# Quantized heat flow in graphene quantum Hall phases: Probing the topological
order

Saurabh Kumar Srivastav  Department of Condensed Matter Physics, Weizmann
Institute of Science
Rehovot, Israel 76001, Israel
<EMAIL_ADDRESS>
https://orcid.org/0000-0002-0498-4217  Anindya Das  Department of Physics,
Indian Institute of Science
Bangalore, 560012, India
<EMAIL_ADDRESS>
https://orcid.org/0000-0002-6310-1576

###### Abstract

Topological quantum numbers are often used to characterise the topological
order of phase having protected gapless edge modes when the system is kept in
a space with the boundary. The famous examples in this category are the
quantized electrical Hall conductance and thermal Hall conductance, which
encodes the topological order of integer and fractional quantum Hall states.
Here, we review the recent thermal transport study of integer and fractional
quantum Hall states realized in graphene-based van der Waals heterostructures.

###### keywords:

Graphene; Quantum Hall; Integer Quantum Hall; Fractional quantum Hall;
Topological order, Jhonson-Nyquist noise, Quantized electrical Hall
conductance, Quantized thermal Hall conductance, Neutral modes

##  1 Introduction

Topological properties of the electronic phases are often encoded in the
quantized physical quantity, such as electrical Hall conductance and thermal
Hall conductance. Although the notion of the topology in condensed matter
physics was introduced long back in 1972 by J. Michael Kosterlitz and David J.
Thouless[1, 2], the first experimental discovery of the topological quantum
liquid was realized by Klaus von Klitzing and coworkers in silicon field-
effect transistor at low temperature and subjected to a high magnetic field.
They found that the Hall conductance was precisely quantized in an integer
multiple of the $G_{0}=\frac{e^{2}}{h}$[3] (where h is Planck’s constant and e
is the elementary electronic charge.), accompanied by the vanishing
longitudinal resistivity. This phenomenon is known as the integer quantum Hall
(IQH) effect and is the first experimentally observed topological phase of
electrons in a solid. For this discovery, Klaus von Klitzing received the
Nobel Prize in Physics in 1985. With improved sample quality, in 1982, Tsui,
Stormer, and Gossard observed that in addition to the integer quantum Hall
plateau, a new plateau appeared beyond the conventional sequence of the
integer quantum Hall state. It was quantized at the value
$\rho_{xy}=\frac{h}{3e^{2}}$ accompanied by a minimum in $\rho_{xx}$[4].
Furthermore, with improvement in sample quality, the plateaus were observed at
$\rho_{xy}=\frac{h}{\nu e^{2}}=\frac{p}{q}\frac{h}{e^{2}}$ accompanied by
vanishing $\rho_{xx}$, where $\nu$ is the filling factor, $p$ and $q$ are
positive integer numbers. A few examples in this series are
$\nu=1/5,1/3,2/5,3/7,4/9,5/9,3/5,4/7,2/3,...$ in the lowest Landau level and
$\nu=4/3,5/3,7/5,5/2,12/5,13/5,...$ in the higher Landau levels[4, 5, 6, 7, 8,
9, 10, 11, 12, 13, 14, 15, 16]. The observation of the quantum Hall plateaus
at the fractional filling is known as fractional quantum Hall (FQH) effect and
lead to the another Nobel prize in physics in 1998. It is one of the most
correlated states observed in condensed matter systems. In contrast to the
integer quantum Hall states, a single-particle phenomenon, the fractional
quantum Hall phase is a highly correlated state of the electrons. It emerges
when the Landau levels are partially filled.

The transition from one IQH or FQH plateau to another IQH or FQH plateau is a
classic example of topological phase transition between two gapped bulk
phases, each characterized by a different topological order. To probe the
topological order of these phases, at first, one would naively expect to use
an experimental technique that directly probes the bulk of these gapped
phases. However, it turns out to be tedious and difficult to probe the bulk.
Thanks to the bulk-edge correspondence principle, the topological order of
bulk can also be accessed by studying the gapless edge modes at the physical
boundary of the system, which is relatively easy to probe compared to the
gapped bulk state. Theoretically, the gapless edge modes can be probed via
quantized electrical Hall conductance and thermal Hall conductance. However,
till now, most of our understanding of the topological order of IQH and FQH
effects comes from electrical Hall conductance measurements, which is quite
successful for the IQH phase and particle-like FQH phases, hosting only
downstream edge modes. However, the transport properties become quite complex
for hole-conjugate states hosting counter-propagating edge modes, which
demands the measurement of quantized thermal Hall conductance.

This article reviews the electronic thermal conductance study of the integer
and several fractional quantum Hall states in graphene-based devices. By
contrast to the conventional two-dimensional electron gas in GaAs/AlGaAs, the
linear, Dirac-like spectrum of low energy excitations and the pseudospin
degeneracy make single-layer graphene (SLG) a unique, truly two-dimensional
“relativistic” electronic system. The $\pi$ Berry’s phase in graphene gives
rise to Landau Levels structure, which is no longer evenly spaced. The Landau
levels in graphene have four-fold internal degeneracy: two for spin and two
for valley degree of freedom. As a result of the Dirac-like dynamics, the Hall
conductivity $\sigma_{xy}$ exhibits an unconventional sequence of the
quantization. Furthermore, the high mobility and gate tunability of the
carrier density in graphene over a wide range provides a huge experimental
phase space for the study of the symmetry-broken integer quantum Hall (IQH)
and fractional quantum Hall phases (FQH). Since the unique band structure and
the spin-valley degeneracy of graphene are quite different from the
conventional GaAs/AlGaAs 2D gas, it is worth starting with the basics of
single-particle band structure, and resulting Landau Levels dispersion in the
magnetic field. Next, we briefly describe the effect of the symmetry breaking
and its signature in the transport experiments, mainly in electrical
conductance. Then, we discuss the emergence of the fractional quantum Hall
states in single-layer graphene. Next, we discuss the topological order of the
QH states and their detection schemes. Later on, we provide a detailed
discussion of the electronic thermal conductance measurement scheme of QH
states, which is the central core of this review article. After the technical
introduction of the measurement scheme, we will describe the results obtained
from the three sets of thermal conductance measurements of integer and
fractional quantum Hall states in single-layer and bilayer graphene. Finally,
we end the review with a discussion on the impact of the experimental work
discussed in this article and future endeavours.

##  2 Anomalous quantum Hall effect in graphene

###  2.1 Single particle band structure of ‘Single layer graphene

In single-layer graphene, carbon atoms are arranged in a honeycomb lattice
structure as shown in 1 (a). To describe its lattice vectors, one can think of
it as a triangular lattice with two sublattice basis A and B. The unit lattice
vectors can be written as

$\displaystyle\mathbf{a}_{1}=\frac{a}{2}\left(3,\sqrt{3}\right)\,,\qquad\mathbf{a}_{2}=\frac{a}{2}\left(3,-\sqrt{3}\right)\,.$
(1)

where $a=1.42$ Å is the carbon-carbon bond length. The corresponding
reciprocal lattice vectors are given by

$\displaystyle\mathbf{b}_{1}=\frac{2\pi}{a}\left(1,\sqrt{3}\right)\,,\qquad\mathbf{b}_{2}=\frac{2\pi}{a}\left(1,-\sqrt{3}\right)\,.$
(2)

The first Brillouin zone (BZ) of the graphene lattice is shown in Fig. 1(b).
There are six corners in the first BZ, out of which only two points, known as
‘Dirac points’, $\mathbf{K}$ and $\mathbf{K}^{{}^{\prime}}$ are non-
equivalent. The positions of $\mathbf{K}$ and $\mathbf{K}^{{}^{\prime}}$
points in momentum space are given by

$\displaystyle\mathbf{K}=\frac{2\pi}{3a}\left(1,1/\sqrt{3}\right)\,,\qquad\mathbf{K^{\prime}}=\frac{2\pi}{3a}\left(1,-1/\sqrt{3}\right)\,.$
(3)

SLG’s low energy band structure can be calculated using the tight-binding
approach. For the scope of this review, we account for only the first three
nearest neighbour sites, which are

$\displaystyle\boldsymbol{\delta_{1}}=\frac{a}{2}\left(1,\sqrt{3}\right)\,,\qquad\boldsymbol{\delta_{2}}=\frac{a}{2}\left(1,-\sqrt{3}\right)\,,\qquad\boldsymbol{\delta_{3}}=-a\left(1,0\right)\,.$
(4)

Figure 1: (a) Carbon atoms are arranged in a honeycomb lattice structure. It
can be thought of as two interpenetrating triangular lattices. Two
nonequivalent sites are named $A$ (red) and $B$ (black), separated by distance
$a=1.42$Å. $\mathbf{a}_{1}$ and $\mathbf{a}_{2}$ are the lattice vectors and
$\delta_{i}$, $i=1,2,3$ are the nearest neighbor vectors. (b) $\mathbf{b}_{1}$
and $\mathbf{b}_{2}$ are the reciprocal lattice vectors. The region bounded by
the hexagon boundary lines is the first Brillouin zone. The gap between the
conduction and valence band vanishes at the Dirac point lattice vectors
$\mathbf{K}$ and $\mathbf{K}^{{}^{\prime}}$. The figure is adapted from
Saurabh Kumar Srivastav’s PhD Thesis (2022). Reprinted with permission from
the Indian Institute of Science[17].

The tight binding Hamiltonian accounting only the first three NN can be
written in the basis of the wave-function amplitudes on the two sublattices
$\Phi_{A}$ and $\Phi_{B}$, as follow;

$\displaystyle\widehat{H}_{SLG}=-\gamma_{0}\left(\begin{array}[]{cc}0&f(\mathbf{k})\\\
f^{*}(\mathbf{k})&0\end{array}\right)$ (7)

where $\gamma_{0}=2.8$ eV is the nearest neighbour hopping between site
$A\leftrightarrow B$. and

$\displaystyle
f(\mathbf{k})=e^{-i\boldsymbol{k.\delta_{1}}}+e^{-i\boldsymbol{k.\delta_{2}}}+e^{-i\boldsymbol{k.\delta_{3}}}$
(8)

The resulting eigenvalues of the Hamiltonian(7) have two eigenvalues, which
become identically zero at $\mathbf{K}$ and $\mathbf{K}^{{}^{\prime}}$ points
in the Brillouin zone. In most cases, only the low-energy bands around these
points ($\mathbf{K}$ and $\mathbf{K}^{{}^{\prime}}$), known as Dirac points
and Valley in the Brillouin zone, are important. The low-energy Hamiltonian
around these Dirac points can be written as

$\displaystyle\widehat{H}_{SLG}=\hbar v_{F}\left(\begin{array}[]{cc}0&\xi
k_{x}-ik_{y}\\\ \xi k_{x}+ik_{y}&0\end{array}\right)$ (11)

where $v_{F}=3\gamma_{0}a/2\hbar\simeq 1\times 10^{6}$ m/s and $\xi=\pm 1$ for
$\mathbf{K/K^{{}^{\prime}}}$ point. The eigenvalues of the Hamiltonian(11)
around $\mathbf{K}$ point is given by;

$\displaystyle E_{\pm}(k)=\pm\hbar v_{F}|\mathbf{k}|$ (12)

The resulting low-energy band dispersion of graphene has a four-fold
degeneracy, 2 for the spin flavour and 2 for the Valley flavour. In the
presence of a magnetic field perpendicular to the graphene plane
$\mathbf{B}=B\hat{\mathbf{z}}$, the band dispersion gets modified; Following
the minimal coupling approach of the vector potential $\mathbf{A}=(-By,0,0)$;
the Hamiltonian(11) around $\mathbf{K}$ point can be written in terms of the
defined operator
$\pi\rightarrow(p_{x}-eBy+ip_{y}=-i\hbar\partial_{x}-eBy+\hbar\partial_{y})$
and $\pi^{\dagger}\rightarrow(p_{x}-eBy-
ip_{y}=-i\hbar\partial_{x}-eBy-\hbar\partial_{y})$. The resulting Hamiltonian
in the magnetic field near the Dirac point $\mathbf{K}$ is given by

$\displaystyle
H_{SLG,\mathbf{K}}=v_{F}\left(\begin{array}[]{cc}0&\pi^{\dagger}\\\
\pi&0\end{array}\right)$ (15)

Operator $\pi$ and $\pi^{\dagger}$ act as a creation and annihilation operator
respectively, such that

$\displaystyle\pi\psi_{N}=-\frac{\sqrt{2}i\hbar}{l_{B}}\sqrt{N}\psi_{N-1},$
(16)
$\displaystyle\pi^{\dagger}\psi_{N}=\frac{\sqrt{2}i\hbar}{l_{B}}\sqrt{N+1}\psi_{N+1},$
(17) $\displaystyle\pi\psi_{0}=0$ (18)

Here, $\psi_{N}(x,y)$ is the scalar orbital Landau level wavefunction;

$\displaystyle\psi_{N}(x,y)=\frac{1}{\sqrt{2^{N}(N!)\sqrt{\pi}}}e^{\frac{ip_{x}x}{\hbar}}H_{N}\bigg{(}\frac{y}{l_{B}}-l_{B}k\bigg{)}exp\bigg{[}-\frac{1}{2}\bigg{(}\frac{y}{l_{B}}-l_{B}k\bigg{)}^{2}\bigg{]}$
(19)

where $l_{B}=\sqrt{\hbar/eB}$ is the magnetic length scale, $N$ is the Landau
level index and $H_{N}$ is $N^{th}$ Hermite polynomial. The resulting
eigenvalues are given by,

$\displaystyle\epsilon_{N}=sign(N)\frac{\sqrt{2}\hbar
v_{F}}{l_{B}}\sqrt{|N|}=sign(N)\sqrt{2\hbar v_{F}^{2}eB|N|}$ (20)

The eigenstates at $\mathbf{K}$ point are given by

$\displaystyle\forall N\geq
1:\Phi_{N,\pm}=\frac{1}{\sqrt{2}}\left(\begin{matrix}\Psi_{N}\\\ \mp
i\Psi_{N-1}\\\ \end{matrix}\right)$ (21) $\displaystyle
N=0:\Phi_{0}=\frac{1}{\sqrt{2}}\left(\begin{matrix}\Psi_{0}\\\ 0\\\
\end{matrix}\right)$ (22)

whereas near $\mathbf{K^{\prime}}$ point, eigenstates are

$\displaystyle\forall N\geq
1:\Phi_{N,\pm}=\frac{1}{\sqrt{2}}\left(\begin{matrix}\pm i\Psi_{N-1}\\\
\Psi_{N}\\\ \end{matrix}\right)$ (23) $\displaystyle
N=0:\Phi_{0}=\frac{1}{\sqrt{2}}\left(\begin{matrix}0\\\ \Psi_{0}\\\
\end{matrix}\right)$ (24)

Figure 2: (a) In the presence of a magnetic field, Landau level forms at
discrete energy intervals. (b) Landau level spectrum is plotted for both
conduction and valence band as a function of the magnetic field $\mathbf{B}$
for different Landau Level index $N$. Here, Landau level energy disperses as
$\epsilon=sgn(N)\sqrt{2\hbar v_{F}^{2}eB|N|}$ and as a result energy gap
reduces as one approaches higher Landau levels. It is in sharp contrast to
conventional two-dimensional electron gas, where the energy gap remains the
same irrespective of the Landau level index. The presence of zero energy
Landau level, equally shared by electron and hole, also makes SLG a unique
two-dimensional material. (c) The corresponding density of states of Landau
levels after incorporating the Lorentzian broadening of the $3meV$ for each
Landau level for $B=10$ T. The figure is adapted from Saurabh Kumar
Srivastav’s PhD Thesis (2022). Reprinted with permission from the Indian
Institute of Science[17].

It should be noted that for all Landau level $N\geq 1$, the electronic density
is shared by both sublattice site $A$ and $B$. However, for $N=0$ state, near
$\mathbf{K/K^{\prime}}$ valley, electronic density only resides on the
sublattice $A/B$. The resulting Landau level energy spectrum and the density
of states are plotted in Fig. 2(b) and (c), respectively. The presence of the
zero energy Landau level (ZLL) and four-fold spin valley degeneracy of SLG,
the Hall conductivity $\sigma_{xy}$ exhibits an unconventional sequence of the
quantization described by[18, 19, 20, 21, 22, 23, 24, 25]

$\sigma_{xy}=\pm 4\bigg{(}N+\frac{1}{2}\bigg{)}\frac{e^{2}}{h}$ (25)

It should be noted that the Landau level’s wavefunction in SLG differs
significantly from the conventional GaAs/AlGaAs. First, here, for SLG, the
wavefunction of the $N^{th}$ Landau level has the orbital component of the
$N^{th}$ and $N^{th}-1$ conventional Landau level, except ZLL. This difference
can lead to different values of Haldane pseudopotentials for the LLs with
different orbital structures and hence affects the sequence of the FQH states
in different LLS. The second major difference, which is more pronounced in ZLL
is the different sublattice structure of wavefunctions near degenerate
$\mathbf{K}$ and $\mathbf{K^{\prime}}$. As a consequence, it affects the
valley and or spin symmetry-breaking of ZLL and the ground state of $\nu=0$
insulating state.

###  2.2 Symmetry breaking and quantum Hall Ferromagnetism

Along with the unconventional sequence of the QH plateaus due to the $\pi$
Berry’s phase, the spin and valley degeneracy makes graphene a wonderful
system for observing the rich physics associated with the multicomponent
quantum Hall phases[26, 27, 23, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38].
In SLG, the relevant energy scales are the Landau quantization energy
$\epsilon_{N}$ (cyclotron energy) and the long-range Coulomb interaction
$E_{C}=\frac{e^{2}}{\epsilon l_{B}}$. At low magnetic fields, $\epsilon_{N}$
and $E_{C}$ do not depend on the spin or valley degrees of freedom. So the
combined spin-valley flavour degeneracy can be described in terms of a single
approximate $SU(4)$ isospin symmetry which corresponds to the invariance of
long-range electron-electron $(E_{C})$ interaction under a rotation within the
fourfold spin/valley internal space. However, the long-range Coulomb
interaction $(E_{C})$ can drive the system through a ferromagnetic
instability, in which the order parameter corresponds to a finite polarization
in a specific direction within the $SU(4)$ isospin space [26, 27, 23, 28, 29,
30, 31, 32, 33, 34, 35, 36, 37]. At the integer fillings within a partially
filled quartet Landau levels, this order parameter is predicted to lead to a
finite gap for charged excitations and a robust quantum Hall effect for
integers outside the sequence described in Eqn. 25. The exact $SU(4)$
polarization for a given sample depends on the interplay between the
anisotropies arising from the $SU(4)$ symmetry-breaking term like Zeeman
splitting, lattice scale interactions and even the disorder (that might not
break the $SU(4)$ symmetries). Fig. 3(c) shows the plot of transverse
conductance as a function of the back gate voltage at $B=9.8$ T and $40$ mK of
bath temperature for a single layer graphene device encapsulated between the
two hexagonal boron nitride substrate (hBN). In addition to the conventional
plateaus at $2\frac{e^{2}}{h},6\frac{e^{2}}{h},10\frac{e^{2}}{h}$, well-
developed symmetry broken plateaus of $N=0$ and $N=1$ LLs are also seen.

Figure 3: (a) In absence of any interaction, the single particle spectrum of
Landau levels gives the quantum Hall plateaus at
$g\big{(}N+\frac{1}{2}\big{)}\frac{e^{2}}{h}$, where $g=4$ is the four fold
spin-valley degeneracy and $N$ is the Landau level index. The distance between
the quantum Hall plateaus along the density axis is defined by the density of
states $\frac{gB}{\phi_{0}}$ on each Landau level, which remarkably does not
depend on the zero field spectrum of the system. Here, $B$ is the magnetic
field and $\phi_{0}=\frac{h}{e}$ is the flux quantum. In addition to the
plateaus in transverse conductance, the corresponding sequence of the Landau
levels are shown in magenta and green for holes and electrons, respectively.
(b) However, in presence of interactions and high magnetic field, the four-
fold spin valley degeneracy breaks, which leads to the observation of the
plateaus at the integer values outside the conventional sequence. The
schematic shows the energy spectrum for the spin (I) or valley(II)
polarization for the Landau levels after spin-valley symmetry breaking. Two
different possible scenarios associated with two sequences of edge states are
labelled by $I$ and $II$, respectively, In the schematic, the edge states’
spin and valley polarization are denoted by $\uparrow/\downarrow$ and $\pm$,
respectively .(c) Transverse conductance is plotted as function of the back
gate voltage at $B=9.8$ T and $40$ mK of bath temperature for a single layer
graphene device. In addition to the conventional plateaus at
$2\frac{e^{2}}{h},6\frac{e^{2}}{h},10\frac{e^{2}}{h}$, one can clearly observe
the well developed symmetry broken plateaus of $N=0$ and $N=1$ Landau levels.
The figure is adapted from Saurabh Kumar Srivastav’s PhD Thesis (2022).
Reprinted with permission from the Indian Institute of Science[17].

##  3 Fractional quantum Hall effect in single-layer graphene

At the partial filling of LLs, when the kinetic energy is completely quenched,
electron-electron interaction leads to the emergence of fractional quantum
Hall (FQH) states. Due to the approximate SU(4) symmetry of SLG, the symmetry
breaking at partial filling of LLS is quite complex. It can lead to a
condition where multiple possible orders, such as single-component and multi-
component FQH states, compete with each other for the ground state. The
experimental appearance or absence of the FQH sequences in SLG can be mainly
explained either via a single-component or multi-component composite fermion
(CF) model. Conventionally, in single component CF model, each electron
captures an even number, say 2p, of the quantized vortices(or flux quanta) and
turns itself into weakly interacting topological particles, known as composite
fermions (CF). Due to the vortex attachment, weakly interacting CFs experience
an effective magnetic field $B^{*}$, which is related to the externally
applied field $B$ via a relation[39, 40];

$\displaystyle B^{*}=B-2p\rho\phi_{0}$ (26)

where $\rho$ is the electron density, $\phi_{0}=h/e$ is the flux quantum and
2p is an even integer. Equivalently, one can write that electrons at filling
factor $\nu$ convert into a composite fermion with filling $\nu^{*}$, related
via

$\displaystyle\nu=\frac{\nu^{*}}{2p\nu^{*}\pm 1}$ (27)

This single-component non-interacting CF model accurately describes the
sequence and hierarchy of FQH states observed between $0<\nu<1$ of SLG[41, 42,
43]. However, FQH states observed experimentally between $1<\nu<2$ raise
concern over the applicability of single component CF model in this branch of
symmetry-broken LLs[44, 45, 46]. Particularly, the absence of $\nu=5/3$ was
quite surprising. Since $5/3=2-1/3$ is closest to the conventional $1/3$
state, naively one would expect to observe well-developed FQH states at
$\nu=5/3$. Furthermore, the emergence of only even numerator states with a
step of 2 between the consecutive FQH states in $1<\nu<2$, suggests the
partial lifting of four-fold spin and valley and a transition into an
approximate SU(2) symmetry phases. Careful observation of the emergent FQH
states suggests that these responses have close analogies with the IQH effect
of real electrons observed in SLG with a four-fold degeneracy of LLs. Within
the CF picture, these states are usually described via a two-component CF
model, similar to what has been used to explain the sequence of multicomponent
FQH states observed in AlAs and strained silicon[47, 48]. To date, many
properties of the ground states and the excitation of FQH states in graphene
are unknown and demand a detailed and systematic investigation.

##  4 Topological order and its Detection in integer and fractional quantum
Hall states

In the topological phase of matter, some of the physical properties remain
insensitive to the local perturbations of the system[49, 50, 51]. These
physical properties may include fractionalization[49], long-range quantum
entanglement[51], and topological degeneracies in the energy spectrum of the
system[49, 50, 51]. Mathematically, the topological order of a quantum phase
is usually characterized by some of the topological invariant numbers.
Topological invariant numbers are often used to characterize the class of
topological order of phase that has protected gapless edge modes when the
system is kept on a space with the boundary[49, 50, 51]. The integer and
fractional quantum Hall phases are the first set of the discovered
topologically ordered phases that have protected gapless edge modes at their
boundary. Usually, the knowledge of the topological order of the IQH and FQH
states demands a detailed understanding of the charge and statistics of the
anyons, which exist in the gapped bulk of the 2D electron liquid. However,
accessing the bulk of the 2D electron liquid in the experiment is challenging.
Thanks to the bulk-edge correspondence principle, the topological order of the
gapped bulk of IQH and FQH phases can be determined by examining the physics
of the gapless edge modes.

###  4.1 Edge states in Integer and fractional quantum Hall states

Similar to the IQH case (Fig. 4(a)), the charge density for the particle-like
FQH states $(\nu<1/2)$ drops monotonically from its bulk values to zero as one
approaches the edge from the bulk[52], as shown in Fig. 4(b, c) for
$\nu=1/3,2/5$. It leads to the emergence of the one-dimensional downstream
edge modes near the physical boundary of the sample[52], as shown in Fig. 4(e,
f) for $\nu=1/3,2/5$. The applied magnetic field’s direction dictates the edge
modes’ downstream chirality[53, 54]. However, the charge density profile does
not decrease monotonically for the hole-conjugate FQH states $(1/2<\nu<1)$.
For example, at $\nu=2/3$, the filling factor increases from $2/3$ to $1$ and
then drops back to zero[52]. In other words, $2/3$ liquids can be considered
an FQH state of $\nu=1/3$ holes in the IQH state at $\nu=1$ electron liquid.
As a consequence, two counter-propagating edge modes of charge $e$ with
conductance $e^{2}/h$ and charge $-e/3$ with conductance $e^{2}/3h$ emerge at
the sample boundary[52, 55]. Fig. 5(a, b) shows the schematics of charge
density profiles and edge states of the $\nu=2/3$ FQH state. However, in the
presence of the disorder-dominated random inter-edge tunnelling between, Kane-
Fisher-Polchinski[56] found a decoupled charge mode of conductance $2e^{2}/3h$
and a neutral mode, which does not carry any charge but can carry the energy.
The schematics of such normalized edge modes are shown in Fig. 5(c, d).

Since the neutral modes do not couple with the charged excitations of the
external probes, it becomes challenging to detect them via the conventional
electrical conductance measurement. Although several proposals were made to
see the neutral modes which involve the measurement of the tunnelling exponent
in constrictions[57], looking for resonance in long constriction[58], and
looking for the heating effects on shot noise[59, 60], the first experimental
proof was demonstrated by Bid et al.;[61] for filling factor $\nu=2/3,3/5,$
and $5/2$ in GaAs/AlGaAs sample. This experiment involved an upstream quantum
point contact (QPC) constriction from an energized ohmic contact. An excited
neutral mode emanates from the ‘hot spot’ at the source, propagates upstream
along the edge, and impinges on a partially pinched QPC. This leads to
observed current fluctuations. Recently, upstream neutral modes for
$\nu=2/3,3/5$ mode have been also observed in bilayer graphene devices[62].

Figure 4: (a) charge density profile of $\nu=1$, (b) $\nu=1/3$, and (c)
$\nu=2/5$ states near the edge. In these states, charge density and potential
profile drops monotonically to zero. The schematic of the downstream edge
modes at the physical boundary of the sample is shown in (d),(e), and (f) for
$\nu=1,1/3,$ and $\nu=2/5$ states, respectively. The conductance of downstream
edge modes in integer ($\nu=1$) and $\nu=1/3$ particle-like states are
$e^{2}/h$ and $e^{2}/3h$, respectively. For $\nu=2/5$ state, two downstream
edge modes with conductance $e^{2}/15h$ (shown with blue dashed colour: inner
mode) and $e^{2}/3h$ (shown with red colour: outer mode), respectively.The
figure is adapted from Saurabh Kumar Srivastav’s PhD Thesis (2022). Reprinted
with permission from the Indian Institute of Science[17].

###  4.2 Detection of topological order

Quantized electrical and thermal Hall conductance in quantum Hall states have
been known for a long time back theoretically[63, 64] since their first set of
experimental observations. Although electrical Hall conductance has been
widely used to understand the topological order of a quantum Hall state, it is
insufficient in the hierarchical fractional quantum Hall states, where the
edge structure is complicated, and transport may occur via the downstream and
upstream modes. The electrical Hall conductance only reveals the number and
the conductance of the downstream charged chiral edge modes. Still, it is
independent of the edge modes’ total number, chirality, and character. By
contrast, the quantized thermal Hall conductance is not only sensitive to the
downstream charged modes, it can also detect the other upstream modes,
including the chargeless neutral modes, which are not detectable in electrical
Hall conductance measurement[63, 64, 65].

Figure 5: (a) charge density profile of $\nu=2/3$ at the sharp edge. (b) A
schematic of the bare edge modes, with conductance $e^{2}/h$ (downstream mode
shown with red colour: outer mode) and $e^{2}/3h$ (upstream mode shown by blue
dashed colour: inner mode), respectively. (c) charge density profile of
$\nu=2/3$ at a disordered edge. In the presence of the disordered dominated
tunnelling between the counter-propagating bare edge modes, one gets a
decoupled charge mode of conductance $2e^{2}/3h$ and an upstream neutral mode,
which does not carry any charge but can support the energy transport. (d) The
schematic of the decoupled downstream charged mode with conductance
$2e^{2}/3h$ (shown with red colour: outer mode) and upstream neutral mode
(shown with blue colour: inner mode). The figure is adapted from Saurabh Kumar
Srivastav’s PhD Thesis (2022). Reprinted with permission from the Indian
Institute of Science[17].

Similar to the quantization of electrical conductance, the quantized thermal
conductance of a single ballistic channel only depends on the fundamental
constants of nature and is given by
$\kappa_{0}T=\frac{\pi^{2}k_{B}^{2}}{3h}T$, where $k_{B}$ and $h$ are the
Boltzmann’s and Planck’s constant, respectively. If multiple downstream edge
modes exist, such as IQH and particle-like FQH states, quantized thermal
conductance becomes $G_{Q}=N_{d}\kappa_{0}T$, where $N_{d}$ is the total
number of downstream edge modes. However, the situation becomes complex in the
inter-mode tunnelling between counter-propagating edge modes at certain
fillings, such as for $2/3$ and $3/5$, where some chiral edge modes propagate
in the upstream direction. In 1997, Kane and Fisher derived the quantized
thermal Hall conductance for the abelian fractional quantum Hall states for an
ideal impurity-free edge[64]. For the FQH states with counter-propagating edge
modes, their analysis assumes a substantially longer edge propagation length
than the thermal equilibration length. They found that quantized thermal Hall
conductance at any filling of abelian FQH state is given by[64]
$G_{Q}=(N_{d}-N_{u})\kappa_{0}T$, where $N_{d}$ and $N_{u}$ are the numbers of
the downstream and upstream edge modes, respectively. However, if the edge
propagation length is smaller than the thermal equilibration length $(l_{H})$,
thermal Hall conductance takes the value of $G_{Q}=(N_{d}+N_{u})\kappa_{0}T$.
Since quantized $G_{Q}$ depends on the $N_{d}$ and $N_{u}$, it can also be
used as a powerful experimental tool to detect the upstream neutral modes and
hence the exact topological order of complex FQH states, which were not
possible in electrical conductance measurements. Furthermore, in contrast to
quantized electrical conductance, quantized thermal conductance remains
independent of the statistics of the carriers. It remains the same for the
fermions, bosons, anyons [64, 66] except in the case of the Majorana mode,
where the quantized thermal conductance becomes half of its quantum limit,[67,
68, 69, 70, 71]. It makes the thermal conductance measurement a powerful
technique which can be used to distinguish non-abelian order of even-
denominator FQH states. Usually, the non-abelian order of FQH state, such as
$5/2$ is believed to host the Majorana modes in its edge structure[72, 73, 74,
75, 76, 77, 78, 79, 80, 81, 82, 83]. Hence the observation of the half-
quantized thermal conductance will be the smoking-gun evidence of the Majorana
mode. Experimentally, the half quantum of the thermal conductance has been
measured at $5/2$ FQH state in GaAs/AlGaAs based two-dimensional electron
gas[84].

###  4.3 Measurement of quantized thermal conductance in IQH and FQH states

Figure 6: (a) Basic principle of thermal conductance measurement of QH states.
A micrometre size metallic floating contact is heated to $T_{M}$ by injecting
a known Joule power $J_{Q}$. The temperature of the floating contact is set by
the heat balance equation $J_{Q}=nJ_{Q}^{e}+J_{Q}^{e-ph}$, where $nJ_{Q}^{e}$
is the electronic contribution to heat flow via $n$ ballistic QH edge channels
from floating contact to cold bath at temperature $T_{0}$ and $J_{Q}^{e-ph}$
is the transfer of heat from the hot electrons in floating contact toward the
cold phonon bath at temperature $T_{0}$. (b) Extracted electronic heat current
factor $\alpha_{n}$ (shown by symbols) defined as
$nJ_{Q}^{e}/(T_{M}^{2}-T_{0}^{2})$ normalized by $\pi^{2}k_{B}^{2}/6h$ is
plotted as a function of the total number of channel $n$. The grey line shows
the predictions for the quantum limit of the heat flow. The figure is adapted
from Jezouin et al. (2013). Reprinted with permission from AAAS.

Different from electrical conductance measurement, quantized thermal
conductance measurement is complicated and tricky. It is evident from the fact
that although the quantization of the electrical and thermal conductance of
the IQH and FQH states has been known for decades, the first experimental
measurement of quantized thermal conductance of quantum Hall states was
reported 33 years after the first experimental demonstration of the IQH states
in the pioneering work of Jezouin et al. [85] in GaAs/AlGaAs based two-
dimensional electron gas. The basic principle of the experimental set-up is
shown in Fig. 6(a). In this measurement scheme, a micrometer-sized metallic
floating reservoir is connected to the cold electronic reservoir via an
adjustable number $n$ of the ballistic QH edge channels and a phonon cold bath
at temperature $T_{0}$. The electron temperature of the floating metallic
reservoir is heated to a temperature $T_{M}$ by injecting a known joule
heating power $J_{Q}$ to it. The temperature of the floating reservoir is
determined by a heat balance equation given by;

$J_{Q}=J_{Q}^{e}+J_{Q}^{e-ph}=0.5N\kappa_{0}(T_{M}^{2}-T_{0}^{2})+J_{Q}^{e-ph}$
(28)

Here, the first and the second terms in this heat balance equation correspond
to the electronic contribution to heat flow via $N$ ballistic edge channel and
heat loss due to electron-phonon coupling from the floating reservoir to the
phonon cold bath. The central result of Jezouin et al. is shown in Fig. 6(b),
where $\alpha_{n}$ measured electronic thermal conductance normalized by
$0.5\kappa_{0}$ is plotted as a function of the $n$ number of the ballistic
edge channel. The unit slope of the experimental fit data corresponds to the
theoretically expected quantized limit on the thermal conductance. Later, a
similar measurement was extended to the FQH regime by Banerjee et al. [86, 84]
in the same system. Banerjee et al. [86] demonstrated the universality of the
quantized thermal conductance for the anyonic heat flow. They further reported
the observation of a half-integer of the quantized thermal conductance for
even-denominator FQH state $5/2$ [84]. The measured thermal conductance value
of $2.5\kappa_{0}T$ was interpreted as the particle–hole Pfaffian (PH-
Pfaffian) topological order of $5/2$ state. As mentioned above, the
experiments by Jezouin et al. and Banerjee et al. were performed in
GaAs/AlGaAs based two-dimensional electron gas.

Figure 7: Schematic of the graphene device with thermal conductance
measurement setup. The device is set in the integer QH regime at filling
factor $\nu=1$, where one chiral edge channel (line with arrow) propagates
along the edge of the sample. The current $I_{S}$ is injected (black line)
through the contact ‘S’, which is absorbed in the floating reservoir (orange
contact). The chiral edge channel (red line) at potential $V_{M}$ and
temperature $T_{M}$ leave the floating reservoir and terminates into two cold
grounds (CGs). The blue lines show the cold edges (without current) at
temperature $T_{0}$. The resulting increase in the electron temperature
$T_{M}$ of the floating reservoir is determined from the measured excess
thermal noise at contact $T$. A resonant (LC) circuit, situated at contact
$T$, with resonance frequency $f_{0}\approx 750$ kHz, filters the signal,
which is amplified by the cascade of amplification chain (cryogenic (cold)
preamplifier placed at 4K plate and a room temperature amplifier). A ceramic
capacitance of 10 nF was introduced between the sample and inductor to block
the DC current along the measurement line. Last, the amplified signal is
measured by a spectrum analyzer. The figure is adapted from Saurabh Kumar
Srivastav’s PhD Thesis (2022). Reprinted with permission from the Indian
Institute of Science[17].

Although The quantization of the electrical Hall conductance in graphene, a
van der Waals material was reported in 2005 by A. Geim and Philip Kim group
separately[87, 88], the first experiment of quantized thermal conductance in
QH states in graphene was reported in 2019[89]. In this experiment, graphene
was encapsulated between two hexagonal boron nitride substrates and was gated
by either the SiO_2/Si gate or the graphite back gate. In addition to having
the Hall probe metallic contacts, a micron-size metallic floating contact was
connected in the middle of the graphene flake via one-dimensional edge
contact. It is important to emphasize that below the metallic floating
reservoir, the graphene region was entirely etched, and after applying the
sufficient magnetic field and the gate voltages, the QH edge channels directly
terminate and leave the floating reservoir from one side of the graphene
chamber to the other side. To heat the floating reservoir, a DC current
$I_{S}$ was injected at contact ‘S’ as shown in Fig. 7. This setup helps to
create a hot reservoir at milli-Kelvin temperature. The cold-grounded (CG)
metal contacts, as shown in Fig. 7, serve as a cold reservoir. Now, one has
two reservoirs of different temperatures. To extract the thermal conductance,
one needs to know the thermal current flowing between the two reservoirs via
the electronic channel and the electron temperatures of the reservoir.

###  4.4 Determination of the thermal current $J_{Q}$:

The thermal current flowing between the two reservoirs can be easily estimated
for integer and particle-like fractional quantum Hall states, where only
downstream edge mode exists. In this scenario, all dissipated power near the
floating contact is used entirely to heat the metallic floating contact.
Hence, the thermal current flowing between two reservoirs will be exactly the
same as the total dissipated power near the floating contact. A DC current
$I_{S}$ injected at the source contact $S$ (Fig. 7) flows from the source
contact $S$ to the floating reservoir along the chiral QH edge channels.

The outgoing current from the floating reservoir splits into two equal parts,
each propagating along the outgoing chiral edge from the floating reservoir to
the cold grounds. The floating reservoir reaches a new equilibrium potential
$V_{M}=\frac{I_{S}}{2\nu G_{0}}=\frac{V_{S}}{2}$ ( where $G_{0}=e^{2}/h$) with
the filling factor $\nu$ of graphene determined by the $V_{BG}$, whereas the
potential of the source contact is $V_{S}=\frac{I_{S}}{\nu G_{0}}$. Thus, the
power input to the floating reservoir is
$P_{in}=\frac{1}{2}(I_{S}V_{S})=\frac{I_{S}^{2}}{2\nu G_{0}}$, where the pre-
factor of $1/2$ results because half power dissipates at the back of source
contact (7). Similarly, the outgoing power from the floating reservoir is
$P_{out}=\frac{1}{2}(2\times\frac{I_{S}}{2}V_{M})=\frac{I_{S}^{2}}{4\nu
G_{0}}$. Thus, the resultant injected power dissipation in the floating
reservoir due to joule heating is $J_{Q}=P_{in}-P_{out}=\frac{I_{S}^{2}}{4\nu
G_{0}}$.

However, the situation may be complicated in hole-conjugate fractional quantum
Hall states like $\nu=2/3$, which also supports the upstream and downstream
modes. The downstream and upstream modes can have different chemical
potentials near the metallic floating contact and the source contact. Due to
the charge tunnelling between the counter-propagating modes, the edge channel
attains the equilibrium chemical potential for the propagation length larger
than the charge equilibration length ($l_{eq}^{c}$). In this scenario, the
total dissipated joule power also remains the same. Still, some portion is
dissipated in the tunnelling regions outside the ohmic reservoirs, and the
rest is dissipated in the floating metallic contact. In this case, one can not
equate the total dissipated power to the thermal current $J_{Q}$. However,
suppose one assumes that all heat from the charge equilibration region returns
to the nearby ohmic reservoirs. In that case, one can safely use the formula
derived for the integer and particle-like fractional quantum Hall states. Such
an assumption is justified if the charge equilibration length is much smaller
than the reservoir size. Fortunately, the charge equilibration length in a
graphene-based system is expected to be quite small; hence, we can safely
equate the total dissipated power near the floating contact to the thermal
current $J_{Q}$. Here we would also like to define the characteristic length
scales like charge and thermal equilibration length, which will be extensively
used in this review.

Charge equilibration length ($l_{eq}^{c}$): Charge equilibration length
($l_{eq}^{c}$) is defined as the minimum propagation length required to
achieve the complete voltage equilibration between the counter-propagating
edge modes. In other words, all counter-propagating edge modes should attain
the same electrostatic voltage after a propagation length of $l_{eq}^{c}$.

Thermal equilibration length ($l_{eq}^{H}$): Thermal equilibration length
($l_{eq}^{H}$) is defined as the minimum propagation length required to
achieve the same temperature between the counter-propagating edge modes. In
other words, all counter-propagating edge modes should attain the same
temperature profile after a propagation length of $l_{eq}^{H}$.

###  4.5 Determination of the electron temperatures $T_{M}$:

The resulting increase in the electron temperature $T_{M}$ of the floating
contact was determined by measuring the excess thermal noise at contact ‘T’.
To avoid the contribution from the 1/f noise, a resonant L//C tank circuit was
used to shift the operating frequencies close to the 0.7 MHz regimes. The
circuit consists of an inductor made of superconducting NbTi wire, and the
capacitance developed along the coaxial line connecting the sample to the
cryogenic amplifier. A ceramic capacitance of 10 nF was also introduced
between the sample and inductor to block the DC current along the measurement
line and isolate the DC voltage at the cryogenic amplifier gate port from the
sample bias voltage. The output voltage from the cryogenic amplifier was
further amplified using a room-temperature voltage amplifier. After the second
amplification stage, the resulting signal was measured using a spectrum
analyzer. The measured excess noise at contact ‘T’ is related to the electron
temperature $(T_{M})$ of the floating contact via the relation,

$S_{I}=\nu k_{B}(T_{M}-T_{0})$ (29)

where $\nu$ is the bulk filling of the QH state. Furthermore, it is worth
emphasising the various conditions that need to be fulfilled to justify the
use of noise thermometry for electron temperature $(T_{M})$ determination.

Continuous energy levels: The energy level spacing of the micron size metallic
floating contact must be negligible compared to the other relevant energy
scales. Since, in typical graphene devices, edge contacts are mainly made via
gold, one can easily estimate the energy level spacing. Using the density of
states ($D_{E}\simeq 1.14\times 10^{47}J^{-1}m^{-3}$), for the gold and the
typical volume of the $\Omega\approx 0.5\mu$m3, the average energy level
spacing becomes $\sim 1/(D_{E}\times\Omega)\sim k_{B}\times 1.45\mu K$, which
is much smaller than the typical electron temperature $8-40$ mK ranged
achieved in thermal conductance measurement set-up. Hence, for micron-sized
floating contact, one can safely ignore the issues related to the discrete
energy levels of the floating metallic contact in our devices.

Quasi-equilibrated electronic distribution function: The metallic floating
contact will act as a hot reservoir only if it has a well-defined quasi-
equilibrated electronic distribution characterized by temperature $T_{M}$. For
the realization of such a quasi-equilibrium regime, the hot electrons must
dwell in metallic floating contact sufficiently longer than the electron-
electron interaction (thermalization) time scale. The dwell time can be
calculated as; $t_{dwell}=\frac{D_{E}\Omega h}{N}$[85, 90], where $D_{E}$ is
the electronic density of states per unit volume per unit energy, $\Omega$ is
the volume of micron-size floating contact, $h$ is the Planck’s constant and
$N$ is the number of channels leaving the floating contact. In typical
graphene devices used for thermal conductance measurement, the floating
contact has a volume $\Omega\approx 0.5\mu$m3, mostly made of gold. Using the
typical density of states for gold $D_{E}\simeq 1.14\times
10^{47}J^{-1}m^{-3}$, estimated dwell time was found to be
$t_{dwell}\simeq\frac{40\mu s}{N}$, which is much larger than the typical
electron-electron interaction time of the order of 10 ns range for gold at a
temperature down to few milli-kelvin[91]. This firmly establishes the quasi-
equilibrium hypothesis that the electron’s energy distribution in the micron
size ohmic contact is a hot Fermi distribution function characterized by a
temperature $T_{M}$.

High-quality ohmic contact between the metallic floating reservoir and
graphene channel: As mentioned, since the electron temperature determination
relies on excess thermal noise measurement, other unwanted noise sources
should be avoided. One such factor is that the metallic floating reservoir
should make good quality ohmic contact with graphene to minimise the current
reflection, which otherwise will generate unwanted Shot noise. Thanks to the
development of a high-quality ohmic one-dimensional edge contact technique for
graphene, the minimal reflection coefficient was obtained, which was
negligible for all devices used for the thermal conductance measurements in
literature.

##  5 Universality of quantized heat flow in graphene

###  5.1 Thermal conductance of Integer quantum Hall states

We will first discuss the results reported for the Integer quantum Hall states
in graphene. Fig. 8(a) shows the quantum Hall response of the device as a
function of back-gate voltage at 9.8 T of the magnetic field at $\sim$ 40 mK
of the bath temperature. In addition to the signature QH plateau of SLG at
$2\frac{e^{2}}{h},6\frac{e^{2}}{h},10\frac{e^{2}}{h}$, QH plateaus at
$1\frac{e^{2}}{h},4\frac{e^{2}}{h},5\frac{e^{2}}{h}$, are also visible, which
are associated with the degeneracy lifting of zeroth and first Landau levels
of the single layer graphene.

To measure the thermal conductance, the central floating contact was heated to
a temperature $T_{M}$ due to the Joule dissipation $J_{Q}$ near the floating
contact. The plot of $T_{M}$ against the $J_{Q}$ is shown in Fig. 8(b). To
extract the electronic contribution to the thermal conductance of the QH
states, $J_{Q}$ can be plotted against the $T_{M}^{2}-T_{0}^{2}$ and the slope
of the linear fit provides the value of the quantized thermal conductance
$G_{Q}$ provided the electron-phonon cooling contribution $(J_{Q}^{e-ph})$ is
negligible. Experimentally, It was found that $J_{Q}^{e-ph}$ is negligible for
the typical device geometry till $\sim T_{M}=100$ mK[89]. However, to
completely rule out any electron-phonon cooling ($J_{Q}^{e-ph}$) contribution,
dissipated joule power between the two QH filling factors was subtracted at
constant $T_{M}$ under the assumption that the $J_{Q}^{e-ph}$ does not depend
on the number of edge channels leaving the floating contact. The resulting
plot between $\lambda=\Delta J_{Q}/0.5\kappa_{0}$ (where $\Delta
J_{Q}=J_{Q}(\nu_{i},T_{M})-J_{Q}(\nu_{j},T_{M})=0.5\Delta
N\kappa_{0}(T_{M}^{2}-T_{0}^{2})$ ) vs $T_{M}^{2}-T_{0}^{2}$ is shown in Fig.
8(c). From the slope of the curve $(m)$, thermal conductance per ballistic
channel was calculated as $G_{Q}=(m/\Delta N)\kappa_{0}T$ and found to be
$\approx 1\kappa_{0}T$ as predicted theoretically. This plot is the first
experimental demonstration of the quantized heat flow for integer quantum Hall
states in graphene since its first mechanical exfoliation isolation. This
experiment also emphasized that the universality of the quantized heat flow is
the same for intrinsic QH states ($\nu=2,6$) and symmetry-broken quantum Hall
states ($\nu=1$) in graphene devices, as expected.

###  5.2 Thermal conductance of particle-like fractional quantum Hall states
$(\nu=4/3)$

The four-fold unique spin valley degeneracy of single-layer graphene
distinguishes it from the conventional GaAs/AlGaAs system. In the regime of
fractional quantum Hall states, the presence of a large magnetic field can
give rise to different possibilities of the symmetry breaking of the internal
four-fold degeneracy, such as the complete lifting of the spin and valley
degeneracy, hence the emergent FQH states will be either spin or valley
polarised. Another possibility includes the degeneracy lifting of only one
flavour, either spin or valley and hence, the emergent FQH phases preserve the
approximate SU(2) symmetry in the remaining flavour sector. In addition to
these two possibilities, there might be a scenario where none of the spin and
valley flavours degeneracy is broken. Consequently, the emergent FQH phases
have mixed spin-valley flavours. The earlier transport experiments and the
local compressibility measurements suggest that while the FQH states with
filling $0<\nu<1$ follow the standard sequence of non-interacting single
component two flux composite fermion model, only the even numerator FQH states
emerge for the filling $1<\nu<2$, suggesting the possibility of two-component
composite fermion model, with an approximately SU(2) symmetric states for the
interacting electrons. This complexity or richness of FQH states in graphene
raised concern over the universality of the quantized heat flow for FQH states
in graphene.

Figure 8:  (a) Gate response of SiO_2/Si gated hBN encapsulated graphene
device at B = 9.8 T. Clear signature plateaus in conductance (marked with red
arrows) at $\nu=2,6,$ and $10$ are observed in the unit of $\frac{e^{2}}{h}$.
In addition to these, well-developed broken symmetric plateaus (marked with
blue arrows) at $1\frac{e^{2}}{h},4\frac{e^{2}}{h},$ and $5\frac{e^{2}}{h}$
are also visible. (b) The increased temperatures $T_{M}$ of the floating
reservoir are plotted (solid circles) as a function of dissipated power
$J_{Q}$ for $\nu$ = 1 ($N$ = 2), 2 ($n$ = 4) and 6 ($N$ = 12), respectively,
where $n=2\nu$ is the total outgoing channels from the floating reservoir. (c)
The $\lambda=\Delta J_{Q}/(0.5\kappa_{0})$ is plotted as a function of
$T^{2}_{M}$ for $\Delta N=2$ (between $\nu$ = 1 and 2), and 8 (between $\nu$ =
2 and 6), respectively in red and black solid circles, where $\Delta
J_{Q}=J_{Q}(\nu_{i},T_{M})-J_{Q}(\nu_{j},T_{M})$. The solid lines are the
linear fittings to extract the thermal conductance values. Slope of these
linear fits are 1.92 and 7.92 for $\Delta N$ = 2, and 8, respectively, which
gives the $G_{Q}=0.96\kappa_{0}T$, and $0.99\kappa_{0}T$ for single edge mode,
respectively.The figure is adapted from Srivastav et al. (2019) [89].
Reprinted with permission from AAAS

The earlier experimental result of the heat flow measurement for the particle-
like FQH state $(\nu=4/3)$[89] in an hBN encapsulated graphite back-gated
single-layer graphene device is shown in Fig. 9. Fig. 9(a) shows the QH
plateau of $\nu=4/3$ FQH state between the integer plateau of $\nu=1$ and $2$.
Fig. 9(b) shows the plot of $J_{Q}$ (solid circles) as a function of
$T^{2}_{M}-T^{2}_{0}$ for $\nu$ = 1, $\frac{4}{3}$, and 2 over the temperature
window where the curve is linear, implying the dominance of the electronic
contribution to the heat flow. The solid lines in Fig. 9(b) represent the
linear fits (in $0.5\kappa_{0}$) and give the values of 2.04, 4.16 and 4.04,
which corresponds to $G_{Q}=1.02,2.08$ and $2.02\kappa_{0}T$ for $\nu$ = 1,
$\frac{4}{3}$, and 2, respectively. Conventionally, $\nu$ = $\frac{4}{3}$, is
thought of as $1+1/3$. As a result, one would expect two downstream charge
modes, one integer and one fractional (inner $\nu$ = $\frac{1}{3}$ with
effective charge, $e^{*}=\frac{e}{3}$). If the universality of quantized heat
flow is preserved, the thermal conductance of $\nu=\frac{4}{3}$ should be the
same as $\nu$ = 2 having two integer downstream charge modes. This is indeed
the same as shown in Fig. 9(b). For $\nu=4/3$, $G_{Q}$ was found to be
$(2.02\pm 0.02)\kappa_{0}T$. This measurement established the universality of
the quantized thermal conductance in graphene for both integer and fractional
QH edges.

Figure 9:  (a) Gate response of SiO_2/Si gated hBN encapsulated graphene
device at B = 9.8 T. Clear signature plateaus in conductance (marked with red
arrows) at $\nu=2,6,$ and $10$ are observed in the unit of $\frac{e^{2}}{h}$.
In addition to these, well-developed broken symmetric plateaus (marked with
blue arrows) at $1\frac{e^{2}}{h},4\frac{e^{2}}{h},$ and $5\frac{e^{2}}{h}$
are also visible. (b) $J_{Q}$ (solid circles) is plotted as a function of
$T^{2}_{M}-T^{2}_{0}$ for $\nu$ = 1, 4/3 and 2 and shown up to $T_{M}$ $\sim$
60-70mK. The solid lines are the linear fit to extract the slopes, which give
the thermal conductance values of 1.02, 2.08 and 2.02$\kappa_{0}T$ for $\nu$ =
1, $\frac{4}{3}$, and 2, respectively. Thermal conductance values are
quantized for $\nu$ = 1 and 2, and, more importantly, the values are the same
for both $\nu$ = 4/3 and 2 plateaus. The inset shows the corresponding
downstream charge modes for integer and fractional edges. The dashed curve
represents the theoretically predicted contribution of heat coulomb blockade
for $\nu$ = 1, showing its negligible contribution to the net thermal current.
The figure is adapted from Srivastav et al. (2019) [89]. Reprinted with
permission from AAAS

##  6 Non-equilibrated heat transport for hole-conjugate fractional quantum
Hall states

Till now, we have mostly discussed the thermal conductance for the integer and
particle-like FQH states, where only downstream edge modes exist. However, the
situation becomes complex for hole-conjugate FQH states, where the edge
structure is complicated and hosts the downstream and upstream edge modes. One
such paradigmatic FQH phase emerges at $\nu=2/3$, consisting of counter-
propagating $1$ (downstream) and $\frac{1}{3}$ (upstream) modes [52]. Although
the thermal conductance for 2/3 state was measured in widely studied
GaAs/AlGaAs structure[86], nothing was known in the case of graphene for hole-
conjugate states. The first experimental report on the measurement of thermal
conductance for hole-conjugate FQH states $(\nu=5/3,8/3)$ in bilayer graphene
was reported in 2021[92], which shows a remarkably different result. Fig.10(a)
shows the plot of the conductance versus gate voltage of an hBN encapsulated
graphite gated bilayer graphene device at $B=10$ T and a bath temperature of
30 mK in the hole-doping region. In addition to the integer QH plateaus at
$\nu=1,2,$ and $3$, well developed FQH plateau emerges at $\nu=4/3,5/3,$ and
$8/3$. Conventionally, 5/3 and 8/3 states can be considered 1+2/3 and 2+2/3
states, respectively.

Figure 10: (a) Conductance $G_{S}$ is plotted as a function of back gate
voltage. The robust fractional plateaus at $\nu=\frac{5}{3}\frac{e^{2}}{h}$,
$\frac{7}{3}\frac{e^{2}}{h}$, $\frac{8}{3}\frac{e^{2}}{h}$ with weaker plateau
$\sim$ $\frac{4}{3}\frac{e^{2}}{h}$ clearly visible along with the integer QH
plateaus at $\nu=1,2,$ and 3. (b) $J_{Q}$ (solid circles) as a function of
$T^{2}_{M}-T^{2}_{0}$ for $\nu$ = 5/3 (red), 7/3 (black) and 8/3 (blue). The
solid magenta, brown, red and blue lines represent $G_{Q}$ = 1$\kappa_{0}T$,
2$\kappa_{0}T$, 3$\kappa_{0}T$ and 4$\kappa_{0}T$, respectively. The linear
fits of the solid circles give $G_{Q}$ = 3.03, 2.96 and 4.03$\kappa_{0}T$ for
$\nu$ = 5/3, 7/3 and 8/3, respectively. (b) $\lambda=\Delta
J_{Q}/(0.5\kappa_{0})$ as a function of $T^{2}_{M}$ for $\Delta\nu=5/3-1$
(red) and $\Delta\nu=8/3-2$ (black), where $\Delta
J_{Q}=J_{Q}(\nu_{i},T_{M})-J_{Q}(\nu_{j},T_{M})$. Solid lines represent linear
fits. Extracted values of $G_{Q}$ of the $2/3$-like FQH states are
$2.02\kappa_{0}T$ and $2.06\kappa_{0}T$ for $\Delta\nu=5/3-1$ and
$\Delta\nu=8/3-2$, respectively. The figure is adapted from Srivastav et al.
(2021) [92]. Reprinted with permission from American Physical Society.

So, normally, one would expect that edge the structure of $\nu=5/3(8/3)$
should host one(two) downstream integer modes of conductance $e^{2}/h$, one
downstream fractional edge mode of conductance $2/3(e^{2}/h)$ and one upstream
neutral modes, which does not carry any charge but can support the heat flow.
Counting of all these edge modes provides $N_{d}=2$ and $N_{u}=1$ for 5/3
state, and $N_{d}=3$ and $N_{u}=1$ for 8/3 state, respectively. If one naively
expects the full thermal equilibration between these edge modes, $G_{Q}$ for
these hole-conjugate FQH states is expected to be $|N_{d}-N_{u}|\kappa_{0}T$,
which should be $1\kappa_{0}T$ and $2\kappa_{0}T$, for 5/3 and 8/3,
respectively. Fig.10(b) shows the plot of $J_{Q}$ with $T^{2}_{M}-T^{2}_{0}$
for $\nu$ = 5/3 (red), 7/3 (black) and 8/3 (blue). The filled circles are the
experimental data points and the solid magenta, brown, red and blue lines
represent the theoretical lines of $J_{Q}$ for $G_{Q}$ = 1$\kappa_{0}T$,
2$\kappa_{0}T$, 3$\kappa_{0}T$ and 4$\kappa_{0}T$, respectively. As clearly
evident from the plots(See Fig. 10(b)), the measured $G_{Q}$ for $\nu=5/3,$
and $8/3$ strikingly matches with $(N_{d}+N_{u})\kappa_{0}T$ ($3\kappa_{0}T$
for 5/3 and $4\kappa_{0}T$ for 8/3), rather than the expected topological
quantum number of $|N_{d}-N_{u}|\kappa_{0}T=1\kappa_{0}T$, and $2\kappa_{0}T$,
respectively. Fig. 10(c) shows the plot $\lambda=\Delta J_{Q}/(0.5\kappa_{0})$
as a function of $T^{2}_{M}$ for two different configurations of
$\Delta\nu=\frac{5}{3}-1$ (red) and $\frac{8}{3}-2$ (black) to extract the
contribution of the partially filled Landau level with $\nu=$ $\frac{2}{3}$
out of the data for $\frac{5}{3}$ and $\frac{8}{3}$. Linear fits give
$2.02\kappa_{0}T$ and $2.06\kappa_{0}T$, respectively, for $G_{Q}$ of the
$\nu=$ $\frac{2}{3}$ state. This value is markedly inconsistent with the
values dictated by the topology (0$\kappa_{0}T$ in long length limit) and the
observation of Banerjee et al. (2017) in GaAs/ AlGaAs sample, which report the
$G_{Q}\approx 0.33\kappa_{0}T$ and interpret the equilibrated heat flow of the
counter-propagating edge modes expected for $2/3$ state, i.e.
$G_{Q}=(N_{d}-N_{u})\kappa_{0}T$. It is important to emphasize that the
measured value of the electrical conductance for 5/3 and 8/3 states in bilayer
graphene in this experiment matches very well with the expected value of the
equilibrated value of the electrical conductance. The theoretical study of
this observed phenomenon in bilayer graphene pointed towards the diverging
thermal equilibration length $l_{eq}^{H}$ while charge equilibration length
$l_{eq}^{C}$ remains finite near the disorder fixed point[92]. The same
experiment was repeated for the fractional states observed in the electron
doping regime, and the observed results were the same. This is particularly
important because the fractional states at the same fillings observed in
electron and hole doping correspond to a different set of the orbital index in
bilayer graphene. This experiment provides the first experimental notion of
the different charges and the heat equilibration length for the counter-
propagating edge modes in QH phases. A similar result was also reported by G.
L. Bretona et al. [93] in the first cool-down of their device.

##  7 Determination of topological edge quantum numbers

Although the measured values of $G_{Q}=$ $3\kappa_{0}T$ and $4\kappa_{0}T$ for
5/3 and 8/3 states, respectively, in bilayer graphene by Srivastav et al.
(2021) are well explained by the non-equilibrated values of the heat flow of
counter-propagating edge modes for 2/3 edge structure, it does not entirely
rule out the possibility of two co-propagating downstream edge modes of
electrical conductance of 1/3 $e^{2}/h$ per edge mode[54]. Measuring the
electrical or thermal conductance quantization is essential to determine its
edge structure. For example, non-equilibrated and equilibrated charge
transport for 2/3 FQH state with counter-propagating edge modes should
correspond to the two terminal electrical conductances of 4/3 $e^{2}/h$ and
2/3 $e^{2}/h$, respectively. However, experimentally, such a crossover of
electrical conductance between two asymptotic limits was never observed for
intrinsic 2/3 bulk filling. This may correspond to either the charge
equilibration length being very short in conventional devices and limiting
such crossover or the 2/3 edge structure comprising two downstream edge modes
of electrical conductance 1/3 $e^{2}/h$ each. This dilemma was answered in the
temperature-dependent study of quantized thermal conductance for 1/3, 2/5,
2/3, and 3/5 FQH states in a single-layer graphene device[94], which is
discussed now. Fig. 11(a) shows the gate response of the extremely clean hBN
encapsulated graphite gated single-layer graphene device at B=10 T. Robust
fractional plateaus at $\nu=\frac{1}{3}\frac{e^{2}}{h}$,
$\frac{2}{5}\frac{e^{2}}{h}$, $\frac{3}{5}\frac{e^{2}}{h}$, and $\sim$
$\frac{2}{3}\frac{e^{2}}{h}$ clearly visible along with the integer QH plateau
at $\nu=1$ (black curve). These QH plateaus are accompanied by the vanishing
longitudinal resistance (red curve), establishing the robustness of the FQH
states. The thermal conductance for the fractional states for $\nu=$ $1/3$
(red), $2/5$ (blue), $2/3$ (magenta), and $3/5$ (black) is plotted as a
function of the bath temperature in Fig. 11(b). As evident from this plot, for
$\nu=$ $1/3$ (red) and $2/5$ (blue) (no counter-propagating edge modes) the
values $G_{Q}$ ($1\kappa_{0}T$ and $2\kappa_{0}T$, respectively) remain
independent of the bath temperature. On the other hand, hole-conjugate states
showed surprising results. $G_{Q}$ for 2/3 and 3/5 (hole-like states with CP
modes) was found to be $2\kappa_{0}T$ and $3\kappa_{0}T$ respectively, at 20
mK of bath temperature and match well with non-equilibrated regimes
$(N_{d}+N_{u})\kappa_{0}T$. Surprisingly, as the temperature increases,
$G_{Q}$ starts decreasing, eventually taking the value of the equilibrated
regime $(N_{d}-N_{u})\kappa_{0}T$. In contrast to these hole-like fillings,
$G_{Q}$ at 1/3 and 2/5 fillings (without CP modes) remains robustly quantized
at $N_{d}\kappa_{0}T$ independent of temperature.

Figure 11: (a) QH response of the device. Robust fractional plateaus at
$\nu=\frac{1}{3}\frac{e^{2}}{h}$, $\frac{2}{5}\frac{e^{2}}{h}$,
$\frac{3}{5}\frac{e^{2}}{h}$, and $\sim$ $\frac{2}{3}\frac{e^{2}}{h}$ clearly
visible along with the integer QH plateau at $\nu=1$ (black curve). The
vanishing longitudinal resistance (red curve) accompanies the QH plateaus at
these fillings. (b)Thermal conductance $G_{Q}$, plotted as a function of the
bath temperature for $\nu=$ 1/3 (red), 2/5 (blue), 3/5 (black), and 2/3
(magenta). The horizontal dashed lines correspond to quantized $G_{Q}$ values.
The solid curves (black and magenta) are theoretical fits of the data to
extract out temperature scaling exponents of thermal equilibration length. (c)
Edge structures of the studied FQH states. Solid black and dashed red arrows
represent downstream and upstream modes, respectively. The two right-most
columns show expected values of the thermal conductance $G_{Q}$ (in units of
$\kappa_{0}T$) in the two limiting regimes of the heat transport. The figure
is adapted from Srivastav et al. (2022) [94]. Reprinted with permission from
Springer Nature.

These results can be understood from the expected edge structures and their
corresponding thermal conductance values for the studied FQH states in Fig.
11(c). For the electron-like $1/3$ and $2/5$ states, there are only downstream
modes with $N_{d}=1$ and $2$, respectively, and thus, the expected $G_{Q}$
should be $1\kappa_{0}T$ and $2\kappa_{0}T$, respectively, and should remain
independent of the temperature. This is seen in the experiment as evident from
Fig. 11(b). This behaviour is analogous to integer QH states, where all edge
modes propagate downstream. In contrast, for the hole-like $3/5$ state, the
temperature dependence crossover of $G_{Q}$ from one quantum value to another
rules out the possibility of having only downstream modes. Furthermore, the
measured values of $3\kappa_{0}T$ and $1\kappa_{0}T$, respectively, perfectly
match with the non-equilibrated $((N_{d}+N_{u})\kappa_{0}T)$ and equilibrated
$(|N_{d}-N_{u}|\kappa_{0}T)$ regimes of $G_{Q}$ with $N_{d}=1$ and $N_{u}=2$.
Similarly, for $2/3$, experimental observation rules out the theoretical model
with only downstream modes and supports the crossover from the non-
equilibrated regime of $G_{Q}$ to the equilibrated regime with
$N_{d}=N_{u}=1$. The equilibrated transport in this situation is diffusive, so
$G_{Q}$ is expected to tend to zero relatively slowly (as $\sim 1/L$) in the
long-length limit. Since the device channel length $L$ used in experiment[94]
is limited to $\sim 5$ $\mu m$, it is presumably the reason for a finite value
of $\sim 0.5\kappa_{0}T$ at $T_{bath}\sim 60$ mK. Thus, measuring the
quantized values of $G_{Q}$ in two regimes determines the exact edge quantum
number and, hence, the topological order of the FQH states. Such a systematic
crossover study of $G_{Q}$ between its asymptotic regime of heat equilibration
opens a new route for finding the topological order of exotic even-denominator
FQH states, like 5/2, 3/2, and so on. Recently, G. L. Bretona et al. [93] also
reported a transition from a non-equilibrated heat regime to an equilibrated
heat regime of $G_{Q}$ for 8/3 states in a graphene device. This crossover was
achieved in a different cooldown of the same sample.

###  7.1 Absence of edge reconstruction

In quantum Hall physics, the issue of edge reconstruction may raise concerns
over the validity of the bulk-edge correspondence principle. Specifically, it
has been proposed that edge reconstruction might occur in various QH
states[95, 96]. Here, it should be emphasized that the experimental crossover
of thermal conductance for hole-conjugate FQH states in two regimes and the
temperature independence of thermal conductance for particle-like FQH states
also suggest the absence of edge reconstruction in graphene devices. For
example, for the $\nu=1/3$ state, the edge reconstruction would increase the
total number of modes from 1 to 3. This would lead to a crossover from
$G_{Q}=3\kappa_{0}T$ at lower temperatures (non-equilibrated regime) to
$G_{Q}=1\kappa_{0}T$ at higher temperatures (equilibrated regime), which is
very different from what was observed ($G_{Q}=1\kappa_{0}T$ at all
temperatures). Similarly, for the $\nu=2/3$ state, edge reconstruction would
increase the number of modes from 2 to 4. This would mean that, at low
temperatures (the non-equilibrated regime), the thermal conductance value will
be $G_{Q}=4\kappa_{0}T$. By the same token, for the $\nu=3/5$ edge (proposed
to have $3$ edge channels), non-equilibrated heat transport would give rise to
$G_{Q}=5\kappa_{0}T$. No traces of these values were observed in our
measurements. These results strongly suggest the absence of edge
reconstruction in the measured graphene devices.

##  8 Conclusions

The quantized thermal conductance measurement is a very powerful technique for
determining the topological order of the complex fractional QH phases. The
experimental verification of the universality of quantized thermal conductance
in graphene is an exciting development. The results covered in this review are
a remarkable manifestation of an interplay of equilibration and topology in
FQH transport. The charge transport in QH in graphene is always in the
equilibrated regime in the existing devices to date, the heat transport shows
a crossover between the non-equilibrated to the equilibrated heat transport
regime. Both asymptotic limits of the thermal equilibration are encoded by
topologically quantized heat conductances determined by the topological edge
quantum numbers. The experimental reports discussed in this review should also
be relevant to other FQH states realized in different host materials.
Particularly, the temperature dependence crossover of quantized thermal
conductance in two asymptotic regimes of the thermal equilibration can be used
to settle the debate on the ground state of non-Abelian $\nu=5/2$ FQH states.
Till now, the interpretation of the measured thermal conductance
$\frac{5}{2}\kappa_{0}T$ at $\nu=5/2$ is based on the assumptions about the
presence, absence, or partial character of thermal equilibration [97, 98, 99,
100]. In particular, the technique discussed in this review can also probe the
thermal conductance of various quantum anomalous Hall phases observed recently
in twisted bilayer graphene and twisted MoTe_2 systems.

##  9 Acknowledgement

S. K. S. acknowledges the financial support from the Prime Minister Research
Fellowship (PMRF) by the Ministry of Education, Govt. of India during the PhD
tenure. S. K. S. further acknowledges the financial support from the Israel
Academy of Sciences and Humanities (IASH) and Council for Higher Education
(CHE) Excellence Fellowship. S. K. S. also thanks the Feinberg Graduate
School, Weizmann Institute of Science for the Dean of Faculty Fellowship and
Weizmann Postdoctoral Excellence Fellowship. A.D. thanks the Department of
Science and Technology (DST) and Science and Engineering Research Board
(SERB), India, for financial support (SP/SERB-22-0387) and acknowledges the
Swarnajayanti Fellowship of the DST/SJF/PSA-03/2018-19. A.D. also thanks
CEFIPRA project SP/IFCP-22-0005.

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