###### Abstract

In this paper we critically discuss several examples of two-dimensional
electronic systems displaying interaction-driven metal-insulator transitions
of the Mott (or Wigner–Mott) type, including dilute two-dimension electron
gases (2DEG) in semiconductors, Mott organic materials, as well as the
recently discovered transition-metal dichalcogenide (TMD) moiré bilayers.
Remarkably similar behavior is found in all these systems, which is starting
to paint a robust picture of Mott criticality. Most notable, on the metallic
side a resistivity maximum is observed whose temperature scale vanishes at the
transition. We compare the available experimental data on these systems to
three existing theoretical scenarios: spinon theory, Dynamical Mean Field
Theory (DMFT) and percolation theory. We show that the DMFT and percolation
pictures for Mott criticality can be distinguished by studying the origins of
the resistivity maxima using an analysis of the dielectric response.

###### keywords:

Mott transition; quantum criticality; resistivity maxima; dielectric response;
dilute 2DEGs; Mott organics; twisted transition-metal dichalcogenide bilayers;
dynamical mean field theory; percolation theory; spinon theory

1 1 0 Academic Editor: Dmitri Donetski 4 May 2022 3 June 2022 28 June 2022
https://doi.org/ How to Recognize the Universal Aspects of Mott Criticality?
How to Recognize the Universal Aspects of Mott Criticality? Yuting Tan 1,*,
Vladimir Dobrosavljević 1,* and Louk Rademaker 2,* Yuting Tan, Vladimir
Dobrosavljević and Louk Rademaker Tan, Y.; Dobrosavljević, V.; Rademaker, L.
Correspondence<EMAIL_ADDRESS>(Y.T<EMAIL_ADDRESS>(V.D.);
<EMAIL_ADDRESS>(L.R.)

##  1 Introduction

The physics of strongly correlated matter has many faces. Still, for a
majority of systems the underlying theme is the role of “Mottness” Phillips
(2006). It is clear that if one aspect of strong correlations should be
understood first, it should be the fundamental nature of the Mott metal-
insulator transition Mott (1990). Its simplest reincarnation is the transition
induced by tuning the bandwidth at half-filling, a setup that produced rather
spectacular advances in recent years. Several systems were identified as
nearly-ideal realizations of this paradigm, allowing systematic study using a
wide arsenal of experimental probes.

In this article we present an overview of three classes of two-dimensional
experimental systems that exhibit bandwidth-controlled Mott criticality:
dilute two-dimensional electron gases in semiconductors, “Mott-organic”
compounds, and transition-metal dichalcogenide moiré systems. Thereby we aim
to present the experimental facts as “bland” as possible, in Section 3,
without favoring one or the other theoretical explanation. The remarkable
similarities between these model systems suggests a robust universality,
including characteristic behavior such as the appearance of resistivity
maxima.

Possible explanations of two-dimensional Mott criticality follow in the
section thereafter (Section 4), where the experimental distinguishable
features of each theory takes the forefront. This is followed by a separate
discussion of the largely-overlooked utility of dielectric spectroscopy in
Section 5, in not only identifying phase segregation and spatial
inhomogeneity, but also in revealing the thermal destruction of coherent
quasiparticles associated with Landau’s Fermi liquid theory.

However, first we need to address the demarcation of our topic. What makes the
metal-to-insulator transition in these systems stand out from ‘traditional’
metal-to-insulator transitions Imada et al. (1998); Dobrosavljević et al.
(2012)?

##  2 In Search of Mott Criticality

Condensed matter physics, or recently for sales purposes re-branded as
“quantum matter physics”, is the study of electric and magnetic properties of
materials that surround us. The grand question that trumps all others is: how
to understand which materials conduct electricity and which are insulating?
Traditionally, a conducting material is called a metal—not to be confused with
the chemical, metallurgical or astronomical meaning of that word. Our main
question (metal or insulator?) has not only tremendous technological
applications (in fact, all modern electronic technology depends on our ability
to rapidly switch materials between metallic and insulating behavior), but
also requires a thorough understanding of the problem of emergent behavior of
many interacting quantum-mechanical electrons and ions.

Only at zero temperature does there exist a sharp difference between
insulators and metals Dobrosavljević et al. (2012). There are three distinct
possibilities: zero conductivity $\sigma(T=0)=0$ means insulating; zero
resistivity $\rho(T=0)=0$ means a superconductor; and anything in between is a
metal $\sigma(T=0)=1/\rho(T=0)\neq 0$. At any nonzero temperature, an
insulator typically has activated behavior $\rho(T)\sim e^{\Delta/T}$ whereas
the standard Fermi liquid theory of a metal predicts a temperature-squared
increase of the resistivity $\rho(T)=\rho_{0}+AT^{2}$. It has therefore become
common-place to use the derivative of the resistivity $d\rho/dT$ as a measure
of whether something is conducting ($d\rho/dT>0$) or insulating
($d\rho/dT<0$)—but this is highly misleading! As we will show later in Section
3, close to a Mott metal-insulator transition we often find non-monotonic
behavior of the resistivity as a function of temperature, making the
‘derivative’ criterion useless. Even worse, there exist cases where the
resistivity has $d\rho/dT<0$ but at zero temperature it does not diverge,
signalling that this is not a true insulator (see e.g., Cao et al. (2018);
Szentpéteri et al. (2021)). Another example is the case of Mooij correlations
Mooij (1973); Ciuchi et al. (2018), where the temperature-derivative of the
resistivity in a metal can become negative. Consequently, since only at zero
temperature the insulator/metal distinction is well-defined, we must stick
with that definition. Regardless of the slope, a material is a metal if its
resistivity does not diverge as $T\rightarrow 0$.

Many materials can be understood within the framework of band theory and its
extensions such as Fermi liquid, Boltzmann transport, and density functional
theory. This framework provides a very simple answer to the metal-or-insulator
question: if the Fermi level lies in the middle of a band gap, the system is
insulating; otherwise, the system behaves as a metal. This concept has the
important consequence that for a crystalline material with (up to some weak
disorder) well-defined unit cells, insulators can only appear when there is an
even number of electrons per unit cell. Consequently, within the band theory
picture, there exist only three possible routes to induce a metal-to-insulator
transition: by changing the electronic density; via spontaneous symmetry
breaking; or via band overlap when the filling is even. An example of the
first is doping a semiconductor, which is the metal-insulator transition we
induce on a daily basis inside transistors. An example of the second is the
transition into antiferromagnetic ordering: when the system is at half-filling
of a band (meaning one electron per unit cell), after antiferromagnetic unit
cell doubling there are two electrons per unit cell, and the system can become
a band insulator. The third case can be realized by for example straining a
system such that the band gap changes from positive to negative.

There are, however, two main exceptions to the paradigm of band theory. On the
one hand, disorder can become so large as to prevent the motion of the charge
carriers—this is known as Anderson localization Evers and Mirlin (2008). On
the other hand, the presence of very strong electron-electron interactions can
force the electrons to become “stuck” like in a traffic jam—this is known as
Mott insulation Mott (1990). The standard model of Mott insulation is the
Hubbard model with a tight-binding Hamiltonian:

$H=-t\sum_{\langle
ij\rangle\sigma}c^{\dagger}_{i\sigma}c_{j\sigma}+U\sum_{i}n_{i\uparrow}n_{j\downarrow},$
(1)

where $t$ is the nearest-neighbor hopping on some lattice and $U$ is the
onsite repulsion. When $U=0$, the system is a metal when half-filled. When
$U\gg t$, it becomes energetically favorable to occupy each site with exactly
one electron rather than to fill bands up to the Fermi level. The resulting
Mott state can therefore not be described by band theory!

Mott insulators have been observed in a wide variety of materials, most
famously transition-metal oxides, including high $T_{c}$ superconducting cup
rates Imada et al. (1998). Observing a clear transition from a standard Fermi
liquid metal to a Mott insulator, however, is quite elusive. This transition
can be induced either by changing the electronic density (“filling-
controlled”) or by changing the ratio $U/t$ (“bandwidth-controlled”). The
filling-controlled Mott transition Lee et al. (2006) notoriously leads to a
whole zoo of different instabilities, pseudogaps, and strange metal behavior,
and is typically masked at low temperatures by superconductivity. The
bandwidth-controlled Mott transition is, in contrast, often masked by
(antiferromagnetic) spin order that hides any Mottness behind the veil of unit
cell doubling.

This might, at first, suggest that Mott criticality is something unattainable.
By “criticality” we mean that approaching the Mott transition we find
vanishing energy scales, and that the resistivity curves display scaling
behavior. There are, however, two clever tricks to realize Mott criticality.
The first trick is dimensionality: a transition that is strongly first-order
in $d=3$ dimensions often becomes continuous or weakly first-order in $d=2$
dimensions. The most striking example of this is, of course, the
solidification of 3He. The second, and perhaps even more important trick is
frustration: if the lattice structure is highly frustrated (with competing
magnetic interactions) one can avoid Balents (2010) antiferromagnetic ordering
altogether—revealing the true Mott transition.

In this review we, therefore, focus on three classes of systems that are
indeed (quasi) two-dimensional as well as frustrated: Wigner crystals in
extremely dilute two-dimensional electron gases; layered Mott organic
compounds; and the more recent addition of transition-metal dichalcogenide
(TMD) moiré bilayers. Indeed, as we will show in Section 3, these systems all
seem to exhibit remarkably similar distinct features, including clear
signatures of critical resistivity scaling. Because these systems all have a
fixed electron density per unit cell of $n=1$ (at least in the insulating
limit), the observed transitions are plausibly within the universality class
of bandwidth-tuned Mott transitions.

A brief side-note is in order: we briefly mentioned superconductivity and
disorder-induced insulators. These phases can also have a continuous
transition between them, the so-called superconductor-to-insulator transition
Sondhi et al. (1997); Saito et al. (2016). This, however, is an interesting
topic that falls outside the scope of this review. Similarly, we also will not
consider disorder-driven metal-insulator transitions Lee and Ramakrishnan
(1985); Belitz and Kirkpatrick (1994), since this regime typically does not
include any Mottness. More general but also somewhat older reviews of metal-
insulator criticality can be found in Refs. Imada et al. (1998); Abrahams et
al. (2001); Spivak et al. (2010); Dobrosavljević et al. (2012); Kravchenko
(2017).

##  3 Experiments

Given that experimental results should always be leading, the aim of this
section is to introduce three material systems that are likely exhibiting a
bandwidth-tuned Mott metal-insulator transition: dilute 2DEGs, organics, and
moiré systems. To support the clarity of interpretation, we will stress
experimental similarities between these systems without much room for
theoretical guesswork—that is the next section’s realm.

While each system has a different tuning parameter (density, pressure, or
field), the electrical resistivity through the transition is the key
observable, see Figure 1. Its behavior reveals how the transport gap $\Delta$
decreases when we approach the transition from the insulating side; as well as
how the resistivity behaves on the metallic side, where Fermi liquid behavior
$\rho(T)=\rho_{0}+AT^{2}$ is typically seen at $T<T_{FL}$ with an enhanced
effective mass $m^{*}$. Remarkably, in all systems one also observes distinct
resistivity maxima at $T\sim T_{max}>T_{FL}$, signalling the breakdown of
coherent transport. Crossover to the quantum critical regime is described by
an additional temperature scale $T_{o}$, which is extracted from the scaling
collapse of the resistivity curves as shown in Figure 2.

Figure 1: The key observable revealing a metal-insulator transition is the
resistivity. Here we show $\rho$ vs. $T$ resistivity curves as a function the
tuning parameter, for representative examples of the three material systems
considered: (a) 2DEG in Si-MOSFET tuned by electronic density (reprinted with
permission from Ref. Melnikov et al. (2019) Copyright 2019 American Physical
Society), (b) Mott organic material $\kappa\text{-(BEDT-
TTF)}_{2}\text{Cu}_{2}\text{(CN)}_{3}$ tuned by pressure Pustogow et al.
(2021), and (c) TMD moiré bilayer MoTe2/WSe2 tuned by displacement field (Data
imported from Li et al. (2021)). In all cases, one observes distinct
resistivity maxima on the metallic side, at a temperature $T_{max}$ that
decreases towards the transition.  Figure 2: Critical scaling has been
observed in all three experimental systems, when the resistivity is plotted
versus $T/T_{o}$ where $T_{o}$ is the characteristic crossover (quantum
critical) energy scale. Note that in all cases a strong “mirror” symmetry
Simonian et al. (1997); Dobrosavljević et al. (1997) exists between the
insulating (upper) and metallic (lower) scaling branch. (a) In a dilute 2DEG,
scaling of the bare resistivity $\rho(T)$ was achieved by simply rescaling $T$
with $T_{o}\sim|\delta|^{1.6}$ (Adapted with permission from Ref. Kravchenko
et al. (1995) Copyright 1995 American Physical Society); (b) In organic
compounds, the normalized resistivity $\tilde{\rho}$ is obtained by
normalizing the resistivity by the critical resistivity along the Widom line.
This leads to excellent scaling collapse with $T_{o}\sim|\delta|^{0.60\pm
0.01}$ (Adapted with permission from Ref. Furukawa et al. (2015) Copyright
2015 Springer Nature); (c) A similar approach was followed in TMD moiré
bilayer MoTe2/WSe2, with similar $T_{o}\sim|\delta|^{0.70\pm 0.05}$ (Data
imported from Li et al. (2021)).

A practical summary of the experimental results is presented in Table 1.

Table 1: A summary of available experimental results for the three classes of systems considered. The sources (references) are given in the text below. Question-marks indicate the lack of reliable data. Fermi liquid ($T^{2}$) transport behavior has not been documented in 2DEG systems, in contrast to strong evidence for it in Mott organics and TMD moiré bilayers. Note that the characteristic energy scales $\Delta$, $(m^{*})^{-1}$, $T_{FL}$, $T_{max}$, as well as $T_{o}$ display similar continuous decrease towards the transition in all three systems, consistent with general expectations for quantum criticality. One should keep in mind that the error bars on the estimated exponent could be substantial, since the results typically depend strongly on the utilized fitting range. System |  Dilute 2DEG |  Mott Organics |  TMD Moiré Bilayers
---|---|---|---
Transition Type |  continuous? |  weakly first order
(at $T<T_{c}\sim 0.01T_{F}$) |  continuous?
$\Delta$ |  $|n-n_{c}|$ |  $|P-P_{c}|^{\nu z}$,
$\nu z\approx 0.7-1$ |  $|E-E_{c}|^{\nu z}$,
$\nu z\approx 0.6$
$\dfrac{1}{m^{*}}$ |  $|n-n_{c}|$ |  ? |  ?
$T_{o}$ |  $|n-n_{c}|^{\nu z}$,
$\nu z\approx 1.6$ |  $|P-P_{c}(T)|^{\nu z}$,
$\nu z\approx 0.5-0.7$ |  $|E-E_{c}|^{\nu z}$,
$\nu z\approx 0.7$
$T_{\mathrm{FL}}$ |  ? |  $|P-P_{c}|$ |  $|E-E_{c}|^{\nu z}$,
$\nu z\approx 0.7$
$T_{max}$ |  $|n-n_{c}|$ |  $|P-P_{c}|$ |  $|E-E_{c}|^{\nu z}$,
$\nu z\approx 0.7$

###  3.1 Dilute 2DEG in Semiconductors

In dilute two-dimensional electron gases (2DEG) Ando et al. (1982), the
electron density can be quantified by the dimensionless parameter
$r_{s}=1/\sqrt{\pi n}a_{B}$ where $n$ is the electron density and $a_{B}$ the
Bohr radius. The ratio of interaction energy versus kinetic energy scales as
$r_{s}$, and therefore at large enough $r_{s}$ (of the order $r_{s}\sim 40$ in
2D) the electrons will spontaneously crystallize into a Wigner solid. In a
two-dimensional Wigner crystal, the electrons form a triangular lattice with
exactly one electron per unit cell—essentially forming a frustrated Mott
insulator. When the electron density $n$ is varied, the size of the unit cell
changes accordingly so that the Wigner crystal always remains fixed at one
electron per unit cell. The transition from an insulating Wigner crystal to a
metal can therefore be plausibly viewed as a bandwidth-tuned Mott transition.
Note that this is counter-intuitive: after all, one tunes the electron
density! However, what matters is the electron density counted per unit cell
and that remains constant. This idea suggests Camjayi et al. (2008); Amaricci
et al. (2010); Radonjić et al. (2012) that the melting of a Wigner solid by
increasing density should be viewed as a Wigner–Mott transition, possibly
bearing many similarities to Mott transitions in narrow-band crystalline
solids such as Mott organics or transition-metal oxides. If this viewpoint is
correct, then the resulting metal above the transition should display resemble
other strongly correlated Fermi liquids, a notion that is starting to gain
acceptance on the base of recent experiments Shashkin et al. (2020, 2022);
Moon et al. (2020).

Experimentally, high-quality 2DEGs can be realized in metal-oxide-
semiconductor field-effect devices (MOSFETs) in various semiconductors
Kravchenko et al. (1995); Popović et al. (1997); Shashkin et al. (2020); Moon
et al. (2020). Through electrostatic gating the electronic density can be
elegantly tuned, typically in the range of $n\sim 10^{10}$–$10^{12}$ cm-2. The
peak electron mobility in ultra-clean samples can be as high as $10^{4}$ cm2/V
s Melnikov et al. (2019), which implies that down to very low temperatures the
transport properties are dominated by electron-electron interactions (like
Wigner crystallization) rather than extrinsic disorder effects. Lower-mobility
devices have also been extensively studied (for a review see Chapter 5 of Ref.
Kravchenko (2017)), displaying different types of metal-insulator transitions
displaying electron glass dynamics Bogdanovich and Popović (2002), which we
will not discuss here.

Indeed, tuning the electronic density leads to insulating transport below a
critical density, typically around $n_{c}\sim 10^{11}$ cm-2 Kravchenko et al.
(1995); Popović et al. (1997), see Figure 1a. Activated behavior is often
observed close to the transition Shashkin et al. (2001); Bogdanovich and
Popović (2002), with the activation energy $\Delta\sim|n-n_{c}|$. Further on
the insulating side disorder effects may become important, where Efros-
Shklovskii hopping (and other effects of disorder) is also observed Kravchenko
et al. (1995), but only at the lowest temperatures. On the metallic side, a
pronounced resistivity drop (often by a factor of 10 or more) is observed
Kravchenko et al. (1995); Shashkin et al. (2020) below the temperature
$T_{max}\sim|n-n_{c}|$ which decreases as the transition is approached.
Characteristic scaling of the resistivity maxima has been reported in several
systems Shashkin et al. (2020, 2022); Moon et al. (2020), see Figure 3, which
has been interpreted as evidence for strong correlation effects. However, the
expected $T^{2}$ dependence of the resistivity has not been observed, despite
the reported effective mass enhancement $(m^{*})^{-1}\sim|n-n_{c}|$ Shashkin
et al. (2002), characteristic of correlated Fermi liquids. Quantum critical
scaling collapse of the resistivity curves has also been demonstrated
Kravchenko et al. (1995) around the critical density, albeit excluding the
lowest temperatures data, as shown in Figure 2a. This is achieved by rescaling
$T$ by a crossover scale $T_{o}\sim|n-n_{c}|^{\nu z}$, with $\nu z\approx
1.6$. The resulting scaling function reveals surprising “mirror symmetry”
Simonian et al. (1997), which was phenomenological interpreted Dobrosavljević
et al. (1997) as evidence “strong-coupling quantum criticality”. Similar
systems to these 2DEGs include the observation of a Wigner crystal in low-
density doped monolayer WSe2 Smolenski et al. (2021), where more detailed
experiments still need to be performed.

Figure 3: Characteristic scaling of the resistivity maxima has been reported
in several 2DEG electron systems in semiconductors: (a) Si-MOSFETs (adapted
with permission from Ref. Radonjić et al. (2012) Copyright 2012 American
Physical Society); (b) p-GaAs/AlGaAs quantum wells (adapted with permission
from Ref. Radonjić et al. (2012) Copyright 2012 American Physical Society);
(c) SiGe/Si/SiGe quantum wells (adapted with permisssion from Ref. Shashkin et
al. (2020) Copyright 2020 American Physical Society); (d) few
layered-$MoS_{2}$ (Adapted with permission from Ref. Moon et al. (2020)
Copyright 2020 American Physical Society). All data collapse to the same
(theoretical) scaling function Radonjić et al. (2012) obtained from the
Hubbard model at half-filling, in the vicinity of the Mott point.

###  3.2 Organic Compounds

An organic compound Dressel and Tomić (2020) refers to a crystalline system
where each unit cell contains an entire molecule, rather than just loosely
bound ions. A particularly interesting class of organic compounds is based on
the molecule bis-(ethylendithio)-tetrathiafulvalen (BEDT-TTF or ET), which can
be fabricated with other ions into quasi-two-dimensional layered systems.
Compounds based on BEDT-TTF exhibit a spectrum of interesting quantum matter
phenomena, ranging from superconductivity Jerome (1991) to electron glass
behavior Kagawa et al. (2013).

Our interest goes out especially to $\kappa$-(ET)2Cu[N(CN)2]Cl and
$\kappa\text{-(ET)}_{2}\text{Cu}_{2}\text{(CN)}_{3}$, where the molecules are
organized in triangular lattice layers Dressel and Tomić (2020). These
materials are strongly correlated, and indeed, despite being half-filled they
are insulating at ambient pressures. Due to the geometric frustration of the
triangular lattice Balents (2010), no magnetic order has been observed in
$\kappa$-Cu2(CN)3 and antiferromagnetic order only at relatively low
temperatures ($T<T_{N}\approx 20K$) in $\kappa$-Cl. The absence of magnetic
order is the strongest indication that $\kappa$-Cu2(CN)3 might realize a spin
liquid ground state Kurosaki et al. (2005); Pustogow et al. (2018).

Upon applying pressure, a zero-temperature first-order phase transition brings
the system into a paramagnetic metallic phase at $p_{c}=122$ MPa
($\kappa$-Cu2(CN)3) or $p_{c}=24.8$ MPa ($\kappa$-Cl), see Figure 1a Furukawa
et al. (2015); Pustogow et al. (2021). The first order phase boundary ends in
a critical point at $T_{c}=20$ K or $T_{c}=38$ K, respectively. It is
important to emphasize that these temperatures are very small compared to the
electronic energy scales. The Hubbard repulsion $U$ and bandwidth $W$ are both
on the order of a fraction of eV Pustogow et al. (2018), which implies
$T_{c}\ll U,W$. As such, even though the observed Mott criticality appears at
nonzero temperatures, much of the observed phenomena above $T_{c}$ can be
described as if the system resides in the vicinity of a quantum critical point
Furukawa et al. (2015).

Above $T_{c}$, a crossover pressure $P_{c}(T)$ can traced Furukawa et al.
(2015) where the measured resistivity exhibits an inflection point, see Figure
4. This defines the “quantum Widom line” (QWL) Vučičević et al. (2013) by
analogy to the standard liquid-gas crossover. Defining the critical
resistivity $\rho_{c}(T)$ to be the resistivity along the QWL, all resistivity
curves collapse onto each other when plotted as $\rho(P,T)/\rho_{c}(T)$ vs.
$T/T_{0}(P)$, as shown in Figure 2b. Here the scale $T_{o}(P)$ reflects a
critical energy scale that vanishes at the critical pressure,
$T_{o}\sim|P-P_{c}|^{0.6}$ for Cu and $T_{o}\sim|p-p_{c}|^{0.5}$ for
$\kappa$-Cl Furukawa et al. (2015). On the insulating side of the transition,
the resistivity is approximately activated $\rho\sim\exp(\Delta/T)$ Pustogow
et al. (2018). On the metallic side, it follows Pustogow et al. (2021) the
standard Fermi liquid behavior at low temperatures $\rho(T)=\rho_{0}+AT^{2}$,
up to a temperature scale $T_{FL}$, see Figure 5a. This destruction of the
Fermi liquid seems to correspond to the appearance of a maximum in the
resistivity Pustogow et al. (2021).

Figure 4: Finite temperature phase diagram of the Mott organic materials. (a)
first-order phase transition line, as observed in $\kappa$-Cu2(CN)3 (adapted
with permission from Ref.Furukawa et al. (2015) Copyright 2015 Springer
Nature) at $T<T_{c}\sim 20K$, displaying “Pomeranchuk” behavior Pomeranchuk
(1950), by “sloping” towards the metallic phase. The corresponding “Quantum
Widom Line” Vučičević et al. (2013) arises at $T>T_{c}$, which is identified
as the center of the quantum critical region Terletska et al. (2011) with
resistivity scaling Furukawa et al. (2015). (b) Phase diagram Pustogow et al.
(2021) for $\kappa$-Cu2(CN)3 over a broader $T$-range, displaying the
convergence of the quantum Widom line (QWL) on the insulating side, and the
“Brinkman-Rice” line ($T_{BR}=T_{max}$, which intersect at the critical end-
point $T=T_{c}$. The Fermi-Liquid line $T_{FL}<T_{BR}$ is also shown. (c) The
universal phase diagram for a series of spin-liquid Mott organics compounds
was established Pustogow et al. (2018) by rescaling the temperature $T$ and
the interaction strength $U$ by the respective electronic bandwidth $W$. The
parameters $W$ and $U$ were independently measured Pustogow et al. (2018) for
each material using optical conductivity.

In addition to transport measurements, and in contrast to other systems we
consider, Mott organics have also been carefully investigated using optical
probes. This allowed to directly identify Pustogow et al. (2018) the quantum
Widom line, which is back-bending towards the insulating side at higher
temperatures following the closing of the Mott gap. In addition, the
“Brinkman–Rice” line traced by $T_{max}$ was identified as marking the thermal
destruction of Landau quasiparticles, as seen by the vanishing of the Drude
peak in the optical conductivity Pustogow et al. (2021), see Figure 6.

Finally, the controversy about the presence or absence of the low-$T$ phase
coexistence region has been resolved in Mott organics, by using dielectric
spectroscopy Pustogow et al. (2021). Its precise location on the phase diagram
has been identified by the observation Pustogow et al. (2021) of colossal
dielectric response, as a smoking gun for percolative phase coexistence. In
addition, the same technique was able to demonstrate the coincidence of the
resistivity maxima in the (uniform) metallic phase, with the thermal
destruction of Landau quasiparticles. This is seen as a dramatic drop and a
change Pustogow et al. (2021) of sign of the dielectric function at
$T<T_{BR}=T_{Max}$. These experimental results are shown in Figure 6, and
discussed in more detail in Section 5.

Figure 5: Fermi liquid behaviour at low temperatures, for (a) Mott organic
material $\kappa-[\text{(BEDT-TTF)}_{1-x}\text{(BEDT-
STF)}_{x}]_{2}\text{Cu}_{2}\text{(CN)}_{3}$ Pustogow et al. (2021) and (b)
MoTe2/WSe2 moiré bilayers (Data imported from Li et al. (2021)). Clear
$\rho=\rho_{0}+AT^{2}$ behavior is observed in both cases, up to a temperature
scale $T_{FL}$ that seems to decrease linearly towards the metal-insulator
transition. The resistivity curves can be collapsed by plotting
$\rho(E,T)/\rho_{c}(T)$ vs. $T/T_{0}$ where $T_{0}\sim|E-E_{c}|^{0.70\pm
0.05}$, see Figure 2c. Note that this crossover scale seems to follow both the
gap size on the insulator, as well as the destruction of the Fermi liquid on
the metallic side.

Figure 6: Transport behavior vs. dielectric response across the phase diagram
of $\kappa$-Cu2(CN)3 Pustogow et al. (2021). (a) DC transport shows only very
gradual change across the BR line (resistivity maxima), and cannot one see any
clear indication of the phase coexistence region. (b) In dramatic contrast,
the low-frequency dielectric function $\epsilon_{1}$ assumes small positive
values in the Mott insulator (pale pink), and large negative values in the
quasiparticle regime (deep blue); we clearly see the boundaries of these
regimes tracing the QWL and the BR line (following $T_{max}$), as observed in
transport. Remarkably, “resilient” quasiparticles Deng et al. (2013) persist
past the Fermi Liquid line, at $T_{FL}<T<T_{BR}=T_{Max}$, where bad metal
behavior Emery and Kivelson (1995) (metallic transport above the Mott-Ioffe-
Regel ‖ et al. (2004) limit is observed). At low temperature, the Mott point
is buried below the phase coexistence dome, which is vividly visualized
through colossal dielectric response ($\epsilon_{1}\sim 10^{3}$–$10^{4}$).

###  3.3 Moiré Materials

The most recent addition to the field of strongly correlated systems are moiré
materials. These are bilayer structures made of Van der Waals materials such
as graphene and transition-metal dichalcogenides (TMDs). A lattice mismatch or
relative twist angle between the layers causes a large-scale geometric “moiré”
pattern. This larger unit cell (typically in the range of 5–10 nm) drastically
reduces the effective electron kinetic energy such that the bandwidth is on
the order of $W\sim 10$ meV. As a result, the systems become strongly
correlated, with $U/W\sim 10$ or larger. Using electrostatic gating one can
tune the electronic density, typically in the range of a few electrons or hole
per moiré unit cell (corresponding to $n\sim 10^{12}$ cm-2).

While the most famous of correlated moiré materials is without a doubt twisted
bilayer graphene, more convincing evidence for Mott correlations has so far
only been observed in TMD bilayers. Here, we focus on one particular system:
the heterobilayer MoTe2/WSe2 Li et al. (2021). The lattice mismatch between
MoTe2 and WSe2 gives rise to a moiré period of $a_{M}\sim 5$ nm. At half-
filling of the first valence band, an insulating phase appears which can be
tuned into a metal by applying a vertical displacement field $E$. This flat
valence band can be described by a spin-orbit coupled triangular lattice
Hubbard model, where the displacement field $E$ tunes the bandwidth Rademaker
(2021).

The temperature-dependent resistivity across the transition is shown in Figure
1c. On the insulating side, the system has well-defined activated behavior of
the resistivity with a gap $\Delta$ continuously vanishes as the critical
displacement-field value is approached, $\Delta\sim|E-E_{c}|^{0.60\pm 0.05}$.
At the critical point, the resistivity is claimed to follow a powerlaw,
$\rho_{c}\sim T^{-1.2}$, although the reliability of the low-$T$ data may be
questionable. On the metallic side, the low-$T$ resistivity follows the Fermi
liquid law $\rho(T)=\rho_{0}+AT^{2}$ where the quadratic prefactor diverges
$A\sim|E-E_{c}|^{-2.8\pm 0.2}$. This clear Fermi liquid does not persist up to
all temperatures, instead, a resistivity maximum appears at
$T_{\mathrm{max}}\sim|E-E_{c}|^{0.70\pm 0.05}$, see Figure 5b. No magnetic
order has been observed, which might be due to the geometric frustration of
the triangular moiré lattice structure. Remarkably, this experiments uses the
recently-developed excitonic sensor Li et al. (2021) , which allows the
measurements of the spin susceptibility across the transition. This reveals
Curie-law behavior over a broad temperature range, thus demonstrating the
presence of localized magnetic moments, as expected for a Mott system.

Finally, among other moiré systems it is worth mentioning the twisted hetero-
bilayer WSe2 Ghiotto et al. (2021). Though it was claimed to exhibit some sort
of Mott-related criticality, it is not certain whether a true insulating phase
has indeed been observed given that the activated transport does not continue
down to the lowest measured temperatures. In addition, insulating behavior
seems to disappear above a relatively low of the order of $T^{*}\sim$ 5–10 K,
which is similar to twisted bilayer graphene, but much smaller than the
estimated bandwidth of the order of $W\sim$ 100–200 K. Furthermore, no clear
resistivity maxima on the metallic side have been observed. Whether or not
tbWSe2 can be classified as a true Mott insulator is therefore quite
controversial. Alternatively, the observed behavior could be a result of some
sort of magnetic order, which may arise close to half filling even in weakly-
coupled systems.

###  3.4 Universal Criticality

The resistivity curves of the three systems, as shown Figures 1 and 2, show
remarkable universality, reflected in the fact that all curves can be
collapsed by scaling with $T_{0}\sim|\delta|^{z\nu}$ where $\delta$ is the
tuning parameter and $z\nu$ the critical exponent. It is important to realize,
however, that the precise scaling procedure applied was not identical in the
three cases, and the resulting critical exponent also somewhat depend on the
system.

So what is different between these systems? Let us first focus on the energy
scale. The typical bandwidth $W$ ranges from $\sim$100 s meV in Mott organics,
to $\sim$ 10 s meV in the moiré systems, to 0.1–1 meV in the dilute 2DEGs. A
finite temperature critical point is only observed in the organics, though at
about $T_{c}\sim 1\%W$—which leaves open the possibility that a finite $T$
critical endpoint exists in the other two systems. Indeed, most experiments
are performed (so far) above the Kelvin range in moiré materials, and above
100 mK in 2DEG, which makes it hardly possible to reliably explore the
$T$-range below few percent of the bandwidth.

Secondly, in order to achieve quantum critical scaling in Mott organics, one
needs to first identify a Widom line as a demarcation of the finite-
temperature crossover from insulator to metal. A similar analysis was carried
out for moiré materials, although the obtained Widom line displayed no
apparent “curving” as a function of temperature. This is manifestly not
performed in the dilute 2DEGs. It might be interesting to see whether better
collapse can be achieved through such a method.

Thirdly, with a bit of good-will, the critical exponents in organics and Moiré
systems are in the same ballpark; whereas the critical exponent in the dilute
2DEGs with $z\nu=1.60$ is significantly larger. It is also important to
realize that in 2DEGs there has not been a clear observation of a Fermi liquid
regime—unlike in organics and Moiré systems, see Figure 5. These ways in which
2DEGs stand out might be related to the fact that there is no underlying
(Wigner) lattice on the metallic side, which could point to a perhaps
nontrivial role of significant charge density fluctuations on the metallic
side, an effect not present in lattice Mott systems.

##  4 Competing Theoretical Pictures

As we mentioned in Section 2, the observation of Mott criticality and scaling
opens big questions on the theoretical front. Currently, there exist three
main different physical pictures to address these issues.

A true Mott transition should not be hidden by some period doubling symmetry
breaking. The Lieb-Schultz-Mattis theorem states that in absence of spin
order, the ground state of Mott insulator must be a spin liquid Savary and
Balents (2016). This leads directly to the first theoretical picture: Mott
criticality can only occur if the Mott phase is a spin liquid, where inter-
site spin correlations play an important role. The theory of Senthil Senthil
(2008) chooses this path, by introducing an explicit spinon theory of the Mott
spin liquid.

Alternatively, one focuses on local electronic processes only, ignoring inter-
site spin correlations. Then the Mott transition at low temperature becomes
first-order; however, it is only weakly first order. A first order transition
line always ends at a critical point $T_{c}$, and as long as $T_{c}$ is
sufficiently low compared to any experimental scale, one still finds
criticality and scaling. This is the picture emerging from Dynamical Mean
Field Theory (DMFT) Georges et al. (1996), a strong-coupling self-consistent
approach to calculate the local electronic self-energy.

The third picture again accepts the first-order nature of a Mott transition,
but this time embraces it. A first-order transition is always accompanied by a
region where both phases coexist. Minor disorder or self-generated pattern
formation Spivak and Kivelson (2004, 2006) can smear this phase coexistence
region into a continuous-looking transition exhibiting nontrivial electron
dynamics. This is the ‘percolation theory’ picture of Mott criticality.

The goal of this review paper is to put the main theoretical predictions next
to the experimental findings. As such, we will not dive into the pros and cons
of each theoretical picture. A summary of the main theoretical predictions is
provided in Table 2.

Table 2: A summary of predictions from competing theoretical pictures. The expected transition type differs between the three pictures, with observable differences in the behavior of the mass enhancement $m^{*}$, the Kadowaki–Woods ratio $A/(m^{*})^{2}$, the destruction of the Fermi liquid at $T_{\mathrm{FL}}$, and the appearance of a resistivity maxima at $T_{max}$. Details are provided in the text below. Theory Predictions |  2D Spinon Theory |  DMFT |  Percolation Theory
---|---|---|---
Transition Type |  continuous |  weakly first order
(at $T<T_{c}\sim 0.01T_{F}$) |  first order
$\Delta$ |  $|g-g_{c}|S^{\nu z}$,
$\nu z=0.67$ |  $|U-U_{c1}|^{\nu z}$,
$\nu z\approx 0.8$ |  remains finite
$m^{*}$ |  weak: $\ln\frac{1}{|g-g_{c}|}$ |  strong: $|U-U_{c2}|^{-1}$ |  no divergence
$A/(m^{*})^{2}$ |  ? |  constant
(KW law obeyed) |  diverges: $(x_{o}-x_{c})^{-t}$; $t=s/m$
$T_{\mathrm{FL}}$ |  $|g-g_{c}|^{2\nu}$ |  $|U-U_{c2}|$ |  $T^{*}\sim|x_{o}-x_{c}|$
$T_{max}$ |  $T_{max}=\infty$ |  $|U-U_{c2}|$ |  $T^{*}\sim|x_{o}-x_{c}|$

###  4.1 Spin Liquid Picture of the Mott Point

A popular approach to describe a spin liquid state is through spin-charge
separation. In Ref. Senthil (2008), the electron is split into a charge-0
spin-$1/2$ fermionic spinon $f$ and a charge-$e$ spin-0 bosonic chargon $b$.
The Mott transition, in this picture, amounts to the condensation of the
chargon field, whose critical behavior falls within the 3D $XY$ universality
class. The Fermi liquid corresponds to the condensed phase of the chargon,
whereas the Mott insulator corresponds to a gapped phase of the charged boson.
The splitting of the electron leads to redundant degrees of freedom described
by an emergent gauge field. Fluctuations of this gauge field lead to a
logarithmic enhancement of the quasiparticle effective mass,

$m^{*}\sim\ln\frac{1}{|g-g_{c}|},$ (2)

where $g$ is the tuning parameter and $g_{c}$ is the critical value. However,
as in any theory with a non-local electronic self-energy, the quasiparticle
residue $Z$ is not simply proportional to the inverse effective mass; instead
$Z\sim|g-g_{c}|^{\beta}/\ln\frac{1}{|g-g_{c}|}$. Furthermore, approaching the
Mott transition from the metallic side the spin susceptibility $\chi$ remains
constant whereas the compressibility $\kappa$ vanishes. Physically, these
effects result from important inter-site spin correlations, where a gapless
spin liquid can be viewed as a certain superposition of spin singlets formed
by pairs of spins in the Mott insulating state. As a result, there emerges a
finite gap $\delta$ to charge excitations, while the rearrangement of singlets
leads to characteristic gapless spin excitations with fermionic
quasiparticles. This picture is a specific realization of the famous RVB
picture of Baskaran and Anderson Baskaran et al. (1987), first proposed in the
context of high-$T_{c}$ superconductors.

Another significant consequence of describing the Mott transition as chargon
condensation, is that the $T=0$ conductivity is not continuous. The electron
resistivity will display a universal jump from a (disorder)-dependent constant
value $\rho=\rho_{0}$ in the Fermi liquid; to $\rho=\rho_{0}+\frac{Rh}{e^{2}}$
(with $R$ of order one) at the critical point; to $\rho=\infty$ in the Mott
insulator. On the metallic side, the Fermi liquid is predicted to break down
above $T_{\mathrm{FL}}\sim|g-g_{c}|^{2\nu}$ and give rise to a marginal Fermi
liquid state, which in turn survives up to
$T_{\mathrm{MFL}}\sim|g-g_{c}|^{\nu}$. In both cases, $\nu=0.67$ is the 3D
$XY$ correlation length exponent. On the insulating side, the boson
condensation picture implies that the charge gap vanishes as
$\Delta\sim|g-g_{c}|^{\nu}$. The spinons, however, remain gapless and form a
spinon Fermi surface, with low-temperature specific heat scaling as $C\sim
T^{2/3}$.

Note that the original work in Ref. Senthil (2008) does not directly provide a
detailed description for finite temperature dependence of the resistivity, and
thus no explicit prediction for a possible deviation from the Kadowaki-Woods
(KW) law ($A/(m^{*})^{2}\approx$ constant) Jacko et al. (2009). On the other
hand, Ref. Jacko et al. (2009) presents arguments that the physical
requirement for the validity of the KW law is the locality of the electronic
self-energy (as in DMFT theory) Georges et al. (1996), a condition which is
not obeyed by the RVB-type spin-liquid theories such as the Senthil’s spinon
picture.

It should be stressed that the spinon theory makes one sharp prediction about
finite temperature transport in the critical regime. Namely, the critical
resistivity curve is predicted to assume a universal power-law form
$\rho_{c}(T)\sim 1/T$ in $d=3$ Podolsky et al. (2009), but remain
$T$-independent in $d=2$, see Figure 7. This therefore leads to distinct
resistivity maxima in 3D, but not in 2D Senthil (2008), where monotonic
$T$-dependence should be found on both sides of the transition, albeit with
opposite slope. Physically, this difference reflects the proposed importance
of “infrared” (IR, long distance) effects due to gauge fields, which should
have strong (spatial) dimensionality dependence. Concerning quantum critical
scaling, it is interesting that this theory proposes the emergence of two
crossover temperature scales, both of which vanish at the transition.

Figure 7: Predictions of the spinon theory (reprinted with permission from
Ref. Podolsky et al. (2009) Copyright 2009 American Physical Society). (a) The
phase diagram features a quantum critical point at $T=0$, and two distinct
finite-$T$ crossover scales $T^{*}$ (above which the system is quantum
critical) and $T^{**}$ (below which the system is either a metal or a gapless
spin liquid). (b) and (c) Resistivity and conductivity along the lines A, B
and C in the phase diagram in (a). Critical resistivity is predicted to
diverge as $\rho_{c}(T)\sim 1/t$ in $d=3$, leading to resistivity maxima on
the metallic side (coductivity minima). In contrast, the same theory predicts
finite critical resistivity $\rho_{c}(T)\sim\rho^{*}$ in $d=2$ Senthil (2008),
hence monotonic behavior on both sides of the transition and no resistivity
maxima.

We finally mention that a similar spin-charge separation theory has been very
recently proposed to also describe the Wigner–Mott transition in TMD bilayers,
where a possible role of charge fluctuations has also been discussed for the
metallic side Xu et al. (2021); Musser et al. (2021).

###  4.2 Dynamical Mean Field Theory Picture of the Mott Point

Our second theory of interest is Dynamical Mean Field Theory (DMFT), which
explicitly ignores all nonlocal (spin or charge) spatial correlations, and
therefore aims to self-consistently calculate the local electronic self-energy
$\Sigma(\omega)$ Georges et al. (1996). Physically, its real part describes
the modifications of the electronic spectra, while its imaginary part encodes
the frequency and temperature dependence of the electron-electron scattering
rate. In this way, this theory is not limited to low-temperature excitations
only, but is able to capture strong inelastic scattering at high temperatures,
and therefore describe both the (coherent) Fermi liquid regime, and also the
incoherent high-temperature transport, for example the famed bad metal
behavior Emery and Kivelson (1995); Vučičević et al. (2015) above the MIR
limit ‖ et al. (2004).

While there exist some limiting cases where an analytic solution is possible,
it is mainly a numerical approach at finite temperature. DMFT is exact in the
limit of large coordination, which physically corresponds to maximal magnetic
frustration. Therefore, in the simplest implementation, DMFT describes Mott
physics in absence of any magnetic order, nor does it include any (inter-site)
spin liquid correlations. We should mention that extensions of DMFT have
recently been proposed Lee et al. (2016) that include spinon effects, based on
an alternative (matrix M,N) rotor representation. This theory, which includes
some dynamical effects even at the saddle-point level, suggest that coherent
spinon excitations are very fragile to charge fluctuations emerging upon the
closing of the Mott gap, suppressing the spin liquid correlations not only on
the metallic side, but also within the critical region. We will not further
discuss these most sophisticated approaches here, but will limit our attention
to the predictions of the simplest single-site DMFT theory.

When applied to the single-band Hubbard model on a frustrated lattice (such as
the triangular lattice), DMFT predicts that on the metallic side the
quasiparticle mass diverges linearly $m^{*}\sim|U-U_{c2}|^{-1}$ at a critical
value $U_{c2}$ (similar to the prediction of the Brinkman–Rice (BR) theory of
the Mott transition Brinkman and Rice (1970)). The quasiparticle weight $Z$ is
inversely proportional to $(m^{*})$. Similarly, other features of the Fermi
liquid such as the Kadowaki-Woods law $A\sim(m^{*})^{2}$ are upheld. This
Fermi liquid behavior persists up to a temperature $T_{\mathrm{FL}}$ that
vanishes linearly when approaching $U_{c2}$. Interesting, at
$T_{\mathrm{max}}\sim T_{\mathrm{FL}}$ the resistivity exhibits a maximum
Radonjić et al. (2012). On the insulating side, there exist no well-defined
quasiparticles as the self-energy diverges, $\Sigma(\omega)\sim 1/\omega$. The
electronic spectrum is split into an upper and lower Hubbard band, separated
by a gap that remains nonzero at $U_{c2}$. The insulating state becomes
unstable at a lower value of the interaction $U_{c1}<U_{c2}$, where the gap
closes $\delta\sim|U-U_{c1}|$, νz, $\nu z\approx 0.8$ Eisenlohr et al. (2019).
As a result, there emerges a low-$T$ first-order metal-insulator transition,
and an associated phase coexistence region at $U_{c1}<U<U_{c2}$ Georges et al.
(1996). These main predictions are summarized in Figure 8.

Figure 8: Predictions of DMFT theory. (a) Phase diagram featuring a phase
coexistence region at $T<T_{c}$, and a Quantum Critical region centered around
the Quantum Widom Line (QWL) (adapted with permission from Ref. Terletska et
al. (2011) Copyright 2011 American Physical Society). (b) Resistivity
(normalized by the Mott-Ioffe-Regel (MIR) limit) as a function of temperature
$T$ across the transition. Note the pronounced resistivity maxima on the
metallic side (adapted with permission from Ref. Radonjić et al. (2011)
Copyright 2010 American Physical Society. (c) scaling collapse of the
resistivity curves, displaying pronounced “mirror symmetry” of the two
branches (adapted with permission from Ref. Terletska et al. (2011) Copyright
2011 American Physical Society).

At nonzero temperature, the first-order transition line ends at a critical
point at a temperature $T_{c}\approx 0.015W$, significantly smaller than the
bare bandwidth $W$. At temperatures $T\gg T_{c}$ the results can be viewed as
effectively quantum critical Terletska et al. (2011); Eisenlohr et al. (2019).
This quantum critical regime is centered around the so-called quantum Widom
line (QWL) Vučičević et al. (2013), which physically represents a finite-
temperature instability trajectory of the insulating phase, as shown in Figure
8a. It extends the first-order line past $T=T_{c}$, and can experimentally be
detected from an inflection point analysis Furukawa et al. (2015) of the
resistivity curves. In this regime, the resistivity satisfies the scaling law
$\rho(T,\delta U)=\rho_{c}(T)f(T/T_{0}(\delta U))$ with a crossover
temperature scale $T_{o}\sim|\delta U|^{\nu z}$ where $\nu z\approx 0.6$, see
Figure 8c. The crossover scale $T_{o}$ is a property of the quantum critical
regime and should not be confused with the low-temperature scaling of the
Fermi liquid temperature $T_{FL}$. DMFT therefore predicts two different
regimes of scaling: the quantum critical regime at $T\gg T_{c}$, and the metal
regime dominated by scaling in $T_{\mathrm{FL}}$.

###  4.3 Percolative Phase Coexistence Picture

In the early theories of both Mott insulators Mott (1990) and Wigner crystals,
the transition from insulator to metal was often assumed to be robustly first
order, at least at sufficiently low temperature. However, even the presence of
weak disorder or medium-ranged interactions will create an “emulsion”
(microscopic phase coexistence) with Mott/Wigner insulating “islands” in
between metallic “rivers” as proposed by Spivak and Kivelson in the context of
2DEG systems in semiconductors Spivak and Kivelson (2004); Jamei et al.
(2005). If so, then tuning bandwidth and/or temperature should produce a
continuous variation of the metallic fraction $x$. As long as it exceeds the
percolation threshold ($x>x_{c}$), the system is conducting. At $x<x_{c}$ the
metallic domains no longer connect across the system, and conduction stops, at
least at $T=0$. Critical behavior now arises because we dealing with a
classical percolation transition.

In this picture, the $T=0$ metal-insulator transition may occur without
actually closing the insulating gap $\Delta$. Similarly, at the percolation
threshold $x=x_{c}$ the metallic Fermi liquid at low $T$ is still stable, and
consequently there is no strict divergence of the effective mass $m^{*}$, nor
the vanishing of $T_{\mathrm{FL}}$ at the percolation threshold. We note,
however, that such a (classical) percolation picture should apply only if the
characteristic domain size is sufficiently larger than the characteristic
correlation (or dephasing) length, therefore strictly speaking not at the
lowest temperatures. However, finite-temperature variation of the transport
properties should be adequately captured, as abundantly documented Dagotto
(2005) in other systems featuring microscopic phase separation, such as
Colossal Magneto-Resistance (CMR) manganites, for example.

There is, however, an interesting and nontrivial feature of the percolation
picture pertaining to finite temperatures. Because of its localized spins, the
entropy of the Mott or Wigner–Mott insulating phase should be higher than that
of the of the metal (Fermi liquid). As a result, when raising the temperature
in the metallic regime, the insulating volume fraction will _increase_ : a
manifestation of the Pomeranchuk effect Pomeranchuk (1950). As a result, the
resistivity should increase up to some $T=T_{\mathrm{max}}$, above which the
metallic domains no longer connect, and resistivity will decrease again Spivak
and Kivelson (2006), leading to resistivity maxima. This qualitative picture
has been advocated in the work of Spivak and Kivelson, but no concrete
prediction of the precise temperature dependence of the resistivity has been
made, nor how the corresponding family of curves should scale as the
transition is approached.

Interestingly, the same physical picture should in fact apply not only to
Wigner–Mott transitions in semiconductors, but also the to conventional Mott
transition, provided that there exists a well-defined metal-insulator phase
coexistence region around the Mott point. Indeed, recent work on Mott organics
Pustogow et al. (2021) revealed precisely such a phase coexistence region,
albeit only at very low temperatures of the order of at most few percent of
the (bare) Fermi energy. Here, careful theoretical modeling Pustogow et al.
(2021) firmly established the validity of the percolation picture, but only
within a well-defined phase coexistence region. In contrast, in all the
systems studied (2DEG, Mott organics, moiré), the pronounced resistivity
maxima persist even much further onto the metallic side, where $T_{max}$ can
reach a substantial fraction of $T_{F}$, where phase coexistence is very
unlikely. Furthermore, recent experimental work on Mott organics by Kanoda and
collaborators demonstrated Urai et al. (2019); Yamamoto et al. (2020) the
extreme fragility of such a phase coexistence region to disorder, as generally
expected in 2D systems Aharony et al. (1976). Nevertheless, it is extremely
useful to have an independent experimental method to distinguish the phase
coexistence region (where percolative effects are likely) from the regimes
where a more uniform electron fluid/solid resides. The possibility to do so
was spectacularly demonstrated in the context of Mott organics. In the next
section we discuss how the dielectric response can tell which mechanism
(quasiparticle destruction or percolation) is at play in a given regime.

We briefly mention that percolation effects have been also discussed in the
context of spinon theory in a recent paper Kim et al. (2022), which does
require however significant disorder. On the other hand, the Spivak-Kivelson
theory does not require disorder as the micro-emulsion of insulators and
metals can be self-generated. This seems to be more in line with the
experiments of Section 3: at least the 2DEGs Melnikov et al. (2019) and the
Mott organics Dressel and Tomić (2020) are displaying Mott criticality in the
cleanest samples possible. It is therefore very plausible that most universal
features observed in all critical Mott systems are not the result of disorder,
but are instead the inherent manifestations of strong correlation physics.

##  5 Interpreting Resistivity Maxima

As we have seen from our brief theory overview above, several scenarios were
proposed, with sometimes similar predictions for characteristic features seen
in experiments. A notable example is the clear emergence of the resistivity
maxima on the metallic side, at a temperature $T=T_{max}\gtrsim T_{FL}$, which
is seen to decrease towards the transition. What is its physical content? The
three theoretical pictures propose very different physical perspectives on
what goes on here.

As we mentioned in Section 4.1, spinon theory Senthil (2008) predicts the
presence of resistivity maxima only in $d=3$, but not in $d=2$. However,
robust resistivity maxima are clearly seen all the material systems of Section
3. An understanding of the resistivity maxima must therefore come from either
the DMFT perspective on Mott physics, or from the percolative scenario. Both
mechanisms provide reasonable albeit very different routes to explain the
resistivity maxima. How should one distinguish them and thus identify the
precise mechanism at play in a given system? Luckily, important clues were
provided by recent experiments on Mott organics Pustogow et al. (2021). Here
one finds two distinct regimes, both featuring similar resistivity maxima, but
with very different dielectric response. One such regime is corresponds to the
(spatially inhomogeneous) metal-insulator phase coexistence region, where
colossal enhancement of dielectric response has been found. The other regime
was found further on the metallic side, where a dramatic drop and a change of
sign the dielectric constant signaled thermal destruction of coherent
quasiparticles due to strong correlation effects.

In the following we show how general scaling arguments can be used within each
of the two proposed scenarios, to demonstrate the general robustness of these
trends, thus providing a new window of what precisely goes on near the Mott
point.

###  5.1 Resistivity Maxima from Thermally Destroying Coherent Quasiparticles

Both experiments and theory provide evidence that a strongly correlated Fermi
liquid forms on the metallic side of the Mott point, with a characteristic
“Brinkman–Rice” (BR) energy scale $T_{BR}\sim 1/m^{*}$, which decreases
towards the transition, thus characterizing the heavy quasiparticles.
Inelastic electron-electron scattering increases with temperature, eventually
leading to the thermal destruction of the quasiparticles around $T\sim
T_{BR}$, and the associated modification of both the single particle (ARPES)
spectra and the optical conductivity. At higher temperatures, transport
assumes incoherent character, which can no longer be understood in terms of
the quasiparticle picture or Fermi Liquid ideas alone. Its precise form
generally depends on band filling and the correlation strength, but more
precise predictions require a specific microscopic model and a theoretical
picture.

Concrete and quantitative results, in this regime, were given by DMFT theory,
which provided first insight into the origin of the resistivity maxima in
certain Mott organic materials at half-filling, as well as in certain oxides.
Subsequent DMFT studies stressed that the characteristic temperature scale for
the resistivity maxima indeed tracks the BR scale of the quasiparticles
($T_{max}=T_{BR}$), while preserving the functional form of the resistivity
curves across this coherence-incoherence crossover. This revealed the scaling
behavior of the resistivity curves in the correlated metallic regime, with a
universal scaling function of $T/T_{max}$. The predicted scaling behavior has
been confirmed by a number of experiments on various systems Radonjić et al.
(2012); Shashkin et al. (2020); Moon et al. (2020), displaying even
quantitative agreement with the theoretical scaling function, with no
adjustable parameters.

Further optical and dielectric studies Pustogow et al. (2021) in Mott organics
also confirmed the predicted destruction of the Drude peak around $T_{BR}$,
again signaling the thermal destruction of quasiparticles. They established
that it dramatically affects not only DC transport, but also the dielectric
response, which in this metallic regime is seen to display a dramatic drop
from moderate positive values at $T>T_{drop}\sim T_{max}$ to very large but
negative values at $T<T_{drop}\sim T_{max}$. These studies, combining
experiments and DMFT theory, have firmly established that the dielectric
response can be used to directly reveal the thermal destruction of
quasiparticles around the BR temperature.

In the following, we extend the systematic studies of Ref.Radonjić et al.
(2012), to stress that within DMFT both DC transport and the dielectric
response display the characteristic crossover behavior across $T_{BR}$, and
the associated scaling behavior upon approaching the Mott point. To do this we
calculate the dielectric function $\epsilon_{1}$ as a function of temperature
and interaction $U$, using the same setup as in our recent work Pustogow et
al. (2021). For simplicity, we focus on a simple semi-circular band model at
half filling, and carry out DMFT calculations using the standard CTQMC
impurity solver with the the Maximum Entropy method for analytical
continuation to the real axis. Just as in Ref.Radonjić et al. (2012), once we
get the single particle self energy from our DMFT equations, we calculate the
real part of optical conductivity $\sigma_{1}(\omega)$ from the standard Kubo
formula, and the imaginary part of optical conductivity $\sigma_{2}(\omega)$
using the Kramers-Kroning tranform; the (complex) dielectric function is then
obtained via Economou (2005)

$\epsilon(\omega)=1+4\pi i\frac{\sigma(\omega)}{\omega}.$ (3)

The results for the single-particle density of states and the optical
conductivity are shown in Figure 9, and results for the DC transport and the
low-frequency dielectric response are displayed in Figure 10, for the
parameter range corresponding to the correlated metallic phase ($U\lesssim
U_{c1}$). Here panels (a) and (b) reproduce the results of Ref. Radonjić et
al. (2012), showing the characteristic scaling behavior of the resistivity
maxima near the Mott point. The analogous behavior for the dielectric function
$\epsilon_{1}$ is shown in panels (c) and (d), firmly establishing that the
observed crossover behavior assumes a universal scaling form in the correlated
metallic regime. The notion that the thermal destruction of quasiparticles
lies behind both phenomena is seen even more clearly in Figure 11, where we
show how the scale $T_{max}$ for the resistivity maxima, and the scale
$T_{drop}$ of the dielectric response, both scale with the quasiparticles
weight $Z=m/m^{*}$, as the transition is approached.

Figure 9: (a) DMFT results for the evolution of the single-particle Density of
States (DOS) for several values of the temperature (reprinted with permission
from Ref. Radonjić et al. (2011) Copyright 2010 American Physical Society), as
well as (b) that of the optical conductivity, in the strongly correlated
metallic regime. Different colors correspond to the four distinctive transport
regimes (inset in (b)). DOS features a distinct quasiparticle peak at low
temperatures, which is thermally destroyed at temperature
$T_{max}=T_{BR}\sim(m^{*})^{-1}$, where the resistivity (inset of right panel)
reaches a maximum. The optical conductivity displays the corresponding
suppression of the low-frequency Drude peak around the same temperature.
Figure 10: (a) DC resistivity as a function of temperature for several
interaction strengths. (b) Scaled resistivity curves. (c) Real part of
dielectric function $\epsilon_{1}$ at $\omega/D=0.01$, as a function of
temperature for several interaction strengths. (d) Scaled dielectric function
curves. Results are obtained for a half-filled Hubbard model solved within
DMFT.

These results establish a way to experimentally recognize the thermal
destruction of quasiparticles as a dominant mechanism behind the resistivity
maxima within a correlated but uniform metallic phase. Since the correlation
processes captured by DMFT are essentially local (i.e. “ultraviolet, UV”),
these effects should not display significant dependency on spatial
dimensional. Indeed, experiments have shown that similar resistivity maxima
are seen within correlated metallic phases both in 2D and in 3D systems.

Figure 11: (a) $T_{drop}$ as a function of $T_{max}$. (b) $T_{max}$ as a
function of $Z$. (c) $T_{drop}$ as a function of $Z$.

Note, however, that DMFT predicts very different behavior closer to the Mott
point, specifically within the phase coexistence region at $T<T_{c}$ and
$U_{c1}<U<U_{c2}$. Here, just as around any first-order phase transition line,
we expect hysteresis phenomena and inhomogeneous phase separation, where
metallic and insulating domains coexist on a nano-scale. As stressed in the
seminal work by Spivak and Kivelson Spivak and Kivelson (2004), thermal
effects can modify the relative volume fraction of the two coexisting phases,
producing under appropriate conditions the characteristic resistivity maxima.
In recent work motivated by experiments, a microscopic “hybrid-DMFT” approach
was developed Pustogow et al. (2021) to quantitatively describe this regime in
the context of Mott organics, resulting in spectacular agreement with
experiments. In the following section, however, we wish to stress that the
qualitative aspects of this regime display a number of universal scaling
features, which can be precisely understood from the perspective of
percolation theory.

###  5.2 Percolation Scenario Due to Phase Coexistence

To focus on the universal scaling aspects of percolative processes within the
metal-insulator phase coexistence region, we follow the seminal ideas of Efros
and Shklovskii Efros and Shklovskii (1976), and set up a two-component random
resistor network model, with characteristic low-frequency form for the
(complex) conductivity for each component:

$\begin{split}\sigma_{I}&=\sigma_{I}^{o}\text{exp}(-\Delta/T)-iC\omega,\\\
\sigma_{M}&=\frac{\sigma_{M}^{o}}{1-i\omega\tau}.\end{split}$ (4)

Here we assumed activated DC transport for the insulating component, with
capacitance $C$ and a standard Drude form for the conducting component, with
finite DC conductivity $\sigma_{M}^{o}$. To leading order near the percolation
point, we ignore the $T$-dependence of $\sigma_{M}^{o}$, $\sigma_{I}^{0}$, and
$C$, since the dominant effects come from the variation of the respective
volume fractions, and the activated form of insulating transport. The
temperature is expressed in the units of the activation energy $\Delta$, which
is also taken to be a constant. The corresponding expressions for the
(complex) dielectric functions of the two components are given by:

$\begin{split}\epsilon_{I}&=1+4\pi C+\frac{4\pi
i}{\omega}\sigma_{I}^{o}\text{exp}(-\Delta/T),\\\
\epsilon_{M}&=1-4\pi\tau\sigma^{o}_{M}+\frac{4\pi
i}{\omega}\sigma^{o}_{M}.\end{split}$ (5)

Here we ignored the capacitance of the metallic domains, which can be
neglected if $\tau\sigma^{0}_{M}/C\gg 1$.

Such a random resistor network model is appropriate for any percolating two
component metal-insulator system. To describe formation of the resistivity
maxima, an additional physical condition has to be met, as emphasized by
Spivak and Kivelson in the context of Wigner–Mott transitions, but which is in
fact valid for any Mott-like system in general. As we mentioned before, this
“Pomeranchuk effect” Pomeranchuk (1950) requires that the first-order line
(and the entire phase coexistence region) be “tilted” towards the metal, so
that the higher-entropy phase emerges at higher temperatures. To schematically
represent such a situation we assume that, within the phase coexistence
region, the volume fraction $x$ of the metallic component decreases with
temperature. As an illustration, we take the following simple model:

$\begin{split}x(x_{o},T)=x_{c}+\frac{1}{2}\text{tanh}[\frac{x_{o}-x^{*}(T)}{w(T)}],\end{split}$
(6)

where $x_{c}$ represents the percolation threshold, $w(T)=a(T_{c}-T)/T_{c}$
defines the width of the coexistence region, and $x^{*}(T)=x_{c}+b(T/T_{c})$,
as illustrated in Figure 12a, for $a=0.4$ and $b=1$. In this model, the
parameter $x_{o}$ controls the metallic volume fraction at $T=0$, which
decreases at $T>0$, and reaches the percolation threshold $x=x_{c}$ at
$T=T^{*}(x_{o})=T_{c}(x_{o}-x_{c})/b$. Physically, the the DC resistivity will
first increase with $T$ as the metallic volume fraction decreases. Past
percolation threshold, however, the metallic domains no longer connect.
Transport then assumes insulating (activated) form, resulting in subsequent
resistivity decrease at $T>T^{*}$, and the emergence of resistivity maxima
around $T\sim T^{*}$. Similarly, the dielectric constant $\epsilon_{1}$ grows
(diverges) as the percolation threshold is approached from the insulating
side, due to the formation of large metallic clusters with increased
polarizability. On the metallic side, however, it displays a rapid decrease,
dropping to large negative values within the metallic phase. As a result,
dielectric response displays colossal enhancement around the percolation
threshold, a phenomenon that can be viewed as a smoking gun for percolative
charge dynamics.

Figure 12: (a) The red line is $x=x(T^{*})$. For $T$ larger than the blue
dashed line, $x=0$. We calculate the percolation results along the grey dashed
lines. (b) $R/R^{o}_{M}$ as a function of $T$ for different $T^{*}$. (c)
Scaled resistivity curves.

To illustrate these ideas, we use the Effective Medium Approximation (EMA) for
percolation, which solves the following nonlinear equations for the complex
dielectric function:

$\begin{split}x(\frac{\epsilon_{M}-\epsilon}{\epsilon_{M}+(z/2-1)\epsilon})+(1-x)(\frac{\epsilon_{I}-\epsilon}{\epsilon_{I}+(z/2-1)\epsilon})=0,\end{split}$
(7)

and for illustration selected
$\sigma_{M}^{o}/\sigma_{I}^{o}=100,\tau\sigma^{o}_{M}=1000$, $C=1$,
$T_{c}/\Delta=0.4$, and $z=4$ corresponding to 2D transport ($x_{c}=0.5$).
Precisely the anticipated behavior is observed from numerically solving the
EMA equation for the corresponding DC resistivity $R=\sigma^{-1}$, as shown in
Figure 12b. Here, we select several values of $x_{o}$, corresponding to
$x>x_{c}$ (low temperature metallic regime), and plot the resistivity as a
function of temperature (following dashed lines in Figure 12a. We observe
distinct resistivity maxima around the temperature $T^{*}(x_{o})$
corresponding to the percolation threshold. Note how the maxima become sharper
and sharper as $T^{*}$ is reduced, corresponding to the exponential
(activated) decrease of the “field” $h\sim\exp\\{-\Delta/T^{*}\\}$. The
expected behavior is also seen in dielectric response, as shown in Figure 13a,
where we observe sharp maxima at $T\sim T^{*}$. Here again we see the
increased “rounding” of these maxima at higher $T^{*}$, corresponding to
larger $h(T^{*})$. This behavior can be seen even more clearly in Figure 13b,
where $\epsilon_{1}$ is plotted as a function of the reduced concentration
$\tau=(x(T)-x_{c})/x_{c}$, which vanishes at $T=T^{*}$.

Figure 13: (a) The dielectric constant $\epsilon_{1}$ as a function of $T$ for
different $T^{*}$. (b) $\epsilon_{1}$ as a function of $\tau$ for different
$T^{*}$. (c) Scaled dielectric function curves.

These qualitative trend can be even more rigorously described within the
scaling theory for percolation, where the DC conductivity as well as the
$\omega=0$ dielectric constant are known to satisfy the following scaling
relations:

$\sigma_{1}(\tau,h)=\sigma_{M}^{o}h^{s}F_{\sigma}(\tau/h^{m});\;\;\;\epsilon_{1}(\tau,h)=h^{s-1}F_{\epsilon}(\tau/h^{m}),$
(8)

where $\tau=(x(T)-x_{c})/x_{c}$ measures the distance to the percolation
threshold, and $h=\sigma_{I}/\sigma_{M}$ plays a role of the “symmetry
breaking field”, which leads to the rounding of the transition. The critical
exponents $s$ and $m$, as well as the crossover scaling functions $F_{\sigma}$
and $F_{\epsilon}$ are universal quantities within percolation theory. To
illustrate this scaling behavior within EMA, we collapse the family of
resistivity curves by plotting $Rh^{s}$ as a function of $\tau/h^{m}$, as
shown in Figure 12c, and $\epsilon_{1}h^{1-s}$ as a function of s a function
of $\tau/h^{m}$, as shown in Figure 13c. Note how a perfect scaling collapse
is observed here, but only around the peak of the dielectric response, i.e.
only close to the percolation threshold. Such behavior is, in fact, not
surprising, since we expect scaling phenomena to arise only within a given
critical region, and not further away from the critical point.

We should stress again that all our qualitative results are rigorously valid
within general percolation theory for our two-component phase coexistence
model, and EMA was simply used as an illustration. EMA correctly captures the
general crossover phenomena associated with percolation, but only introduces
approximate values for the critical exponents $s_{EMA}=0.5$ and $m_{EMA}=0.5$,
which are otherwise know even more accurately from numerical simulations.
These details, however, are not of direct relevance for our purposes. What is
important is the result that, within our “Pomeranchuk” model for phase
coexistence, the percolation scenario predicts distinct resistivity maxima but
also striking colossal dielectric anomalies, at the same temperature scale of
$T=T^{*}$ which decreases towards the MIT. This behavior is in distinct
contrast to the behavior we found from the DMFT picture of a correlated but
uniform metallic phase, which also leads to resistivity maxima, but very
different behavior of the dielectric response. This observation, which was
quantitatively validated in recent experiments on Mott organics Pustogow et
al. (2021), thus reveals a distinct criterion to settle the long-lasting
controversies between the origin of the resistivity maxima in different
systems.

##  6 Conclusions

In this paper we discussed three different classes of physical systems which
all display very similar phenomenology expected for Mott-like metal-insulator
transitions. We stressed that most qualitative features are clearly seen in
all these examples, including the continuous decrease of the characteristic
energy scales $T_{FL}$, $T_{BR}=T_{max}$, $T_{o}$, $\Delta$ towards the
transition, the phenomenon of quantum critical scaling seen in transport, as
well as the emergence of distinct resistivity maxima on the metallic side.
These observations, which are starting to portray a robust and consistent
phenomenology of Mott criticality, is putting serious constraints on theory.
We discussed which of these features seem compatible with various proposed
theoretical pictures of the Mott point, and which ones do not.

In the final section of this paper we also presented new theoretical results,
which open the possibility to precisely determine, from experiments, which
mechanism dominates in which regime. We argue that the dielectric response
offers unique insights, which so far have not been appreciated enough, as a
powerful tool to distinguish between different phase coexistence and the
thermal destruction of quasiparticles.

A class of issues we did not discuss in any detail in this paper is the
(explicit) role of disorder around the Mott point. Given the fact that new
classes of ultra-clean material are starting to emerge, with even more
pronounced salient features of Mott criticality, it is becoming possible to
plausibly minimize the role of disorder on experimental grounds. On the other
hand, new experimental efforts are starting Urai et al. (2019); Yamamoto et
al. (2020) to emerge in the opposite direction: to systematically add and to
control the level of disorder, for example by high-energy X-ray irradiation.
These fascinating research directions are guaranteed to open entirely new
chapters in the study of metal-insulator quantum criticality. This will
require theorists to rekindle the efforts to understand the interplay of
strong correlation with disorder Miranda and Dobrosavljević (2005), and
perhaps to develop new ideas in the process.

Y.T. carried out theoretical calculations and performed the analyses. V.D. and
L.R. designed the project. All authors discussed the data, interpreted the
results, and wrote the paper. All authors have read and agreed to the
published version of the manuscript.

This work was supported by the NSF Grant No. 1822258, and the National High
Magnetic Field Laboratory through the NSF Cooperative Agreement No. 1644779
and the State of Florida. LR acknowledges support by the Swiss National
Science Foundation via Ambizione grant PZ00P2_174208.

Not applicable. Not applicable.

Data are provided: https://github.com/yutingtanphysics/ (accessed on 3 May
2022) How-to-Recognize-the-Universal-Aspects-of-Mott-Criticality.

###### Acknowledgements.

We thank Pak Ki Henry Tsang for providing technical support in doing computer
cluster calculations. The authors declare no conflict of interest. The funders
had no role in the design of the study; in the collection, analyses, or
interpretation of data; in the writing of the manuscript, or in the decision
to publish the results. References

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