# Exploring quantum criticality in a 4D quantum disordered system

Farid Madani  Univ. Lille, CNRS, UMR 8523 – PhLAM – Laboratoire de Physique
des Lasers Atomes et Molécules, F-59000 Lille, France  Maxime Denis  Univ.
Lille, CNRS, UMR 8523 – PhLAM – Laboratoire de Physique des Lasers Atomes et
Molécules, F-59000 Lille, France  Pascal Szriftgiser  Univ. Lille, CNRS, UMR
8523 – PhLAM – Laboratoire de Physique des Lasers Atomes et Molécules, F-59000
Lille, France  Jean Claude Garreau  0000-0003-4155-4459 Univ. Lille, CNRS, UMR
8523 – PhLAM – Laboratoire de Physique des Lasers Atomes et Molécules, F-59000
Lille, France  Adam Rançon  Univ. Lille, CNRS, UMR 8523 – PhLAM – Laboratoire
de Physique des Lasers Atomes et Molécules, F-59000 Lille, France  Radu
Chicireanu  Univ. Lille, CNRS, UMR 8523 – PhLAM – Laboratoire de Physique des
Lasers Atomes et Molécules, F-59000 Lille, France

Phase transitions are prevalent throughout physics, spanning thermal phenomena
like water boiling to magnetic transitions in solids Herbut (2007). They
encompass cosmological phase transitions in the early universe and the
transition into a quark-gluon plasma in high-energy collisions Mazumdar and
White (2019). Quantum phase transitions, particularly intriguing, occur at
temperatures near absolute zero and are driven by quantum fluctuations rather
than thermal ones Sachdev (1999). The strength of the fluctuations is very
sensitive to the dimensionality of the physical systems, which determines the
existence and nature of phase transitions. Low-dimensional systems often
exhibit suppression of phase transitions, while high-dimensional systems tend
to exhibit mean-field-like behavior Kardar (2007). The localization-
delocalization Anderson transition stands out among quantum phase transitions,
as it is thought to retain its non-mean-field character across all dimensions
Evers and Mirlin (2008). This work marks the first observation and
characterization of the Anderson transition in four dimensions using ultracold
atoms as a quantum simulator with synthetic dimensions. We characterize the
universal dynamics in the vicinity of the phase transition. We measure the
critical exponents describing the scale-invariant properties of the critical
dynamics, which are shown to obey Wegner’s scaling law Wegner (1976). Our work
is the first experimental demonstration that the Anderson transition is not
mean-field in dimension four.

_This work is dedicated to the memory of Dominique Delande, collaborator and
friend, who passed away in September 2023_

Dimensionality plays a key role in physical phenomena, especially in the
context of phase transitions characterized by an order parameter, e.g. in
magnets, liquid crystals, superconductors, and solids, as well as in the
chiral symmetry-breaking phase transition of quantum chromodynamics and the
electroweak transition in the early universe. Quantum and thermal fluctuations
increase as the dimension is lowered so that ordered phases –and thus the
corresponding phase transitions– disappear below a so-called lower critical
dimension. Conversely, large dimensionality tends to increase the number of
neighbors, so that mean-field theories become a good description above a
certain upper critical dimension (UCD). Dimension four is the UCD for many
systems, for instance magnets, superfluids, and superconductors Herbut (2007)
and the paradigmatic classical Ising model Fröhlich (1982); Aizenman (1982);
Aizenman and Duminil-Copin (2021); quantum systems tend to have lower UCD
Sachdev (1999), but adding disorder can increase it Imry and Ma (1975). There
are however models for which the UCD is not known, e.g. that of the Kardar-
Parisi-Zhang (KPZ) equation is strongly debated Castellano  _et al._ (1998);
Colaiori and Moore (2001); Marinari  _et al._ (2002); Fogedby (2006); Oliveira
(2022).

Another case where the UCD has been controversial is the Anderson transition
of waves in a disordered medium Evers and Mirlin (2008). For dimensions larger
than the lower critical dimension $d=2$ Abrahams  _et al._ (1979), the system
presents a transition from a localized phase, where subtle destructive
interferences completely suppress propagation in the medium, to a
delocalized/diffusive phase where transport is possible. The Anderson
transition has been observed in three dimensions with all kinds of waves,
classical or quantum, e.g. electrons in disordered electronic systems
Katsumoto  _et al._ (1987), cold-atom quantum-chaotic systems Chabé  _et al._
(2008), atomic matter waves in disordered optical potentials Jendrzejewski
_et al._ (2012); Semeghini  _et al._ (2015), acoustic Hu  _et al._ (2008) and
electromagnetic waves in random media Lahini  _et al._ (2008); Chabanov  _et
al._ (2000); Schwartz  _et al._ (2007). Remarkably, all these various and very
different systems are all described by the same “universal” functions and
critical exponents, independently of microscopic details and depending only on
dimensionality and the symmetries of the system, namely, in the present case,
the time reversal invariance, defining the so-called _orthogonal universality
class_. This universal behavior is intimately related to the fact that, close
to the phase transition, the physics is expected to become effectively scale
invariant Wilson (1983). On the theory side, Anderson transition in high
dimensions is still an incredibly hard hurdle, since it is a non-standard
phase transition, i.e. not captured by Landau theory in terms of an order
parameter. While the UCD has been predicted to be four by the so-called self-
consistent theory Vollhardt and Wölfle (1982), it has also been argued to be
infinite Mirlin and Fyodorov (1994), as also suggested by numerical
simulations up to dimension six Ueoka and Slevin (2014); Tarquini  _et al._
(2017) Furthermore, the presumed infinite UCD prevents an expansion around a
mean-field-like solution, as is usually done in the context of second-order
phase transitions. In particular, quantitative theoretical descriptions of the
Anderson transition exist only in the vicinity of dimension two (the so-called
$2+\epsilon$ expansion) and on the Bethe lattice Efetov (1997), but there are
none in dimensions greater or equal to three.

Figure 1:  Experimental quantum simulation of the 4D Anderson transition. a)
The red curve shows the kick potential’s amplitude modulation envelop
$\mathcal{K}(t)$ used to synthesize extra dimensions. The potential is pulsed
periodically (black line). b) Absorption images of the kicked atomic cloud at
different times at the critical point. The color scale represents the optical
depth. The sinusoidal potential is applied along the $x$ direction (red
arrows). The relevant dynamics is along the $x$ direction, which is
decorrelated from the others. c) Momentum distributions in the $x$ direction
in semi-log scale, extracted from the absorption images [shown as colored
lines in panel b)]. Darker colors represent later times. d) Scale invariance
of the momentum distribution at the critical point: after rescaling the
distribution and momenta by $t^{1/d}$ with $d=4$, Eq. (4), all curves collapse
into a single one.

In this work, we experimentally realize a quantum simulation of the celebrated
Anderson model Anderson (1958) in four dimensions and study its metal-
insulator (or localization-delocalization) phase transition. We find that
close to the transition, the 4D scaling relations are obeyed and we determine
critical exponents compatible with numerical estimates. We therefore
experimentally rule out that the Anderson transition is mean-field in 4D. We
also measure the full two-parameter scaling function of the momentum
distribution in terms of rescaled momentum and distance to the critical point,
which gives a stringent test of the scaling properties close to the Anderson
transition.

Recently, various ways have been devised to create synthetic dimensions Ozawa
and Price (2019). They can be engineered using “parametric dimensions” Lohse
_et al._ (2018); Zilberberg  _et al._ (2018) or degrees of freedom like spin
Boada  _et al._ (2012); An  _et al._ (2017); Yuan  _et al._ (2018); Mancini
_et al._ (2015); Chalopin  _et al._ (2020), frequency Yuan  _et al._ (2016);
Ozawa  _et al._ (2016), linear An  _et al._ (2021) and angular Luo  _et al._
(2015) momentum, time bins Regensburger  _et al._ (2012); Chalabi  _et al._
(2019) or carefully designed lattice Maczewsky  _et al._ (2020). Here, we use
a time-modulated driven gas of ultracold atoms to engineer both synthetic
dimensions and disorder Shepelyansky (1987); Casati  _et al._ (1989). This
allows us to probe experimentally, for the first time, the Anderson transition
in four dimensions. The system is an atomic kicked rotor Moore  _et al._
(1995) of dilute potassium atoms, submitted to a quasi-periodically modulated
pulsed laser standing wave (SW), so that only the dynamics along the SW
direction is relevant, Fig. 1-b). The corresponding Hamiltonian reads

$H=\frac{p^{2}}{2}+\mathcal{K}(t)\cos(x)\sum_{n}\delta(t-n),$ (1)

where $x$ is the particle position (along the laser axis, measured in units of
the inverse of the kick-potential wavenumber $k_{K}$), $p$ is the conjugate
momentum, normalized such that $[x,p]=i{\mathchar 22\relax\mkern-9.0muk}$,
with ${\mathchar 22\relax\mkern-9.0muk}=\hbar k_{K}^{2}T_{1}/M$ ($M$ is the
mass of the atoms, and $T_{1}$ the kick period) and $\mathcal{K}$ is
proportional to the ratio of SW intensity to its detuning. Time is measured in
in units of $T_{1}$ and the pulse duration is short enough to be considered
instantaneous, modeled as Dirac deltas in Eq. (1). If the kick strength
$\mathcal{K}(t)$ is constant, the system is effectively 1D and displays
dynamical localization, that is, Anderson localization in momentum space
Casati  _et al._ (1979); Moore  _et al._ (1995). Indeed, the kicks change the
atoms’ momentum, allowing them to “hop” in momentum space, while kinetic
energy term give them a pseudo-random phase in momentum space, playing the
role of disorder Fishman  _et al._ (1982). By carefully crafting the time-
dependence of the $\mathcal{K}(t)$, one can change the system’s effective
dimensionality Shepelyansky (1987); Casati  _et al._ (1989). This has allowed
for a quite thorough investigation of Anderson physics: observation of the
localization in one Moore  _et al._ (1995) and two dimensions Manai  _et al._
(2015), observation of the 3D transition Chabé  _et al._ (2008); Lemarié  _et
al._ (2009), characterization of its critical properties Lemarié  _et al._
(2010); Lopez  _et al._ (2013) and of its universality Lopez  _et al._ (2012).

For our present purpose, following Casati  _et al._ (1989); Chabé  _et al._
(2008); Lemarié  _et al._ (2009), we realize a quasiperiodic quantum kicked
rotor (QpQKR) by taking
$\mathcal{K}(t)=K\left(1+\varepsilon\cos(\omega_{2}t+\varphi_{2})\cos(\omega_{3}t+\varphi_{3})\cos(\omega_{4}t+\varphi_{4})\right)$,
see Fig. 1-a), with frequencies $\omega_{2}=\sqrt{5}$, $\omega_{3}=\sqrt{13}$
and $\omega_{4}=\sqrt{19}$ (the frequencies $\omega_{i}$, ${\mathchar
22\relax\mkern-9.0muk}$ and $2\pi$ must be incommensurate to avoid
resonances). This makes the driving quasiperiodic, but by defining the
synthetic dimensions $x_{i}=\omega_{i}t+\varphi_{i}$ ($i=2,3,4$) and their
conjugate momenta $p_{i}$ we can map this Hamiltonian onto a periodic one

$\mathcal{H}=\frac{p^{2}}{2}+\omega_{2}p_{2}+\omega_{3}p_{3}+\omega_{4}p_{4}+K\left(1+\varepsilon\cos(x_{2})\cos(x_{3})\cos(x_{4})\right)\cos(x)\sum_{n}\delta(t-n).$
(2)

Here again, the kicks play the role of a hopping term of the (four-
dimensional) momentum while the kinetic energy (both quadratic and linear)
plays the role of disorder. One can show Fishman  _et al._ (1982); Casati  _et
al._ (1989) that the Floquet eigenstates of the _periodic_ Hamiltonian
$\mathcal{H}$ are eigenstates of a 4D disordered tight-binding Hamiltonian in
momentum space, thus able to display an Anderson transition, see Methods. It
turns out that there is no mobility edge, meaning that, for given values of
$K$ and $\varepsilon$, all eigenstates are of the same nature, either
localized, critical, or diffusive, making a detailed study of the critical
properties possible. For $\varepsilon=0$ the system is 1D, dynamics arises in
the synthetic dimensions as $\varepsilon$ increases Lemarié  _et al._ (2010);
Lopez  _et al._ (2013).

The experiment uses a thermal could of $\sim 10^{6}$ potassium atoms (isotope
41K) prepared by evaporative cooling in a crossed optical dipole trap at a
temperature $T=1.8$ $\mu$K. The cloud is kicked along the horizontal direction
by a far-detuned optical standing wave created by a retro-reflected Gaussian
beam with a waist of $2.8$ mm, issued from a pulsed laser at $770.229$ nm,
corresponding to a detuning of $\Delta=-61.2$ GHz to the D1 line ($770.108$
nm). The repetition frequency of the laser is $142.8$ kHz, corresponding to
${\mathchar 22\relax\mkern-9.0muk}=2.89$. The maximum laser power is $170$ W,
and the pulse duration is 20 ns ($\sim 0.3\,\%$ of the pulse period) short
enough that the motion of the atoms can be neglected during the pulse and the
$\delta-$kicked potential approximation is well verified. The SW amplitude
modulation is generated by the RF drive of an acousto-optic modulator, thus
realizing the synthetic dimensions. After a given number of kicks, the
momentum distribution is measured by absorption imaging, after a time-of-
flight of typically 20 ms. Throughout the experiments, atomic densities are
kept below $10^{12}$ cm-2, so that interaction effects, which might alter the
characteristics of the phase transition Cherroret  _et al._ (2014), are
negligible at the time scale of the kick sequence. To confirm that, we
performed experiments with densities up to three times lower, without
noticeable change in the measured momentum distributions.

Figure 2:  Locating the critical point of the Anderson transition. a) Phase
diagram of the QpQKR as a function of $K$ and $\varepsilon$ (numerical
simulation). The color scale represents the slope of
$\log\left(t^{2\alpha}\langle p^{2}\rangle_{t}\right)$, $\alpha=-1/4$, with
the blue (red) region corresponding to the (de)localized phase. The critical
line is in bright red along the green region. The dashed line represents the
path in phase space studied experimentally. b) Binder parameter
$B(t,K)=\Pi_{t}(0)^{2}\langle p^{2}\rangle_{t}$ in the localized ($K<K_{c}$),
critical ($K\simeq K_{c}$) and diffusive ($K>K_{c}$) regimes (from top to
bottom). After a short transient time ($t\lesssim t_{s}$, gray area), $B(t,K)$
is stationary at the critical point. c) The slope of the linear part of the
Binder parameter after $t_{s}$ [shown as dashed lines on panel b)] as a
function of $K$. The critical point $K_{c}$ corresponds to a vanishing slope.
In b) and c), the colors of the symbols match that of the corresponding dots
in a).

The phase diagram in the $(K,\varepsilon)$ space is shown in Fig. 2-a), see
Methods, with a color code corresponding the behavior of the kinetic energy,
proportional to the variance of the momentum distribution $\Pi_{t}(p)$,
$\langle p^{2}\rangle_{t}=\int dp\,p^{2}\Pi_{t}(p)$. For small $K$ and
$\varepsilon$ the system is localized (blue region): the atomic gas momentum
distribution freezes at long times with a finite width, $\langle
p^{2}\rangle_{t}\to p_{\mathrm{loc}}^{2}$ with localization length
$p_{\mathrm{loc}}$ (in momentum space). In the opposite limit, the system is
delocalized/diffusive (red region): the variance of the momentum distribution
increases linearly with time, $\langle p^{2}\rangle_{t}\to D\,t$, with
diffusion coefficient $D$. The Anderson transition takes place on a critical
line $\varepsilon_{c}(K)$ [red line in Fig. 2-a)], and the dynamics is
universal and scale invariant in the vicinity of the transition Abrahams  _et
al._ (1979), see the green region in the phase diagram. This implies that the
atomic momentum distribution $\Pi_{t}(p)$ obeys a universal finite-time
scaling law, allowing for studying the critical behavior of the Anderson
transition. The scaling law reads

$\Pi_{t}(p)=t^{\alpha}\mathcal{F}(p\,t^{\beta},\delta_{K}\,t^{\gamma}),$ (3)

with $\delta_{K}=K-K_{c}$, $\mathcal{F}$ a universal function, and critical
exponents $\alpha$, $\beta$ and $\gamma$. This universal scaling behavior
holds for sufficiently long times (at short times, typically below $t_{s}=50$
kicks in the present setup, the dynamics is not universal) and close enough to
the critical point. It is the analog of the finite-size scaling in standard
phase transitions, with time playing the role of the finite size that the
system has been able to explore. The scaling form Eq. (3) can be interpreted
as follows. After a short transient dynamics ($t<t_{s}$), and as long as
$\delta_{K}\,t^{\gamma}$ is small enough, the momentum distribution at the
critical point takes a universal shape $\Pi_{t}(p)\simeq
t^{\alpha}\mathcal{F}(p\,t^{\alpha},0)$, see Fig. 1-c) and d). If
$\delta_{K}=0$, it stays critical. Otherwise, it will display this critical
shape until $\delta_{K}t^{\gamma}$ is of order one, after which it will tend
to a localized ($\delta_{K}<0)$ or diffusive ($\delta_{K}>0)$ shape.

Figure 3:  Determination of the critical exponents $\nu$ and $s$. a) Kinetic
energy $\langle p^{2}\rangle_{t}$ as a function of the distance to the
critical point $K-K_{c}$ at various times, with $K_{c}=5.62$. Each value of
$K-K_{c}$ is encoded in a different color, which goes from lighter to darker
as time increases. b) Same data after rescaling, the black line corresponds to
the optimal scaling function $\Lambda(\tilde{\delta})$ using $\nu$ and $s$ as
fitting parameters, here shown at the optimal values $\nu_{\rm exp}$ and
$s_{\rm exp}$, see Methods. c) Joint probability distribution of the exponents
$\nu$ and $s$ obtained by bootstrap (see Methods). We obtain $\nu_{{\rm
exp}}=1.07\pm 0.16$ and $s_{{\rm exp}}=2.22\pm 0.38$ (blue ellipse corresponds
to the $95\%$ confidence region, green dot with error bars corresponding to
the most probable value and $95\%$ confidence intervals). The best numerical
values for the critical exponents for the Anderson transition in $d=4$ are
$\nu_{\mathrm{num}}=1.156\pm 0.014$ and $s_{\mathrm{num}}=2.312\pm 0.028$
(using Wegner’s scaling law), shown as a red dot. The red cross represents the
self-consistent theory prediction $\nu_{\rm sc}=1/2$ and $s_{\rm sc}=1$.

The exponents in Eq. (3) are not independent but are related by the
normalization of the momentum distribution and by the scaling of
$p_{\mathrm{loc}}$ and $D$ Evers and Mirlin (2008): in the localized phase
($\delta_{K}<0$), $p_{\mathrm{loc}}\propto|\delta_{K}|^{-\nu}$, defining the
critical exponent $\nu$; in the diffusive phase ($\delta_{K}>0$), with
diffusion coefficient $D\propto\delta_{K}^{s}$. As shown in the Methods, this
implies $\alpha=\beta=-\nu/(s+2\nu)$ and $\gamma=1/(s+2\nu)$. Moreover, the
exponents $s$ and $\nu$ have been predicted to obey Wegner’s scaling law
$s=(d-2)\nu$ for $d$-dimensional systems Wegner (1976). It has been verified
in three-dimensional electronic systems Itoh  _et al._ (2004), although with
critical exponents in disagreement with the best numerical estimates Ueoka and
Slevin (2014) and state-of-the-art cold atoms experiments Chabé  _et al._
(2008). Studying the critical behavior Eq. (3) for the 4D Anderson transition
allows us to experimentally determine the scaling exponents $\nu$ and $s$, and
test Wegner’s scaling law.

We experimentally studied the momentum distribution across the Anderson
transition by following the path in the phase diagram indicated as a dashed
black line in Fig. 2-a). Our first goal is to locate the critical kick
strength $K_{c}$; for that, we take advantage of the above-mentioned scale
invariance of the distribution shape at the critical point. For this task, it
is convenient to introduce a Binder-like parameter that does not scale at the
transition Binder (1981). We observe that on the one hand, $\langle
p^{2}\rangle_{t}=t^{-2\alpha}\Lambda(\delta_{K}\,t^{\gamma})$, with
$\Lambda(\tilde{\delta})=\int
d\tilde{p}\,\tilde{p}^{2}\mathcal{F}(\tilde{p},\tilde{\delta})$ a universal
scaling function, with $\tilde{\delta}=\delta_{K}t^{\gamma}$ and
$\tilde{p}=pt^{\alpha}$. On the other hand, we have
$\Pi_{t}(0)^{2}=t^{\alpha}\mathcal{F}(0,\tilde{\delta})$. We thus define
$B(t,K)=\Pi_{t}(0)^{2}\langle p^{2}\rangle_{t}=b(\delta_{K}\,t^{\gamma})$ with
$b(\tilde{\delta})=\mathcal{F}(0,\tilde{\delta})^{2}\Lambda(\tilde{\delta})$ a
universal scaling function. Being the product of the distribution’s amplitude
squared by its second moment, this Binder parameter is a constant if the shape
of the distribution remains unchanged during its time evolution. The critical
point $K_{c}$ can thus be identified as the value of $K$ at which the Binder
parameter is constant, while for other values of $K$ it evolves towards the
asymptotic localized (exponential) or diffusive (Gaussian) shapes.
Importantly, using the Binder parameter allows us to locate the critical point
without any prior assumption on the critical scaling (namely, here, on the
critical exponents $\nu$ and $s$).

Figure 2-b) displays the Binder parameter calculated from the experimental
data for three values of $K$. The curve in green dots is almost horizontal
after a time $t>t_{s}=50$, the Binder parameter is about constant, signaling
the proximity with the critical point. The curves in blue and red dots vary
almost linearly with time, indicating, respectively, a value of $K$ smaller or
larger than $K_{c}$. Hence, we can use the slope of such curves as an
indicator of the distance to the critical point. Adding more values of $K$ we
obtain the plot displayed in the panel c), from which we can extract the value
$K_{c}\simeq 5.62\pm 0.15$, in good agreement with numerical simulations
$K_{c}\simeq 5.45\pm 0.05$ (error bars correspond to one standard deviation).

Figure 4:  Two-parameter scale invariance near criticality of the 4D Anderson
transition. Top panels: raw (top) and rescaled (bottom) momentum
distributions, averaged over 20 experimental realizations, in the localized
$\delta_{K}<0$ [a) and b)] and diffusive ($\delta_{K}>0$) [c) and d)] regimes.
Colors encode the value of the scaling parameter $\tilde{\delta}$, darker
tones representing longer times. e) Reconstruction of the two-parameter
scaling function. Data obtained for various values of $K$ and $t$, properly
rescaled, all fall on a smooth surface corresponding to the scaling function
$\mathcal{F}(\tilde{p},\tilde{\delta})$. The data of the bottom plot in panels
a), b), c), and d), as well as the data at the critical point [Fig. 1-d)], are
shown as thick lines.

After finding the location of the critical point, we determine the critical
exponents $\nu$ and $s$, see Methods for details. Fig. 3-a) shows the values
of the kinetic energy $\langle p^{2}\rangle_{t}$, measured at various times as
a function of $\delta_{K}$. Rescaling this data with trial values of the
exponents allows us to optimize the collapse of $\langle p^{2}\rangle_{t}$
into a single curve, see Fig. 3-b), using a least-square Zhang  _et al._
(2012). To assert the uncertainty of the critical exponents, we use a
bootstrap method Dogra  _et al._ (2023), which gives the joint probability
distribution of the critical exponents shown in Fig. 3-c). The optimal values
of the exponents are $\nu_{{\rm exp}}=1.07\pm 0.16$ and $s_{{\rm exp}}=2.22\pm
0.38$ (error bars correspond to $95\%$ confidence intervals), with Wegner’s
scaling law $s=(d-2)\nu$ well satisfied for $d=4$. This is in good agreement
with the best numerical estimate $\nu_{\mathrm{num}}=1.156\pm 0.014$ and
$s_{\mathrm{num}}=2.312\pm 0.028$ (using Wegner’s scaling law) for a dimension
four Anderson model Ueoka and Slevin (2014), marked as a red dot. The so-
called self-consistent theory Vollhardt and Wölfle (1982) predicts a UCD equal
to four, with $\nu_{\rm sc}=1/2$ and $s_{\rm sc}=1$ for any $d\geq 4$, marked
as a red cross, while numerical simulations indicate non-trivial $\nu>1/2$ at
least up to $d=6$ Ueoka and Slevin (2014); Tarquini  _et al._ (2017). Our
measurement is about $7$ standard deviations away from the mean-field value
$\nu=1/2$, and thus constitutes the first experimental demonstration that
$d=4$ is _not_ the Anderson transition’s upper critical dimension.

We now verify experimentally the two-parameter scaling law Eq. (3), which we
find convenient to reexpress as

$\Pi_{t}(p)=t^{-1/d}\mathcal{F}(p\,t^{-1/d},\delta_{K}\,t^{1/\nu d}),$ (4)

where we use Wegner’s scaling law $s=(d-2)\nu$, with $d=4$ and $\nu=1.156$. We
have shown in Fig. 1-d) that, at criticality $K=K_{c}$, the momentum
distribution is indeed scale-invariant. This property also holds for $K\neq
K_{c}$ provided $\tilde{\delta}=\delta_{K}t^{1/\nu d}$ is kept constant. This
is shown in the top plots of Figs. 4-a) and 4-b) on the localized side of the
transition $\tilde{\delta}<0$ and in the top plots of Figs. 4-c) and 4-d) on
the diffusive side $\tilde{\delta}>0$ (the corresponding values of $t$ and
$\delta_{K}$ are given in Methods). Having tested the two-parameter scale
invariance, we can experimentally reconstruct the shape of the scaling
function $\mathcal{F}(\tilde{p},\tilde{\delta})$. While one-parameter scaling
functions at criticality have been measured experimentally, e.g. for the 3D
Ising Damay  _et al._ (1998); Bonetti  _et al._ (2000) and 1D KPZ Fontaine
_et al._ (2022) universality classes, here we measured a _two-parameter_
scaling function, which fully characterizes the dynamics close to the critical
point. For that, we applied the scaling procedure to all the data displayed in
Figs. 2 and 3; the result is shown in Fig. 4-e). We observe that the data form
a smooth surface corresponding to $\mathcal{F}(\tilde{p},\tilde{\delta})$,
evolving from a localized, exponential shape for $\tilde{\delta}<0$ to a
delocalized, Gaussian shape for $\tilde{\delta}>0$.

In conclusion, we presented the first experimental study of a quantum phase
transition in four spatial dimensions, three of them being synthetic
dimensions added to a physically 1D system. The critical scaling properties of
the Anderson localization-delocalization transition were explicitly exhibited,
including the full two-parameter scaling function, and the resulting critical
exponents were found in good agreement with the numerical simulations of the
Anderson model (_not_ that of the QpQKR). These results also show that $d=4$
is not the upper critical dimension and thus may serve as a benchmark for the
still missing quantitative theory of the Anderson transition. The techniques
used here might be extended to higher dimensions and other kinds of systems,
thus opening new ways to test theories with varying dimensionality.

_Acknowledgements:_ We thank C. Cherfan for his contributions to the early
stages of the experiment, and V. Vuatelet for preliminary numerical
simulations on the problem. We thank C. Hainaut for useful discussions, and A.
Amo, C. Hainaut, and G. Lemarié for a careful reading of the manuscript. This
work was supported by the Agence Nationale de la Recherche (ANR) through
Research Grants MANYLOK No. ANR-18-CE30-0017 and Labex CEMPI (GrantNo.
ANR-11-LABX-0007-01), by CPER Wavetech, and also by the PHC Cogito and CNRS
IEA programs. The Contrat de Plan Etat-Region (CPER) WaveTech is supported by
the Ministry of Higher Education and Research, the Hauts-de-France Regional
council, the Lille European Metropolis (MEL), the Institute of Physics of the
French National Centre for Scientific Research (CNRS) and the European
Regional Development Fund (ERDF).

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##  I Methods

Measurement procedure and experimental uncertainties. The average kinetic
energy is determined by fitting the quantity $\Pi_{t}(p)\times p^{2}$ with a
functional form (Lobkis-Weaver distribution), following Hainaut  _et al._
(2018). Experiments, corresponding to a given kick number and
$(K,\varepsilon)$ couple, were typically repeated and averaged between three
and six times. The relative dispersion of $\langle p^{2}\rangle_{t}$ has a
standard deviation of approximately $10\%$ in the experiments, and is almost
independent of the experimental parameters. The precise control of the kick
strength is also crucial for measuring the critical exponents. The kick laser
power is controlled and modulated with an acousto-optical modulator, whose
transfer function was regularly remeasured (typically every ten experimental
cycles) using the a beam pick-off on a photodiode, to ensure long-term
stability. While the QpQKR experiments were performed with a thermal cloud and
at low density, we used a 41K BEC in order to precisely determine the value of
the kick strength. This was achieved by pulsing the lattice beams and
observing an atom diffraction pattern in time-of-flight vs. the power of the
lattice beams. We estimate that the uncertainty of the measurement of the kick
amplitude strength, as well as the achieved long-term stability, are on the
order of $1\%$.

Mapping on a disordered model. Following Lemarié  _et al._ (2009), the
dynamics of the system can be mapped onto an effectively periodic one by
introducing three extra dimensions, represented by canonically conjugated
position and momentum operators $x_{i}$ and $p_{i}$, $i=2,3,4$. Defining the
effective Hamiltonian
$\mathcal{H}=\frac{p^{2}}{2}+\omega_{2}p_{2}+\omega_{3}p_{3}+\omega_{4}p_{4}+K\left(1+\varepsilon\cos(x_{2})\cos(x_{3})\cos(x_{4})\right)\cos(x)\sum_{n}\delta(t-n).$
The effective Hamiltonian is thus four-dimensional, with linear kinetic energy
for the extra dimensions. Furthermore, the four-dimensional wavefunction with
a “plane source” initial condition
$\Psi(x,x_{2},x_{3},x_{4})=\psi(x)\delta(x_{2})\delta(x_{3})\delta(x_{4})$ has
the same dynamics as that of the one-dimensional quasiperiodic system with
initial condition $\psi(x)$. The Floquet eigenstates $\Phi$ of the periodic
Hamiltonian $\mathcal{H}$ with eigenvalue $e^{i\epsilon/{\mathchar
22\relax\mkern-9.0muk}}$ are eigenstates of a 4D tight binding model
$\epsilon_{\mathbf{p}}\Phi_{\mathbf{p}}+\sum_{\mathbf{q}}t_{\mathbf{q}}\Phi_{\mathbf{p}-\mathbf{q}}=0$,
where $\mathbf{p}=(p,p_{2},p_{3},p_{4})$ and $\mathbf{q}$ are the four-
dimensional momenta on a hypercube. The on-site energies are
$\epsilon_{\mathbf{p}}=\tan\left[\frac{1}{2{\mathchar
22\relax\mkern-9.0muk}}\left(\epsilon-\frac{p^{2}}{2}-\omega_{2}p_{2}-\omega_{3}p_{3}-\omega_{4}p_{4}\right)\right]$
and the short-range hopping amplitude $t_{\mathbf{q}}$ is the four-fold
Fourier transform of $\tan\left(\frac{K}{2{\mathchar
22\relax\mkern-9.0muk}}(1+\varepsilon\cos(x_{2})\cos(x_{3})\cos(x_{4}))\cos(x)\right)$.
If ${\mathchar 22\relax\mkern-9.0muk}$ is incommensurate with $2\pi$,
$\epsilon_{\mathbf{p}}$ is a pseudo-random sequence, and the tight-binding
model is disordered Anderson model in momentum space Lemarié  _et al._ (2009).
Note that since the effective Hamiltonian is periodic in $x$ with period
$2\pi$, momentum $p$ can be split into a quasi-momentum $\beta{\mathchar
22\relax\mkern-9.0muk}$, $\beta\in[0.5,0.5]$, and an “integer” part
$n{\mathchar 22\relax\mkern-9.0muk}$, $n\in\mathbb{Z}$. The quasi-momentum
$\beta$ is conserved during the dynamics and plays the role of a disorder
realization in $\epsilon_{\bf p}$. Furthermore the sum $\sum_{\mathbf{q}}$
above is only over the integer part of the momentum.

Numerical determination of the phase diagram and $K_{c}$. The phase diagram
and the location of the critical point are determined numerically by analysing
the variation of the kinetic energy’s scaling function $\Lambda=\langle
p^{2}\rangle_{t}t^{-2/d}$. We performed numerical simulations of the QpQKR
model and computed the time evolution of $\langle p^{2}\rangle_{t}$ for
different values of the $(K,\varepsilon)$ couple. The results were obtained by
averaging the momentum distributions corresponding to $10^{4}$ values of the
quasi-momentum $\beta$, uniformly distributed in $[-0.5,0.5]$, and kick
numbers up to 5000. The phase diagram was obtained by computing the slope of
$\Lambda=\langle p^{2}\rangle_{t}t^{-1/2}$ vs. $pt^{-1/4}$ (corresponding to
$d=4$). The critical region in Fig. 2 corresponds to the $(\epsilon,K)$
couples for which $\Lambda$ is constant, yielding a zero-slope. The
intersection between the critical line and the $(K,\varepsilon)$ path used in
the experiment is obtained at $K=K_{c}=5.45\pm 0.05$.

Scaling form and critical exponents. From Eq. (3), the normalization of the
momentum distribution $\int dp\,\Pi_{t}(p)=1$ for all $\delta_{K}$ and all
times implies $\alpha=\beta$. Furthermore, this implies $\langle
p^{2}\rangle_{t}=t^{-2\alpha}\Lambda(\delta_{K}t^{\gamma})$. On the one hand,
in the localized phase ($\delta_{K}<0$), $\langle p^{2}\rangle_{t}$ will
become time independent at very long times, $\lim_{t\to\infty}\langle
p^{2}\rangle_{t}=p_{\rm loc}^{2}$, defining the localization length $p_{\rm
loc}$. It diverges at the transition as $p_{\rm
loc}\propto|\delta_{K}|^{-\nu}$, defining the exponent $\nu$. This implies
$\Lambda(\tilde{\delta})\propto|\tilde{\delta}|^{2\alpha/\gamma}$ for
$\tilde{\delta}<0$ and $|\tilde{\delta}|\gg 1$, and thus
$2\alpha/\gamma=-2\nu$. On the other hand, in the diffusive phase
($\delta_{K}>0$) $\langle p^{2}\rangle_{t}$ grows linearly in time at long
times, $\lim_{t\to\infty}\langle p^{2}\rangle_{t}=Dt$. The diffusion
coefficient $D$ vanishes as $\delta_{K}^{s}$ close to the transition, defining
the critical exponent $s$. This implies
$\Lambda(\tilde{\delta})\propto\tilde{\delta}^{s}$ for $\delta_{K}\gg 1$ and
$s\gamma-2\alpha=1$. Finally, assuming Wegner’s scaling law, which relates the
exponents $s$ and $\nu$ as $s=(d-2)\nu$, we would find $\alpha=-1/d$ and
$\gamma=1/\nu d$. Our experimental measurements of the critical exponents
($\nu_{\rm exp}=1.07\pm 0.16$ and $s_{\rm exp}=2.22\pm 0.38$, see text) are in
good agreement with Wegner’s law for $d=4$.

Figure 5:  Determination of the critical exponents a) Color plot showing the
variance of the dispersion of the scaled kinetic energy around $\Lambda_{0}$,
as a function of the critical exponents $\nu$ and $s$. The minimum of this
variance corresponds to the white point. The corresponding cross sections are
shown in panels b) and c).

Fitting procedure for $\nu$ and $s$: Our optimal scaling procedure uses the
exponents $\nu$ and $s$ as fit parameters and consists in searching for the
best collapse of the kinetic energy $\langle p^{2}\rangle_{t}$ into a single
scaling function $\Lambda$, for all values of $(K,\varepsilon)$ and $t$. We
use a set of values of $\langle p^{2}\rangle_{t}$ obtained along the
$(K,\varepsilon)$ path shown in Fig. 2.a), in the vicinity of $K_{c}$
(typically $|K-K_{c}|\leq 1$), and at times $t>30$ kicks. We use trial values
of the two critical exponents $\nu$ and $s$ to compute the two scaled
quantities $t^{2\alpha}\langle p^{2}\rangle_{t}$ and
$\tilde{\delta}=\delta_{K}t^{\gamma}$, with $\alpha=-\nu/(s+2\nu)$ and
$\gamma=1/(s+2\nu)$, see text. We then compute an average curve
$\Lambda_{0}(\tilde{\delta};\nu,s)$, using a sliding average with a window of
typically 1/20 of the total range of $\tilde{\delta}$. We interpolate this
average curve and calculate the total variance of the quantity $\langle
p^{2}\rangle_{t}t^{2\alpha}-\Lambda_{0}(\tilde{\delta};\nu,s)$. The result is
shown in Fig. 5.a), and the location of the minimum corresponds to the best-
guess values of the exponents, $\nu_{\rm fit}$ and $s_{\rm fit}$. The
corresponding horizontal and vertical slices are shown in Figs. 5.b-c). Note
that the shape of the reconstructed scaling function
$\Lambda_{0}(\tilde{\delta};\nu,s)$ does depend on the trial exponents $\nu$
and $s$. Our best estimate of the scaling function $\Lambda$ shown in Fig.
3-b) is obtained from $\Lambda_{0}(\tilde{\delta};\nu_{\rm fit},s_{\rm fit})$.

Bootstrap procedure To determine the optimal values of the critical exponents,
$\nu_{\rm exp}$ and $s_{\rm exp}$, as well as the confidence intervals, we use
a bootstrap method to take into account the impact of the statistical
uncertainties of the experimental $\langle p^{2}\rangle_{t}$ data, as well as
that of the critical point location $K_{c}$ William  _et al._ (1992). Assuming
a Gaussian probability distribution for these quantities, and given the
standard deviations estimated above, we resample $\langle p^{2}\rangle_{t}$
and $K_{c}$ and repeat the fitting procedure described in the previous
paragraph to determine values of the $\nu$ and $s$ exponents. This procedure
is repeated $10^{4}$ times, which provides the ($\nu,s$) samples shown in Fig.
3-c). We use these samples to determine the best estimates $\nu_{\rm exp}$ and
$s_{\rm exp}$ of the critical exponents, as well as the corresponding
confidence region (blue ellipse) and the confidence intervals (shown as error
bars). Due to the limited amount of experimental data, we estimate that our
numerical resampling method provides realistic error estimations. Another
option, standard bootstrapping (random sampling of the data with replacement),
tends to underestimate confidence intervals in our case due to the small
dataset (three to six averages per determination of $\langle
p^{2}\rangle_{t}$).

Parameters used for testing the two-parameter scale invariance (Fig. 4).

$\tilde{\delta}$ | $(\delta_{K},t)$
---|---
$-3$ | $(-1.2,69)$ | $(-1.1,103)$ | $(-1,161)$ | $(-0.9,262)$ | $(-0.8,451)$
$-1.4$ | $(-0.6,50)$ | $(-0.5,117)$ | $(-0.45,190)$ | $(-0.41,293)$ | $(-0.37,470)$
$+1.4$ | $(+0.6,50)$ | $(+0.5,117)$ | $(+0.45,190)$ | $(+0.41,293)$ | $(+0.37,470)$
$+3$ | $(+1.2,69)$ | $(+1.1,103)$ | $(+1,161)$ | $(+0.9,262)$ | $(+0.8,451)$