###### Abstract

We study electronic and thermoelectric transport in a type I Weyl semimetal
nanojunction, with a torsional dislocation defect, in the presence of an
external magnetic field parallel to the dislocation axis. The defect is
modeled in a cylindrical geometry, as a combination of a gauge field
accounting for torsional strain, and a delta-potential barrier for the lattice
mismatch effect. In the Landauer formalism, we find that due to the
combination of strain and magnetic field, the electric current exhibits chiral
valley-polarization, and the conductance displays the signature of Landau
levels. We also compute the thermal transport coefficients, where a high
thermopower and a large figure of merit are predicted for the junction.

###### keywords:

Weyl semimetals; transport; torsion; dislocation; magnetic field.

1 1 0 https://doi.org/ Thermo-magneto-electric transport through a torsion
dislocation in a type I Weyl Semimetal Thermo-magneto-electric transport
through a torsion dislocation in a type I Weyl Semimetal Daniel Bonilla 1,
Enrique Muñoz 1,2,∗ and Rodrigo Soto-Garrido 1 Daniel Bonilla, Enrique Muñoz
and Rodrigo Soto-Garrido Bonilla, D.; Muñoz, E.; Soto-Garrido, R.
Correspondence<EMAIL_ADDRESS>Tel.: +56-2-2354-7625

##  1 Introduction

Since the experimental discovery of topological insulators, there has been an
increasing interest in the search for other materials that may exhibit non-
trivial topological properties Hasan and Kane (2010); Qi and Zhang (2011);
Bansil et al. (2016); Vanderbilt (2018); Moessner and Moore (2021). A
remarkable example of three-dimensional gapless topological materials are Weyl
semimetals (WSMs). First proposed theoretically, Wan et al. (2011); Fang et
al. (2012); Ruan et al. (2016); Vafek and Vishwanath (2014); Yan and Felser
(2017); Armitage et al. (2018); Burkov (2018) WSMs were recently discovered
experimentally on TaAs crystalsXu et al. (2015) and observed in photonic
crystals Lu et al. (2015). In a WSM, the conduction and valence bands touch
each other in an even number of points with linear dispersion, referred as
Weyl nodes. These nodes are protected from being gapped because they are
monopolar sources of Berry curvature, and hence their charge (chirality) is a
topological invariant Burkov (2018). In the vicinity of these nodes, low
energy conducting states can be described as Weyl fermions, i.e. massless
quasi-particles with pseudo-relativistic Dirac linear dispersion Vafek and
Vishwanath (2014); Yan and Felser (2017); Armitage et al. (2018, 2018); Burkov
(2018). In addition to their intrinsic electronic spin, in Weyl fermions
chirality determines the projection of the spin over their momentum direction,
a condition often referred to as “spin-momentum locked states”. While Type I
WSMs fully respect Lorentz covariance, such condition is not satisfied in Type
II WSMs, where the Dirac cones are strongly tilted Vanderbilt (2018).

The presence of Weyl nodes in the bulk spectrum determines the emergence of
Fermi arcsXu et al. (2015), the chiral anomaly, and the chiral magnetic
effect, among other remarkable properties Vanderbilt (2018). Perhaps the most
studied is the chiral anomaly, which is the non-conservation of the
independent chiral currents in the presence of non-orthogonal electric and
magnetic fields. Therefore, considerable attention has been paid to understand
the electronic transport properties of WSMs Hosur and Qi (2013); Hu et al.
(2019); Nagaosa et al. (2020). For instance, there are recent works on charge
transport Hosur et al. (2012) in the presence of spin-orbit coupled impurities
Liu et al. (2017), electrochemicalFlores-Calderón and Martín-Ruiz (2021) and
nonlinear transport induced by Berry curvature dipoles Zeng et al. (2021).
Regarding thermoelectric transport in WSMs, it is known that the linear Dirac-
type dispersion induces a non-trivial dependence on the chemical potential
Lundgren et al. (2014). Somewhat less explored are the effects of mechanical
strain and deformations in WSMs. From the theory perspective, it has been
proposed that different sorts of elastic strains can be modeled as gauge
fields in WSMs Cortijo et al. (2015, 2016); Arjona and Vozmediano (2018),
similar to the case of graphene. In previous works, we have studied the
effects of strain and magnetic field on the electronic Soto-Garrido and Muñoz
(2018); Soto-Garrido et al. (2020) and the thermoelectric Muñoz and Soto-
Garrido (2019) transport properties of WSMs, using the Landauer ballistic
formalism in combination with the quantum mechanical scattering cross-
sectionsMuñoz and Soto-Garrido (2017). The study of thermoelectric transport
properties is a field of permanent interest, not only regarding WSMs but in a
wide range of materials. For instance, there is recent literature involving
the experimental determination of the thermoelectric properties (in particular
the figure of merit ZT) of Cu-Sn–based thiospinel compounds Bourgès et al.
(2020), and SnTe-based materials Muchtar et al. (2021).

This work focuses on the effect of a Repulsive Delta-Shell potential (RDSP),
in addition to the torsional strain and the external magnetic field studied
early on in Refs. Muñoz and Soto-Garrido (2019); Soto-Garrido and Muñoz
(2018), on the thermoelectric transport properties of type I WSMs. The RDSP is
a toy model for the surface repulsion produced by the mismatch between the
lattices of the strained and the non-strained WSMs. The effect of the delta
potential in the context of the Dirac equation is to produce a chiral rotation
between the spinors on either side of the boundary that represents the support
of the delta function Calkin et al. (1987); Benguria et al. (2000). The
rotation angle is proportional to the strength of the delta barrier and
depends on the chirality of the fermion scattered. This RDSP model for the
lattice mismatch of the dislocation is combined with a gauge field
representation of the torsional strain in a cylindrical geometry. In addition,
an external magnetic field directed along the axis of the dislocation is
imposed at the junction, as depicted in Fig. 1.

The paper is organized as follows. In Sec. Theory we establish the Hamiltonian
for the model, and describe each of its contributions. Then, we proceed with
the Landauer formulation for transport accross the junction, first analyzing
the sole effect of the RDSP that describes the lattice mismatch, and finally
for the full system that includes the torsional strain and the external
magnetic field at the WSM junction, with mathematical details presented in the
Supplementary Material’s file. The analysis and discussion of the results are
presented in Sec. Results, with a final summary and conclusions presented in
Sec. Discussion.

Figure 1: A pictorial description of the system under consideration: A WSM
slab of dimensions $L\times W$, with a cylindrical region of radius $a$
submitted to a combination of torsional strain and an external magnetic field
$\mathbf{B}^{\xi}=(B+\xi B_{S})\mathbf{\hat{z}}$ and an RDSP on the boundary
surface of the cylinder.

##  2 Theory

As a minimal model for a WSM, we start by considering a free Hamiltonian
describing Weyl quasiparticles in the vicinity of each of the nodal points
with opposite chirality $\xi=\pm 1$,

$\displaystyle
H_{\xi}(\mathbf{k})=v_{F}\left(\sigma_{1}k_{x}+\sigma_{2}k_{y}+\xi\sigma_{3}k_{z}\right)$
(1)

with $\sigma_{j}$ ($j=1,2,3$) the Pauli matrices. The spectrum of this “free”
WSM Hamiltonian is given by (for $\lambda=\pm$ the band index)

$\displaystyle E_{\lambda,k}=\lambda\hbar v_{F}|\mathbf{k}|.$ (2)

As depicted in Fig. 1, we consider a nanojunction where the WSM is submitted
to torsional strain in a cylindrical region of radius $a$, and we further
assume that the axial length $L$ satisfies $L\gg a$. As discussed in Ref.
Arjona and Vozmediano (2018), the mechanical strain effect can be incorporated
as a gauge field
$\mathbf{A}_{S}=B_{S}/2\left(-y\hat{e}_{1}+x\hat{e}_{2}\right)$, where the
constant $B_{S}$ plays the role of a pseudo-magnetic field. Moreover, if a
true magnetic field is imposed upon the junction along the axis of the
dislocation, i.e. $\mathbf{B}=\hat{e}_{3}B_{0}$, then the combination is
described by a node-dependent gauge field
$\mathbf{A}_{\xi}=B_{\xi}/2\left(-y\hat{e}_{1}+x\hat{e}_{2}\right)$, with
$B_{\xi}=\left(B_{0}+\xi B_{S}\right)$ an effective pseudo-magnetic field. In
addition to this combined effect, already discussed in our previous workSoto-
Garrido and Muñoz (2018); Muñoz and Soto-Garrido (2019), we here consider also
the lattice mismatch near the boundary of the dislocation. As a simple model
for this effect, we include a RDSP potential of the form
$V_{RD}(r)=V_{0}\delta(r-a)$. Therefore, the quasi-particle states inside the
dislocation region correspond to the solutions of the eigenvalue problem

$\displaystyle\left[H_{\xi}\left(\mathbf{k}+\mathbf{A}_{\xi}\right)+V_{RD}(r)\right]|\Psi_{n,m}^{(\lambda,\xi)}\rangle=E_{\lambda,n}^{\xi}|\Psi_{n,m}^{(\lambda,\xi)}.\rangle$
(3)

The spectrum inside the cylindrical regionMuñoz and Soto-Garrido (2017)
corresponds to relativistic Landau levels with an effective magnetic field
$B_{\xi}$ that is node-dependent

$\displaystyle E_{\lambda,n}^{\xi}=\lambda\hbar
v_{F}\sqrt{2n|B_{\xi}|/\tilde{\phi}_{0}+k_{z}^{2}},$ (4)

with $\tilde{\phi}_{0}=(v_{F}/c)\hbar/e$ a modified magnetic flux quantum
expressed in terms of the carrier velocity $v_{F}$. The effect of the RDSP
potential (see Sec. 1 of the Supplementary Material for mathematical details)
is to introduce a rotation in the pseudo-spinor components across the
dislocation boundary $r=a$, with an “angle” $\alpha=V_{0}/(\hbar v_{F})$

$\displaystyle\left.\Psi_{n,m}^{(\lambda,\xi)}(\mathbf{r})\right|_{r\rightarrow
a^{+}}=\begin{pmatrix}\cos\alpha&-\sin\alpha\\\
\sin\alpha&\cos\alpha\end{pmatrix}\left.\Psi_{n,m}^{(\lambda,\xi)}(\mathbf{r})\right|_{r\rightarrow
a^{-}}.$ (5)

###  2.1 Transmission and Landauer conductance

In the Landauer formalism, we define an energy-dependent transmission
coefficient along the $x$-direction based on the scattering differential
cross-section of the junction,

$\displaystyle\bar{T}(E)=\int_{-\pi/2}^{\pi/2}d\phi\cos\phi\frac{1}{\sigma(E)}\frac{d\sigma}{d\phi},$
(6)

where $\sigma(E)$ is the total scattering cross-section at energy $E$. In what
follows, we shall assume that the cylindrical dislocation satisfies $L\gg
1/k_{F}$. For instanceNeupane et al. (2014), in TaAs where $b\sim 0.08$
$\mathring{\text{A}}^{-1}$ and $v_{F}\sim 1.3\times 10^{5}$ m/s, we have
$1/k_{F}\sim 9$ $\mathring{\text{A}}$, so even a slab of a few microns is
already in the range of validity of this assumption. Moreover, for Cd3As2,
$b\sim 0.2$ $\mathring{\text{A}}^{-1}$ and $v_{F}\sim 1.5\times 10^{6}$ m/s,
$1/k_{F}\sim 0.8$ $\mathring{\text{A}}$ Neupane et al. (2014), and hence the
applicability of this criteria is even more striking in this second example.
Therefore, for $L\gg 1/k_{F}$, the differential cross-section is given in
terms of the scattering phase-shift $\delta_{m}$ for each angular momentum
channel $m$Muñoz and Soto-Garrido (2017); Soto-Garrido and Muñoz (2018), and
integrating over the scattering angle (see Sec. 2 of the Supplementary
Material for mathematical details) we obtain the corresponding total cross-
sectionMuñoz and Soto-Garrido (2017); Soto-Garrido and Muñoz (2018)
$\sigma/L=\frac{4}{k_{\perp}}\sum_{m=-\infty}^{\infty}\sin^{2}\delta_{m}$.

Let us first consider the effect of the RDSP only. For this case, the current
is expressed in terms of the transmission function $\mathcal{T}(E)$, evaluated
at the free energy eigenvalues $E_{\lambda,k_{\perp}}$ defined in Eq. (2)

$\displaystyle
I=2ev_{F}\sum_{\lambda}\int_{0}^{\infty}dk_{\perp}\mathcal{T}(E_{\lambda,k_{\perp}})\left[f_{L}(E_{\lambda,k_{\perp}})-f_{R}(E_{\lambda,k_{\perp}})\right],$
(7)

where $f_{L/R}(E)=\left(\exp[(E-\mu_{L/R})/(k_{B}T_{L/R})]+1\right)^{-1}$ are
the Fermi-Dirac distributions at the chemical potential $\mu_{L/R}$ and
temperature $T_{L/R}$ of the left (L) and right (R) metallic contacts,
respectively (see Sec. 3 of the Supplementary Material for mathematical
details). The factor of 2 accounts for the (symmetric) contribution from each
chiral node $\xi=\pm$ (see Fig. 2). The corresponding expression for the
differential conductance $G(T,V)=\left.\partial I/\partial V\right|_{T}$
through the junction is

$G(T,V)=2\frac{e^{2}v_{F}}{k_{B}T}\sum_{\lambda}\int_{0}^{\infty}dk_{\perp}\mathcal{T}(E_{\lambda,k_{\perp}})f_{L}(E_{\lambda,k_{\perp}})\left[1-f_{L}(E_{\lambda,k_{\perp}})\right].$
(8)

Let us now consider the transmission through the junction in its full level of
complexity, i.e. including the RDSP for the lattice mismatch, as well as the
torsional strain (included via the gauge field model) and the external
magnetic field along the axis of the cylindrical dislocation. For this case,
scattering is no longer symmetric for each chirality, as seen in the Landau
level spectrum $E_{\lambda,n}^{\xi}$ defined in Eq. (4) and in the
corresponding scattering phase shift (Fig. 2). Therefore, the current for each
chirality $\xi=\pm$ is expressed by the transmission function
$\mathcal{T}(E)$,

$\displaystyle
I_{\xi}=ev_{F}\sum_{n,\lambda}\mathcal{T}(E_{\lambda,n}^{\xi})\left[f_{L}(E_{\lambda,n}^{\xi})-f_{R}(E_{\lambda,n}^{\xi})\right],$
(9)

with the total current defined by the superposition of both chiral
contributions $I=I_{+}+I_{-}$. As before, the differential conductance through
the junction is obtained as the voltage-derivative of the expression above,

$\displaystyle
G(T,V)=\frac{e^{2}v_{F}}{k_{B}T}\sum_{\lambda,n,\xi}\mathcal{T}(E_{\lambda,n}^{\xi})f_{L}(E_{\lambda,n}^{\xi})\left[1-f_{L}(E_{\lambda,n}^{\xi})\right].$
(10)

###  2.2 Thermoelectric transport coefficients

The energy current accross the junction arising from each chiral node
contribution $\xi=\pm$ is also expressed in terms of the transmission function
$\mathcal{T}(E)$ as followsMuñoz and Soto-Garrido (2019)

$\displaystyle\dot{U}_{\xi}=v_{F}\sum_{n,\lambda}E_{\lambda,n}^{\xi}\mathcal{T}(E_{\lambda,n}^{\xi})\left[f_{L}(E_{\lambda,n}^{\xi})-f_{R}(E_{\lambda,n}^{\xi})\right].$
(11)

On the other hand, according to the basic thermodynamic relation $TdS=dU-\mu
dN$ between entropy $S$, internal energy $U$ and particle number $N$, the net
heat current transmitted across the junction arising from the node
$\mathbf{K}_{\xi}$ (for $\xi=\pm$) is

$\dot{Q}_{\xi}=\dot{U}_{\xi}-\left(\mu_{L}\dot{N}_{L}^{\xi}-\mu_{R}\dot{N}_{R}^{\xi}\right).$
(12)

The thermal conductance is defined, as usual, under the condition that the net
electric current vanishes ($I=0$)

$\displaystyle\kappa(T,V)=-\left.\frac{\partial\dot{Q}}{\partial\Delta
T}\right|_{I=0}=-\left.\frac{\partial\dot{U}}{\partial\Delta T}\right|_{I=0},$
(13)

where $\Delta T=T_{R}-T_{L}$ is the temperature difference between the
contacts and the total heat flux is given by the superposition from both Weyl
nodes $\dot{Q}=\dot{Q}_{+}+\dot{Q}_{-}$, and similar relations hold for the
total energy flux $\dot{U}$ and the total electric current $I$. The condition
of a vanishing electric current defines an implicit relation between the
voltage difference and the thermal gradient across the junction, by $I(\Delta
T,V,T)=0$. Therefore, we obtain the Seebeck coefficient by applying the
implicit function theoremMuñoz and Soto-Garrido (2019)

$\displaystyle S(T,V)=-\left.\frac{\partial V}{\partial\Delta
T}\right|_{I=0,T}=\frac{\left.\displaystyle\frac{\partial I}{\partial\Delta
T}\right|_{T,V}}{\left.\displaystyle\frac{\partial I}{\partial
V}\right|_{T,\Delta T}},$ (14)

where the temperature difference accross the junction $\Delta T(V,T)$ is
obtained as the solution of the equation $I(T,V,\Delta T)=0$. Following the
argument above, the thermal conductance defined in Eq.(13) is calculated by
means of the chain rule and in terms of the Seebeck coefficientMuñoz and Soto-
Garrido (2019)

$\kappa(T,V)=-\left.\frac{\partial\dot{U}}{\partial\Delta
T}\right|_{T,V}+S(T,V)\left.\frac{\partial\dot{U}}{\partial
V}\right|_{T,\Delta T}.$ (15)

From the general relations discussed above among the thermoelectric transport
coefficients, we obtain the explicit formulae (see Sec. 4 of the Supplementary
Material for mathematical details) for the thermal conductance

$\displaystyle\kappa(T,V)$ $\displaystyle=$
$\displaystyle\frac{v_{F}}{k_{B}(T+\Delta
T)^{2}}\sum_{\xi,\lambda,n}\mathcal{T}(E_{\lambda,n}^{\xi})E_{\lambda,n}^{\xi}\left[E_{\lambda,n}^{\xi}-\mu\right]f_{R}(E_{\lambda,n}^{\xi})\left[1-f_{R}(E_{\lambda,n}^{\xi})\right]$
(16)
$\displaystyle\quad+\,\,S(T,V)\frac{ev_{F}}{k_{B}T}\sum_{\lambda,n,\xi}\mathcal{T}(E_{\lambda,n}^{\xi})E_{\lambda,n}^{\xi}\,f_{L}(E_{\lambda,n}^{\xi})\left[1-f_{L}(E_{\lambda,n}^{\xi})\right],$

and for the Seebeck coefficient

$\displaystyle
S(T,V)=-\frac{T\sum_{\lambda,n,\xi}\mathcal{T}(E_{\lambda,n}^{\xi})\left(E_{\lambda,n}^{\xi}-\mu\right)f_{R}(E_{\lambda,n}^{\xi})\left[1-f_{R}(E_{\lambda,n}^{\xi})\right]}{e(T+\Delta
T)^{2}\sum_{\lambda,n,\xi}\mathcal{T}(E_{\lambda,n}^{\xi})f_{L}(E_{\lambda,n}^{\xi})\left[1-f_{L}(E_{\lambda,n}^{\xi})\right]}.$
(17)

##  3 Results

In this section we will apply the analytical results derived in Section Theory
to study the response of the transport coefficients to the relevant physical
parameters of the model, such as the external magnetic field $B_{0}$, the
torsion angle $\theta$, the temperature $T$, and the applied bias voltage $V$
Soto-Garrido and Muñoz (2018); Muñoz and Soto-Garrido (2019). In particular,
we will analyze the effect of the RDSP, as a model for the lattice mismatch,
by varying the $V_{0}$ parameter that characterizes the strength of the
repulsive barrier, expressed in terms of the “spinor rotation” angle
$\alpha=V_{0}/\hbar v_{F}$.

By considering first the case where only the lattice mismatch effect is
present (RDSP) (see Eq. (29) in the Sec. 2 of the Supplementary Material), we
notice that the phase shifts depend on the parameter $V_{0}$ through
$\tan\alpha$. Therefore, the results depend on $\alpha$ periodically, with
period $\pi$, as seen in Fig. 2. It is also clear (from Eq. (29) in the Sec. 2
of the Supplementary Material) that if the only scattering mechanism is the
RDSP, the transmission is maximum for $\alpha=n\pi$, with $n$ an integer. At
these particular “magic” values, despite the presence of the lattice mismatch,
the corresponding interfacial energy barrier becomes transparent to the Weyl
fermions of both chiralities $\xi=\pm$.

In order to study the additional effect of torsion and magnetic field for
TaAs, we estimate Pikulin et al. (2016) $B_{S}\approx 1.8\times 10^{-3}T$ per
angular degree of torsion. Furthermore, we have that the modified flux quantum
in this material is approximately
$\displaystyle\tilde{\phi}_{0}\equiv\frac{\hbar
v_{F}}{e}=\frac{1}{2\pi}\frac{v_{F}}{c}\frac{hc}{e}=\frac{1}{2\pi}\frac{1.5}{300}\cdot
4.14\times 10^{5}$ T$\mathring{\text{A}}^{2}\approx 330$
T$\mathring{\text{A}}^{2}$. Using these values, we obtain the simple relation
between the torsional angle $\theta$ (in degrees) and the pseudo-magnetic
field $B_{S}$ representing strain

$B_{S}a^{2}=1.36\,\theta\,\tilde{\phi}_{0}.$ (18)

In this case, the analytical expression for the scattering phase shift is
given by Eq. (32) in the Sec. 2 of the Supplementary Material. We notice that
the effect of the barrier is again given by $\tan\alpha$, and hence it becomes
minimal at “magic” values of $\alpha=n\pi$, i.e. integer multiples of $\pi$.
However, in this second case the scattering phase-shift does not vanish, due
to the residual combined effect of torsion and magnetic field. This can be
seen in Fig. 3, where for $\alpha=0,\,\pi,\,2\pi$, $\tan\delta_{m}\neq 0$, in
contrast to Fig. 2. Actually, the value of $\tan\delta_{m}$ for $\alpha=n\pi$
($n$ integer) and the consequences of the scattering by the combined magnetic
field and torsion, but in the absence of the lattice mismatch barrier, was
extensively discussed in our previous worksMuñoz and Soto-Garrido (2017);
Soto-Garrido and Muñoz (2018); Muñoz and Soto-Garrido (2019).

Another important aspect to notice is that, when we only consider the lattice
mismatch effect, the scattering phase shift is symmetric for both chiral nodes
$\xi=\pm 1$, as seen in Subfigs. (a) and (b) in Fig. 2. In contrast, when the
magnetic field and torsion are present, this symmetry is broken, as displayed
in Subfigs. (a) and (b) in Fig. 3. As we explained in Ref. Soto-Garrido and
Muñoz (2018), this occurs because the magnitude of the pseudo-field that
combines torsion and magnetic field $B_{\xi}=B_{0}+\xi B_{S}$ depends on the
sign of the node chirality, a manifestation of the chiral anomaly which can be
also observed in the electric current (see Fig. 8).

(a) $\xi=+1$

(b) $\xi=-1$

Figure 2: (Color online) Analytical expression for $\tan\delta_{m}$ (in
Eq.(29) in Supporting Information) plotted as a function of $\alpha$. The
plots are computed for a wave vector $k_{\perp}\sim 1/a$ and an orbital
angular momentum $m=1$. The subfig. (a) is for a node index $\xi=1$; the red
(solid) line corresponds to a band index $\lambda=1$ and the blue (dashed)
line is for $\lambda=-1$. The subfig. (b) is for a node index $\xi=-1$; the
orange (solid) line corresponds to a band index $\lambda=1$ and the purple
(dashed) line is for $\lambda=-1$.

(a) $\xi=-1$

(b) $\xi=+1$

Figure 3: (Color online) Analytical expression for $\tan\delta_{m}$ (in
Eq.(32) of Supporting Information) plotted as a function of $\alpha$. The
plots are computed for a quantum number $n=1$, orbital angular momentum $m=1$,
an external magnetic field $B_{0}a^{2}=25\tilde{\phi}_{0}$ and a torsion angle
$\theta=10^{\circ}$. The Subfig. (a) is for a node index $\xi=1$; the red
(solid) line corresponds to a band index $\lambda=1$ and the blue (dashed)
line is for $\lambda=-1$. The Subfig. (b) is for a node index $\xi=-1$; the
orange (solid) line corresponds to a band index $\lambda=1$ and the purple
(dashed) line is for $\lambda=-1$.

###  3.1 Electronic transport

The electric current (in units of $ev_{F}/a$) is computed from Eq. (7) for the
case of the RSDP only, in the absence of torsion and magnetic field. Fig. 4
shows the periodic dependence of the total current as a function of the
dimensionless parameter $\alpha=V_{0}/(\hbar v_{F})$ that characterizes the
magnitude of the lattice mismatch barrier, and for a temperature $T=0.2\,\hbar
v_{F}/k_{B}a$. As expected, the maxima of transmission occur for the “magic
angles” $\alpha=n\pi$ ($n$ integer), and the overall effect of the barrier is
to slightly reduce the current, reaching minimal values near $\alpha=\pi/4$
and $\alpha=3\pi/4$, respectively. The behavior of the current, for the same
temperature, as a function of the bias voltage is presented in Fig. 4. As can
be seen, for low temperatures the current across the junction displays an
approximately quadratic dependence on the applied bias voltage $eV$ (in units
of $\hbar v_{F}/a$), that leads to an approximately linear dependence of the
differential conductance (in units of $e^{2}/\hbar$) on the bias voltage in
Fig. 5.

Figure 4: (Color online) Electric current (in units of $ev_{F}/a$) computed
from the analytical expression in Eq. (7) for the case of the RDSP barrier
alone and $T=0.2\,\hbar v_{F}/k_{B}a$: (a) Plotted as a function of the
applied bias $eV$ (in units of $\hbar v_{F}/a$) and (b) plotted as a function
of $\alpha$ (dimensionless).  Figure 5: (Color online) Differential
conductance (in units of $e^{2}/\hbar$), for the RDSP barrier alone, plotted
as function of applied bias $eV$ (in units of $\hbar v_{F}/a$) for
$\alpha=3\pi/4$ and $T=0.2\,\hbar v_{F}/k_{B}a$, computed from Eq. (8).

Figure 6: (Color online) (a) Electric current (in units of $ev_{F}/a$) as
function of applied bias $eV$ (in units of $\hbar v_{F}/a$), computed from the
analytical expression in Eq. (9) at zero temperature, for an external magnetic
field $B_{0}a^{2}=25\tilde{\phi}_{0}$ and $\alpha=3\pi/4$. The blue line
corresponds to a twist angle $\theta=0^{\circ}$, red is for
$\theta=5^{\circ}$, green is for $\theta=10^{\circ}$ and the orange line
corresponds to $\theta=15^{\circ}$. (b) Comparison of electric currents at
zero temperature, for $B_{0}a^{2}=25\tilde{\phi}_{0}$ and $\theta=15^{\circ}$:
the purple line is for $\alpha=0$, and the orange line corresponds to
$\alpha=3\pi/4$.

Figure 7: (Color online) (a) Electric current (in units of $ev_{F}/a$) plotted
as function of applied bias $eV$ (in units of $\hbar v_{F}/a$), computed from
the analytical expression in Eq. (9) at $T=0.4\,\hbar v_{F}/k_{B}a$, for an
external magnetic field $B_{0}a^{2}=25\tilde{\phi}_{0}$ and $\alpha=3\pi/4$.
The blue line corresponds to a twist angle $\theta=0^{\circ}$, red is for
$\theta=5^{\circ}$, green is for $\theta=10^{\circ}$ and the orange line
corresponds to $\theta=15^{\circ}$. (b) Comparison of electric currents at
$T=0.4\,\hbar v_{F}/k_{B}a$, for $B_{0}a^{2}=25\tilde{\phi}_{0}$ and
$\theta=15^{\circ}$: the purple line is for $\alpha=0$ and orange corresponds
to $\alpha=3\pi/4$.

Figure 8: (Color online) (a) Node-polarized components of the currents
computed for an external magnetic field $B_{0}a^{2}=25\tilde{\phi}_{0}$, a
torsion angle $\theta=15^{\circ}$ and $\alpha=3\pi/4$: the magenta line
corresponds to the contribution of $I_{-}$ arising from the $\mathbf{K}_{-}$
node and the green line corresponds to the contribution of $I_{+}$ arising
from the $\mathbf{K}_{+}$ node. (b) Comparison of conductance (in units of
$e^{2}/\hbar$) as a function of the bias voltage $eV$ (in units of $\hbar
v_{F}/a$) for the case of an external magnetic field
$B_{0}a^{2}=25\tilde{\phi}_{0}$, $T=0.1\hbar v_{F}/k_{B}a$, and a torsion
angle $\theta=15^{\circ}$. The purple line is for $\alpha=0$, orange
corresponds to $\alpha=3\pi/4$ and the magenta line corresponds to the
difference between both $\Delta G=G(\alpha=0)-G(\alpha=3\pi/4)$.

Now, when we include the combined effect of the delta barrier, the external
magnetic field, and the torsion strain, the current is calculated from the
analytical expression in Eq. (9). Fig. 6 presents the total current as a
function of voltage at zero temperature, an external field
$B_{0}a^{2}=25\tilde{\phi}_{0}$, a value of $\alpha=3\pi/4$, and different
values of the torsion angle $\theta$. A remarkable feature at zero temperature
is the appearance of plateaus in the current; the elastic scattering condition
explains it because the incident particle energy must be resonant to one of
the pseudo-Landau levels inside the cylinder, and hence each subsequent
plateau corresponds to the transmission of an additional Landau level. Such
plateaus tend to be smoothed with increasing temperature, as can be seen in
Fig. 7. As we discussed in our previous work in the absence of the RDSP
contributionSoto-Garrido and Muñoz (2018); Muñoz and Soto-Garrido (2019), for
a fixed external magnetic field the electric current increases with the
torsion angle $\theta$. This effect is due to an enhanced transmission of the
Weyl fermions arising from the $\mathbf{K}_{-}$ node, since for this
particular chirality $\xi=-1$ the magnitude of the effective pseudo-magnetic
field $|B^{-}|=|B_{0}-B_{S}|$ is smaller, thus increasing the spectral density
of pseudo-Landau levels ($\sim\sqrt{|B^{\xi}|n}$) for chirality $\xi=-1$, and
consequently an increase in the number of channels available for transmission.
Fig. 8 presents the difference between the currents originated at each node.
Furthermore, for a fixed torsion angle, the transmitted current decreases as
the external magnetic field increasesSoto-Garrido and Muñoz (2018). This
effect occurs because, by increasing the external field $B_{0}$ (for a fixed
torsion field $B_{S}$), the magnitude of the effective pseudo-magnetic field
$|B^{\xi}|=|B_{0}+\xi B_{S}|$ increases for both chiralities $\xi=\pm$, thus
reducing the density of Landau levels available for transmission. Fig. 6 and
Fig. 7 present a comparison of the current, for $\alpha=0$ and
$\alpha=3\pi/4$, at $T=0$ and $T=0.4\,\hbar v_{F}/k_{B}a$, respectively. We
see that the magnitude of the current is reduced while the position of the
plateaus remains fixed. This effect is more significant at higher bias
voltage, and is due to the repulsive effect of the RDSP barrier that reduces
the transmission. Finally, Fig. 8 compares the conductance (in units of
$e^{2}/\hbar$) as a function of the bias voltage $eV$ (in units of $\hbar
v_{F}/a$) for the case of an external magnetic field
$B_{0}a^{2}=25\tilde{\phi}_{0}$, a torsion angle $\theta=15^{\circ}$,
$T=0.1\hbar v_{F}/k_{B}a$, and two different values of the lattice mismatch
RDSP barrier, $\alpha=0$ and $\alpha=3\pi/4$, respectively. As expected, the
conductance shows peaks as a consequence of the plateaus observed in the
current. The effect of the RDSP barrier is to reduce the conductance without
affecting the position of the peaks.

###  3.2 Thermal transport

Let us now analyze the thermoelectric transport coefficients. Fig. 9 presents
the electric conductance (in units of $e^{2}/\hbar$) as a function of
temperature (in units of $\hbar v_{F}/k_{B}a$) for an external field
$B_{0}a^{2}=25\tilde{\phi}_{0}$, a bias voltage $eV=0.5\,\hbar v_{F}/a$,
$\alpha=3\pi/4$, and different torsion angles $\theta$. On the other hand,
Fig. 10 presents the thermal conductance (in units of $e^{2}/\hbar$) as a
function of temperature, for the same set of parameters as in Fig. 9. Both
transport coefficients show a monotonic increase with temperature. This effect
occurs because Weyl fermions are the same entities transporting current and
energy, since as we explained in Sec. Theory, in the present work we only
consider the electronic contribution to the transport. Other effects, such as
phonons, will be analyzed in future work.

From Fig. 9 and Fig. 10, it is clear that both transport coefficients, i.e.
the thermal and the electric conductance, increase with torsion. This effect,
already observed in our previous work in the absence of the lattice mismatch
barrier contributionMuñoz and Soto-Garrido (2019), is due to the enhancement
of the pseudo-Landau levels density of states arising from the $\xi=-1$ chiral
node, as already discussed in the previous section.

Figure 9: (Color online) (a) Conductance (in units of $e^{2}/\hbar$) as a
function of temperature (in units of $\hbar v_{F}/k_{B}a$) for external
$B_{0}a^{2}=25\tilde{\phi}_{0}$, a bias $eV=0.5\,\hbar v_{F}/a$ and
$\alpha=3\pi/4$. The blue line corresponds to $\theta=0^{\circ}$, red is for
$\theta=5^{\circ}$, green is for $\theta=10^{\circ}$ and the orange line
corresponds to $\theta=15^{\circ}$. (b) Comparison of conductance for
$\theta=15^{\circ}$: the purple line is for $\alpha=0$ whereas the orange line
is for $\alpha=3\pi/4$.

Figure 10: (Color online)(a) Thermal conductance (in units of $k_{B}v_{F}/a$)
as a function of temperature (in units of $\hbar v_{F}/k_{B}a$), computed from
the analytical expression in Eq.(16), for external
$B_{0}a^{2}=25\tilde{\phi}_{0}$, a bias $eV=0.5\,\hbar v_{F}/a$ and
$\alpha=3\pi/4$. The blue line corresponds to $\theta=0^{\circ}$, red is for
$\theta=5^{\circ}$, green is for $\theta=10^{\circ}$ and the orange line
corresponds to $\theta=15^{\circ}$. (b) Comparison of the thermal conductance
for $\theta=15^{\circ}$: the purple line is for $\alpha=0$ whereas the orange
line is for $\alpha=3\pi/4$.

Fig. 11 shows the Seebeck coefficient (in units of $k_{B}/e$) as a function of
temperature (in units of $\hbar v_{F}/k_{B}a$), for the same set of parameters
as in Figs. 9 and 10. We have chosen the chemical potential as $\mu=1.0\,\hbar
v_{F}/a>0$, such that the negative charge carriers dominate the transport,
which explains the negative sign of the Seebeck coefficient. As can be seen,
the slope of $S$ is very steep at low temperatures and varies monotonically.
On the other hand, the absolute value of $S$ increases with the torsion angle
$\theta$.

Now, let us discuss the effect of the RDSP barrier representing the lattice
mismatch via the parameter $\alpha$. Figs. 9 and 10 present a comparison of
the $\alpha=0$ and $\alpha=3\pi/4$ cases for the electric and thermal
conductance, respectively, as a function of temperature. For the case of
electric conductance the effect is hardly noticeable, with a tiny decrease of
the conductance for the case with the RDSP barrier present, $\alpha\neq 0$. On
the contrary, the effect is most notorious for the case of the thermal
conductance, which increases when the lattice mismatch barrier is present. In
both cases, the effect tends to be more significant at high temperatures.

Figure 11: (Color online)(a) Seebeck coefficient (in units of $k_{B}/e$)
computed from the analytical expression in Eq.(17) as a function of
temperature $T$ (in units of $\hbar v_{F}/k_{B}a$). The plot is for fixed
$B_{0}a^{2}=25\tilde{\phi}_{0}$, a bias $eV=0.5\,\hbar v_{F}/a$ and
$\alpha=3\pi/4$. The blue line corresponds to $\theta=0^{\circ}$, red is for
$\theta=5^{\circ}$, green is for $\theta=10^{\circ}$ and the orange line
corresponds to $\theta=15^{\circ}$. (b) Comparison of the Seebeck coefficient
for $\theta=15^{\circ}$: the purple line is for $\alpha=0$ whereas the orange
line is for $\alpha=3\pi/4$.

For the characterization of the thermoelectric performance of this WSM
junction, a useful quantity is the magnitude of the figure of merit $ZT$,
defined by the well known formula

$ZT=S^{2}\frac{T\,G(T,V)}{\kappa(T,V)}.$ (19)

Fig. 12 presents the figure of merit $ZT$ (dimensionless), as a function of
temperature and for various torsion angles $\theta$. As we showed in our
previous work in the absence of the lattice mismatch effect Muñoz and Soto-
Garrido (2019), it is important to notice that extremely high values of $ZT$
can be achieved through the combination of external magnetic field and
torsional strain. The value of $ZT$ increases with the torsion angle $\theta$,
and the effect is more appreciable at low temperatures. The effect of the RDSP
barrier representing lattice mismatch by the parameter $\alpha$ is shown in
Fig. 12. The presence of the barrier produces a small reduction of the figure
of merit at high temperatures.

It is also pertinent to explore the deviation from the metallic behavior by
studying the Lorenz number as a function of temperature. The Lorenz number is
defined by the formula

$L=\frac{\kappa(T,V)}{TG(T,V)}.$ (20)

The Lorenz number is represented, at fixed bias and magnetic field, as a
function of temperature for different values of torsion in Fig. 13. Strong
deviations from the Wiedemann-Franz law are observed at low temperatures. This
effect occurs because the electronic conductance exhibits a non-metallic
behavior at low temperatures, due to the discrete pseudo-Landau level
spectrum, as can be seen in the staircase pattern in Fig. 6. It is precisely
this effect that explains the extremely high $ZT$ values at low temperatures,
in agreement with the experimental evidence reported Skinner and Fu (2018)
that suggested values as high as $ZT\sim 10$. In contrast with the $ZT$
behavior, the presence of the delta barrier is to increase the Lorenz number
at high temperatures, as can be seen in Fig. 13. This trend is explained
since, as we discussed previously, at high temperatures the thermal
conductance increases with the delta barrier, while the electric conductance
slightly decreases.

Figure 12: (Color online)(a) The figure of merit ZT (dimensionless) as a
function of temperature (in units of $\hbar\,v_{F}/k_{B}a$), calculated for
fixed $B_{0}a^{2}=25\tilde{\phi}_{0}$, a bias $eV=0.5\,\hbar v_{F}/a$ and
$\alpha=3\pi/4$. The blue line corresponds to $\theta=0^{\circ}$, red is for
$\theta=5^{\circ}$, green is for $\theta=10^{\circ}$ and the orange line
corresponds to $\theta=15^{\circ}$. (b) Comparison of the figure of merit ZT
for $\theta=15^{\circ}$: the purple line is for $\alpha=0$ whereas the orange
line is for $\alpha=3\pi/4$.

Figure 13: (Color online) (a) The Lorenz number (in units of
$k_{B}^{2}/e^{2}$) as a function of temperature (in units of
$\hbar\,v_{F}/k_{B}a$), calculated for fixed $B_{0}a^{2}=25\tilde{\phi}_{0}$,
a bias $eV=0.5\,\hbar v_{F}/a$ and $\alpha=3\pi/4$. The blue line corresponds
to $\theta=0^{\circ}$, red is for $\theta=5^{\circ}$, green is for
$\theta=10^{\circ}$ and the orange line corresponds to $\theta=15^{\circ}$.
(b) Comparison of the Lorenz number for $\theta=15^{\circ}$: the purple line
is for $\alpha=0$ whereas the orange line is for $\alpha=3\pi/4$.

##  4 Discussion

In this work, we studied the thermoelectric transport properties of a type I
Weyl semimetal with a torsional defect, in the presence of an external
magnetic field along the axis of the dislocation in a cylindrical geometry.
Moreover, the effect of torsion was modeled by a combination of a gauge field
representation, and a repulsive delta-shell potential (RDSP) representing the
lattice mismatch at the edge of the cylindrical region. We remark that the
mechanical gauge field, in combination with the external magnetic field
imposed upon the region, combine into an effective node-dependent pseudo-
magnetic field $\mathbf{B}^{\xi}=\mathbf{B}+\xi\mathbf{B}_{S}$ (for $\xi=\pm)$
that breaks time-reversal symmetry and hence the nodal symmetry. Therefore,
our analysis shows that the electronic states within the region correspond to
effective node-polarized Landau levels, leading to a node-polarization effect
of the total electric current $I=I_{+}+I_{-}$. In particular, the low-
temperature differential conductance displays the corresponding characteristic
trend of discrete peaks corresponding to each of such Landau levels. We also
demonstrated that the effect of the lattice-mismatch, represented by the RDSP,
is periodic in the strength of the repulsive barrier $V_{0}$, in the form
$\tan(V_{0}/\hbar v_{F})$, thus revealing the presence of "magic angles" (the
zeroes of the tangent) where the barrier becomes transparent. This somewhat
surprising effect is a manifestation of the Klein-tunneling effect of Dirac’s
theory, observed in this particular context and geometry. Finally, we also
studied the thermoelectric transport coefficients, thermal conductivity and
Seebeck, as a function of temperature, external magnetic field, torsion and
strength of the lattice mismatch (RDSP).

We would like to emphasize that our analytical equations, and the
corresponding figures presented in the Results section, are expressed in terms
of dimensionless groups involving structural parameters (such as the radius
$a$ of the torsional defect and the dimensions $W$ and $L$ of the WSM slab) as
well as the material’s specific parameters (such as the Fermi velocity
$v_{F}$). This has the advantage that the equations presented are quite
general, and hence our theoretical predictions for the transport coefficients
can be compared with specific experimental measurements by choosing the
appropriate material-dependent parameters. For instance, choosing the
dimensions of the slab as $W\sim L\sim 50\,\,\text{nm}$ and the radius of the
cylindrical strip as $a\sim 15\,\,\text{nm}$, we obtain an electrical
resistivity $\rho\sim 2.15\times 10^{-4}\,\,\Omega\text{m}$ which is within
the range reported in Ref. Han et al. (2020) ($\rho\sim 2\times
10^{-2}\,\,\Omega\text{m}$ for Bi and $\rho\sim 10^{-5}\,\,\Omega\text{m}$ for
TaP). On the other hand, for the case of the thermal conductivity, using the
Fermi velocity $v_{F}\sim 1.5\times 10^{6}\,\,\text{m}/\text{s}$ for the
material Cd3As2 Neupane et al. (2014), and the same values for $a$, $L$, and
$W$ as before we found a value of $\kappa\sim 6.6\,\,\text{W}/\text{mK}$ which
is of the same order of magnitude to those reported in Ref. Skinner and Fu
(2018) ($\sim 3\,\,\text{W}/\text{mK}$ for Pb1-xSnxSe) and in Ref. Han et al.
(2020) ($\sim 5-25\,\,\text{W}/\text{mK}$ for TaP).

Finally, we point out that our theoretical calculations suggest that a very
high figure of merit can be obtained from such configuration (torsional strain
+ RDSP), thus constituting a very interesting candidate for thermoelectric
applications in energy harvesting.

This research was funded by Fondecyt grants number 1190361 and 1200399, as
well as by ANID PIA Anillo ACT/192023.

The authors declare no conflict of interest.

The following abbreviations are used in this manuscript:

WSM | Weyl semimetal
---|---
RDSP | Repulsive delta-shell potential

References

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The method based on the Rayleigh quotient (7.13) is inconclusive if
$\mathcal{V}^{\prime\prime}(s)>0$ because both the numerator and the
denominator attain negative values for some $v\in\mathcal{H}_{\varepsilon}$.

##  8\. Conclusion

This work can be summarized as follows. We have considered a new NLS model
with intensity-dependent dispersion (1.6) as an alternative of the cubic
defocusing NLS equation (1.1). We have shown that the same black soliton has
better stability properties in the NLS model (1.6) in several aspects. The NLS
model provides a natural Hilbert $L^{2}$ space with exponential weights, where
the linearized stability problem admits only isolated eigenvalues of finite
multiplicity and no finite accumulation points. The energetic stability
argument is obtained with four rather than three orthogonality conditions,
where the additional constraint is due to the new scaling symmetry (1.13). The
black soliton is continued to the family of traveling dark solitons for every
speed compared to the finite speed cutoff in the cubic NLS equation (1.1).
Finally, the black soliton remains orbitally stable in the presence of a small
decaying potential if it is pinned to the minimum point of the effective
potential compared to either oscillatory or monotone instability of the black
soliton of the cubic NLS equation with a small decaying potential.

We end this paper with a list of further open problems.

1. We have not attempted to address local well-posedness of the new NLS model (1.6) in $\Sigma\cap\mathcal{H}$, where $\Sigma$ is the energy space (3.2) and $\mathcal{H}$ is the exponentially weighted $L^{2}(\mathbb{R})$ space (3.12). We conjecture that the initial-value problem is locally well-posed in the set of functions $\mathcal{F}$ in (1.7) for which $|u(t,x)|\leq 1$ for $(t,x)\in\mathbb{R}\times\mathbb{R}$, but this problem is a subject on its own. Recent results on local well-posedness of such NLS models with zero boundary conditions can be found in [18].

2. Another related problem is the asymptotic stability of black solitons. By using ideas of [15], the asymptotic stability analysis can be developed provided that the local well-posedness problem for the NLS model (1.6) is solved first.

3. Orbital stability of the continuous family of dark solitons with any speed $c\in\mathbb{R}$ can be considered by using the exponentially weighted Hilbert space $\mathcal{H}_{c,\omega=1}$ in Remark 6.1. According to the standard orbital stability criterion [5, 23], the dark soliton with profile $U_{c}$ is orbitally stable if the mapping $c\mapsto P(U_{c})$ is monotonically increasing. It follows from the expansion (5.24) that the stability is satisfied for small $c\in(-c_{0},c_{0})$ with some $c_{0}>0$. However, since $U_{c}$ is only available implicitly, the precise stability criterion can only be checked numerically for larger values of $c\in\mathbb{R}$ or verified with computer-assisted proofs [27].

4. It would be interesting to justify the new NLS model (1.6) as a reduction of the Maxwell equations in the framework of either temporal or spatial nonlinear optics. Justification of the extended NLS models of the type

$i(1-\partial_{x}^{2})\psi_{t}+\psi_{xx}+2(1-|\psi|^{2})\psi=0$

was considered in [10, Section 4.1.4] in the framework of the NLS models with
a full dispersion relation. However, the relevant analysis did not incorporate
the intensity-dependent dispersion. Justification of the NLS model (1.6) is
open for further studies.

5. One can think of the regularization of the NLS model (1.6) with bounded intensity-dependent dispersion, similar to the one used in [31] for the NLS model (1.14). Local well-posedness of the regularized model in the space of functions with nonzero boundary conditions at infinity can be shown by using the general analysis of [12]. It is expected that the stability properties of the black solitons analyzed in our work will persist in the time evolution of the regularized version of the NLS model (1.6).

Acknowledgement. The work of D. E. Pelinovsky is partially supported by
AvHumboldt Foundation as Humboldt Reseach Award. The work of M. Plum is
supported by Deutsche Forschungsgemeinschaft (German Research Foundation) -
Project-ID 258734477 - SFB 1173.

##  Appendix A Integration by parts in $\mathcal{H}$

We start with the following technical estimates.

######  Lemma A.1.

For every $w\in L^{2}(\mathbb{R})$, it is true that

$\displaystyle\left\|\cosh(\cdot)\int_{\cdot}^{\infty}{\rm
sech}(t)w(t)dt\right\|_{L^{2}(0,\infty)}\leq 2\|w\|_{L^{2}(0,\infty)},$ (A.1)
$\displaystyle\left\|\cosh(\cdot)\int_{-\infty}^{\cdot}{\rm
sech}(t)w(t)dt\right\|_{L^{2}(-\infty,0)}\leq 2\|w\|_{L^{2}(-\infty,0)},$
(A.2)

and

$\displaystyle\left\|{\rm
sech}(\cdot)\int_{0}^{\cdot}\cosh(t)w(t)dt\right\|_{L^{2}(\mathbb{R})}\leq
2\|w\|_{L^{2}(\mathbb{R})}.$ (A.3)

###### Proof.

In order to prove (A.1), we integrate by parts and obtain for each $y>0$:

$\displaystyle\int_{0}^{y}\cosh^{2}(x)\left|\int_{x}^{\infty}{\rm
sech}(t)w(t)dt\right|^{2}dx$
$\displaystyle=\left[\frac{1}{2}(x+\cosh(x)\sinh(x))\left|\int_{x}^{\infty}{\rm
sech}(t)w(t)dt\right|^{2}\right]\biggr{|}_{x=0}^{x=y}$
$\displaystyle+\int_{0}^{y}(x{\rm sech}(x)+\sinh(x)){\rm
Re}\left(\bar{w}(x)\int_{x}^{\infty}{\rm sech}(t)w(t)dt\right)dx$

The first and second terms are estimated separately with Cauchy–Schwarz
inequality as

$\displaystyle\frac{1}{2}(y+\cosh(y)\sinh(y))\left|\int_{y}^{\infty}{\rm
sech}(t)w(t)dt\right|^{2}$
$\displaystyle\leq\frac{1}{2}(y+\cosh(y)\sinh(y))(1-\tanh(y))\int_{y}^{\infty}|w(t)|^{2}dt$
$\displaystyle\leq\int_{y}^{\infty}|w(t)|^{2}dt$

and

$\displaystyle\int_{0}^{y}(x{\rm
sech}(x)+\sinh(x))|\bar{w}(x)|\left|\int_{x}^{\infty}{\rm
sech}(t)w(t)dt\right|dx$ $\displaystyle\leq
2\int_{0}^{y}\cosh(x)|\bar{w}(x)|\left|\int_{x}^{\infty}{\rm
sech}(t)w(t)dt\right|dx$ $\displaystyle\leq
2\left(\int_{0}^{y}\cosh^{2}(x)\left|\int_{x}^{\infty}{\rm
sech}(t)w(t)dt\right|^{2}dx\right)^{1/2}\left(\int_{0}^{y}|w(x)|^{2}dx\right)^{1/2}$
$\displaystyle\leq\frac{1}{2}\int_{0}^{y}\cosh^{2}(x)\left|\int_{x}^{\infty}{\rm
sech}(t)w(t)dt\right|^{2}dx+2\int_{0}^{y}|w(x)|^{2}dx,$

where we have used $2ab\leq\frac{1}{2}a^{2}+2b^{2}$ in the last line.
Combining the estimates, we obtain

$\displaystyle\int_{0}^{y}\cosh^{2}(x)\left|\int_{x}^{\infty}{\rm
sech}(t)w(t)dt\right|^{2}dx\leq
2\int_{y}^{\infty}|w(x)|^{2}dx+4\int_{0}^{y}|w(x)|^{2}dx,$

which yields the bound (A.1) in the limit $y\to\infty$. A similar bound (A.2)
follows analogously for $y<0$.

In order to prove (A.3), we again integrate by parts and obtain for each
$y>0$:

$\displaystyle\int_{0}^{y}{\rm
sech}^{2}(x)\left|\int_{0}^{x}\cosh(t)w(t)dt\right|^{2}dx$
$\displaystyle=\left[(\tanh(x)-1)\left|\int_{0}^{x}\cosh(t)w(t)dt\right|^{2}\right]\biggr{|}_{x=0}^{x=y}$
$\displaystyle+2\int_{0}^{y}(\cosh(x)-\sinh(x)){\rm
Re}\left(\bar{w}(x)\int_{0}^{x}\cosh(t)w(t)dt\right)dx$

The first term is negative while the second term is again estimated with
Cauchy–Schwarz inequality as

$\displaystyle 2\int_{0}^{y}{\rm
sech}(x)|w(x)|\left|\int_{0}^{x}\cosh(t)w(t)dt\right|dx$ $\displaystyle\leq
2\left(\int_{0}^{y}{\rm
sech}^{2}(x)\left|\int_{0}^{x}\cosh(t)w(t)dt\right|^{2}dx\right)^{1/2}\left(\int_{0}^{y}|w(x)|^{2}dx\right)^{1/2}.$

In the limit $y\to\infty$, this gives the bound

$\displaystyle\int_{0}^{\infty}{\rm
sech}^{2}(x)\left|\int_{0}^{x}\cosh(t)w(t)dt\right|^{2}dx\leq
4\int_{0}^{\infty}|w(t)|^{2}dt.$

A similar is obtained analogously for $y<0$, which yields together the bound
(A.3). ∎

Using the estimates in Lemma A.1, we prove the integration by parts formula
for both function spaces. Here $\langle\cdot,\cdot\rangle$ denotes the
standard inner product in $L^{2}(\mathbb{R})$.

######  Lemma A.2.

For every $f\in\mathcal{H}^{2}_{-}$ and $g\in\mathcal{H}^{1}_{-}$, it is true
that $\langle f^{\prime},g^{\prime}\rangle=-\langle
f^{\prime\prime},g\rangle$.

###### Proof.

First, we check that if $f\in\mathcal{H}^{2}_{-}$, then
$\cosh(\cdot)f^{\prime}\in L^{2}(\mathbb{R})$ and so
$\sinh(\cdot)f^{\prime}\in L^{2}(\mathbb{R})$. Indeed, if
$f\in\mathcal{H}^{2}_{-}$, then $f^{\prime}\in L^{2}(\mathbb{R})$ and
$\cosh(\cdot)f^{\prime\prime}\in L^{2}(\mathbb{R})$. It is then clear that
$f^{\prime}\in H^{1}(\mathbb{R})$ so that $f^{\prime}\in C(\mathbb{R})$ and
$f^{\prime}(x)\to 0$ as $|x|\to\infty$ by Sobolev’s embedding. Therefore, we
write

$f^{\prime}(x)=-\int_{x}^{\infty}f^{\prime\prime}(t)dt=-\int_{x}^{\infty}{\rm
sech}(t)\left[\cosh(t)f^{\prime\prime}(t)\right]dt,$

so that $\cosh(\cdot)f^{\prime}\in L^{2}(0,\infty)$ by the bound (A.1) with
$w:=\cosh(\cdot)f^{\prime\prime}\in L^{2}(\mathbb{R})$. Similarly, we obtain
$\cosh(\cdot)f^{\prime}\in L^{2}(-\infty,0)$ by the bound (A.2).

Next, since

$\frac{d}{dx}\left[\cosh(x)f^{\prime}(x)\right]=\cosh(x)f^{\prime\prime}(x)+\sinh(x)f^{\prime}(x),$
(A.4)

we infer that if $f\in\mathcal{H}^{2}_{-}$, then $\cosh(\cdot)f^{\prime}\in
H^{1}(\mathbb{R})$, which implies that $\cosh(x)f^{\prime}(x)\to 0$ as
$|x|\to\infty$. Similarly, if $g\in\mathcal{H}^{1}_{-}$, we have ${\rm
sech}(\cdot)g\in H^{1}(\mathbb{R})$ since ${\rm sech}(\cdot)g\in
L^{2}(\mathbb{R})$, ${\rm sech}(\cdot)g^{\prime}\in L^{2}(\mathbb{R})$, and

$\frac{d}{dx}\left[{\rm sech}(x)g(x)\right]={\rm sech}(x)g^{\prime}(x)-{\rm
sech}(x)\tanh(x)g(x).$ (A.5)

This implies that ${\rm sech}(x)g(x)\to 0$ as $|x|\to\infty$ so that

$f^{\prime}(x)g(x)=\left[\cosh(x)f^{\prime}(x)\right]\left[{\rm
sech}(x)g(x)\right]\to 0\quad\mbox{\rm as}\;\;|x|\to\infty$

and integration by parts yields $\langle f^{\prime},g^{\prime}\rangle=-\langle
f^{\prime\prime},g\rangle$. ∎

######  Lemma A.3.

For every $f\in\mathcal{H}^{2}_{+}$ and $g\in\mathcal{H}^{1}_{+}$, it is true
that $\langle f^{\prime},g^{\prime}\rangle=-\langle
f^{\prime\prime},g\rangle$.

###### Proof.

Here we recall that $\mathcal{H}^{1}_{+}\equiv H^{1}(\mathbb{R})$ and that if
$f\in\mathcal{H}^{2}_{+}$, then $f\in H^{1}(\mathbb{R})$ and
$\cosh(\cdot)(-f^{\prime\prime}+4f)\in L^{2}(\mathbb{R})$. This implies that
$(-f^{\prime\prime}+4f)\in L^{2}(\mathbb{R})$ and hence $f\in
H^{2}(\mathbb{R})$ so that $f(x),f^{\prime}(x)\to 0$ as $|x|\to\infty$ and the
integration by parts holds. ∎

##  Appendix B Proof of the expansion (6.13) with the bound (6.14)

We follow the ideas in the proof of Propositions 4 and 5 in [15] and the proof
of Lemma 4.2 in [11] but we give a self-contained presentation.

It is trivial to see that if $u\in\mathcal{H}^{1}_{-}$, then

$\|\sqrt{1-\varphi^{2}}u\|_{L^{\infty}}\leq\|u\|_{\mathcal{H}^{1}_{-}}.$ (B.1)

Indeed, since
$\|\sqrt{1-\varphi^{2}}u^{\prime}\|_{L^{2}}\leq\|u^{\prime}\|_{L^{2}}$, the
product rule (A.5) implies that

$\|\sqrt{1-\varphi^{2}}u\|_{H^{1}}\leq\|u\|_{\mathcal{H}^{1}_{-}}$

which yields (B.1) by Sobolev’s embedding of $H^{1}(\mathbb{R})$ into
$L^{\infty}(\mathbb{R})$. On the other hand, it is rather nontrivial that the
decomposition

$\psi=e^{i\theta}\left[U_{c,\omega}(\cdot+\zeta)+u(\cdot+\zeta)+iv(\cdot+\zeta)\right],$
(B.2)

with

$|c|+|\omega-1|+\|u\|_{\mathcal{H}^{1}_{-}}+\|v\|_{\mathcal{H}^{1}_{-}}+\|\eta\|_{L^{2}}\leq
C_{0}\epsilon,$ (B.3)

where

$\eta:=|\psi|^{2}-|U_{c,\omega}|^{2}=2u{\rm Re}(U_{c,\omega})+2v{\rm
Im}(U_{c,\omega})+u^{2}+v^{2},$ (B.4)

controls the $L^{\infty}$ norm of the perturbation. Nevertheless, this result
is given in the following lemma used for the proof of the bound (6.14). In
what follows, the constant $C>0$ may change from one line to another line.

######  Lemma B.1.

For every $c,\omega\in\mathbb{R}$ and $\psi\in\Sigma\cap\mathcal{H}$
satisfying (B.2) and (B.3), there exists $C>0$ such that

$\|u\|_{L^{\infty}}+\|v\|_{L^{\infty}}\leq C,\quad\|\eta\|_{L^{\infty}}\leq
C\epsilon.$ (B.5)

###### Proof.

Since

$(u+{\rm Re}(U_{c,\omega}))^{2}+(v+{\rm
Im}(U_{c,\omega}))^{2}=|U_{c,\omega}|^{2}+\eta,$

there exists $C>0$ such that

$\|u\|_{L^{\infty}}+\|v\|_{L^{\infty}}\leq C(1+\|\eta\|_{L^{\infty}})$ (B.6)

Using the bound

$\|\eta\|^{2}_{L^{\infty}}\leq\|\eta\|_{L^{2}}\|\eta^{\prime}\|_{L^{2}},$

the product rule for $\eta$ in (B.4), and the triangle inequality yields

$\displaystyle\|\eta\|^{2}_{L^{\infty}}$ $\displaystyle\leq
2\|\eta\|_{L^{2}}\left(\|{\rm
Re}(\bar{U}_{c,\omega}(u^{\prime}+iv^{\prime}))\|_{L^{2}}+\|{\rm
Re}(\bar{U}^{\prime}_{c,\omega}(u+iv))\|_{L^{2}}+\|uu^{\prime}\|_{L^{2}}+\|vv^{\prime}\|_{L^{2}}\right)$
$\displaystyle\leq
C\|\eta\|_{L^{2}}\left(\|u\|_{\mathcal{H}^{1}_{-}}+\|v\|_{\mathcal{H}^{1}_{-}}+\|u\|_{L^{\infty}}\|u^{\prime}\|_{L^{2}}+\|v\|_{L^{\infty}}\|v^{\prime}\|_{L^{2}}\right)$
$\displaystyle\leq
C\left(1+\|u\|_{L^{\infty}}+\|v\|_{L^{\infty}}\right)\left(\|u\|^{2}_{\mathcal{H}^{1}_{-}}+\|v\|^{2}_{\mathcal{H}^{1}_{-}}+\|\eta\|^{2}_{L^{2}}\right),$

where we have used the properties of $U_{c,\omega}:=U_{c}(\omega\cdot)$ from
Theorem 6 and the proximity of the norm in $\mathcal{H}_{c,\omega}$ and
$\mathcal{H}$ due to Remark 5.5 for $(c,\omega)$ near $(0,1)$. Using the
bounds (B.3) and (B.6) yields

$\displaystyle\|\eta\|^{2}_{L^{\infty}}\leq
C(1+\|\eta\|_{L^{\infty}})\epsilon^{2}$

Since $\epsilon>0$ is small, this is equivalent to $\|\eta\|_{L^{\infty}}\leq
C\epsilon$, which yields (B.5) due to (B.6). ∎

We are now ready to give the proof of the expansion (6.13) with the bound
(6.14). Since $P(\psi)$ is invariant under the two symmetries (1.5), we
translate $\psi\to e^{-i\theta}\psi(\cdot-\zeta)$. After translation, we fix
$R>0$ and split the integral in $P(\psi)$ on
$(-\infty,-R]\cup[-R,R]\cup[R,\infty)$. The outer and inner integrals are
treated differently based on the two representations (2.3) and (2.5).

###  B.1. Outer integrals on $(-\infty,-R]\cup[R,\infty)$

We use the following expression for this part of $P(\psi)$ denoted by
$P_{R}(\psi)$:

$P_{R}(\psi):=\frac{1}{2i}\int_{(-\infty,-R]\cup[R,\infty)}\frac{(1-|\psi|^{2})^{2}}{|\psi|^{2}}(\bar{\psi}\psi_{x}-\bar{\psi}_{x}\psi)dx.$

The integrand is nonsingular because $|\psi|^{2}\geq|U_{c,\omega}|^{2}-|\eta|$
is bounded away from zero on $(-\infty,-R]\cup[R,\infty)$ which follows from
the boundedness of $|U_{c,\omega}|^{2}$ away from zero and smallness of
$\|\eta\|_{L^{\infty}}$ according to the bound (B.5). We expand

$\displaystyle\bar{\psi}\psi_{x}-\bar{\psi}_{x}\psi=\bar{U}_{c,\omega}U^{\prime}_{c,\omega}-\bar{U}^{\prime}_{c,\omega}U_{c,\omega}+\nu+2i(uv^{\prime}-u^{\prime}v),$

where

$\displaystyle\nu:=2i{\rm Im}(U_{c,\omega}^{\prime})u-2i{\rm
Re}(U_{c,\omega}^{\prime})v-2i{\rm Im}(U_{c,\omega})u^{\prime}+2i{\rm
Re}(U_{c,\omega})v^{\prime},$

so that

$\displaystyle P_{R}(U_{c,\omega}+u+iv)$
$\displaystyle=P_{R}(U_{c,\omega})+P_{1}+P_{2},$

where

$\displaystyle
P_{1}:=\frac{1}{2i}\int\left[\frac{(1-|U_{c,\omega}|^{2})^{2}}{|U_{c,\omega}|^{2}}\nu-\frac{(1-|U_{c,\omega}|^{4})}{|U_{c,\omega}|^{4}}(\bar{U}_{c,\omega}U^{\prime}_{c,\omega}-\bar{U}^{\prime}_{c,\omega}U_{c,\omega})\eta\right]dx$

and

$\displaystyle P_{2}$
$\displaystyle:=\int\frac{(1-|U_{c,\omega}|^{2}-\eta)^{2}}{|U_{c,\omega}|^{2}+\eta}(uv^{\prime}-u^{\prime}v)dx$
$\displaystyle\quad+\frac{1}{2i}\int\left[\frac{(1-|U_{c,\omega}|^{2}-\eta)^{2}}{|U_{c,\omega}|^{2}+\eta}-\frac{(1-|U_{c,\omega}|^{2})^{2}}{|U_{c,\omega}|^{2}}\right]\nu
dx,$
$\displaystyle\quad+\frac{1}{2i}\int\left[\frac{(1-|U_{c,\omega}|^{2}-\eta)^{2}}{|U_{c,\omega}|^{2}+\eta}-\frac{(1-|U_{c,\omega}|^{2})^{2}}{|U_{c,\omega}|^{2}}+\frac{(1-|U_{c,\omega}|^{4})}{|U_{c,\omega}|^{4}}\eta\right](\bar{U}_{c,\omega}U^{\prime}_{c,\omega}-\bar{U}^{\prime}_{c,\omega}U_{c,\omega})dx,$

and the integration is performed over $(-\infty,-R]\cup[R,\infty)$.
Substituting $\nu$ into $P_{1}$ and integrating $u^{\prime}$ and $v^{\prime}$
by parts yield

$\displaystyle P_{1}$
$\displaystyle=\frac{(1-|U_{c,\omega}|^{2})^{2}}{|U_{c,\omega}|^{2}}\left(-{\rm
Im}(U_{c,\omega})u+{\rm
Re}(U_{c,\omega})v\right)\left(\biggr{|}^{x\to+\infty}_{x=R}+\biggr{|}^{x=-R}_{x\to-\infty}\right)$
$\displaystyle\quad+4\int(1-|U_{c,\omega}|^{2})(-{\rm
Im}(U^{\prime}_{c,\omega})u+{\rm Re}(U^{\prime}_{c,\omega})v)dx$
$\displaystyle\quad+\frac{i}{2}\int\frac{(1-|U_{c,\omega}|^{4})}{|U_{c,\omega}|^{4}}(\bar{U}_{c,\omega}U^{\prime}_{c,\omega}-\bar{U}^{\prime}_{c,\omega}U_{c,\omega})(u^{2}+v^{2})dx,$

where the second term in $P_{1}$ is obtained as follows:

$\displaystyle\qquad
2\int\frac{(1-|U_{c,\omega}|^{2})^{2}}{|U_{c,\omega}|^{2}}\left[{\rm
Im}(U_{c,\omega}^{\prime})u-{\rm Re}(U_{c,\omega}^{\prime})v\right]dx$
$\displaystyle\quad-2\int\frac{(1-|U_{c,\omega}|^{4})}{|U_{c,\omega}|^{4}}\left[{\rm
Im}(U_{c,\omega})u-{\rm Re}(U_{c,\omega})v\right]\left[{\rm
Re}(U_{c,\omega}){\rm Re}(U^{\prime}_{c,\omega})+{\rm Im}(U_{c,\omega}){\rm
Im}(U^{\prime}_{c,\omega})\right]dx$
$\displaystyle\quad+2\int\frac{(1-|U_{c,\omega}|^{4})}{|U_{c,\omega}|^{4}}\left[{\rm
Re}(U_{c,\omega})u+{\rm Im}(U_{c,\omega})v\right]\left[{\rm
Im}(U_{c,\omega}){\rm Re}(U^{\prime}_{c,\omega})-{\rm Re}(U_{c,\omega}){\rm
Im}(U^{\prime}_{c,\omega})\right]dx$
$\displaystyle=4\int(1-|U_{c,\omega}|^{2})(-{\rm
Im}(U^{\prime}_{c,\omega})u+{\rm Re}(U^{\prime}_{c,\omega})v)dx$

The first term in $P_{1}$ is zero in the limits $x\to\pm\infty$ due the
boundnedness of $u$ and $v$ in (B.5). The limits $x=\pm R$ are combined with
the inner intergrals on $[-R,R]$. The second term in $P_{1}$ combined with a
similar term on $[-R,R]$ is zero due to the second and fourth orthogonality
conditions (6.11). The third term in $P_{1}$ is bounded by
$\|u\|^{2}_{\mathcal{H}}+\|v\|^{2}_{\mathcal{H}}$ due to proximity of norms in
$\mathcal{H}_{c,\omega}$ and $\mathcal{H}$ given by (5.22) for $(c,\omega)$
near $(0,1)$ according to (B.3).

The terms in $P_{2}$ are analyzed by using the facts that $|U_{c,\omega}|^{2}$
is bounded away from zero on $(-\infty,-R]\cup[R,\infty)$,
$\|\eta\|_{L^{\infty}}$ is small, and the perturbation $u+iv$ is bounded
according to (B.5). The first term in $P_{2}$ is estimated by using
Cauchy–Schwarz inequality:

$\displaystyle\int\frac{(1-|U_{c,\omega}|^{2}-\eta)^{2}}{|U_{c,\omega}|^{2}+\eta}(uv^{\prime}-u^{\prime}v)dx\leq
C\int(1-|U_{c,\omega}|^{2}-\eta)^{2}(|u||v^{\prime}|+|u^{\prime}||v|)dx$
$\displaystyle\quad\leq
C\left((1+\|\eta\|_{L^{\infty}})\|u+iv\|_{\mathcal{H}_{c,\omega}}\|u^{\prime}+iv^{\prime}\|_{L^{2}}+\|\eta\|_{L^{\infty}}\|u+iv\|_{L^{\infty}}\|u^{\prime}+iv^{\prime}\|_{L^{2}}\|\eta\|_{L^{2}}\right)$
$\displaystyle\quad\leq
C\left(\|u^{\prime}+iv^{\prime}\|^{2}_{L^{2}}+\|u+iv\|^{2}_{\mathcal{H}}+\|\eta\|^{2}_{L^{2}}\right),$

where bounds (B.5) and the proximity between the norms in
$\mathcal{H}_{c,\omega}$ and $\mathcal{H}$ (Remark 6.1) have been used. The
second and third terms in $P_{2}$ are estimated similarly:

$\displaystyle\int\frac{\eta^{2}|U_{c,\omega}|^{2}-\eta(1-|U_{c,\omega}|^{4})}{|U_{c,\omega}|^{2}(|U_{c,\omega}|^{2}+\eta)}\left({\rm
Im}(U_{c,\omega}^{\prime})u-{\rm Re}(U_{c,\omega}^{\prime})v-{\rm
Im}(U_{c,\omega})u^{\prime}+{\rm Re}(U_{c,\omega})v^{\prime}\right)dx$
$\displaystyle\quad\leq
C\left(\|u^{\prime}+iv^{\prime}\|^{2}_{L^{2}}+\|u+iv\|^{2}_{\mathcal{H}}+\|\eta\|^{2}_{L^{2}}\right)$

and

$\displaystyle\frac{1}{2i}\int\frac{\eta^{2}}{|U_{c,\omega}|^{4}(|U_{c,\omega}|^{2}+\eta)}(\bar{U}_{c,\omega}U^{\prime}_{c,\omega}-\bar{U}^{\prime}_{c,\omega}U_{c,\omega})dx\leq
C\|\eta\|^{2}_{L^{2}}.$

###  B.2. Inner integrals on $[-R,R]$

We use the following expression for this part of $P(\psi)$ denoted by
$\hat{P}_{R}(\psi)$:

$\hat{P}_{R}(\psi):=\frac{i}{2}\int_{[-R,R]}(2-|\psi|^{2})(\bar{\psi}\psi_{x}-\bar{\psi}_{x}\psi)dx+\theta(R)-\theta(-R),$

where $\theta(x)=\arg(\psi(x))$. Expanding $\psi=U_{c,\omega}+u+iv$ and
$\theta=\arg(U_{c,\omega})+\theta_{1}+\theta_{2}$ with

$\displaystyle\theta_{1}$ $\displaystyle=\frac{{\rm Re}(U_{c,\omega})v-{\rm
Im}(U_{c,\omega})u}{|U_{c,\omega}|^{2}},$ $\displaystyle\theta_{2}$
$\displaystyle=\arg(U_{c,\omega}+u+iv)-\arg(U_{c,\omega})-\frac{{\rm
Re}(U_{c,\omega})v-{\rm Im}(U_{c,\omega})u}{|U_{c,\omega}|^{2}}$

yields the expansion

$\displaystyle\hat{P}_{R}(U_{c,\omega}+u+iv)$
$\displaystyle=\hat{P}_{R}(U_{c,\omega})+\hat{P}_{1}+\hat{P}_{2},$

where

$\displaystyle\hat{P}_{1}:=\frac{i}{2}\int\left[(2-|U_{c,\omega}|^{2})\nu-(\bar{U}_{c,\omega}U^{\prime}_{c,\omega}-\bar{U}^{\prime}_{c,\omega}U_{c,\omega})\eta\right]dx+\theta_{1}(R)-\theta_{1}(-R)$

and

$\displaystyle\hat{P}_{2}:=-\int(2-|U_{c,\omega}|^{2})(uv^{\prime}-u^{\prime}v)dx+\frac{1}{2i}\int\eta\nu
dx+\theta_{2}(R)-\theta_{2}(-R),$

and the integration is performed over $[-R,R]$. Substituting $\nu$ into
$\hat{P}_{1}$ and integrating $u^{\prime}$ and $v^{\prime}$ by parts yield
after simplifications:

$\displaystyle\hat{P}_{1}$ $\displaystyle=(2-|U_{c,\omega}|^{2})\left({\rm
Im}(U_{c,\omega})u-{\rm Re}(U_{c,\omega})v\right)\biggr{|}^{x=R}_{x=-R}$
$\displaystyle\quad+4\int(1-|U_{c,\omega}|^{2})(-{\rm
Im}(U^{\prime}_{c,\omega})u+{\rm Re}(U^{\prime}_{c,\omega})v)dx$
$\displaystyle\quad+\frac{1}{2i}\int(\bar{U}_{c,\omega}U^{\prime}_{c,\omega}-\bar{U}^{\prime}_{c,\omega}U_{c,\omega})(u^{2}+v^{2})dx+\theta_{1}(R)-\theta_{1}(-R).$

The quadratic term in $\hat{P}_{1}$ is bounded by
$\|u\|^{2}_{\mathcal{H}}+\|v\|^{2}_{\mathcal{H}}$ because there is $C_{R}>0$
such that

$\int_{-R}^{R}\left|\bar{U}_{c,\omega}U^{\prime}_{c,\omega}-\bar{U}^{\prime}_{c,\omega}U_{c,\omega}\right|(u^{2}+v^{2})dx\leq
C_{R}\int_{\mathbb{R}}(1-\varphi^{2})\left|\bar{U}_{c,\omega}U^{\prime}_{c,\omega}-\bar{U}^{\prime}_{c,\omega}U_{c,\omega}\right|(u^{2}+v^{2})dx,$

and $U_{c,\omega},U_{c,\omega}^{\prime}\in L^{\infty}(\mathbb{R})$. The linear
integral term in $\hat{P}_{1}$ is combined together with a similar term in
$P_{1}$ yields

$4\int_{\mathbb{R}}(1-|U_{c,\omega}|^{2})(-{\rm
Im}(U^{\prime}_{c,\omega})u+{\rm Re}(U^{\prime}_{c,\omega})v)dx,$

which vanish due to the second and fourth orthogonality conditions (6.11).
Finally, the terms at $x=\pm R$ are combined together in $P_{1}$ and
$\hat{P}_{1}$ to yield

$\displaystyle\frac{{\rm Im}(U_{c,\omega})u-{\rm
Re}(U_{c,\omega})v}{|U_{c,\omega}|^{2}}\biggr{|}^{x=R}_{x=-R}+\theta_{1}(R)-\theta_{1}(-R)=0.$

All integral terms in $\hat{P}_{2}$ are bounded by
$\|u^{\prime}+iv^{\prime}\|^{2}_{L^{2}}+\|u+iv\|^{2}_{\mathcal{H}}+\|\eta\|^{2}_{L^{2}}$
because integration on $[-R,R]$ is bounded by integration on $\mathbb{R}$ with
the weight $(1-\varphi^{2})$. Similarly, by using the Sobolev embedding of
$H^{1}([-R,R])$ to $L^{\infty}([-R,R])$ and the Taylor expansion with
$|U_{c,\omega}(\pm R)|^{2}$ being bounded away from zero, we obtain

$|\theta_{2}(\pm R)|\leq C_{R}(|u(\pm R)|^{2}+|v(\pm R)|^{2})\leq
C_{R}\|u+iv\|^{2}_{H^{1}([-R,R])},$

which is bounded by
$\|u^{\prime}+iv^{\prime}\|^{2}_{L^{2}}+\|u+iv\|^{2}_{\mathcal{H}}$ since
$[-R,R]$ is compact and $1-\varphi^{2}$ is bounded away from zero on $[-R,R]$.

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