the $\mathcal{H}(2g-2)$ stratum: $X_{\mathcal{Q}}$ will
have exactly 1 singular point of cone angle $(4g-2)\pi$ and, consequently, of
order $2g-2$. Note that all sectors $C_{n}$ get mapped to a single circle $C$
centered at the singular point of $X_{\mathcal{Q}}$. The number of self-
tangencies of $C$ is equal to the number of unique tangency points between
members of $\\{C_{n}|1\leq n\leq 4g\\}.$ Note that each of the $2g$ pairs of
tangency points in the complex plane that are at midpoints of pairs of
opposite sides of $\mathcal{Q}$ get identified to 1 point on the surface,
contributing $2g$ to the number of self-tangencies of $C$. The tangency points
at midpoints of the $2g$ sides of $\mathcal{Q}_{0}$ are in the interior of
$\mathcal{Q}$, and so contribute another $2g$ to $C$’s self-tangencies. Thus,
it is possible for a circle packing realized on a surface in
$\mathcal{H}(2g-2)$ to have $4g$ self-tangencies, as desired. ∎

######  Theorem 4.4.

It is possible to realize $4g$ multi-edges on a contacts graph on at least 1
surface of the genus $g$ stratum $\mathcal{H}(2g-2).$

###### Proof.

Consider the regular $4g$-gon $\mathcal{Q}_{\text{reg}}=A_{1}A_{2}\cdots
A_{4g}.$ Identification of opposite sides of $\mathcal{Q}_{\text{reg}}$ yields
a genus $g$ surface in $\mathcal{H}(2g-2).$ At each vertex $A_{i}$ of
$\mathcal{Q}_{\text{reg}},$ center a circular sector of angle measure
$\frac{2g-1}{2g}\pi$ and radius $r<\frac{A_{1}A_{2}}{2}.$ Next, center a
circle $C_{1}$ with radius
$\frac{A_{1}A_{2}}{2\sin\left(\frac{\pi}{4g}\right)}-r$ at the center of
$\mathcal{Q}_{\text{reg}}.$

After identification of opposite side lengths, each circle centered at a
vertex of $\mathcal{Q}_{\text{reg}}$ gets mapped to a single circle $C_{2},$
and there will be $4g$ tangency points between $C_{1}$ and $C_{2}$
corresponding to $4g$ vertices of $\mathcal{Q}_{\text{reg}}.$ Thus, we have
constructed a packing whose contacts graph has $4g$ multi-edges on a surface
in $\mathcal{H}(2g-2).$ ∎

Similarly, on genus $g$ translation surfaces with 2 singular points of equal
order, the possibility of certain packing complexities is demonstrated in
Theorems 4.5 and 4.6.

######  Theorem 4.5.

It is possible to realize $2g+1$ multi-loops on a contacts graph on at least 1
surface of the genus $g$ stratum $\mathcal{H}(g-1,g-1).$

###### Proof.

First, note that the surface formed by identifying opposite edges of a regular
polygon $\mathcal{Q}_{\text{reg}}=A_{1}A_{2}\cdots A_{4g+2}$ is in
$\mathcal{H}(g-1,g-1).$ Now, center $2g+1$ circular sectors with radii
$A_{1}A_{2}\cdot\sin\left(\frac{g}{2g+1}\pi\right)$ at each even-numbered
vertex $A_{2i}.$ This will yield $2g+1$ tangency points in the interior
$\mathcal{Q}_{\text{reg}}$, which correspond to $2g+1$ multi-loops on the
contacts graph of the $\mathcal{H}(g-1,g-1)$ surface, as desired. ∎

######  Theorem 4.6.

It is possible to realize $2g+2$ multi-edges on a contacts graph on at least 1
surface of the genus $g$ stratum $\mathcal{H}(g-1,g-1).$

###### Proof.

Consider 2 regular $(2g+2)$-sided polygons, $\mathcal{Q}_{1}$ and
$\mathcal{Q}_{2}$. Choose a side of each $(2g+2)$-gon, glue $\mathcal{Q}_{1}$
and $\mathcal{Q}_{2}$ together along those 2 sides, and then remove the edge
along which they were glued. The result is a $(4g+2)$-gon $\mathcal{Q}$,
which, as discussed in the proof of Theorem 4.5, becomes a surface in
$\mathcal{H}(g-1,g-1)$ upon identification of opposite edges.

Now, inscribe a circle $C_{1}$ in $\mathcal{Q}_{1},$ and likewise inscribe a
circle $C_{2}$ in $\mathcal{Q}_{2}$. Identifying opposite edges of
$\mathcal{Q}$ yields $2g+2$ mutual tangencies between $C_{1}$ and $C_{2}$,
corresponding to the $2g+2$ edges of $\mathcal{Q}_{1}.$ Thus, the contacts
graph of the packing consisting of $C_{1}$ and $C_{2}$ on the surface formed
by identifying opposite sides of $\mathcal{Q}$ has $2g+2$ multi-edges,
satisfying the desired condition. ∎

######  Corollary 4.7.

Up to $5$ multi-loops and $6$ multi-edges can be realized on a contacts graph
on at least 1 surface in $\mathcal{H}(1,1).$

###### Proof.

Substitute $g=2$ in Theorems 4.5 and 4.6. ∎

##  5 Research Directions

Theorems 4.1 and 4.2 provide a bound on the complexity of contacts graphs in
$\mathcal{H}(2)$, while no such bound is presented for contacts graphs on
$\mathcal{H}(1,1)$ surfaces in Corollary 4.7. A natural research direction is
therefore establishing such a bound in order to complete our results on
complexity in the genus $2$ stratum. Progress can also be made in bounding the
numbers of possible multi-loops and multi-edges on surfaces in the genus $g$
strata $\mathcal{H}(2g-2)$ and $\mathcal{H}(g-1,g-1),$ continuing the progress
made in Theorems 4.3, 4.4, 4.5, and 4.6.

Additionally, only partial progress was made towards describing equivalent
packings that can be realized on distinct, same-stratum surfaces. It would be
particularly interesting to investigate the existence of packings that can be
realized on any surface in certain strata, and to give the most complex such
packings in $\mathcal{H}(2),$ $\mathcal{H}(1,1),$ and potentially higher-genus
strata.

##  6 Acknowledgements

I would like to sincerely thank my PRIMES-USA mentor, Professor Sergiy
Merenkov, for his guidance during the reading and research phases of the
program. Without him, this work would not have been possible. I am grateful to
MIT PRIMES-USA for giving me the invaluable opportunity to conduct this
research. I am also thankful to Doctor Tanya Khovanova for her comments on my
initial reports and this paper.

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