SciPost Phys. 11, 075 (2021)

Nonadiabatic nonlinear optics and quantum geometry —Application to the twisted Schwinger effect
Shintaro Takayoshi1,2 , Jianda Wu3 and Takashi Oka1,4,5?
1 Max Planck Institute for the Physics of Complex Systems, Dresden 01187, Germany
2 Department of Physics, Konan University, Kobe 658-8501, Japan
3 Tsung-Dao Lee Institute & School of Physics and Astronomy,
Shanghai Jiao Tong University, Shanghai 200240, China
4 Max Planck Institute for Chemical Physics of Solids, Dresden 01187, Germany
5 The Institute for Solid State Physics, The University of Tokyo,
Kashiwa, Chiba 277-8581, Japan? oka@issp.u-tokyo.ac.jp

Abstract
We study the tunneling mechanism of nonlinear optical processes in solids induced bystrong coherent laser fields. The theory is based on an extension of the Landau-Zenermodel with nonadiabatic geometric effects. In addition to the rectification effect known
previously, we find two effects, namely perfect tunneling and counterdiabaticity at fast
sweep speed. We apply this theory to the twisted Schwinger effect, i.e., nonadiabatic
pair production of particles by rotating electric fields, and find a nonperturbative generation mechanism of the opto-valley polarization and photo-current in Dirac and Weyl
fermions.
Copyright S. Takayoshi et al.
This work is licensed under the Creative Commons
Attribution 4.0 International License.
Published by the SciPost Foundation.

Received 17-05-2021
Accepted 15-09-2021
Check forupdatesPublished 12-10-2021
doi:10.21468/SciPostPhys.11.4.075

Contents
1 Introduction

22 Nonadiabatic geometric effects in quantum tunneling
2.1 A detailed derivation of the tunneling formula3
5

3 Twisted Schwinger effect in 2D: Nonadiabatic opto-valleytronics
3.1 Valley polarization via tunneling creation
3.2 Crossover in the production rate

68
10

4 Twisted Schwinger effect in 3D: Nonadiabatic photo-current
4.1 Expression of the total and chiral current
4.2 Novel crossover between weak-to-strong field behaviors

11
12
14

5 Conclusion

15

1

SciPost Phys. 11, 075 (2021)

A Detailed calculations for the 2D Dirac fermions
A.1 Low frequency region
A.2 High frequency region

16
16
17

B Detailed calculations for the 3D Dirac fermions
B.1 Weak field regime
B.2 Strong field regime

18
18
19

References

20

1

Introduction

Today, geometric effects [1] in electron dynamics have become a central research topic in condensed matter [2]. In adiabatic processes, it is known that electrons acquiring a geometricphase provoke exotic effects such as quantum Hall effect [3, 4]. On the other hand, the importance of geometric effects in nonadiabatic processes have been overlooked except for a few
examples such as the geometric amplitude factor [5–7] and counterdiabatic driving [8–10] as
well as the modification of the adiabaticity condition [11, 12]. In an example of nonadiabatic
dynamics governed by a time-dependent Hamiltonian, M.V. Berry showed that the tunnelingprobability can depend on the direction of the parameter sweep due to the geometric amplitude factor [5].
We revisit the problem of nonadiabatic geometric effects with a motivation to apply it to
the twisted Schwinger effect in Dirac and Weyl Fermions. The Schwinger effect is fermionantifermion pair production in strong electric fields [13–15] and is known to originate from
nonadiabatic tunneling in the momentum space [16–20]. Previously, AC extensions of the
Schwinger effect were studied for linearly polarized fields E x = E cos(Ωt) [21–23]. The results
have common nature as the problems of strong-field ionization [24] and a particle escaping
from an oscillating trap [25]. For low frequency, the tunneling is exponentially suppressedwith a threshold known as the Schwinger limit [13–15]. For higher frequency (but still lower
than the excitation gap), multiphoton excitation is activated and the excitation probability
obeys a power law. Nonadiabatic geometric effects kicks in when we study pair production
induced by rotating electric fields (or circularly polarized laser fields) E x +i E y = Ee iΩt [26,27],
which we coin as the “twisted Schwinger effect”. If we assume momentum conservation, theproblem of the twisted Schwinger effect can be recast to the Landau-Zener problem with acurved trajectory in the parameter space. This effective model is reminiscent of the twisted
Landau-Zener model studied by M.V. Berry mentioned above [5]. We perform a numericalanalysis of the effective model dynamics and find three geometric effects. The first is the sweep
direction dependence, which we call rectification. This is the same phenomenon indicated inref. [5] in terms of the geometric amplitude factor. The two other effects are perfect tunnelingand counterdiabaticity at fast sweep speed. In order to clarify the origin of the effects, we“untwist” the model with a unitary transformation, and obtain the standard Landau-Zenermodel with an effective gap parameter depending on the geometric amplitude factor (seeEq.(8) below). We can understand the three nonadiabatic geometric effects in a unified way
through the modulation of the effective gap. Recently, rectification in quantum tunneling has
been studied in solid-state systems [28]. However, as far as we know, the perfect tunnelingand counterdiabaticity at fast sweep has not been argued in previous studies.
In a condensed matter framework, a rotating electric field is created by a circularly polar2

SciPost Phys. 11, 075 (2021)

ized laser [29–31], or shaking an optical lattice [32], while in high energy physics, it mimics
the field created by ions passing by each other in heavy-ion collision experiments [33]. The
rotating electric fields are known to induce valley polarization [34, 35] and photo-currents in
2D and 3D Dirac/Weyl materials [36–38], respectively. Second order perturbation [39] hasserved as a theoretical framework to describe these phenomena. Due to the development ofstrong coherent laser sources, an extension of the theory to the nonperturbative regime is being awaited. We show that the three nonadiabatic geometric effects, i.e., rectification, perfecttunneling and counterdiabaticity, play an important role in understanding the nonperturbative
versions of the opto-valley polarization and photo-currents in 2D and 3D Dirac/Weyl materials
which are microscopically caused by the twisted Schwinger effect. On the other hand, if thesesymmetries are broken, it is possible to realize finite U(1) photocurrent in a similar way as in
the optical absorption mechanism proposed in [38, 40, 41].

2

Nonadiabatic geometric effects in quantum tunnelingWe demonstrate the nonadiabatic geometric effects in a two level Hamiltonian with a parameter q defined by
1
Ĥ(q) = mσ̂z + vqσ̂ x + κk v 2 q2 σ̂ y ,2(1)

where σ̂ j ( j = x, y, z) is the Pauli matrix, m is the gap, and v (> 0) the energy slope. We
use the unit ħ
h = c = 1. The model Eq. (1) is a quadratic expansion of the twisted LZ model
introduced by M. V. Berry [5]. We consider the diabatic tunneling problem in this HamiltonianĤ(q) which is formulated as follows.
1. We select an initial time t = t i (< 0). This is selected to be far enough from the time
t = 0 when anti-crossing occurs.
2. Between the two eigenstates Ĥ(qi )| ± (qi )〉 = E± (qi )| ± (qi )〉 with qi = q(t i ), we selectthe initial state to be the lower energy eigenstate | − (qi )〉.

Gap minimum(c)Tunneling probability

LZ withLZ curvature(b)
Instantaneous energy(a)Figure 1: Nonadiabatic geometric effects: (a) Schematic picture of the LZ tunnelingwith curvature in parameter space. (b) Instantaneous energy of the Hamiltonian
Eq. (1) with (m, v, κk ) = (0.1, 1, 1) and schematic picture for quantum tunneling. (c)Tunneling probability P(F ) for Eq. (1) with a parameter sweep q = −F t obtained
numerically (marks) compared with the tunneling formula Eq. (2) (lines).

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SciPost Phys. 11, 075 (2021)

3. The state evolves from the initial time t = t i to the final time t = t f ≡ |t i | according to
the Hamiltonian Ĥ(q(t)).
4. The solution is given as |ψ(t f )〉 = α| − (q f )〉 + β| + (q f )〉 where | ± (q f )〉 are the two
eigenstates of Ĥ(q f ) with q f = q(t f ). The tunneling probability is given by P(F ) = |β|2 .
If we regard the coefficients of the Pauli matrices x (q) = (vq, 12 κk v 2 q2 , m) as a trajectory inthe three-dimensional (3D) space, it defines a curve and κk is the geodesic curvature aroundthe gap minimum in the parameter space [Fig. 1(a)]. The case of κk = 0 corresponds to theLandau-Zener (LZ) Hamiltonian [16, 42]. The instantaneous energy of this Hamiltonian isplotted in Fig. 1(b). The tunneling probability P(F ) for a linear parameter sweep q = −F t in
Eq. (1) can be evaluated and becomes (see subsection 2.1 for derivation)(m + κk v F /4)2
P(F ) = exp − π
.
(2)
v|F |
Comparing this expression with the LZ formula, we notice that the effective tunneling gap is
modified by the geodesic curvature.
The nonadiabatic geometric effects in the tunneling probability Eq.(2) can be related to the
Berry connection and quantum geometry. Using the instantaneous eigenstates of the Hamiltonian satisfyingĤ(q)|ψm (q)〉 = Em (q)|ψm (q)〉 ,(3)with m = ±, we define the Berry connection

Anm (q) = 〈ψn (t)|i∂q |ψm (t)〉.

(4)

The Berry connection relates the basis sets |ψm (q)〉 spanned by the instantaneous eigenstatesat different parameter points q. We can define a gauge independent quantity
R nm (q) = −Ann (q) + Amm (q) + ∂q arg Anm (q) ,(5)

known as the geometric amplitude factor [5] or the quantum geometric potential [11, 12]. In
the Berry phase theory of polarization [43], where q is regarded as the momentum in solids,
R nm (q) is known as the shift vector that corresponds to the difference of the electric polarization between the n and m-th bands [39].
Quantum tunneling in the presence of the geometric amplitude factor has been studied [5, 11, 12, 28] and it was pointed out that this factor strongly affects the adiabaticity condition [11, 12]. We can see this by rewriting the tunneling probability using the geometricamplitude factor. The parameter κk in our quadratic Hamiltonian (1) is related to the geometric amplitude factor by
R+− (q = 0) = vκk ,and we can write the tunneling probability as (∆ = E+ − E− at q = 0) 
F R+− 2πP(F ) = exp −∆+.
4v|F |
2(6)(7)

This expression shows how quantum geometry affects the nonadiabatic tunneling process,where the effective tunneling gap is modified to
F R+−
.
(8)
2
This expression reproduces the generalized adiabaticity condition obtained by one of thepresent authors in Refs. [11, 12].
The tunneling formula (7) predicts several interesting phenomena as we list below.
∆eff = ∆ +4

SciPost Phys. 11, 075 (2021)

Rectification Although the instantaneous band structure is symmetric in q → −q, the tunneling probability depends on the sign of F and rectification happens [5]. The ratio∆Rγ(F ) ≡ P(|F |)/P(−|F |) = exp −π 2v+− deviates from unity for m 6= 0 [Fig. 1(c)].
Perfect tunneling In conventional LZ tunneling, the tunneling probability monotonically increase from 0 (adiabatic) to 1 (diabatic limit or perfect tunneling) as the sweep speed
increase. However, in the presence of nonadiabatic geometric effects, perfect tunnelingis realized at finite sweep speed. For m 6= 0, P(F ) peaks out and becomes unity at aperfect tunneling sweeping speed FPT = −2∆/R+− indicated by an arrow in Fig. 1(c),which is determined from the condition ∆eff = 0.
Counterdiabaticity at fast sweep For large |F |, P(F ) decreases as exp(−πR2+− |F |/16v). In
the extreme case of m = 0, the tunneling probability is a monotonically decreasing
function of speed.
We have performed a numerical calculation of the tunneling probability using the Hamiltonian Eq. (1) and compared it with the tunneling formula Eq. (2) as depicted in Fig. 1(c).
The results show good agreement and the above three nonadiabatic geometric effect is clearlyseen.
For convenience, we also consider the two-band Hamiltonian with general operators up to
q2 order,H = Â + B̂q + Ĉq2 /2 .

(9)

The gap minimum and velocity extremum conditions at q = 0 require {Â, B̂} = 0 and {B̂, Ĉ} = 0,
respectively. This Hamiltonian is equivalent to the case of Eq. (11) with the parametersm = kÂk ,where kÔk ≡ 122.1

Æv = kB̂k ,κk v 2 = −i Tr{[Â, B̂], Ĉ},
8 kÂkkB̂k(10)

Tr{Ô, Ô}.

A detailed derivation of the tunneling formula

In this subsection, we explain the derivation of the tunneling formula (Eq. (2)) for the Hamiltonian1Ĥ(q) = mσ̂z + vqσ̂ x + κk v 2 q2 σ̂ y ,2(11)

where σ̂ j ( j = x, y, z) is the Pauli matrices, m is the gap, v the energy slope, and κk is the curvature around the gap minimum in the parameter space. The idea is to move to a local framewith trivial geometry, which we call the “LZ frame”, and use the LZ formula or its extension:
the Dykhne-Davis-Pechukas (DDP) (also known as the Landau-Dykhne or the imaginary time)
method [44, 45] (see Ref. [20] for an extended discussion of the method).
Let us start from a general two-band HamiltonianĤ(q) = d(q) · σ̂ ,(12)where d(q) defines a curve in the Euclidean space. We consider tunneling at the gap minimumq = 0, and define the unit directional, tangential, and normal vectors as
r =d(0)/|d(0)| ,t =∂q d(0)/|∂q d(0)| ,n =r × t .
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SciPost Phys. 11, 075 (2021)

Note that t ⊥ r . We move to the LZ frame, where the curve d(q) is transformed to a curveθ (q)on the plane spanned by r and t using a unitary operator Û = e i 2 r ·σ̂ . The angle θ (q) is
determined asÛ † Ĥ(q)Û = [a(q)r + b(q)t ] · σ̂ ,pd(q)·nwhere a(q) = d(q) · r , b(q) = |d(q)|2 − a(q)2 , and θ (q) = − arctan d(q)·t . Then the Hamiltonian in the LZ frame becomesĤLZ (q) =Û † Ĥ(q)Û − i Û † ∂ t Û
θ 0 (q) dq 
= a(q) +r + b(q)t · σ̂ .
2 dt

(13)

In the case of the model Eq. (11), the parameters are a(q) = m, b(q) = vq, and θ 0 (q) = −κk v/2.
Through the transformation, the additional quadratic term is eliminated and the gap is effectively modified from m to meff = m+κk v F /4. The above formulation shows that the geometricmeaning of κk is the curvature of d(q) in the plane spanned by t and n at q = 0.
With the application of the DDP method [44, 45] for Eq. (13), the tunneling probability isexpressed as
Z qc∆(q) 
P ' exp − 2Im
dq ,
(14)
0 |F (q)|dqwhere ∆(q) = 2[(a(q) − θ 0 (q)F (q)/2)2 + b(q)2 ]1/2 is the energy difference and F (q) = − d t is
the Jacobian (expressed as function of q). In the DDP method, the integration path is deformed
from the real axis, and the singular point closest to the real axis governs the tunneling probability. In Eq. (14), the integration is performed to qc (on the imaginary axis), which is defined
as a point in complex plane where the gap vanishes ∆(qc ) = 0 (the branching point of squaredqroot). For the linear sweep q = −F t, the Jacobian is just F (q) = − d t = F . Applying Eq. (14) toÆthe model Eq. (11), and noticing a(q) = m, b(q) = (vq)2 + (κk v 2 q2 /2)2 = vq + O(q3 ), anddθ 0 (q) = − dq
arctan(κk vq/2) = −κk v/2 + O(q2 ), we can calculate the tunneling probability as
Z 1 (m+κk vF /4)

|v|
q
4
P(F ) = exp −(m + κk v F /4)2 − (vq)2 dq|F | 02
π (2m + κk v F /2),= exp −
4
v|F |


(15)

as given in Eq. (2).

3Twisted Schwinger effect in 2D: Nonadiabatic opto-valleytronics

In the following sections, we study how nonadiabatic geometric effects in the tunneling probability Eq. (7) lead to nontrivial dynamics of electrons in Dirac and Weyl semimetals driven bystrong electric laser fields.
We begin our analysis with the dynamics of 2D Dirac fermions in rotating electricfields. We introduce the field as gauge potential A = A(− sin(Ωt), cos(Ωt)) [electric field
E = E(cos(Ωt), sin(Ωt)) (E = AΩ > 0)], and the effective Hamiltonian for the fermions withchirality ξ = ± is given as

Ĥ = v[ξ(k x + eA x )σ̂ x + (k y + eA y )σ̂ y ] + mσ̂z ,6(16)

SciPost Phys. 11, 075 (2021)

LZ withcurvaturekyEnergy

k+eA

K

k

K

kx

gap minimum

Figure 2: Mapping from the twisted Schwinger effect to the twisted Landau Zener
problem: In rotating electric fields, the electron-hole pairs have a covariant momentum k + eA(t) which performs a rotating motion in the momentum space. During
this dynamics, the energy gap minimizes when k + eA(t) is closest to the K-point.
We focus on this gap minimum point as depicted in the right box. By performing aquadratic expansion of the Hamiltonian Ĥ(t) in the time variable (q = Ωt) aroundthe gap minimum time, we obtain the twisted Landau Zener problem defined by
Eq. (9).
where e (> 0) is the elementary charge, v is the Fermi velocity, and m (> 0) is the mass
parameter. This model has implication to valleytronics in 2D materials such as monolayertransition metal dichalcogenide (TMD) and graphene [46, 47], where laser-induced valley
polarization is demonstrated [34, 35, 48–52]. In these materials, the chirality ξ corresponds
to the valley index specifying the two Dirac points Kξ in the dispersion.
We assume that the Fermi energy is zero, and the time evolution starts from a zerotemperature ground state. After the field is switched on at t = 0, nonadiabatic processes
take place creating fermion-antifermion pairs. The tunneling process in momentum space canbe mapped to a twisted Landau Zener problem discussed in the previous section as depicted
in Fig. 2. We note that, in this mapping, we use a quadratic approximation. In this system,
the laser frequency Ω plays the role of the speed parameter F in the twisted LZ model. We
allow Ω to be positive or negative which corresponds to the helicity specifying left or rightcircular polarization. The fermion-antifermion production probability per cycle of the laser
field is given byPξ (k) = exp − πξΩm 2 M − 4M
veE

,

(17)

p
where we defined M = v 2 (|k| − eE/(|Ω|))2 + m2 . To derive this expression, we have expanded the Hamiltonian (16) around the time that minimizes the energy gap up to quadraticorder obtaining the form (10) and used the tunneling formula Eq. (2). We note that the remaining analysis is based on this approximate treatment (quadratic expansion) and the results
are not exact.
In Fig. 3, we schematically compare the pair production induced by (a) standard opticalabsorption process, and by (a) tunneling with nonadiabatic geometric effects.
(a) Perturbative optical absorption process In the case of standard optical absorption
process, a perturbative picture of optical absorption is employed, where we consider the eigenstates |ψn (k)〉 of the single body Hamiltonian Ĥ(k) satisfying
Ĥ(k)|ψn (k)〉 = En |ψn (k)〉. Electrons in the occupied bands are excited to the unoc7

SciPost Phys. 11, 075 (2021)(a) Optical absorption(b) Tunneling creation
weak
field

strongfield

Figure 3: Perturbative versus Nonadiabatic opto-valley polarization in 2D gapped
Dirac fermion: Schematic picture of pair excitations at the two valleys through two
mechanisms. (a) In optical absorption, the pairs are concentrated on an equal energy
curve ∆E(k) = Ω due to energy conservation. (b) In tunneling creation, the pairs
are produced according to Eq. (17) (see Fig. 4).
cupied bands, and the energy difference of the electron and hole is given by the photon energy. The momentum dependence of the excitation density is determined by
the optical selection rule encoded in the transition dipole moment. In electrons insolids, thePoptical transition between bands m and n (m 6= n) is given by the perjjturbation
j=x, y,z E (t)Amn (k) to the Hamiltonian Ĥ(k). The transition dipole moj

ment is given by the Berry connection Amn (k) = 〈ψm |i∂ j |ψn 〉 ( j = x, y, z) [39].
The photo absorption rate of circularly polarized laser in the Kξ valley becomesyxPAξ=± ∝ |Aξmn (k)|2 |E(Ω)|2 δ(∆E −Ω), where we defined Aξ=±mn (k) = Amn (k)±i Amn (k)and ∆E is the energy difference between the occupied and unoccupied bands [34, 35].
(b) Nonadiabatic optical absorption process In the case of tunneling excitations, the properties of the excited pairs are different from the perturbative case. Energy is no longer
conserved since the Hamiltonian Eq. (16) is depends on time. Electron and hole pairs
can be created even when their energy difference is not equal to the photon energy. The
role of the photo absorption rate PAξ=± is now played by the production probabilityPξ (k) given in Eq. (17) or more generically in Eq. (7).
We summarize the comparison in Table 1.
Table 1: Perturbative versus Nonadiabatic opto-valley polarization.
Perturbative nonlinear optics [34, 35]
Optical absorptionPAξ ∝ |Aξmn (k)|2 |E(Ω)|2 δ(∆E − Ω)
Optical selection rule (transition dipole)
Energy momentum conservation modulo photon

Nonadiabatic nonlinear optics (this work)
Tunneling creation of electron-hole pairs
Tunneling probability Pξ (k)Geometric amplitude factor
Non-conservation of energy and momentum

PA (k)P (k)Valley polarization γ = PA+ (k)Valley polarization γ = P+ (k)

−

3.1

−

Valley polarization via tunneling creation

In Fig. 4, we plot the production probability for several Ω. We see that there is a strong chirality dependence, and the sign of ξΩ determines whether excitations are “optically allowed”(ξΩ > 0) or “optically forbidden” (ξΩ < 0). This difference originates from the geometric
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SciPost Phys. 11, 075 (2021)

amplitude factor. In this sense, the optical selection rule [34, 35] in perturbative optics is replaced by the nonadiabatic geometric effects when nonperturbative strong field excitations are
considered. The ratio of the production rates between the two chiralitiesγ= πΩm P+ (k)= exp
P− (k)
veE(18)

is independent of the wavenumber. In the gapless case, as in graphene, γ is unity and there isno valley dependence. When the gap parameter m is finite, as in monolayer TMD, imbalance
becomes finite and the ratio exponentially grows or decays with increasing |Ω|/E.
Next, let us study the wavenumber dependence of the production probability as depicted
in Fig. 4. The distribution is rotationally symmetric and only depends on |k|a (a: lattice
constant). They have peaks as shown in Fig. 4 at |k| = kpeak , where

eE




|Ω|


 eE 1 Æ±m(ξΩ/4 − m)kpeak =
|Ω| v




eE 1 Æ+m(ξΩ/4 − m)|Ω| v(Ω < 4m) ,(19)(Ω ≥ 4m; E ≥ Eco ) ,(20)(Ω ≥ 4m; E < Eco ) .

(21)

The crossover field is defined by
Eco =

|Ω| Æ
m(ξΩ/4 − m) .
ev

(22)

We can understand the peak structure from the wavenumber dependent effective mass parameter in Eq. (17) defined bymeff = M − ξΩm/(4M ) .

(23)

The peaks are dictated by the wavenumber minimizing the effective mass and their properties
qualitatively change depending on whether the frequency Ω is below or above 4m. For Ω < 4m,
the distributions have a single peak at the wavenumber where meff > 0 is minimized. On the

Figure 4: Tunneling probability in 2D gapped Dirac fermion: The wavenumber dependence of the production probability Pξ (k).
The parameters are2(Ω/m, eEa /v) = (1, 1), (5, 1), (5, 0.01) and ma/v = 0.5, where a is the lattice constant.
9

SciPost Phys. 11, 075 (2021)

(a)

(b)

(c)

exponential
suppression

exponential
suppression

Figure 5: 2D gapped Dirac fermion: (a) The total pair production rate per unit of timeand volume. We fix ma/v = 0.5 and eEa2 /v = 1. (b) The electric field dependenceof the total production rate Pξtot . (c) (E, Ω)-phase diagram of the twisted Schwinger
effect.
other hand, for higher frequencies Ω ≥ 4m, perfect tunneling takes place at the opticallyallowed valley (ξΩ > 0) when the effective gap meff close. There is a crossover when theelectric field is increased, The number of perfect tunneling peaks changes from one for E < Ecoto two for E ≥ Eco . This field strength Eco characterizes a crossover of the total production
rate which we will explain below.

3.2

Crossover in the production rate

We define the total fermion-antifermion production rate per unit of time and volume asR|Ω|Γξ ≡ (2π)3 d k Pξ (k) and plot it against frequency in Fig. 5(a). We see clearly the rectification
effect where the imbalance ratio Γ+ /Γ− = γ increases exponentially for large Ω/E following
Eq. (18). In the low-frequency region, it takes the form (Appendix A)v
ES,ξ 
eE t eE
Γξ '
exp
−
π
.
(24)
(2π)2
v
E
Here we define the Schwinger limit of field strength as
ES,ξ ≡ (meff,ξ )2 /(ve) = (m − ξΩ/4)2 /(ve) ,(25)

where meff,ξ is the effective mass at the gap minimizing wavenumber Eq. (19). Equation (24)
is an extension of Schwinger’s production rate evaluated originally for a DC electric field to thecase of rotating electric field. For Ω = 0, Eq. (24) coincides with the 2D version of Schwinger’s
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SciPost Phys. 11, 075 (2021)

result [13,19] with the QED Schwinger limit ES = m2e c 3 /(ħhe) obtained by replacing m → me c 2
and v → ħhc.
Figure 5(b) shows the electric field dependence of the production rate with the opticallyallowed chirality (ξΩ >q0) for several frequencies. For strong fields, all curves converge to theeEeEdashed line Γξ → (2π)
2v described by the asymptotic form of Eq. (24) independent of Ω.
For weak fields, we observe two different behaviors. The low frequency (Ω < 4m) curves drop
below the dashed line following Eq. (24) due to the exponential suppression of tunneling at
weak fields. In contrast, curves for high frequency (Ω ≥ 4m) turn above the dashed line andconverge to a Γ+ ∝ E 1/2 behavior. In Fig. 5(c), we summarize the tunneling behaviors into a(E, Ω)-phase diagram, which we explain below.

Low frequency (|Ω| < 4m) (Appendix A.1) The Schwinger limit E = ES,ξ [Eq. (25)] characterizes the crossover from the weak field exponentially suppressed regime to the Γξ ∝ E 3/2
behavior at strong field. Increasing Ω from zero, the Schwinger limit ES,ξ for the opticallyallowed chirality (ξΩ > 0) decreases and becomes zero at |Ω| = 4m, where perfect tunnelingstarts to happen. In contrast, for the optically forbidden chirality (ξΩ < 0), ES,ξ monotonicallyincrease against |Ω|. This suppression of tunneling is the consequence of counterdiabaticity inthe twisted LZ tunneling.
High frequency (|Ω| ≥ 4m) (Appendix A.2) For the optically allowed chirality ξΩ > 0,
the effective gap closes and the Schwinger limit vanishes due to perfect tunneling. There is acrossover taking place around E = Eco defined in Eq. (22) where the number of the peaks in the
distribution function changes (Fig. 4). The production rate shows the Γξ ∝ E 3/2 behavior atstrong field E > Eco and changes to a Γξ ∝ E 1/2 behavior at weak fields E < Eco . In particular,
in the weak field regime, the production rate shows an asymptotic form
v
t eE
|Ω| Æ
Γξ '
|Ω|m
,
(26)
(4π)2 vv
for optically allowed ξ, which is evaluated in the appendix A.2. On the other hand, for the
optically forbidden chirality ξΩ < 0, the Schwinger limit monotonically increases as |Ω| increases.
Before closing this section, we give an estimate of the Schwinger limit in solid-state materials. The Fermi velocity v and mass parameters m = ∆/2 for a typical TMD material MoS2 isgiven by v = 3.5 ÅeV and m = ∆2 = 0.83 eV where ∆ is the optical gap [35]. The Schwinger
limit of MoS2 is given by ES,ξ (0) = m2 /(ve) = 0.20 V/Å = 2.0 × 109 V/m for Ω = 0. For finite
photon energy Ω, the Schwinger limit decreases and vanish at Ω = 4m = 2∆ = 3.3 eV for one
valley. Experimentally realizable fields using THz laser (Ω ∼ 0) is around ETHz = 108 V/m [53],while it exceeds ENI = 109 V/m [54] in the near-infrared region Ω = 0.6−1 eV. Thus, the fieldstrength of near-infrared lasers is comparable to the Schwinger limit at finite photon energyΩ and can be used to verify our predictions.

4

Twisted Schwinger effect in 3D: Nonadiabatic photo-current

Next, we proceed to an analysis of 3D massless Dirac fermions subject to rotating electric fields
described by the Hamiltonian PX−v j=x, y,z (q j + eA j )σ̂ j
0
0 j
PĤ3D =vγ̂ γ̂ (q j + eA j ) =, (27)
0v j=x, y,z (q j + eA j )σ̂ jj=x, y,z11

SciPost Phys. 11, 075 (2021)

with the 3D wave number q = (k, kz ), A = A(− sin(Ωt), cos(Ωt), 0) and the gamma matrices




0 I
0
σ̂ j
0
j
γ̂ =
, γ̂ =( j = x, y, z) .
I 0
−σ̂ j 0We can recast this Hamiltonian to the 2D Dirac Hamiltonian studied in the previous section.
By performing the unitary transform
exp(i π2 σ̂ x ) 0Û =0Ito the Hamiltonian Eq. (27), we obtain


Ĥ− 0
,
Û Ĥ3D Û =
0 Ĥ+
†

where

Ĥξ = v[ξ(k x − eAsin(Ωt))σ̂ x + (k y + eAcos(Ωt))σ̂ y + kz σ̂z ](28)is the Weyl Hamiltonian with chirality ξ = ±. This Hamiltonian is equivalent to the 2D Dirac
Hamiltonian Eq. (16) studied in the previous section with the replacement of the mass m byvkz . Thus, the fermion-antifermion production probability per cycle of the laser field is givenbyPξ (k) = exp − πM−ξΩvkz 2 
4M
,
veE

(29)

Æ
where we defined M = v (|k| − eE/|Ω|)2 + kz2 .
Below, we assume that the Fermi energy is at the Dirac point and exploit the scaling symmetry rewriting the model with variables t̃ = |Ω|t and q̃ = vq /|Ω|. The Schrödinger equationˆ = ξ(k̃ −sgn(Ω)Ãsin t̃)σ̂ x +(k̃ + Ãcos t̃)σ̂ y + k̃ σ̂z .
ˆ |Ψ ( t̃)〉 with H̃
is recast to i∂ |Ψ ( t̃)〉 = H̃
t̃

ξ

ξ

ξ

ξ

x

y

z

Then we can set the frequency |Ω| to unity andÃ = veA/|Ω| = veE/Ω2(30)

is the unique scaling parameter that characterizes the field strength.
In Fig. 6, we schematically compare the pair production in the 3D Dirac systems inducedby (a) standard optical absorption process to (b) tunneling creation. Similarly to the 2D casesummarized in Fig. 3 and Table 1, the nonadiabatic geometric effects take the place of the
optical selection rule [38]. Circularly polarized laser field propagating along the z axis induces
vertical transitions that are imbalanced between ±kz . The imbalance in the created pairs
result in a photocurrent Jξz for each Weyl component ξ = ± as we will see below. In Diracsystems with chiral and mirror reflection symmetries, the total photocurrent cancels since theproduction rate Pξ (q̃ ) is symmetric under ξ → −ξ, kz → −kz . On the other hand, if thesesymmetries are broken, it is possible to realize finite U(1) photocurrent in a similar way as in
the optical absorption mechanism proposed in [38, 40, 41].

4.1

Expression of the total and chiral current

The total U(1) and chiral current operators are represented as


σ̂z 0
z
† 0 z
Ĵ = −ve Û γ̂ γ̂ Û = −ve
,
0 σ̂z
12

SciPost Phys. 11, 075 (2021)

(a)

(b)

Optical absorption

weak
field

Tunneling creationstrongfield

Figure 6: Nonadiabatic versus Perturbative photo-current generation in 3D massless
Dirac fermion: Schematics of pair production and resulting photocurrent in the two
Weyl components induced by the (a) tunneling creation and (b) optical excitations.
(a) In tunneling creation, the pairs are produced according to Eq. (29). See Fig. 7
for the results. (b) In optical absorption, the pairs are concentrated on equal energy
curves due to energy conservation.
andĴ5z = −ve Û † γ̂5 γ̂0 γ̂z Û = −ve


−σ̂z
0

0
σ̂z


,



−I 0
where γ̂5 =. To evaluate the expectation value of the currents, we need to esti0 I
mate the distribution of the electron-hole (e-h) pairs. This can be done by calculating thetime evolution of the density matrix in the presence of relaxation. One of the schemes is
to employ the Liouville von Neumann equation in the momentum space within the relaxation time approximation [55]. However, for simplicity, here we assume that the system is
on-shell, i.e., the density matrix is diagonal in the eigenstate basis and the distribution is
obtained by the balance between the creation process characterized by the tunneling probability and the relaxation time. This can be done by first representing the density matrix asρq,ξ (t) = nq,ξ (t)|Ψq,ξ,1 〉〈Ψq,ξ,1 | + [1 − nq,ξ (t)]|Ψq,ξ,2 〉〈Ψq,ξ,2 |, where |Ψq,ξ,1 〉 and |Ψq,ξ,2 〉 are
the states for upper and lower bands with chirality ξ. The master equation isd nq ,ξ (t)dt= [1 − nq,ξ (t)]Pξ (q )|Ω|− nq,ξ (t)/τ ,
2π

(31)

|Ω|

where τ is the relaxation time. Note that we have the factor 2π (= inverse of the time period) in the first term on the r.h.s. since Pξ (q ) is defined as the tunneling probability per
cycle. In the steady state d nq ,ξ (t)/d t = 0, if we assume that the relaxation time is short|Ω|τPξ (q)/(2π)  1, we obtain nq,ξ (t) = |Ω|τPξ (q )/(2π). The current density for the component of chirality ξ is provided as
Z
Z
kz
k̃z
2eτ|Ω|4|Ω|τ  a 3zJξ = −2vedq ÆPξ (q ) = −d q̃ qPξ (q̃ ) . (32)342
2πa 2π(2π) v|k|2 + k2|k̃|2 + k̃2zz

We can calculate the total and chiral (spin) currents as J = J+z + J−z and J5z = J+z − J−z .
3DWe also define total (chiral) production rates as Γtot
= Γ+3D + Γ−3D (Γ53D = Γ+3D − Γ−3D ) usR4|Ω|
ing Γξ3D = (2π)4 v 3 d q̃ Pξ (q̃ ). Due to the symmetry of Pξ (q̃ ) under ξ → −ξ, kz → −kz ,Γ53D = J z = 0 holds.
z13

SciPost Phys. 11, 075 (2021)

0.3

(a)

(b)

(c)

1

5

0.4

(d)

0.2
0.2

0.1

0

0

0
0

laser

0.1

0.2

0.3

0

laser

0.5

-5
0.2

0.4

0.6

0

5

10

0

laser

Figure 7: 3D massless Dirac fermion: (a)-(c) The production probability Pξ (q̃ ) for
chirality ξ = + plotted for several field strength parameters (a) Ã = veE/Ω2 = 0.001,(b) Ã = 1/8, and (c) Ã = 10. They are rotationally symmetric around the k̃z axis andthe probability for particles with chirality ξ = − is reflected as k̃z → −k̃z . The solid
black curve denotes wavenumber at which perfect tunneling occurs [Eqs. (20) and(21) with m replace byp kz ]. The lower panels show the fermion-antifermion pairs on
the Weyl cone E = ± |q̃ | for fixed k̃ y = 0. (d) The total production rate and chiralcurrent are plotted as blue and red solid curves while the dashed lines represent their
asymptotic power-law behavior Eqs. (34) and (35).

4.2

Novel crossover between weak-to-strong field behaviors

Now, let us discuss the physical consequence of the geometric nonadiabatic effect in the tunneling creation in 3D Dirac fermions. In the massless Dirac and Weyl fermions, there is notunneling threshold and we expect that the total production rate shows a power-law behavior
against the electric field strength. We show that there is a crossover between the weak andstrong field regimes accompanied by a change in power.
In Figs. 7(a)-7(c), we plot the production probability Pξ (q̃ ) for ξ = + and Ω > 0 obtainedin Eq. (29). It is rotationally symmetric around the k̃z axis. The production probability forthe other chirality ξ = − is a reflection of ξ = + around the k̃z = 0 plane. The production
probability shows peaks around the wavenumber satisfying the perfect tunneling conditions
Eqs. (20) and (21) with m replace by kz . In the plane of (|k̃|, k̃z ), the perfect tunneling peaks
define a circle centered at (|k̃|, k̃z ) = (Ã, 1/8) with a radius 1/8 and are plotted as black solid
curves. We find a crossover in the shape of the perfect tunneling peaks that occurs atÃco = 1/8.

(33)

For Ã < Ãco the circle is incomplete and approaches a semicircle in the small Ã limit, and forlarge field Ã ≥ Ãco the circle becomes complete. Remembering the definition of Ã given in(ħhΩ)2

Eq. (30), the crossover field strength is Eco = 18 eħh v , where we have temporally recovered the
Planck constant. For example, in the case of Cd3 As2 , the velocity parameter is of the order ofv ∼ 105 m/s and using ħ
h = 6.6 × 10−16 eVs, the crossover fields for photon energies ħhΩ = 1eVand ħhΩ = 1meV are Eco ∼ 2 × 109 V/m and Eco ∼ 2 × 103 V/m, respectively. We stress thatthese parameters for the laser strength are experimentally feasible.
Next, we investigate how this crossover is seen in the physically observable quantities. We
plot the total production rate Γξ3D and the chiral current in the z direction J5z in Fig. 7(d). The
quantities show a power-law behavior in the weak and strong field limits with different powers. The change of the power occurs around the crossover field Ã = Ãco and their asymptotic
14

SciPost Phys. 11, 075 (2021)

behaviors are given by
1
Ã1/2 (Ã/Ãco  1)3(4π)32
v3
Ã2(Ã/Ãco  1) ,
(2π)3
¨
Á
sgn(Ω) 1/2
Ã
(Ã/Ãco  1)−eτ|Ω|4 
2(4π)3J5z→sgn(Ω)1
v2
Ã
(Ã/Ãco  1) .
(2π)3

3D
Γtot

Á 4 
|Ω|

¨

→

(34)

(35)

It is possible to analytically evaluate the asymptotic behaviors using the fact that the distribution around the peak is a Gaussian with a width scaling as Ã1/2 . The detailed calculation is
given in appendix B.
This is a novel nonperturbative crossover that originated from the the nonadiabatic geometric effect that has no perturbative analogue. Let us discuss how we can measure thecurrent as well as the crossover in solid-state experiments. We have discussed a general theory based on Weyl and Dirac Hamiltonians. There are various material realizations of Weyl
and Dirac Hamiltonians [56], where the chirality ξ may correspond to degrees of freedom
such as orbitals and spins as well as their mixtures. For example, in Co3 Sn2 S2 [57] the chirality ξ corresponds to spin [58] and the chiral current J5z can be detected as a spin current. The
generation of U(1) photocurrent due to optical absorption in Weyl semimetals with broken
symmetry have been studied in refs. [37, 38, 40, 41, 59]. If the Fermi energy is non-zero andthe system has finite carrier density, the nonlinear anomalous Hall current [60] can also contribute to the current generation in the z-direction [61]. The three mechanisms, i.e., tunnelingcreation, optical absorption, nonperturbative Hall current, have different dependencies on thelaser and material parameters such as Field strength, photon energy, and Fermi energy. The
asymptotic behaviors of the physical observables in Eq. (34), (35) is useful in identifying theorigin of the photoinduced current.

5Conclusion

We studied the nonadiabatic geometric effects in quantum tunneling and found that they provoke anomalous phenomena such as rectification, perfect tunneling and counterdiabaticity.
We derived the tunneling formula describing these effects through the modulation of the effective mass. We studied the implication of nonadiabatic geometric effects in the Schwinger
effect, i.e., tunneling creation of carriers, induced by rotating electric fields. Two condensed
matter applications are mentioned. One is the valley polarization that can be induced in 2D
Dirac materials, and the other is the generation of spin (and charge) current in 3D Dirac (andWeyl) materials. Our finding adds another example to the rich nonperturbative phenomena
induced by circularly polarized laser in electronic systems [62–68]. Finally, we comment thatthe interplay between the nonadiabatic geometric effects and interaction is an open problem
calling for further study. We point out that there is an interesting resemblance between thephase diagram of the twisted Schwinger effect [Fig. 5(c)] and that of a strongly interacting
holographic model [69, 70].

Acknowledgements
We would like to thank Masamitsu Hayashi, Ryo Shimano, Sota Kitamura, Takahiro Morimoto,
Masafumi Udagawa, Francesco Peronaci, Alexandra Landsman, Hamed Koochaki Kelardeh,and Lisa Ortmann for fruitful discussions.

15

SciPost Phys. 11, 075 (2021)

Funding information This work was supported by JSPS KAKENHI Grant No. JP21K03412
and JST CREST Grant No. JPMJCR19T3, Japan. J. W. acknowledges additional support from a
Shanghai talent program. The work at Shanghai Jiao Tong University is sponsored by Natural
Science Foundation of Shanghai with Grant No. 20ZR1428400 and Shanghai Pujiang Programwith Grant No. 20PJ1408100 (JW).

A

Detailed calculations for the 2D Dirac fermions

We consider the HamiltonianĤ = v[ξ(k x − eAsin q)σ̂ x + (k y + eAcos q)σ̂ y ] + mσ̂z ,where q = Ωt. It is expanded as to q up to the second order and can be written in the form ofEq. (9) withÂ =mσ̂z + ξvk x σ̂ x + v(k y + eA)σ̂ y ,B̂ = − ξveAσ̂ x ,Ĉ = − veAσ̂ y .
Let us consider the tunneling at k x = 0, k y < 0 for Ω > 0 and k x = 0, k y > 0 for Ω < 0 (i.e.,k y = −sgn(Ω)|k|) in the time interval of −π/|Ω| ≤ t ≤ π/|Ω|. Note that the sign of A is thesame as that of Ω. The parameters in Eq. (11) are given asÆm → v 2 (−sgn(Ω)|k| + eA)2 + m2 ,v →veA,κk v 2 → p

ξmveAv 2 (−sgn(Ω)|k| + eA)2 + m2.

Since F = −Ω and E = AΩ, the tunneling probability for twisted Schwinger effect in 2D Diracfermions is given asPξ (k) = exp − πξΩm 2 
4M
,
veE

M−

(36)

where we definedM=Æv 2 (|k| − eE/|Ω|)2 + m2 .

Za2
d k Pξ (k) =2π

We investigate the total probabilityPξtot ≡

 a 22π

Z∞

d kkPξ (k) ,0below focusing on the case of ξΩ > 0.

A.1

Low frequency region

When the laser frequency is smaller than double the gap |Ω| < 4m, Pξ shows apeak at |k| = eE/|Ω| in the momentum space.
Let us expand Eq. (36) around0k = eE/|Ω|. We represent k = k − eE/|Ω|, and since vk0  m, we can approximate as(m/M )2 = (1 + v 2 k02 /m2 )−1 ' 1 − v 2 k02 /m2 . Hence, in the low frequency region,
16

SciPost Phys. 11, 075 (2021)



π  2
ξΩm Ω2Pξ (k) ' exp −+(1 − v 2 k02 /m2 )m + v 2 k02 −
veE
2
16

π ¦ 2
ξΩ 2 ©
Ω2  02 = exp −
k
+
m
−
v 1−
,
(37)
veE
16m2
4
p
which pis the normal distribution with the standard deviation eE/(2πv)(1−Ω2 /(16m2 ))−1/2 .
When eE/(2πv)(1 − Ω2 /(16m2 ))−1/2  eE/(|Ω|), Pξtot can be calculated asπ ¦ 2
Ω2  02 ξΩ 2 ©Pξtot ' exp −
v 1−
k
+
m
−
veE
16m2
4
v
2 t
2

π 
eEa
eE 
Ω −1/2
ξΩ 2 
exp
−
=
1−
m
−
.
2π|Ω|
v
16m2
veE
4
Therefore the e-h production rate per unit of time and volume is provided asv
|Ω| tot
eE t eEπ ξΩ 2 
−
m
−
.
Γξ ≡
P
'
exp
2πa2 ξ
(2π)2
v
veE
4

A.2

(38)

High frequency region

In the high frequency region |Ω| > 4m, Pξ (k) have peaks at the perfect tunneling pointsk=eE 1 Æ±m(ξΩ/4 − m) ,|Ω| v

instead of k = eE/|Ω|.
In the case of strong electric field, however, the broadening of Pξ (k) is much largerppthan the distance between the perfect tunneling points eE/(2πv)pm(ξΩ/4 − m)/v
and the contribution to Pξtot mainly comes from k < eE/|Ω| − m(ξΩ/4 − m)/v andpk > eE/|Ω| + m(ξΩ/4 − m)/v, where the approximation Eq. (37) is still valid. Hence the
e-h production rate per unit of time and volume is provided by Eq. (38).
In the case of weak electric field, the contribution to Pξtot comes from the wavenumberparound the perfect tunneling point k = eE/(|Ω|) + m(ξΩ/4 − m)/v. By expanding Pξ (k)paround this wave number, i.e., k = eE/|Ω| + m(ξΩ/4 − m)/v + k0 , we obtain 
4π  0 Æ
2 2
Pξ (k) ' exp − ξ
vk m(ξΩ − 4m) + v 2 k0
veEΩm


4πv
2
' exp − ξ(ξΩ − 4m)k0 .
eEΩpThus, by noting eE/|Ω|  m(ξΩ/4 − m)/v, the total production rate is given as
v
2 p
t eE
a
Pξtot '
ξΩm.
8πv
v
Thus the e-h production rate per unit of time and volume is given as
v
t eE
|Ω| p
Γξ =
ξΩm
.
(4π)2 v
v

17

(39)

(40)

(41)

SciPost Phys. 11, 075 (2021)

B

Detailed calculations for the 3D Dirac fermions

In the same way as the 2D case, the tunneling probability for twisted Schwinger effect in 3D
Dirac fermions is given asPξ (q̃ ) = exp − πM−ξsgn(Ω)k̃z 2 
4M
,
Ã

(42)

with
M=

Ç

(|k̃| − Ã)2 + k̃z2 .

Then the e-h production rate per unit time and volume for each chirality is given asZZ|Ω|  a 3
|Ω|4
3D
Γξ =
d q̃ Pξ (q̃ ) .
dq Pξ (q ) =2πa3 2π(2π)4 v 3

(43)

3D
We can calculate the total and chiral e-h production rates as Γtot
= Γ+3D +Γ−3D and Γ53D = Γ+3D −Γ−3D .
The main contribution to the production rates and currents come from the wavenumbers
around the perfect tunneling pointsq(44)|k̃| = Ã ± k̃z (ξsgn(Ω)/4 − k̃z ) (ξsgn(Ω)/8 − 1/8 ≤ k̃z ≤ ξsgn(Ω)/8 + 1/8) .

Equation (44) is rewritten as(|k̃| − Ã)2 + (k̃z − ξsgn(Ω)/8)2 = (1/8)2 ,which forms a circle with the center (Ã, ξsgn(Ω)/8) and the radius 1/8 or a part of it in the|k|-kz space.

B.1

Weak field regime

In the weak field regimeÆ Ã  1/8, the contribution mainly comes from the positive sign branchof Eq. (44) |k̃| = Ã+ k̃z (ξsgn(Ω)/4 − k̃z ). The tunneling probability can be approximated as4πPξ (q̃ ) ' exp − ξsgn(Ω) (ξsgn(Ω) − 4k̃z )k̃02 ,
(45)
Ã
Æ
where k̃0 = |k̃|− Ã− k̃z (ξsgn(Ω)/4 − k̃z ). Then we can calculate the production rate Eq. (43)asÁ 4 
|Ω|
3D
Γξ
v3
Z


1
4π
02
'
d
q̃
exp
−
ξsgn(Ω)(ξsgn(Ω)
−
4
k̃
)
k̃
z
(2π)4
Ã
v
Z ξsgn(Ω)/8+1/8
u
q


1
Ã
t
=
d k̃z Ã + k̃z (ξsgn(Ω)/4 − k̃z )32(2π) ξsgn(Ω)/8−1/8ξsgn(Ω)(ξsgn(Ω) − 4k̃z )Z ξsgn(Ω)/8+1/8
q
1
A1/2
'
d
k̃
ξsgn(Ω)
Ã
k̃
=
.
(46)
z
z
4(2π)3 ξsgn(Ω)/8−1/86(4π)318

SciPost Phys. 11, 075 (2021)

Therefore3DΓtotÁ 4 
|Ω|
v3

A1/2
=
,
3(4π)3

Γ53D

Á 4 
|Ω|
v3

= 0.

(47)

For the calculation of currents, noting thatq2|k̃|2 + k̃z2 ' Ã + k̃z (ξsgn(Ω)/4 − k̃z ) + k̃z2 ' ξsgn(Ω)k̃z /4,
we can deriveJξzÁ−eτ|Ω|4 
v2

4
'
(2π)4
1
'
(2π)3

Z

q
d q̃ ξsgn(Ω) ξsgn(Ω)k̃z Pξ (q̃ )Z ξsgn(Ω)/8+1/8

d k̃z Ã1/2 k̃z = ξsgn(Ω)ξsgn(Ω)/8−1/8Ã1/2.
4(4π)3(48)

Therefore
Jz

B.2

Á
−eτ|Ω|4 
v2

= 0,

J5z

Á
−eτ|Ω|4 v2= sgn(Ω)Ã1/2.
2(4π)3(49)

Strong field regime

In the strong field regime Ã  1/8, the contribution mainly comes from both sign branches ofEq. (44). The tunneling probability can be approximated as π¦ξsgn(Ω) 2 ©
Pξ (q̃ ) ' exp −,k̃02 + k̃z −4Ã
where k̃0 = |k̃| − Ã. Then we can calculate the production rate Eq. (43) asZÁ 4  π¦ξsgn(Ω) 2 ©
|Ω|
1
3D
02
Γξ
'
d
q̃
exp
−
k̃
+
k̃
−
z
v3
(2π)4
4
Ã
Z∞
3/2


ξsgn(Ω) 2 
Ã
π
A2
=
=
.
d
k̃
exp
−
k̃
−
z
z
(2π)3 −∞
4
(2π)3
Ã

(50)

(51)

Therefore3DΓtotÁ 4 
|Ω|
v3

=

2A2
,
(2π)3

Γ53D

Á 4 
|Ω|
v3

= 0.

(52)

For the calculation of currents, noting that |k̃|2 + k̃z2 ' Ã2 , we can deriveJξzÁ−eτ|Ω|4 v2

2Zd q̃ k̃z Pξ (q̃ )(2π)4 ÃZ∞ π
ξsgn(Ω) 2 Ã
2Ã1/2
'
d
k̃
k̃
exp
−
k̃
−
= ξsgn(Ω).
zzz
3(2π) −∞
42(2π)3Ã

'(53)

ThereforeJzÁ−eτ|Ω|4 
v2

= 0,

J5z

Á
−eτ|Ω|4 v2

19= sgn(Ω)

Ã
.
(2π)3

(54)

SciPost Phys. 11, 075 (2021)

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392, 45 (1984), doi:10.1098/rspa.1984.0023.
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