# Formation and Fluctuation of Two-dimensional Dodecagonal Quasicrystal

Uyen Tu Lieu<EMAIL_ADDRESS>Mathematics for Advanced Materials-OIL,
AIST, 2-1-1 Katahira, Aoba, 980-8577 Sendai, Japan  Natsuhiko Yoshinaga
<EMAIL_ADDRESS>Mathematics for Advanced Materials-OIL, AIST, 2-1-1
Katahira, Aoba, 980-8577 Sendai, Japan  WPI-Advanced Institute for Materials
Research (WPI-AIMR), Tohoku University, 2-1-1 Katahira, Aoba, 980-8577 Sendai,
Japan

###### Abstract

The self-assembly of two-dimensional dodecagonal quasicrystal (DDQC) from
patchy particles are investigated by Brownian dynamics simulations. The patchy
particle has a five-fold rotational symmetry pattern described by the
spherical harmonics $Y_{55}$. From the formation of the DDQC obtained by an
annealing process, we find the following mechanism. The early stage of the
dynamics is dominated by hexagonal structures. Then, nucleation of dodecagonal
motifs appears by particle rearrangement, and finally the motifs expand whole
system. The transition from the hexagonal structure into the dodecagonal motif
is made by the collective rational motion of the particles. The DDQC consists
of clusters of dodecagonal motifs, which can be classified into several
packing structures. By the analyses of the DDQC under fixed temperature, we
find the fluctuations are characterised by changes in the network of the
dodecagonal motifs. Finally we compare the DDQC assembled from the patchy
particle system and isotropic particle system. The two systems both share a
similar mechanism of the formation and fluctuation of DDQC.

##  I Introduction

The quasicrystals (QCs) are ordered structures lacking periodic translational
symmetry [1, 2]. Different from the crystals which possess 2-, 3-, 4-, 6-fold
rotational symmetry, the quasicrystals may have 5-, 8-, 10-, 12-, 18-fold
rotational symmetry. The quasicrystal can be applied in various applications,
such as advanced coatings, reinforced composites, optics, photovoltaics,
magnetism [3], superconductivity [4], bandgap materials in photonic devices
[5]. The appearance of quasicrystals and approximants have been observed in
different length scales, from intermetallics [1, 6] to meso-scale [7, 8, 9,
10, 11]. Recently, quasicrystalline structures have been found in soft
materials, such as block copolymers [12, 13], surfactants [14], colloids [15],
and dendrimers [16].

The quasiperiodic self-assembly has been obtained mainly by three types of
mechanisms: isotropic interactions between particles, anisotropic
interactions, or polydispersed systems. In the first case, the isotropic
interaction potential depends only on the distance between particles, but has
at least two length scales [17, 18, 19, 20, 21, 22]. For example, the Lennard-
Jones-Gauss potentials can produce decagonal QC, dodecagonal QC [17], or the
hard-core/square-shoulder potentials [22] create a family of QC structures
with 10-, 12-, 18- and 24-fold bond orientational order. Continuum description
has also been proposed in ref.[23]. The three-well oscillating pair potential
is able to form a three-dimensional icosahedral QC in non-atomic systems [18].

In the second case of anisotropic interactions between particles, the
interactions depend on their mutual orientation while the distance-dependence
has only one length scale. For example, the five-patch particles, whose five
sticky patches are equally distributed around the equator of the sphere, can
arrange into a two-dimensional dodecagonal QC [24, 25, 26, 27]; hard
tetrahedra create three-dimensional dodecagonal QC [28].

In the third type of systems, DDQC and approximants structures are realised in
polydispersed systems [8, 29, 30], such as binary nanoparticle [8], bi-
dispersed pentavalent and hexavalent patchy particles [29].

Despite the intensive studies on the structures of quasicrystals, kinetics and
dynamics of quasicrystal growth, however, are still incomplete [31, 32, 33,
34]. It is not clear how the quasicrystals appear from a liquid or crystalline
state, what is happening at the periphery of quasicrystals duing their growth,
and how the structures fluctuate. It is also of interest whether the dynamics
and kinetics of quasicrystals are dependent or not on the three aforementioned
mechanisms. Most of the previous studies focused on the analysis of the
structures. In ref. [24, 25], stable DDQC was generated via Monte-Carlo
simulations of the five-patch particles. In ref. [26], the growth of DDQC was
studied by particle deposition on a prepared quasicrystalline substrate. The
growth of a three-dimensional DDQC was studied in ref. [31], and it was
suggested that characteristic structures (icosahedrons in that system)
preferentially appear around the nucleus of DDQC. The growth of a two-
dimensional DDQC was also studied in ref. [35]. Nevertheless, what kind of
local structures preferentially appear near the nucleus of quasicrystals in
other systems remains unknown. Also, it is of relevance to clarify how those
structures are transformed as a part of DDQC by particle displacements.

In this study, we investigate the kinetics and dynamics of DDQC by computer
simulations. The majority of reported quasicrystals in soft materials are
DDQC. Therefore, we focus on the formation of DDQC. We have developed a model
for the interaction of a pair of anisotropic particle whose surface pattern is
described by spherical harmonics $Y_{lm}$ [36, 37]. Using the five-fold
symmetry patchy particles, we are able to assemble the dodecagonal
quasicrystal. We investigate the formation of the DDQC from a bulk phase, and
the fluctuation after the DDQC is formed. The same procedure is applied to the
DDQC assembled from the isotropic particle system. We compare the kinetics and
dynamics of the DDQCs from the patchy particle system and isotropic particle
system.

##  II Methods

###  II.1 Numerical simulation

We consider the patchy particle with five-fold symmetry, composing of ten
alternating patches of two different types (Fig. 1). The pattern on a
spherical particle is described by spherical harmonics $Y_{55}$ [36, 37].

The pairwise interaction of the particle include a Weeks-Chandler-Andersen
(WCA) term preventing the overlap and a Morse-like, orientation-dependent
term. The details of the anisotropic potential is given in Appendix A. Figure
2 illustrates the potential used in the study at different particle
configurations.

Figure 1: Illustration of a patchy particle with five-fold symmetry expressed
by the pattern of $Y_{55}$ and the corresponding patchiness. The patches in
the same colours are attractive, while those in the different colours are
repulsive. Figure 2: Pairwise potential of patchy particles at different
configurations. The red arrows indicate the orientations of the patchy
particles.

The assembly of patchy particles is performed by Brownian dynamics simulations
[38]. The positions $\mathbf{r}$ and orientations $\bm{\Omega}$ of the
particles after the time step $\Delta t$ are updated according to the
equations:

$\mathbf{r}(t+\Delta
t)=\mathbf{r}(t)+\frac{D^{\text{T}}}{k_{\text{B}}T}\mathbf{F}(t)\Delta
t+\bm{\delta}^{\text{G}}\sqrt{2D^{\text{T}}\Delta t},$ (1)
$\bm{\Omega}(t+\Delta
t)=\bm{\Omega}(t)+\frac{D^{\text{R}}}{k_{\text{B}}T}\mathbf{T}(t)\Delta
t+\bm{\delta}^{\text{G}}\sqrt{2D^{\text{R}}\Delta t},$ (2)

where $D^{\text{T}},D^{\text{R}}$ are the translational and rotational
diffusion coefficients, respectively; the force $\mathbf{F}$ and torque
$\mathbf{T}$ are derived from the pair potential $V$; and each component of
$\bm{\delta}^{\text{G}}$ is a Gaussian distribution with zero mean and unit
variance. The simulations are conducted in dimensionless form where the
characteristic length, energy, time, and temperature, are the particle radius
$a$, Lennard-Jones potential well-depth $\epsilon$, the Brownian diffusion
time $\tau_{\text{B}}=a^{2}/D^{\text{T}}$, and $\epsilon/k_{B}$,
respectivetly.

The spherical particles are confined to a two-dimensional plane while rotate
freely in three dimensions. The initial condition of the simulation is random
distribution of positions in a periodic box of the size $L_{x}\times L_{y}$
with random orientations. The number of particles is $N=1024$. The density is
defined as area fraction $\rho_{\text{a}}=\pi N/(L_{x}L_{y})$.

We consider the systems undergoing annealing and at fixed temperature. In the
annealing, the temperature is varied from $T_{max}=1.2$ to $T_{min}=0.4$ with
intervals of $\Delta T=0.0125$. At each temperature the number of steps is
$0.1\times 10^{6}$, and the total steps are $6.5\times 10^{6}$. The time step
is set at $\Delta t=0.5\times 10^{-3}T$, thus the total time of the annealing
simulation is $t=2600$. We have checked the annealing schedule is slow enough
to make stable quasicrystals. For the system at fixed temperature, the system
at the initial condition is suddenly quenched at the temperature $T$, and is
relaxed at the fixed $T$. We conduct simulations at fixed temperature
$T\in[0.5,1.1]$, the number of simulation steps is $5\times 10^{6}$, the
simulation time at each temperature is $t=2500T$.

To analyse the fluctuation of the DDQC, simulations are performed with the
parameters: area fraction $\rho_{a}=0.75$, fixed temperature $T=0.8$ and
$50\times 10^{6}$ steps, which are approximately 10 times longer and slightly
less dense than the annealing simulation. For five independent simulations,
the DDQC is formed from a rich hexagonal phase after the first $5\times
10^{6}$ steps. The fluctuation of the DDQC is reported after it is formed. We
also investigate the DDQC assembled from the isotropic interacting particles.
The details of the potential and simulation condition are given in Appendix B.

###  II.2 Structural analysis

The types of local structure of each particle are determined by the connection
of the nearest neighbours of a particle. The local environments in a
dodecagonal quasicrystal are $\sigma$, $H$, $Z$ [24] as given in Fig. 3. These
local structures are based on the number of common neighbours of the
neighbouring particles analogial to the Frank-Kasper phases. For example, a
$\sigma$ particle has five neighbours. One of them has two common neighbours,
and four of them have only one common neighbour. Therefore the local structure
of a $\sigma$ particle is identified as $\\{21111\\}$. The other local
structures are $H$ $\\{$22110$\\}$, $Z$ $\\{$22222$\\}$, $D_{1}$
$\\{$22211$\\}$, $D_{2}$ $\\{$222211$\\}$. In this study, the new local
structures $D_{1}$, $D_{2}$ result from fluctuating $Z$ particles, where they
form a diamond shape shown in Fig. 3. The particle which does not fall into
these categories is considered as undefined $U$. The number of unidentified
particles is only $\lesssim 5\%$ of the total number of the particles in the
system.

Figure 3: Local structure $\sigma$, $Z$, $H$, $D_{1}$, $D_{2}$ and its nearest
neighbours $r\leq 2.5a$, and the illustration of the diamond structure
$D_{1}$, $D_{2}$ inside a cluster of $Z$ particles. The common neighbours are
also given.

The local structure of the particle varies with time. The interchange of a
given local structure to/from other local structures is determined between two
two consecutive snapshots different by $10^{4}$ time steps, for each local
structure. The purpose of this calculation is to identify the relevant
“reaction” of the local structures during the growth of DDQC. For example, we
can calculate the interchange rate between $Z$ and $\sigma$. If the rate of
$Z\rightarrow\sigma$ is higher than that of $\sigma\rightarrow Z$, then the
speed of $Z\rightarrow\sigma$ is faster than $\sigma\rightarrow Z$.

The DDQC has a characteristic dodecagonal motif [25, 7] (see Fig. 5(e)). The
motif is made of one $Z$ particle at the centre, six $\sigma$ particles on the
first ring, and twelve $\sigma$ particles on the second ring. Figure 4 shows
several ways of packing the motifs. These dodecagonal motifs are identified by
the network of $Z$ particles in each snapshot of simulations. The $Z$
particles are centres of each dodecagonal ring. We describe the motif packing
by the triangular tiles in the network of $Z$ particles obtained from Delaunay
triangulation. Different clusters of motifs are shown in different colours.
Each cluster is characterised by the distance between the dodecagonal motifs.
For example, the dodecagonal motifs can interpenetrate in Fig. 4(a), share a
common edge in Fig. 4(d-e); or connect via extra $\sigma$ particles in Fig.
4(e). The criteria of the clusters are based on the edge lengths of the
triangular tiles; e.g. the edge lengths of the perfect cluster in Fig. 4(a)
are $l=(1+\sqrt{3})2a$; a simulated cluster is of this type if the edge
lengths $l_{\text{sim}}$ satisfy $|l_{\text{sim}}-l|<0.1l$. We will discuss
how the structure of the clusters of motifs changes in time. The time-
dependent network structure during $[t,t+\delta t]$ is visualised by
overlapping the images at $[t,\ldots,t+\delta t]$ (see Fig. 11, 12).

Figure 4: Typical packing of dodecagonal motifs. The packing is characterised
by the Delaunay triangulation of the dodecagonal motif centres. Particle
colours are given in Fig. 3.  Figure 5: Change in local structures during
annealing simulations at $\rho_{\text{a}}=0.78$. (a) Ratio of the local
structure in time; the line represents the data of a simulation, the shade
area corresponds to the 95$\%$ confidence intervals of 10 independent
simulations. The growth of the dodecagonal quasicrystal is divided into three
stages. (b-d) The snapshots and their Fourier transformation; the colour of
particle indicates the local structure as shown in the legend and Fig. 3. (e)
Illustration of two kinds of dodecagonal motifs in the boxes in (d), they are
interchangeable by rotating the particles on the first ring $\pm 30^{\circ}$.

##  III Results of patchy particle system

###  III.1 The formation of dodecagonal quasicrystal

The dynamical self-assembly using the five-fold symmetric patchy particles
under annealing is investigated. Figure 5 shows the change in the ratios of
local structures during the formation of DDQC and the representative snapshots
with the colour coded according to local structures $\sigma$, $Z$, $D_{1}$,
$D_{2}$ (Fig. 3). Dynamical formation of a dodecagonal quasicrystal can be
decomposed into three stages. At stage I, the majority of particles are the
$Z$ type, with a ratio of around 0.7. Both the ratios of $D_{1}$ and $D_{2}$
are around 0.1. The snapshot of this stage is a $Z$-rich structure with clear
hexagonal spots in its Fourier transform. The $D_{1}$ and $D_{2}$ particles
often appear in pairs as a diamond shape surrounding by $Z$ particles. Stage
II corresponds to the growth of the QC where the ratios of $Z$, $D_{1}$,
$D_{2}$ decrease, while that of $\sigma$ increases. A $\sigma$-rich region is
formed in the bulk of $Z$ particles. The $\sigma$-rich region contains a few
$Z$ particles inside. The $D_{1}$ and $D_{2}$ particles accumulate at the
boundary between this region and the ‘bulk’ $Z$ particles. Finally at stage
III, when the expansion of $\sigma$ reaches the system size, all the ratios
maintain constant values with fluctuations. The $\sigma$-rich region expands
whole in the system and a dodecagonal quasicrystal (DDQC) is assembled, as we
can observe the twelve-fold rotational symmetry in the Fourier space [24, 7].

There are several dodecagonal motifs in the QC structure. The motif contains
one $Z$ particle at the centre and six $\sigma$ particles on the first ring
and twelved $\sigma$ particles on the second ring (see Fig. 5(e)). There are
two types of motifs which are different by 30∘ degrees of the first ring. The
QC structures consists of the two types of dodecagonal motifs packed in
different ways as shown in Fig. 4. The four structures in which the $Z$
particles of the motif centres form either equillateral triangle (Fig. 4(a,d))
or right angle triangle (Fig. 4(b,e)), were obtained in [24] using patchy
particles with five-fold or seven-fold symmetry. In our simulations, we also
obtain the structure in which the $Z$ particles form isosceles triangle (Fig.
4(c)). It is noted that the term “dodecagonal quasicrystal” here is loosely
used for a structure whose Fourier transform has spots with clear twelve-fold
symmetry. The obtained structures are not clear enough to be identified as
quasicrystal or approximants. The structure contains both defects and ordered
dodecagonal motifs.

Figure 6: Interchanges between the local structures during the growth of DDQC
at $\rho_{\text{a}}=0.78$ by decomposition of the “in” and “out” amount of
selected pairs of local structures (a). (b-e) The corresponding data inside
the box in (a). The graph show moving average of 15 data points.

The ratio in Fig. 5(a), however, can not show the “reaction” rate of each
local structure change to/from other ones. Figure 6 shows the interchange
among $Z,D_{1},D_{2},\sigma$. During stage I, a majority of $Z$ particles
changes to and from $D_{1},D_{2}$ as the “reactions” are comparable in both
forward and backward directions. In stage II, the particles $Z,D_{1},D_{2}$
change to $\sigma$ faster than the backward direction. It suggests that the
relevant ingredients for the formation of the $\sigma$ particle, or the
dodecagonal motif, are the $Z,D_{1},D_{2}$ particles.

We propose that there is a relation between $Z,D_{1},D_{2}$, and $\sigma$.
Specifically, the particles $D_{1}$, $D_{2}$ play as an intermediate role for
the formation and expansion of the dodecagonal quasicrystal from a $Z$-rich
structure. Figure 7(a) shows the decomposition of the $D_{1}$, $D_{2}$
particles in terms of the bulk $D_{\text{bulk}}$ and the boundary
$D_{\text{boundary}}$ amounts. The $D_{\text{boundary}}$ are defined as the
$D$ particles ($D_{1}$ and $D_{2}$) neighbouring to the $\sigma$ particles,
while the rest is defined as $D_{\text{bulk}}$. The bulk $D$ fluctuates around
a constant value, then decreases to zero as the QC occupies all the space.
Figure 7(b) show the relation between the bulk $D$ particles and $Z$
particles. For the bulk particles, the amount of $D_{1}$ and $D_{2}$ is almost
same as they often appear as pairs (see snapshots in Fig. 5, 7), and linearly
increase with $Z$. We fit the data in Fig. 7(b) by linear regression, and find
the slope of $6.6^{-1}$. In Fig. 3 we illustrate a diamond shape made of four
$D_{\text{bulk}}$ in twelve $Z$. In this case, the ratios of $D_{1}$ and
$D_{2}$ to $Z$ is equal as $6^{-1}$. This argument supports a picture in which
$D_{1}$ and $D_{2}$ particles appear as fluctuations of $Z$ particles and
appear in pairs.

Figure 7: Decomposition of the local structure $D_{1}$, $D_{2}$ into the
corresponding $D_{1}$, $D_{2}$ in bulk and in boundary. (a) The composition at
bulk and boundary; the illustration shows the bulk particles and the boundary
particles. (b) Scaling of bulk $D$ with $Z$ for simulation steps $[3000\times
10^{3},4100\times 10^{3}]$. (c) Scaling of boundary $D$ with $\sigma$ for
simulation steps $[3000\times 10^{3},3600\times 10^{3}]$. The dashed lines are
regression data.

Regarding the $D$ particles at the boundary, as given in Fig. 7(a), these
particles occupy a small amount before the growth of the $\sigma$-rich domain,
then increases during the growth, and eventually decreases to a plateau.
Before the QC is formed, the number of stable $D$ particles is small and they
appear stochastically associated with $\sigma$ particles; then when the QC is
expanded, the number of boundary $D$ particles increases. The decreases of the
boundary $D$ is because of the limit of the finite system size. Moreover, the
number of $D_{1}$ particles is higher than $D_{2}$ particles, suggesting that
there is an extra $D_{1}$ which may contribute to the transformation from
$D_{1}$ to $\sigma$. Figure 7(c) shows that the $D_{\text{boundary }}$ grows
linearly with $\sigma^{0.5}$. This is because the growing QC has the area
$\propto N_{\sigma}$ and the circumference $\propto\sqrt{N_{\sigma}}$.

We seek for a simple mechanism of DDQC growth. In the simulation, the
particles fluctuate locally and change their local structures continuously.
However, we observe a typical local rotation of the particle during growth of
DDQC. Figure 8 suggests a mechanism from a $Z$-rich structure (stage I) to a
DDQC (stage III). In the schematic of Fig. 8(a), from a hexagonal lattice, the
first ring of the hexagonal lattice rotates by 15∘ either clockwise or
anticlockwise. Simultaneously some particles of the second ring displace and
eventually a dodecagonal motif is formed. The $D_{1}$, $D_{2}$ particles also
appear during this local rotation, and become $\sigma$ or $Z$ as the
dodecagonal motif grows. We can calculate the rotation of the first neighbours
of a centre particle (see Appendix C) from the displacements of the particles
between two snapshots in Fig. 8(b). Figure Figure 8(c) shows that around the
centres of the DDQC motif (roughly the $Z$ particles in the DDQC), there are
rotations for the particles on the first rings. The histogram in Fig. 8(d)
approximately has two peaks at $\pm 15^{\circ}$.

Figure 8: Local rotation during the growth of a DDQC. (a) Illustration of
local rotation from a hexagonal structure to a dodecagonal motif by rotating
the particles on the first ring in clockwise direction $15^{\circ}$. (b) Two
snapshots before and after the DDQC is formed. (c) Corresponding displacement
for the snapshots in (b). The yellow dots are the $Z$ particles in the DDQC.
The size of $Z$ particles indicates the “strength” of the rotation of the
first ring. The $Z$ particles of very low rotation are not shown. The
displacement of the first ring of the $Z$ particle reveals a rotation. (d)
Histogram of the rotations of the neighbouring particles of the $Z$ particles
in (c). The vertical dashed lines show rotations at $\pm 15^{\circ}$.

To summarise, the mechanism of the self-assembled DDQC from patchy particles
can be proposed as follows: (i) initially the structure is rich in $Z$
particle and thermal fluctuation induces the pairs of $D_{1}$ and $D_{2}$ in
the bulk of $Z$, (ii) at the right temperature, the $\sigma$ particles are
formed and they cluster together to dodecagonal motifs. (iii) the DDQC rapidly
expands at the boundary where the transformation
$Z\rightarrow(D_{1},D_{2})\rightarrow\sigma$ occurs. Around the centre of the
dodecagonal motifs, local rotations of the nearest neighbours take place.

###  III.2 Temperature dependent

In this section, the effect of temperature on the DDQC is investigated. Figure
9 shows the dependence of local structure ratio on temperature for annealing
simulations and fixed temperature simulations. In both cases, there are two
phases: $Z$-rich and $\sigma$-rich, separated by a critical temperature
$T^{*}\approx 0.8$. The $Z$-rich phase corresponds to $T>T^{*}$, where the
structure is dominated by $Z$ particles and some $D_{1}$, $D_{2}$ particles
due to the strong thermal fluctuation. In this case, the global structure is
hexagonal. The $\sigma$-rich region appears at $T<T^{*}$, where the structure
is dominated by $\sigma$ particles. There is a slight difference in $T^{*}$ of
the annealing case and fixed temperature case. This is expected because the
onset of the quasicrystal, i.e. the growth from the first dodecagonal motif,
depends on both temperature and time, and these two parameters can not be
directly compared in the two simulation procedures. For example, at $T\approx
T^{*}$, the onset of annealing QC is in a uniform manner, whereas it is more
scattered for fixed temperature QC. As a result, among ten independent
simulations at fixed temperature $T=0.825$, three of them are DDQC, and the
remaining seven are $Z$-rich. Regarding the quality of the QC of fixed
temperature simulation, the ratio of $\sigma$ particle decreases as $T$
decreases. This is because the simulation is equivalent to very fast quenching
before fixing the temperature, the particles are kinetically trapped; hence
the structure is more disordered and contains more defects. Such a behaviour
is not observed in the slow annealing case, as the ratio of $\sigma$ is
independent of temperature. The results reveal that the patchy particle DDQC
can be obtained by annealing easier than fixed temperature when $T\leq T^{*}$.
The self-assemblies of high quality DDQC corresponds to the annealing scheme
or fixed temperature at $T=T^{*}$.

Figure 9: Dependence of ratio of local structure on temperature in (a)
annealing simulations, (b) fixed temperature simulations at
$\rho_{\text{a}}=0.78$. The dashed lines estimate the critical temperature
$T^{*}$.

###  III.3 Local fluctuation after DDQC is formed

Figure 10: Example of fluctuation of dodecagonal motif network with $\delta
t=10^{5}$ steps at $T=0.8$ when the network (a) locally maintains or (b)
changes. Left panels: network of $Z$ particles and the displacements of the
particles between two snapshots. Right panels: illustration of the network in
the left; colours corresponding to different packed dodecagonal motifs in Fig.
4, the white triangles are ill-defined motif packings. The coloured dots
represent particle positions and particle types like Fig. 3. The red arrows
shows the displacements of the particles with $|r(t+\delta t)-r(t)|\geq 0.5a$.
In the overlapping snapshots in (b), bold and clear edges/faces suggest the
structure maintains, whereas blur ones indicate a temporal change of
structure. Figure 11: Fluctuation of dodecagonal motif network after a long
time scale $\delta t=10^{6}$ steps at $T=0.8$. (a) Collective movement of the
particle as rearrangement of the network. (b) Histogram of the rotation angle
of the particles in (a) with peaks at $\pm 30^{\circ}$ (dashed lines). (c)
Energy of the whole simulation process, the displacement is calculated for the
time steps marked by the vertical dot lines.

After the DDQC is formed, we have observed that there are still structural
fluctuation of the DDQC when it is subjected to thermal fluctuation. The QC
structures consist of two types of dodecagonal motifs packed in different ways
shown in Fig. 4. We focus on a network of the dodecagonal motifs, which is
evaluated from the network of the $Z$ particles because the $Z$ particles are
the centres of the dodecagonal motifs. Here we illustrate the fluctuation of
the network in a short and a long timescales.

Figure 10 illustrates the $Z$ particle network by Delaunay triangulation of
two snapshots within a “short” time scale of $\delta t=10^{5}$ steps.  Two
cases are observed: (i) the network does not change regardless the local
displacement (Fig. 10(a)). In the dodecagonal motif, the whole first ring of
the centre $Z$ can rotate 30∘ and eventually the dodecagonal motif is still
maintained (see Fig. 5 for the two motifs). (ii) As shown in Fig. 10(b), a few
particles on the first ring rotate around $Z$, and cause changes in the
network of the dodecagonal motif. In both cases, the displacements of the
particles mostly occur for the first neighbours of the $Z$ particle as shown
in Fig.10. The displacement may occur as a chain in Fig.10. In this case, the
rearrangement of the network is coincident with the displacement.

In Figure 11, we show fluctuations in a longer time scale ($\delta t=10^{6}$
steps) by means of the packing of motifs for the whole QC structure. The
darker colours indicate that the network of $Z$ particles is maintained. The
lighter regions indicate network rearrangement. There is a collective motion
of the particles along the place where rearrangement of the network of the
dodecagonal motifs occurs. Readers consult Fig. 12 for clearer visualisation
of the maintained network and network rearrangement. We perform the same
analysis of rotational displacement in Sec. III.1. The rotation angles of
those particles are around $\pm 30^{\circ}$ as depicted in Fig. 11(b). The
structural change of the dodecagonal network is associated with the local
rotation of the first neighbours. Such rotations seem to appear at the border
of different types of the packing of dodecagonal motifs. During the structural
changes associated with network rearrangement of dodecagonal motifs, the
energy is fluctuating around the constant value as shown in Fig. 11(c).

##  IV Results of isotropic particle system

As mentioned in Introduction, the DDQC can be assembled also from isotropic
particles. We perform same analysis to the isotropic DDQC. The details of the
potential and simulation parameters can be found in Appendix B.

The formation and fluctuation of the isotropic DDQC is presented in Fig.
12(a-b). Similar to the formation of the annealed patchy particle DDQC, the
isotropic DDQC also start with a $Z$-rich structure with some $D_{1},D_{2}$
particles generated due to fluctuation of $Z$. The dodecagonal motif is then
organised and expanded, in which the $D_{1},D_{2}$ are accumulated at the
boundary of the cluster of dodecagonal motif and the $Z$-rich bulk. The
resulted DDQC consists of dodecagonal motifs packed in different ways (Fig.
4). Regarding the mechanism of the transformation from $Z$-rich structures to
DDQCs, the first neighbours of the dodecagonal motif centres usually rotate by
$\pm 15^{\circ}$ (Fig. 12(c-d)). The fluctuation of the DDQC is investigated
by performing simulations at fixed temperature at $T=0.7$. Figure 12(e)
illustrates the fluctuations of isotropic DDQC in terms of the network of the
dodecagonal motifs. The change in the network is also associated with the
collective rotation by $\pm 30^{\circ}$ of the first neighbours of the motif
centres. For a detailed comparison with Fig. 5, 6, 9 in patchy particle
system, the corresponding analyses for isotropic particle system are given in
Fig. S1-S3 in Supplementary Information.

However, there are differences between the DDQCs made of patchy particles and
isotropic particles. In general, the isotropic DDQC (Fig. 12(e)) has less
defects as the network has more well-defined packing structures than that of
patchy particle DDQC in Fig. 11.

In a dodecagonal motif of the DDQC, the $Z$ particle of the isotropic system
has lowest energy whereas that of patchy particle is highest (see Fig. 15).
For patchy particle case, the orientation of the $Z$ particle at the center is
incompatible with the six $\sigma$ neighbours. The energy of this hexagonal
particle is expected higher than that of the $\sigma$ particle, and hence
$\sigma$-dominent structures are expected at low density. The patchy particle
system requires a fine tuning of density to assemble a DDQC, whereas the DDQC
in isotropic particle system works even at low density condition (Fig. 16).
Additional data can be found in Fig. 15 and Fig.16 in Appendix D.

Although the DDQCs in both systems consist of dodecagonal motifs, the network
of the motifs is different. In isotropic DDQC (Fig. 12), the motifs are
arranged into the dense packing types as illustrated in Fig. 4(a,b,d). The
network of the isotropic DDQC is more compacted so as the energy is minimised.

Figure 12: Typical rotations of particles in the (a-d) formation and (e-f)
fluctuation of the DDQCs. (a-b) The two snapshots for the growth of the DDQC.
(c) Corresponding particle displacement of the two snapshots in (a-b). The
yellow dots are the $Z$ particles in (b). The size of $Z$ particles indicates
the “strength” of the rotation of the first ring. The $Z$ particles of very
low rotation are not shown. (d) Histogram of the rotations of the neighbouring
particles of the $Z$ particles in (c); the vertical dashed lines show
rotations at $\pm 15^{\circ}$. (e) Change in network of the dodecagonal motifs
during the fluctuation of a DDQC. (f) Histogram of the rotation of the
particles in (e); the vertical dashed lines show rotations at $\pm
30^{\circ}$.

##  V Discussions and Conclusions

We have investigated the formation of dodecagonal quasicrystals from the
patchy particles with five-fold symmetry. From a $Z$-rich hexagonal structure,
fluctuations of positions of patchy particles generate intermediate structures
$D_{1}$, $D_{2}$. After the dodecagonal motif is formed, the motif is rapidly
expanded and the $D_{1},D_{2}$ are accumulated as a boundary between the
cluster of dodecagonal motif and the hexagonal structure. The DDQC contains
many dodecagonal motifs packed in different ways. The formation of the DDQC is
driven from the local fluctuations of the particles. The fluctuation is
characterised by rotation of the first ring in the dodecagonal motif. After
the DDQC is formed, the dodecagonal motif network changes when thermal
fluctuation makes large displacements of the particles. The change of the
network is associated with rotation of the nearest neighbours of the centres
of the motif.

Regarding the formation of the DDQC, the transformation from a $Z$-rich
structure to a DDQC is observed in both patchy particle system and isotropic
particle system. Superposition of a hexagonal lattice and a dodecagonal motif
reveals the difference in the position of the first ring, which is manifested
by a circular rotation of $15^{\circ}$ around the motif’s centre. Such
displacements of the particles on the first rings of the dodecagonal motifs
are confirmed in our simulations. This maybe the reason why the DDQCs of
patchy particles and isotropic particles both share same mechanism although
the interactions are completely different.

The DDQC comprises different types of the packing of dodecagonal motifs (see
Fig.4). These packing structure can be constructed from the network of centres
of dodecagonal motifs. The packing is divided into several characteristic
structures (Fig.4). In the obtained DDQCs, different types of the packing
structures coexist, and they do not show periodic unit. We have observed that
the DDQC continues to fluctuate in the sense that the relative positions of
some particles change. Two types of displacement are considered. (i) In the
case of individual displacement, the particles on the first ring of the
dodecagonal motif quickly rotate and swap their positions. Despite the
particle displacements, the whole dodecagonal motif remains intact. (ii)
Collective displacement of particles can bring about the rearrangement of the
DDQC network. Such changes appear at the border of different types of packing
structures, i.e. some dodecagonal motifs are disintegrated and new motifs are
generated. The displacement of the particle mostly occur at the first ring of
the dodecagonal motifs, and at short lengthscale like (i). However such
displacement can propagate to the neighbouring motifs, and the whole DDQC
network eventually changes.

There is another type of DDQC made of isotropic particles [17] where the the
high-symmetry dodecagonal motif has five particles are on the inner ring
(compared to six particles for the DDQC in our study) and twelve particles are
on the second ring. Kinetics and dynamics of those DDQCs may be different from
those in this study. We leave the analyses of such different DDQCs as a future
study.

The patchy particle DDQC in our study is in agreement with other works using
five-fold symmetry patchy particle system, such as particle with five equally
distributed patches on the equator [24]. The $Y_{55}$ patchy particle in our
study also has five-fold symmetry, but the number of patches and the
interaction are different. This particle has ten patches in red and blue,
where the interaction of red-red/blue-blue patches are assigned attractive and
red-blue patches are repulsive. In the assembled DDQC, the orientation of the
patchiness of the $\sigma$ particle and its neighbours (Fig. 3) is found
similar to the one in ref. [24]. Moreover, the ratios of the $Z$ and $\sigma$
particles of the DDQC in the Monte-Carlo simulations in ref. [24] are about
0.07 and 0.8, respectively, which are comparable to our system in Fig. 5.

The DDQC can be formed by bi-dispersed DNA system in simulation [29] or
experimentation [30]. The two DNA tiles can be considered as 5- and 6-point-
star particles. The self-assemblies of patchy particle DDQC at different
densities in our study is somewhat analogical to that of bi-dispersed DNA
systems [30]. When the ratio of the 5- to 6-point-star particle increases, the
resulted structures change. Such a low ratio leads to $\sigma$ phase
structures. Then medium ratio forms DDQC of which many dodecagonal motif are
observed. At high ratio, the hexagonal structures dominate the dodecagonal
motifs. Such changes are also observed in our study when the density of the
patchy particle increases.

We have investigated the formation and fluctuation of dodecagonal
quasicrystals in patchy particle system and isotropic particle system. It may
help to predict and design the two-dimensional dodecagonal quasicrystals in
different soft-matter systems. The fluctuation of the DDQC may relate to the
properties of the QC, e.g. heat capacity. An analogical study is still
required for, for example, octagonal QC and decagonal QC to know how the other
quasicrystals behave.

###### Acknowledgements.

The authors acknowledge the support from JSPS KAKENHI Grant number JP20K14437
to U.T.L., and JP20K03874 and JP20H05259 to N.Y.

##  Appendix A Potential of patchy particle system

The potential for a pair of patchy particle is

$V=V_{WCA}(r)+V_{M}(r)\Xi(\bm{\Omega}),$ (3)

The detail of the isotropic Week-Chandler-Anderson potential $V_{WCA}$
preventing the overlapping of particles, and the Morse potential $V_{M}$ in
Eq. (3) is given as

$V_{WCA}=\begin{cases}4\epsilon\left[(\frac{2a}{r})^{12}-(\frac{2a}{r})^{6}+\frac{1}{4}\right],&r\leq
2a\sqrt[6]{2}\\\ 0,&r>2a\sqrt[6]{2}\end{cases}$ (4)

$V_{M}=\epsilon
M_{d}\left\\{\left[1-e^{\left(-\frac{r-r_{eq}}{M_{r}}\right)}\right]^{2}-1\right\\},$
(5)

where $\mathbf{r}=\mathbf{r}^{ij}=\mathbf{r}^{j}-\mathbf{r}^{i}$ is the
distance vector between particle centres, $r=\left|\mathbf{r}\right|$, and
$\hat{\mathbf{r}}=\mathbf{r}/r$, $\epsilon$ is the potential well depth,
$r_{eq}$ is the Morse potential equilibrium position ($r_{eq}=1.878a$),
$M_{d}=2.294a$, $M_{r}=a$ is the Morse potential depth and range, respectively
[39, 36, 37].

The anisotropic interaction is calculated based on the mutual orientation of a
pair of particle $i$ and $j$. Let the $\hat{\bf n}^{(m)}_{i}$, $\hat{\bf
n}^{(m)}_{j}$, $m=1,2,3$ are local bases of particle $i$ and $j$, and
$\hat{\bf r}$ is the unit distance vector between particle center. The
anisotropic interaction
$\Xi_{lm}\propto\\{\mathbf{C}_{(i)}^{(l,m)}\\}\odot\nabla_{\mathbf{r}}^{2l}\frac{1}{r}\odot\\{\mathbf{C}_{(j)}^{(l,m)}\\}$
estimates the angular dependence of a pair of particles $Y_{lm}$ as
$\Xi_{lm}\propto\\{\hat{\bf n}_{0}^{l-m}\hat{\bf
n}_{+}^{m}\\}_{(i)}\odot\\{\hat{\bf r}^{2l}\\}\odot\\{\hat{\bf
n}_{0}^{l-m}\hat{\bf n}_{+}^{m}\\}_{(j)}$, where $\hat{\bf n}_{0}=\hat{\bf
n}^{(3)}$ and $\hat{\bf n}_{+}=\frac{1}{\sqrt{2}}(\hat{\bf n}^{(1)}+i\hat{\bf
n}^{(2)})$. For example, $\Xi_{10}\propto\\{\hat{\bf
n}_{0}\\}_{(i)}\odot\\{\hat{\bf r}\hat{\bf r}\\}\odot\\{\hat{\bf
n}_{0}\\}_{(j)}$ for a pair of particles $Y_{10}$, and
$\Xi_{20}\propto\\{\hat{\bf n}_{0}\hat{\bf n}_{0}\\}_{(i)}\odot\\{\hat{\bf
r}\hat{\bf r}\hat{\bf r}\hat{\bf r}\\}\odot\\{\hat{\bf n}_{0}\hat{\bf
n}_{0}\\}_{(j)}$ for a pair of particles $Y_{20}$.

##  Appendix B Isotropic particle system

The isotropic potential is a Lennard-Jones Gauss potential [17, 19, 26, 40] as
follows

$V_{\text{LJG}}=\epsilon\left[\left(\frac{2a}{r}\right)^{12}-2\left(\frac{2a}{r}\right)^{6}-\epsilon^{{}^{\prime}}\exp\left(\frac{(r-r_{G})^{2}}{2(2a)^{2}\sigma^{2}}\right)\right],$
(6)

where the particle radius is $a$, the potential parameters $r_{G}$,
$\epsilon^{{}^{\prime}}$, $\sigma^{2}$ determine the position, depth, and
width of the second well. The values of those parameters are $r_{G}=1.95\times
2a$, $\epsilon^{{}^{\prime}}=2.0$, $\sigma^{2}=0.02$. Figure 13 shows the
potential.

Figure 13: Isotropic potential for the creation of DDQC.

The simulation parameters are almost identical to that of patchy particle
system. The number of particles is $N=1024$. In the annealing simulations, the
temperature decreases from $T=1.2$ to $T=0.4$ with an interval of $\Delta
T=0.0125$. There are 100,000 steps at each value of temperature, making a
total of $65\times 100,000$ steps.

##  Appendix C Determination of neighbouring rotation

The particle configuration is known at every time step. Let
$\mathbf{r}(t_{1})$, $\mathbf{r}(t_{2})$ be the particle position at time
$t_{1}$, $t_{2}$, respectively, and
$\mathbf{d}=\mathbf{r}(t_{2})-\mathbf{r}(t_{1})$ the particle displacement
from $t_{1}$ to $t_{2}$. Consider that the particle $i$ has $N_{i}$ nearest
neighbours $\\{j_{1},j_{2},...,j_{N_{i}}\\}$ in clockwise order. At $t_{1}$,
we define $\hat{\mathbf{e}}_{j_{k}}$ the unit vector from $j_{k}$ to
$j_{(}k+1)$ of the polygon made of $\\{j_{1},j_{2},...,j_{N_{i}}\\}$. The
displacement of the neighbours particle is determined from $t_{1}$ to $t_{2}$,
for example, $\mathbf{d}_{j_{1}}$ is the displacement of the particle $j_{1}$
(Fig. 14). The rotation of the neighbours of particle $i$ is defined as

$\omega_{i}=\frac{1}{N_{i}}\sum_{j}\mathbf{d}_{j_{k}}.\hat{\mathbf{e}}_{j_{k}}.$
(7)

Note that the relative displacement to the center particle is considered
$\mathbf{d}\mapsto\mathbf{d}-\mathbf{d}_{i}$.

Figure 14: Calculation of local rotation

##  Appendix D Energy of local structure and density dependence

Figure 15: Energy of local structures during annealing simulation from $T=1.2$ to $T=0.4$ for (a) patchy particle system and (b) isotropic particle system. The colours of local structures are given in Fig. 3, and the black dotted line shows the average energy of all particles. The vertical dash lines define the three stages of the growth of the DDQC (see Fig. 5). $\\#$ | $\rho_{\text{a}}$ | $T_{\text{max}}$ | $T_{\text{min}}$ | $\Delta T$ | Total simulation steps
---|---|---|---|---|---
Patchy particles |  |
(a) | 0.45 | 1.0 | 0.2 | 0.0125 | $13000\times 10^{3}$
(b) | 0.84 | 1.2 | 0.4 | 0.0125 | $6500\times 10^{3}$
Isotropic particles |  |
(c) | 0.30 | 1.2 | 0.2 | 0.0125 | $8500\times 10^{3}$
(d) | 0.87 | 1.2 | 0.2 | 0.0125 | $8500\times 10^{3}$
Table 1: Parameters of annealing simulations at different densities. The
results are shown in Fig.16 Figure 16: Changes in the ratio of local
structures of patchy particles (a-b) and isotropic particles (c-d) self-
assemblies at low and high values of area fractions. The snapshots are taken
at the last steps of the annealing simulations. The area fractions
$\rho_{\text{a}}$ of the simulations (a-d) are 0.45, 0.84, 0.30, 0.87,
respectively. Details of simulation parameters are given in Table 1.

We compare the energy of each local structure during the growth of the DDQC
assembled from the patchy particles and isotropic particles. In each snapshot,
for example, the energy of the $Z$ particle is the mean energy of all $Z$
particles. The data are calculated from the annealing simulations of Fig. 5
and Fig. S1 in Supplementary Information.

The energy of each local structure during annealing is given in Fig. 15. The
DDQC is steadily formed when the ratio of the local structure is in stage III.
Regarding the local structure of DDQC, the lowest energy is the $\sigma$
particle in patchy DDQC, while that is $Z$ particle in isotropic DDQC. This is
due to the difference in two particle systems. The five-fold patchy particle
prefers five nearest neighbours. Therefore, the local structure $\sigma$ is
expected to have lowest energy. Then self-assembly of patchy particles at low
densities result in $\sigma$-rich structure instead of DDQC. In contrast, the
$Z$ particle in isotropic DDQC has lowest energy, and therefore the DDQC
structure is formed at low densities. The decrease of energy with time when
the DDQC reaches stable stage III is mainly due to lowering temperature in
annealing. As a result, the thermal fluctuation is reduced and the energy
decreases.

We investigate the effect of density on the formation of DDQC by patchy
particles and isotropic particles. Annealing simulations are conducted with
varying the area fractions of the particles. The simulation parameters and
results are given in Table 1 and Fig. 16.

There are different phases depending on the area fraction. For the patchy
particle system, the suitable density for the assembly of DDQC is limited
around $0.69\lesssim\rho_{\text{a}}\lesssim 0.81$. At low density
$\rho_{\text{a}}<0.69$, the structure contains clusters of $\sigma$ particles
and undefined particles at the interfaces. The dodecagonal motif is hardly
observed (Fig.16(a)). At high density $\rho>0.81$ the hexagonal structure is
dominant (Fig. 16(b)).

For the isotropic particle system, the DDQC more easily as
$\rho_{\text{a}}\lesssim 0.84$. Even at the low density, clusters of the
dodocagonal motif are clearly observed inside the interfaces (Fig. 16(c)). At
high density $\rho>0.84$ the hexagonal structure is also dominant (Fig.
16(d)).

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# Mixed Order Phase Transitions

David Mukamel  Department of Physics of Complex Systems, Weizmann Institute of
Science
Rehovor 7610001, Israel

###### Abstract

Mixed order phase transitions are transitions which have common features with
both first order and second order transitions. I review some results obtained
in the context of one of the prototypical models of mixed order transitions,
the one-dimensional Ising model with long-range coupling that decays as
truncated inverse square distance between spins. The correspondence between
this model and the Poland Scheraga model of DNA denaturation, a subject to
which Michael Fisher made substantial contributions, is then outlined.

##  I Joining Michael Fisher as a postdoc at Cornell

I was a postdoctoral fellow of Michael Fisher at Cornell in the years
1975-1977. Before arriving in Cornell I was a research fellow at Brookhaven
National Lab where I applied symmetry considerations of Landau theory to argue
that $n$-vector spin models with $n\geq 4$ are in fact realized in a large
variety of physical systems. Renormalization group analysis of these models
thus provides their classification into universality classes. Furthermore, it
suggests that when an accessible stable fixed point is lacking they should
display a first order transition.

At the time Michael was skeptical about the use of Landau theory, beyond it
being perhaps a useful but uncontrolled approximation. So upon my arrival in
Cornell he asked me to present my work in one of his famous Tuesday seminars.
The seminar turned into a tutorial series of three lengthy and grueling
lectures where Michael kept posing questions and making infinitely many
comments and suggestions. After the third seminar he asked me to his office
for a chat in which at some point he said, you know Landau theory is under
appreciated in the West, and then he added candidly ”…and maybe I am to
blame”. We ended up collaborating on a number of studies combining Landau’s
symmetry considerations with renormalization group calculations together with
my old friend and his then student Eytan Domany. We have, for example,
suggested a physical realization of the three state Potts model 111For an
impressive piece of art generated by Michael for this project where he
displays his artistic drawing skills, see PRL 37, 565 (1976) and in the
contribution of Eytan Domany in this volume, arXiv:2301:11086. and
demonstrated how a symmetry breaking field can turn a first order transition
into a continuous one.

Cornell at the time was an exciting and intelectually stimulating place to be,
and working with Michael was a unique experience. I have learned a lot from
Michael’s non-compromising approach to science, how to try to select and
pursue meaningful scientific problems and how to conduct research. My stay
with Michael has deeply influenced my research throughout my career, and I am
grateful for that.

##  II Introduction

In the present article I briefly review some results on mixed order phase
transitions, a subject to which Michael has made important contributions.
Mixed order phase transitions, also known as first order critical transitions
or hybrid transitions, particularly in the percolation community, are phase
transitions which have common features with both first and second order phase
transitions. In particular, they display a discontinuity of the order
parameter as is common in first order phase transitions, and a diverging
correlation length as is typical in second order phase transitions. They have
been observed theoretically in a variety of models and found experimentally in
some physical systems. Examples include the denaturation transition of DNA [2,
3, 4, 5, 6], jamming transitions [7, 8, 9, 10, 11], $k$-core percolation [12,
13, 14, 15], non-enclave percolation [16], models of wetting [17, 18],
networks dynamics [19], models of moving condensates [20], the one dimensional
Ising model with long range interactions that decay as $1/r^{2}$ with the
distance between the interacting spins [21, 22, 23, 24, 25, 26, 27, 28] and a
truncated interaction modification of this model [29, 30, 31].

While in some of the models the correlation length $\xi$ was found to diverge
exponentially rapidly with the deviation, $\epsilon$, from criticality as
$\xi\sim e^{\epsilon^{-\mu}}$, in other models the divergence follows a power
law, $\epsilon^{-\nu}$, with $\mu$ and $\nu$ some positive exponents. The
former class includes models of jamming transitions [8] and the $1d$ Ising
model with $1/r^{2}$ interaction [25, 27], while the latter includes models of
DNA denaturation [3, 4, 5, 6], percolation transitions [16, 13] and a
truncated version of the $1d$ Ising model with long range interactions [29,
30, 31].

One of the prototypical models which exhibits a mixed order transition is the
$1d$ Ising model with long-range interactions which decay as $\sim 1/r^{2}$ at
large distance $r$. The model, later termed the inverse distance square Ising
model (IDSI), was introduced by Thouless [21] who argued that it possesses a
phase transition with a jump in the magnetization at the transition. This
magnetization discontinuity became known as the “Thouless effect”. A closely
related hierarchical model was then introduced and solved by Dyson [22] who
demonstrated that magnetization discontinuity indeed takes place. An exact
proof of the discontinuous nature of the transition in the IDSI model was
eventually given by Aizenman et. al. [28]. In addition, scaling arguments [25]
and renormalization group analysis [27] have shown that the correlation length
$\xi$ diverges at the transition with essential singularity, $\xi\sim
e^{b/\sqrt{T-T_{c}}}$, where $b$ is a constant and $T_{c}$ is the critical
temperature. Although a lot is known about the thermodynamic behavior of this
model, it is not exactly solvable. In $d=2$ and higher dimensions the model
with $1/r^{2}$ interaction results in energy which scales super linearly with
the volume. When properly scaled with the volume the model exhibits an
ordinary continuous transition.

More recently, a modified version of the IDSI model was introduced [29, 30,
31]. This model is exactly solvable and was shown to display a mixed order
transition which is closely related to that of IDSI although with some
differences. In what follows, this model is introduced and its main propertied
are reviewed. I will then comment on the relation of this model to the Poland
Scheraga model of DNA denaturation. I will also highlight the contributions of
Michael to the understanding of the nature of mixed order transitions in
general and to DNA denaturation in particular.

##  III The truncated inverse distance square Ising (TIDSI) model

The TIDSI model is a truncated version of the IDSI model. Here, as in IDSI,
the long range interaction between spins at distance $r$ apart decay as
$1/r^{2}$ with the distance. However the interaction is confined to spins
which lie within the same domain of either all up or all down spins. The
Hamiltonian of the model is

$\mathcal{H}=-\Sigma_{i<j}J(i-j)\sigma_{i}\sigma_{j}I(i\sim
j)-J_{NN}\Sigma_{i}\sigma_{i}\sigma_{i+1},$ (1)

where $\sigma_{i}=\pm 1$, $I(i\sim j)=1$ as long as sites $i$ and $j$ are in
the same domain of either all up or all down spins and $I(i\sim j)=0$
otherwise. The long range coupling is taken to be

$J(r)\approx\frac{C}{r^{2}}\quad\,\text{for}\,\,r\gg 1,$ (2)

and, for simplicity, one may take periodic boundary conditions
$\sigma_{i}=\sigma_{i+L}$ in a lattice of length $L$. Note that the indicator
function $I(i\sim j)$ may be expressed in terms of the spin variables as

$I(i\sim
j)=\prod^{j-1}_{k=i}\delta_{\sigma_{k}\sigma_{k+1}}=\prod^{j-1}_{k=i}\frac{1+\sigma_{k}\sigma_{k+1}}{2},$
(3)

where $\delta_{\sigma_{k}\sigma_{k+1}}$ is the Kronecker delta function. The
Hamiltonian (1) is thus not of the usual binary spin interaction form but
rather it involves multi-spin interactions.

Figure 1:  Microscopic configuration of the Ising model with the domain
lengths indicated.

It is convenient to express the Hamiltonian in terms of the domain length
representation, where a domain is defined as a stretch of successive parallel
spins (see Fig. 1). The long-range interaction in the first term operates only
between pairs of spins which belong to the same domain, while the nearest
neighbor interaction in the second term results in an energy cost for each
domain wall. A typical configuration $\mathcal{C}$ is thus described by a set
of domains with lengths $\\{l_{1},\,l_{2},\,\cdots,\,l_{N}\\}$ where the
number of domains $N$ can vary from one configuration to another. In terms of
these variables the Hamiltonian can be expressed as

$\mathcal{H}=\sum^{N}_{n=1}\mathcal{H}_{n}$ (4)

where

$\mathcal{H}_{n}=-J_{NN}(l_{n}-2)-\sum^{l_{n}}_{r=1}(l_{n}-r)J(r)\,.$ (5)

The variables $l_{n}$ are constrained by the system size $L$ to satisfy

$\sum^{N}_{n=1}l_{n}=L\,.$ (6)

Using the form of $J(r)$ in Eq. (2), one can estimate the sum by replacing it
by an integral as

$\displaystyle\sum J(r)$ $\displaystyle\approx$ $\displaystyle
a_{0}-\frac{C}{l_{n}}$ $\displaystyle\sum rJ(r)$ $\displaystyle\approx$
$\displaystyle b_{0}+C\,\ln l_{n}\,,$ (7)

where we have assumed that $l_{n}$ is large and kept the two leading order
terms for large $l_{n}$. This is justified since we are interested in
phenomena close to the critical line where domains are typically large.
Dropping an overall unimportant constant, one obtains the effective
Hamiltonian

$\mathcal{H}=C\sum_{n}\ln l_{n}+\Delta N$ (8)

where the constant $C$ is the amplitude of the long-range interaction and
$\Delta=2J_{NN}+C+b_{0}$ acts as a chemical potential for the number of
domains. The resulting Hamiltonian is that of a Coulomb gas of alternating
sign charges in $1d$, where the interaction is confined to nearest neighbor
charges only, and it is logarithmic in the distance, as is the case of charges
in $2d$. In this representation the charges correspond to the domain
boundaries, and the domain lengths, $l_{n}$, are the distance between nearest
neighbor charges.

The free energy and the resulting thermodynamic properties of the model have
been calculated in recent studies and the mixed order nature of the transition
has been clarified [29, 30, 31]. In addition, a discussion of this model
within the context of a different class of models exhibiting fluctuation-
dominated phase ordering has been given by Barma et. al. [31] and in the
contribution of Mustansir Barma in the present volume. Let us briefly outline
this analysis. The probability of a configuration $\mathcal{C}$, defined by
the domain lengths $(l_{1},\dots,l_{N})$ and their number $N$ is given by its
Boltzmann weight

$P(l_{1},\,l_{2},\,\cdots,\,l_{N},\,N|L)=\frac{y^{N}}{Z_{y}(L)}\,\prod^{N}_{n=1}\,\frac{1}{l_{n}^{c}}\,\delta_{\sum^{N}_{n=1}l_{n},\,L}$
(9)

where

$c=\beta C\,\,\,\mbox{ ,}\,\,\,y=e^{-\beta\Delta}\,,$ (10)

and $\delta_{i,j}$ is the Kronecker delta function that enforces the sum rule.
The normalization constant $Z_{y}(L)$ is the partition function given by

$Z_{y}(L)=\sum^{\infty}_{N=1}y^{N}\sum^{\infty}_{l_{1}=1}\ldots\sum_{l_{N}=1}^{\infty}\prod^{N}_{n=1}\frac{1}{l_{n}^{c}}\delta_{\sum^{N}_{n=1}l_{n},\,L}\,.$
(11)

Here we have ignored a geometrical factor which arises from the number of ways
of placing the sequence of domains on the lattice. This factor results in a
subleading contribution to the partition sum and hence it does not affect the
thermodynamic properties of the model.

Figure 2:  The $(c,y)$ phase diagram of the TIDSI model. The red solid line
represents a continuous transition with $\langle n\rangle$ vanishing
continuously at the transition. The dashed line represents a mixed order
transition with a discontinuous order parameter $\langle n\rangle$ and
diverging length. Note that the $m$ order parameter is discontinuous at the
transition for any $c>1$.

In order to analyze the behavior of the model and particularly the phase
diagram and the probability distribution of the domains length, we proceed by
considering the grand canonical ensemble. The grand partition sum, $Q_{y}(z)$
is given by

$Q_{y}(z)=\sum^{\infty}_{L=1}z^{L}Z_{y}(L)\,,$ (12)

where $z$ is the fugacity. Carrying out the sums in Eqs. (11, 12) one obtains

$Q_{y}(z)=\frac{y\phi_{c}(z)}{1-y\phi_{c}(z)}$ (13)

with

$\phi_{c}(z)=\sum^{\infty}_{l=1}\frac{z^{l}}{l^{c}}\equiv{\rm
Li}_{c}\left(z\right)\,,$ (14)

where ${\rm Li}_{c}(z)$ is the polylogarithmic function. The sum in Eq.(14)
converges for $z<1$, diverges for $z>1$ and it exhibits a singularity at
$z=1$.

The phase diagram and the nature of the phase transition can be deduced by
direct inspection of $Q_{y}(z)$. The partition sum of the model, $Z_{y}(L)$ is
determined by the pole of $Q_{y}(z)$ when it exists. The pole is obtained at
$z^{*}$ which satisfies

$y\phi_{c}(z^{*})=1\,,$ (15)

and it exists as long as $z^{*}<1$. In this case the partition sum is given by

$Z_{y}(L)\propto z^{*-L}\,.$ (16)

A phase transition takes place in the model at $z^{*}=1$, namely at $y$ and
$c$ satisfying

$y\phi_{c}(1)=1\,.$ (17)

The nature of the phases and the phase transition are simply found by
calculating the distribution function , $P(l)$, of a single domain. This is
done by summing over the lengths of all other domains in (9) yielding

$P(l)=\frac{Z_{y}(L-l)}{Z_{y}(L)}\frac{y}{l^{c}}\,,$ (18)

which, given (16), results in

$P(l)=\frac{yz^{*l}}{l^{c}}\propto\frac{e^{-l/\xi}}{l^{c}}\,,$ (19)

as long as $z^{*}<1$. Here $\xi\equiv-1/{\ln z^{*}}$ is a length scale which
controls the typical domain size. For finite $\xi$ (namely for $z^{*}<1$), the
equilibrium domains are of finite typical length controlled by $\xi$, and
their average number $N$ scales with the system’s length $L$. As a result, the
domain’s density $n=N/L$, is finite. As $z^{*}\rightarrow 1$ the correlation
length $\xi$ diverges and the density of domains (or charges) vanishes. One
can think of two order parameters involved in this transition: the
magnetization $m$ and the average number of domains $\langle n\rangle$. The
two order parameters yield somewhat different results as to the nature of the
phase transition, as is outlined below.

In the spin representation, a typical configuration in the paramagnetic phase
is composed of a finite density of domains of alternating up and down spins,
so that the overall magnetization vanishes. At the transition point,
$z^{*}=1$, the domain’s density vanishes, leading to a phase which is composed
of a single domain of length $L$ with some small domains whose total length is
subleading in $L$. This is a ferromagnetic phase with average magnetization
per spin $m=\pm 1$ for any $c>1$. The transition from the paramagnetic to this
ferromagnetic phase was thus termed ”extreme Thouless effect”. The fact that
the discontinuous order parameter is accompanied by a diverging correlation
length, makes the transition of mixed order nature for any $c>1$.

Figure 3: Typical schematic $(T,H)$ phase diagram for (a) second order
transition at $H=0$ and (b) first order transition at $H=0$ with three first
order lines intersecting at a triple point. The two fork branches terminate at
critical points marked x in the figure. Scaling arguments suggest that the
phase diagram of the IDSI model is of the form of (a).

Figure 4: Schematic $(T,H)$ phase diagram of the TIDSI model with $c>2$.
Dashed lines represent lines of mixed order phase transition of the TIDSI type
with $H$ dependent parameter $c(H)>2$, while red thin lines represent
continuous transitions with $1<c(H)<2$. On the thin lines the magnetization
$m$ approaches $m=\pm 1$ continuously while on the dashed lines it exhibits a
discontinuous jump. The thick solid line at $H=0$ is a first order transition
line on which the magnetization changes discontinuously from $+1$ to $-1$.

In order to explore the nature of the phase transition using the $\langle
n\rangle$ order parameter let us consider the domain size distribution at the
transition, $P(l)\propto 1/l^{c}$, which exists as a normalizable distribution
for $c>1$. For $1<c\leq 2$ the average domain length, $\langle l\rangle$,
diverges, and thus the the average density of domains, $\langle n\rangle$
vanishes. This indicates that the order parameter of the transition, $\langle
n\rangle$, goes continuously to zero at the transition. The transition is thus
a usual second order transition with diverging correlation length and
continuously vanishing order parameter. On the other hand for $c>2$, $\langle
n\rangle$ is finite at the transition, indicating that the transition into the
ordered phase, at which $\langle n\rangle$ vanishes, involves a discontinuous
change of the order parameter. Thus for $c>2$ the model is of a mixed order
nature, with a diverging length $\xi$ and a discontinuous order parameter
$\langle n\rangle$. Note that the parameter $c$ of the model, given in (10),
is temperature dependent, and the nature of the transition is determined by
the value of $c$ at the transition temperature, namely $c=\beta_{c}C$. The
phase diagram of the model in the $y-c$ plane is given in Fig. 2.

The critical behavior of the model can be extracted from the singularity of
the polylogarithmic function at $z=1$. In the disordered phase, where the
domains density $\langle n\rangle$ is non vanishing and $z<1$, the fugacity is
related to temperature variable $y$ ($\equiv\beta\Delta$) via
$y\phi_{c}(z)=1$. Close to the transition point, and for $1<c<2$ the
polylogarithmic function satisfies $\phi_{c}(1-\delta
z)-\phi_{c}(1)\propto\delta z^{c-1}$. Thus close to the transition, where
$y=y_{crit}+\delta y$, one has $\delta y\propto\delta z^{c-1}$. The
correlation length $\xi=-1/{\ln(\delta z)}$ thus satisfies $\xi\propto\delta
z^{-1}$, or $\xi\propto\delta y^{-\frac{1}{c-1}}$. Since $y$ is basically the
temperature variable , $\delta y$ is proportional to the reduced temperature
$t=(T-T_{c})/T_{c}$ resulting in

$\xi\propto t^{-\nu}\quad{\rm with}\quad\nu=1/(c-1)\,.$ (20)

For $c>2$ the singular term in $\phi_{c}(z)$ is sub leading with
$\phi_{c}(1-\delta z)-\phi_{c}(1)\propto z$ which yields $\nu=1$.

The order parameter of the transition is $\langle n\rangle=1/{\langle
l\rangle}$. Using the domain length distribution (19) one finds that for
$1<c<2$

$\langle n\rangle=1/\xi^{2-c}\propto t^{\frac{2-c}{c-1}}\,.$ (21)

For $c>2$ , the average domain length $\langle l\rangle$ is finite and the
order parameter is discontinuous at the transition.

As mentioned above, in the IDSI model the correlation length diverges at the
transition with an essential singularity rather than a power law. An insight
into this behavior is gained by considering the model within the framework of
the charge configurations. Here, as in the TIDSI model, a microscopic
configuration is expressed as a sequence of alternating charges marking the
domain boundaries. However, unlike TSIDI, the logarithmic interaction between
the charges is not restricted to nearest neighbor charges only but is rather
all to all. This makes the transition of a Kosterlitz-Thouless type, with its
characteristic essential singularity. This has first been studied by Anderson
et al. [25] using scaling arguments and later elaborated by more detailed
renormalization group approach by Cardy [27].

An interesting question is what is the effect of an ordering field $H$ on a
mixed order phase transition. In ordinary second order phase transition an
ordering field removes the transition, as, for example is the case for the
Ising model. On the other hand first order transitions are retained in the
presence of an ordering field, resulting in a fork shape phase diagram in the
($T,H$) plane where three lines of first order transitions meet at a triple
point (see Fig. 3). In the case of the IDSI model it was first conjectured by
Anderson et al. [25] that the ($T,H$) phase diagram of this mixed order
transition is like that of an ordinary second order transition (see Fig. 3a).
Fisher, who has always been intrigued by first order transitions, developed
together with Berker a scaling approach to first order transitions and argued
that this is indeed the case and termed the transition “first order critical
point” [26]. The phase diagram of the TIDSI model is quite different. In
Figure 4 a schematic phase diagram of a model with $c>2$ is illustrated. In
this figure the $H=0$ mixed order transition survives the ordering field. The
phase diagram has a fork shape with two lines of mixed order phase transitions
meeting the $H=0$ first order transition line. The mixed order lines are
characterized by a continuously varying parameter $c(H)$. This parameter
decreases with $H$ until it reaches the value $c=2$ where the transition
becomes continuous. On the mixed order sections of the branches the
magnetization changes discontinuously from some value in the paramagnetic
phase to $|m|=1$ in the ferromagnetic phase. However on the continuous section
of the branches the magnetization changes continuously to its $|m|=1$ value in
the ferromagnetic phase (for details see [30]).

Figure 5:  Schematic configuration of DNA molecule with alternating bound
segments and open loops.

##  IV Comment on DNA denaturation

DNA denaturation corresponds to melting of double stranded DNA, a phase
transition at which the two bound strands of the molecule become separated.
This transition has been extensively studied experimentally and theoretically
for over sixty years by now. The canonical model used to study the transition
is the Poland-Scheraga (PS) model introduced in the 1960’s [3]. Within this
model a DNA molecule is schematically viewed at an alternating sequence of
segments of bound strands and denatured loops of unbound single strands (see
Fig. 5). The statistical weight of a microscopic configuration, which is
characterized by the number of segments and length of each one of them is
determined by the following simplifying assumptions: each bound pair carries a
(negative) binding energy $u$ so that the statistical weight of a bound
segment of length $l$ is $\omega=e^{-\beta ul}$. For simplicity all bound
pairs carry the same energy and the bound segment is non-flexible so that it
carries no configurational entropy. A segment of length $l$ of unbound pairs
constitutes a loop of length $2l$. The loop carries no energy, however it is
viewed as composed of a flexible string and thus it possesses configurational
entropy which has to be taken into account is calculating the statistical
weight of the DNA configuration. In the PS model as introduced by Poland and
Scheraga, loops are considered as fully flexible and they are modeled as non-
interacting random loops. In the large loop limit the statistical weight of a
loop of length $l$ in $d$ dimensions is

$\Omega(l)\propto\frac{s^{l}}{l^{c}}\,\,,$ (22)

where $s$ is some $d$ dependent constant and $c=d/2$. Here $1/l^{c}$
represents to probability of a random walk in dimension $d$ to return to the
starting point after $l$ steps.

Equation (22) implies that as in the TIDSI model, the statistical weight of a
loop may be expressed by two opposite charges interacting logarithmically as
$c\ln l$, leading to the same type of transition as in TIDSI. In particular,
the loop length distribution takes the form of Eq. 19, leading to no
transition for $c\leq 1$, continuous transition for $1<c<2$ and a mixed order
transition for $c\geq 2$. A major distinction between the two models is that
while the parameter $c$ in TIDSI is temperature dependent, and thus
nonuniversal, in the PS model $c$ is universal. It is independent of
temperature and of any type of short range interactions that may exist between
the two strands. The prediction of the PS model for DNA denaturation is that
$c=3/2$ in $d=3$, implying a continuous transition. This result disagrees with
the experimental observations which suggest that the transition is
discontinuous.

An important step in resolving this discrepancy was made by Michael Fisher in
1966 [4], who observed that exclusion interaction between the strands in a
loop, which was neglected in the PS analysis, should be taken into account.
This interaction is long ranged and can thus affect the value of $c$. Noting
that the end-to-end distance $R$ of a self avoiding random walk of length $l$
scales as $R\sim l^{\nu}$, the probability of the walk to return to the origin
becomes $1/l^{d\nu}$, making $c=d\nu$. Given that in $d=3$ the Flory exponent
is estimated to be $\nu\approx 0.583$, the parameter $c$ increases from $1.5$
in the noninteracting model to $\approx 1.75$ when exclusion is taken into
account. While the predicted transition is still continuous, the inclusion of
the exclusion interaction is a step in the right direction.

This approach was later taken a step forward by Kafri et al.[5, 6] who noted
that the exclusion interaction between a loop and the rest of the chain
modifies $c$ even further. Modeling the rest of the chain as a linear flexible
polymer one can apply a renormalization group approach developed for
evaluating the entropy of a general polymer network [32, 33, 34] to study the
length distribution of a self avoiding loop embedded in a chain. It was found
that the loop size distribution is of the same form as that of an isolated
loop but with a modified parameter $c$ which takes the value $c\approx 2.11$
in $d=3$ dimensions. This makes the denaturation transition of mixed order
type with a discontinuous order parameter, in line with what experiments
suggest.

##  V Summary

Mixed order phase transitions have been suggested to exist in a variety of
theoretical models of physical systems. In this paper some properties of this
intriguing class of phase transitions are reviewed and their derivation is
outlined in the framework of the TIDSI model. This model is exactly solvable
and thus provides useful insight into the mechanism leading to a transition
with discontinuous order parameter together with a diverging correlation
length. The model can be expressed in terms of charges of alternating sign in
$1d$ whose interaction is restricted to nearest neighbor only. This feature is
found to lead to a power law divergence of the correlation length. When the
restriction is removed and the interaction becomes all-to-all, the model
represents the IDSI model, which displays a Kosterlitz-Thouless type mixed
order transition with exponential divergence of the correlation length. It
would be of great interest to explore the relevance of this picture of
interacting charges to models in higher dimensions, such as models of jamming
transitions.

## Acknowledgments

I thank Amir Bar, Mustansir Barma, Yariv Kafri, Satya N Majumdar, Luca Peliti
and Gregory Schehr with whom I have collaborated over the years on studies of
properties and peculiarities of mixed order transitions. I also thank Eytan
Domany and Martin Evans for comments on the manuscript. This work was
supported by a research grant from the Center for Scientific Excellence at the
Weizmann Institute of Science.

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