# Time Glass: A Fractional Calculus Approach

R. C. Verstraten1, R. F. Ozela1,2, C. Morais Smith1 1Institute for Theoretical
Physics, Utrecht University, Princetonplein 5, 3584CC Utrecht, The Netherlands
2Faculdade de Física, Universidade Federal do Pará, 66075-110 Belém, PA,
Brazil

###### Abstract

Out of equilibrium states in glasses and crystals have been a major topic of
research in condensed-matter physics for many years, and the idea of time
crystals has triggered a flurry of new research. Here, we provide the first
description for the recently conjectured Time Glasses using fractional
calculus methods. An exactly solvable effective theory is introduced, with a
continuous parameter describing the transition from liquid through normal
glass, Time Glass, into the Gardner phase. The phenomenological description
with a fractional Langevin equation is connected to a microscopic model of a
particle in a sub-Ohmic bath in the framework of a generalized Caldeira-
Leggett model.

Introduction. The concept of time crystals, i.e., materials with an emergent
periodicty in time shapere2012classical ; wilczek2012quantum  has attracted
much attention recently khemani2019brief . After no-go theorems were proven
for several systems bruno2013impossibility ; watanabe2015prl , it was
understood that time crystals may actually occur in open iemini2018boundary ;
buvca2019non , driven khemani2016phase , and long-range interaction
valerii2019longrange  systems, and many experimental and theoretical
activities followed this novel idea. By now, time crystals have been conceived
and observed zhang2017observation ; rovny2018observation ; li2020discrete ;
yao2020classical ; medenjak2019isolated . Recently, a novel system was
conjectured, a Time Glass, which would have periodic intermediate glass states
wilczek2019timecrystals . It was claimed that they should appear in static
many-body localized systems exhibiting incommensurate local frequencies and no
long-range spatiotemporal order khemani2019brief .

The microscopic states of glasses have been puzzling researchers for many
years debenedetti2001supercooled ; lubchenko2015theory ;
schoenholz2016structural ; berthier2017active ; ninarello2017models ;
liao2019hierarchical . Microscopically, glasses look like liquids because
their molecules do not show any kind of structural order. These amorphous
materials exhibit completely different phase transitions in comparison to
ordered solids and often they do not even have a thermodynamic ground state.
Hence, equilibrium physics cannot be used for their description
cugliandolo1994out . A common understanding is that they correspond to the
occupation of a set of metastable states in the free-energy landscape.

One promising approach to describe the glass transition is the random first-
order transition theory biroli2008thermodynamic . It describes a hard-sphere
model, in which the spheres can get caged by their neighbors, thus restricting
their movement. This leads to a mean square displacement (MSD) that
corresponds to a free particle at small time scales, until it hits the cage
size (see Fig. 1), where the MSD saturates, indicating a glassy state
berthier2011theoretical ; flores2012msd  (for a discussion on glass states,
see SM). In the free energy landscape, this is the moment at which the
particle has explored the entire basin. This theory, however, can only be
solved exactly in infinite dimension, with a questionable connection to its
finite-dimensional counterpart biroli2012random .

More recently, a richer phase diagram was proposed for glasses, including the
so-called Gardner phase berthier2019gardner ; li2021determining . Although the
Gardner phase was first discovered in the description of spin glasses as a
solution that breaks one replica symmetry gardner1985spin , it was later
understood to occur in many materials seguin2016experimental . This marginal
glass phase has a free energy in which basins transform into metabasins
charbonneau2017glass , and is known to have a fractal structure
charbonneau2014fractal  (see Fig. 1). In the Gardner phase, there is a
hierarchy of cages inside cages charbonneau2015numerical , reminiscent of the
fractal structure in the energy landscape. Therefore, as time goes by, the
system explores larger cages, thus triggering an infinite staircase-like
behavior of the MSD.

Fig. 1: Left: Sketch of a free energy landscape with fractal-like metastable
minima. Right: A hierarchy of cages inside cages. The energy landscape
(purple) has four local minima, corresponding to the total number of large
orange cages. Zooming in on one of these minima, we see five smaller local
minima (green cages) inside this orange cage. Moreover, inside one green cage
there are three red cages. Depending on the number of nested cages we find 4
differing phases: Liquid (no cage), normal glass (1 cage), Time Glass (finite
number of cages), and marginal glass (infinite number of cages).

Here, we propose a unified picture where liquid, glass, Time Glass, and the
Gardner phase can be understood as a (sub-)diffusive Brownian motion described
by a fractional Langevin equation with white noise. Previously, the fractional
Langevin equation had been studied for colored noise metzler2000random ;
hilfer2000applications ; lutz2001langevin ; barkai2001fractional ;
burov2008critical ; deng2009ergodic ; pottier2003aging ; metzler2014anomalous
; jeon2010fractional ; jeon2013anomalous ; lizana2010foundation ;
vojta2019probability , but the features described here occur exclusively for
the case of white noise. At very low temperatures, the equipartition theorem
breaks down and a white-noise fractional Langevin equation arises as the
semiclassical description of a particle coupled to a two-level systems (TLS)
bath in the subohmic regime. Our unified procedure has a single varying
generic $s$-derivative friction, with $s$ integer or fractional, to describe
the different states of matter. Moreover, we show that for $0<s\lesssim 0.1$,
a Time Glass emerges, with a periodicity of $\pi(M/\eta)^{1/(2-s)}$, where $M$
is the mass of the “Brownian” particle and $\eta$ is a type of
viscoelasticity. Within our mean-field description, the Time Glass shows an
emergent frequency and periodically passes through many metastable states
before it gets to a final frozen glassy state. Therefore, our work sets a
mathematical framework for the description and realization of a Time Glass,
thus furthering our understanding of “time materials”.

Fractional Calculus. Fractional Differential Equations have been used in
physics, engineering, material science, control systems, protein folding
metzler2000random , and more sun2018new , but there are still many new
opportunities to be explored hilfer2000applications . Different definitions
were proposed by Riemann-Liouville, Caputo, Weyl, etc. (see SM for a short
historical overview). From a mathematical perspective, there is still a
discussion on which of the various fractional derivative definitions longlist
should be used for each kind of problem. Concerning the Caputo definition, the
idea is to rewrite a repeated integral into a generalizable form. As
factorials often appear in conventional integrals, one uses the Gamma function
as their non-integer generalization. The Caputo derivative is then given by
taking an integer derivative before doing a fractional integral;

$\mathbf{D}_{t}^{\alpha}f(t)=\frac{1}{\Gamma(n-\alpha)}\int_{0}^{t}(t-\tau)^{n-\alpha-1}f^{(n)}(\tau)\text{
d}\tau,$ (1)

where $n$ is an integer such that $n-1\leq\alpha<n$. Since the Caputo
definition is non-local, we have chosen the left-handed definition for the
boundary of the integral to retain causality, once we apply this time
derivative to our system. One benefit of the Caputo definition compared to
other definitions is that we can keep integer-order boundary conditions;
however, continuity in the order is lost on the integers. When this non-
integer derivative is applied to an exponential, we find the Mittag-Leffler
function, defined by

$E_{\alpha,\beta}(t)=\sum_{k=0}^{\infty}\frac{t^{k}}{\Gamma(\alpha k+\beta)},$
(2)

which is a generalized exponential that appears regularly in solutions of
fractional differential equations. For different parameters, this function can
show many features related to exponentials, such as damped oscillations and
exponential-like growth.

Fractional Langevin Equation. The fractional Langevin equation was recently
used to describe a system exhibiting Lévy flights vojta2019probability . Lévy
flights are often used for modeling the spreading of viruses, as they include
a description of the long (and fast) journeys that people make by plane, as
well as a more local random motion. Brownian motion only has one typical time
and length scale associated to it, while Lévy flights have many different time
and length scales metzler2000random ; eliazar2013motions . The equipartition
theorem implies a relation between the fluctuation and dissipation terms
kubo1966fluctuation ; hence, previous references used either colored noise
with the fractional Langevin equation metzler2000random ;
hilfer2000applications ; lutz2001langevin ; barkai2001fractional ;
burov2008critical ; lizana2010foundation ; vojta2019probability  or a white
noise associated to a fractional kinetic term and normal friction
kobelev2000fractional  which, after fractional integration, is equivalent to
the fractional Langevin equation with colored noise. However, none of these
models exhibit the plateaus in the MSD characterizing a glassy behavior, which
we will describe below. The features that we will discuss here are inherent to
a fractional Langevin equation with white noise, and occur only at low
temperatures, when the equipartition theorem breaks down. Later, in the
Microscopic Model Section, we present a microscopic description corresponding
to a physical realization of our phenomenological model.

Let us start by considering the fractional Langevin equation

$M\frac{\text{d}^{2}x(t)}{\text{d}t^{2}}+\eta\mathbf{D}_{t}^{s}x(t)=f(t),$ (3)

with $f(t)$ a white-noise force with average $\langle f(t)\rangle=0$ and
correlation $\langle f(t)f(t^{\prime})\rangle=K\delta(t-t^{\prime})$, where
$K=2\sin\left(\frac{\pi s}{2}\right)\eta t_{s}^{1-s}k_{B}T$ and
$t_{s}=\left(M/\eta\right)^{\frac{1}{2-s}}$ is the time scale of the system.
Introducing these fractional derivatives as friction, scaling with the
$s^{th}$ order derivative, yields a model for both sub-diffusion $(s<1)$ and
super-diffusion $(s>1)$, in addition to the usual Langevin Equation, which is
retrieved when $s=1$. The motivation for this change of friction compared to
the Langevin equation is firstly to introduce a general non-local operator
that knows about the history of a system and allows one to study non-Markovian
processes. The choice to take fractional derivatives can then be illustrated
with a thought experiment: Suppose a particle is moving at a constant speed
for a certain time. Then, ordinary friction is constant in time, while
friction of this fractional form scales as $t^{1-s}$. This means that for
$s>1$ the friction will fall off quickly, allowing for Lévy flights, while for
$s<1$ the friction will increase in time, thus reducing the probability for
large jumps bovet2015introduction .

Fig. 2: The exact MSD for different values of $s$, regularized by the typical
time and length scales $t_{s}$ and $L_{s}$. The analytical asymptotes for
$t\to 0$ and $t\to\infty$ are drawn next to the plots. The delocalized states
are shown at the top and the localized states at the bottom, while the
labeling order follows the decreasing value of $s$. (a) A liquid, described by
regular Brownian motion. (b) A normal glass, exhibiting ballistic motion at
short times and localization at longer times. (c) A Time Glass, displaying
ballistic motion at short times, but also an intermediate regime with a set of
small plateaus, before the long-time plateau sets in. (d) A marginal glass,
characterized by a driven particle in a quadratic potential. Here, the
friction term vanishes and an infinite collection of plateaus appear. The
overall slope is however finite, reminiscent of a liquid behavior.

The strength of this theory is that it is exactly solvable, which makes
possible the calculation of statistical properties, such as the MSD. For
$0\leq s<1$, the MSD is given by (see SM for details)

$\displaystyle\langle x(t)^{2}\rangle$
$\displaystyle=\frac{K}{M^{2}}\int_{0}^{t}\left[\tau
E_{2-s,2}\left(-\frac{\eta}{M}\tau^{2-s}\right)\right]^{2}\;d\tau$
$\displaystyle\qquad+\left[v_{0}L_{s}\frac{t}{t_{s}}E_{2-s,2}\left(-\frac{\eta}{M}t^{2-s}\right)\right]^{2},$
(4)

where $v_{0}=\langle x^{\prime}(0)\rangle t_{s}/L_{s}$ is dimensionless. The
short-time expansion for $t\ll t_{s}$ yields a ballistic behavior,

$\langle x(t)^{2}\rangle=(v_{0}L_{s})^{2}\left(\frac{t}{t_{s}}\right)^{2},$
(5)

while the long-time expansion is logarithmic for $s=0.5$, and otherwise given
by

$\langle
x(t)^{2}\rangle=\Delta_{s}^{2}+\frac{Kt^{2s-1}}{(2s-1)\eta^{2}\Gamma(s)^{2}},$
(6)

where, for $s<1/2$, the exponent of the second term in the MSD becomes
negative and, therefore, the MSD converges to the typical final cage size
$\Delta_{s}^{2}$ for $t\to\infty$.

In Fig. 2, the MSD has been plotted for several values of $s$ from one to
zero. We introduced the typical length scale of the system, which can be found
by dimensional analysis to be $L_{s}=\sqrt{Kt_{s}^{3}/M^{2}}$. The MSD shows
ballistic short-time behavior in all cases. For $s=1$, we retrieve the
conventional Langevin equation, which describes Brownian motion. The MSD also
shows a crossover from a ballistic ($\sim t^{2}$) to a linear dependence in
time, characteristic of a liquid [Fig. 2(a)]. For $s\lesssim 0.5$, instead,
the MSD saturates at large times, thus describing a glass [Fig. 2(b)]. We find
that a particularly interesting regime is provided by small values of $s$, in
the interval $0<s\lesssim 0.1$. In this case, a sequence of small metastable
plateaus characterizes a finite-depth fractal glass phase, before the
conventional glass regime is reached at larger times [Fig. 2(c)]. For $s=0$,
the “marginal glass” phase, proposed by Gardner, is realized, with an infinite
number of metastable plateaus and finite average slope ($\sim t$), typical of
liquids. This is an asymptotic phase, in which the fractal glass acquires
infinite depth [Fig. 2(d)].

Fig. 3: The MSD for several small values of $s$, regularized by the typical
time and length scales $t_{s}$ and $L_{s}$. The inset shows a linear scale
plot to highlight the periodicity in the anomalous glass phase, characterizing
a Time Glass.

Time Glass. Recently conjectured wilczek2019timecrystals , Time Glass is a
phase in which the MSD periodically repeats a glass-like plateau in an overall
glassy phase (see SM for a comparison between definitions of Time Glass). Now,
we concentrate on the region $0<s\lesssim 0.1$, which describes a finite-depth
fractal glass, reminiscent of the Gardner phase (see Fig. 1). The evolution of
the MSD upon varying $s$ is depicted in Fig. 3 and discloses many interesting
traits: i) At short times ($0<t<\pi t_{s}$), there exists a universal regime,
in which all curves collapse into a single one; ii) The small plateaus regime
sets in afterwards, but the overall slope of the intermediate-time behavior
increases as $s$ is reduced, thus showing a gradual transition from an overall
glass to liquid phase; iii) At sufficiently long times, there is saturation
for all $s\neq 0$, but this freezing occurs on increasingly longer timescales
as $s$ is reduced; iv) These plateaus appear at fixed intervals in time, as
promptly visualized in a linear scale plot (inset of Fig. 3). Put together,
all these features closely resemble (see SM) the marginal glass phase that has
recently been observed in colloidal glass experiments hammond2020experimental
and the emergent frequency that we obtain indicates that we are describing a
Time Glass phase. These results are further corroborated by calculations of
the position and velocity auto-correlation functions, which show the same
periodicity (see SM).

To find the coefficient of the time scale such that we have the period of the
system, we concentrate on the case $s=0$ and put $v_{0}=0$ for simplicity. As
this case is just a driven harmonic oscillator, we get

$\displaystyle\langle x(t)^{2}\rangle$
$\displaystyle=\frac{K}{M^{2}}\int_{0}^{t}\underbrace{\left[\tau
E_{2,2}\left(-\frac{\eta}{M}\tau^{2}\right)\right]^{2}}_{=\sin^{2}\left(\sqrt{\frac{\eta}{M}}\tau\right)}\;d\tau$
$\displaystyle=\frac{K}{2M\eta}\left[t+\frac{1}{2}\sqrt{\frac{M}{\eta}}\sin\left(2\sqrt{\frac{\eta}{M}}t\right)\right],$
(7)

where the relation between the Mittag-Leffler function and the sine can be
seen using their Taylor expansions. This yields a periodicity of
$\pi\sqrt{M/\eta}$ in time, thus providing the general period
$P=\pi(M/\eta)^{1/(2-s)}=\pi t_{s}$ of the system. For non-zero $s$, this
period applies only in a finite time-window before freezing.

Microscopic Model. Now, we aim at identifying an underlying microscopic model,
which is phenomenologically described by the fractional Langevin equation. We
will consider an open quantum system, upon which we perform a mean-field
approximation to obtain an effective model. Since Hamiltonian dynamics relies
on conservation of energy, we have to couple a system undergoing friction to a
bath, which exerts that force. Inspired by a generalization of the Caldeira-
Leggett model caldeira1993dissipative ; ferrer2007dynamical ; caldeira1983path
; caldeira1983tunneling ; caldeira1985influence , we consider a Hamiltonian

$H=H_{S}+H_{B}+H_{int},$ (8)

where

$H_{S}=\frac{\hat{p}^{2}}{2M}+V(x)$ (9)

is a Hamiltonian describing a “Brownian” particle, with mass $M$, momentum
$p$, coordinate $x$ and subject to a potential $V(x)$;

$H_{B}=\sum_{j}\frac{\hbar\omega_{j}}{2}\sigma_{zj}$ (10)

describes a TLS bath (i.e. truncated harmonic oscillators) with natural
frequencies $\omega_{j}$ and

$H_{int}=-x\sum_{j}J_{j}\sigma_{xj}$ (11)

is the interaction between the bath and the “Brownian” system
caldeira1993dissipative . The degrees of freedom in the bath are then
integrated out to describe quantum dissipation in the system. The spectral
function is given by the imaginary part of the Fourier transform of the
retarded dynamical susceptibility of the TLS bath, namely,

$J(\omega):=\operatorname{Im}\mathcal{F}\left\langle-i\Theta(t-t^{\prime})\left[F(t),F(t^{\prime})\right]\right\rangle$
(12)

and it is crucial for connecting the microscopic parameters of the Hamiltonian
with the phenomenological viscoelasticity coefficient $\eta$ appearing in the
Langevin equation. Here, $F(t)$ is the force produced by the particle on the
bath. For an Ohmic bath of harmonic oscillators, the spectral function is
given by $J(\omega)=\eta\omega$ for $\omega<\Omega$, where $\Omega$ denotes a
cutoff frequency, and is zero otherwise caldeira2014introduction . Such a bath
will then be effectively described by the Langevin equation, where the
friction is proportional to the velocity (first derivative of position) of the
system. However, for our choice of sub-Ohmic TLS bath ferrer2006optical ;
ferrer2007dynamical , the spectral function is given by

$J(\omega,T)=\eta\sin\left(\frac{\pi
s}{2}\right)\omega^{s}\tanh\left(\frac{\hbar\omega}{2kT}\right)\Theta(\Omega-\omega),$
(13)

where $0<s<1$ and $\Omega$ is a cutoff frequency. The hyperbolic tangent is
important for the quantum description where it allows the temperature to
select which bath components are more relevant, namely, the ones with
frequencies $\omega>2k_{B}T/\hbar$. In our semi-classical approach, however,
we take the limit $kT\ll\hbar\Omega$, which reduces it to the Caldeira-Leggett
model, apart from losing equipartition. We can therefore rescale the TLS
energies by $\omega^{1-s}t_{s}^{1-s}$ to retain a white-noise correlation. The
friction force is then given by

$F_{fr}=\frac{2}{\pi}\frac{d}{dt}\left\\{\int_{0}^{t}\int_{0}^{\infty}\frac{J(\omega)}{\omega}\cos[\omega(t-t^{\prime})]q(t^{\prime})\text{
d}\omega\text{ d}t^{\prime}\right\\},$ (14)

which, after a reparametrization $\omega\to\omega/(t-t^{\prime})$, becomes
proportional to a fractional derivative in the limit $kT\ll\hbar\Omega$. The
proportionality constant is then calculated by a complex contour integral,
which results in a finite value for $0<s<1$ (see SM for details of the
calculation). This then leads to a friction term given by

$F_{fr}=\eta\mathbf{D}_{t}^{s}q(t).$ (15)

Therefore, a particle interacting with a sub-Ohmic two-level systems bath is
well described by the fractional Langevin equation with white noise in the
low-temperature limit.

Conclusion. Making use of fractional Caputo derivatives, we have shown that a
semi-classical system coupled to a low-temperature sub-Ohmic bath of two-level
systems can be described by the fractional Langevin equation with white noise.
We have solved this equation analytically and analyzed the anomalous
diffusion. Different behaviors were observed, depending on $s$, from ordinary
Brownian motion for $s=1$ to glassy behavior for $s\lesssim 0.5$, a Time Glass
for $0<s\lesssim 0.1$, and a marginal glass for $s=0$. Our work extends the
use of fractional derivatives to the realm of sub-diffusion, by linking the
formalism to the description of the Gardner transition. We identified a new
regime between the Gardner phase and the usual glass, and showed that it is a
realization of the long sought Time Glass. Further extensions of this model
using the techniques applied to the fractional Langevin equation with colored
noise deng2009ergodic ; pottier2003aging ; metzler2014anomalous  are
anticipated.

###### Acknowledgements.

We are grateful to L. M. C. Janssen for introducing us to the Gardner phase
and to P. A. Zegeling for giving us insight into fractional calculus. We also
thank J. de Graaf, S. Franz, M. Katsnelson, W. van Saarloos, J. Wettlaufer, T.
H. Hansson, E. Barkai, B. Buča, and F. Wilczek for fruitful discussions. R. F.
O. is partially supported by Coordenação de Aperfeiçoamento de Pessoal de
Nível Superior (CAPES), finance code 001.

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See pages 1,,,2,,3,,4,,5,,6,,7,,8,,9,,10, of arXiv_SM_Verstraten.pdf