# Efficient Control of High Harmonic Terahertz Generation in Carbon Nanotubes
using the Aharonov-Bohm Effect

Yuta Murakami  Center for Emergent Matter Science, RIKEN, Wako, Saitama
351-0198, Japan  Kohei Nagai  Department of Physics, Tokyo Institute of
Technology, Meguro, Tokyo 152-8551, Japan  Akihisa Koga  Department of
Physics, Tokyo Institute of Technology, Meguro, Tokyo 152-8551, Japan

###### Abstract

We show that high-harmonic generation (HHG) in carbon nanotubes (CNTs) can be
efficiently controlled using the Aharonov-Bohm (AB) effect. When a static
magnetic field ($B$) is applied along the tube, electronic wave functions
acquire complex phases along the circumferential direction (the AB effect),
which modifies the band structure. When the magnetic field is applied to
metallic CNTs, which can be regarded as one-dimensional massless Dirac
systems, realistic values of $B$ lead to a nonzero gap in the THz regime. We
demonstrate that such change from gapless to gapped Dirac systems drastically
increases the HHG intensity in the THz regime. In the gapless Dirac system,
the velocity of each electron never changes under the electric field, and thus
there is no HHG. On the other hand, the gap opening activates both the
interband and itraband currents, which strongly contribute to HHG. Our work
demonstrates a unique way to manipulate HHG in nanotubes by tuning electronic
wave functions using the magnetic field and the tube structure.

The recent development of strong lasers in the terahertz (THz) and mid-
infrared regimes enables us to study various nonlinear and nonequilibrium
phenomena in condensed matter Kruchinin  _et al._ (2018); Giannetti  _et al._
(2016); Basov  _et al._ (2017); Cavalleri (2018); Oka and Kitamura (2019); de
la Torre  _et al._ (2021); Koshihara  _et al._ (2022). Among them, high-
harmonic generation (HHG), where higher harmonics of the laser injected to
materials are radiated, is a fundamental and technologically important example
Corkum and Krausz (2007); Krausz and Ivanov (2009); Ghimire and Reis (2019).
HHG was originally studied in gas systems Ferray  _et al._ (1988); Corkum
(1993); Lewenstein  _et al._ (1994). Recently, HHG in condensed matter has
been observed, and the scope of the HHG research is widely extended to
conventional semiconductors, band insulators Ghimire  _et al._ (2011);
Schubert  _et al._ (2014); Vampa  _et al._ (2014); Luu  _et al._ (2015); Vampa
_et al._ (2015); Langer  _et al._ (2016); Hohenleutner  _et al._ (2015);
Ndabashimiye  _et al._ (2016); Otobe (2016); Tancogne-Dejean  _et al._ (2017);
Ikemachi  _et al._ (2017); Liu  _et al._ (2017); You  _et al._ (2017);
Kaneshima  _et al._ (2018); Sekiguchi  _et al._ (2022), topological materials
Giorgianni  _et al._ (2016); Yoshikawa  _et al._ (2017); Hafez  _et al._
(2018); Silva  _et al._ (2019); Chacón  _et al._ (2020); Cheng  _et al._
(2020); Schmid  _et al._ (2021); Baykusheva  _et al._ (2021a) and strongly
correlated systems Silva  _et al._ (2018); Murakami  _et al._ (2018);
Tancogne-Dejean  _et al._ (2018); Imai  _et al._ (2020); Murakami  _et al._
(2021); Orthodoxou  _et al._ (2021); Udono  _et al._ (2022); Bionta  _et al._
(2021); Shao  _et al._ (2022); Hansen  _et al._ (2022); Uchida  _et al._
(2022); Murakami  _et al._ (2022); Hansen and Madsen (2022); Grånäs  _et al._
(2022).

Condensed matter is considered as an interesting playground of the HHG
research, at least, for two good reasons. Firstly, the periodic potential from
atoms, which is absent in gaseous systems, can lead to peculiar motion of
electrons under strong electric field. An interesting example is the Dirac
system, which is characterized by the non-parabolic band dispersion. Due to
the nonlinear dynamics of electrons in the vicinity of the Dirac node, such
systems are theoretically expected to exhibit large nonlinearity Slepyan  _et
al._ (1999); Mikhailov (2007); Wright  _et al._ (2009); Al-Naib  _et al._
(2014, 2015); Cheng  _et al._ (2015); Chizhova  _et al._ (2017); Mrudul and
Dixit (2021); Sato  _et al._ (2021); Baykusheva  _et al._ (2021b). HHG was
experimentally studied for two-dimensional (2D) Dirac systems realized in
graphene Yoshikawa  _et al._ (2017); Hafez  _et al._ (2018) and in the surface
of topological insulators Giorgianni  _et al._ (2016); Schmid  _et al._
(2021); Baykusheva  _et al._ (2021a), and for the 3D Dirac system in Cd3As2
Cheng  _et al._ (2020); Kovalev  _et al._ (2020). It was reported that the
efficiency of HHG is indeed large compared to other systems Hafez  _et al._
(2018); Cheng  _et al._ (2020); Schmid  _et al._ (2021), although the origin
of HHG is not necessary processes theoretically expected Paul  _et al._
(2013); Hafez  _et al._ (2018). The second reason is the controllability of
HHG. Generally, properties of condensed matter can be tuned by system
parameters such as temperature and doping level, which can be also applicable
to HHG. For example, peculiar HHG dependence on crystal orientations Ghimire
_et al._ (2011); You  _et al._ (2017), electronic phases Silva  _et al._
(2018); Uchida  _et al._ (2022); Murakami  _et al._ (2022); Bionta  _et al._
(2021); Shao  _et al._ (2022), physical pressure Tamaya and Kato (2021) and
doping Nishidome  _et al._ (2020); de Vega  _et al._ (2020); Hansen and Madsen
(2022) has been proposed theoretically or been observed experimentally.

In this work, we propose a unique way to efficiently control HHG in carbon
nanotubes (CNTs) combining physics of the Dirac system and the Aharonov-Bohm
(AB) effect, see Fig. 1(a). The proposed setup provides an ideal platform to
study HHG in Dirac systems, systematically. When the static magnetic field
($B$) is applied along the tube, electrons in the CNT gain complex phases
along the circumferential direction, which modifies the band structure (the AB
effect) Ajiki and Ando (1993, 1994); Zaric  _et al._ (2004); Matsunaga  _et
al._ (2008). In particular, metallic CNTs, which can be regarded as 1D
massless Dirac systems, become massive Dirac systems when $B\neq 0$. This
change qualitatively modifies the electron dynamics, and drastically enhances
HHG.

Figure 1: (a) A CNT with the static magnetic field
(${\bm{B}}=B{\bm{e}}_{\parallel}$) and the AC electric field
(${\bm{E}}(t)=E(t){\bm{e}}_{\parallel}$) along the tube. (b) A graphite sheet.
${\bm{b}}_{i}$ represents the vector from a site of the $A$ sublattice to
neighboring sites of the $B$ sublattice. ${\bf a}_{1}$ and ${\bf a}_{2}$ are
primitive translation vectors. ${\bm{L}}\equiv n{\bf a}_{1}+m{\bf
a}_{2}$($\propto{\bm{e}}_{\perp}$) is the chiral vector charactering CNTs.
${\bm{e}}_{\parallel}$ and ${\bm{e}}_{\perp}$ are unit vectors of the tube and
circumferential directions. (c) Correspondence of the momentum of an armchair
CNT and that of graphene. The blue hexagon is the BZ of graphene, where $K$
and $K^{\prime}$ are the Dirac points. (d) The band structure of the armchair
CNT with $(n,m)=(15,15)$ with and without the static magnetic field. The sold
line indicates the bands closest to $K$ and $K^{\prime}$, which form 1D Dirac
systems.

In the dipolar gauge Li  _et al._ (2020); Schüler  _et al._ (2021); Murakami
and Schüler (2022), the tight-binding Hamiltonian for the CNT with the
magnetic field and the AC electric field ($E(t)$) along the tube is expressed
as

$\displaystyle\hat{H}(t)=-t_{\rm hop}\sum_{\langle
ij\rangle}e^{i\frac{q}{\hbar}{\bm{A}}(t)\cdot{\bf
r}_{ij}}\hat{c}^{\dagger}_{i}\hat{c}_{j}-\mu\sum_{i}\hat{n}_{i}~{}.$ (1)

$\hat{c}^{\dagger}_{i}$ is the creation operator of an electron at the $i$th
site and we neglect the spin degrees of freedom. $\langle ij\rangle$ indicates
the nearest neighbor sites, $t_{\rm hop}$ is the hopping parameter, $q$ is the
charge of an electron, $\hbar$ is the Dirac constant, $\mu$ is the chemical
potential and $\hat{n}_{i}=\hat{c}^{\dagger}_{i}\hat{c}_{i}$. The model is
defined on the graphite sheet imposing the periodic boundary condition with
respect to the chiral vector ${\bm{L}}$, which is characterized by the chiral
index $(n,m)$ as ${\bm{L}}=n{\bm{a}}_{1}+m{\bm{a}}_{2}$, see Fig. 1(b) Hamada
_et al._ (1992). The effects of the magnetic and electric fields are included
via the Peierls phase through the vector potential
${\bm{A}}(t)={\bm{A}}_{B}+{\bm{A}}_{E}(t)$. The contribution from the static
magnetic field is ${\bm{A}}_{B}=A_{B}{\bm{e}}_{\perp}$ and that from the
electric field is ${\bm{A}}_{E}(t)=A_{E}(t){\bm{e}}_{\parallel}$.
$A_{B}=\frac{B\cdot|{\bm{L}}|}{4\pi}$ and $E(t)=-\partial_{t}A_{E}(t)$.
${\bm{e}}_{\parallel}$ and ${\bm{e}}_{\perp}$ are unit vectors of the tube and
circumferential directions, respectively, see Fig. 1(b). In the following, we
focus on the armchair CNT with the chiral index $(n,m)$ with $n=m$.

The Fourier transform of the Hamiltonian (6) is
$\hat{H}(t)=\sum_{\bm{k}}\hat{\bm{\psi}}^{\dagger}_{\bm{k}}{\bf
h}({\bm{k}}(t))\hat{\bm{\psi}}_{\bm{k}}$, where

$\displaystyle{\bf h}({\bm{k}})=\begin{bmatrix}-\mu&F({\bm{k}})\\\
F^{*}({\bm{k}})&-\mu\end{bmatrix},$ (2)

and ${\bm{k}}(t)\equiv{\bm{k}}-\frac{q}{\hbar}{\bm{A}}(t)$, see the
Supplementary Material (SM) 111See Supplemental Material at [URL will be
inserted by publisher] for the details of the model and numerical method; the
derivation of Eqs. (4) and (5); the doping effects; the dependence of the
excitation frequency; the dependence on the chiral index; the effects of the
relaxation times. The Supplemental Material also contains Refs. Khosravi  _et
al._ (2009); Dimitrovski  _et al._ (2017); Dvuzhilova  _et al._ (2021);
Wilhelm  _et al._ (2021). for details. Here,
$\hat{\bm{\psi}}^{\dagger}_{\bm{k}}=[\hat{c}^{\dagger}_{{\bm{k}}A},\hat{c}^{\dagger}_{{\bm{k}}B}]$,
and $\hat{c}^{\dagger}_{{\bm{k}}A}$ and $\hat{c}^{\dagger}_{{\bm{k}}B}$ are
the Fourier transform of $\hat{c}^{\dagger}_{i}$ on the A and B sublattices,
respectively. $F({\bm{k}})=-t_{\rm
hop}[e^{i{\bm{k}}\cdot{\bm{b}}_{1}}+e^{i{\bm{k}}\cdot{\bm{b}}_{2}}+e^{i{\bm{k}}\cdot{\bm{b}}_{3}}]$,
see Fig. 1(b) for ${\bm{b}}_{i}$. The Hamiltonian (16) is the same as that of
graphene, and the set of
${\bm{k}}_{B}(\equiv{\bm{k}}-\frac{q}{\hbar}{\bm{A}}_{B})$ for the CNT
corresponds to some lines along ${\bm{k}}_{\perp}$ on the BZ of graphene, see
Fig. 1(c). Note that
${\bm{k}}(t)={\bm{k}}_{B}-\frac{q}{\hbar}{\bm{A}}_{E}(t)$, which moves along
${\bm{k}}_{\parallel}$ in time. For $B=0$, ${\bm{k}}_{B}$’s for the CNT pass
the Dirac points $K$ and $K^{\prime}$ of graphene, hence the CNT bands host
gapless Dirac bands, see Figs. 1(c) and (d). Meanwhile, for $B\neq 0$,
${\bm{k}}_{B}$’s are shifted by $-\frac{q}{\hbar}{\bm{A}}_{B}$ and do not
passes the Dirac points. Hence the CNT bands host gapful Dirac bands. Around
the Dirac points, the Hamiltonian (16) is effectively described by the Dirac
model, see Fig. 1 (d) and SM Note (1) for details. In our setup, the dynamics
of electrons around the Dirac points leads to HHG, which is captured by this
model.

Diameters of single-wall CNTs are typically $0.7\sim 2$nm Yanagi  _et al._
(2008). The static magnetic field applied to CNTs in optical experiments can
be several tens of Tesla Zaric  _et al._ (2004). In the following, we set
$t_{\rm hop}=3$eV and $a=0.246$nm and focus on the CNT with $(n,m)=(15,15)$,
whose diameter is $2.04$nm. The band gap for $B=30$T becomes $29.7$meV
($\simeq 7.5$THz), see Fig. 1 (d). We mainly focus on half filling ($\mu=0$)
and apply the gaussian electric field described by
$A_{E}(t)=\frac{E_{0}}{\hbar\Omega}\exp[-\frac{(t-t_{0})^{2}}{2\sigma^{2}}]\sin(\Omega(t-t_{0}))$.
We set the electric-field frequency $\hbar\Omega=10$meV ($\simeq 2.5$THz), the
pulse center $t_{0}=3.95$ps and the pulse width $\sigma={658}$fs, in the main
text.

We simulate the time evolution of the tight-binding and Dirac models using the
semiconductor Bloch equation (SBE) Huttner  _et al._ (2017); Yue and Gaarde
(2022)

$\displaystyle\partial_{t}\bm{\rho}_{\bm{k}}(t)$
$\displaystyle=\frac{i}{\hbar}[\bm{\rho}_{\bm{k}}(t),{\bf
h}({\bm{k}}(t))]-\frac{\bm{\rho}_{\bm{k}}(t)-\bm{\rho}_{{\rm
eq},{\bm{k}}(t)}}{T_{1}}$
$\displaystyle+\Bigl{(}\frac{1}{T_{1}}-\frac{1}{T_{2}}\Bigl{)}{\bf
U}({\bm{k}}(t))\bm{\rho}^{\rm H}_{{\rm off},{\bm{k}}}(t){\bf
U}^{\dagger}({\bm{k}}(t)),$ (3)

where $\bm{\rho}_{\bm{k}}(t)$ is the single-particle density matrix consisting
of
$\rho_{\alpha\beta,{\bm{k}}}(t)=\langle\hat{c}^{\dagger}_{{\bm{k}}\beta}(t)\hat{c}_{{\bm{k}}\alpha}(t)\rangle$
with $\alpha,\beta=A,B$. The last two terms express the relaxation processes
within the relaxation-time approximation. $\bm{\rho}_{{\rm eq,}{\bm{k}}}$ is
the equilibrium density matrix for the initial temperature ($T$). ${\bf
U}({\bm{k}})$ is a unitary matrix diagonalizing ${\bf h}({\bm{k}})$ as ${\bf
U}^{\dagger}({\bm{k}}){\bf h}({\bm{k}}){\bf
U}({\bm{k}})={\bm{\epsilon}}({\bm{k}})$ with ${\bm{\epsilon}}({\bm{k}})={\rm
diag}[\epsilon_{0}({\bm{k}}),\epsilon_{1}({\bm{k}})]$. It offers the
descrition in the band (Houston) basis. In this basis, the density matrix
becomes $\bm{\rho}^{\rm H}_{{\bm{k}}}(t)\equiv{\bf
U}^{\dagger}({\bm{k}}(t))\bm{\rho}_{\bm{k}}(t){\bf U}({\bm{k}}(t))$, whose
off-diagonal components are $\bm{\rho}^{\rm H}_{{\rm off},{\bm{k}}}(t)$
Murakami and Schüler (2022). The diagonal components of $\bm{\rho}^{\rm
H}_{\bm{k}}$ indicate the occupation of the bands, and the off-diagonal
components indicate the interband hybridization. Thus, $T_{1}$ represents the
relaxation time of the band occupation, while $T_{2}$ represents the dephasing
time of the interband coherence. We set $T_{1}=98.8$fs and $T_{2}=19.8$fs to
be consistent with previous studies for graphene and CNTs Habenicht  _et al._
(2007); George  _et al._ (2008); Tani  _et al._ (2012); Heide  _et al._
(2021); Sato  _et al._ (2021).

The light radiation originates from the oscillation of electric polarization
(or corresponding current) Jackson (1998). Since the current along the
circumferential direction does not yield polarization, the HHG spectrum
(normalized by the number of sites) is evaluated from $J_{\parallel}(t)$ (the
current along ${\bm{e}}_{\parallel}$) as $I_{\rm HHG}(\omega)=|\omega
J_{\parallel}(\omega)|^{2}$. Here, $J_{\parallel}(\omega)=\int dte^{i\omega
t}J_{\parallel}(t)$, and $J_{\parallel}(t)$ is evaluated from
$\rho_{\alpha\beta,{\bm{k}}}(t)$ using the current operator
$\hat{J}_{\parallel}(t)\equiv-\frac{1}{N}\frac{\delta\hat{H}(t)}{\delta
A_{E}(t)}$. $2N$ is the number of atoms (sites) for the CNT. In the conditions
studied here, HHG mainly originates from the bands closest to the Dirac
points, i.e. solid lines in Fig. 1(d).

Figure 2: (a) HHG spectra $I_{\rm HHG}$ of the armchair CNT with and without
the static magnetic field simulated with the tight-binding model. $I_{\rm
HHG}$ for graphene is also shown. The inset show the bandgap induced by the
magnetic field $B$. (b)The intensity at the HHG peaks $I_{\rm peak}(l)$ as a
function of $B$. We set the chiral index $(n,m)=(15,15)$, $\mu=0$, $T=11.6$K,
$T_{1}=98.8$fs and $T_{2}=19.8$fs. The parameters of the electric field are
$\hbar\Omega=10$meV and $E_{0}=30$kV/cm. Figure 3: (a)(b) Comparison of the
HHG spectra of the armchair CNT obtained in different ways for (a) $B=0$T and
(b) $B=30$T. $I_{\rm HHG}(\omega)$ is evaluated from the tight-binding model,
while $I_{\rm HHG}^{\rm Dirac}(\omega)$, $I_{\rm intra}(\omega)$ and $I_{\rm
inter}(\omega)$ are evaluated from the Dirac model considering only bands
closest to $K$ or $K^{\prime}$. The parameters of the system and the electric
field are the same as Fig.2. (c) The dynamics of electrons with and without
the magnetic field. The momentum shift originates from the intraband
acceleration by the electric field.

Figure 2(a) shows $I_{\rm HHG}$ of the armchair CNT with and without the
magnetic field obtained from the tight-binding model. Only odd harmonics
components appear in $I_{\rm HHG}(\omega)$, because the system (6) is
symmetric under a time-space operation, $t\rightarrow t+\frac{\pi}{\Omega}$
and ${\bf e}_{\parallel}\rightarrow-{\bf e}_{\parallel}$ during the time-
periodic excitation, see Ref. Neufeld  _et al._ (2019) for general argument.
When $B=0$ the intensity for $l(\geq 3)$th harmonics is very small. When
$B\neq 0$ the HHG intensity for $l\geq 3$ is strongly enhanced by more than
several orders of magnitude. Figure 2(b) shows the intensity at the HHG peaks
$I_{\rm peak}(l)$ as a function of $B$ for $l=3,5,7$. $I_{\rm peak}(l)$ is
defined as the maximum value of $I_{\rm HHG}(\omega)$ for
$\omega\in[l\Omega-\Omega,l\Omega+\Omega]$. $I_{\rm peak}(l)$ is non-monotonic
against $B$, since a gap too large compared to $\Omega$ suppresses the
excitation and thus HHG, see the inset of Fig. 2(a). Namely, there exists the
optimal value of $B$ maximizing $I_{\rm HHG}(\omega)$ for a given $E_{0}$.
Furthermore, the HHG intensity in the CNT per atom for this optimal $B$ is
much larger than that in graphene per atom, see Fig. 2(a). This is understood
as follows. In the CNT, we can tune $B$ such that the bands closest to the
Dirac points yield HHG most efficiently for a given $E_{0}$, and the
contributions from the rest bands are negligible. Meanwhile, in graphene,
which is identical to a CNT with $n,m\rightarrow\infty$, the number of bands
along ${\bm{k}}_{\parallel}$ contributing to HHG increases. However, the
contribution from each band are not optimal for a given electric-field
condition, and they may interfere and cancel each other on average. We also
note that the drastic enhancement of HHG with $B$ can be observed even for
doped CNTs or for higher-frequency excitations, see SM Note (1).

The efficiency of the enhancement depends on the chiral index $n(=m)$.
Firstly, the momentum shift due to the AB effect is $\frac{q}{\hbar}A_{B}$,
which is proportional to $B$ and $L(\equiv|{\bm{L}}|\propto n)$. Therefore, in
order to realize a given value of $A_{B}$ (or the gap), smaller $B$ is
required for larger $n$, see SM Note (1) for examples. Thus, as far as only
the bands closest to the Dirac points are involved in HHG, larger $n$ is
favorable for the efficient control of HHG. Meanwhile, with increasing $n$,
the number of bands along ${\bm{k}}_{\parallel}$ increases, and many bands
start to contribute to HHG. Then, the impact of the AB effect on HHG becomes
less prominent. In the extreme limit of $n\rightarrow\infty$ (graphene), the
AB effect has no impact on physical properties.

Figure 4: The intensity at the HHG peaks $I_{\rm peak}(l)$ for the armchair
CNT as a function of the strength of the AC electric field $E_{0}$ for (a)
$l=3$, (b) $l=5$ and (c) $l=7$. The tight-binding model is used. The dashed
lines are proportional to $(E_{0})^{2l}$ for each panels. The parameters of
the system and the electric field are the same as Fig.2.

To understand the origin of the enhancement of HHG with the magnetic field, we
compare the HHG intensity evaluated from the tight-binding model ($I_{\rm
HHG}$) and that from the Dirac model ($I_{\rm HHG}^{\rm Dirac}$), see Fig.
3(a)(b). We only consider the contributions from the bands closest to $K$ and
$K^{\prime}$ for the Dirac model. The results from the Dirac model match well
those from the tight-binding model both for $B=0$ and $B=30$T. The only
exception is the 3rd harmonic generation at $B=0$, which originates from the
nonlinear part of the band dispersion. Namely, the 1D gapless Dirac model
shows no HHG at $B=0$. Simply speaking, this is because the transition of
electrons between bands in the 1D gapless Dirac system under electric field is
always perfect and electrons never change the velocity, see Fig. 3(c). Without
the relaxation, this leads to $J_{||}(t)\propto A_{E}(t)$ Ishikawa (2010);
Óscar Zurrón-Cifuentes  _et al._ (2020). If we take into account the
relaxation, we obtain

$\displaystyle\partial_{t}J_{||}(t)=\frac{1}{n}\frac{t_{\rm
hop}}{\hbar}\sigma_{0}E(t)-\frac{1}{T_{1}}J_{||}(t)~{},$ (4)

from the SBE (21), where $\sigma_{0}=\frac{\sqrt{3}a^{2}q^{2}}{2\pi\hbar}$,
see SM Note (1). This is a linear differential equation, leading to no HHG.
The last term of Eq.(4), which we call $[\partial_{t}J_{\parallel}(t)]_{\rm
corr}$, originates from the relaxation term of the SBE. The present form
originates from the relaxation-time approximation in Eq. (21). In principle,
if $[\partial_{t}J_{\parallel}(t)]_{\rm corr}$ depends on $J_{\parallel}$
nonlinearly, the 1D gapless Dirac system can show HHG. Still, in a previous
experiment of HHG in CNTs Nishidome  _et al._ (2020), metallic CNTs show very
weak HHG, which is consistent with the consequence from Eq. (4).

To obtain further insight, we evaluate the contribution from the intraband
current ($I_{\rm intra}=|\omega J_{\parallel,\rm intra}(\omega)|^{2}$) and
that from the interband current ($I_{\rm inter}=|\omega J_{\parallel,\rm
inter}(\omega)|^{2}$) using the Dirac model. We define the intraband current
as $J_{\parallel,\rm intra}(t)\equiv\frac{q}{\hbar N}\sum_{\bm{k}}{\rm
Tr}[\bm{\rho}^{H}_{\bm{k}}(t)\partial_{k_{\parallel}}{\bm{\epsilon}}({\bm{k}}(t))]$
and the interband current as $J_{\parallel,\rm inter}(t)\equiv
J_{\parallel}(t)-J_{\parallel,\rm intra}(t)$. The former is associated with
the change in the band occupation, while the latter is associated with that of
the interband polarization (the off-diagonal component of
$\bm{\rho}^{H}_{\bm{k}}(t)$), see SM Note (1) for details. When $B=0$, the
perfect transition between the conduction and valence bands happens without
inducing the interband polarization, leading to $I_{\rm HHG}=I_{\rm intra}$,
see Fig. 3(c). Meanwhile, when $B\neq 0$, the gap opens and the dipole moments
between the bands become finite, which allows electrons to stay in the same
band and tunnel to the other band, see Fig. 3(c). Thus, both the intraband and
interband currents are activated to contribute to HHG, see Fig. 3(b). In the
present case, the gap and the excitation frequency are comparable, which makes
the contributions from these currents comparable. The HHG contributions from
$J_{\parallel,\rm intra}(t)$ and $J_{\parallel,\rm inter}(t)$ cancel each
other due to their interference.

Finally, we discuss the dependence on the strength of the applied AC electric
field $E_{0}$, see Fig. 1(a). In Fig. 4, we show the intensity at the HHG
peaks $I_{\rm peak}(l)$ as a function of $E_{0}$ for $l=3,5,7$. In the
perturbative regime with respect to $E_{0}$, we expect $I_{\rm
peak}(l)\propto(E_{0})^{2l}$. (For small $B$, this perturbative behavior is
recovered with $E_{0}$ below the range shown in Fig. 4.) For all harmonics,
the deviation from the perturbative regime happens at larger $E_{0}$ for
larger $B$, i.e. larger gap. $I_{\rm peak}(l)$ tends to be saturated due to
the higher order corrections. Because of this, the value of $B$ maximizing
$I_{\rm peak}(l)$ for a given $E_{0}$ depends on the value of $E_{0}$. Namely,
when $E_{0}$ is small, the optimal $B$ is small, while for larger $E_{0}$, the
optimal $B$ becomes large. We note that these properties are insensitive to
the choice of $T_{1}$ and $T_{2}$, see SM Note (1).

One interesting observation is that the optimal value of $B$ continuously
decreases with decreasing $E_{0}$ and it becomes a few Tesla for $E_{0}\sim 1$
kV/cm. This indicates that $\sigma^{(l)}$ monotonically increases with
decreasing $B$. Here, $\sigma^{(l)}$ represents the $l$th order optical
conductivity, i.e. $J_{\parallel}(l\Omega)\simeq\sigma^{(l)}(E_{0})^{l}$.
However, this peculiar behavior turns out to be sensitive to the choice of
$T_{1}$ and $T_{2}$, see SM Note (1). When $T_{1}=T_{2}$, the optimal $B$ for
small $E_{0}$ remains around $20$T, where the gap is almost twice of $\Omega$.
The crucial dependence on $T_{1}$ and $T_{2}$ can be attributed to the change
in cancelation between the interaband and intraband currents. This point can
be directly seen in the analytic expression of $\sigma^{(3)}$. If we express
contributions from $J_{\rm inter}$ and $J_{\rm intra}$ as $\sigma^{(3)}_{\rm
inter}$ and $\sigma^{(3)}_{\rm intra}$, the leading terms in the limit of
$B\rightarrow 0$ read

$\displaystyle\sigma^{(3)}_{\rm
intra}\simeq-F\frac{1}{A_{B}^{2}}\frac{1}{\zeta_{3}},\;\;\sigma^{(3)}_{\rm
inter}\simeq F\frac{1}{A_{B}^{2}}\frac{1}{\xi_{3}},$ $\displaystyle F\equiv
i\sigma_{0}t_{\rm
hop}\hbar^{2}\frac{1}{n}\frac{1}{\xi_{1}}\Bigl{(}\frac{\lambda_{0}}{2\zeta_{2}}+\frac{\lambda_{2}}{\xi_{2}}\Bigl{)}$
(5)

at $T=0$. Here $\zeta_{m}=m\hbar\Omega+i\hbar/T_{1}$ and
$\xi_{m}=m\hbar\Omega+i\hbar/T_{2}$ and $\lambda_{m}=\int
dx\frac{x^{m}}{\sqrt{1+x^{2}}^{7}}$. The expression tells that $\sigma^{(3)}$
diverges for $B\rightarrow 0$ when $T_{1}\neq T_{2}$, while the divergence is
absent when $T_{1}=T_{2}$ due to the cancelation between $\sigma^{(3)}_{\rm
inter}$ and $\sigma^{(3)}_{\rm intra}$. Our numerical simulation suggests that
the same happens also for higher harmonics with $l\geq 5$. Namely, for
$T_{1}=T_{2}$, the contributions from the interband and intraband currents
cancel each other efficiently, while for $T_{1}\neq T_{2}$, the cancelation
becomes weaker, see SM Note (1). Although we used $T_{1}=98.8$fs and
$T_{2}=19.8$fs to be consistent with previous studies, to fully determine the
realistic choice of $T_{1}$ and $T_{2}$ for the present energy scale,
microscopic evaluation of relaxation and dephasing processes is required Malic
_et al._ (2011); Kemper  _et al._ (2013); Stefanucci and Leeuwen (2013);
Schüler  _et al._ (2020); Ridley  _et al._ (2022); Rostami and Cappelluti
(2021).

To summarize, we showed that HHG in metallic CNTs can be drastically
manipulated in the THz regime using the AB effect. Tuning the strength of the
magnetic field, the HHG intensity can be maximized, where HHG per atom is more
efficient than in graphene. Such controllability of HHG may be useful to
develop an efficient HHG device in the THz regime. In addition, with realistic
parameters of CNTs and external fields, a metallic CNT under magnetic fields
can be regarded as a tunable 1D Dirac system for THz excitations. Therefore,
it serves as a useful platform to systematically study HHG in Dirac systems.
Furthermore, the unique idea of manipulating HHG via the AB effect can be
applied for all kinds of nanotubes including semiconducting CNTs, boron
nitride nanotubes Rubio  _et al._ (1994); Chopra  _et al._ (1995) and
transition-metal dichalcogenide nanotubes Tenne  _et al._ (1992).

###### Acknowledgements.

We would like to acknowledge fruitful discussions with Hiroyuki Nishidome and
Kazuhiro Yanagi. This work is supported by Grant-in-Aid for Scientific
Research from JSPS, KAKENHI Grant Nos. JP20K14412, JP21H05017 (Y. M.),
JP22K03525, JP21H01025, JP19H05821 (A.K.), JST CREST Grant No. JPMJCR1901 (Y.
M.).

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##  I Detail of numerical simulation

###  I.1 Tight Binding Model

As explained in the main text, the tight-binding (TB) model for the armchair
CNT with the static magnetic field and the AC electric field along the tube
can be constructed on a honeycomb (graphite) sheet;

$\displaystyle\hat{H}(t)=-t_{\rm hop}\sum_{\langle
ij\rangle}e^{i\frac{q}{\hbar}{\bm{A}}(t)\cdot{\bm{r}}_{ij}}\hat{c}^{\dagger}_{i}\hat{c}_{j}-\mu\sum_{i}\hat{n}_{i}~{},$
(6)

where the periodic boundary condition (PBC) with respect to the chiral vector
${\bm{L}}(=n{\bm{a}}_{1}+n{\bm{a}}_{2})$ is imposed. We set
${\bm{A}}(t)={\bm{A}}_{B}+{\bm{A}}_{E}(t)$. In this TB model for CNTs, the
effects of the curvature of the tube are ignored for simplicity Hamada  _et
al._ (1992). Due to the curvature of the tube, the Dirac points may shift.
Still, for the armchair CNT, which we focus on in this paper, the shift is
along $k_{\parallel}$ and the system remains metallic Hamada  _et al._ (1992).
Thus, the physics of the manipulation of HHG discussed in this paper is hardly
affected. The primitive cell is defined by

$\displaystyle{\bm{L}}_{0}\equiv{\bm{a}}_{1}+{\bm{a}}_{2},\;\;{\bf
T}_{0}\equiv{\bm{a}}_{2}-{\bm{a}}_{1},$ (7)

see Fig. 5. Note that ${\bm{L}}=n{\bm{L}}_{0}\propto{\bm{e}}_{\perp}$ and
${\bf T}_{0}\propto{\bm{e}}_{\parallel}$. We have four sites (atoms) in the
primitive cell. Due to the PBC with respect to ${\bm{L}}$, we have $n$
primitive cells along ${\bm{e}}_{\perp}$. We also impose the PBC along
${\bm{e}}_{\parallel}$ and take $N_{T}(\rightarrow\infty)$ primitive cells
along ${\bm{e}}_{\parallel}$. Using the reciprocal vectors ${\bm{l}}_{0}\equiv
2\pi\frac{{\bm{L}}_{0}}{|{\bm{L}}_{0}|^{2}}$ and ${\bm{t}}_{0}\equiv
2\pi\frac{{\bf T}_{0}}{|{\bf T}_{0}|^{2}}$, the wave vectors can be defined as

$\displaystyle{\bm{k}}=\frac{i_{L}}{n}{\bm{l}}_{0}+\frac{i_{T}}{N_{T}}{\bm{t}}_{0},$
(8)

where $i_{L}=0,1\cdots n-1$ and $i_{T}=0,1\cdots N_{T}-1$. We name the four
sites in the primitive cell as $A_{1},B_{1},A_{2},$ and $B_{2}$ as in Fig. 5.

Now, we consider the expression of the Hamiltonian in the momentum (Fourier)
space. We introduce the electron operators in the momentum space as
$\hat{c}^{\dagger}_{{\bm{k}}\alpha}=\frac{1}{\sqrt{N_{c}}}\sum_{i\in\alpha}e^{i{\bm{k}}\cdot{\bm{r}}_{i}}\hat{c}^{\dagger}_{i}$
for $\alpha=A_{1},A_{2},B_{1},B_{2}$. Here ${\bf r}_{i}$ is the position
vector of the $i$th site, $N_{c}=n\cdot N_{T}$ is the total number of the
cells. The inverse tranform is
$\hat{c}^{\dagger}_{\bm{r}_{i}}=\frac{1}{\sqrt{N_{c}}}\sum_{\bm{k}}e^{-i{\bm{k}}\cdot{\bf
r}_{i}}\hat{c}^{\dagger}_{{\bm{k}}\alpha}$ for ${\bm{r}}_{i}\in\alpha$. Then
the Hamiltonian (6) is expressed as

$\displaystyle\hat{H}(t)=\tilde{\sum}_{\bm{k}}\begin{bmatrix}\hat{c}^{\dagger}_{{\bm{k}}A_{1}}&\hat{c}^{\dagger}_{{\bm{k}}A_{2}}&\hat{c}^{\dagger}_{{\bm{k}}B_{1}}&\hat{c}^{\dagger}_{{\bm{k}}B_{2}}\end{bmatrix}\tilde{\bf
h}({\bm{k}}(t))\begin{bmatrix}\hat{c}_{{\bm{k}}A_{1}}\\\
\hat{c}_{{\bm{k}}A_{2}}\\\ \hat{c}_{{\bm{k}}B_{1}}\\\
\hat{c}_{{\bm{k}}B_{2}}\end{bmatrix},$ (9)

where

$\displaystyle\tilde{\bf
h}({\bm{k}})=\begin{bmatrix}-\mu&0&\alpha({\bm{k}})^{*}&\gamma({\bm{k}})^{*}\\\
0&-\mu&\gamma({\bm{k}})^{*}&\alpha({\bm{k}})^{*}\\\
\alpha({\bm{k}})&\gamma({\bm{k}})&-\mu&0\\\
\gamma({\bm{k}})&\alpha({\bm{k}})&0&-\mu\end{bmatrix},$ (10)

${\bm{k}}(t)={\bm{k}}-\frac{q}{\hbar}{\bm{A}}(t)$ and

$\displaystyle\alpha({\bm{k}})=-t_{\rm
hop}[e^{-i{\bm{b}}_{1}\cdot{\bm{k}}}+e^{-i{\bm{b}}_{3}\cdot{\bm{k}}}],\;\;\gamma({\bm{k}})=-t_{\rm
hop}e^{-i{\bm{b}}_{2}\cdot{\bm{k}}}.$ (11)

Here, $\tilde{\sum}_{\bm{k}}$ means to calculate the sum over ${\bm{k}}$
expressed as Eq. (8) with $i_{L}=0,1\cdots n-1$ and $i_{T}=0,1\cdots N_{T}-1$.

Figure 5: Schematic figure of a graphite sheet. The orange shaded area
corresponds to the primitive cell for armchair CNTs. We also show the name of
four sites in the cell.

Next, we show that the Hamiltonian for each ${\bm{k}}$ can be block
diagonalized into $2\times 2$ matrices, each of which corresponds to the
Hamiltonian of graphene at a certain momentum. We introduce the new basis for
electrons at each ${\bm{k}}$ as Murakami and Schüler (2022)

$\displaystyle\begin{bmatrix}\hat{c}_{{\bm{k}}A_{+}}\\\
\hat{c}_{{\bm{k}}B_{+}}\\\ \hat{c}_{{\bm{k}}A_{-}}\\\
\hat{c}_{{\bm{k}}B_{-}}\end{bmatrix}\equiv\frac{1}{\sqrt{2}}\begin{bmatrix}1&1&0&0\\\
0&0&1&1\\\ e^{i\frac{\pi}{6}}&-e^{i\frac{\pi}{6}}&0&0\\\
0&0&e^{-i\frac{\pi}{6}}&-e^{-i\frac{\pi}{6}}\end{bmatrix}\begin{bmatrix}\hat{c}_{{\bm{k}}A_{1}}\\\
\hat{c}_{{\bm{k}}A_{2}}\\\ \hat{c}_{{\bm{k}}B_{1}}\\\
\hat{c}_{{\bm{k}}B_{2}}\end{bmatrix}.$ (12)

Then, we have

$\displaystyle\hat{H}(t)=$
$\displaystyle\tilde{\sum}_{\bm{k}}\begin{bmatrix}\hat{c}^{\dagger}_{{\bm{k}}A_{+}}&\hat{c}^{\dagger}_{{\bm{k}}B_{+}}&\hat{c}^{\dagger}_{{\bm{k}}A_{-}}&\hat{c}^{\dagger}_{{\bm{k}}B_{-}}\end{bmatrix}{\bf
h}^{\prime}({\bm{k}}(t))\begin{bmatrix}\hat{c}_{{\bm{k}}A_{+}}\\\
\hat{c}_{{\bm{k}}B_{+}}\\\ \hat{c}_{{\bm{k}}A_{-}}\\\
\hat{c}_{{\bm{k}}B_{-}}\end{bmatrix}$ (13)

with

$\displaystyle{\bf
h}^{\prime}({\bm{k}})=\begin{bmatrix}-\mu&F({\bm{k}})&0&0\\\
F({\bm{k}})^{*}&-\mu&0&0\\\ 0&0&-\mu&F({\bm{k}}+{\bm{l}}_{0})\\\
0&0&F({\bm{k}}+{\bm{l}}_{0})^{*}&-\mu\end{bmatrix}.$ (14)

We note that each $2\times 2$ matrix in ${\bf h}^{\prime}({\bm{k}})$ has the
same form as that of graphene at momentum ${\bm{k}}$ and
${\bm{k}}+{\bm{l}}_{0}$ Murakami and Schüler (2022). If we extend the region
of ${\bm{k}}$ for CNTs as Eq. (8) with $i_{L}=0,1\cdots 2n-1$ and
$i_{T}=0,1\cdots N_{T}-1$ and remove $\pm$ in $\hat{c}_{{\bm{k}}A_{\pm}}$ and
$\hat{c}_{{\bm{k}}B_{\pm}}$, we have the expression used in the main text;

$\displaystyle\hat{H}(t)=\sum_{\bm{k}}\begin{bmatrix}\hat{c}^{\dagger}_{{\bm{k}}A}&\hat{c}^{\dagger}_{{\bm{k}}B}\end{bmatrix}{\bf
h}({\bm{k}}(t))\begin{bmatrix}\hat{c}_{{\bm{k}}A}\\\
\hat{c}_{{\bm{k}}B}\end{bmatrix}$ (15)

with

$\displaystyle{\bf h}({\bm{k}})=\begin{bmatrix}-\mu&F({\bm{k}})\\\
F^{*}({\bm{k}})&-\mu\end{bmatrix}.$ (16)

In the following, we express the Hamiltonian (15) for each ${\bm{k}}$ as
$\hat{H}_{\bm{k}}(t)$. The expression tells that for CNTs one needs to choose
2n lines along ${\bm{t}}_{0}$ (${\bm{k}}_{\parallel}$ in the main text) from
the momentum space of graphene and the corresponding Hamiltonian for each
point is the same as graphene. The magnetic field ${\bm{A}}_{B}$ shifts the
position of lines in the direction of ${\bm{l}}_{0}$ (${\bm{k}}_{\perp}$ in
the main text).

The explicit expression of the current is

$\displaystyle\hat{J}_{\parallel}(t)$ $\displaystyle=\frac{qt_{\rm hop}}{\hbar
N}\sum_{\alpha=1,2,3}i({\bm{e}}_{||}\cdot{\bm{b}}_{\alpha})\Bigl{\\{}$
$\displaystyle\sum_{{\bm{k}}}\begin{bmatrix}\hat{c}^{\dagger}_{{\bm{k}}A}&\hat{c}^{\dagger}_{{\bm{k}}B}\end{bmatrix}\begin{bmatrix}0&-e^{i{\bm{k}}(t)\cdot{\bm{b}}_{\alpha}}\\\
e^{-i{\bm{k}}(t)\cdot{\bm{b}}_{\alpha}}&0\end{bmatrix}\begin{bmatrix}\hat{c}_{{\bm{k}}A}\\\
\hat{c}_{{\bm{k}}B}\end{bmatrix}\Bigl{\\}},$ (17)

where $N=2nN_{T}=2N_{c}$. In the following, we express the contribution from
each ${\bm{k}}$ as $\hat{J}_{\parallel,{\bm{k}}}(t)$, i.e.
$\hat{J}_{\parallel}(t)=\frac{1}{N}\sum_{\bm{k}}\hat{J}_{\parallel,{\bm{k}}}(t)$.

###  I.2 Dirac model around $K$ and $K^{\prime}$

The TB model for graphene hosts two Dirac points ${\bm{K}}$ and
${\bm{K}}^{\prime}$, where $F({\bm{K}})=0$ and $F({\bm{K}}^{\prime})=0$. To be
specific, ${\bm{K}}={\bm{l}}_{0}+\frac{1}{3}{\bm{t}}_{0}$ and
${\bm{K}}^{\prime}=\frac{2}{3}{\bm{t}}_{0}$. The Hamiltonian (15) around these
points can be approximated by the Dirac model. By expanding $F({\bm{k}})=0$
around ${\bm{K}}$ or ${\bm{K}}^{\prime}$ and regarding
$e^{-i{\bm{K}}\cdot{\bm{b}}_{1}}\hat{c}^{\dagger}_{{\bm{k}}B}$
($e^{-i{\bm{K}}^{\prime}\cdot{\bm{b}}_{1}}\hat{c}^{\dagger}_{{\bm{k}}B}$ ) as
new $\hat{c}^{\dagger}_{{\bm{k}}B}$, $\hat{H}_{\bm{k}}(t)$ is approximated as

$\displaystyle\hat{H}^{({\bm{K}},{\bm{K}}^{\prime})}_{{\bm{k}}}(t)=\begin{bmatrix}\hat{c}^{\dagger}_{{\bm{k}}A}&\hat{c}^{\dagger}_{{\bm{k}}B}\end{bmatrix}{\bf
h}_{\pm}({\bm{k}}(t))\begin{bmatrix}\hat{c}_{{\bm{k}}A}\\\
\hat{c}_{{\bm{k}}B}\end{bmatrix}$ (18)

with

$\displaystyle{\bf
h}_{\pm}({\bm{k}})=\begin{bmatrix}-\mu&-\frac{\sqrt{3}at_{\rm hop}}{2}[\pm
k_{x}+ik_{y}]\\\ -\frac{\sqrt{3}at_{\rm hop}}{2}[\pm
k_{x}-ik_{y}]&-\mu\end{bmatrix}.$ (19)

Here, ”+” is for ${\bm{K}}$ and ”-” is for ${\bm{K}}^{\prime}$, $k_{x}$ and
$k_{y}$ are the x and y components of ${\bm{k}}$. In this expression, we shift
the origin of the momentum to ${\bm{K}}$ or ${\bm{K}}^{\prime}$. The
corresponding expression of $\hat{J}_{\parallel,{\bm{k}}}$ is

$\displaystyle\hat{J}^{({\bm{K}},{\bm{K}}^{\prime})}_{\parallel,{\bm{k}}}=\mp\frac{\sqrt{3}aqt_{\rm
hop}}{2\hbar}({\bm{e}}_{\parallel}\cdot{\bm{e}}_{x})\;\begin{bmatrix}\hat{c}^{\dagger}_{{\bm{k}}A}&\hat{c}^{\dagger}_{{\bm{k}}B}\end{bmatrix}\begin{bmatrix}0&1\\\
1&0\end{bmatrix}\begin{bmatrix}\hat{c}_{{\bm{k}}A}\\\
\hat{c}_{{\bm{k}}B}\end{bmatrix}$ $\displaystyle\;\;-\frac{\sqrt{3}aqt_{\rm
hop}}{2\hbar}({\bm{e}}_{\parallel}\cdot{\bm{e}}_{y})\;\begin{bmatrix}\hat{c}^{\dagger}_{{\bm{k}}A}&\hat{c}^{\dagger}_{{\bm{k}}B}\end{bmatrix}\begin{bmatrix}0&i\\\
-i&0\end{bmatrix}\begin{bmatrix}\hat{c}_{{\bm{k}}A}\\\
\hat{c}_{{\bm{k}}B}\end{bmatrix}.$ (20)

Here, ${\bm{e}}_{x}$ and ${\bm{e}}_{y}$ are the unit vectors for the x and y
directions.

###  I.3 Semiconductor Bloch Equation

In the dipolar gauge Li  _et al._ (2020); Schüler  _et al._ (2021); Murakami
and Schüler (2022), the semiconductor Bloch equation (SBE, i.e. the von
Neumann equation for the single particle density matrix) is expressed as

$\displaystyle\hbar\partial_{t}\bm{\rho}_{\bm{k}}(t)$
$\displaystyle=i[\bm{\rho}_{\bm{k}}(t),{\bf
h}({\bm{k}}(t))]+\hbar[\partial_{t}\bm{\rho}_{\bm{k}}(t)]_{\rm corr},$ (21)

where $[\partial_{t}\bm{\rho}_{\bm{k}}(t)]_{\rm corr}$ indicates the collision
integral. The collision integral expresses the scattering, relaxation and
dephasing processes originating from electron-electron interactions, electron-
phonon couplings and/or impurity scatterings. In principle, these terms can be
evaluated microscopically Stefanucci and Leeuwen (2013); Schüler  _et al._
(2020); Ridley  _et al._ (2022). However, the direct microscopic evaluation is
computationally expensive. Therefore, in this paper, we employ the relaxation
time approximation as

$\displaystyle[\partial_{t}\bm{\rho}_{\bm{k}}(t)]_{\rm
corr}=-\frac{\bm{\rho}_{\bm{k}}(t)-\bm{\rho}_{{\rm eq},{\bm{k}}(t)}}{T_{1}}$
$\displaystyle\;\;\;\;\;+\Bigl{(}\frac{1}{T_{1}}-\frac{1}{T_{2}}\Bigl{)}{\bf
U}({\bm{k}}(t))\bm{\rho}^{\rm H}_{{\rm off},{\bm{k}}}(t){\bf
U}^{\dagger}({\bm{k}}(t)).$ (22)

Here $T_{1}$ corresponds to the relaxation of the band population, while
$T_{2}$ represents the interband dephasing time. We assume that initially
($t=0$) the system is in equilibrium at temperature $T$ and there is no
electric field ${\bm{A}}_{E}(0)=0$. $\bm{\rho}_{{\rm eq},{\bm{k}}}$ indicates
the equilibrium single particle density matrix for ${\bm{k}}$ at this initial
temperature. The meaning of these terms becomes clear if one expresses the SBE
using the basis that diagonalizes the Hamiltonian at each time (the Houston
basis). For the expression, we introduce a unitary matrix ${\bf U}({\bm{k}})$
that diagonalizes ${\bf h}({\bm{k}})$ as ${\bf U}^{\dagger}({\bm{k}}){\bf
h}({\bm{k}}){\bf U}({\bm{k}})={\bm{\epsilon}}({\bm{k}})$ with
${\bm{\epsilon}}({\bm{k}})={\rm
diag}[\epsilon_{0}({\bm{k}}),\epsilon_{1}({\bm{k}})]$. The single-particle
density matrix in the new basis is $\bm{\rho}^{\rm H}_{{\bm{k}}}(t)\equiv{\bf
U}^{\dagger}({\bm{k}}(t))\bm{\rho}_{\bm{k}}(t){\bf U}({\bm{k}}(t))$. The
diagonal component of $\bm{\rho}^{\rm H}_{\bm{k}}$ indicates the occupation of
the bands, and the off-diagonal component means the interband hybridization
(coherence). In equilibrium, $\bm{\rho}^{\rm H}_{{\rm eq},{\bm{k}}}$ only
hosts the diagonal terms. In this basis, the SBE equivalent to Eq. (22) is

$\displaystyle\hbar\partial_{t}\bm{\rho}^{\rm H}_{{\bm{k}}}(t)$
$\displaystyle=i[\bm{\rho}^{\rm
H}_{{\bm{k}}}(t),\bm{\epsilon}({\bm{k}}(t))-q\sum_{a=x,y}E_{a}(t){\bf
d}_{a}({\bm{k}}(t))]$ $\displaystyle+\hbar[\partial_{t}\bm{\rho}_{\bm{k}}^{\rm
H}(t)]_{\rm corr},$ (23)

with

$\displaystyle[\partial_{t}\bm{\rho}_{\bm{k}}^{\rm H}(t)]_{\rm corr}$
$\displaystyle=-\frac{\bm{\rho}^{\rm H}_{{\rm
diag},{\bm{k}}}(t)-\bm{\rho}^{\rm H}_{{\rm
eq},{\bm{k}}(t)}}{T_{1}}-\frac{\bm{\rho}^{\rm H}_{{\rm
off},{\bm{k}}}(t)}{T_{2}}.$ (24)

Here, $\bm{\rho}_{{\rm diag}}$ indicates a matrix consisting of diagonal
components of $\bm{\rho}$, while $\bm{\rho}_{{\rm off}}$ indicates a matrix
consisting of off-diagonal components of $\bm{\rho}$. ${\bf
d}_{a}({\bm{k}})=i{\bf U}^{\dagger}({\bm{k}})[\partial_{a}{\bf U}({\bm{k}})]$
is the (non-abelian) Berry connection for the direction $a$, which plays the
role of dipole matrix elements. $\partial_{a}=\partial/\partial k_{a}$ is the
momentum derivate for the direction $a$. Note that Eqs. (21) and (23) are
equivalent to the conventional expression of SBE in the length gauge Wilhelm
_et al._ (2021); Yue and Gaarde (2022); Murakami and Schüler (2022).

As explain in the main text, the origin of the radiation is the time-dependent
change in the electric polarization $P\;(\equiv
q\sum_{i}{\bm{e}}_{\parallel}\cdot{\bm{r}}_{i}\langle\hat{n}_{i}\rangle)$, or
the corresponding current Huttner  _et al._ (2017); Yue and Gaarde (2022). As
in previous works Ishikawa (2010); Dimitrovski  _et al._ (2017), we introduce
the intraband and interband currents based on the Houston basis as

$\displaystyle J_{\parallel,\rm intra}(t)$ $\displaystyle=\frac{q}{\hbar
N}\sum_{\bm{k}}{\rm Tr}\Bigl{[}\bm{\rho}^{\rm
H}_{\bm{k}}(t)\partial_{k_{\parallel}}{\bm{\epsilon}}({\bm{k}}(t))\Bigl{]},$
(25) $\displaystyle J_{\parallel,\rm inter}(t)$
$\displaystyle=-i\frac{q}{\hbar N}\sum_{{\bm{k}},a}e_{\parallel,a}{\rm
Tr}\Bigl{[}\bm{\rho}^{\rm H}_{\bm{k}}(t)\;[{\bf
d}_{a}({\bm{k}}(t)),\bm{\epsilon}({\bm{k}}(t))]\Bigl{]}.$ (26)

The intraband current is related to the diagonal components of $\bm{\rho}^{\rm
H}_{\bm{k}}(t)$. Hence this current originates from the change in the band
occupation. On the other hand, the interband current originates from the off-
diagonal components of $\bm{\rho}^{\rm H}_{\bm{k}}(t)$ (the interband
components). Given that the current is the time derivative of the
polarization, the interband current is closely related the change in the
interband polarization $P_{\rm inter}$. It is the polarization originating
from the interband components as

$\displaystyle P_{\rm inter}=\frac{q}{\hbar
N}\sum_{{\bm{k}},a}e_{\parallel,a}{\rm Tr}\Bigl{[}\bm{\rho}^{\rm
H}_{\bm{k}}(t){\bf d}^{\rm off}_{a}({\bm{k}}(t))\Bigl{]},$ (27)
$\displaystyle{\bf d}^{\rm
off}_{a}({\bm{k}})=\begin{bmatrix}0&d_{01,a}({\bm{k}})\\\
d_{10,a}({\bm{k}})&0\end{bmatrix}.$

In the actual implementation, we evaluate currents
($J_{\parallel},J_{\parallel,\rm intra},J_{\parallel,\rm inter}$) using the
difference of the single-particle density matrix from equilibrium
$\delta{\bm{\rho}}_{\bm{k}}(t)=\bm{\rho}_{\bm{k}}(t)-\bm{\rho}_{{\rm
eq},{\bm{k}}(t)}$. Note that these currents are zero in equilibrium. This
strategy has two benefits. Firstly, one can save the calculation cost since
away from the Dirac points $\delta{\bm{\rho}}_{\bm{k}}(t)$ is essentially zero
and we do not need to evaluate them. Secondly, this can be directly applied
for the Dirac model. Namely, the currents for CNTs and graphene can be
evaluated from the Dirac models using $\delta\bm{\rho}_{{\bm{k}}}(t)$ at
${\bm{K}}$ and ${\bm{K}}^{\prime}$ and summing up these contributions. If we
evaluate the currents from the Dirac models using $\bm{\rho}_{{\bm{k}}}(t)$
the results depend on the momentum cutoff.

###  I.4 Comments on previous studies

Although the manipulation of HHG in a single-wall CNT using the Aharanov-Bohm
effect is our main point, there have been some studies on the effects of the
static magnetic field on HHG in CNTs Khosravi  _et al._ (2009); Dvuzhilova
_et al._ (2021). In Ref. Khosravi  _et al._ , 2009, they consider the mangetic
field applied to the perpendicular direction of the tube and numerically show
the sensitivity of the HHG spectrum for the excitation with the light of the
wave length 800nm. Thus, the Aharanov-Bohm effect and the Dirac physics are
not relevant here. In Ref. Dvuzhilova  _et al._ , 2021, they study the
propagation of the Airy beam with 1 fs duration in the photonic crystal
consisting of many CNTs under magnetic field. We note that the setup and the
timescale involved are very different from our case.

##  II Derivation of Eq. (4) in the main text

We focus on the case of $B=0$ and the contribution from the bands passing the
Dirac points. The AC electric field is applied along
${\bm{e}}_{\parallel}(=\begin{bmatrix}\cos\theta\\\ \sin\theta\end{bmatrix})$.
The momentum on the line passing $K$ or $K^{\prime}$ can be expressed as
${\bm{k}}=k{\bm{e}}_{\parallel}$. The corresponding Hamiltonian of the Dirac
model is

$\displaystyle\hat{H}_{\bm{k}}(t)=$ (28)
$\displaystyle\begin{bmatrix}\hat{c}_{{\bm{k}}A}^{\dagger}&\hat{c}_{{\bm{k}}B}^{\dagger}\end{bmatrix}\begin{bmatrix}-\mu&\frac{\sqrt{3}at_{\rm
hop}}{2}e^{i\theta_{\pm}}k(t)\\\ \frac{\sqrt{3}at_{\rm
hop}}{2}e^{-i\theta_{\pm}}k(t)&-\mu\end{bmatrix}\begin{bmatrix}\hat{c}_{{\bm{k}}A}\\\
\hat{c}_{{\bm{k}}B}\end{bmatrix},$

where $k(t)\equiv k-\frac{q}{\hbar}A_{E}(t)$, $\theta_{+}=\theta+\pi$ for
${\bm{K}}$ and $\theta_{+}=-\theta$ for ${\bm{K}}^{\prime}$. We introduce a
new basis (the Houston basis) as

$\displaystyle\begin{bmatrix}\hat{c}_{kR}\\\
\hat{c}_{kL}\end{bmatrix}\equiv\frac{1}{\sqrt{2}}\begin{bmatrix}e^{-i\frac{\theta_{\pm}}{2}}&e^{i\frac{\theta_{\pm}}{2}}\\\
-e^{-i\frac{\theta_{\pm}}{2}}&e^{i\frac{\theta_{\pm}}{2}}\end{bmatrix}\begin{bmatrix}\hat{c}_{kA}\\\
\hat{c}_{kB}\end{bmatrix}.$ (29)

Then we can express the Hamiltonian as

$\displaystyle\hat{H}_{k}(t)=$ (30)
$\displaystyle\begin{bmatrix}\hat{c}_{kR}^{\dagger}&\hat{c}_{kL}^{\dagger}\end{bmatrix}\begin{bmatrix}\frac{\sqrt{3}at_{\rm
hop}}{2}k(t)-\mu&0\\\ 0&-\frac{\sqrt{3}at_{\rm
hop}}{2}k(t)-\mu\end{bmatrix}\begin{bmatrix}\hat{c}_{kR}\\\
\hat{c}_{kL}\end{bmatrix},$

and the current as

$\displaystyle\hat{J}_{k,\parallel}(t)=\frac{\sqrt{3}aqt_{\rm
hop}}{2\hbar}\begin{bmatrix}\hat{c}_{kR}^{\dagger}&\hat{c}_{kL}^{\dagger}\end{bmatrix}\begin{bmatrix}1&0\\\
0&-1\end{bmatrix}\begin{bmatrix}\hat{c}_{kR}\\\ \hat{c}_{kL}\end{bmatrix}.$
(31)

We introduce the single-particle density matrix in the present basis as
$\rho^{R}_{k}(t)=\langle\hat{c}_{kR}^{\dagger}(t)\hat{c}_{kR}(t)\rangle$ and
$\rho^{L}_{k}(t)=\langle\hat{c}_{kL}^{\dagger}(t)\hat{c}_{kL}(t)\rangle$. The
corresponding SBE becomes

$\displaystyle\partial_{t}\rho_{k}^{X}(t)=[\partial_{t}\rho_{k}^{X}(t)]_{\rm
corr}$ (32)

with

$\displaystyle[\partial_{t}\rho_{k}^{X}(t)]_{\rm
corr}=-\frac{\rho_{k}^{X}(t)-\rho^{X}_{\rm eq}(k(t),T(t))}{T_{1}}.$ (33)

Here $X=R,L$ and $\rho^{X}_{\rm eq}(k(t),T(t))$ represents the equilibrium
single-particle density matrix at the momentum $k(t)$ and the temperature
$T(t)$. Note that this expression is slightly extended from Eq. (22). Namely,
we consider the time-dependent temperature for the equilibrium state that the
system is relaxed to at each time. Furthermore, the SBE tells that no
interband component in the density matrix is induced by the electric field,
which already indicates the absence of the interband current. Using the
integrated form of the SBE, the deviation of the single-particle density
matrix from equilibrium
$\delta\rho_{\bm{k}}^{X}(t)=\rho_{\bm{k}}^{X}(t)-\rho^{X}_{\rm eq}(k(t),T(t))$
can be expressed as

$\displaystyle\delta\rho_{k}^{X}(t)$ $\displaystyle=\rho^{X}_{\rm
eq}(k,T(0))-\rho^{X}_{\rm
eq}(k(t),T(t))+\int^{t}_{0}dt[\partial_{t}\rho_{k}^{X}(t)]_{\rm corr}.$ (34)

Note that, at $t=0$, the system is in equilibrium at the temperature $T(0)$
and there is no electric field.

Now, if we focus on the contributions to the current from the line passing $K$
and $K^{\prime}$ points, we have

$\displaystyle J_{\parallel}(t)$
$\displaystyle=\frac{1}{N}\sum_{\bm{k}}J_{{\bm{k}}\parallel}(t)$
$\displaystyle=\frac{1}{n}\frac{\sqrt{3}a^{2}qt_{\rm hop}}{4\pi\hbar}\int
dk[\delta\rho_{k}^{R}(t)-\delta\rho_{k}^{L}(t)],$ (35)

where

$\displaystyle\int dk[\delta\rho_{\bm{k}}^{R}(t)-\delta\rho_{\bm{k}}^{L}(t)]=$
$\displaystyle-\int dk[\rho^{R}_{\rm eq}(k(t),T(t))-\rho^{R}_{\rm
eq}(k,T(0))]$ $\displaystyle+\int dk[\rho^{L}_{\rm
eq}(k(t),T(t))-\rho^{L}_{\rm eq}(k,T(0))]$ $\displaystyle+\int
dk\int^{t}_{0}dt\\{[\partial_{t}\rho_{k}^{R}(t)]_{\rm
corr}-[\partial_{t}\rho_{k}^{L}(t)]_{\rm corr}\\}$ (36)
$\displaystyle=-2\frac{q}{\hbar}A_{E}(t)+\int
dk\int^{t}_{0}dt\\{[\partial_{t}\rho_{k}^{R}(t)]_{\rm
corr}-[\partial_{t}\rho_{k}^{L}(t)]_{\rm corr}\\}.$

Note that the temperature effect vanishes in the integral of the first two
terms in the second line. Finally, we obtain

$\displaystyle J_{\parallel}(t)=-\frac{1}{n}\frac{\sqrt{3}a^{2}q^{2}t_{\rm
hop}}{2\pi\hbar}A_{E}(t)+\int^{t}_{0}dt[\partial_{t}J_{\parallel}(t)]_{\rm
corr}$ (37)

with $[\partial_{t}J_{\parallel}(t)]_{\rm
corr}\equiv\frac{1}{n}\frac{\sqrt{3}a^{2}qt_{\rm hop}}{4\pi\hbar}\int
dk\\{[\partial_{t}\rho_{k}^{R}(t)]_{\rm
corr}-[\partial_{t}\rho_{k}^{L}(t)]_{\rm corr}\\}$. Taking the derivative and
substituting Eq. (33) to $[\partial_{t}\rho_{k}^{X}(t)]_{\rm corr}$, we have

$\displaystyle\partial_{t}J_{\parallel}(t)=\frac{1}{n}\frac{\sqrt{3}a^{2}q^{2}t_{\rm
hop}}{2\pi\hbar^{2}}E(t)-\frac{J_{\parallel}(t)}{T_{1}}.$ (38)

This is Eq. (4) in the main text. Our derivation tells that, even if we
improve the relaxation time approximation considering the time-dependent
temperature, HHG is absent in the gapless 1d Dirac system.

##  III Derivation of $\sigma^{(3)}$

We derive the third-order optical conductivity by perturbatively solving the
SBE for the Dirac model as in Ref. Cheng  _et al._ , 2015. To this end, we
change the gauge from the dipolar gauge, which is mainly used in this paper,
to the length gauge expressed with the band basis Murakami and Schüler (2022).
From the Dirac Hamiltonian Eq. (3) in the main text, we reach the SBE in the
length gauge

$\displaystyle\partial_{t}\bm{\rho}^{\rm LB}_{k}(t)$
$\displaystyle=-\frac{q}{\hbar}E(t)\partial_{k}\bm{\rho}^{\rm
LB}_{k}(t)+\frac{i}{\hbar}[\bm{\rho}^{\rm
LB}_{k}(t),{\bm{\epsilon}}(k)-qE(t){\bf d}(k)]$
$\displaystyle-\frac{\bm{\rho}^{\rm LB}_{{\rm diag},k}(t)-\bm{\rho}^{\rm
LB}_{{\rm eq},k}}{T_{1}}-\frac{\bm{\rho}^{\rm LB}_{{\rm off},k}(t)}{T_{2}},$
(39)

where

$\displaystyle{\bm{\epsilon}}(k)$ $\displaystyle=\begin{bmatrix}\tilde{t}_{\rm
hop}\sqrt{k^{2}+K_{B}^{2}}-\mu&0\\\ 0&-\tilde{t}_{\rm
hop}\sqrt{k^{2}+K_{B}^{2}}-\mu\end{bmatrix},$ $\displaystyle{\bf d}(k)$
$\displaystyle=-\frac{K_{B}}{2(k^{2}+K_{B}^{2})^{2}}\begin{bmatrix}0&-i\\\
i&0\end{bmatrix}.$ (40)

Here, $k\in[-\infty,\infty]$ represents the momentum along ${\bf
k}_{\parallel}$, $\bm{\rho}^{\rm LB}_{k}$ is the $2\times 2$ single-particle
density matrix in the length gauge with the band basis, $\tilde{t}_{\rm
hop}\equiv\frac{\sqrt{3}at_{\rm hop}}{2}$ and
$K_{B}\equiv\frac{q}{\hbar}A_{B}$. In the following, we set
$\kappa\equiv\sqrt{k^{2}+K_{B}^{2}}$, $r_{k}\equiv\frac{K_{B}}{2\kappa^{2}}$,
denote the band index of the matrix as $s=0,1$, and omit ”LB” from
$\bm{\rho}^{\rm LB}_{k}$.

To solve the SBE perturbatively, we expand $\bm{\rho}_{k}$ as

$\displaystyle\rho_{s_{1}s_{2},k}(t)=\sum_{n=0}^{\infty}\rho_{s_{1}s_{2},k}^{(l)}(t)$
(41)

with $\rho^{(l)}_{s_{1}s_{2},k}(t)\propto E^{l}$. Note that we have
$\rho_{s_{1}s_{2},k}^{(0)}(t)=\delta_{s_{1},s_{2}}n_{s_{1},k}$, where
$n_{s,k}=[e^{\epsilon_{s,k}/{k_{B}T}}+1]^{-1}$ and
$\epsilon_{s,k}=(-)^{s}\tilde{t}_{\rm hop}\kappa-\mu$.

By substituting Eq. (41) into Eq. (39), we have

$\displaystyle\begin{split}\hbar\partial_{t}\rho^{(l)}_{ss,k}(t)&=-(-)^{s}qE(t)r_{k}(\rho^{(l-1)}_{\bar{s}s,k}+\rho^{(l-1)}_{s\bar{s},k})\\\
&-qE(t)\partial_{k}\rho^{(n-1)}_{ss,k}-\Gamma_{1}\rho^{(l)}_{ss,k},\\\
\hbar\partial_{t}\rho^{(l)}_{s\bar{s},k}(t)&=-i(-)^{s}\epsilon_{g,k}\rho_{s\bar{s},k}^{(l)}(t)-(-)^{s}qE(t)r_{k}(\rho^{(n-1)}_{\bar{s}\bar{s},k}-\rho^{(l-1)}_{ss,k})\\\
&-qE(t)\partial_{k}\rho^{(l-1)}_{s\bar{s},k}-\Gamma_{2}\rho^{(l)}_{s\bar{s},k},\end{split}$
(42)

where $\Gamma_{1}\equiv\hbar/T_{1}$, $\Gamma_{2}\equiv\hbar/T_{2}$ and
$\epsilon_{g,k}\equiv\epsilon_{0,k}-\epsilon_{1,k}$. We express the SPDM as

$\displaystyle\begin{split}\rho^{(1)}_{s_{1}s_{2},k}(t)&=\int\frac{d\omega_{3}}{2\pi}(-q)E_{\omega_{3}}e^{-i\omega_{3}t}\mathcal{P}^{(1)}_{s_{1}s_{2},k}(\omega_{3}),\\\
\rho^{(2)}_{s_{1}s_{2},k}(t)&=\int\frac{d\omega_{2}d\omega_{3}}{(2\pi)^{2}}(-q)^{2}E_{\omega_{2}}E_{\omega_{3}}e^{-i\omega_{0}t}\mathcal{P}^{(2)}_{s_{1}s_{2},k}(\omega_{2},\omega_{3}),\\\
\rho^{(3)}_{s_{1}s_{2},k}(t)&=\int\frac{d\omega_{1}d\omega_{2}d\omega_{3}}{(2\pi)^{3}}(-q)^{3}E_{\omega_{1}}E_{\omega_{2}}E_{\omega_{3}}\times\\\
&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;e^{-i\omega
t}\mathcal{P}^{(3)}_{s_{1}s_{2},k}(\omega_{1},\omega_{2},\omega_{3}),\end{split}$
(43)

where $\omega_{0}\equiv\omega_{2}+\omega_{3}$ and
$\omega\equiv\omega_{1}+\omega_{2}+\omega_{3}$. Using Eq. (42), one can
evaluate $\mathcal{P}^{(l)}$ sequentially from $l=1$. For $l=1$, we have

$\displaystyle\begin{split}\mathcal{P}^{(1)}_{ss,k}(\omega_{3})&=\frac{i}{\nu_{3}}\partial_{k}n_{s,k},\\\
\mathcal{P}^{(1)}_{s\bar{s},k}(\omega_{3})&=\frac{-ir_{k}\Delta
n_{k}}{\vartheta_{3}-(-)^{s}\epsilon_{g,k}},\end{split}$ (44)

where $\nu_{3}\equiv\hbar\omega_{3}+i\Gamma_{1}$,
$\vartheta_{3}\equiv\hbar\omega_{3}+i\Gamma_{2}$, $\Delta n_{k}\equiv
n_{0,k}-n_{1,k}$. For $l=2$, we have

$\displaystyle\begin{split}\mathcal{P}^{(2)}_{ss,k}(\omega_{2},\omega_{3})&=\frac{i}{\nu_{0}}\Bigl{[}\frac{i}{\nu_{3}}\frac{\partial^{2}n_{s,k}}{\partial
k^{2}}\\\ &\;\;\;\;\;\;\;\;\;\;\;\;-i(-)^{s}r_{k}^{2}\Delta
n_{k}\bigl{(}\frac{1}{\vartheta_{3}+\epsilon_{g,k}}+\frac{1}{\vartheta_{3}-\epsilon_{g,k}}\bigl{)}\Bigl{]},\\\
\mathcal{P}^{(2)}_{s\bar{s},k}(\omega_{2},\omega_{3})&=\frac{1}{\vartheta_{0}-(-)^{s}\epsilon_{g,k}}\Bigl{[}\frac{r_{k}}{\nu_{3}}\frac{\partial\Delta
n_{k}}{\partial k}+\partial_{k}\Bigl{(}\frac{r_{k}\Delta
n_{k}}{\vartheta_{3}-(-)^{s}\epsilon_{g,k}}\Bigl{)}\Bigl{]},\end{split}$ (45)

where $\nu_{0}\equiv\hbar\omega_{0}+i\Gamma_{1}$ and
$\vartheta_{0}\equiv\hbar\omega_{0}+i\Gamma_{2}$.

For $l=3$, we have

$\displaystyle\mathcal{P}^{(3)}_{ss,k}(\omega_{1},\omega_{2},\omega_{3})$
$\displaystyle=\frac{1}{\nu\nu_{0}\nu_{3}}P_{1;s,k}+\frac{1}{\nu\nu_{0}}P_{2;s,k}(\vartheta_{3})$
(46)
$\displaystyle+\frac{1}{\nu\nu_{3}}P_{3;s,k}(\vartheta_{0})+\frac{1}{\nu}P_{4;s,k}(\vartheta_{0},\vartheta_{3})$

with

$\displaystyle\begin{split}P_{1;s,k}&=-i\frac{\partial^{3}n_{s,k}}{\partial
k^{3}},\\\
P_{2;s,k}(\vartheta_{3})&=i(-)^{s}\partial_{k}\Bigl{[}r_{k}^{2}\Delta
n_{k}\Bigl{(}\frac{1}{\vartheta_{3}+\epsilon_{g,k}}+\frac{1}{\vartheta_{3}-\epsilon_{g,k}}\Bigl{)}\Bigl{]},\\\
P_{3;s,k}(\vartheta_{0})&=i(-)^{s}r_{k}^{2}\frac{\partial\Delta
n_{k}}{\partial
k}\Bigl{(}\frac{1}{\vartheta_{0}+\epsilon_{g,k}}+\frac{1}{\vartheta_{0}-\epsilon_{g,k}}\Bigl{)},\\\
P_{4;s,k}(\vartheta_{0},\vartheta_{3})&=i(-)^{s}r_{k}\Bigl{[}\frac{1}{\vartheta_{0}+\epsilon_{g,k}}\partial_{k}\Bigl{(}\frac{r_{k}\Delta
n_{k}}{\vartheta_{3}+\epsilon_{g,k}}\Bigl{)}\\\
&\;\;\;\;\;\;\;\;\;\;\;+\frac{1}{\vartheta_{0}-\epsilon_{g,k}}\partial_{k}\Bigl{(}\frac{r_{k}\Delta
n_{k}}{\vartheta_{3}-\epsilon_{g,k}}\Bigl{)}\Bigl{]},\end{split}$ (47)

and

$\displaystyle\mathcal{P}^{(3)}_{s\bar{s},k}(\omega_{1},\omega_{2},\omega_{3})$
$\displaystyle=\frac{1}{\nu_{0}\nu_{3}}P_{5;s,k}(\vartheta)+\frac{1}{\nu_{0}}P_{6;s,k}(\vartheta,\vartheta_{3})$
(48)
$\displaystyle+\frac{1}{\nu_{3}}P_{7;s,k}(\vartheta,\vartheta_{0})+P_{8;s,k}(\vartheta,\vartheta_{0},\vartheta_{3})$

with

$\displaystyle\begin{split}P_{5;s,k}(\vartheta)&=\frac{ir_{k}}{\vartheta-(-)^{s}\epsilon_{g,k}}\frac{\partial^{2}\Delta
n_{k}}{\partial k^{2}},\\\
P_{6;s,k}(\vartheta,\vartheta_{3})&=-2i\frac{r_{k}^{3}\Delta
n_{k}}{\vartheta-(-)^{s}\epsilon_{g,k}}\Bigl{(}\frac{1}{\vartheta_{3}+\epsilon_{g,k}}+\frac{1}{\vartheta_{3}-\epsilon_{g,k}}\Bigl{)},\\\
P_{7;s,k}(\vartheta,\vartheta_{0})&=\frac{i}{\vartheta-(-)^{s}\epsilon_{g,k}}\partial_{k}\Bigl{[}\frac{r_{k}}{\vartheta_{0}-(-)^{s}\epsilon_{g,k}}\frac{\partial\Delta
n_{k}}{\partial k}\Bigl{]},\\\
P_{8;s,k}(\vartheta,\vartheta_{0},\vartheta_{3})&=\frac{i}{\vartheta-(-)^{s}\epsilon_{g,k}}\times\\\
&\partial_{k}\Bigl{[}\frac{1}{\vartheta_{0}-(-)^{s}\epsilon_{g,k}}\Bigl{\\{}\partial_{k}\Bigl{(}\frac{r_{k}\Delta
n_{k}}{\vartheta_{3}-(-)^{s}\epsilon_{g,k}}\Bigl{)}\Bigl{\\}}\Bigl{]}.\end{split}$
(49)

Here $\nu\equiv\hbar\omega+i\Gamma_{1}$ and
$\vartheta\equiv\hbar\omega+i\Gamma_{2}$.

Now we evaluate the third order conductivity. By taking into account the
contribution from the two Dirac points ($K$ and $K^{\prime}$), the third order
current is expressed as

$\displaystyle
J^{(3)}_{\parallel}(t)=\frac{qa}{n\hbar}\sum_{s_{1},s_{2}}\int\frac{dk}{2\pi}v_{s_{2}s_{1},k}\rho^{(3)}_{s_{1}s_{2},k}(t),$
(50)

where $v_{s_{2}s_{1},k}$ denotes an element of the velocity operator

$\displaystyle{\bm{v}}_{k}\equiv\frac{\tilde{t}_{\rm
hop}}{\kappa}\begin{bmatrix}k&-K_{B}\\\ -K_{B}&-k\end{bmatrix}.$ (51)

We define the unsymmetrized third order conductivity as

$\displaystyle\sigma^{(3)}(\omega_{1},\omega_{2},\omega_{3})\equiv-\frac{q^{4}a}{n\hbar}\sum_{s_{1},s_{2}}\int\frac{dk}{2\pi}v_{s_{2}s_{1},k}\mathcal{P}^{(3)}_{s_{1}s_{2},k}(\omega_{1},\omega_{2},\omega_{3}),$
(52)

where we have

$\displaystyle
J^{(3)}_{\parallel}(t)=\int\frac{d\omega_{1}d\omega_{2}d\omega_{3}}{(2\pi)^{3}}E_{\omega_{1}}E_{\omega_{2}}E_{\omega_{3}}e^{-i\omega
t}\sigma^{(3)}(\omega_{1},\omega_{2},\omega_{3}).$ (53)

Note that in Eq. (52) the contribution from $s_{1}\neq s_{2}$ corresponds to
the interband current ($\sigma^{(3)}_{\rm er}$), while that from $s_{1}=s_{2}$
corresponds to the intraband current ($\sigma^{(3)}_{\rm ra}$).

In the following, we focus on $T=0$ and half filling ($\mu=0$). In this case,
we have $\Delta n_{k}=-1$ and $P_{1;s,k}=P_{3;s,k}=P_{5;s,k}=P_{7;s,k}=0$.
Then, we have $\sigma^{(3)}_{\rm ra}=S_{2}+S_{4}$ and $\sigma^{(3)}_{\rm
er}=S_{6}+S_{8}$, where

$\displaystyle\begin{split}S_{2}(\omega_{1},\omega_{2},\omega_{3})&\equiv-\frac{q^{4}a}{n\hbar}\frac{1}{\nu\nu_{0}}2\int\frac{dk}{2\pi}v_{00,k}P_{2;0,k}(\vartheta_{3}),\\\
S_{4}(\omega_{1},\omega_{2},\omega_{3})&\equiv-\frac{q^{4}a}{n\hbar}\frac{1}{\nu}2\int\frac{dk}{2\pi}v_{00,k}P_{4;0,k}(\vartheta_{3}),\\\
S_{6}(\omega_{1},\omega_{2},\omega_{3})&\equiv-\frac{q^{4}a}{n\hbar}\frac{1}{\nu_{0}}\int\frac{dk}{2\pi}v_{01,k}[P_{6;0,k}(\vartheta,\vartheta_{3})+P_{6;1,k}(\vartheta,\vartheta_{3})],\\\
S_{8}(\omega_{1},\omega_{2},\omega_{3})&\equiv-\frac{q^{4}a}{n\hbar}\int\frac{dk}{2\pi}v_{01,k}[P_{8;0,k}(\vartheta,\vartheta_{0},\vartheta_{3})+P_{8;1,k}(\vartheta,\vartheta_{0},\vartheta_{3})].\end{split}$
(54)

The concrete expressions for these terms are

$\displaystyle
S_{2}(\omega_{1},\omega_{2},\omega_{3})=-\frac{iq^{4}a\tilde{t}_{\rm
hop}}{4\pi n\hbar}\frac{1}{\nu\nu_{0}}\int
dk\frac{K_{B}^{4}}{\kappa^{7}}\Bigl{(}\frac{1}{\vartheta_{3}+2\tilde{t}_{\rm
hop}\kappa}+\frac{1}{\vartheta_{3}-2\tilde{t}_{\rm hop}\kappa}\Bigl{)},$ (55a)
$\displaystyle
S_{4}(\omega_{1},\omega_{2},\omega_{3})=-\frac{iq^{4}a\tilde{t}_{\rm
hop}}{2\pi n\hbar}\frac{1}{\nu}\int
dk\frac{K_{B}^{2}k^{2}}{\kappa^{7}}\Bigl{[}\frac{(\vartheta_{3}+3\tilde{t}_{\rm
hop}\kappa)}{(\vartheta_{0}+2\tilde{t}_{\rm
hop}\kappa)(\vartheta_{3}+2\tilde{t}_{\rm
hop}\kappa)^{2}}+\frac{(\vartheta_{3}-3\tilde{t}_{\rm
hop}\kappa)}{(\vartheta_{0}-2\tilde{t}_{\rm
hop}\kappa)(\vartheta_{3}-2\tilde{t}_{\rm hop}\kappa)^{2}}\Bigl{]},$ (55b)
$\displaystyle
S_{6}(\omega_{1},\omega_{2},\omega_{3})=\frac{iq^{4}a\tilde{t}_{\rm hop}}{8\pi
n\hbar}\frac{1}{\nu_{0}}\int
dk\frac{K_{B}^{4}}{\kappa^{7}}\Bigl{(}\frac{1}{\vartheta+2\tilde{t}_{\rm
hop}\kappa}+\frac{1}{\vartheta-2\tilde{t}_{\rm
hop}\kappa}\Bigl{)}\Bigl{(}\frac{1}{\vartheta_{3}+2\tilde{t}_{\rm
hop}\kappa}+\frac{1}{\vartheta_{3}-2\tilde{t}_{\rm hop}\kappa}\Bigl{)},$ (55c)
$\displaystyle
S_{8}(\omega_{1},\omega_{2},\omega_{3})=\frac{iq^{4}a\tilde{t}_{\rm hop}}{2\pi
n\hbar}\int dk\frac{K_{B}^{2}k^{2}}{\kappa^{7}}\times$ (55d)
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\Bigl{[}\frac{(\vartheta+4\tilde{t}_{\rm
hop}\kappa)(\vartheta_{3}+3\tilde{t}_{\rm
hop}\kappa)}{(\vartheta+2\tilde{t}_{\rm
hop}\kappa)^{2}(\vartheta_{0}+2\tilde{t}_{\rm
hop}\kappa)(\vartheta_{3}+2\tilde{t}_{\rm
hop}\kappa)^{2}}+\frac{(\vartheta-4\tilde{t}_{\rm
hop}\kappa)(\vartheta_{3}-3\tilde{t}_{\rm
hop}\kappa)}{(\vartheta-2\tilde{t}_{\rm
hop}\kappa)^{2}(\vartheta_{0}-2\tilde{t}_{\rm
hop}\kappa)(\vartheta_{3}-2\tilde{t}_{\rm hop}\kappa)^{2}}\Bigl{]}.$

The third harmonic generation corresponds to the case of
$\omega_{1}=\omega_{2}=\omega_{3}=\Omega$. In the main text, we denote
$\sigma^{(3)}(\Omega,\Omega,\Omega)$ simply as $\sigma^{(3)}$.

Now we consider the limit of $B\rightarrow 0$ ($K_{B}\rightarrow 0$). One can
show that the leading terms are

$\displaystyle
S_{2}(\omega_{1},\omega_{2},\omega_{3})=-\frac{iq^{4}a\tilde{t}_{\rm
hop}}{2\pi n\hbar}\frac{1}{\nu\nu_{0}\vartheta_{3}}\int
dk\frac{K_{B}^{4}}{\kappa^{7}},$ (56a) $\displaystyle
S_{4}(\omega_{1},\omega_{2},\omega_{3})=-\frac{iq^{4}a\tilde{t}_{\rm hop}}{\pi
n\hbar}\frac{1}{\nu\vartheta_{0}\vartheta_{3}}\int
dk\frac{K_{B}^{2}k^{2}}{\kappa^{7}},$ (56b) $\displaystyle
S_{6}(\omega_{1},\omega_{2},\omega_{3})=\frac{iq^{4}a\tilde{t}_{\rm hop}}{2\pi
n\hbar}\frac{1}{\nu_{0}\vartheta\vartheta_{3}}\int
dk\frac{K_{B}^{4}}{\kappa^{7}},$ (56c) $\displaystyle
S_{8}(\omega_{1},\omega_{2},\omega_{3})=\frac{iq^{4}a\tilde{t}_{\rm hop}}{\pi
n\hbar}\frac{1}{\vartheta_{0}\vartheta\vartheta_{3}}\int
dk\frac{K_{B}^{2}k^{2}}{\kappa^{7}}.$ (56d)

These terms diverge as $B\rightarrow 0$, and lead to the expression of
$\sigma^{(3)}$ shown in the main text. The rest contributions do not diverge
(regular). The expression of $\sigma^{(3)}$ tells that the intraband component
strongly depends on $T_{1}$, while the interband component strongly depends on
$T_{2}$. This can be understood by the following points. The intraband
component is related to the occupation of electrons in the conduction and
valence bands, and $T_{1}$ is the time scale of the occupation returning to
its equilibrium value. On the other hand, the interband component is related
to the modification of the interband polarization, i.e. the off-diagonal
(interband) component of the density matrix in the Houston basis. These off-
diagonal components are zero in equilibrium, and $T_{2}$ is the time scale
that the components vanish.

Figure 6: (a) HHG spectra $I_{\rm HHG}$ of the doped armchair CNT with and
without the static magnetic field simulated with the tight-binding model. (b)
The intensity at the HHG peaks $I_{\rm peak}(l)$ as a function of $B$. We set
the chiral index $(n,m)=(15,15)$ and consider doped systems with $\mu=100$meV.
We set $T=11.6$K, $T_{1}=98.8$fs and $T_{2}=19.8$fs. The parameters of the AC
electric field are $\hbar\Omega=10$meV, $E_{0}=30$kV/cm, $t_{0}=3.95$ps and
$\sigma={658}$fs.

##  IV Supplementary results

Figure 7: (a) HHG spectra $I_{\rm HHG}$ of the armchair CNT with and without
the static magnetic field simulated with the tight-binding model. (b) The
intensity at the HHG peaks $I_{\rm peak}(l)$ as a function of $B$. We set the
chiral index $(n,m)=(15,15)$ and consider half filling ($\mu=0$). We set
$T=11.6$K, $T_{1}=98.8$fs and $T_{2}=19.8$fs. The parameters of the AC
electric field are $\hbar\Omega=40$meV, $E_{0}=120$kV/cm, $t_{0}=986$fs and
$\sigma={164}$fs.

In this section, we show the supplementary results to support the discussion
in the main text.

###  IV.1 Doping effects

In Fig. 6(a), we show the HHG spectra of the armchair CNT with $(n,m)=(15,15)$
away from half filling ($\mu=0.1$eV) with and without the magnetic field $B$.
The excitation condition is the same as that in the main text. Although the
position of the chemical potential can be much larger than the gap and the
excitation frequency, the drastic enhancement of HHG caused by the magnetic
field is observed. In Fig. 6(b), we show the intensity of the HHG peaks as a
function of $B$. Unlike in the half-filled case, there is no clear optimal
value of $B$ which maximizes the peak intensity. This may be attributed to
that, in the doped system, charge carriers contributing to the intraband
current exist from the beginning. This situation is different from the half-
filled system, where charge carriers need to be created by the AC electric
field.

###  IV.2 Excitation with higher frequency

In Fig. 7(a), we show the HHG spectra of the armchair CNT with $(n,m)=(15,15)$
at half filling for the excitation frequency $\hbar\Omega=40$meV. Figure 6(b)
shows the corresponding intensity of the HHG peaks as a function of $B$. The
drastic enhancement of HHG can be still observed even when the gap is smaller
than $\hbar\Omega$. For example, the gap for $B=10$T is about $10$meV. Note
that the peaks in the HHG spectrum is less sharp for $\hbar\Omega=40$meV than
for $\hbar\Omega=10$meV, since the relaxation and dephasing times become
relatively large with respect to the period of the excitation
$\frac{2\pi}{\Omega}$. In the range of $B$ in Fig. 7(b) , we cannot observe
the optimal value of $B$ unlike in the case of $\hbar\Omega=10$meV. However,
it should exist in the larger-$B$ regime. Note that, for $\hbar\Omega=10$meV,
the optimal $B$ is around $30\sim 40$T, where the gap is about $30\sim 40$meV.

Figure 8: HHG spectra $I_{\rm HHG}\times n^{2}$ of the armchair CNT simulated
with the tight-binding model for indicated values of the chiral index $n$ and
magnetic field $B$. Note that the cases of $(n,B)=(15,30)$, $(n,B)=(5,90)$ and
$(n,B)=(30,15)$ overlap each other. We set $T=11.6$K, $T_{1}=98.8$fs and
$T_{2}=19.8$fs. The parameters of the electric field are $\hbar\Omega=10$meV,
$E_{0}=30$kV/cm, $t_{0}=3.95$ps and $\sigma={658}$fs. Figure 9: The intensity
at the peaks of the HHG spectra $I_{\rm peak}(l)$ as a function of the
strength of the AC electric field $E_{0}$ for (a) $l=3$, (b) $l=5$ and (c)
$l=7$. The dashed lines are proportional to $(E_{0})^{2l}$ for each panels. We
set $(n,m)=(15,15)$, $\mu=0$, $T=11.6$K, $T_{1}=98.8$fs and $T_{2}=98.8$fs.
The parameters of the electric field are $\hbar\Omega=10$meV, $t_{0}=3.95$ps
and $\sigma={658}$fs.  Figure 10: Comparison of the HHG spectra obtained in
different ways for (a) $(T_{1},T_{2})=(98.8{\rm fs},98.8{\rm fs})$ and (b)
$(T_{1},T_{2})=(98.8{\rm fs},19.8{\rm fs})$. All of $I_{\rm HHG}^{\rm Dirac}$,
$I_{\rm intra}$ and $I_{\rm inter}$ are evaluated from the Dirac model
considering only bands closest to $K$ or $K^{\prime}$. We set $(n,m)=(15,15)$,
$\mu=0$, $T=11.6$K, and $B=2$T. The parameters of the electric field are
$\hbar\Omega=10$meV, $E_{0}=1.37$kV/cm, $t_{0}=3.95$ps and $\sigma={658}$fs.

###  IV.3 Dependence on chiral index $n$

As we discussed in the main text, in order to achieve the same amount of
momentum shift due to the AB effect and thus to open the same size of the gap,
an armchair CNT with larger $n$ requires weaker magnetic field $B$. To
exemplify this, we compared the HHG spectra for different values of the chiral
index $n$ and magnetic field in Fig. 8. To compare the structure of the HHG
spectra for different $n$, $I_{\rm HHG}(\omega)$ is multiplied by $n^{2}$.
Remember that $I_{\rm HHG}(\omega)$ is defined as the HHG intensity normalized
by the number of atoms of a CNT. In Fig. 8, we take the case of $n=15$ and
$B=30$T as a reference. For $n=5$, the case with the magnetic field $B=90$T
shows essentially the same HHG spectrum as the reference. As we discussed in
the main text, this is because i) in the present excitation condition, only
the bands closest to $K$ and $K^{\prime}$ contributes to HHG and ii) the gap
of the corresponding massive Dirac system is controlled by $A_{B}\propto
B\cdot n$. With $B=30$T, the gap is smaller compared to the reference case.
For $n=30$, the case with $B=15$T shows essentially the same HHG spectrum as
the reference due to the same reason. With $B=30$T, the gap becomes larger
than the reference case.

###  IV.4 Effects of $T_{1}$ and $T_{2}$

In Fig. 9, we show the intensity at the HHG peaks $I_{\rm peak}(l)$ as a
function of the electric-field strength $E_{0}$ for $(T_{1},T_{2})=(98.8{\rm
fs},98.8{\rm fs})$. This figure can be directly compared with Fig. 4 in the
main text, which shows the results for $(T_{1},T_{2})=(98.8{\rm fs},19.8{\rm
fs})$. The generic behavior is similar to Fig. 4 in the main text. Namely, the
higher order corrections lead to the saturation of the HHG intensity. (The
additional increase of $I_{\rm peak}(3)$ around $E_{0}\simeq 100$kV/cm for
$B=2$T originates from the deviation from the linear band dispersion.) The
deviation from the perturbative regime happens at smaller $E_{0}$ for smaller
$B$, i.e. for smaller gap. The optimal value of $B$ that maximizes $I_{\rm
peak}$ shifts to larger values with increasing $E_{0}$. On the other hand,
unlike the results for $(T_{1},T_{2})=(98.8{\rm fs},19.8{\rm fs})$, the
optimal value of $B$ does not decrease continuously with decreasing $E_{0}$.
For small $E_{0}$, the HHG intensity is largest around $B=20$T, where the gap
is almost twice of $\Omega$.

In order to understand the origin of the different behavior in the weak-
electric-field regime between the cases of $(T_{1},T_{2})=(98.8{\rm
fs},98.8{\rm fs})$ and $(T_{1},T_{2})=(98.8{\rm fs},19.8{\rm fs})$, we compare
the contributions to HHG from the interaband and intraband currents, i.e.
$I_{\rm inter}$ and $I_{\rm intra}$, for a weak magnetic field, see Fig. 10.
When $T_{1}=T_{2}$, $J_{\rm inter}$ and $J_{\rm intra}$ cancel each other very
efficiently, and thus the total intensity $I_{\rm HHG}$ becomes much smaller
than $I_{\rm inter}$ and $I_{\rm intra}$, see Fig. 10(a). On the other hand,
when $T_{1}>T_{2}$, the cancelation between $J_{\rm inter}$ and $J_{\rm
intra}$ is less efficient and $I_{\rm HHG}$ becomes closer to $I_{\rm inter}$
and $I_{\rm intra}$, see Fig. 10(b). Thus, the total HHG intensity becomes
much larger in the latter case, although both of $I_{\rm inter}$ and $I_{\rm
intra}$ in the former case can be larger than those in the latter case. This
result is consistent with the expression of $\sigma^{(3)}$. Namely, for
$T_{1}\neq T_{2}$ it diverges with $B\rightarrow 0$ due to the absence of the
cancelation between $J_{\rm inter}$ and $J_{\rm intra}$.