# Femtosecond field-driven on-chip unidirectional electronic currents in
nonadiabatic tunnelling regime

Liping Shi  These two authors contributed equally  School of Engineering,
Westlake University, 18 Shilongshan Road, Hangzhou, 310024, China  Institute
of Advanced Technology, Westlake Institute for Advanced Study, 18 Shilongshan
Road, Hangzhou, 310024, China  Ihar Babushkin  These two authors contributed
equally<EMAIL_ADDRESS>Institute of Quantum Optics, Leibniz
University Hannover, Welfengarten 1, 30167 Hannover, Germany  Cluster of
Excellence PhoenixD (Photonics, Optics, and Engineering-Innovation Across
Disciplines), 30167 Hannover, Germany  Max Born Institute, Max-Born-Str. 2a,
12489 Berlin, Germany  Anton Husakou  Max Born Institute, Max-Born-Str. 2a,
12489 Berlin, Germany  Oliver Melchert  Institute of Quantum Optics, Leibniz
University Hannover, Welfengarten 1, 30167 Hannover, Germany  Cluster of
Excellence PhoenixD (Photonics, Optics, and Engineering-Innovation Across
Disciplines), 30167 Hannover, Germany  Bettina Frank  Universität Stuttgart,
Pfaffenwaldring 57, 70569 Stuttgart, Germany  Juemin Yi  Institute of Physics
and Center of Interface Science, Carl von Ossietzky University Oldenburg,
26129 Oldenburg, Germany  Gustav Wetzel  Institute of Electronic Materials and
Devices, Leibniz University Hannover, Schneiderberg 32, 30167 Hannover,
Germany  Ayhan Demircan  Institute of Quantum Optics, Leibniz University
Hannover, Welfengarten 1, 30167 Hannover, Germany  Cluster of Excellence
PhoenixD (Photonics, Optics, and Engineering-Innovation Across Disciplines),
30167 Hannover, Germany  Christoph Lienau  Institute of Physics and Center of
Interface Science, Carl von Ossietzky University Oldenburg, 26129 Oldenburg,
Germany  Harald Giessen  Universität Stuttgart, Pfaffenwaldring 57, 70569
Stuttgart, Germany  Misha Ivanov  Max Born Institute, Max-Born-Str. 2a, 12489
Berlin, Germany  Uwe Morgner  Institute of Quantum Optics, Leibniz University
Hannover, Welfengarten 1, 30167 Hannover, Germany  Cluster of Excellence
PhoenixD (Photonics, Optics, and Engineering-Innovation Across Disciplines),
30167 Hannover, Germany  Milutin Kovacev  Institute of Quantum Optics, Leibniz
University Hannover, Welfengarten 1, 30167 Hannover, Germany  Cluster of
Excellence PhoenixD (Photonics, Optics, and Engineering-Innovation Across
Disciplines), 30167 Hannover, Germany

###### Abstract

Recently, asymmetric plasmonic nanojunctions [Karnetzky et. al., Nature Comm.
2471, 9 (2018)] have shown promise as on-chip electronic devices to convert
femtosecond optical pulses to current bursts, with a bandwidth of multi-
terahertz scale, although yet at low temperatures and pressures. Such
nanoscale devices are of great interest for novel ultrafast electronics and
opto-electronic applications. Here, we operate the device in air and at room
temperature, revealing the mechanisms of photoemission from plasmonic
nanojunctions, and the fundamental limitations on the speed of optical-to-
electronic conversion. Inter-cycle interference of coherent electronic
wavepackets results in a complex energy electron distribution and birth of
multiphoton effects. This energy structure, as well as reshaping of the
wavepackets during their propagation from one tip to the other, determine the
ultrafast dynamics of the current. We show that, up to some level of
approximation, the electron flight time is well-determined by the mean
ponderomotive velocity in the driving field.

##  I Introduction

As known from atomic physics [1, 2, 3, 4], with increase of the incident laser
intensity, the ionization dynamics undergoes a transition from a relatively
slow multiphoton process to fast sub-cycle bursts, referred to as tunneling
ionization. The dimensionless Keldysh parameter $\gamma=2\pi\tau_{T}/T$
characterizes, relative to the optical cycle duration $T$, a typical time
$\tau_{T}=\sqrt{2m\phi}/|eE|$, required for an electron to leave the atom with
the ionization potential $\phi$ in an external field of the amplitude $E$
(here $m$ and $e$ are electron mass and charge). In the multiphoton regime
($\gamma\gg 1$), electrons require many optical cycles to be ionized. It is
easier to describe such a process in the frequency-domain as an absorption of
$n=\phi/\hbar\omega$ photons of energy $\hbar\omega$. But in the tunneling
regime ($\gamma\ll 1$) electrons escape from the nucleus during a small
fraction of an optical cycle, and thus the description in the time-domain is
more convenient. Nevertheless, time- and frequency-domain descriptions
represent the same process. In particular, multiphoton dynamics can be also
described in time-domain, as an interference among the electronic wavepackets
[3, 5] created at different optical cycles. This fact was utilized in the
Yudin-Ivanov model [6], where both multiphoton and tunneling regimes were
described in a single formula. In this unified description, the time-domain
tunneling picture is used. Yet, the electron ionization dynamics differ in
these two regimes, demonstrating sub-cycle features in the tunneling regime
and much slower dynamics in the multiphoton one. In the intermediate regime,
$\gamma\sim 1$, both fast and slow components appear.

This tunneling picture arose from atomic physics, and has also been proven
valid for photoemission at surfaces of metallic nanotips [7, 8, 9, 10, 11,
12]. Although the electronic wavefunctions inside the metal are not localized,
an approximation of localized wavefunctions still yields good results when
considering ionization from metallic surfaces [9, 8, 11, 12]. Strong near-
field enhancement near the nanotips triggers electrons near the Fermi level to
tunnel through the surface on a sub-cycle timescale [7, 8, 13, 14, 15, 16, 17,
18, 19, 20, 12, 21], which is of particular interest in ultrafast time-
resolved electron nanoscopy [22, 23, 24, 25, 12, 21]. Interference of the
wavepackets tunneled at different cycles gives rise to pronounced peaks in the
electron spectra separated by the energy of the pump photons [8, 11, 12, 21].

Recently, ultrafast electron emission from gold dimer nanoantennas
(nanojunctions) with gap sizes down to the few nanometer scale has attracted
great interest for on-chip petahertz electronics [25, 26, 27]. When embedding
such a nanojunction into a closed circuit, unidirectional electronic optically
controllable currents bursts, which vary on the sub-cycle scale, may arise
[26, 27], if few-cycle driving pulses with controlled carrier-envelope phase
(CEP) are used. Especially interesting in this respect is the recent proposal
to create ultrafast unidirectional currents in asymmetric nanostructures [28],
which enables a DC bias without necessity of CEP control of the driving pulse.

Since gold nanoantennas cannot withstand strong electric fields due to the
limitations of photothermal damage as well as near-field nonthermal ablation
[29, 30, 31, 32], the quasistatic tunneling regime ($\gamma\ll 1$) is rather
impractical. In contrast, the intermediate regime ($\gamma\sim 1$) should be
much more attractive. However, in this regime, the nature of photoemission and
the corresponding electron dynamics are still poorly understood, despite being
of crucial importance for investigating the limitations on the speed and
bandwidth of on-chip ultrafast electronic devices. The pioneering works in
Ref.[28] left several important questions open. For instance, they suggested
that dynamical lowering of the barrier (Schottky effect) can significantly
reduce the scaling of multiphoton photoemission with intensity. However, this
assumption has no analogies in other systems in strong optical fields like
atoms or molecules. Besides, typical velocities of electrons in the nanogap
remain unclarified, although they are crucial to understand the bandwidth
limits of the plasmonic electronic devices.

Here we employ asymmetric plasmonic nanojunctions to produce ultrafast
unidirectional currents in a way similar to used in Ref. [28]. In contrast to
Ref. [28], we operate the device at room temperature and in ambient
conditions. Furthermore, we develop a modification of Yudin-Ivanov
nonadiabatic tunneling model adapted for gold nanostructures. It allows us to
clarify the origin of the photoemission currents. Moreover, we investigate
ultrafast dynamics of electronic wavepackets in the nanojunctions, and
reconsider fundamental limitations in speed and bandwidth of such optical
field-driven electronic devices.

Figure 1:  (a) An overview SEM image of the plasmonic nanodevice. $J$,
current; $A$, ammeter; $E_{0}$, optical electric field of incident laser;
$\theta$, laser polarization direction with respect to y-axis. (b) A high-
resolution SEM image of two representative nanojunctions, with numerical
simulation of electric near-field distribution at the gold-air interface (d).
(e) Electric field enhancement factor along the dashed line in (d). (c)
Measured photoemission current (blue curve) and third harmonic signal (red
curve) at various thickness of inserted pair of silica wedges, which controls
the laser pulse duration. (f) Temporal evolution of dark current (blue curve)
of the device, photoemission current (black curve) and third-harmonic
generation (red curve) from the device.  Figure 2:  (a) Experimentally
measured photoemission current as a function of incident electric field
strength $E_{0}$, F-N, Fowler-Nordheim model; Y-I, Yudin-Ivanov model. Inset:
photoemission rate versus laser field strength according to the Yudin-Ivanov
model (red solid curve) in comparison to multiphoton model (MPI, green solid
line, rate proportional to $I^{4}$, where $I$ is the field intensity), F-N
model (black dashed-dotted curve) and Keldysh formula (blue dashed curve)[1].
(b) Photoemission current $J$ (blue squares) versus the laser polarization
direction $\theta$ in a good agreement with the function
$J\sim\cos^{4}\theta$, where $\theta$ is the polarization angle (black curve).
Inset: Instantaneous photoemission rate over a half cycle of laser pulse. Gray
curve shows the electric field. Blue solid and black dashed curves depict
nonadiabatic and quasistatic tunneling rates, respectively.

##  II Experiment

Femtosecond pulses from a Ti:sapphire oscillator with a repetition rate of
$r=100$ MHz are tightly focused onto an array of asymmetric plasmonic
nanojunctions. The plasmonic nanojunctions are fabricated by focused ion beam
milling of a 100 nm thick Au film on a sapphire substrate. Figure 1(a) depicts
an overview scanning electron microscopy (SEM) image of our nanostructure. The
laser spectrum spans from 650 to 1000 nm with a central wavelength of 840 nm.
Two broadband double-chirped mirrors and a pair of fused silica wedges are
employed to control the dispersion of the pulses. We characterize the pulses
by dispersion-scan method [33] and retrieve the shortest duration to be
$\simeq 7.6$ fs. The laser beam diameter on the sample is estimated to be
around 7 $\mu$m, which corresponds to a simultaneous illumination of about
$N\simeq 100$ unit cells of the nanojunctions. Therefore, for the highest
incident pulse energy of 1 nJ, we estimate the peak electric field of the pump
laser to be $E_{0}\sim 1.6$ V/nm. Figure 1(b) displays a representative unit
cell of the nanojunctions, which consists of a tip-to-tip triangular Au
needles with a gap of 50 nm. However, the upper tip is much sharper than the
bottom one, leading to an asymmetric distribution of the near-field
enhancement, as shown in Fig. 1(d) and Fig. 1(e). The sharper tip exhibits
much higher near-field strength. As a result, an effective negative bias
arises from the upper tip to the bottom one, breaking the symmetry of
electronic transport. The upper Au needle acts as an emitter electrode and the
bottom one functions as a collector. We, therefore, expect to observe a net
photoemission current when integrating over the entire pulse width.

We measure the time-integrated photoemission current $J$ by a low-noise
amplifier. Meanwhile, the plasmon-enhanced third harmonic generation [34, 35]
is employed to optimize the laser dispersion and the spatial overlap between
laser focus plane and the plasmonic nanojunctions. Figure 1(c) shows the
measured photoemission current (blue curve) and third harmonic signal (red
curve) at various thickness of the inserted silica wedges. The sensitive
dependence of the current on the pulse duration confirms that the electron
emission is induced by nonthermal processes, because the thermal effects do
not depend on the pulse duration for femtosecond laser [36]. At the shortest
pulse duration, the integrated current reads $J=0.6$ nA, which translates to
40 electrons per pulse in average. As shown in Fig. 1(f), the nanodevice
withstands a long-term exposure.

In Fig. 2(a) a log-log dependence of the photoemission current on the incident
laser electric field strength ($J-E$ curve) is shown, with an observed slope
of $\sim 2$. We also study the integrated current versus the polarization of
the incident laser [$J-\theta$ curve, cf. Fig. 2(b)]. Here $\theta$ is defined
as the crossing angle between electric field direction and tip-to-tip
orientation of the nanojunction. The current follows a fourth-order power of
$\cos\theta$, corresponding to a $J-E$ curve with a coefficient $n=2$, i.e.,
$J\propto(E^{2})^{2}=I^{2}$, consistent with directly measured $J-E$ curve in
Fig. 2(a). Considering that the electric near-field enhancement factor at a
nanotip is inverse proportional to its radius of curvature [37], as shown by
the SEM in Fig. 1 (b), the ratio of photoemission rate from the emitter with
respect to the collector is estimated to be $(R/r)^{4}\sim$40\. Therefore, we
ignore the photoemission from the collector in the below theoretical sections.
It should be pointed out that these results also agree with those of Ref.
[28]. However, we investigate the photoemission process using the time-domain
approach rather than the frequency-domain approach, and draw a conclusion
opposite to that of Ref. [28], as discussed later.

##  III Theoretical description

###  III.1 Yudin-Ivanov approach

For the near-field enhancement in our structure the Keldysh parameter is
estimated to be in the intermediate range [c.f., gray area in the inset of
Fig. 2(a)]. In the framework of atomic physics, a model named Yudin-Ivanov
(Y-I) formula [6] works well in the range from multiphoton to tunneling regime
[red curve in the inset of Fig. 2(a)] and keeps correct inter-and intra-cycle
ionization dynamics. Assuming the driving field in the form of
$E(t)=\mathcal{E}(t)\cos\omega t$, where $\cos\omega t$ denotes the fast
oscillating component, and $\mathcal{E}(t)$ the slow-varying envelope, the
cycle-resolved ionization rate $\Gamma$ is given by (in atomic units, that is,
frequency $\omega$, time $t$, ionization potential $\phi$, and field
$\mathcal{E}$ are measured in the corresponding Hartree units
$\omega_{a}=0.26$ rad/as, $t_{a}=24.2$ as, $\phi_{a}=27.21$ eV, and
$\mathcal{E}_{a}=514.2$ V/nm):

$\Gamma(t)=\frac{\pi}{\tau_{T}}\exp\left(-\sigma_{0}\frac{\mathcal{E}(t)^{2}}{\omega^{3}}\right)\left[\frac{2\kappa^{3}}{\mathcal{E}(t)}\right]^{2Z/\kappa}\exp\left[-\frac{\mathcal{E}(t)^{2}}{2\omega^{3}}\sigma_{1}\sin^{2}(\omega
t)\right].$ (1)

Here $Z$ is the effective atomic charge, $\kappa=\sqrt{2\phi}$,
$\sigma_{0}=\frac{1}{2}(\gamma^{2}+\frac{1}{2})\ln
C-\frac{1}{2}\gamma\sqrt{1+\gamma^{2}}$,
$C=1+2\gamma\sqrt{1+\gamma^{2}}+2\gamma^{2}$, and $\sigma_{1}=\ln
C$-$2\gamma/\sqrt{1+\gamma^{2}}$. The averaged photoemission rate over a
single optical cycle of Eq. (1) reads (in atomic units):

$\Gamma_{\mathrm{av}}=\frac{\pi}{\tau_{T}}\exp\left(-\sigma_{0}\frac{\mathcal{E}(t)^{2}}{\omega^{3}}\right)\left[\frac{2\kappa^{3}}{\mathcal{E}(t)}\right]^{2Z/\kappa}\left[\frac{2\omega^{3}}{\pi\mathcal{E}(t)^{2}\sigma_{1}}\right]^{1/2}.$
(2)

To proceed further, it is important to understand the basic idea behind the
approach leading to Eq. (1) and Eq. (2). The population in continuum $W$ is
represented as an integral over all partial amplitudes
$a_{\mathrm{\mathbf{p}}}$ of the ionized electron with the momentum
$\mathrm{\mathbf{p}}$:
$W=\int_{\mathrm{\mathbf{p}}}|a_{\mathrm{\mathbf{p}}}|^{2}d^{3}p$. The
corresponding amplitude $a_{\mathrm{\mathbf{p}}}$, under reasonable
approximations, in particular if the influence of the Coulomb potential after
tunneling is neglected, can be obtained as

$a_{\mathrm{\mathbf{p}}}\sim\int_{-\infty}^{t}e^{-iS(t,t^{\prime})/\hbar}dt^{\prime},$
(3)

where $S(t,t^{\prime})$ is the action:

$S(t,t^{\prime})=(\phi+\frac{p^{2}}{2m})(t-t^{\prime})+\frac{1}{2m}\int_{t^{\prime}}^{t}\left(\mathrm{\mathbf{p}}+e\mathrm{\mathbf{A}}(t^{\prime\prime})\right)^{2}dt^{\prime\prime},$
(4)

$\mathrm{\mathbf{A}}(t)$ being the vector potential corresponding to the
driving electric field $E(t)$. Following Eq. (3), every amplitude
$a_{\mathrm{\mathbf{p}}}$ is the result of summation over all partial
amplitudes, each having the phase $S(t,t^{\prime})$ defined by Eq. (4). The
first term in Eq. (4) corresponds to phase shift gained by overcoming the
barrier, and the second term, so-called Volkov phase, corresponds to the phase
electron gains in the electric field. Action $S(t,t^{\prime})$ changes quickly
for all points except the stationary ones, that is, obeying $\partial
S/\partial t^{\prime}=0$. As a result, integration over the fast oscillating
argument yields zero everywhere except at the stationary points (this fact
constitutes the essence of so-called stationary phase, or saddle-point
method). The stationary condition results in a complex value of $t^{\prime}$
equal to $t^{\prime}_{s}$, which can be found analytically. The ionization
rate can be, up to the insignificant prefactor, calculated as
$\Gamma(t)\sim\exp(-2\textrm{Im}\,{[S(t,t^{\prime}_{0})]}/\hbar)$. Under
reasonable approximations, this expression can be calculated analytically and
gives rise to Eq. (1).

The Y-I model, as mentioned above, is written for a rather general case,
without the details of the potential. The only parameters referring to a
particular system are the effective charge $Z$ and the ionization potential
$\phi$. For gold nanostructures, we should take into account the ability of
electron density on the surface to redistribute, on a femtosecond time scale,
to “screen” the ionized electron. This screening is in a good approximation
described by the mirror-charge model [38, 39]. In this model, if we consider
an ideal metal, the outgoing electron “feels” a charge at the position
mirrored relative to the metal surface, but with the charge sign inverted. The
distance from electron to this effective ”ion”, represented by mirrored
charge, is twice the distance to the true parent ion which is positioned
exactly on the surface. Thus the attraction force becomes 4 times smaller than
for the case of an electron and a single atom. This attraction force reduction
can be taken into account by introducing the effective charge $1/4$ instead of
$1$ in the Y-I model. Furthermore, in a non-ideal metal, the effective charge
value is modified by a factor $|(1-\epsilon)/(1+\epsilon)|$, where $\epsilon$
is the complex susceptibility of the metal. As a result, the effective charge
in Eq. (1) is governed by the equation:

$Z=\frac{1}{4}\left|\frac{1-\epsilon}{1+\epsilon}\right|.$ (5)

We fit the experimental data in Fig. 2a by the modified Y-I model and obtain
the field enhancement factor to be $g\approx 14.2$. This is a very reasonable
value which is in a good agreement with the numerical simulation [c.f. Fig.
1(d, e)]. Accordingly, the Keldysh parameter $\gamma$ in our experiments is
evaluated to be in the range from 0.8 to 1.7. The fitting was made without
considering the spatial profile of the pulse. It is however easy to see that
in the present case, taking into account the Gaussian profile leads to only a
constant pre-factor 1/4. Indeed, since we have, with a good precision $J\sim
E_{0}^{4}$, spatial integrating of the current $J(r)$ (were $r$ is the radial
coordinate) taking into the Gaussian profile of the field
$E_{0}\sim\exp(-r^{2}/2\sigma^{2})$ ($\sigma$ is the pulse width) will give
the constant factor 1/4 in comparison with the same integration, performed
over the constant-field distribution with the same area.

Figure 3:  Dynamics of ionization at the emitter, and current at the
collector, according to numerical simulations of the single-electron problem
Eq. (6) for $gE_{0}=12.3$ V/nm and 7 fs pulse duration. (a) The the modulus of
the wavefunction of the ionized electron $|\psi(x,t)|$ in dependence on $x$
and $t$. White line shows the driving electric field; as the electronic
wavepacket arises in continuum and propagates, it is separated into the well-
visible electronic “beams” labeled by $n=0,1,\ldots$. Each beam [see also (c)]
corresponds to an electron absorbed approximately $n$ photons from the driving
field and thus having energy $\approx n\hbar\omega$. (b) The electron current
$J(t)$ created on the collector surface in dependence on time $t$. Dashed red
line shows the current created by a homogeneous field ($\tilde{\alpha}=0)$.
The upper inset shows $J(t)$ on a larger times scale, and the lower inset
depicts the spectrum of $J(t)$. The vertical lines show the energies
corresponding to $n\hbar\omega$. (c) Temporal evolution of the free electron
energy. White curve shows again the driving electric field, whereas the
horizontal white lines show the energies corresponding to $n\hbar\omega$. (d)
Energy spectra at the (somewhat arbitrarily selected) times $t_{i}$ [marked in
(c)]. In (c) and (d) one clearly sees multiphoton peaks $n=1,2,\ldots$ as they
arise on the inter-cycle time scale due to interference, and become more and
more pronounced towards the end of the pulse.

As shown in the inset of Fig. 2(a), the Y-I formula approaches the multiphoton
limit at low intensities, and approximates the tunneling limit as given by the
F-N equation at high intensities. We remark that there is another analytical
expression, so-called Keldysh formula [1, 40], which works in the tunneling,
multiphoton and intermediate regimes as well. It is obtained using the same
stationary phase method as the Y-I formula, but the corresponding integrals
are taken in the frequency space. However, the advantages of the Y-I formula
are the applicability to very short (yet single-color) pulses and a
possibility to describe the dynamics inside the laser cycle [c.f. the inset in
Fig. 2(b)], whereas the Keldysh equation was written for the quasi-
monochromatic wave and gives only cycle-averaged ionization rate. For the
particular parameters presented here there is no possibility to reliably
differentiate between Y-I and Keldysh formulas. This differentiation can be
done, for instance, for even shorter pulses in the case if the CEP control is
implemented. In this situation the change of the CEP leads to change of the
result in the case of Y-I formula but in the framework of the Keldysh formula
the result should be CEP-independent.

As depicted in Fig. 2, the current $J$ is nearly proportional to $I^{2}$.
According to the Y-I model, we identify this $I^{2}$-law as an indication that
we are in the intermediate regime, and thus no Schottky effect is necessary to
explain this behaviour as it is done in Refs. [28, 40]. In the following
paragraphs, we return to this point in details.

###  III.2 Srödinger equation

To get deeper insight into the dynamics of the photoemission and subsequent
electron propagation between the tips, we simulate the emission by numerical
solution of the following one-dimensional time-dependent Schrödinger equation
(TDSE) in the Coulomb gauge ($\partial_{x}A=0$):

$i\hbar\frac{\partial\psi(x,t)}{\partial
t}=\frac{1}{2m}\left[(p+eA(x,t))^{2}+V(x)\right]\psi(x,t)$ (6)

where $\psi(x,t)$ is the electronic wavefunction,
$p=-i\hbar\frac{\partial}{\partial x}$, $A(x,t)$ is the vector potential which
takes into account spatial field inhomogeneity as a pre-factor
$e^{-\tilde{\alpha}x}$, $\tilde{\alpha}=1.0$ ns-1, $V(x)$ is a rectangular
asymmetric potential

$V(x)=\begin{cases}-V_{0},&\text{if $|x|<a$;}\\\ 0,&\text{if $x>a$;}\\\
\infty,&\text{if $x<-a$,}\end{cases}$ (7)

where $a=0.106$ nm, $V_{0}=16.94$ eV are selected in such a way that i) the
potential has exactly one bound state and ii) the ionization potential of this
bound state equals to the work function of gold (5.1 eV). We note that this
potential assumes that the wavefunction inside the metal is localized. It was
shown, however, that this assumption does not significantly influence the
ionization rate [9]. Moreover, the experimentally observed electron spectra
[8, 10] and even electron dynamics [41] are good described by this rather
simple approach. Therefore, this approach is widely used to model the
ionization [42, 21, 12, 8, 10, 11, 43, 44]. In the potential defined by Eq.
(7), ionization can occur only in the positive direction of $x$. The electrons
leaving the emitter are accelerated by the field and propagate towards the
collector. These electrons are considered to be fully absorbed, that is,
reflection on the potential of the collector is neglected. We model this by
adding to the potential the soft absorbing boundary $V(x)\to V(x)-i\alpha(x)$
with
$\alpha=\alpha_{0}\left(1+\textrm{erf}\,{\left((x-s)/\delta\right)}\right)/2$,
where $\delta=2.65$ pm, $\alpha_{0}=1$ au, and $s=50$ nm is the distance
between the emitter and collector. The simulation was made by a split-step
method, with separate evaluation of the terms $\sim p^{2}$, $pA+Ap$, $A^{2}$
and $V$; the action of $p$ was calculated using the fast Fourier transform.

The resulting dynamics of the electronic wavefunction $\psi(x,t)$ is shown in
Fig. 3(a) for the peak driving field amplitude $gE_{0}=12.3$ V/nm
(corresponding to $\gamma=1.68$, clearly in the transient regime) and pulse
duration of 7 fs, according to the experiment. One can see from Fig. 3(a),
that at every positive-field subcycle of the driving electric field, an
ionization event takes place: noticeable part of the electron is released
close to extremum of the electric field. As one can see from Fig. 3(c), the
free electron density increases starting from the maximum of the electric
field, achieves its maximum one quarter of the cycle later, and then decreases
because some part of electrons returns back. After being ionized, the part of
electrons, which does not return, propagate towards the collector.
Interestingly, upon the propagation the electron wavepacket is separated into
distinct well-visible “beams” marked by $n=1,2,\ldots$, every of them
propagating with velocity clearly different from the others. The field
inhomogeneity plays only relatively minor role in this dynamics. If it is
removed [$\tilde{\alpha}=0$, see dashed line in Fig. 3(b)] the beam structure
remains the same, only most of the electrons move faster to the collector.

Ionized electrons reach, after some propagation, the collector, producing the
current $J(t)$ given as

$J(t)=\frac{\hbar}{2im}\left(\psi^{*}\frac{\partial\psi}{\partial
x}-\psi\frac{\partial\psi^{*}}{\partial x}\right).$ (8)

Here, $\psi$ is taken at the surface of the collector. The above mentioned
“beams”, which are visible in Fig. 3(a), manifest themselves as the short
spikes of $J(t)$ as can be seen in Fig. 3(b).

The nature of this dynamics becomes apparent if we consider the picture in
energy space, see Fig. 3(c), where the energy spectrum of the electrons in
dependence on time is shown, as well as in Fig. 3(d), where energy spectra at
specific times $t_{i}$, $i=0\ldots 3$ are presented. One can see [cf. for
instance $t_{0},t_{1}$] that in the beginning of the pulse there is no visible
structure in the energy distribution of electrons. Electrons are born with a
broad energy spectrum of more than 10 eV width. The energy structure sets up
gradually during the next few cycles (see the time events $t_{2}$, $t_{3}$),
and, finally close to the end of the pulse (the time event $t_{3}$) it settles
to be peaked around the multiples of the photon energy $n\hbar\omega$. This
very clearly shows that appearance of multiphoton effects (absorption of $n$
photons) appears only on the inter-cycle scale, as an interference between
newly-born parts of electron wavepacket and the ones which are already present
in the continuum. These prominent inter-cycle effects in energy space are well
nown in atomic physics [3, 5] but were demonstrated also for nanostructures,
see [8, 11].

Figure 4:  Schematic representation of different photon absorption channels by
electron ionization. This picture is to be compared to Fig. 3(c). At every
ionization event the electrons are born with a broad unstructured energy
spectrum (rectangles with black-white gradient), and phase $\Phi$ (colored
gradients) which depends on the temporal position of the ionization event
($\Phi\sim t-t_{0}$, here $t_{0}$ is set so that $t-t_{0}=0$ at the first of
two depicted events). The interference of these wavepackets leads to visible
periodic modulations of the energy distribution after the pulse, which can be
interpreted as multiphoton channels $n=0,n=1,\ldots$. Thus, after the end of
the pulse one can distinguish between the channels, for instance direct
ionization channel (green horizontal line) and ionization via Schottky effect
(red dashed line). The amplitudes of the channels allow to estimate the
relative importance of each channel.

In even more clear form this inter-cycle interference is schematically shown
in Fig. 4, which illustrates the physics behind the results in Fig. 3(c). As
it is presented in Fig. 4, the newborn electrons have a broad spectrum (black-
white gradient) and different phases (color gradient - in the figure we
deliberately set the time origin so that the first phase is constant in
energy). The interference between parts of the electronic wavepacket created
at different cycles leads to clearly visible channels $n=0,n=1,\ldots$ after
the end of the pulse. Note that in Fig. 3(c) these dynamics start to set up
already in the middle of the pulse since, unless the wavepacket is not the
first one [$t_{0}$ in Fig. 3(c)], the newborn parts of the wavepacket start to
interfere with the previously born ones [$t_{1},t_{2}$ in Fig. 3(c)].

This condition of constructive interference can be easily obtained as follows:
the phase shift $\Phi$ of the electron in the continuum (in the presence of
the field) over the optical period $T$ is $\Phi=T(p^{2}/2m+U_{p})/\hbar$,
where $U_{p}=e^{2}\mathcal{E}^{2}/4m\omega^{2}$ is the ponderomotive energy.
At the same time, the electrons in the metal (near the Fermi energy) will
experience the relative phase shift $\Phi_{F}=-T\phi/\hbar$. Thus, the
relative phase of two partial wavepackets born in the continuum at the two
time instants separated by $T$ is $\Phi+\Phi_{F}$. The corresponding
constructive interference condition $\Phi+\Phi_{F}=2\pi n$ gives:
$p^{2}/2m+U_{p}+I_{p}=n\hbar\omega$, that is, the subsequent peaks in electron
energy $p^{2}/2m$ are separated by the photon energy $\hbar\omega$. This
demonstrates that the multiphoton peaks corresponding to absorption of $n$
photons by an electron appear from the interference between the electronic
wavepackets created by different optical cycles. Inside the cycle, no such
multiphoton effects can be identified. One can explain this also in the terms
of the Heisenberg uncertainty relation $\Delta p_{x}\Delta x\geq\hbar/2$: in
every ionization event, the electrons are born in the very small region
surrounding the surface, of the order of $\Delta x\approx 0.1$ nm, which means
uncertainty in momentum $\Delta p_{x}$, corresponding to the kinetic energy in
the range from zero to several eV. This uncertainty in momentum is partially
”regularized” on the longer, intercycle scale as described above, giving rise
to the multiphoton energy structure.

From Fig. 4 one can clearly see how to distinguish the impacts from different
ionization channels (accomplished by absorption of different number of
photons), and to estimate their relative importance: if a certain channel is
present in the ionization process, there must be a corresponding peak in the
electron energy distribution after the end of this pulse. The “intensity” of
every particular peak allows to estimate the relative impact of different
channels. To distinguish different channels is important in view of Refs. [28,
40] where so called Schottky effect is proposed to describe the current-vs-
intensity behaviour. The Schottky-effect-based explanation suggests that the
potential barrier is lowered by the external field (see Fig. 4, red dashed
line) so that two instead of four photons are sufficient for ionization. This
channel corresponds to $n=2$ in Fig. 4 and in Fig. 3, in contrast to “direct”
tunneling which corresponds to $n=0$ (green line in Fig. 4).

As one can see from Fig. 3(c) and Fig. 4, the peak at $2\hbar\omega$ ($n=2$)
which could correspond to Schottky effect indeed appears, but it is also
clearly not a dominating one. The peak corresponding to “direct” tunneling
$n=0$ has even higher amplitude. The “anomalous” dependence of the current $J$
on intensity $I$ ($J\sim I^{2}$) is thus not explained by the Schottky effect
but by the fact that we are in the transition region from tunnel to
multiphoton ionization, and the impacts from different channels $n$ have close
amplitudes and add up to give the observed scaling.

Interestingly, the emergence of the multiphoton energy structure of the
electronic wavepacket allows to understand, what happens with an electron as
it propagates between the tips [Fig. 3(a)] and thus to interpret the resulting
current $J(t)$ in Fig. 3(b). Indeed, different electronic “beams” in Fig. 3(a)
and different peaks in $J(t)$ in Fig. 3(b) correspond to different peaks in
the electron energy in Fig. 3(c). The slowest beam $n=0$ corresponds to the
lowest-energy electrons, which absorbed just enough energy to get through the
barrier; $n=1$ corresponds to the electrons which absorbed one photon more,
and so on. Every beam corresponds to electrons which have a mean velocity
around $v=\sqrt{2n\hbar\omega/m}$. Because of diffractive spreading of the
wavepacket, its width grows as $\sqrt{t}$. The slowest wavepackets spend
longer time to overcome the nanogap, and thus spread stronger. The resulting
structure of the current $J(t)$ in Fig. 3(b) is thus the collection of peaks,
every of them having increasing width due to increasing diffractive spreading.
Although the peak with $n=0$ carriers the largest part of the whole electron
probability, it is also broadened at strongest, so that as it arrives to the
collector, it has a relatively small amplitude (see inset to Fig. 3(b) where
this peak is visualized). The same is true for other, not too high values of
$n$: the diffractive spreading significantly decreases the corresponding
amplitude in $J(t)$. On the other hand, the amount of the electrons contained
in the subsequent $n$ decreases as $n$ increases. The global maximum of $J(t)$
is thus the result of this interplay between the diffractive spreading and
energy balance of individual beams. In our configuration, the most intense
peaks appear at around $n=7-11$.

As follows from Fig. 3(b), although the shape of the current bunch occurring
at the collector is rather complicated, one can define some quantities
characterizing it, in particular i) the time delay in respect to the center of
the pump pulse, and ii) the duration of the bunch. Here we define the above
mentioned delay via the position (in time) of the highest peak of $J(t)$, and
the width is defined by full-width half-maximum. The defined delay and
duration are shown in Fig. 4 for different sizes $s$ of the nanogap. For the
particular case of Fig. 3(b) with $s=50$ nm, the current $J(t)$ peaks at
around 40 fs (that is, delayed by around 30 fs from the pulse center) and the
duration of the current bunch is of the order of 15 fs.

Figure 5:  The position of the maximum of the current $J(t)$ at the collector
(red points and red solid line) relative to the pump pulse center (thus
denoting the arriving time of the electron at the collector), as well as the
width of $J(t)$ (blue points and blue dashed line), as a function of the size
of the nanogap, obtained by simulations of Eq. (6). Red dotted line shows the
arrival time of an electron at the collector, if the electron’s energy is
equal to $U_{p}$, the ponderomotive energy.

One can see from Fig. 5 that the delay of the current spike does not,
generally speaking, grow linearly with the gap size. This is explained by the
reshaping of the electronic wavepacket described above. This reshaping is, as
already mentioned, a result of interplay between the diffractive spreading and
electron energy balance. Nevertheless, if we compare the delay of $J(t)$ with
the naïve estimation for the flight time $t_{f}=s/v_{p}$, where $s$ is the
size of the nanogap, and $v_{p}=\sqrt{2U_{p}/m}$ is the mean velocity
corresponding to the ponderomotive energy
$U_{p}=e^{2}\mathcal{E}^{2}/4m\omega^{2}$ (red dotted line in Fig. 4), we
observe quite a good correspondence. This indicates that, although the
electron wavepacket has a complicated shape, its propagation can be described
with a velocity which corresponds to the mean ponderomotive energy of the
electron in the field.

##  IV Conclusion

In conclusion, we utilized an array of spatially asymmetric nanojunctions to
break the symmetry of the ionization process and to generate an ultrafast
optically switchable on-chip electronic current at room temperature and under
the standard conditions. Generation of currents up to 0.6 nA by a few-cycle
driver pulse with a random carrier-envelope phase, without using a dc bias, is
possible in this way. Introducing CEP control to this scheme could
additionally increase the efficiency, but for relatively long pulse durations
we used here such increase is only minor. We have extended the Y-I model,
which works well for the tunneling, multiphoton and intermediate regimes of
atoms, to gold nanostructures. The excellent agreement of the Y-I model
allowed to establish the leading current formation mechanism: the observed
data can be well explained by assuming by nonaddiabatic tunneling through the
barrier. This conclusion is supported by direct simulations of electron
ionization dynamics using the time-dependent Schrödinger equation, which
demonstrated that any effects manifesting as an absorption of several photons
occur by the inter-cycle interference of electronic wavepackets, and are
undefined on the sub-cycle time scale. No signatures of prevailing influence
of the Schottky barrier lowering was found. In contrast, the unusual current
scaling $J\sim I^{2}$ is explained via the joint influence of all multiphoton
channels taking place in the nonaddiabatic tunneling regime, that is, in the
transition region between tunneling and multiphoton ionization. Our detailed
view of the electron dynamics allowed to determine the limits on the speed of
such devices. We observe that the shape of the electron wavepacket is rather
complicated: the flying electrons are separated into “beams”, each of them
having the velocity corresponding to certain number of absorbed photons.
Nevertheless, in average, the flight time of electrons in the nanogap is
determined, to a good precision, by the ponderomotive velocity of electrons in
the driving field. This suggests that the primary way to increase the speed
could be not only to decrease the gap but also to increase the ponderomotive
energy, which does not automatically mean increasing the peak field:
multicolor driving fields could also help at this [45].

Acknowledgements

The authors acknowledge support from Deutsche Forschungsgemeinschaft (DFG) (KO
3798/4-1, BA 4156/4-2, MO 850-19/2, MO 850-23/1) and from German Research
Foundation under Germany’s Excellence Strategy EXC-2123 and Germany’s
Excellence Strategy within the Cluster of Excellence PhoenixD (EXC 2122,
Project ID 390833453), Lower Saxony through ’Quanten und Nanometrologie’
(QUANOMET, Project Nanophotonik). H. G. and B. F. acknowledge funding by ERC
(ComplexPlas and 3D Printedoptics) and DFG (SPP1839). A.H. acknowledges
funding from MSCA RISE project ID 823897. C. L. gratefully fully acknowledges
the DFG (SPP 1839 and SPP1840) for financial support.

## References

  * Keldysh  _et al._ [1965] L. Keldysh _et al._ , Ionization in the field of a strong electromagnetic wave, Sov. Phys. JETP 20, 1307 (1965).
  * Mevel  _et al._ [1993] E. Mevel, P. Breger, R. Trainham, G. Petite, P. Agostini, A. Migus, J.-P. Chambaret, and A. Antonetti, Atoms in strong optical fields: evolution from multiphoton to tunnel ionization, Phys. Rev. Lett. 70, 406 (1993).
  * Lindner  _et al._ [2005] F. Lindner, M. G. Schätzel, H. Walther, A. Baltuška, E. Goulielmakis, F. Krausz, D. Milošević, D. Bauer, W. Becker, and G. G. Paulus, Attosecond double-slit experiment, Physical review letters 95, 040401 (2005).
  * Babushkin  _et al._ [2017] I. Babushkin, C. Brée, C. M. Dietrich, A. Demircan, U. Morgner, and A. Husakou, Terahertz and higher-order brunel harmonics: from tunnel to multiphoton ionization regime in tailored fields, Journal of Modern Optics 64, 1078 (2017).
  * Zimmermann  _et al._ [2017] H. Zimmermann, S. Patchkovskii, M. Ivanov, and U. Eichmann, Unified time and frequency picture of ultrafast atomic excitation in strong laser fields, Phys. Rev. Lett. 118, 013003 (2017).
  * Yudin and Ivanov [2001] G. L. Yudin and M. Y. Ivanov, Nonadiabatic tunnel ionization: Looking inside a laser cycle, Phys. Rev. A 64, 013409 (2001).
  * Bormann  _et al._ [2010] R. Bormann, M. Gulde, A. Weismann, S. Yalunin, and C. Ropers, Tip-enhanced strong-field photoemission, Phys. Rev. Lett. 105, 147601 (2010).
  * Krüger  _et al._ [2011] M. Krüger, M. Schenk, and P. Hommelhoff, Attosecond control of electrons emitted from a nanoscale metal tip, Nature 475, 78 (2011).
  * Yalunin  _et al._ [2011] S. V. Yalunin, M. Gulde, and C. Ropers, Strong-field photoemission from surfaces: Theoretical approaches, Phys. Rev. B 84, 195426 (2011).
  * Krüger  _et al._ [2012a] M. Krüger, M. Schenk, M. Förster, and P. Hommelhoff, Attosecond physics in photoemission from a metal nanotip, Journal of Physics B: Atomic, Molecular and Optical Physics 45, 074006 (2012a).
  * Krüger  _et al._ [2012b] M. Krüger, M. Schenk, P. Hommelhoff, G. Wachter, C. Lemell, and J. Burgdörfer, Interaction of ultrashort laser pulses with metal nanotips: a model system for strong-field phenomena, New Journal of Physics 14, 085019 (2012b).
  * Krüger  _et al._ [2018] M. Krüger, C. Lemell, G. Wachter, J. Burgdörfer, and P. Hommelhoff, Attosecond physics phenomena at nanometric tips, Journal of Physics B: Atomic, Molecular and Optical Physics 51, 172001 (2018).
  * Herink  _et al._ [2012] G. Herink, D. Solli, M. Gulde, and C. Ropers, Field-driven photoemission from nanostructures quenches the quiver motion, Nature 483, 190 (2012).
  * Dombi  _et al._ [2013] P. Dombi, A. Hörl, P. Rácz, I. Máron, A. Tügler, J. R. Krenn, and U. Hohenester, Ultrafast strong-field photoemission from plasmonic nanoparticles, Nano Lett. 13, 674 (2013).
  * Piglosiewicz  _et al._ [2014] B. Piglosiewicz, S. Schmidt, D. J. Park, J. Vogelsang, P. Groß, C. Manzoni, P. Farinello, G. Cerullo, and C. Lienau, Carrier-envelope phase effects on the strong-field photoemission of electrons from metallic nanostructures, Nature Photonics 8, 37 (2014).
  * Racz  _et al._ [2017] P. Racz, Z. Papa, I. Márton, J. Budai, P. Wróbel, T. Stefaniuk, C. Prietl, J. R. Krenn, and P. Dombi, Measurement of nanoplasmonic field enhancement with ultrafast photoemission, Nano Lett. 17, 1181 (2017).
  * Park  _et al._ [2012] D. J. Park, B. Piglosiewicz, S. Schmidt, H. Kollmann, M. Mascheck, and C. Lienau, Strong field acceleration and steering of ultrafast electron pulses from a sharp metallic nanotip, Phys. Rev. Lett. 109, 244803 (2012).
  * Vogelsang  _et al._ [2015] J. Vogelsang, J. Robin, B. J. Nagy, P. Dombi, D. Rosenkranz, M. Schiek, P. Groß, and C. Lienau, Ultrafast electron emission from a sharp metal nanotaper driven by adiabatic nanofocusing of surface plasmons, Nano Lett. 15, 4685 (2015).
  * Schertz  _et al._ [2012] F. Schertz, M. Schmelzeisen, M. Kreiter, H.-J. Elmers, and G. Schönhense, Field emission of electrons generated by the near field of strongly coupled plasmons, Phys. Rev. Lett. 108, 237602 (2012).
  * Robin  _et al._ [2016] J. Robin, J. Vogelsang, B. J. Nagy, P. Dombi, P. Groß, and C. Lienau, Strong field above threshold ionization of Rydberg electrons localized to a gold nanotip, in  _CLEO: QELS_Fundamental Science_ (Optical Society of America, 2016) pp. FTh4B–1.
  * Dombi  _et al._ [2020] P. Dombi, Z. Pápa, J. Vogelsang, S. V. Yalunin, M. Sivis, G. Herink, S. Schäfer, P. Groß, C. Ropers, and C. Lienau, Strong-field nano-optics, Reviews of Modern Physics 92, 025003 (2020).
  * Priebe  _et al._ [2017] K. E. Priebe, C. Rathje, S. V. Yalunin, T. Hohage, A. Feist, S. Schäfer, and C. Ropers, Attosecond electron pulse trains and quantum state reconstruction in ultrafast transmission electron microscopy, Nature Photonics 11, 793 (2017).
  * Feist  _et al._ [2017] A. Feist, N. Bach, N. R. da Silva, T. Danz, M. Möller, K. E. Priebe, T. Domröse, J. G. Gatzmann, S. Rost, J. Schauss, _et al._ , Ultrafast transmission electron microscopy using a laser-driven field emitter: Femtosecond resolution with a high coherence electron beam, Ultramicroscopy 176, 63 (2017).
  * Zhou  _et al._ [2019] S. Zhou, K. Chen, M. T. Cole, Z. Li, J. Chen, C. Li, and Q. Dai, Ultrafast field-emission electron sources based on nanomaterials, Advanced Materials , 1805845 (2019).
  * Schoetz  _et al._ [2019] J. Schoetz, Z. Wang, E. Pisanty, M. Lewenstein, M. F. Kling, and M. Ciappina, Perspective on petahertz electronics and attosecond nanoscopy, ACS Photonics 6, 3057 (2019).
  * Rybka  _et al._ [2016] T. Rybka, M. Ludwig, M. F. Schmalz, V. Knittel, D. Brida, and A. Leitenstorfer, Sub-cycle optical phase control of nanotunnelling in the single-electron regime, Nature Photonics 10, 667 (2016).
  * Ludwig  _et al._ [2019] M. Ludwig, G. Aguirregabiria, F. Ritzkowsky, T. Rybka, D. C. Marinica, J. Aizpurua, A. G. Borisov, A. Leitenstorfer, and D. Brida, Sub-femtosecond electron transport in a nanoscale gap, Nature Physics , 1 (2019).
  * Karnetzky  _et al._ [2018] C. Karnetzky, P. Zimmermann, C. Trummer, C. D. Sierra, M. Wörle, R. Kienberger, and A. Holleitner, Towards femtosecond on-chip electronics based on plasmonic hot electron nano-emitters, Nature Commun. 9, 2471 (2018).
  * Pfullmann  _et al._ [2013] N. Pfullmann, C. Waltermann, M. Noack, S. Rausch, T. Nagy, C. Reinhardt, M. Kovačev, V. Knittel, R. Bratschitsch, D. Akemeier, _et al._ , Bow-tie nano-antenna assisted generation of extreme ultraviolet radiation, New J. Phys. 15, 093027 (2013).
  * Putnam  _et al._ [2017] W. P. Putnam, R. G. Hobbs, P. D. Keathley, K. K. Berggren, and F. X. Kärtner, Optical-field-controlled photoemission from plasmonic nanoparticles, Nature Phys. 13, 335 (2017).
  * Shi  _et al._ [2018a] L. Shi, B. Iwan, Q. Ripault, J. R. Andrade, S. Han, H. Kim, W. Boutu, D. Franz, R. Nicolas, T. Heidenblut, _et al._ , Resonant-plasmon-assisted subwavelength ablation by a femtosecond oscillator, Phys. Rev. Appl. 9, 024001 (2018a).
  * Shi  _et al._ [2018b] L. Shi, R. Nicolas, J. R. Andrade, W. Boutu, D. Franz, T. Heidenblut, C. Reinhardt, U. Morgner, H. Merdji, and M. Kovacev, Impact of plasmon-induced atoms migration in harmonic generation, ACS Photonics 5, 1208 (2018b).
  * Shi  _et al._ [2019] L. Shi, J. R. Andrade, A. Tajalli, J. Geng, J.-M. Yi, T. Heidenblut, F. Segerink, I. Babushkin, M. Kholodtsova, H. Merdji, _et al._ , Generating ultrabroadband deep-uv radiation and sub-10 nm gap by hybrid-morphology gold antennas, Nano Lett. (2019).
  * Hentschel  _et al._ [2012] M. Hentschel, T. Utikal, H. Giessen, and M. Lippitz, Quantitative modeling of the third harmonic emission spectrum of plasmonic nanoantennas, Nano Lett. 12, 3778 (2012).
  * Shi  _et al._ [2017] L. Shi, B. Iwan, R. Nicolas, Q. Ripault, J. R. Andrade, S. Han, H. Kim, W. Boutu, D. Franz, T. Heidenblut, _et al._ , Self-optimization of plasmonic nanoantennas in strong femtosecond fields, Optica 4, 1038 (2017).
  * Plech  _et al._ [2006] A. Plech, V. Kotaidis, M. Lorenc, and J. Boneberg, Femtosecond laser near-field ablation from gold nanoparticles, Nature Physics 2, 44 (2006).
  * Novotny and Hecht [2012] L. Novotny and B. Hecht,  _Principles of nano-optics_ (Cambridge university press, 2012).
  * Jackson [1962] J. D. Jackson,  _Classical Electrodynamics_ (John Wiley & Sons, Inc., New York, 1962).
  * Sahni and Bohnen [1985] V. Sahni and K.-P. Bohnen, Image charge at a metal surface, Phys. Rev. B 31, 7651 (1985).
  * Zimmermann  _et al._ [2019] P. Zimmermann, A. Hötger, N. Fernandez, A. Nolinder, K. Müller, J. J. Finley, and A. W. Holleitner, Toward plasmonic tunnel gaps for nanoscale photoemission currents by on-chip laser ablation, Nano Lett. 19, 1172 (2019).
  * Dienstbier [2021] P. Dienstbier, personal communication (2021).
  * Hommelhoff  _et al._ [2006] P. Hommelhoff, C. Kealhofer, and M. A. Kasevich, Femtosecond laser meets field emission tip-a sensor for the carrier envelope phase, in  _2006 IEEE International Frequency Control Symposium and Exposition_ (IEEE, 2006) pp. 470–474.
  * Yalunin  _et al._ [2013] S. V. Yalunin, G. Herink, D. R. Solli, M. Krüger, P. Hommelhoff, M. Diehn, A. Munk, and C. Ropers, Field localization and rescattering in tip-enhanced photoemission, Annalen der Physik 525, L12 (2013).
  * Keathley  _et al._ [2013] P. D. Keathley, A. Sell, W. P. Putnam, S. Guerrera, L. Velásquez-García, and F. X. Kärtner, Strong-field photoemission from silicon field emitter arrays, Annalen der Physik 525, 144 (2013).
  * Martínez  _et al._ [2015] P. G. d. A. Martínez, I. Babushkin, L. Bergé, S. Skupin, E. Cabrera-Granado, C. Köhler, U. Morgner, A. Husakou, and J. Herrmann, Boosting terahertz generation in laser-field ionized gases using a sawtooth wave shape, Phys. Rev. Lett. 114, 183901 (2015).


# Non-Abelian and Abelian descendants of vortex spin liquid: fractional
quantum spin Hall effect in twisted MoTe2

Ya-Hui Zhang  William H. Miller III Department of Physics and Astronomy, Johns
Hopkins University, Baltimore, Maryland, 21218, USA

###### Abstract

The recent experimental observation of a potential fractional quantum spin
Hall (FQSH) state in the twisted MoTe2 system has sparked theoretical
explorations at total filling $\nu_{T}=1$ of a pair of $C=\pm 1$ Chern bands
from the two spins (locked to valley). One intriguing candidate is a vortex
spin liquid (VSL), which can be viewed as an exciton version of composite
fermi liquid (CFL). The VSL insulator is incompressible in the charge channel,
while compressible in the spin channel. Here we investigate fully gapped
descendants of the VSL phase at the total filling
$\nu_{T}=\nu_{\uparrow}+\nu_{\downarrow}=1$. At zero magnetization, a non-
Abelian state with both FQSH and thermal Hall effect emerges from weak $p+ip$
pairing of the neutral Fermi surface, hosting 12 anyons (up to addition of
physical electron) including two independent Ising anyons separately carrying
charge and spin. Strong pairing leads to a Z4 topological order with only FQSH
effect. At non-zero magnetization $m=2S_{z}$, there is a Jain sequence of
magnetic plateaus with $m=\frac{1}{p},p\in Z$, exhibiting both half FQSH
effect and spin fractional quantum Hall effect (SFQH). Our work highlights the
VSL’s potential as a parent state to organize numerous FQSH insulators with
non-trivial inter-valley correlations at $\nu_{T}=1$. The quantized FQSH
behavior remains robust even in the presence of ferromagnetism, thanks to the
spin-charge separation nature inherent in the parent VSL phase. Future
experimental investigations are crucial to validate or rule out spontaneous
magnetization and time reversal symmetry breaking.

###### pacs:

Valid PACS appear here

##  I Introduction

Strong correlated physics in the moiré Chern bandsZhang  _et al._ (2019); Wu
_et al._ (2019) have recently attracted many attentions. In the moiré system
one can find a pair of Chern bands with opposite Chern numbers in the two
valleys. Due to strong Coulomb exchange, fully valley polarized state is often
favoredZhang  _et al._ (2019); Repellin  _et al._ (2020), leading to a single
Chern band similar to the Landau level at high magnetic field. Therefore we
can expect these moiré chern bandsZhang  _et al._ (2019) to be wonderful
platforms for fractional quantum anomalous Hall effect(FQAH) from fractional
Chern insulator(FCI) at zero magnetic fieldSun  _et al._ (2011); Sheng  _et
al._ (2011); Neupert  _et al._ (2011a); Wang  _et al._ (2011); Tang  _et al._
(2011); Regnault and Bernevig (2011); Bergholtz and Liu (2013); Parameswaran
_et al._ (2013); Neupert  _et al._ (2011b). Indeed FQAH states have been
observed in twisted MoTe2 homobilayer Cai  _et al._ (2023); Zeng  _et al._
(2023); Park  _et al._ (2023); Xu  _et al._ (2023) and in rhombohedrally
stacked multilayer graphene aligned with hBNLu  _et al._ (2023), which also
attracted many theoretical investigations for the twisted MoTe2 systemWu  _et
al._ (2019); Yu  _et al._ (2020); Devakul  _et al._ (2021); Li  _et al._
(2021); Crépel and Fu (2023); Wang  _et al._ (2023); Reddy  _et al._ (2023);
Xu  _et al._ (2023); Yu  _et al._ (2023); Goldman  _et al._ (2023); Dong  _et
al._ (2023a); Morales-Durán  _et al._ (2023); Song  _et al._ (2023) and the
pentalayer graphene systemDong  _et al._ (2023b); Zhou  _et al._ (2023); Dong
_et al._ (2023c); Guo  _et al._ (2023); Kwan  _et al._ (2023).

One interesting question is whether new physics can be realized different from
the familiar quantum Hall systems. One unique feature of the moiré system is
that the Chern bands always exist as a pair with opposite Chern numbers due to
the time reversal symmetry. For example, in the twisted MoTe2 system, one
always finds a $C=1$ Chern band in one valley and a conjugate $C=-1$ Chern
band in the other valley. Therefore we can simulate a completely new Coulomb
coupled quantum Hall bilayer with opposite chiralities in the two layers by
treating the valley as a ‘layer’. The author already considered the total
filling $\nu_{T}=\frac{1}{2}+\frac{1}{2}$ case of this conjugate bilayer in a
previous paper in 2018Zhang (2018) and predicted a new incompressible state
called composite fermion insulator. Such a state has a neutral Fermi surface
and gapless exciton excitations.

More recently, a state with vanishing charge Hall conductivity and fractional
quantum spin Hall (FQSH) effect was reported experimentally in twisted MoTe2
at the hole filling $\nu_{T}=3$ with twist angle $\theta=2.1^{\circ}$Kang  _et
al._ (2024). In this state, the first Chern band in both valleys is fully
occupied and we can consider the total filling $\nu_{T}=1$ of the second Chern
band. This is exactly the conjugate Chern band case considered by the author
in Ref. Zhang, 2018. Actually, the previously proposed composite fermion
insulatorZhang (2018) also exhibits quantized FQSH and is a strong candidate
for the FQSH state observed in the experimentZhang (2024). In the context of
the twisted MoTe2 system, as the spin is locked to valley, the internal flux
of this composite fermion insulator carries gapless spin excitation and we dub
the phase as vortex spin liquid (VSL).

Figure 1: The relationship of different descendants of the conjugate composite
Fermi liquid (cCFL) at total filling
$\nu_{T}=\nu_{\uparrow}+\nu_{\downarrow}=1$. We can reach various phases from
Cooper pairing or exciton pairing of the composite fermions. Due to inter-
valley Coulomb repulsion, we postulate the vortex spin liquid (VSL) from
exciton pairing to be favored among the second generation. Below the charge
gap, the VSL phase is an exciton version of composite Fermi liquid (CFL) and
is a parent state for the descendants at different exciton density decided by
the magnetizaiton $m=2\langle S_{z}\rangle$. At zero magnetization, weak and
strong p+ip pairing leads to a non-Abelian topological order with 12 anyons
and an Abelian $Z_{4}$ topological order. The non-Abelian pfaffian-like state
also has a quantized thermal Hall effect with chiral central charge
$c=\frac{1}{2}$. At non-zero $m=\frac{1}{p},p\in Z$, there emerges a Jain
sequence of maganetic plateaus with both quantized fractional quantum spin
Hall effect(FQSH) with $\frac{1}{4\pi}A_{c}dA_{s}$ response and spin
fractional quantum Hall effect (SFQH) with
$\frac{1}{4p}\frac{1}{4\pi}A_{s}dA_{s}$ response. It also hosts thermal Hall
effect with chiral central charge $c=-(p-1)$.

VSL can be constructed from inter-valley exciton pairingZhang (2018, 2024);
Shi  _et al._ (2024) of the composite fermions (CF) from a decoupled composite
Fermi liquids at half filling of the conjugate $C=\pm 1$ Chern bandsZhang
(2018); Myerson-Jain  _et al._ (2023). It can be viewed as an exciton version
of the composite Fermi liquid (CFL)Halperin  _et al._ (1993); Son (2015) with
also particle-hole symmetry. The VSL has a bulk charge gap with gapless
helical edge modes. The gapless bulk excitations are purely neutral and should
not influence any charge transport measurements. In the familiar Landau level
system, it is known that the CFL phase is a parent state for gapped
topological orders at half-filling or fillings doped away from half
fillingJain (2000, 2007, 1989). Following the same spirit, in this paper we
study possible fully gapped descendants of the VSL phase at zero magnetization
or non-zero magnetization $m=2\langle S_{z}\rangle$ (see Fig. 1). At zero $m$,
we can obtain a gapped state from $p+ip$ pairing of the neutral Fermi surface
in the VSL phase. In the weak pairing case, this leads to a non-Abelian state
with 12 anyons (up to addition of the physical electron), including two
independent Ising anyons carrying charge and spin separately. The non-Abelian
state also has a quantized thermal Hall effect. In the strong pairing case we
obtain a Z4 topological order with only FQSH effect. At finite magnetization,
there is a Jain sequence of Abelian state at $m=\frac{1}{p},\ p\in Z$, where
the neutral fermion forms a Chern insulator with $C=-p$. These states exhibit
both quantized half FQSH and a quantized spin fractional quantum Hall
effect(SFQH). Note SFQH is defined as a spin-spin Hall conductivity and can
not be accessed in any charge transport measurements. All of these descendants
share the property of the half quantized FQSH and spin-charge separation from
the parent VSL state and they are all consistent with the current
experimentKang  _et al._ (2024). We suggest future experiments to measure the
valley polarization and time reversal symmetry breaking to further distinguish
them.

##  II Low energy field theory of the VSL phase

We can constructZhang (2018, 2024) the VSL phase from inter-valley exciton
pairing $\Phi f^{\dagger}_{+}f_{-}$ of the composite fermion $f_{\pm}$ from
the conjugate CFL. Here $\pm$ labels the two valleys with $C=\pm 1$. Depending
on the value of $\Phi$ and also lattice superlattice potential, there are many
possibilities for the neutral fermi surface formed by $f_{+}\sim f_{-}\sim f$.
Here we assume a single Fermi surface with volume $1$ in units of the moire
BZ. We ignore the moiré superlattice potential which is appropriate in the
Landau level limit.

The low energy field theory of the VSL phase isZhang (2024):

$\displaystyle\mathcal{L}_{\text{VSL}}$
$\displaystyle=\mathcal{L}_{FS}[f,a]-\frac{1}{4\pi}A_{s}da+\frac{1}{4\pi}A_{c}dA_{s}$
(1)

with

$\displaystyle\mathcal{L}_{\text{FS}}[f,a]$
$\displaystyle=f^{\dagger}(t,\mathbf{x})(i\partial_{t}+a_{0})f(t,\mathbf{x})$
$\displaystyle~{}~{}~{}-\frac{\hbar^{2}}{2m}f^{\dagger}(t,\mathbf{x})(-i\partial_{\mu}-a_{\mu})^{2}f(t,\mathbf{x})$
(2)

Here $a_{\mu}$ is an internal U(1) gauge field.
$A_{c;\mu}=\frac{1}{2}(A_{+;\mu}+A_{-;\mu})$ and
$A_{s;\mu}=A_{+;\mu}-A_{-;\mu}$ are the probing U(1) gauge fields in the
charge and spin channel. $A_{\pm;\mu}$ is the U(1) probing field coupled to
the valley $\tau=\pm$ which has Chern number $C=\pm$ respectively. $adb$ is an
abbreviation of the Chern-Simons term
$\epsilon_{\mu\nu\sigma}a_{\mu}\partial_{\nu}b_{\sigma}$ with $\epsilon$ the
anti-symmetric tensor.

The above action resembles Son’s Dirac theorySon (2015) of the CFL in half
filled Landau level. Actually the VSL phase can be viewed as a CFL for the
neutral excitons charged under $A_{s}$. The microscopic time reversal symmetry
$\mathcal{T}$ acts as an effective particle-hole symmetry CT for the excitons
and forbids the interal Chern-Simons term. For bosonic excitons CT${}^{2}=1$
and the neutral Fermi surface can be from ordinary dispersion instead of Dirac
fermion.

In the following we will consider the fully gapped topological orders as
descendants of the VSL phase. The above action loses the information of the
gapped excitations carrying the charge under $A_{c}$, which is encoded in the
vortex of the composite fermion exciton order $\Phi$. For the purpose to
obtain the full data of the topological order, we need to consider the
complete action:

$\displaystyle\mathcal{L}_{\text{Full-VSL}}$
$\displaystyle=\mathcal{L}_{\text{FS}}[f,\frac{a_{+}+a_{-}}{2}]+\frac{1}{2\pi}b_{1}d(a_{+}-a_{-})$
$\displaystyle~{}~{}~{}-\frac{2}{4\pi}\alpha_{+}d\alpha_{+}+\frac{1}{2\pi}(A_{+}-a_{+})d\alpha_{+}$
$\displaystyle~{}~{}~{}+\frac{2}{4\pi}\alpha_{-}d\alpha_{-}+\frac{1}{2\pi}(A_{-}-a_{-})d\alpha_{-}$
(3)

In the first line, we introduce another U(1) gauge field $b_{1;\mu}$ to
represent the CF exciton condensation order $\Phi$. The charge of $b_{1}$
represents the vortex of $\Phi$ and carries the physical charge.
$\alpha_{\pm}$ are introduced to provide the Chern-Simons term in the
conjugate CFLZhang (2024). If we do not care about the gapped charge
excitations, we can integrate $b_{1}$ which locks $a_{+}=a_{-}=a$. Then
integration of $\alpha_{\pm}$ gives the term
$-\frac{1}{4\pi}A_{s}da+\frac{1}{4\pi}A_{c}dA_{s}$.

##  III Non-Abelian FQSH insulator at zero valley polarization

We consider a $p+ip$ pairing of the neutral Fermi surface $f$. The resulting
phase is a non-Abelian topological order with 12 anyons in the weak pairing
case and a Z4 topological order in the strong pairing case.

###  III.1 Strong pairing limit: Z4 topological order

First let us consider the strong pairing limit so we do not need to worry
about the Majorana mode. As we already have an exciton order $\Phi
f^{\dagger}_{+}f_{-}$, we now introduce the pairing term
$\Delta(\mathbf{k})f(\mathbf{k})f(-\mathbf{k})\sim\Delta(\mathbf{k})f_{+}(\mathbf{k})f_{-}(-\mathbf{k})$.
Starting from the full action of the VSL in Eq. 3, we introduce a U(1) gauge
field $b_{2}$ as the dual field of $\Delta$, the resulting theory is:

$\displaystyle\mathcal{L}$
$\displaystyle=\frac{1}{2\pi}b_{1}d(a_{+}-a_{-})+\frac{1}{2\pi}b_{2}d(a_{+}+a_{-})$
$\displaystyle~{}~{}~{}-\frac{2}{4\pi}\alpha_{+}d\alpha_{+}+\frac{2}{4\pi}\alpha_{-}d\alpha_{-}$
$\displaystyle~{}~{}~{}+\frac{1}{2\pi}(A_{+}-a_{+})d\alpha_{+}+\frac{1}{2\pi}(A_{-}-a_{-})d\alpha_{-}$
(4)

Integration of $a_{\pm}$ locks $\alpha_{+}=b_{1}+b_{2}$ and
$\alpha_{-}=b_{2}-b_{1}$. The final action is:

$\displaystyle\mathcal{L}_{Z_{4}}=-\frac{4}{2\pi}b_{1}db_{2}+\frac{2}{2\pi}A_{c}db_{2}+\frac{1}{2\pi}A_{s}db_{1}$
(5)

This is a Z4 topological order with K matrix $K=\begin{pmatrix}0&4\\\
4&0\end{pmatrix}$. We have the charge vector $q_{c}=(0,2)^{T}$ for $A_{c}$ and
$q_{s}=(1,0)^{T}$ for $A_{s}$. It is easy to get $\sigma_{cs;xy}=\frac{1}{2}$
while $\sigma_{cc;xy}=\sigma_{ss;xy}=0$. We have two minimal anyons: (1) a
boson generated by $l_{1}=(1,0)^{T}$ with charge $Q=\frac{1}{2},S_{z}=0$. This
corresponds to the vortex of the exciton order $\Phi$. (2) a boson generated
by $l_{2}=(0,1)^{T}$ with charge $Q=0,S_{z}=\frac{1}{4}$. This corresponds to
the vortex of the cooper pairing order $\Delta$. Therefore we have spin-charge
separation in this topological order as is inherited from the parent VSL
phase. We expect that the spin gap is smaller than the charge gap as the later
is decided by the Coulomb interaction scale while the former is from spin
interaction.

The edge theory of the Z4 topological order is:

$\displaystyle S$ $\displaystyle=\int
dtdx\frac{4}{2\pi}\partial_{x}\varphi_{1}\partial_{t}\varphi_{2}-\upsilon(\partial_{x}\varphi_{1})^{2}-\upsilon(\partial_{x}\varphi_{2})^{2}$
$\displaystyle~{}~{}~{}-g\partial_{x}\varphi_{1}\partial_{x}\varphi_{2}$ (6)

One can see that it is in the form of the usual Luttinger liquid. But now
$e^{i\varphi_{1}}$ creates an excitation with $Q=\frac{1}{2},S_{z}=0$ and
$e^{i\varphi_{2}}$ creates an excitation with $Q=0,S_{z}=\frac{1}{4}$.
$e^{i2(\varphi_{1}+\varphi_{2})}$ creates an excitation with $Q=1,S_{z}=1/2$,
similar to physical electron. However it is a bosonic field and should not be
identified with the physical electron operator. The Z4 topological order is
essentially a bosonic state with single electron trivially gapped. As a
result, there is still single electron gap at the edge.

###  III.2 Weak pairing limit: non-abelian order with $12$ anyons

In the weak pairing limit, the vortex of the cooper pairing hosts a majorana
zero modeRead and Green (2000); Moore and Read (1991). Actually the vortex of
the exciton order $\Phi$ also has a majorana zero mode. In the vortex core of
$\Phi$, $\Phi$ vanishes and we have two fermi surfaces from $f_{+},f_{-}$.
Then the $p+ip$ pairing of
$\Delta(\mathbf{k})f_{+}(\mathbf{k})f_{-}(-\mathbf{k})$ gives a Chern number
$C=2$, while the bulk has $C=1$ in the large $\Phi$ region. As a consequence,
there is a domain wall separating $C=2$ and $C=1$ around the vortex of the
$\Phi$ order, leading to a majorana zero mode.

The final topological order is (Z4 $\times$ Ising)/$Z_{2}$, where the Z4
corresponds to the action in Eq. 5. Ising means the topological order with
three anyons $I,\psi,\sigma$ and fusion rule: $\sigma\times\sigma=I+\psi$,
$\psi\times\psi=I$, $\psi\times\sigma=\sigma$. $I$ is the trivial anyon,
$\psi$ is a neutral fermion (the bogoliubov quasiparticle) and $\sigma$ is the
Ising anyon (bound to the elementary vortex of $\Phi$ or $\Delta$). There is a
$\pi$ mutual statistics between the fermion $\psi$ and the Ising anyon
$\sigma$. $\sigma$ has self statistics $\theta_{\sigma}=e^{i\frac{\pi}{8}}$.

We can label the anyons of this topological order as $(l_{1},l_{2},I)$,
$(l_{1},l_{2},\psi)$ and $(l_{1},l_{2},\sigma)$. $(l_{1},l_{2})$ are the
charges under $b_{1},b_{2}$ in the Z4 topological order in Eq. 5.
$(l_{1},l_{2})$ corresponds to $l_{1}$-vortex of $\Phi$ and $l_{2}$-vortex of
$\Delta$. The majorana mode (the $\sigma$ anyon) is bound to odd $l_{1}+l_{2}$
and the $1,\psi$ anyons are bound to even $l_{1}+l_{2}$. Then there are only
$(4\times 4\times 3)/2=24$ choices. Besides, we also find that
$(2,2,I),(2,2,\psi)$ are equivalent to $I=(0,0,I),\psi=(0,0,\psi)$ up to
addition of the physical electron. It is easy to find that $(2,2,I)$ and
$(2,2,\psi)$ have charges $(Q,S_{z})=(1,\frac{1}{2})$ simply from the Z4 part.
They have self statistics $0,\pi$ respectively. $(2,2,\psi)$ has trivial
mutual statistics with all of the other anyons and we can identify
$(2,2,\psi)=e$ as the physical electron. Then $(2,2,I)=e\psi$. Given that we
are considering a fermionic system formed by electrons, $e$ should be viewed
as a trivial particle: $e\simeq I$. Then we have $(2,2,\psi)\simeq(0,0,I)$ and
$(2,2,I)\simeq(0,0,\psi)$. As a result, $(l_{1},l_{2})$ is equivalent to
$(l_{1}+2,l_{2}+2)$, which further reduces the number of anyons to 12. To our
best knowledge, this is the minimal topological order with smallest ground
state degeneracy on torus consistent with the quantized half FQSH, zero charge
Hall conductivity and zero magnetization. In contrast, if we consider the
decoupled FQSH insulator with a Pfaffian state in one valley and its conjugate
in the other valley, the number of anyons is $6\times 6=36$ and is much
larger.

Among the 12 anyons, there are two elementary Ising anyons:
$\sigma_{1}=(1,0,\sigma)$ and $\sigma_{2}=(0,1,\sigma)$. One can find that
$\sigma_{1}$ has charge $(Q,S_{z})=(\frac{1}{2},0)$ and $\sigma_{2}$ has
charge $(Q,S_{z})=(0,\frac{1}{4})$. Thus we have spin-charge separation: the
gap of $\sigma_{1}$, $\sigma_{2}$ set the charge gap and spin gap
respectively. We again expect the spin gap to be smaller than the charge gap.

This non-Abelian state breaks time reversal symmetry and has a chiral central
charge $c=\frac{1}{2}$ and thus thermal Hall effect. The edge theory is the
same theory in Eq. 6 with an additional chiral Ising conformal field theory
(CFT). In this time we have a physical electron operator as
$c^{\dagger}_{\uparrow}\sim\psi e^{i2(\varphi_{1}+\varphi_{2})}$ and thus
there is no single electron gap at the edge.

Lastly we comment on the difference between the strong pairing and weak
pairing state. Their relationship is quite similar to that between the strong
pairing $U(1)_{8}$ Abelian order and weak pairing Pfaffian state from the CFL
in half-filled Landau level. In the previous numerical studies of the half-
filled Landau level, the weak pairing seems to be more likely than the strong
pairing state. Therefore, for our case, we also expect the non-Abelian state
to be more relevant than the Z4 state. The non-Abelian state necessarily
breaks the time reversal symmetry. But the detection of the time reversal
symmetry breaking is subtle because the charge Hall effect and valley
polarization both vanish. The state is definitely compatible with the current
experiment and may be considered seriously in future experimental
investigations.

##  IV Jain sequence at non-zero magnetization $m=\frac{1}{p}$: quantized FQSH
and SFQH

Now we move to the non-zero magnetization corresponding to the filling
$\nu_{\uparrow}=\frac{1}{2}+\frac{m}{2}$ and
$\nu_{\downarrow}=\frac{1}{2}-\frac{m}{2}$. This is equivalent to tuning the
density of the excitons and we expect a Jain sequence of gapped states at
different magnetization $m=\frac{1}{p},p\in Z$.

First, from action of the VSL in Eq. 1, we have the following relationship
between the internal magnetic field $b=\partial_{x}a_{y}-\partial_{y}a_{x}$
and the magnetization: $\frac{b}{2\pi}=-m$. Thus at non-zero magnetization
$m$, the neutral fermion $f$ feels an effective flux and forms Landau levels.
Its effective filling is at $\nu=\frac{n_{f}}{b/(2\pi)}=-\frac{1}{m}$ where we
used $n_{f}=1$ per moiré unit cell. Therefore the neutral fermion $f$ is in an
integer quantum Hall state with $C=-p$ when $m=\frac{1}{p}$, $p=2,3,4,...$.
Note that $p=1$ is impossible a we must have $m<1$. At $m=\frac{1}{p}$, there
is a self Chern-Simons term $-\frac{p}{4\pi}ada$. Integration of $a$ gives the
final response
$\mathcal{L}_{\text{response}}=\frac{1}{4p}\frac{1}{4\pi}A_{s}dA_{s}+\frac{1}{4\pi}A_{c}dA_{s}$.
Hence these gapped states have both spin fractional quantum Hall effect (SFQH)
with $\sigma_{ss;xy}=\frac{1}{4p}$ and the FQSH with
$\sigma_{cs;xy}=\frac{1}{2}$. There is also quantized thermal Hall effect
corresponding to a chiral central charge $c=-(p-1)$. Note that the $C=-p$
Chern insulator gives $p$ number of edge modes, but one will be removed by the
coupling to the gauge field $a_{\mu}$. This will become clearer in the
following K matrix description.

To obtain the precise data of the Abelian topological order, we should start
from the full action of the VSL phase in Eq. 3. We introduce $p$ number of
U(1) gauge fields $\beta_{I}$ to capture the $C=-p$ integer quantum Hall state
of the neutral fermion $f$. The final action is:

$\displaystyle\mathcal{L}$
$\displaystyle=\sum_{I=1}^{p}\frac{1}{4\pi}\beta_{I}d\beta_{I}+\frac{1}{2\pi}\sum_{I=1}^{p}\frac{a_{+}+a_{-}}{2}d\beta_{I}+\frac{1}{2\pi}b_{1}d(a_{+}-a_{-})$
$\displaystyle~{}~{}~{}-\frac{2}{4\pi}\alpha_{+}d\alpha_{+}+\frac{1}{2\pi}(A_{+}-a_{+})d\alpha_{+}$
$\displaystyle~{}~{}~{}+\frac{2}{4\pi}\alpha_{-}d\alpha_{-}+\frac{1}{2\pi}(A_{-}-a_{-})d\alpha_{-}$
(7)

We can integrate $b_{1}$ to lock $a_{+}=a_{-}=a$. Then integration of $a$
leads to the constraint $\alpha_{+}+\alpha_{-}=\sum_{I=1}^{p}\beta_{I}$. In
the end we can use $\beta_{p}=\alpha_{+}+\alpha_{-}-\sum_{I=1}^{p-1}\beta_{I}$
to rewrite the action. This leads to a K matrix with dimension
$(p+1)\times(p+1)$. In the order of
$(\alpha_{+},\alpha_{-},\beta_{1},...,\beta_{p-1}$, the $K$ matrix is
$K=\begin{pmatrix}1&-1&1\\\ -1&-3&1\\\ 1&1&-2\end{pmatrix}$ for $p=2$ and
$K=\begin{pmatrix}1&-1&1&1\\\ -1&-3&1&1\\\ 1&1&-2&-1\\\
1&1&-1&-2\end{pmatrix}$ for $p=3$. Larger $p$ can be easily generalized: the
left top $2\times 2$ block remains the same; the right bottom
$(p-1)\times(p-1)$ block has $-2$ in the diagonal and $-1$ in the other entry.
Other entries in the right-top and left-bottom $2\times(p-1)$ block are all
$1$. The charge vector is $q_{c}=(1,1,0,...,0)^{T}$ under $A_{c}$ and
$q_{s}=(\frac{1}{2},-\frac{1}{2},0,...,0)^{T}$. It may also be interesting to
note that the right-bottom $(p-1)\times(p-1)$ block is the same as the K
matrix of the $SU(p)_{1}$ chiral spin liquid. One can verify that the chiral
central charge $c=-(p-1)$.

There is a more simplified form to obtain the charge and statistics of the
anyon. We can simply integrate all of $\beta_{I}$ field and reach an action in
terms of $(\alpha_{+},\alpha_{-},a)$. In this approach we lose the information
of the thermal Hall effect and also we need to keep in mind that the self
statistics from charge under $a$ should be shifted by $\pi$ because the
elementary charge $f$ is a fermion. The $K$ matrix is
$K=\begin{pmatrix}2&0&1\\\ 0&-2&1\\\ 1&1&p\end{pmatrix}$ in the order of
$(\alpha_{+},\alpha_{-},a)$. We have charge vector $q_{c}=(1,1,0)^{T}$ and
$q_{s}=(\frac{1}{2},-\frac{1}{2},0)^{T}$. We find
$K^{-1}=\frac{1}{4p}\begin{pmatrix}2p+1&-1&-2\\\ -1&1-2p&2\\\
-2&2&4\end{pmatrix}$. There are $4p$ number of anyons because $\det K=-4p$. It
is easy to find that $l=(0,0,-1)^{T}$ generates an anyon with charge
$(Q,S_{z})=(0,\frac{1}{2p})$. It has statistics $\theta=\frac{\pi}{p}-\pi$.
$l=(1,0,0)^{T}$ generates an anyon with charge
$(Q,S_{z})=(\frac{1}{2},\frac{1+p}{4p})$ and statistics
$\theta=\frac{2p+1}{4p}\pi$. Similarly $l=(0,-1,0)^{T}$ creates an anyon with
charge $(Q,S_{z})=(\frac{1}{2},\frac{1-p}{4p})$ and statistics
$\theta=\frac{1-2p}{4p}\pi$.

Just from the Hall response we can read that there is an anyon with charge
$Q=\frac{1}{2},S_{z}=\frac{1\pm p}{4p}$ which can be generated from threading
a $dA_{c}=\pi,dA_{s}=\pm 2\pi$ flux (for example, from $dA_{+}=2\pi$,
$dA_{-}=0$). Meanwhile from threading a flux of $dA_{c}=2\pi,dA_{s}=0$ one can
create a different anyon with $Q=0,S_{z}=\frac{1}{2}$. The above analysis from
the K matrix actually shows that there is a neutral anyon with even smaller
$S_{z}=\frac{1}{2p}$. We expect this neutral anyon to be energetically cheaper
than the charged anyon and sets a spin gap smaller than the charge gap.

##  V Discussion on experiment

In this paper we show that there are many descendant fully gapped topological
orders from the vortex spin liquid in the conjugate moiré Chern band at the
total filling $\nu_{T}=1$. In the current experimentKang  _et al._ (2024), we
only know that there is a half FQSH effect and the charge Hall conductivity
vanishes. This is actually consistent with the VSL phase and all of its
descendants discussed here. Actually it is also consistent with the more naive
decoupled phase with one quantum Hall state in one valley and its conjugate
partner in the other valley. Compared to the decoupled phase, the VSL phase
and its descendants all have the property of spin charge separation. We expect
a finite charge gap $\Delta_{c}$ set by the Coulomb interaction and a much
smaller or even zero spin gap $\Delta_{s}$. In contrast, in the decoupled
phase $\Delta_{c}=\Delta_{s}$.

Another important observation is that the quantized half FQSH is compatible
with time reversal symmetry breaking and even a finite valley polarization
$m=2\langle S_{z}\rangle$. This is unprecedented in the quantum spin Hall band
insulator where a ferromagnetism of $S_{z}$ would make the system metallic.
Due to the spin-charge separation nature of the VSL phase, one can deform the
spin part and change the magnetization continuously without influencing the
charge transport and the helical edge modes. We suggest the future experiment
to measure the magnetization $m$ at zero magnetic field $B$ through magnetic
circular dichroism (MCD). If $m=\frac{1}{p},p\in Z$ at $B=0$, then it may
likely be in one of the Jain sequence state with both quantized FQSH and spin
fractional quantum Hall effect (SFQH) with $\sigma_{ss;xy}=\frac{1}{4p}$. If
$m$ is finite but not equal to $\frac{1}{p}$, it may be in a gapless state
with charge gap, but zero spin gap. In this case it still has quantized FQSH,
but a non-quantized SFQH. Even if $m=0$ when $B=0$, one can push the system
into the state with non-zero $m$ by applying a Zeeman field. Observing
quantized FQSH with the same charge transport behavior under varying
magnetization would be a strong evidence of spin-charge separation. Actually
the existence of the Jain sequence at $m=\frac{1}{p}$ suggests quantum
oscillations under the Zeeman field. We expect certain quantity like the
density of states oscillates in the form $\cos(2\pi
p)=\cos(\frac{2\pi}{m})\sim\cos(\frac{2\pi}{\chi_{s}gB})$ with $\chi_{s}$ the
spin susceptibility. Thus there is a quantum oscillation in terms of
$\frac{1}{B}$, but now with a potentially very different coefficient from the
usual quantum oscillation of a Fermi liquid under the orbital magnetic field.

If a FQSH insulator with zero magnetization is realized, it may be in the time
reversal invariant VSL phase or in the time reversal broken non-Abelian
pfaffian-like state from p+ip pairing. One way to distinguish them is to
measure the spin gap. The time reversal breaking of the later state is quite
subtle and may be revealed only through the challenging thermal Hall
experiment. We expect the cheapest anyon in our non-Abelian descendant to be a
neutral Ising anyon with $Q=0$ and $S_{z}=\frac{1}{4}$. We leave to future
work to design a protocal to detect and control such a neutral Ising anyon.

##  VI Summary

In summary, we studied the possible fully gapped descendants from the recently
proposed vortex spin liquid (VSL)Zhang (2024) in the context of the
experimental discovered FQSH state in twisted MoTe2. We find a non-Abelian
state with 12 anyons (including two independent Ising anyons) at zero
magnetization and a Jain sequence of Abelian states at magnetic plateau
$m=2\langle S_{z}\rangle=\frac{1}{p},\ p\in Z$. Therefore the VSL phase can be
viewed as a powerful parent state to organize FQSH insulators at total filling
$\nu_{T}=1$ under continuously varying magnetizations.

Note added:  Recently there is a preprintJian and Xu (2024) which constructed
a Z4 topological order and a non-Abelian state with Ising anyons for FQSH
insulator from a different perspective. They appear to be the same topological
orders as the states from strong and weak pairing of the VSL phase in the
current paper.

Acknowledgement YHZ thanks Chao-Ming Jian and Cenke Xu for discussions. This
work was supported by the National Science Foundation under Grant No.
DMR-2237031.

## References

  * Zhang  _et al._ (2019) Y.-H. Zhang, D. Mao, Y. Cao, P. Jarillo-Herrero, and T. Senthil, Physical Review B 99, 075127 (2019).
  * Wu  _et al._ (2019) F. Wu, T. Lovorn, E. Tutuc, I. Martin, and A. MacDonald, Physical review letters 122, 086402 (2019).
  * Repellin  _et al._ (2020) C. Repellin, Z. Dong, Y.-H. Zhang, and T. Senthil, Physical Review Letters 124, 187601 (2020).
  * Sun  _et al._ (2011) K. Sun, Z. Gu, H. Katsura, and S. D. Sarma, Physical review letters 106, 236803 (2011).
  * Sheng  _et al._ (2011) D. Sheng, Z.-C. Gu, K. Sun, and L. Sheng, Nature communications 2, 389 (2011).
  * Neupert  _et al._ (2011a) T. Neupert, L. Santos, C. Chamon, and C. Mudry, Physical review letters 106, 236804 (2011a).
  * Wang  _et al._ (2011) Y.-F. Wang, Z.-C. Gu, C.-D. Gong, and D. Sheng, Physical review letters 107, 146803 (2011).
  * Tang  _et al._ (2011) E. Tang, J.-W. Mei, and X.-G. Wen, Physical review letters 106, 236802 (2011).
  * Regnault and Bernevig (2011) N. Regnault and B. A. Bernevig, Physical Review X 1, 021014 (2011).
  * Bergholtz and Liu (2013) E. J. Bergholtz and Z. Liu, International Journal of Modern Physics B 27, 1330017 (2013).
  * Parameswaran  _et al._ (2013) S. A. Parameswaran, R. Roy, and S. L. Sondhi, Comptes Rendus Physique 14, 816 (2013).
  * Neupert  _et al._ (2011b) T. Neupert, L. Santos, S. Ryu, C. Chamon, and C. Mudry, Phys. Rev. B 84, 165107 (2011b).
  * Cai  _et al._ (2023) J. Cai, E. Anderson, C. Wang, X. Zhang, X. Liu, W. Holtzmann, Y. Zhang, F. Fan, T. Taniguchi, K. Watanabe, Y. Ran, T. Cao, L. Fu, D. Xiao, W. Yao, and X. Xu, Nature (2023), 10.1038/s41586-023-06289-w.
  * Zeng  _et al._ (2023) Y. Zeng, Z. Xia, K. Kang, J. Zhu, P. Knüppel, C. Vaswani, K. Watanabe, T. Taniguchi, K. F. Mak, and J. Shan, Nature (2023), 10.1038/s41586-023-06452-3.
  * Park  _et al._ (2023) H. Park, J. Cai, E. Anderson, Y. Zhang, J. Zhu, X. Liu, C. Wang, W. Holtzmann, C. Hu, Z. Liu, T. Taniguchi, K. Watanabe, J.-h. Chu, T. Cao, L. Fu, W. Yao, C.-Z. Chang, D. Cobden, D. Xiao, and X. Xu, Nature (2023), 10.1038/s41586-023-06536-0.
  * Xu  _et al._ (2023) F. Xu, Z. Sun, T. Jia, C. Liu, C. Xu, C. Li, Y. Gu, K. Watanabe, T. Taniguchi, B. Tong, J. Jia, Z. Shi, S. Jiang, Y. Zhang, X. Liu, and T. Li, Phys. Rev. X 13, 031037 (2023).
  * Lu  _et al._ (2023) Z. Lu, T. Han, Y. Yao, A. P. Reddy, J. Yang, J. Seo, K. Watanabe, T. Taniguchi, L. Fu, and L. Ju, arXiv e-prints , arXiv:2309.17436 (2023), arXiv:2309.17436 [cond-mat.mes-hall] .
  * Yu  _et al._ (2020) H. Yu, M. Chen, and W. Yao, National Science Review 7, 12 (2020).
  * Devakul  _et al._ (2021) T. Devakul, V. Crépel, Y. Zhang, and L. Fu, Nature communications 12, 6730 (2021).
  * Li  _et al._ (2021) H. Li, U. Kumar, K. Sun, and S.-Z. Lin, Physical Review Research 3, L032070 (2021).
  * Crépel and Fu (2023) V. Crépel and L. Fu, Phys. Rev. B 107, L201109 (2023).
  * Wang  _et al._ (2023) C. Wang, X.-W. Zhang, X. Liu, Y. He, X. Xu, Y. Ran, T. Cao, and D. Xiao, arXiv preprint arXiv:2304.11864 (2023).
  * Reddy  _et al._ (2023) A. P. Reddy, F. F. Alsallom, Y. Zhang, T. Devakul, and L. Fu, arXiv preprint arXiv:2304.12261 (2023).
  * Xu  _et al._ (2023) C. Xu, J. Li, Y. Xu, Z. Bi, and Y. Zhang, arXiv e-prints , arXiv:2308.09697 (2023), arXiv:2308.09697 [cond-mat.str-el] .
  * Yu  _et al._ (2023) J. Yu, J. Herzog-Arbeitman, M. Wang, O. Vafek, B. A. Bernevig, and N. Regnault, arXiv e-prints , arXiv:2309.14429 (2023), arXiv:2309.14429 [cond-mat.mes-hall] .
  * Goldman  _et al._ (2023) H. Goldman, A. P. Reddy, N. Paul, and L. Fu, Phys. Rev. Lett. 131, 136501 (2023).
  * Dong  _et al._ (2023a) J. Dong, J. Wang, P. J. Ledwith, A. Vishwanath, and D. E. Parker, Phys. Rev. Lett. 131, 136502 (2023a).
  * Morales-Durán  _et al._ (2023) N. Morales-Durán, N. Wei, and A. H. MacDonald, arXiv preprint arXiv:2308.03143 (2023).
  * Song  _et al._ (2023) X.-Y. Song, Y.-H. Zhang, and T. Senthil, arXiv preprint arXiv:2308.10903 (2023).
  * Dong  _et al._ (2023b) Z. Dong, A. S. Patri, and T. Senthil, “Theory of fractional quantum anomalous hall phases in pentalayer rhombohedral graphene moiré structures,” (2023b), arXiv:2311.03445 [cond-mat.str-el] .
  * Zhou  _et al._ (2023) B. Zhou, H. Yang, and Y.-H. Zhang, arXiv preprint arXiv:2311.04217 (2023).
  * Dong  _et al._ (2023c) J. Dong, T. Wang, T. Wang, T. Soejima, M. P. Zaletel, A. Vishwanath, and D. E. Parker, “Anomalous hall crystals in rhombohedral multilayer graphene i: Interaction-driven chern bands and fractional quantum hall states at zero magnetic field,” (2023c), arXiv:2311.05568 [cond-mat.str-el] .
  * Guo  _et al._ (2023) Z. Guo, X. Lu, B. Xie, and J. Liu, arXiv preprint arXiv:2311.14368 (2023).
  * Kwan  _et al._ (2023) Y. H. Kwan, J. Yu, J. Herzog-Arbeitman, D. K. Efetov, N. Regnault, and B. A. Bernevig, arXiv preprint arXiv:2312.11617 (2023).
  * Zhang (2018) Y.-H. Zhang, arXiv preprint arXiv:1810.03600 (2018).
  * Kang  _et al._ (2024) K. Kang, B. Shen, Y. Qiu, K. Watanabe, T. Taniguchi, S. J. Shan, and M. K. Fai, arXiv preprint arXiv:2402.03294 (2024).
  * Zhang (2024) Y.-H. Zhang, arXiv preprint arXiv:2402.05112 (2024).
  * Shi  _et al._ (2024) Z. D. Shi, H. Goldman, Z. Dong, and T. Senthil, arXiv preprint arXiv:2402.12436 (2024).
  * Myerson-Jain  _et al._ (2023) N. Myerson-Jain, C.-M. Jian, and C. Xu, arXiv preprint arXiv:2311.16250 (2023).
  * Halperin  _et al._ (1993) B. I. Halperin, P. A. Lee, and N. Read, Physical Review B 47, 7312 (1993).
  * Son (2015) D. T. Son, Physical Review X 5, 031027 (2015).
  * Jain (2000) J. K. Jain, Physics Today 53, 39 (2000).
  * Jain (2007) J. K. Jain,  _Composite fermions_ (Cambridge University Press, 2007).
  * Jain (1989) J. K. Jain, Physical review letters 63, 199 (1989).
  * Read and Green (2000) N. Read and D. Green, Physical Review B 61, 10267 (2000).
  * Moore and Read (1991) G. Moore and N. Read, Nuclear Physics B 360, 362 (1991).
  * Jian and Xu (2024) C.-M. Jian and C. Xu, arXiv preprint arXiv:2403.07054 (2024).