# A unified non-interacting-electron theory of the integer
and fractional quantum Hall conductance

Eugenio DelRe  Dipartimento di Fisica, Università di Roma “La Sapienza”, 00185
Rome, Italy  ISC-CNR, Università di Roma “La Sapienza”, 00185 Rome, Italy
Paolo Di Porto  Dipartimento di Fisica, Università di Roma “La Sapienza”,
00185 Rome, Italy

###### Abstract

We generalize the Laughlin argument for the integer quantum Hall effect of
non-interacting two-dimensional electrons by taking into account the full
three-dimensional nature of gauge symmetry. This naturally leads to the
prediction of the principal experimental results on the quantum Hall effect,
including the integer and fractional values and plateau widths of quantized
Hall conductance. The approach includes, in a fundamental way, the influence
of spin, allowing the description of both single electrons and Cooper pairs.
Not requiring the introduction of strongly correlated multi-particle states
nor a key role for disorder, the unified analysis sheds light on the profound
connection between dimensionality, symmetry, and quantum behavior.

###### pacs:

Valid PACS appear here

††preprint: APS/123-QED

At low temperatures, electrons bound to a plane and subject to a magnetic
field normal to it manifest the so-called quantum Hall effect Klitzing1980 ;
Tsui1982 ; Klitzing1985 ; Prange1990 ; Chakraborty1995 ; DasSarma1997 ;
Stormer1999 ; Jacak2003 . The Hall conductance shows strikingly precise
plateaus versus magnetic field intensity at the values $G_{xy}=I/V=\nu
e^{2}/h$, where $I$ is the current intensity along the plane, $V$ is the
voltage difference measured across the plane in the direction normal to the
current itself, and $\nu$ is the so-called integer or fractional filling
factor. While the integer effect, where $\nu=n$ is an integer, can be
interpreted using a general single-electron argument based on gauge invariance
in a two-dimensional system Laughlin1981 , fractional values of $\nu$ are
presently believed to be connected to multi-electron strongly-correlated
states Laughlin1983 ; Laughlin1983_1 ; Laughlin1998 ; Jain2007 ; Feldman2021 .
The situation is further complicated by the fact that even-denominator values
of $\nu$, observed in very cold samples, do not easily fit into the strongly
correlated electron picture Willett1987 ; Jiang1989 ; Kang1993 ; Pan1999 ;
Pan2002 ; Martin2004 ; Lin2014 .

In what follows, we deduce both integer and fractional values of the Hall
conductance filling factor from a generalized version of the single-electron
argument formulated by Laughlin. The key idea is to consider both electron
spin and the three-dimensional nature of the underlying gauge symmetry. The
model predicts, down to the very fine details, the observed behavior of the
Hall resistance $R_{xy}=1/G_{xy}$, this both for the odd denominator values of
$\nu$ in cold samples, and for the even denominator values observed in ultra-
cold conditions. These details include the presence and width of the
conductance plateaus, features that are presently considered to be
fundamentally connected to disorder.

We begin by recalling Laughlin’s single-electron argument to explain the
observed integer Hall effect. The key idea is to represent the two-dimensional
metal as a ribbon of length $\ell$ bent into a loop, with the magnetic field
$\mathbf{H}_{0}$ normal to the ribbon, the current $I$ flowing along the
ribbon, and the Hall potential $V$ measured between its edges (see Fig.1A).
Taking as a candidate vector potential $\mathbf{A}=-H_{0}y\mathbf{u}_{x}$, the
integer Hall effect then descends from the added constraint $\Delta
A=nhc/e\ell$ on the allowed values of the gauge transformation
$\mathbf{A}\rightarrow\mathbf{A}+\Delta A\mathbf{u}_{x}$ that ensures the
single-valued nature of extended wavefunctions around the loop itself. This
leads to a condition on the magnetic flux $\phi$ through the loop, that can
change only by integer multiples of the flux quantum $\Delta\phi=hc/e$. The
result is the quantization of the Hall conductance $G_{xy}=I/V=ne^{2}/h$,
where the integer $n$ refers to the nth-Landau level Landau1930 .

We next note that the universality in the experimental results, where only
fundamental constants appear, does not solely suggest the involvement of a
general principle, as Laughlin put it, but also that any explanatory argument
should not be connected to specific geometric or circumstantial constraints.
More precisely, the closure condition on $A$ around the loop must be valid for
any direction of the current $I$ on the plane normal to $\mathbf{H}_{0}$, for
any specific value of $H_{0}$, $I$, and $\ell$, and even for any direction of
$\mathbf{H}_{0}$ (Fig.1B). In short, the condition in the ribbon loop
representation actually refers to a property of the entire set of all possible
loop orientations, amounting to a statement of the fact that the extended
electron spatial wavefunction must be single-valued under a full three-
dimensional rotation. In these terms, the closure condition is, formally,
equivalent to the quantization of the angular momentum of a point particle in
3D. Performing then a Hall conductance measurement, where the orientations of
$\mathbf{H}_{0}$ and $I$ are actually fixed, implies, according to quantum
mechanics, the projection of the wavefunction onto an a-priori unknown
eigenstate of $\mathbf{J}^{2}$, $J_{z}$, where
$\mathbf{J}=\mathbf{L}+\mathbf{S}$ is the sum of orbital angular momentum and
spin (see Fig.1C).

Figure 1: Generalized Laughlin argument. (a) Laughlin argument, the plane of
the Hall experiment loop is identified by the unit vector $\mathbf{u}_{m}$
parallel to the direction of the electromotive Hall force
$\mathbf{F}=(q/c)\mathbf{v}\wedge\mathbf{H}_{0}$, where $q=-e$ is the electron
charge and $\mathbf{v}$ its drift velocity. (b) Quantum-mechanically possible
loops associated to different directions of $q\mathbf{v}$ and $\mathbf{H}_{0}$
and corresponding $S^{2}$ manifold for $\mathbf{u}_{m}$. (c) Act of
measurement on a specific Hall experiment loop orientation $\mathbf{u}_{z}$.
(d) Altered flux quantum
$\Delta\phi(\mathbf{u}_{z})=\Delta\phi/\cos{\theta}=(J/J_{z})(hc/e)$ for
$\mathbf{u}_{m}\neq\mathbf{u}_{z}$.

More precisely, performing an experiment that corresponds to projecting an a
priori unknown $\mathbf{J}$ in a specific direction identified by the
associated orthogonal loop, will select one out of all possible eigenstates of
$J_{z}$, from the maximum value $J$ to the minimum value, i.e., $(1/2)$ (in
units of $\hbar$) for single electrons (negative values of $J_{z}$ are
equivalent to the positive ones for obvious symmetry reasons). Consequently,
limiting our analysis solely to the actual plane of the experiment and
relative direction of the current, as done in the Laughlin argument (see
Fig.1A), amounts to considering only cases in which $J_{z}$ is maximum, i.e.,
$J_{z}=J$. The resulting $n$-th Landau level then corresponds to the integer
Hall effect $I/V=ne^{2}/h$. For a generic value of $J_{z}$, we note that the
magnetic flux quantum $\Delta\phi=hc/e$ refers to the loop orthogonal to the
specific direction in which the act of measurement has projected $\mathbf{J}$.
This is a necessary consequence of the single-valued nature of extended
wavefunctions on every possible plane in space. Therefore, the apparent flux
quantum through the actual loop that lies on the plane normal to
$\mathbf{u}_{z}$ (orthogonal to $\mathbf{H}_{0}$) is enhanced with respect to
$hc/e$, which refers to the tilted loop, i.e.,
$\Delta\phi(\mathbf{u}_{z})=(J/J_{z})\Delta\phi$ (see Fig.1D). The result is

Figure 2: Spectrum sequence. (a) Illustration of the number sequence of Eq.(3)
relative to electrons for $n_{max}=20$ and $p_{max}=20$. Zoom into (b) the gap
at $\nu_{g}=1/2$ and (c) the gap-center state at $\nu_{gc}=1$. Illustration
(d) of the approximate self-similarity as expressed by
$\nu^{\prime}=\nu/(2l+1)$ for $l=1$ in the spectral region $0<\nu<1$ (top
panel) and $0<\nu^{\prime}<1/3$ (bottom panel). (e) Illustration of the number
sequence of Eq.(4) relative to bosons (Cooper pairs) for $n_{max}=20$ and
$p_{max}=50$ and (f) rescaled histogram showing finer details.

$\frac{I}{V}=\frac{ne^{2}}{h}\frac{J_{z}}{J},$ (1)

that is, setting $\nu=nJ_{z}/J$,

$\frac{I}{V}=\frac{\nu e^{2}}{h}.$ (2)

Equation (2) is the central result of this paper.

For an electron, with its half-integer spin, Eq.(2) implies

$\nu=\frac{nJ_{z}}{J}=n\frac{q+1/2}{p+1/2},$ (3)

with $q$ and $p$ integers ($q\leq p$). Hence, in conditions typical of a
quantum Hall effect experiment, Eq.(3) with $J=1/2$ leads to $J_{z}=1/2$ and
$\nu=n$; for $J=3/2$, $J_{z}=3/2$ leads to $\nu=n$, while $J_{z}=1/2$ leads to
$\nu=(1/3)n$; for $J=5/2$, $J_{z}=5/2$ gives $\nu=n$, while $J_{z}=3/2$ gives
$\nu=(3/5)n$ and $J_{z}=1/2$ gives $\nu=(1/5)n$, and so on.

For an integer spin system, such as a Cooper pair, Eq.(2) implies

$\nu=\frac{nJ_{z}}{J}=n\frac{q}{p},$ (4)

with $q$ and $p$ integers (and again, $q\leq p$). Using Eq.(4) for $J=1$ we
have $J_{z}=1$ and $\nu=n$; for $J=2$, $J_{z}=2$ leads to $\nu=n$, while
$J_{z}=1$ leads to $\nu=(1/2)n$; for $J=3$, $J_{z}=3$ leads to $\nu=n$,
$J_{z}=2$ gives $\nu=(2/3)n$, while $J_{z}=1$ gives $\nu=(1/3)n$; for $J=4$,
$J_{z}=4$ leads to $\nu=n$, $J_{z}=3$ gives $\nu=(3/4)n$, $J_{z}=2$ gives
$\nu=(1/2)n$, while $J_{z}=1$ leads to $\nu=(1/4)n$, and so on.

Figure 3: Comparing to experiments. (a) Prediction based on Eq.(3) (black
curve in top panel) pitted against sample Hall resistance measurements taken
from Ref.Willett1987  (blue curve). Gap-center states $\nu_{gc}$ and gaps
$\nu_{g}$ are referenced back to the appropriate spectrum ($p_{max}=10$),
reported in the bottom panel, dark-shaded regions referring to gap-center
plateaus, light-shaded regions referring to gaps. For clarity, the
illustration is halted for high values of $\nu>2$. The inset in the top panel
underlines how the observed plateau at $\nu_{gc}=3/7$ emerges only for
$p_{max}\geq$ 18 (see text). (b) Prediction based on Eq.(4) pitted against
measurements from Ref.Paalanen1982  (top panel, black and blue curves,
respectively). The absence of gaps signals the fundamental role of Cooper
pairs (see text). Note the hysteresis and its relationship to the underlying
Cooper-pair spectrum (central panel) and governing electron spectrum (bottom
panel) (see text). Again, for the sake of clarity, values of $\nu>8$ are not
illustrated.

We begin our analysis of Eq.(3), that refers to non-interacting electron
states, by reporting in Fig.2A an example of a sequence of $\nu$ with
$n=1,...,n_{max}$ and $p=0,...,p_{max}$. A first important feature is the
appearance of recurrent gaps in the spectrum, that is, regions for which no
quantum Hall states form, this for an arbitrary $n_{max}$ and $p_{max}$. The
paradigmatic example at $\nu=1/2$ is reported in Fig.2B. The gaps are the
result of a peculiar property of Eq.(3), that is, that
$\nu\neq\nu_{g}\equiv(2s+1)/(2r)$ with $s$ and $r$ integers. This is because
$\nu=(2s+1)/(2r)$ would imply, through Eq.(3), $2nr(2q+1)=(2s+1)(2p+1)$, that
is manifestly impossible. Hence, for example, no Hall states and plateaus will
be observed in and around $\nu_{g}=1/2$ ($s=0$, $r=1$),$\nu_{g}=1/4$ ($s=0$,
$r=2$), $\nu_{g}=3/4$ ($s=1$, $r=2$). The width of the gaps depends
specifically on $p_{max}$. To illustrate this, consider that Eq.(3) implies
that the spectrum is explored in steps of $1/(2p+1)$, with $p\leq p_{max}$.
Since the spectrum necessarily includes the integer values $\nu=m$ as long as
$n_{max}\geq m$, as follows from the fact that $n(2q+1)=m(2p+1)$ is always
satisfied for $n=m$, each of these sequences will employ an odd number of
steps to go from $\nu=m-1$ to $\nu=m$ ($m\geq 1$). It follows that half-
integer values of the $\nu_{g}$ set, such as 1/2, 3/2, 5/2, and so on, lie
exactly at a half-point of the minimum step $\Delta\nu=1/(2p_{max}+1)$,
meaning that the gap itself is of width $\Delta\nu$. Congruently, in the
example of Fig.2B, $p_{max}=20$ and the gap at $\nu_{g}=1/2$ is
$\Delta\nu=1/(2p_{max}+1)=1/41\simeq 0.024$ wide.

A second feature is the recurrent appearance of a quite remarkable spectral
structure: a gap with a single isolated gap-center state, the isolated state
positioned at the center of its gap. Calling these states $\nu_{gc}$, one
example is reported in Fig.2C, at $\nu_{gc}=1$. Also these gaps, that form
symmetrically around $\nu_{gc}$, depend on $p_{max}$. To illustrate this,
consider once again the integer states $\nu=m$. These will be part of all
sequences explored through steps of $1/(2p+1)$, with $p\leq p_{max}$. Since
the minimum step is $\Delta\nu=1/(2p_{max}+1)$, it follows that for a
sufficient $n_{max}$, the state at $\nu_{gc}$ will be at the center of a
$2\Delta\nu$ gap, as found in the example reported in Fig.2C.

A third fundamental property of the spectrum described by Eq.(3) is self-
similarity. In fact, for any $\nu$ in the spectrum there will always be a
$\nu^{\prime}=\nu/(2l+1)$ for $l$ integer, as long as the condition
$(2p^{\prime}+1)=(2p+1)(2l+1)$ can be satisfied. The latter condition implies
that self-similarity is ultimately limited by $p_{max}$. An example of this is
reported in Fig.2D for $n_{max}=20$, $p_{max}=20$, and $l=1$, for the interval
$0<\nu<1$. Evidently, while the overall features follow a self-similar
behavior, the details, such as the widths of the gaps and gap-center states
will not be exactly self-similar. To oberve the full mathematical realization
of the self-similarity would require a larger $p^{\prime}_{max}$ (i.e.,
$(2p^{\prime}_{max}+1)=(2p_{max}+1)(2l+1)$).

Self-similarity allows us to provide a general statement on the expected
values of $\nu_{g}$ and $\nu_{gc}$. For example, $\nu_{g}=1/2$ implies the set
of $\nu_{g}^{\prime}=(1/2)/(2l+1)$, this including 1/6, 1/10, and so on (the
gaps at 1/6 and 1/10 are visible in Fig.2D). The $\nu_{g}=1/4$ implies
$\nu_{g}^{\prime}=(1/4)/(2l+1)$, this including 1/12, 1/20, and so on. For the
all-important gap-center states, $\nu_{gc}=1$ leads to the self-similar gap-
center states at 1/3, 1/5, and so on, $\nu_{gc}=2$ leads to 2/3, 2/5, and so
on, $\nu_{gc}=3$ leads to 3/5, 3/7, and so on, $\nu_{gc}=4$ leads to 4/3, 4/5,
4/7, 4/9, and so on, $\nu_{gc}=5$ leads to 5/3, 5/7, 5/9, and so on,
$\nu_{gc}=6$ leads to 6/5, 6/7, and so on, and $\nu_{gc}=7$ leads to 7/3, 7/5,
and so on.

For the spectrum available to Cooper pairs described by Eq.(4), in Fig.2E,F we
report the spectrum $2\leq\nu\leq 3$ for $n_{max}=20$, $p_{max}=50$, where the
principal gap-center states are 15/7, 9/4, 16/7, 7/3, 12/5, 5/2, 18/7, 8/3,
14/5, 20/7 (see Fig.2F), while no gaps like the ones in Fig.2B are present, in
agreement with the fact that in Eq.(4) $p$ and $q$ have no definite parity.
Self-similarity holds for $\nu^{\prime}=\nu/l$, with $l$ integer, as long as
the condition $n^{\prime}q^{\prime}/p^{\prime}=nq/pl$ is satisfied.

For electrons, Eq.(3) naturally contains the similarity between fractional
states and those of the integer effect for an appropriately shifted magnetic
field $H_{0}^{*}=H_{0}-H_{0}^{\prime}$ Lin2014 . Indeed, since
$G_{xy}=C/H_{0}$, with $C=\rho c$, $\rho$ being the sample charge density
Laughlin1998 , $\nu=C/H_{0}$ and $\nu^{*}=C/(H_{0}-H_{0}^{\prime})$. Imposing
$\nu^{*}=n$ with $\nu=n(2q+1)/(2p+1)$ (Eq.(3)) requires that
$H^{\prime}_{0}/C=(2p+1)/(n(2q+1))-1/n$, an equality that holds only if $q=0$
and $p=nr$, where $r$ is an integer. Consequently, the correspondence holds
for $H^{\prime}_{0}=2rC$, $\nu=n/(2nr+1)$, and $\nu^{*}=n$. In particular, for
$r=1$, we have the observed $H^{\prime}_{0}=H(1/2)=2C$, $\nu=n/(2n+1)$, and
$\nu^{*}=n$, while for $r=2$ we have $H^{\prime}_{0}=4C$, $\nu=n/(4n+1)$, and
$\nu^{*}=n$. An analogous similarity occurs for the mirror reflected
$\nu^{*}=-n$, in which case each state of spectrum $-n$ is superimposed with
the state $n/(2nr-1)$, a similarity typically reported for $r=1$ Lin2014 . The
similarity does not, in turn, hold for Cooper pairs described by Eq.(4), since
an equivalent superposition of the integer states with fractional ones would
imply the condition $1/n=p/nq-H_{0}^{\prime}/C$, that holds for any $n$ only
if $p=q$, i.e., for the integer states themselves.

In terms of the spectrum, each of the possible states associated to an allowed
value of $J_{z}$ leads to an equivalent Landau-level harmonic oscillator
spectrum $E(n,J_{z}/J)=\hbar(J_{z}/J)\omega_{c}(n+1/2)$, where
$\omega_{c}=eH_{0}/mc$ and $m$ is the effective mass of the electrons. It
follows that as $H_{0}$ is increased, different extended states with ever
lower values of $n$ and $(J_{z}/J)$ reach the electron Fermi level $E_{F}$.
The structure of the spectrum, with its characteristic gap-center states and
gaps (Fig.2), emerges for a finite value of $p_{max}$, which is fixed by the
maximum available $J_{z}$, $J^{max}_{z}=p_{max}+1/2$ for electrons (Eq.(3)),
and $J^{max}_{z}=p_{max}$ for Cooper-pairs (Eq.(4)). The $J^{max}_{z}$ is, in
turn, fixed by the uncertainty principle for a given sample, since
$(1/2)\langle J^{max}_{z}\rangle\leq\sigma_{J_{x}}\sigma_{J_{y}}$ Cohen-
Tannoudji1977 , where the maximum value of the uncertainties $\sigma_{J_{x}}$
and $\sigma_{J_{y}}$ in the in-plane components of $\mathbf{J}$ is determined
by how strong the electrons and pairs are confined to the metallic plane. For
a given set of explored values $H_{0}\leq H_{0}^{max}$, the actual features of
the spectrum that are observed depend on the spread of the electron
distribution around $E_{F}$, i.e., on the electron temperature $T$. As $T$ is
lowered, the first observable effects will be associated to widest gap-center
states. As $H_{0}$ causes the gap-center state spectrum to sweep through the
electron Fermi level, the states above and below the gaps that flank the gap-
center state form the equivalent of band structures, implying that the Fermi
level automatically coincides with the gap-center state itself. It follows
that for all values of $H_{0}$ that correspond to values of $\nu$ that would
fall inside the gap-center spectrum, $R_{xy}=\nu_{gc}e^{2}/h$, a plateau whose
width $\Delta H_{0}$ then provides a direct measurement of $p_{max}$ for the
given sample. For $m$ integer, since the plateau forms from
$\nu_{-}=m-1/(2p_{max}+1)$ to $\nu_{+}=m+1/(2p_{max}+1)$, then $\Delta
H_{0}=(2C/(2p_{max}+1))(m^{2}-1/(2p_{max}+1)^{2})^{-1}$ ($p_{max}\geq 1$). In
turn, for these values of $H_{0}$, since the electrons are pinned to the
highly degenerate extended gap-center state, $R_{xx}=0$. We note that the
electron temperature at which plateaus will emerge is determined by the
condition that $k_{B}T$ be smaller than the energy spread that corresponds to
the specific plateau. For $\Delta H_{0}(\nu_{gc}=1)$, considering the energy
of the corresponding Landau levels $E(n,J_{z}/J)$, the condition becomes
$k_{B}T\ll\hbar e\Delta H_{0}(1)/(mc(2p_{max}+1))\simeq 2\hbar
eC/(mc(2p_{max}+1))$ (corresponding to a temperature of several degrees K in
typical experiments). It goes without saying that to observe the spectrum of
Eq.(4), the generally stronger constraint that $k_{B}T$ be smaller than the
Cooper pair gap must be met Abrikosov1988 . Incidentally, for values of $T$
and $H_{0}^{max}$ that allow the observation of the $\nu=\nu_{gc}=1$ plateau
(at $R_{xy}=e^{2}/h$), the value of $C$, that depends on the density of charge
carriers of the specific sample, can be experimentally determined from the
value of magnetic field $H_{0}(1)$ at which the $\nu_{gc}=1$ plateau begins to
develop as $T$ is lowered. For a given $p_{max}$ and a sufficiently low $T$,
universal curves will emerge plotting $R_{xy}$ versus $h_{0}=H_{0}/H_{0}(1)$
($\nu=1/h_{0}$). The situation at the gaps at $\nu_{g}$ is fundamentally
different. Here no extended state is present in the spectrum at the Fermi
level, so that the material behaves like a standard metal, $R_{xy}$ increasing
in proportion to $H_{0}$, and $R_{xx}$ being approximately constant, apart
from the inevitable changes associated to standard magnetoresistance. In turn,
for temperatures low enough as to allow Cooper pairs to form, Eq.(4) implies
that a gap-center state right at the center of the single-electron gap forms.
Since the Cooper pair gap is pinned to form below the single-electron Fermi
level Abrikosov1988 , as long as the superconducting gap is much smaller than
the Hall spectrum gap of Eq.(4), the Cooper pair superconducting state will
coincide with the gap-center state for the whole development of the gap,
implying that also in this case we expect a plateau.

In Fig.3 we analyze specific experimental results that have been amply
presented in literature. We begin in Fig.3A with the data of Fig. 1 in
Ref.Willett1987 . Direct inspection indicates that the $R_{xy}$ predicted on
the basis of the spectrum described by Eq.(3) with $p_{max}=10$ is able to
reproduce the principal experimental findings. The spectrum, reported in the
bottom panel of Fig.3A in the form of the normalized state degeneracy versus
$1/\nu$, manifests a sequence of gap-center states (blue shade) and gaps
(yellow shade). As shown in the top panel of Fig.3A, the corresponding
predicted and detected behavior are found to be in agreement. Increasing the
value of $p_{max}$ does not alter the picture in any fundamental way, aside
from reducing progressively the actual width of the plateaus and causing the
full observed spectrum, that includes the $\nu_{gc}=3/7$ state, to emerge. An
illustration of this can be found in the inset of Fig.3A for $R_{xy}$ with
$p_{max}=18$. An analogous situation holds for $\nu_{gc}=2/5$, that manifests
rigorously a gap-center state only for $p_{max}\geq 18$, while for the
$p_{max}=10$, this plateau is only a very good approximation (the gap forms
from $\nu_{-}=5/13$ to $\nu_{+}=7/17$, so that the
$(\nu_{-}+\nu_{+})/2=88/221$ that differs from $\nu_{gc}=2/5$ by 0.002). The
implication of this is two-fold: first, $p_{max}$ may not be constant
throughout the $H_{0}$ scan, a fact that would then question the idea that
$p_{max}$ is principally connected to the specific fabrication features of the
sample and its ability to confine the electrons to the plane. Second, that
when we assume that an extended state is populated, we cannot, in all cases,
wholly neglect the role of thermal agitation, that will cause the Fermi level
to be affected, even though very weakly, by details of the degeneracy of the
spectrum.

While the analysis detailed in Fig.3A can be repeated for a large part of
experimental scans reported in literature, some very well-established results
cannot be accomodated into the electron picture of Eq.(3). For example, in
Fig.3B we report the data from Fig.1 of Ref.Paalanen1982 . It is immediately
apparent that the behavior of $R_{xy}$ does not include segments corresponding
to the classical Hall effect, i.e., for which $R_{xy}=1/\nu$ (apart from the
asymptotic $\nu\rightarrow 0$ regime), as in turn occurs for the data analyzed
in Fig.3A. The absence of these classical-like regimes means that the system
does not manifest the gap states $\nu_{g}$ that characterize the electron
spectrum. The data, in turn, can be analyzed in terms of Cooper-pairs obeying
Eq.(4). The predicted $R_{xy}$ on the basis of the Cooper-pair spectrum
(central panel in Fig.3B) for increasing and decreasing values of $H_{0}$
(bottom and top curves respectively), as reported in Fig.3B, is in reasonable
agreement with data. We should note that the behavior is expected, as
discussed above, in regimes of specifically low temperatures (50mK in the
experiment analyzed), and the Cooper pair gap levels follow the Fermi level of
the corresponding condensing electrons (bottom panel of Fig.3B). Since two
electrons form a Cooper pair, the correspondence between the electron spectrum
and that of the pairs, reported in the central panel of Fig.3B, is such that
each electron state at $\nu$ is associated to a Cooper pair state at
$\nu^{\prime}=\nu/4$. Hysteresis is the product of the basic difference in how
electrons and Cooper pairs participate in transport. Specifically, as $H_{0}$
is increased so that the corresponding electron level is pinned to the center
of the gap at a given $\nu_{g}$ (yellow shaded regions), the fact that no
extended electron Hall state exists implies that no corresponding Cooper-pair
state exists, so that the populated boson state that participates in transport
refers to the higher $\nu_{+}$ boundary state (or lower $1/\nu_{+}$ state in
the plots). If, on the other hand, $H_{0}$ is decreased to the electron
$\nu_{g}$ gap, the corresponding populated boson state is the lower $\nu_{-}$
boundary state (i.e., the corresponding higher $1/\nu_{-}$ state in the
plots). The correspondence discussed in Fig.3B is for an electron spectrum
with $p_{max}=1$ and a Cooper pair spectrum with $p^{\prime}_{max}=6$
($2(2p_{max}+1)=p^{\prime}_{max}$). We note that while hysteresis is now known
to occur in specific low-temperature $H_{0}$ scans, original papers did not
explicitly mention this behavior Zhu2000 . One might speculate that Cooper
pairs could condense from a virtual electron state, even though this state
does not exist, a mechanism that could lead to the absence of hysteresis and
the formation of extra plateaus.

We show how the physical underpinnings of quantum Hall effects lie more with
quantum-mechanically projecting 3D gauge symmetry in 2D than with many-body
correlations. Apart from allowing the understanding of a large variety of
effects and results that are not all explained in the particle-interaction
picture, the shift brings the entire field back into the folds of how symmetry
and topology affect quantum behavior Dirac1931 ; Berry1984 . Apart from
clarifying the fundamental relationship between fractional and integer states
Jain1990 , findings provide a more profound assessment of how 2D physics can
play a role in applicative scenarios, such as the emerging field of
topological quantum computing Jacak2003 ; Ma2019 . This said, our analysis is
here focused on the conundrum of Hall effects in cold and ultra-cold
semiconductors. In different 2D systems, such as graphene, behavior will also
depend on the specific charge carrier physics Zhang2005 ; Novoselov2006 .

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