# Spin and valley degrees of freedom in a bilayer graphene quantum point
contact: Zeeman splitting and interaction effects

Vanessa Gall  Institute for Quantum Materials and Technologies, Karlsruhe
Institute of Technology, 76021 Karlsruhe, Germany Institut für Theorie der
Kondensierten Materie, Karlsruhe Institute of Technology, 76128 Karlsruhe,
Germany Rainer Kraft  Physikalisches Institut, Karlsruhe Institute of
Technology, 76131 Karlsruhe, Germany  Igor V. Gornyi  Institute for Quantum
Materials and Technologies, Karlsruhe Institute of Technology, 76021
Karlsruhe, Germany Institut für Theorie der Kondensierten Materie, Karlsruhe
Institute of Technology, 76128 Karlsruhe, Germany Ioffe Institute, 194021 St.
Petersburg, Russia  Romain Danneau  Institute for Quantum Materials and
Technologies, Karlsruhe Institute of Technology, 76021 Karlsruhe, Germany

###### Abstract

We present a study on the lifting of degeneracy of the size-quantized energy
levels in an electrostatically defined quantum point contact in bilayer
graphene by the application of in-plane magnetic fields. We observe a Zeeman
spin splitting of the first three subbands, characterized by effective Landé
$g$-factors that are enhanced by confinement and interactions. In the gate-
voltage dependence of the conductance, a shoulder-like feature below the
lowest subband appears, which we identify as a $0.7$ anomaly stemming from the
interaction-induced lifting of the band degeneracy. We employ a
phenomenological model of the $0.7$ anomaly to the gate-defined channel in
bilayer graphene subject to in-plane magnetic field. Based on the qualitative
theoretical predictions for the conductance evolution with increasing magnetic
field, we conclude that the assumption of an effective spontaneous spin
splitting is capable of describing our observations, while the valley degree
of freedom remains degenerate.

##  I Introduction

Exploiting the quantum degrees of freedom of charge carriers offers a
potential route for designing new types of quantum electronic devices. While
most studied systems involve the electron’s spin degree of freedom aiming at
spintronic applications [1, 2], more recently the additional valley isospin in
a variety of materials has attracted a growing interest for use in
valleytronics [3]. However, irrespective of the system of choice, the
implementation of spin- or valley-based functionalities into electronic
devices requires a full control of the quantum state itself. A quantum point
contact that confines charge carriers into one dimension [4], is one of the
basic building blocks for efficient injection, control, and read-out measures.

Recently, we have reported [5] on an electrostatically-induced quantum point
contact (QPC) in bilayer graphene (BLG) [6, 7, 8, 9, 10, 11, 12, 13, 14], i.
e., a system with four-fold spin and valley degeneracy, where the constriction
is realized by local band gap engineering with a displacement field
perpendicular to the BLG plane. We observed confinement with well-resolved
conductance quantization in steps of $4\,e^{2}/h$ down to the lowest one-
dimensional (1D) subband, as well as a peculiar valley subband splitting and
merging of $K$ and $K^{\prime}$ valleys from two non-adjacent subbands in an
out-of-plane magnetic field (see also Ref. [10]).

In the present paper, we investigate the same system in an in-plane magnetic
field. In this context, we became aware of the publication [11] that reported
on conductance measurements in a similar setup and found certain features
additional to the expected conductance quantization. These features were
attributed [11] to the substrate-induced Kane-Mele spin-orbit coupling [15]
below the lowest plateau. Since the reported values of the spin-orbit coupling
in monolayer graphene is of the order of $40\,\mu$eV [16] (corresponding to
temperatures of the order of $0.5$ K) and there is no clear mechanism that
would lead to an enhancement of spin-orbit coupling by hexagonal boron nitride
(hBN), we expect another mechanism behind such features. Here, we explore
alternative possibilities for the explanation of the appearance of additional
features in the conductance.

A very natural guess is that the lifting of degeneracy is due to interaction
effects. While renormalization-group studies show that the Coulomb interaction
in clean graphene becomes marginally irrelevant [17], BLG behaves more like a
typical two-dimensional (2D) electron gas. Non-perturbative approaches to the
effects of long-range interactions show that graphene may feature interaction-
induced instabilities. These effects are expected to be particularly important
in very clean samples, at very low densities, and in high magnetic fields.
Proposed theories include superconducting instabilities [17, 18, 19, 20],
(anti-)ferromagnetic instabilities [21, 22, 23], excitonic instabilities [24,
25, 26], and whole lot of others [27, 28, 29, 30, 31, 32, 33]. For a summary
or comparison see, e.g., Refs. [34, 35, 36, 37, 38, 39, 40].

Figure 1: (a) Schematic illustration of the setup with the QPC formed in BLG
subject to the in-plane magnetic field $B$. In experiment, the magnetic field
$B$ is oriented in plane with an angle of approximately $45^{\circ}$ with
regard to the black dashed line. (b) Cross-section of the setup along the
dashed line in left panel. The QPC is tunable by the split gates (SG), back
gate (BG), and top gate (TG).

One rather notorious phenomenon, where interaction effects show up in
transport measurements, is the appearance of an additional shoulder in the
quantized conductance of QPCs below the lowest plateau. It is commonly known
as the $0.7$ conductance anomaly, since, in systems with spin degeneracy, it
is usually observed close to the value of the conductance

$G\approx 0.7\times 2\,\frac{e^{2}}{h}.$

This feature was first recognized in a GaAs QPC in Ref. [41] and, while it is
fair to say that there is still no commonly accepted theory explaining all
aspects of the anomaly, there are several microscopic theories that are
capable of capturing some salient features of the phenomenon. These theories
invoke various distinct physical mechanisms driven by electron-electron
correlations, such as variants of the Kondo effect [42, 43, 44, 45, 46, 47],
Wigner crystallization [48, 49, 50], and other interaction-based mechanisms
[51, 52, 53, 54, 55, 56, 57]. In particular, there are studies investigating
the influence of the QPC barrier on electron-electron interaction effects
perturbatively. On a very simplistic level, considering local interaction,
only the Hartree type processes involving electrons with opposite spin
contribute, leading to an effective blocking of the channel for one spin
species for a certain amount of time and thus lowered conductance. Since
interaction effects are enhanced at low densities, such type of effects would
be strongest in the lowest quantization subband.

In this work, we study the conductance of a BLG QPC for in-plane magnetic
field orientation. We start with presenting our experimental results (Sec.
II), which were obtained in the same sample as in Ref. [5], but in another
cool down for changing the sample orientation within the magnet. In
particular, we demonstrate the importance of interaction effects in the lowest
size-quantized subbands by measuring the renormalized Landé $g$-factor
governing the Zeeman splitting of the subbands. This motivates us to employ
the picture based on the interaction-induced spontaneous polarization of spin
or valley degrees of freedom to describe the shoulder-like features in the
conductance. After a short reminder on the band structure of BLG and,
especially, the influence of external gating on the gap and the densities
(Sec. III), we discuss the conductance of the BLG QPC. In Sec. IV we detail an
extension of a phenomenological model for the $0.7$ anomaly proposed in Ref.
[58] to BLG. Within this framework, we investigate all possible scenarios in
order to find the one most likely to be present in this experiment. We do not
explicitly consider any microscopic model of the anomaly, but, instead, assume
that some sort of interaction-induced spin and/or valley splitting is present
at zero magnetic field and investigate the consequences of possible types of
splitting on the conductance in increasing magnetic field. In fact, the
assumed polarization does not need to be static, it just needs to fluctuate
slowly compared to the traveling time through the constriction, which
according to Ref. [59] is indeed fulfilled. By comparing our experimental
results with these scenarios (Sec. V), we conclude that our sample shows
spontaneous spin polarization but no valley splitting. Our findings are
summarized in Sec. VI, and technical details are described in Appendices.

##  II Experimental results

###  II.1 Fabrication and characterization

For this experiment, we have used the same BLG device as presented in Ref.
[5], see Fig. 1. The chosen gate configurations is $V_{\text{BG}}=10$ V (back-
gate voltage) and $V_{\text{SG}}=-12$ V (split-gate voltage). This setup
differs from the one used in Ref. [8] for the study of the supercurrent
confinement in BLG QPC by an addition of an overall top gate. The device
consists of a hBN-BLG-hBN heterostructure, which is edge contacted with
$\mathrm{Ti}$/$\mathrm{Al}$ electrodes. The thickness of the top and bottom
hBN layers of the sandwich are $38$ nm and $35$ nm, respectively. The sandwich
is placed onto a pre-patterned back gate, which is designed on a sapphire
substrate that is, in turn, covered by an additional layer of the dielectric
$\mathrm{Al_{2}O_{3}}$. The magnetic field was applied in the plane of the BLG
layer. The measurements were performed under the same experimental condition
as in Ref. [5], but in a different cool down, with the magnetic field oriented
in the plane of the BLG (at approximately 45° from the current direction).

The QPC in BLG is engineered electrostatically by means of the split gate
placed on top of the device and the whole sample is covered in an extra layer
of $\mathrm{Al_{2}O_{3}}$ with $30$ nm thickness before adding the overall top
gate made from $\mathrm{Ti}$/$\mathrm{Cu}$. The measurements were performed at
either $20$mK or $4$K in a $\mathrm{{}^{3}He/{}^{4}He}$ dilution refrigerator
BF-LD250 from BlueFors. A two-terminal configuration was used employing the
standard low-frequency ($\approx 13$Hz) lock-in technique, with an AC-
excitation ranging from $1$ to $20\mu$V. For further details of the
characterization of the sample the reader is referred to the Supplemental
Material in Ref. [5]. Figure 8 of the Supplemental Material in Ref. [5] also
shows the finite-bias measurements used to extract the gate-coupling
parameter.

To the best of our knowledge, there are two papers by other groups that have
investigated similar setups, namely Ref. [60] and Ref. [12]. While both these
papers also studied transport through a BLG QPC, the confinement conditions
there were different from those in our setup. This difference might be crucial
for observation interaction effects, including the 0.7 anomaly. Specifically,
in the present work, the QPC is formed by split gates of a physical width
$w\approx$ 65 nm. Because of the additional layers of $\mathrm{Al_{2}O_{3}}$,
the distances between the channel and the global back and top gate are 55 nm
and 68 nm, respectively. In Ref. [60], the physical width of the split gates
is 120 nm, while the distance to the back gate and split gate is not
specified. Since Ref. [60] did specify that the BLG is encapsulated in hBN,
the distance to the back gate and the split gate is likely of the order of 30
nm, with an additional 35 nm of $\mathrm{Al_{2}O_{3}}$ between split gates and
local top gate. Similarly, Ref. [12] stated a width of 250 nm, a distance of
25 nm to the back gate (and, probably, a similar one to the split gates), and
additionally 25 nm of $\mathrm{Al_{2}O_{3}}$ between split gates and a local
top gate. This means, that our channel is a lot narrower, confinement a lot
stronger, and, thus, the density of states way larger, which enhances all
interaction effects.

Moreover, interaction effects in Refs. [60, 12] should be more strongly
suppressed by the top and bottom gate, which are closer than the typical
distance of interacting electrons within the constriction. It is worth noting
that Ref. [61] stated that gates need to be closer than a few nanometer, to
fully suppress electron-electron interaction in graphene and BLG. At this
point, it should be mentioned that, depending on the exact shape of the
constriction, the 0.7 shoulder can appear at different conductance values (for
example, at 0.5 $e^{2}/h$ [55, 56]), which would fit with the alleged spin-
orbit gap of Ref. [12].

The global back gate that covers also parts of the leads in our device leads
to a smoother coupling in the QPC region, while also modifying the band
structure and gap in the non-QPC regions. As has been shown, for example, in
Refs. [55, 56], both the presence and shape of the 0.7 anomaly depend rather
strongly on the exact constriction profile, so that a smoother constriction
region might be necessary for its appearance. This also applies to the larger
parameter space we explore by varying our split gate and back gate not only
along the direction of zero displacement field. Lastly, we want to point out
that most of our reported results are based on the three lowest size quantized
levels, which are not even resolved in Ref. [60], while Ref. [12] does not
reach full pinch-off.

Figure 2: Measured conductance of the QPC in BLG for $V_{\text{BG}}=10$ V and
$V_{\text{SG}}=-12$ V. (a): Differentiated differential conductance as a
function of the top-gate voltage $V_{\mathrm{TG}}$ and in-plane magnetic field
$B$ for temperature $20$ mK. Plateaus of the conductance corresponds to bright
regions, steps correspond to dark region. The map scale is cut at
$0\,e^{2}/hV$ and $8\,e^{2}/hV$ to bring out the details. The black dashed
line corresponds to $B=0.2$ T, the lines of blue tones to $2,4,6$ T. The dots
of different shades of pink mark the development of the spin subbands used to
extract the Zeeman splitting and the effective Landé $g$-factors. (b) and (c):
Cubic spline fit of the differential conductance $G$ as a function of the
$V_{\mathrm{TG}}$ in elevating $B$ for $20$ mK and $4$ K, respectively. The
curves are shifted vertically with $\alpha=2{e^{2}}/{h\mathrm{T}}$ and colored
according to their first derivative. (d): Differential conductance $G$ as a
function of $V_{\mathrm{TG}}$ at $20$ mK for $B=0.2$ T (black curve) and $B=6$
T (light blue). The arrow marks the additional shoulder, which we identify as
a 0.7 anomaly. (e): Differentiated differential conductance as a function of
$B$ and conductance $G$ for $20$ mK. Plateaus of the conductance correspond to
bright regions, slopes correspond to dark regions.

###  II.2 Conductance

We start by investigating the dependence of the conductance on the magnetic
field and top-gate voltage. Figures 2(a)-(d) show the experimental data at
temperature $20$ mK. In Fig. 2(d), the conductance is shown as a function of
the top-gate voltage $V_{\mathrm{TG}}$ for two different values of in-plane
magnetic fields $B_{\parallel}$. The black curve corresponds to
$B_{\parallel}=0.2$ T and the light-blue one to $B_{\parallel}=6$ T as marked
in Fig. 2(a). The light-blue curve highlights the appearance of additional
half-step conductance plateaus in high in-plane magnetic fields. The black
curves contains a shoulder marked by the arrow, which we will attribute to the
0.7 conductance anomaly. We note that the valley degeneracy is apparently not
affected by the application of the in-plane magnetic field, and the Zeeman
spin-split subbands remain degenerate in the two valleys $K$ and $K^{\prime}$.
Since the aluminum leads are superconducting at $20$ mK, a finite magnetic
field is needed to kill this effect and curves below $0.2$ T show influence of
the superconducting leads, cf. Appendix A.

Cubic spline fits of the conductance for all measured values of magnetic field
between $0.2$ T and $6$ T are shown in Fig. 2(b) and Fig. 2(c) for
temperatures $20$ mK and $4$ K, respectively. Curves in both figures are
shifted vertically for clarity and colored according to their first
derivative. For both temperatures, there are two regions of steep incline
(orange-red) for high magnetic field, corresponding to the chemical potential
crossing through the spin-split bands. The splitting is both sharper and
higher for the lower temperature, and plateaus are flatter there as well. The
lower spin-subband stays roughly at the same value of $V_{\mathrm{TG}}$.

Figure 3: (a) and (b): Zoom-in view for the lowest plateau of the differential
conductance $G$ at $20$ mK and $4$ K, respectively, as a function of
$V_{\mathrm{TG}}$ with horizontal shifts for different values of magnetic
field, $\alpha=0.5$ V/T. (c) and (d): Differentiated differential conductance
as a function of $V_{\mathrm{TG}}$ for $20$ mK and $4$ K respectively. Curves
for different magnetic field values are shifted vertically with
$\tilde{\alpha}=4\mathrm{e^{2}/(hVT)}$. In all panels, the colored curves
correspond to the magnetic field values introduced in Fig. 2(a).

Figure 2(a) shows a grayscale map of the differentiated differential
conductance $dG/dV_{\mathrm{TG}}$ as a function of top gate voltage
$V_{\mathrm{TG}}$ and in-plane magnetic field $B_{\parallel}$ for $T=20$ mK.
Transitions across 1D subband edges appear as dark lines, while conductance
plateaus are visible as light regions in between. One clearly sees the four
well-resolved conductance plateaus. These are separated by the three regions
corresponding to the 1D subbands, which are split roughly symmetrically with
the applied in-plane field for higher bands. This corresponds to the evolution
from the spin-degenerate into spin-split energy levels. The lifting of the
spin degeneracy occurs for the lowest three subbands, where the confinement
and interactions are the strongest.

Figure 2(e) shows the same data as 2(a), but as a function of $B_{\parallel}$
and $G$. The bright horizontal lines at multiples of $4e^{2}/h$ correspond to
the spin- and valley-degenerate conductance quantization plateaus for zero
magnetic field, the additional half-integer multiples for higher magnetic
fields correspond to the spin-split plateaus due to the Zeeman effect.

###  II.3 Extra features of the conductance

Additionally, we note the presence of a shoulder-like feature below the lowest
conductance plateau at about $G=2.5\,e^{2}/h$, similar to the $0.7$ structure
described in many other materials [62], which develops into the lowest spin-
split subband at $G=2\,e^{2}/h$. This feature is well visible in the black
curves in Figs. 2(a) and 2(d). Since flatter parts of the conductance
correspond to brighter color in Figs. 2(c) and 2(f), it corresponds to a
bright region in between the zeroth and first plateau, i.e., within the darker
region to the left of $V_{\text{TG}}=-12$ V, making it look like a spin
splitting of the 1D subbands at zero magnetic field.

This additional feature is also visible in Fig. 3, which shows cadence plots
of the conductance at $20$ mK and $4$ K in Fig. 3(a) and (b), respectively,
and of the derivative of the conductance at $20$ mK and $4$ K in Fig. 3(c) and
(d), respectively. In all cases, only the lowest band is shown. The cadence
plots for a larger range of conductance variation are shown in Appendix A. The
colored curves correspond to the values of magnetic field marked in Figure
2(a). In the black curves in both Figure 3(a) and (b) the is an additional
shoulder at around $2.5e^{2}/h$, which develops into the spin split plateau
for higher magnetic fields. In the cadence plots of the conductance Figure
3(c) and (d) this shoulder corresponds to an additional peak. which clearly
develops into the spin split peak for $4$ K whereas this transition is
somewhat obscured by yet another feature at $20$ mK. We identify this
obscuring feature as part of a larger oscillation pattern discussed later.
Similar plots are shown in Ref. [45] for GaAs, where the observed behavior was
attributed to the $0.7$ structure.

The extra feature cannot be an effect caused by the finite magnetic field
needed to kill superconductivity, since it is not located on the imaginary
line extending the Zeeman splitting down to small magnetic fields. Instead, a
finite magnetic field is needed to bring this feature down to the spin-split
value. Moreover, this feature is seen already at zero magnetic field in Figs.
2(c) and 3(b) and (d) at higher temperature, where the contacts are not
superconducting. At stronger magnetic fields, $B_{\parallel}\gtrsim 4\,$T,
this feature merges with the shoulder that, at the lowest magnetic fields,
splits off the lowest main conductance quantization plateau at $G=4\,e^{2}/h$
and goes down to form a plateau slightly below $G=2\,e^{2}/h$. This behavior
is clearly observed as the evolution of the red region above
$V_{\mathrm{TG}}\approx-12$ V in Fig. 2(b). The merging of the two shoulders
is also evident in Fig. 2(a) as an intersection of the two bright regions at
$B_{\parallel}\approx 4\,$T and $V_{\mathrm{TG}}\approx-12$ V.

Finally, there are additional oscillations in the conductance (of which the
obscuring feature in Fig. 3(c) is one), which are most visible close to
conductance plateaus in Figs. 2 (d). These appear as vertical lines in Fig.
2(a) and are less visible for the higher temperature in Fig. 3(b). Most
notably, a maximum of such an oscillation is seen to go straight through one
of the spin-split bands of the lowest 1D subband in Fig. 2(a) and (b) and Fig.
3(c), starting at around $-12$V and $0$T in the lowest plateau, crossing one
spin subband at around $3$T, and ending up in the $0.5\,e^{2}/h$ plateau for
higher magnetic field. Similar oscillations appear at other voltages in a
regular fashion.

###  II.4 Effective Landé g-factor

From the spin splitting of the 1D subbands marked in pink in Fig. 2(a) we
extract the Zeeman energy splitting $\Delta E_{Z}$ by converting the top-gate
voltage $V_{\mathrm{TG}}$ into energy, using the splitting rate of the energy
levels in source-drain bias measurements [5], as described in Ref. [63, 64,
65, 66]. The confinement in this cooldown, $V_{\text{BG}}=10$ V and
$V_{\text{SG}}=-12$ V, does not exactly correspond to the setup in the source-
drain measurement, where $V_{\text{SG}}=-11.6$ V. We observed a good agreement
between the two measurements in Ref. [5], which had a bigger difference in the
confining potentials. Most importantly, the extracted gate coupling is the
same for all nine visible subbands. Thus, we expect this value to be a very
good fit here as well and use

$E=\alpha_{\text{TG}}\,e\left(V_{\text{TG}}-V_{\text{TG}}^{(0)}\right),\quad\alpha_{\text{TG}}=3.8\times
10^{-3}.$

The obtained value of $\Delta E_{Z}$ for each of the three lowest subbabnds is
plotted in Fig. 4(a) as a function of magnetic field, revealing linearly
increasing Zeeman energy splittings. Remarkably, in case of the $N=0$ subband,
the Zeeman splitting shows a linear behavior only for $B_{\parallel}\gtrsim 5$
T, whereas at smaller fields an almost constant splitting is observed.

This saturation effect can be linked to the observed additional shoulder in
the conductance curves in Figs. 2(b) and 2(c) at not too strong magnetic
fields. The plateau in the Zeeman splitting corresponds to the magnetic fields
below $4$ T in Fig. 2(a), where the bright region to the left of
$V_{\text{TG}}=-12$ V disappears. One can either fit the dependence of $\Delta
E_{Z}$ on the magnetic field requiring a vanishing splitting extrapolated to
zero $B$ or not (using then the best linear fit at high magnetic field). In
the latter case, a finite intercept at $\Delta E_{Z}\approx 1.7$ meV is
observed for $N=0$ subband at $B_{\parallel}=0$, unlike the cases $N=1$ and
$N=2$, which extrapolate to close to zero energy splitting. This suggests that
a spontaneous spin splitting occurs for the $N=0$ subband, where the effects
of the interaction and confinement are expected to be the most prominent.
Fitting with a finite intercept, as was done, e.g., in Ref. [45], establishes
a bound on zero-field splitting without interaction effects. One should note
that this splitting is fully obscured by the much larger, interaction-induced
0.7 anomaly that produces a much larger value of the zero-$B$ splitting.

Figure 4: (a) Zeeman energy splitting $\Delta E_{Z}$ as a function of magnetic field $B$ for various 1D subbands. Dashed lines show best linear fits of high-field data points, dotted ones are linear fits going through zero at zero magnetic field. (b) Extracted effective Landé $g$-factors $|g^{*}|$ for the three quantization subbands [solid dots of the colors corresponding to legend in panel (a)], obtained for the in-plane magnetic field. The gray line indicates the value bare 2D $g$-factor $g=2$ for BLG. The error bars mark the 1$\sigma$ intervals from the two performed fits shown in panel (a). The dotted points correspond to the dotted lines above, the crosses to the dashed lines. All parameters are given in Table 1.  subband | 0 | 1 | 2
---|---|---|---
$g^{*}$ (no offset) | 6.04(6) | 4.22(4) | 3.73(4)
$g^{*}$ (finite offset) | 4.91(40) | 4.14(17) | 3.72(22)
offset in meV | 0.438(154) | 0.0288(546) | 0.00506(755)
Table 1: Parameters extracted from the fit of the splitting in magnetic field.
The uncertainty corresponds to the uncertainty of the fit, i.e., a $1\sigma$
interval.

From the slopes of the Zeeman splitting in Fig. 4(a), we find the (independent
of the magnetic field) values of effective Landé $g$-factors $|g^{*}_{N}|$ for
each of the subbands, shown in Fig. 4(b), see Table 1. These values are
obtained by taking the linear fit to the splitting with and without a finite
intercept. For both fits we use the 1$\sigma$ intervals to obtain error bars.
The obtained $|g^{*}_{N}|$ values are increasingly enhanced for lower subbands
compared to the bare 2D $g$-factor $g=2$, with a maximum enhancement by a
factor of about 2–3 for $N=0$. This observation also supports the idea of an
enhanced role of interaction effects for the $N=0$ subband. Independent of the
exact value of the gate coupling, this enhancement only relies on a gate
coupling that is the same for the three subbands. This enhancement is seen
both for a fit with finite intercept or without. Since the reported Kane-Mele
spin-orbit gap of $0.04-0.08$ meV is in between the finite and the vanishing
intercept, it would also not change the resulting enhancement of the Landé
$g$-factor by more than a few percent.

##  III Theoretical model

The quantization of conductance in a QPC is a well-known experimental proof of
the possibility of confining charge carriers and it clearly shows their
quantum nature [67]. What makes BLG an interesting platform for such
measurements, is its additional valley degree of freedom and the high
electrostatic tunability of its band gap [68, 6]. In this section, we discuss
the effects of the applied gate voltages on the band structure and, thus, on
the observed conductance within the essentially non-interacting model
(interaction here is taken into account only through the self-consistent
screening of the gate potentials).

###  III.1 Effective Hamiltonian and dispersion of BLG

We describe the low-energy properties of BLG relevant for the transport
measurements in the QPC geometry by the effective two-band Hamiltonian, see
Ref. [69]. The details of this approximation are given in Appendix C. The two-
band matrix Hamiltonian, acting in the space of the pseudospin degree of
freedom (Pauli matrices $\hat{\sigma}$) combined with the Zeeman interaction
in the spin space (Pauli matrices $\hat{s}$), has the form

$\displaystyle\hat{H}$
$\displaystyle=\left(\hat{H}_{0}+\hat{H}_{M}\right)\otimes\hat{s}_{0}+\frac{1}{2}\Delta
E_{Z}\hat{\sigma}_{0}\otimes\hat{s}_{z},$ (1) $\displaystyle\hat{H}_{0}$
$\displaystyle=-\frac{1}{2m}\begin{pmatrix}0&(\pi^{\dagger})^{2}\\\
\pi^{2}&0\end{pmatrix}-\frac{U}{2}\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix},$
(2) $\displaystyle\hat{H}_{M}$
$\displaystyle=\frac{Uv^{2}}{\gamma_{1}^{2}}\begin{pmatrix}\pi^{\dagger}\pi&0\\\
0&-\pi\pi^{\dagger}\end{pmatrix}.$ (3)

Here, $\pi=\xi p_{x}+ip_{y}$ is the kinetic momentum, with ${\xi=\pm}$
referring to the $K_{\pm}$ valley. Here, we disregard possible spin-orbit
coupling, which is a small effect at the energy scales of the experiment and
not capable of explaining the zero-field splitting or the magnetic field
behavior we observe, as seen by the obtained zero-field splitting of Fig.
4(a). We will return to this issue again below.

In what follows, we disregard the Mexican-hat term $\hat{H}_{M}$ that develops
for finite layer asymmetry $U$ as discussed in Ref. [70]. We also neglect the
skew interlayer hopping, which leads to trigonal warping [70, 71]. The effect
of these subtle features of the BLG spectrum on the conductance of a QPC in
in-plane magnetic field will be discussed elsewhere. Here, we adopt the
simplest model that, as we demonstrate below, is capable of describing the
salient features of the conductance.

Clearly, we have to distinguish the two spatial regions in our physical
sample. Away from the split gates there is no confinement and electrons feel
an approximately constant top-gate and back-gate voltage. Close to the split
gate, the shape of the confinement leads to a non-trivial, spatially dependent
effective top-gate voltage.

The dispersion of the spin $\sigma=\uparrow,\downarrow$ band for the low-
energy Hamiltonian (1) without the Mexican-hat feature (3) is given by

$\displaystyle
E_{\sigma}=\pm\sqrt{\frac{U^{2}}{4}+\frac{\hbar^{4}k^{4}}{4m^{2}}}+\frac{\sigma}{2}\Delta
E_{Z}.$ (4)

This corresponds to a 2D density for spin projection $\sigma$:

$\displaystyle n^{\mathrm{2D}}_{\sigma}(\mu)$
$\displaystyle=2\frac{m}{8\pi\hbar^{2}}\sqrt{4\left(\mu-\frac{\sigma}{2}\Delta
E_{Z}\right)^{2}-U^{2}},$ (5)

where the factor of $2$ accounts for the valley degree of freedom and the
chemical potential $\mu$ is measured with respect to the middle of the
asymmetry gap. For a small Zeeman splitting, $\Delta
E_{Z}\ll\sqrt{4\mu^{2}-U^{2}}$, one can use the expansion

$\displaystyle n^{\mathrm{2D}}_{\sigma}(\mu)$
$\displaystyle\approx\frac{m}{4\pi\hbar^{2}}\left(\sqrt{4\mu^{2}-U^{2}}-\frac{2\sigma\mu\Delta
E_{Z}}{\sqrt{4\mu^{2}-U^{2}}}\right).$ (6)

This expansion tells us that the effect of the Zeeman splitting on the density
is enhanced when the chemical potential is close to the gap. The total density
$n^{2D}=\sum_{\sigma=\pm}n^{\text{2D}}_{\sigma}$ is, to first order in $\Delta
E_{Z}$, independent of magnetic field, and we get for the chemical potential
in weak fields:

$\displaystyle\mu(n^{\text{2D}})$
$\displaystyle=\frac{\sqrt{m^{2}U^{2}+4\pi^{2}(n^{\text{2D}})^{2}\hbar^{4}}}{2m}$
$\displaystyle-\frac{m^{3}U^{2}(\Delta
E_{Z})^{2}}{16\pi^{2}(n^{\text{2D}})^{2}\hbar^{4}\sqrt{m^{2}U^{2}+4\pi^{2}(n^{\text{2D}})^{2}\hbar^{4}}}.$
(7)

###  III.2 Controlling BLG with gates

In the 2D regions away from the QPC, the effect of a constant back-gate and
top-gate voltage is described by the self-consistent gap equation [72, 70].
The total density $n=n_{\uparrow}+n_{\downarrow}$ is electrostatically
determined by the gates and given by

$\displaystyle
n=\frac{\varepsilon_{0}\varepsilon_{\text{BG}}V_{\mathrm{BG}}}{eL_{\text{BG}}}+\frac{\varepsilon_{0}\varepsilon_{\text{TG}}V_{\mathrm{TG}}}{eL_{\text{TG}}}.$
(8)

Here, $\varepsilon_{0}$ is the vacuum permittivity, $L_{\text{BG}}$
($L_{\text{TG}}$) is the distance from the BLG plane to the back gate (top
gate), and $\varepsilon_{\text{BG}}$, $\varepsilon_{\text{TG}}$ are the
relative dielectric constants of the material between BLG and the back gate
and top gate, respectively. In the absence of screening, the interlayer
asymmetry factor $U$ is given by

$\displaystyle
U_{\mathrm{ext}}=\frac{ec_{0}}{2\varepsilon_{r}}\left(\frac{\varepsilon_{\text{BG}}}{L_{\text{BG}}}V_{\text{BG}}-\frac{\varepsilon_{\text{TG}}}{L_{\text{TG}}}V_{\text{TG}}\right),$
(9)

where $c_{0}$ is the distance between the two BLG planes and $\varepsilon_{r}$
is the relative permittivity between these sheets.

Figure 5: (a) and (b): Band structure in the 2D regions (dark gray) and inside
the constriction (light gray) for zero magnetic field, with and without
spontaneous splitting (s.s.), respectively. Here, the spontaneous splitting is
between spin subbands shown as dashed or dotted lines. Since it requires the
proximity of the chemical potential, there is no s.s. in the valence band.
Because of the split-gate stray fields, the asymmetry gap inside the
constriction is larger. Energies are measured with respect to the middle of
the gap in the 2D region. (c): The magnetic field introduces a spin splitting
of the bands with respect to the $B=0$ band bottom, which leaves the middle of
the gap at the same value. The splitting inside the constriction is larger,
since the Landé $g$-factor is enhanced. As observed experimentally, the Zeeman
splitting inside the constriction is not symmetric. To the lowest order in a
weak magnetic field, the chemical potential for a fixed total 2D density is
independent of magnetic field, see Eq. (6).

Since the two layers of BLG screen the effect of the closer gate for the other
BLG plane depending on their density and thus the felt voltage, the actual
asymmetry as a function of the density is given by the self-consistent
equation [70]

$\displaystyle U(n)$
$\displaystyle=\dfrac{U_{\mathrm{ext}}}{1-\frac{\Lambda}{2}\ln\left(\frac{|n|}{2n_{\perp}}+\frac{1}{2}\sqrt{\frac{n^{2}}{n^{2}_{\perp}}+\frac{U^{2}}{4\gamma_{1}^{2}}}\right)}$
$\displaystyle\approx
U_{\mathrm{ext}}\left[1-\frac{\Lambda}{2}\ln\left(\frac{|n|}{n_{\perp}}\right)\right]^{-1},$
(10) $\displaystyle n_{\perp}$
$\displaystyle=\frac{\gamma_{1}^{2}}{\pi\hbar^{2}v^{2}},\qquad\Lambda=\frac{c_{0}e^{2}n_{\perp}}{2\gamma_{1}\varepsilon_{0}\varepsilon_{r}}.$
(11)

Thus, changing the top-gate voltage tunes the density $n$ according to Eq.
(8), which, in turn, influences the asymmetry factor $U$ according to Eq. (10)
and hence the dispersion (4) and the chemical potential according to Eq. (7).
This chemical potential remains constant over the whole sample, including the
QPC constriction, where the density is no longer given by Eq. (5):

$\displaystyle\mu(V_{\text{BG}},V_{\text{TG}})$
$\displaystyle\simeq\sqrt{\left(\frac{U_{\text{ext}}}{2-\Lambda\ln\frac{|n^{\text{2D}}|}{n_{\perp}}}\right)^{2}+\left(\frac{\pi\hbar^{2}n^{\text{2D}}}{m}\right)^{2}}.$
(12)

Here, the chemical potential depends on $V_{\text{BG}}$ and $V_{\text{TG}}$
through the corresponding dependence of the 2D density, Eq. (8), and the
dependence of $U_{\text{ext}}$, Eq. (9).

In the experiment, the combination of back-gate and split-gate voltages is
used to open a gap $U$ under the constricted region and tune the chemical
potential inside this gap, as shown in Ref. [5], and thus form the QPC, see
Fig. 5. The overall top gate is used to tune into the low-density regime,
where the observation of conductance quantization is possible [5].
Importantly, for fixed back-gate and split-gate voltages, like in the
experimental setup, the top-gate voltage tunes the electronic density in the
sample linearly [70].

As proposed in Ref. [73], we model the QPC by projecting the 2D problem onto a
one-dimensional one. The procedure for a standard Schrödinger equation is
described in Appendix E.1. A generalization of the method to BLG is discussed
in Appendix E.2. The quantization of conductance is already visible in the
simplest approximation of hard-wall boundary conditions, as we will show now.
In the case of a channel of width $W$, the dispersion relation for the
longitudinal wavevector $k$ resulting from Eq. (1) takes the form [5]

$\displaystyle E_{N,\sigma}(k)$
$\displaystyle=\pm\sqrt{\frac{U^{2}}{4}+\frac{\hbar^{4}}{4m^{2}}q_{N}^{4}(k)}+\frac{\sigma}{2}\Delta
E_{Z},$ (13) $\displaystyle q_{N}^{2}(k)$
$\displaystyle=k^{2}+\left(\frac{N\pi}{W}\right)^{2},$ (14)

where $N=0,1,2\ldots$ labels the size-quantized bands. While the case $N=0$,
strictly speaking, requires a different choice of boundary condition, we still
chose to investigate the effect of the resulting $k^{4}$ dispersion, which one
would also get in the 2D setup. It will turn out, that the choice of any non-
linear dispersion does not have qualitative consequences for the 0.7 effect.
Note that $U$ in Eq. (13) differs from the 2D expression (10), since the
screening in a 1D channel differs from that in the unconfined regions of BLG.
We also note that the channel width is affected in a non-trivial way by
$V_{\text{BG}}$ and $V_{\text{TG}}$.

The lowest band is, to leading order, quartic in the momentum, so that the
zero-temperature density resulting from Eq. (13) is given by

$\displaystyle
n^{\mathrm{1D}}_{\sigma}(\mu)=\frac{2\sqrt{2m}}{\pi\hbar}\left[(\mu-\sigma\Delta
E_{Z}/2)^{2}-U^{2}\right]^{1/4},$ (15)

as opposed to the square-root dependence of the 2D density (5). The total
density in the constriction is again determined electrostatically by the
gates, but the stray fields of the split gates make the evaluation of the
dependence of the density on the gate voltages harder. Since the split-gate
voltage is applied additionally in the constricted region, the gap there is
larger and the density inside the QPC is lower than away from the barrier
(Fig. 5), enabling the observation of the very lowest size-quantized bands.

###  III.3 Conductance quantization

We describe the conductance of the system by means of the Landauer-Büttiker
formula,

$\displaystyle
G=\frac{e^{2}}{h}\sum_{\sigma,\xi}\int_{-\infty}^{\infty}\mathrm{d}\epsilon\mathcal{T}_{\sigma,\xi}(\epsilon)\left[-f^{\prime}(\epsilon-\mu)\right],$
(16)

where $\mathcal{T}_{\sigma,\xi}(\epsilon)$ is the transmission of a subband
with spin $\sigma$ and valley $\xi$ and $f^{\prime}(x)$ is the derivative of
the Fermi function $f(x)=[1+\exp(x/k_{B}T)]^{-1}$. Assuming an idealized step-
function transmission coefficient, where a band contributes to $G$ as soon as
it is starting to get filled, the Landauer-Büttiker conductance is given by

$\displaystyle G(T,B)$
$\displaystyle=2\frac{e^{2}}{h}\sum_{N,\sigma}f\left(E_{N}^{0}+\frac{1}{2}\sigma
g^{*}\mu_{B}B-\mu\right),$ (17)

where the factor of $2$ accounts for the valley degeneracy,

$\displaystyle E_{N}^{0}$
$\displaystyle=\sqrt{\frac{U^{2}}{4}+\left(\frac{\hbar^{2}}{2m}\right)^{2}\left(\frac{N\pi}{W}\right)^{4}}$
(18)

is the lower band edge of band $N$ at zero magnetic field, and the Zeeman
interaction is written explicitly.

Every time the chemical potential crosses another lower band edge at finite
magnetic field, the conductance makes a step of $\Delta G=2\,e^{2}/h$ and, for
zero magnetic field, a step of $\Delta G=4\,e^{2}/h$. Each step has the shape
of the Fermi function. The steps are separated by conductance plateaus, thus
giving rise to a staircase structure seen in Fig 2 and Fig 3. This is the
conventional conductance quantization for a QPC, with an appropriate
degeneracy of the bands. In contrast to the case of an out-of-plane magnetic
field [5, 71], the in-plane magnetic field does not couple to the valley
degree of freedom. As discussed in Ref. [74], the direct effect on the band
structure is also negligible at experimentally accessible magnetic fields.
Therefore, at arbitrary fields, the steps of non-interacting conductance have
a factor of two corresponding to the two valleys of BLG.

###  III.4 Screening and electron-electron correlations

Electrons in the device are subject to Coulomb interaction, which is screened
by the electrons themselves, by the metallic gates, and by the dielectric
material. Let us first discuss the screening effect of the gates. There are
three relevant length scales in the system. The first one is the physical
distance between the split gate fingers is $w\approx 65$ nm and the
electrostatically induced channel is smaller than that. The width of the
split-gate fingers is of the order of $L\approx 300$ nm, so that we can
distinguish two ranges of length scales relevant to electrostatic screening in
our device.

On scales smaller or of the order of $w$, the system is truly 2D, only for
larger distances it crosses over to 1D. Another relevant scale is the distance
to the back gate and top gate, which are both of the order of $d\approx 55$
nm. Here, we also take into account the dielectric screening by further
assuming, for simplicity, that the insulating layers in between have the same
dielectric constant $\epsilon_{r}$ (the vacuum dielectric constant is denoted
below by $\epsilon_{0}$). The bare, only dielectrically screened Coulomb
interaction is given by its Fourier component at wave vector $\mathbf{q}$
(different in 1D and 2D cases):

$\displaystyle
V(q)=\begin{cases}\dfrac{e^{2}}{2\epsilon_{0}\epsilon_{r}}\dfrac{1}{q},&\text{2D},\\\\[14.22636pt]
\dfrac{e^{2}}{2\pi\epsilon_{0}\epsilon_{r}}\ln\dfrac{1}{qw},&\text{1D}.\end{cases}$
(19)

The gate-screened interaction can be found by summing up the infinite series
of mirror charges. In the 2D case, this leads to

$\displaystyle V(q)$
$\displaystyle=\frac{e^{2}}{2\epsilon_{0}\epsilon_{r}}\sum_{k=-\infty}^{\infty}\left(\frac{e^{-2l|k|q}}{q}-\frac{e^{-2|l^{\prime}-kl|q}}{q}\right)$
(20)
$\displaystyle=\frac{e^{2}}{2\epsilon_{0}\epsilon_{r}}\frac{2}{q}\frac{\sinh(ql-
ql^{\prime})\sinh(ql^{\prime})}{\sinh(ql)}$ (21)
$\displaystyle=\frac{e^{2}}{2\epsilon_{0}\epsilon_{r}}\frac{\tanh(qd)}{q},$
(22)

where $l=L_{\text{TG}}+L_{\text{BG}}$, $l^{\prime}=L_{\text{BG}}$ and in the
last line we assumed $L_{\text{TG}}=L_{\text{BG}}=d$. This means that
screening strongly alters the interaction if $qd\ll 1$. But in the 2D case we
require $r<w$, i.e., $q>1/w$, and thus $qd\gtrsim d/w\gtrsim 1$, so that the
interactions are not strongly altered by the screening of the gates.

A closer look, including the screening effects on the interaction for
monolayer graphene, is discussed in Ref. [61] and reveals, that gates need to
be closer than a few nanometer to really alter the interaction, which is not
experimentally accessible and certainly not the case here. There, it has also
been stressed that for BLG distances need to be even closer. In the 1D case
the presence of the gates is relevant only on scales $x>d$ and $q<1/d$. In
this case, we get a constant interaction strength, which is in agreement with
our phenomenological model.

One effect of electron-electron interaction is an enhancement of both the
Landé $g$-factor and spin-orbit coupling, as discussed in Refs. [75, 76, 77].
By introducing the Fermi-liquid constants $F_{0},F_{1}$ we can express the
Landé $g$-factor enhancement as $\tilde{g}=g/(1-|F_{0}|)$. Spin-orbit coupling
has an additional linear momentum dependence, which means that $|F_{1}|$
enters instead of $|F_{0}|$. Since $|F_{0}|>|F_{1}|$, this means that the
$g$-factor will always be more strongly enhanced than the spin-orbit coupling.
The enhancement is largest for large density of states, so that a strong
confinement further enhances this effect.

##  IV 0.7 anomaly in BLG QPC

As mentioned in the introduction, the 0.7 feature in the QPC conductance is
seen as an additional kink below the lowest plateau sitting at around
$0.7\times 2\,e^{2}/h$ [41] for systems without additional degeneracies of the
spectrum. It has been observed and extensively studied in quantum point
contacts in GaAs/AlGaAs heterostructures for both electrons [41, 78, 79, 43,
80, 81, 82, 83, 84, 85, 55, 86, 47, 57, 87, 88], and holes [89, 90, 91, 92,
93, 94, 95]; signatures of the anomaly were also observed in Si/SiGe
heterostructures [96].

This phenomenon cannot be explained within a single-particle picture [97, 62],
and it is commonly admitted that it is directly linked to spin [41, 78]. In
addition, this effect appears to be thermally activated and therefore not a
ground-state property [79]. Moreover, experiments show that the confinement
potential seems to play a crucial in the strength and the position of this
conductance feature [87, 88]. Recently, various explanations have been
suggested to capture the physical origin of the 0.7 anomaly, such as dynamical
spin polarization or spin gap models due to electron-electron interaction [58,
79, 98, 80, 99, 95], Kondo effect [100, 43, 42, 101, 102, 103, 44, 94], Wigner
crystallization [104, 105, 106], or charge density waves [49] To our
knowledge, no comprehensive study of the interaction-induced 0.7 anomaly in
systems where both spin and valley degrees of freedom are degenerate has been
reported so far.

In contrast to the conductance quantization, the shoulder-like feature
appearing in the conductance cannot be explained by the non-interacting theory
presented in Sec. III. In this section, we explore the possibility of
explaining this special feature in the context of the interaction-induced
$0.7$ conductance anomaly. As already discussed above, several microscopic
theories were used to describe the $0.7$ anomaly. Here, we do not specify any
microscopic mechanism behind the anomaly, but, instead, just assume that there
is some that effectively leads to spin and/or valley polarization. Based on
this assumption, we extend the phenomenological model of Ref. [58] to four
bands and the BLG band structure. The required polarization does not have to
be static, it can fluctuate slowly compared to the typical traveling time
through the constriction. For simplicity we nevertheless describe the model
for a static situation. The “classic” $0.7$ effect is only seen in the lowest
conductance step, Fig. 2, so that below we restrict our consideration to the
lowest size-quantized band shown in Fig. 3.

###  IV.1 Phenomenological model

Following the general idea of [58], we again use the Landauer-Büttiker formula
(16) for the conductance, here for the quantized band $N=0$ from Eq. (13). The
$0.7$ effect requires a finite temperature. Assuming that the energy scale for
the variation of the transmission probability is smaller than that of the
thermal distribution function, we approximate the former as a step function. A
spin-valley subband contributes to the conductance as soon as the chemical
potential reaches its lower band edge $\epsilon_{\sigma,\xi}^{0}$ within the
temperature window.

In this section, we develop a phenomenological model to describe how
interaction effects may influence these lower band edges beyond the self-
consistent screening. There are two ways in which these can differ from the
non-interacting single-particle ones. The first one is the spontaneous
polarization mentioned above, which is assumed to be arbitrary in the space of
spin and valleys. Already when the chemical potential is way below any of the
relevant subbands, these subbands may be spontaneously split to different
values of energy. The arrangement of these values, which are acquired for very
low chemical potential and zero magnetic field, is referred to as the initial
subband configuration. All subbands that are above the lowest subband are
called minority bands; those that are characterized by the lowest band edge
are majority bands.

The second effect is the dependence of the subbands on the chemical potential
when it is close to the band edge. A particular type of this
dependence—pinning of the band edge to the chemical potential—gives rise to
additional plateaus in the conductance. It is this interaction-induced
dependence of the lower band edge of minority bands on the chemical potential
that our phenomenological model describes for any assumed initial
configuration. We then consider the corresponding evolution of the conductance
with increasing in-plane magnetic field, and, by comparing the resulting
behavior with the experimentally observed one (Sec. II), infer the initial
splitting configuration.

Figure 6: (a) Subfigures a to h show every possible initial subband structure.
The $x$ axis labels spin and valley degree of freedom and the $y$ axis shows
the position of the lower band edges in energy space with the values used for
all plots. The assignment of spin $\sigma$ and valley $\xi$ to the $x$ axis is
not unique, yet. In a there is no initial splitting, all lower band edges are
equivalent. In case b, three bands are fixed and there is one minority band.
Whether the single minority band is spin up/down or valley $K$/$K^{\prime}$
can only be distinguished once we include a magnetic field. In c to d and e to
h there are two and three split subbands, respectively. (b) Exemplary
conductance for the initial configuration shown in (a) for zero magnetic
field. The parameters are chosen to suppress the plateaus at multiples of
$1\,e^{2}/h$, where the majority band is filled and minority bands are not yet
populated. The majority bands are fixed at $0.1$ meV, the next levels
correspond to $\epsilon^{0}_{1}=0.2$ meV, $\epsilon^{0}_{2}=0.6$ meV,
$\epsilon^{0}_{3}=1$ meV. The temperature is $T=0.7$ K and $C=6$ meV-3 for all
minority bands [see Eq. (30)].

####  IV.1.1 General four-band model

For a system with four degrees of freedom, like BLG, we label the subbands by
their spin and valley index, i.e., $\epsilon_{\sigma,\xi}^{0}$. Moreover, we
assume that the lower band edges of minority bands start to depend on the
chemical potential once it reaches a certain value $\mu_{\sigma,\xi}$, i.e.,
$\epsilon_{\sigma,\xi}^{0}=\epsilon_{\sigma,\xi}^{0}(\mu)$.

All possible initial spontaneously polarized configurations of the band edges
are shown in Fig. 6(a). For the analysis of the dependence of the band edges
on $\mu$, we look at the local spin-valley energy-density functional in the
form

$\displaystyle\mathcal{F}=\mathcal{E}[\\{n_{\sigma,\xi}\\}]-\mu\sum_{\sigma,\xi}n_{\sigma,\xi}.$
(23)

Here, $\mathcal{F}$ is the free energy of the system and $\mathcal{E}$ is its
internal energy. A diagrammatic approach to obtaining such a free energy and a
corresponding analysis of possible instabilities in models with multiple
species of quasiparticles is discussed, e.g., in Refs. [107, 108].

The lowest bands in Fig. 6 are majority bands with a fixed lower band edge and
we decompose their density into $n=n_{0}+\delta n$. All changes with the
chemical potential are included in $\delta n$. For minority bands, we do not
make this decomposition, but assume that $n=0$ for $\mu<\mu_{\sigma,\xi}$. We
approximate the free energy functional $\mathcal{F}$ as bilinear in all
partial majority density contributions $\delta n$ and minority densities $n$,
i.e.

$\displaystyle\mathcal{F}=\sum_{i=\sigma,\xi}\left[(\alpha_{i}-\mu)n_{i}+\frac{\beta_{i}}{2}n_{i}^{2}\
+\\!\sum_{j=\sigma^{\prime},\xi^{\prime}}\gamma n_{i}n_{j}\right],$ (24)

where $\alpha_{\sigma,\xi}$, $\beta_{\sigma,\xi}$, and $\gamma$ are
phenomenological constants to be determined experimentally and
$n_{\sigma,\xi}$ is understood as $\delta n_{\sigma,\xi}$ for majority bands.
The minimum of the energy functional is achieved when

$\displaystyle\frac{\partial\mathcal{F}}{\partial
n_{\sigma,\xi}}=\alpha_{\sigma,\xi}-\mu+\beta_{\sigma,\xi}n_{\sigma,\xi}+\gamma\sum_{\sigma^{\prime},\xi^{\prime}}n_{\sigma^{\prime},\xi^{\prime}}=0,$
(25)

which leads to solutions of the form

$n_{\sigma,\xi}\propto(\mu-\mu_{\sigma,\xi})$ (26)

for minority bands, where $\mu_{\sigma,\xi}$ is the critical chemical
potential of the minority subband that depends on the parameters of the free
energy Eq. (24).

####  IV.1.2 Application to BLG

At this point, we have to specify the band structure in order to get access to
the single-particle densities entering the energy functional (24). For this
purpose, we use the results of Sec. III.2 for the non-interacting 1D
dispersion in the BLG QPC, and modify them to include the interaction effects
at the phenomenological level.

Without a magnetic field, we consider a quartic dispersion relation of the
form

$\displaystyle\epsilon_{\sigma,\xi}(k)=ak^{4}+\epsilon_{\sigma,\xi}^{0}(\mu),$
(27)

where $a$ is a constant. This form of the effective—modified by
interactions—form of single-particle energies in BLG QPC is based on the
fourth-order expansion of dispersion relation (13) for $N=0$, namely

$\displaystyle
E_{N=0,\sigma}(k)\approx\left(\frac{\hbar^{2}}{2m}\right)^{2}\frac{k^{4}}{U}+\frac{U}{2}.$
(28)

As shown above, the gap magnitude $U$ depends on the chemical potential
through the self-consistent electrostatic screening. Specifically, $U=U(n)$,
according to Eq. (10) with additional effects of the split gates, and
$n=n(\mu)$ according to Eq. (15). The chemical potential is set by the 2D
density according to Eq. (5) and Eq. (8). Within the lowest band, these
dependencies are very smooth and do not lead to any additional features. The
main role in our consideration is played by the interaction-induced bandgap
that determines the band edge $\epsilon_{\sigma,\xi}^{0}(\mu)$. For this
reason, we neglect all electrostatic contributions [effectively fixing
$U=U(\mu_{\sigma,\xi})$] and introduce a new bandgap instead of $U/2$. One
could just as well assume that this bandgap is applied on top of the fixed gap
$U(\mu_{\sigma,\xi})/2$, since this would only lead to an overall shift and a
redefinition of the origin.

For the dispersion (27), we get a one-dimensional density in the form

$\displaystyle
n_{\sigma,\xi}(\mu)=\frac{2}{\pi}\left\\{\frac{1}{a}\left[\epsilon_{\sigma,\xi}^{0}(\mu)-\mu\right]\right\\}^{1/4}.$
(29)

By combining this with Eq. (26), we thus get the dependence

$\displaystyle\epsilon_{\sigma,\xi}^{0}(\mu)=\begin{cases}\mu-
C_{\sigma,\xi}(\mu-\mu_{\sigma,\xi})^{4},&\mu>\mu_{\sigma,\xi}\\\
\mu_{\sigma,\xi},&\mu<\mu_{\sigma,\xi}\end{cases}.$ (30)

where $C_{\sigma,\xi}$ is a phenomenological constant depending on the
parameter $a$ as well as the parameters of the free energy functional
$\mathcal{F}$. This means that once the chemical potential reaches the lower
band edge of a minority band they become pinned together over a certain energy
range. For continuity reasons we require
$\epsilon_{\sigma,\xi}^{0}=\mu_{\sigma,\xi}$, i.e., the initial configuration
determines the critical chemical potentials. It is worth emphasizing that the
enhanced density of states at the bottom of the almost flat (quartic in
momentum) band in BLG QPC (27) is expected to enhance the role of interactions
compared to the case of conventional parabolic bands.

####  IV.1.3 Resulting conductance

With the step-function transmission probabilities, the conductance reads:

$\displaystyle
G(T)=\frac{e^{2}}{h}\sum_{\sigma,\xi}f(\epsilon_{\sigma,\xi}^{0}(\mu)-\mu).$
(31)

At this point, one should note that the Fermi function $f(x)$ is close to
$1/2$ for $|x|\ll k_{B}T$. For fixed lower band edges, this corresponds to a
very small region and does not lead to conductance anomalies, but for the
pinned band edges and finite temperature considered here it does. If we tune
the chemical potential through all band edges, the crossing of a fixed
majority band corresponds to a plateau of $1\,e^{2}/h$, while for every
minority band we get an additional less sharp one at $0.5\,e^{2}/h$. Any
additional plateau from a majority band at $1\,e^{2}/h$ can be smeared by
temperature. If we have several minority band edges at different initial
energies, the distance between the bands compared to the temperature
determines whether lower minority bands already contribute fully or not, cf.
Ref. [58].

The conductance corresponding to the initial splitting configurations from
Fig. 6(a) is shown in Fig. 6(b). The values of the additional shoulders are
summarized in Table 2. In experimental conductance curves in Fig. 2(d), there
is one additional shoulder at around $2.5\,e^{2}/h$ and another one around
$3.5\,e^{2}/h$ for zero magnetic field. Thus we can rule out case a, because
it does not have any additional shoulder and cases f, g, and h, which have a
too low first shoulder from the very beginning. Only cases b, c,d and e from
Fig. 6 are relevant here.

case |  conductance shoulders in units of $e^{2}/h$
---|---
a |  none
b |  3.5
c |  3
d |  2.5 and 3.5
e |  2.5
f |  1.5 and 3
g |  2 and 3.5
h |  1.5 and 2.5 and 3.5
Table 2: Values of conductance shoulders for initial configurations shown in Fig. 6 case |  spin configurations
---|---
a |  $(\uparrow\uparrow\downarrow\downarrow)$
b |  $(\uparrow\uparrow\downarrow)\downarrow,\quad(\uparrow\downarrow\downarrow)\uparrow$
c |  $\uparrow\uparrow\downarrow\downarrow,\quad(\uparrow\downarrow)(\downarrow\uparrow),\quad\downarrow\downarrow\uparrow\uparrow$
d |  $\uparrow\uparrow\downarrow\downarrow,\quad(\uparrow\downarrow)\downarrow\uparrow,\quad(\uparrow\downarrow)\uparrow\downarrow,\quad\downarrow\downarrow\uparrow\uparrow$
e |  $\downarrow(\downarrow\uparrow\uparrow),\quad\uparrow(\uparrow\downarrow\downarrow)$
f |  $\uparrow\uparrow\downarrow\downarrow,\quad\downarrow\uparrow(\uparrow\downarrow),\quad\uparrow\downarrow(\uparrow\downarrow),\quad\downarrow\downarrow\uparrow\uparrow$
g |  $\uparrow\downarrow\downarrow\uparrow,\quad\downarrow\uparrow\uparrow\downarrow,\quad\downarrow(\uparrow\downarrow)\uparrow,\quad\uparrow(\uparrow\downarrow)\downarrow$
h |  $\uparrow\uparrow\downarrow\downarrow,\quad\uparrow\downarrow\downarrow\uparrow,\quad\uparrow\downarrow\uparrow\downarrow,\quad\downarrow\downarrow\uparrow\uparrow,\quad\downarrow\uparrow\uparrow\downarrow,\quad\downarrow\uparrow\downarrow\uparrow$
Table 3: Different spin configurations for the arrangements shown in Fig. 6;
brackets “$(\ldots)$” denote all possible permutations. The order corresponds
to the subband ordering shown in Fig. 6. For each permutation we can assign
the valley indices in four different ways, which does not lead to different
behavior in this model and is thus not distinguishable. For example, the last
arrangement in case g corresponds to the eight valley-resolved cases:
${\uparrow(\uparrow\downarrow)\downarrow}=\\{{\uparrow_{1}\uparrow_{2}\downarrow_{1}\downarrow_{2},}\
{\uparrow_{1}\uparrow_{2}\downarrow_{2}\downarrow_{1},}\
{\uparrow_{2}\uparrow_{1}\downarrow_{1}\downarrow_{2},}\
{\uparrow_{2}\uparrow_{1}\downarrow_{2}\downarrow_{1},}\
{\uparrow_{1}\downarrow_{1}\uparrow_{2}\downarrow_{2},}\
{\uparrow_{1}\downarrow_{2}\uparrow_{2}\downarrow_{1},}$
${\uparrow_{2}\downarrow_{1}\uparrow_{1}\downarrow_{2},}\
{\uparrow_{2}\downarrow_{2}\uparrow_{1}\downarrow_{1}}\\}$, where the index
$1,2$ corresponds to the $K,K^{\prime}$ valley, respectively.

###  IV.2 Behavior of conductance in magnetic field

In order to distinguish between the cases of spin, valley or spin-valley
splitting, we consider the behavior of the conductance in parallel magnetic
field $B$. This is incorporated by the following replacement

$\displaystyle\epsilon_{\uparrow,\xi}^{0}$
$\displaystyle\rightarrow\epsilon_{\uparrow,\xi}^{0}-\frac{1}{2}g\mu_{B}|B|,$
(32) $\displaystyle\epsilon_{\downarrow,\xi}^{0}$
$\displaystyle\rightarrow\epsilon_{\downarrow,\xi}^{0}+\frac{1}{2}g\mu_{B}|B|.$
(33)

There are in total 6 possibilities of assigning $2\times 2$ spins to four
subbands. Since one cannot distinguish different valley indices this way,
after this assigning, the spins are still four-fold degenerate in their valley
index. We only know that each valley has two opposite spins. The permutation
of spins within subbands with the same lower band edge does not change the
outcome. From this we get in total 26 different arrangements with distinct
development with magnetic field for the eight initial cases shown in Fig. 6.
These are shown in Table 3.

The magnetic field behavior of the relevant cases b, c, d, e is shown in Fig.
7 in analogy to Fig. 3(a). Here one should note, that the initial spontaneous
splitting is a spontaneous symmetry breaking and if the magnetic field is
tuned adiabatically, it will always favor the initial splitting in the
direction of the magnetic field. Behavior like case c2, where the initial
spontaneous splitting is opposite to the Zeeman splitting will only be
observed if the magnetic field is switched on very fast.

Figure 7: Behavior of the four different initial splitting cases a, b, c and e
from Fig. 6 in an in-plane magnetic field. Conductance curves for magnetic
fields between $0$T and $8$T are shown with a horizontal shift parametrized by
$\alpha=1.5\mathrm{meV}/\mathrm{T}$. The thick black curve corresponding to
$B=0$ T is non-shifted. The blue lines correspond to $2$ T , $4$ T and and $6$
T, as in Fig. 3. Without any initial splitting, there is no continuous
development of a shoulder in a, the additional plateau appears, as soon as it
can be resolved. The particular assignment of spin to the subbands is
irrelevant: all six possibilities are indistinguishable. In case c, three
different scenarios are possible. Each case happens for all four possible
valley assignments. For cases b and e, there are two distinguishable spin
configurations; for case d four. Same parameters as in Fig. 6; according to
the measured Landé $g$-factor, $g$=4 was chosen.

A comparison of the experimental data and theoretical ones for symmetric
splitting Eqs. (32) and (33) and a phenomenological asymmetric one, where we
use

$\displaystyle\epsilon_{\uparrow,\xi}^{0}$
$\displaystyle\rightarrow\epsilon_{\uparrow,\xi}^{0}-g\mu_{B}|B|,$ (34)
$\displaystyle\epsilon_{\downarrow,\xi}^{0}$
$\displaystyle\rightarrow\epsilon_{\downarrow,\xi}^{0},$ (35)

assuming that the spin up band is energetically higher is shown in Fig. 8.
From this we see that an asymmetric splitting, Eqs. (34) and (35), yields a
better agreement with the experimental observations in this particular case,
that will turn out to be the most relevant one. However, owing to the special
dependence of minority bands on the chemical potential, this asymmetric
replacement rule may lead to unphysical half-integer plateaus in high magnetic
fields for some initial configurations. Therefore, we have used the
symmetrical splitting introduced in Eqs. (32) and (33) to produce Fig. 7.

Figure 8: Zoom into the evolution of the lowest conductance step with
increasing magnetic fields (without shifts). The colored curves correspond to
those marked in Fig. 2a. (a): Experimental data for $20$ mK. Curves for higher
magnetic fields show lower conductance at a fixed value of the top-gate
voltage. (b): Theoretical curve for case c1 in Fig. 7 with the non-symmetric
splitting (sp.) introduced in Eqs. (34) and (35). Same parameters as in Fig.
6; according to the measured Landé $g$-factor, $g$=4 was chosen. The behavior
of (a) is qualitatively replicated. (c): Same as in (b), but with a symmetric
Zeeman splitting introduced in Eqs. (32) and (33). There is a fixed(crossing)
point clearly absent in the experimental data. As is apparent by comparing the
distance of the blue plateaus in chemical-potential space in (b) and (c) a
symmetric splitting enhances the $g$-factor even more.

##  V Discsussion

Let us now compare the results of our phenomenological model with the
experimental results reported in Sec. II. Many, but not all, features in Fig.
2, e.g., conventional conductance quantization, can be explained without
considering interaction effects. Other features, e.g., additional shoulders in
the conductance curves and behavior of the $g$-factor, are compatible with the
phenomenological model presented in Sec. IV.

###  V.1 Conductance plateaus

Every time the chemical potential, tuned by the top gate voltage, reaches a
new lower band edge, the conductance makes a step of $1\,e^{2}/h$ per spin and
valley. For zero magnetic field, the plateaus are at multiples of
$4\,e^{2}/h$, which can be clearly seen in the cadence plots in Fig. 3. This
is in contrast to Ref. [13], where the valley splitting was observed in a
similar setup with changing the split-gate voltage, but at much higher back-
gate voltage. The Zeeman coupling of the spin to the in-plane magnetic field
leads to appearance of steps at multiples of $2\,e^{2}/h$ for higher magnetic
field. The additional plateaus become visible when the Zeeman-split bands have
a spacing that can be experimentally resolved, which occurs in our case above
2 T.

###  V.2 Effective g-factor

The Zeeman splitting in the first three subbands shown in Fig. 4(a) for $20$mK
is very close to linear at sufficiently high magnetic fields. For the lowest
subband, a nearly constant splitting is observed up to nearly $4$T. The
extracted $g$-factors show a strong enhancement compared to the bare value of
$g=2$ for BLG. We attribute this enhancement to the strong confinement and
interaction effects, similar to those discussed in Ref. [109].

These effects are strongest for the lowest subbands because of lower densities
in the almost flat (quartic) band, which is consistent with the increase of
the enhancement with lowering the band index. This effect should only be
present for electrons going through the constriction. Electrons that bounce
back and stay in the 2D region are at too high densities for the interaction-
induced enhancement to be visible. This effect combined with the peculiar low-
field behavior of the lowest subband strongly hints at the importance of
interaction effects in this experiment.

###  V.3 Additional features

There are two additional features below the conventional conductance
quantization step at $G=4\,e^{2}/h$, which thus only involve the lowest
quantized subband. The first one starts at around $3\,e^{2}/h$ at zero
magnetic field, and the second one at around $2.5\,e^{2}/h$.

####  V.3.1 0.7 anomaly

One might try to identify both these features with case d in Fig. 6, where
there are two additional shoulders at zero magnetic field. However,
considering the conductance behavior in magnetic field shown in Fig. 7, one
sees that when these two additional shoulder move down with magnetic field, as
in case d4, the additional plateau at $3\,e^{2}/h$ persists for higher
magnetic fields. We do not see such a plateau in the experimental data. The
feature starting around $2.5\,e^{2}/h$ in Fig. 3(a) moving down into the
$2\,e^{2}/h$ plateau is visible in Fig. 2(a) as a splitting of the spin-valley
subbands at vanishing magnetic field, which makes it a strong contender for
the $0.7$ effect. It also leads to the non-linear behaviour of the Zeeman
splitting in the first subband in Fig. 4.

Since we have ruled out case d, only the cases b, c and e in Table 2 are still
possible. Upon comparing the magnetic field behavior with Fig. 7, we conclude
that we are in either case c1 or e1. While one might not be convinced by the
value of the theoretical shoulder of case c, that is $3\,e^{2}/h$ compared to
the experimental one at $2.5\,e^{2}/h$, which is exactly the value in case e,
as shown in Fig. 6, this would require the spin-up state of one valley split
to the same energy as the non-spin split bands of the other valley, which
implies an accidental fine-tuning. Instead, if only one valley was spin split,
cases d, f and g would be way more probable, but these were already ruled out.
Thus, we identify the experimentally observed behavior as case c1, which
assumes an initial spin splitting, but no valley splitting.

It is also clear that, in contrast to [12], we do not see any crossing of
Zeeman-split bands. A shoulder similar to ours but at $2\,e^{2}/h$ was
attributed to the substrate-enhanced Kane-Mele spin-orbit coupling in Ref.
[12]. We note that such effects of the weak spin-orbit coupling can be
observed only at very low temperatures, but we still see a similar effect at
$4$K. Finally, the proposed Kane-Mele spin-orbit splitting would lead to
opposite spin splitting in the two valleys, so that there is no net spin
splitting, as detailed in Appendix C. However, the observed Zeeman splitting
at low magnetic fields suggests the presence of spontaneous net spin splitting
in our case, while the enhancement of the effective $g$-factor points towards
rather strong interaction effects. We thus identify this feature in the
conductance as an interaction-induced $0.7$ anomaly. As mentioned in Ref.
[56], the exact value of the shoulder may depend on the exact QPC geometry, so
that it may also appear very close to the value of $0.5\times 4\,e^{2}/h$.

####  V.3.2 Fabry-Pérot resonances

We identify the upper feature in the lowest subband conductance at low
magnetic field, corresponding to an additional peak in the low-temperature
plot in Fig. 3(c) at around $-11.8$ V that goes vertically through the right
spin-split band, as a Fabry-Pérot resonance [110, 111, 112, 113, 114, 115,
116]. In Fig. 2(e), this additional feature is seen as a faint bright curve
moving down from the $4\,e^{2}/h$ plateau (at weak fields) and merging with
the 0.7-feature to from the spin-split $2\,e^{2}/h$ plateau at magnetic field
around $4$ T. Note that at this same value of magnetic field, the Zeeman
splitting of the lowest subband starts growing linearly with magnetic field,
see Fig. 4(a). With increasing temperature, this feature disappears, in
contrast to the 0.7 anomaly, see Fig. 3(d).

The Fabry-Pérot resonances in our geometry emerge from interferences of
electronic waves in the 2D region, which are back-scattered from the interface
with the contact on one hand, and the barrier created by the split gates, see
Ref. [5]. In a parallel magnetic field, there are two Fermi-wavevectors, one
for each spin, so that the minima and maxima of these oscillations disperse
with the magnetic field. Since Fabry-Pérot resonances correspond to electrons
bouncing back and forth between the contacts and the split gate, these
electrons are inherently two-dimensional and are not affected by the
enhancement of the $g$-factor in the QPC region. A closer look reveals
additional Fabry-Pérot peaks at other values of the top-gate voltage, which do
not move to different plateaus in the considered range of magnetic fields. For
a discussion of this effect, see App. B.

##  VI Conclusions

In conclusion, we have studied an electrostatically defined QPC in BLG which
shows a zero field quantized conductance in steps of $4\,e^{2}/h$ owing to the
spin and valley degeneracy. In an in-plane magnetic field, a splitting of the
first three subbands at $20$ mK is observed that results from the Zeeman spin
splitting, while the valley degeneracy is not affected. Additionally, a
$0.7$-like structure is located below the lowest size-quantized energy level
which develops into the lowest spin split subband at $2\,e^{2}/h$. This
additional feature is also observed in the $4$K data, where only the splitting
of the lowest band is clearly resolved. On top of the quantized conductance we
observe Fabry-Pérot resonances. Because of the higher densities in the 2D
region and the relatively small bare Zeeman splitting in BLG, these stay at
fixed top gate values with increasing magnetic field.

From the Zeeman energy splitting, the effective 1D $g$-factors in an in-plane
magnetic field are found to be increasingly enhanced for lower subbands
compared to the bare 2D Landé $g$-factor $g=2$ in BLG. Moreover, the fact that
the linear fitting of the Zeeman energy splitting for $N=0$ does not
extrapolate to zero at $B=0$ further indicates the spontaneous spin
polarization of the lowest subband. The behavior of the Zeeman spitting is a
clear sign of the importance of interaction effects and confinement in this
experiment. Based on this, we also attribute the observed shoulder below the
lowest subband to the $0.7$ anomaly stemming from the interaction-induced
lifting of the band degeneracy.

We employ a phenomenological model to qualitatively describe the behavior of
this feature in the applied in-plane magnetic field. In this model we assume,
that each spin-valley subband can be spontaneously split by the electron-
electron correlations. By comparing the development of resulting features in a
magnetic field, Fig. 7, with the experimental conductance curves in Fig. 3, we
conclude, that the observed behavior can be explained by the assumption of an
effective spontaneous spin splitting, while the valley degree of freedom is
not affected. This is in a full agreement with the picture of spontaneous spin
polarization inferred from the measured Zeeman splitting.

Our experimental findings, supported by phenomenological calculations and
combined with those of Ref. [5] for out-of-plane magnetic field, establish the
exquisite tunability of spin and valley degree of freedom by the application
of gates or external magnetic fields. Furthermore, our results also
demonstrate relevance of electron-electron correlations in BLG QPC geometry,
as well as a possibility to control the effective strength of interactions by
means of electrostatic spatial confinement by a combination of external gates.

Apart from developing the microscopic theory of 0.7 anomaly in BLG QPC,
several questions regarding the phenomenological model still remain. While it
is straightforward to obtain a free energy of the form of Eq. (24) for a 2D
model with quadratic band structure (cf. Ref. [107]), the corresponding
calculation for a hybrid 2D-1D geometry is much more involved. Within such a
derivation, it would be especially interesting to show the microscopic origin
of the parameters of the phenomenological description. In particular, such a
calculation would yield their explicit dependence on the applied magnetic
field.

There are several additional ingredients that could be combined with this sort
of setup. In particular, in Ref. [117] it was observed that, at least if there
is a substantial gap and the trigonal warping is relevant, electrons might
predominately orient along the lattice directions and not take the shortest
path, which is expected to affect the conductance of the QPC in the
corresponding regime of gate voltages. Further, while the intrinsic spin-orbit
coupling in BLG is very weak, using an additional layer with strong spin-orbit
coupling, e.g., a layer of a transition metal dichalcogenide, should induce
noticeable proximity spin-orbit related effects [118, 119] and may lead to
topologically nontrivial states. In addition, the introduction of a finite
twist between the layers may also lead, at certain fillings and twist angles,
to topological states [120, 119]. In order to open a gap in such a system,
spin-orbit coupling has to be added as well. To what extent these states can
be manipulated with gates and external magnetic fields and what role
interaction effects play in such engineered sample are questions worth
exploring. The analysis of the present paper serves as the starting point for
further studies in this direction.

## Acknowledgments

We thank Alexander Dmitriev, Angelika Knothe, Ralph Krupke, Alex Levchenko,
Christoph Stampfer, and Xavier Waintal for discussions and comments. This work
was partly supported by Helmholtz society through program STN, by the DFG via
the projects DA 1280/3-1 and GO 1405/5 within FLAGERA Joint Transnational Call
2017 (Project GRANSPORT), and by the RFBR (grant No. 20-02-00490).

Figure 9: (a) and (b): Differential conductance $G$ as a function of the top
gate voltage $V_{\mathrm{TG}}$ for different values of magnetic field $B$ and
for $20$ mK and $4$ K, respectively. The curves are shifted with
$\alpha=2{\mathrm{V}}/{\mathrm{T}}$. Figures 3(a) and 3(d) are a zoom into the
lowest plateau shown here.

##  Appendix A Additional plots

The $0.7$ anomaly discussed in the main text is only visible in the lowest
conductance plateau. The other discussed features, including the Fabry-Pérot
resonances, are visible in other plateaus as well. In this Appendix, we show
cadence plots of the measured conductance for larger ranges of conductances
and top-gate voltages, than in Figs. 3(a) and 3(b). Figures 9(a) and 9(b) show
the differential conductance as a function of the applied top-gate voltage
with horizontal shift linear in the applied in-plane magnetic field. The
additional $0.7$-shoulder is seen in the lowest step for both temperatures.
The main difference between the two temperatures is the smoother and flatter
behavior for higher temperatures. Moreover, there are two additional features
that are only visible in the $20$ mK case, Fig. 9(a). For magnetic fields
below $0.2$T, the aluminum leads are still superconducting, so that the
conductance is affected by superconducting fluctuations. Additionally, one
sees Fabry-Pérot resonances, which are most clear on top of plateaus.

Since the aluminum leads are superconducting at $20$ mK, a finite magnetic
field is needed to kill this effect and curves below $0.2$T show a higher
conductance than the quantized values. This should be contrasted with the data
shown in Fig. 3(b) for $4$K, where there are no superconducting effects even
at vanishing magnetic field. The superconducting proximity effect for the QPC
in BLG is out of scope of the present paper; the analysis of conductance
curves affected by superconducting fluctuations is an interesting task from
both the experimental and theoretical points of view (for a related analysis
of the supercurrent in this geometry, see Ref. [8]).

Figure 10:  Cubic spline fit of the differential conductance $G$ of the same
sample in a different cooldown for $V_{\text{BG}}=10$ V and a perpendicular
magnetic field of $20$ mT as a function of the $V_{\mathrm{TG}}$ in elevating
$V_{\text{SG}}$ for $20$ mK. The curves are shifted vertically with
$\alpha=10{e^{2}}/{h\mathrm{V}}$ and colored according to their first
derivative. The onset of the conductance plateau (black solid lines serving as
guides for the eye) shows a clear dependence on the exact gate configuration.
The black dotted line shows the dispersion of a feature, that we identify as a
0.7 shoulder, with a dispersion parallel to the onset of the plateau, in
agreement with its quasi-1D nature. Fabry-Pérot oscillations are marked by
black dashed lines and show a different (weaker) split-gate voltage
dependence, since they are generated by the 2D lead modes.

Since it is well known that the 0.7 anomaly is very sensitive to the exact
shape of the constriction, we include data of the same sample in a different
cool down at $T=20$ mK and with a perpendicular magnetic field of $20$ mT. The
back-gate voltage is again $V_{\text{BG}}=10$ V and the split-gate voltage
ranges between $-12$ V and $-11.5$ V. Figure 10 shows a cubic spline fit of
the obtained conductance data in a form similar to Fig. 2 (b) and (c) but the
vertical shifts correspond to different split-gate voltages, starting at
$V_{\text{SG}}=-12$ V for the lowest curve and ending with
$V_{\text{SG}}=-11.5$ V as a function of $V_{\text{TG}}$. The curves are
colored according to their derivative. The thick solid lines mark the onset of
the conductance plateaus, showing their dependence on the exact confinement
condition, i.e., the split-gate voltage. The lowest curve corresponds to the
same split-gate and back-gate configuration as the data in the main text,
where we have identified the 0.7 anomaly by its magnetic field dependence, see
Fig. 2 (b) and (e).

In this cooldown, we see a similar feature, marked by the arrow. When
following the split-gate dependence of this feature (black dotted line), one
observes that it stays parallel to the onset of the lowest plateau, which
verifies that it is a feature of the QPC modes. Additionally, we again see
Fabry-Pérot oscillation on top of the $4\,e^{2}/h$ and $8\,e^{2}/h$ plateaus
(black dashed lines). Since they are generated by the lead modes, they show a
different dispersion with the split-gate voltage. They always appear at the
same electronic 2D density, which is only slightly tuned by the split-gate
voltage. The onset of the conductance steps (and the 0.7 anomaly) is much more
strongly dependent on the exact gate configuration, which makes the two
effects clearly distinct.

##  Appendix B Fabry-Pérot resonances

In our experiment (Fig. 2), the 0.7 shoulder in the conductance of the lowest
subband merged at magnetic field of about $4$ T merges with an additional
conductance feature, which was identified with the Fabry-Pérot resonances in
the main text. Here, we present additional details supporting this
identification. The transmission coefficient accounting for the Fabry-Pérot
resonance can be described by

$\displaystyle\mathcal{T}=\frac{1}{1+F\sin^{2}\left[Lk\cos(\theta)\right]},$
(36)

where $F$ is the finesse and $\theta$ is the angle of incidence of the
electron wave. At very low temperatures, the contribution of the resonance to
the conductance is given by

$\displaystyle G=\frac{e^{2}}{h}\frac{1}{1+F\sin^{2}\left(Lk_{F}\right)}.$
(37)

One should note that, according to Ref. [121], the Zeeman splitting in BLG is
around $1.1$ meV for $10$T. Using the conversion formula for top-gate voltages
to energies from the Supplemental Materials of [5] for the same device at
slightly different voltages, the distance between $V_{\mathrm{TG}}=-12$ V and
$V_{\mathrm{TG}}=-8$ V corresponds to the band splitting of $15.2$ meV.
Moreover, the density in the 2D region is not as low as in the constriction,
since the split gates do not cover this region. Therefore, the total spin
polarization of these 2D bands cannot be achieved and, since we only observe
faint oscillations on top of the plateaus, the finesse $F$ is small. As a
result, this dependence of the conductance on the magnetic field is not
experimentally resolved. These Fabry-Pérot oscillations are clearly visible in
the differentiated differential conductance in Fig. 11, where they appear as
small oscillations over the full top-gate and magnetic field range.

Figure 11: Differentiated differential conductance at $20$ mK for different
magnetic fields with a vertical shift with $\alpha=4\mathrm{e^{2}/(hVT)}$.
Only magnetic fields with magnitudes of multiples of $0.2$ T are shown for
clarity. The blue curves correspond to the magnetic field values given in Fig.
2(a). For all magnetic field values and over the full plotted voltage range
repetitive behavior corresponding to Fabry-Pérot oscillations is visible.

An experimental example of the dependence of this conductance contribution on
$V_{\text{TG}}$ and magnetic field is shown in Fig. 12(a) for the case of
vanishing back-gate and split-gate voltage and a theoretical plot based on Eq.
(37) in Fig. 12(b). The Fabry-Pérot resonances are seen for all magnetic field
values and over the whole top-gate voltage range. These peaks, in contrast to
the Zeeman-split subbands, only weakly depend on magnetic field, since the
Zeeman splitting in the 2D bulk is smaller than in the QPC region.

Figure 12: (a) Oscillations of conductance $\delta G=G-\bar{G}$for
$V_{\text{BG}}=V_{\text{SG}}=0$. The smooth background conductance $\bar{G}$
is obtained by means of a Savitzky-Golay Filter of the measured data. (b)
Fabry-Pérot oscillations given by Eq. (37) in a magnetic field. The Fermi
vectors in magnetic field were obtained by calculating the gap from Eq. (10)
and the chemical potential, Eq. (7), for the given voltage and $n=c(V-V_{0})$.
The value $c=0.8\times 3.9\times 10^{15}$ was determined in Quantum Hall
measurements and $V_{0}=-1.2$ V is the position of the Dirac point for
$V_{\text{BG}}=V_{\text{SG}}=0$. The chosen length $L=230$ nm corresponds to
the distance between the leads and the split-gate fingers. The maxima and
minima of oscillations stay nearly parallel over a large range of magnetic
field.

While the plots in Fig. 12 agree qualitatively, there are two points to keep
in mind. The theoretical plot was obtained using no residual density, which is
certainly not the case in the experiment, and does not account for the
peculiarity of the screening in the experimental setup in the presence of the
split gates. Even if the split-gate voltage is zero, the split gates affect
the electrostatics of the setup by locally screening the top gate and
developing mirror charges for carriers in this region. In addition, the
dielectric layer in the split-gate region is noticeably thinner. This
introduces inhomogeneity in the middle of the sample, and the length scale
corresponding to the distance between the leads and the split gates naturally
appears.

In order to fully reproduce the experimentally observed pattern one would need
a full electrostatic simulation. One very apparent difference is the fact that
the period stays nearly constant in the experiment but not in the theory.
While we did take into account the influence of the top-gate voltage on the
density and the gap, the voltage also changes the boundary condition at the
contacts and close to the split-gate fingers. Moreover we neglected any
present residual density, which might change the position within the spectrum
and thus the top-gate dependence. Finally, the presence of the split gates
will naturally induce a breaking of translational symmetry, as the side
boundaries do, which were not taken into account. For additional Fabry-Pérot
interference data in the same sample, the reader is referred to the
Supplemental Material of Ref. [5]. A more thorough study of Fabry-Pérot
interferences in BLG can be found in Ref. [115].

##  Appendix C Effective low-energy theory

In order to derive the conductance of the system considered in the main text,
we start with the effective two-band model for BLG, Eq. (1). In this Appendix,
we discuss how to obtain this low-energy approximation and how it is affected
by the magnetic field, as well as by possible terms describing spin-orbit
interaction.

Close to the $K$ point, the full four-band Hamiltonian is given by

$\displaystyle\mathcal{H}_{0}$
$\displaystyle=\begin{pmatrix}\epsilon_{A1}&v\pi^{\dagger}&-v_{4}\pi^{\dagger}&v_{3}\pi,\\\
v\pi&\epsilon_{B1}&\gamma_{1}&-v_{4}\pi^{\dagger},\\\
-v_{4}\pi&\gamma_{1}&\epsilon_{A2}&v\pi^{\dagger},\\\
v_{3}\pi^{\dagger}&-v_{4}\pi&v\pi&\epsilon_{B2}\end{pmatrix},$ (38)

where

$\displaystyle\epsilon_{A1}$ $\displaystyle=\frac{1}{2}(-U+\delta_{AB}),$ (39)
$\displaystyle\epsilon_{B1}$
$\displaystyle=\frac{1}{2}(-U+2\Delta^{\prime}-\delta_{AB}),$ (40)
$\displaystyle\epsilon_{A2}$
$\displaystyle=\frac{1}{2}(-U+2\Delta^{\prime}+\delta_{AB}),$ (41)
$\displaystyle\epsilon_{B2}$ $\displaystyle=\frac{1}{2}(U-\delta_{AB}).$ (42)

It acts on the four-component wave-function according to

$\displaystyle\mathcal{H}\begin{pmatrix}\psi_{A1}\\\ \psi_{B1}\\\ \psi_{A2}\\\
\psi_{B2}\end{pmatrix}$ $\displaystyle=E\begin{pmatrix}\psi_{A1}\\\
\psi_{B1}\\\ \psi_{A2}\\\ \psi_{B2}\end{pmatrix}.$ (43)

The spin degree of freedom is included through

$\displaystyle\mathcal{H}_{i}=\mathcal{H}_{0}\otimes\hat{s}_{0}$ (44)

in the spin-degenerate sector, and the Zeeman splitting is introduced via

$\displaystyle\mathcal{H}_{Z}=\Delta
E_{Z}\,\hat{\mu}_{0}\otimes\hat{\sigma}_{0}\otimes\hat{s}_{z},$ (45)

where $\mu,\sigma\text{ and }s$ are the Pauli matrices in layer, sublattice,
and spin space, respectively. According to [122], the two types of intrinsic
spin-orbit interaction allowed by the symmetry of the problem are

$\displaystyle\mathcal{H}^{\text{so}}_{1}$
$\displaystyle=\xi\lambda_{1}\hat{\mu}_{0}\otimes\hat{\sigma}_{z}\otimes\hat{s}_{z},$
(46) $\displaystyle\mathcal{H}^{\text{so}}_{2}$
$\displaystyle=\xi\lambda_{2}\hat{\mu}_{z}\otimes\hat{\sigma}_{0}\otimes\hat{s}_{z},$
(47)

where $\xi=\pm$ corresponds to valley $K$ ($K^{\prime}$). Following [12], we
additionally introduce an extrinsic spin-orbit interaction of the form

$\displaystyle\mathcal{H}^{\text{so}}_{3}=\xi\,\mathrm{diag}(\lambda_{u},-\lambda_{u},\lambda_{d},-\lambda_{d})\otimes\hat{s}_{z}.$
(48)

For the effective low-energy theory, $|E|\ll\gamma_{1}$, we follow the
derivation of Ref. [69]. The basis states are reordered to
$(\psi_{A1},\psi_{B2},\psi_{A2},\psi_{B1})\otimes(\uparrow,\downarrow)$, where
the first half corresponds to the low-energy, non-dimer states and the second
half to the dimer states, which are coupled by the large energy $\gamma_{1}$.
From now on, all terms in the Hamiltonian are reordered according to this
basis. Then, we can define the Greens function of the total Hamiltonian
$\mathcal{H}=\mathcal{H}_{i}+\mathcal{H}_{Z}+\mathcal{H}^{\text{so}}_{i}$ as
follows:

$\displaystyle\mathcal{H}$ $\displaystyle=\begin{pmatrix}H_{11}&H_{12}\\\
H_{21}&H_{22}\end{pmatrix},$ (49) $\displaystyle G$
$\displaystyle=\begin{pmatrix}G_{11}&G_{12}\\\
G_{21}&G_{22}\end{pmatrix}=(\mathcal{H}-E)^{-1}$
$\displaystyle=\begin{pmatrix}G_{11}^{(0)-1}&H_{12}\\\
H_{21}&G_{22}^{(0)-1}\end{pmatrix}^{-1},$ (50) $\displaystyle
G_{\alpha\alpha}^{(0)}$ $\displaystyle=(H_{\alpha\alpha}-E)^{-1}.$ (51)

The goal is to find a closed expression for $G_{11}$ which is then used to
define the new Hamiltonian $\mathcal{H}_{2}$ according to

$\displaystyle G_{11}$
$\displaystyle=(\mathcal{H}_{2}-E)^{-1}\quad\Leftrightarrow\quad\mathcal{H}_{2}=G_{11}^{-1}+E.$
(52)

We find

$\displaystyle
G_{11}=\left(1-G_{11}^{(0)}H_{12}G_{22}^{(0)}H_{21}\right)^{-1}G_{11}^{(0)}$
(53)

and, thus,

$\displaystyle\mathcal{H}_{2}=G_{11}^{-1}+E=H_{11}-H_{12}G_{22}^{(0)}H_{21}.$
(54)

We now expand $G_{22}^{(0)}=(H_{22}-E)^{-1}$ in ${E/\gamma_{1}}\ll 1$.
Applying this procedure to the reordered full Hamiltonian produces, to linear
order in $U,\Delta^{\prime},\delta_{AB},v_{4},v_{3},\lambda,\Delta E_{Z}$, the
effective two-band Hamiltonian:

$\displaystyle\mathcal{H}_{2}=h_{0}+h_{U}+h_{3}+h_{\text{AB}}+h_{4}+h_{\Delta^{\prime}}+h_{Z}+h_{\text{so}},$
(55)

where

$\displaystyle
h_{0}=-\frac{v^{2}}{\gamma_{1}}\begin{pmatrix}0&(\pi^{\dagger})^{2}\\\
\pi^{2}&0\end{pmatrix}\otimes\hat{s}_{0},$ (56) $\displaystyle
h_{U}=-\frac{U}{2}\left[\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}-\frac{v^{2}}{\gamma_{1}^{2}}\begin{pmatrix}\pi^{\dagger}\pi&0\\\
0&-\pi\pi^{\dagger}\end{pmatrix}\right]\otimes\hat{s}_{0},$ (57)
$\displaystyle h_{3}=v_{3}\begin{pmatrix}0&\pi\\\
\pi^{\dagger}&0\end{pmatrix}\otimes\hat{s}_{0},$ (58) $\displaystyle
h_{\text{AB}}=\frac{\delta_{AB}}{2}\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}\otimes\hat{s}_{0},$ (59) $\displaystyle
h_{\Delta^{\prime}}=2\Delta^{\prime}\frac{v^{2}}{\gamma_{1}^{2}}\begin{pmatrix}\pi^{\dagger}\pi&0\\\
0&\pi\pi^{\dagger}\end{pmatrix}\otimes\hat{s}_{0},$ (60) $\displaystyle
h_{4}=2v_{4}\frac{v^{2}}{\gamma_{1}^{2}}\begin{pmatrix}\pi^{\dagger}\pi&0\\\
0&\pi\pi^{\dagger}\end{pmatrix}\otimes\hat{s}_{0},$ (61) $\displaystyle
h_{Z}=\frac{\Delta E_{Z}}{2}\begin{pmatrix}1&0\\\
0&1\end{pmatrix}\otimes\hat{s}_{z},$ (62) $\displaystyle
h_{\text{so}}=\xi\begin{pmatrix}\lambda_{1}+\lambda_{2}+\lambda_{u}&0\\\
0&-\lambda_{1}-\lambda_{2}-\lambda_{d}\end{pmatrix}\otimes\hat{s}_{z}.$ (63)

In the main text, we restrict ourselves to the terms $h_{0}$, $h_{U}$, and
$h_{Z}$. This is exactly equation (1) and (3). For all calculations, we
furthermore neglect the second term of $h_{U}$ that describes the Mexican-hat
feature of the spectrum. The only terms capable of lifting spin degeneracy are
$h_{Z}$ and the spin-orbit term

$\displaystyle\xi\begin{pmatrix}\lambda_{u}&0\\\
0&-\lambda_{d}\end{pmatrix}\otimes\hat{s}_{z}$ (64)

for asymmetry between the layers $\lambda_{u}\neq\lambda_{d}$, which can be
caused by the lack of mirror symmetry of the whole stack [123]. Because of the
valley index $\xi$ in this expression, the splitting is opposite in the two
valleys, so that there is no net spin splitting due to spin-orbit interaction
at all. If such a term is present in the Hamiltonian, it would lead to full
spin-valley splitting in an applied magnetic field, i.e., four steps of
$1\,e^{2}/h$. This is, however, not seen in the experiment. This type of
effect of spin-orbit coupling on the first conductance plateau in in-plane
magnetic fields for the parameters specified in Ref. [12] is shown in Fig. 13.

Figure 13:  Influence of a finite Kane-Mele spin-orbit coupling on the first
conductance plateau in in-plane magnetic field, including the effect of layer-
asymmetry encoded in $\lambda_{u}\neq\lambda_{d}$. In all plots we chose $g=2$
and $\alpha=0.1$ meV/T, thus the rightmost curve corresponds to $B=8$ T and
the temperature is $20$ mK. (a) No spin-orbit coupling, there is no splitting
at zero magnetic field. (b) and (d) Different spin-orbit couplings in the two
layers, finite splitting at zero magnetic field, full lifting of spin and
valley degeneracy at higher magnetic field. (c) Same spin-orbit coupling in
both layer, no splitting at zero magnetic field.

##  Appendix D Effect of tilted magnetic field

We have investigated the effect of a perpendicular magnetic field on the
quantized conductance in the same device in Ref. [5]. Large out-of-plane
magnetic fields lead to a valley splitting, similar to the Zeeman spin
splitting, with characteristic braiding behavior. Since we see neither a
lifting of the valley degree of freedom, which would lead to a full resolution
of conductance steps of $1\,e^{2}/h$ for large magnetic fields, nor any hint
at a non-linear splitting, we can exclude a large out-of-plane component of
the magnetic field. The presence of an appreciable out-of-plane component of
the magnetic field would also show up in a curving of the Fabry-Pérot
oscillations, which is also not observed here.

For small out-of-plane components the valley splitting is roughly linear [71]
and can be easily included into our model by adding a term $\tau g_{v}B$ to
the energy spectrum, where $\tau=\pm 1$ corresponds to the valleys $K$ and
$K^{\prime}$, respectively, and $g_{v}$ contains both the angular dependence
on the tilt angle and the magnetic moment due to the non-trivial Berry
curvature. The expected effect of the tilt on the conductance traces is shown
in Fig. 14. We see that a very small tilt does not lead to any noticeable
difference, while a bigger one leads to a full lifting of all degeneracies at
strong magnetic fields. This is in contrast to quantum dots in BLG, see, e.g.,
Ref. [124], where all four single-particle energies can be extracted at all
values of magnetic field due to their additional charging energies and one can
construct an effective $g$-factor by combining spin and valley splitting in a
specific way, that would get enhanced over the bare spin Landé $g$-factor for
one combination, while reducing it for the other combination. Since we do not
observe a valley splitting, this effect would exactly average out in our case.

Figure 14:  Influence of finite tilt angle on the conductance traces for the
cases $\mathbf{a}$ and $\mathbf{c1}$. In all plots, we chose $g_{s}=4$ and
$\alpha=1.3$ meV/T, thus the rightmost curve corresponds to $B=8$ T and the
temperature is $20$ mK. The configurations corresponding those in Fig. 6 are
marked inside the panels; the index 1,2 corresponds to the $K,K^{\prime}$
valley respectively. Panels (a) and (b): No finite tilt and thus no valley
splitting. Panels (c) and (d): Finite tilt angle resulting in $g_{v}=0.04$. No
change compared to $g_{v}$=0 is visible. Panels (e) and (f): Finite tilt angle
resulting in $g_{v}=0.4$. At a certain magnetic field this would lead to the
appearance of additional conductance steps at $1\,e^{2}/h$ and $3\,e^{2}/h$,
which are not observed in our experiment.

##  Appendix E QPCs in BLG

In the experiment, we electrostatically induce a constriction in the shape of
a QPC. While the full self-consistent treatment of this electrostatic problem
is very involved, the main features can be described by the inclusion of a
local potential in the corresponding Schrödinger equation, thus neglecting the
coupling between the Schrödinger and the Poisson equations. There are two
slightly different ways to include this potential profile. We start by
summarizing the most common one for a normal two-dimensional Schrödinger
equation, and proceed with an alternative method for BLG. The obtained energy
spectrum would extend dispersion (13) in the main text and thus also the
density (15).

###  E.1 Projection procedure for the Schrödinger equation

In this section, we briefly recapitulate the projection procedure for a QPC in
a conventional 2D electron gas, as discussed in Ref. [73]. The Hamiltonian is
given by

$\displaystyle\mathcal{H}$
$\displaystyle=-\frac{\hbar^{2}}{2m}(\partial_{x}^{2}+\partial_{y}^{2})+V(x,y),$
(65) $\displaystyle V(x,y)$
$\displaystyle=V_{\text{QPC}}(x,y)+V_{\text{lead}}(x),$ (66) $\displaystyle
V_{\text{QPC}}(x,y)$
$\displaystyle=\frac{V_{0}}{\cosh(x/a)^{2}}+\frac{m\omega_{y}^{2}}{2}y^{2},$
(67) $\displaystyle V_{\text{lead}}(x)$
$\displaystyle=V_{l}\left[\Theta(x-d)+\Theta(-d-x)\right],$ (68)

where one models the QPC as a harmonic potential in the transverse direction
and $V_{l}$ corresponds to the potential difference at the leads. We expand
the wavefunction in transverse modes $\chi_{nx}(y)$ according to

$\displaystyle\Psi(x,y)=\sum_{n}\phi_{n}(x)\chi_{nx}(y),$ (69)

where the complete orthonormal basis $\chi_{nx}(y)$ satisfies

$\displaystyle\left[-\frac{\hbar^{2}}{2m}\partial_{y}^{2}+V(x,y)\right]\chi_{nx}(y)=\epsilon_{n}(x)\chi_{nx}(y).$
(70)

For the given potential, these eigenvalues are given by

$\displaystyle\epsilon_{n}(x)$
$\displaystyle=\hbar\omega_{y}\left(n+\frac{1}{2}\right)+\frac{V_{0}}{\cosh(x/a)^{2}}$
$\displaystyle+V_{l}\left[\Theta(x-d)+\Theta(-d-x)\right]$

and from this we get the projected 1d problem

$\displaystyle\left[-\frac{\hbar^{2}}{2m}\partial_{x}^{2}+\epsilon_{n}(x)\right]\phi_{n}(x)=E\phi_{n}(x).$
(71)

At low energies, only the lowest transverse modes contribute, and we can
approximate the full solution as $\Psi(x,y)=\phi_{0}(x)\chi_{0x}(y)$. The
exact form of the top of the effective 1D potential $\epsilon_{n}(x)$
determines at what position the additional $0.7$ shoulder appears; for the
usual parabolic barrier top it appears very close to $0.7\times 2\,e^{2}/h$
[56].

###  E.2 Procedure for BLG

In the present case of BLG it is easier to include the constriction by means
of boundary conditions than a real local potential. The Hamiltonian (1) acts
as

$\displaystyle\hat{H}\begin{pmatrix}\psi_{A1\uparrow}\\\
\psi_{A1\downarrow}\\\ \psi_{B2\uparrow}\\\ \psi_{B2\downarrow}\end{pmatrix}$
$\displaystyle=E\begin{pmatrix}\psi_{A1\uparrow}\\\ \psi_{A1\downarrow}\\\
\psi_{B2\uparrow}\\\ \psi_{B2\downarrow}\end{pmatrix}$ (72)

on the four-component wave-function in the spin and sublattice space. These
four coupled second-order equations can be decoupled into a fourth-order one
and we get for the first two components:

$\displaystyle\left[\left(\frac{U}{2}-U\frac{v^{2}}{\gamma_{1}^{2}}\pi^{\dagger}\pi\right)^{2}+\frac{(\pi^{\dagger}\pi)^{2}}{(2m)^{2}}\right]\psi_{A1\sigma}$
$\displaystyle=\left(E-\sigma\frac{\Delta
E_{Z}}{2}\right)^{2}\psi_{A1\sigma}.$ (73)

Here, we have used that, without a magnetic field, the momentum operators
commute.

We expand the wave-function $\psi_{A1\sigma}$ in transverse modes
$\chi_{nx\sigma}(y)$ as

$\displaystyle\psi_{A1\sigma}(x,y)$
$\displaystyle=\sum_{n}\phi_{n\sigma}(x)\chi_{nx\sigma}(y),$ (74)

which leads to a new differential equation, where the $x$ and $y$ component
are still coupled. We already assume, that
$\chi_{nx\sigma}(y)\propto\sin(k_{nx\sigma}y)[\cos(k_{nx\sigma}y)]$ if the
solution is antisymmetric [symmetric] when describing the QPC by imposing
hard-wall boundary conditions along the $y$ direction. The width of the
channel $W(x)$ depends smoothly on $x$ and we get standing waves with
wavevector

$k_{n}(x)=\frac{n\pi}{W(x)}.$

We decouple the components by neglecting all $x$ derivatives of
$\chi_{nx\sigma}(y)$, which leads to the effective 1D equation:

$\displaystyle\left[\left(\frac{U}{2}+\hbar^{2}U\frac{v^{2}}{\gamma_{1}^{2}}[\partial_{x}^{2}-k_{n}^{2}(x)]\right)^{2}+\frac{\hbar^{4}[\partial_{x}^{2}-k_{n}^{2}(x)]^{2}}{(2m)^{2}}\right]\phi_{n\sigma}(x)$
$\displaystyle=(E-\sigma\frac{\Delta E_{Z}}{2})^{2}\phi_{n\sigma}(x),$ (75)

where the constriction $W(x)$ acts as an effective 1D potential
$E_{n}(x)=\frac{\hbar^{2}n^{2}\pi^{2}}{2mW(x)^{2}}$. At low energies, only the
lowest transverse mode contributes and we can approximate the full solution as
$\psi_{A1\sigma}=\phi_{1\sigma}(x)\chi_{1x\sigma}(y).$ Choosing, for example,
$W(x)=\cosh(x/L)$ we get a very realistic 1D potential containing terms of the
form $1/\cosh(x/L)^{2}$.

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