Modulation of quantum geometry and its coupling to pseudo-electric field by dynamic strain

Domain: physics
Published: 2026-03-24
Editorial outcome: accept

Abstract
Two-dimensional materials are a fertile ground for exploring quantum geometric phenomena, with
Berry curvature and its first moment, the Berry curvature dipole, playing a central role in their
electronic response. These geometric properties influence electronic transport and result in the
anomalous and nonlinear Hall effects, and are typically controlled using static electric fields or
strain. However, the possibility of modulating quantum geometric quantities in real-time remains
unexplored. Here, we demonstrate the dynamic modulation of Berry curvature and its moments,
as well as the generation of a pseudo-electric field and their coupling. By placing heterostructures
on a membrane, we introduce oscillatory strain together with an in-plane AC electric field and
measure Hall signals that are modulated at linear combinations of the frequencies of strain and
electric field. We also present direct experimental and theoretical evidence for coupling between
pseudo-electric field and quantum geometry that results in an unusual dynamic strain-induced Hall
response. This approach opens up a new pathway for controlling quantum geometry on demand,
moving beyond conventional static perturbations. The coupling of pseudo-electric field with Berry
curvature provides a framework for external electric field-free anomalous Hall response and opens
new avenues for probing the topological properties.

Introduction
Two-dimensional (2D) materials have emerged as an exciting platform in condensed matter physics
and material sciences, offering unique opportunities to study, tune, and probe fundamental quantum
phenomena. From graphene to transition metal dichalcogenides (TMDC) and onwards to moiré
superlattices, these atomically thin systems exhibit properties such as tunable band gaps [1, 2],
high carrier mobility, strong spin-orbit coupling [3], and pronounced excitonic effects, making them
ideal candidates for next-generation quantum devices. Among the various approaches to tailor
their properties, the application of mechanical strain stands out as a versatile and non-invasive
tool. Strain enables modification of band structures, valley polarization [4, 5], and induces phase
transitions [6, 7], thereby significantly altering the electronic and optical responses.

A particularly intriguing aspect of 2D materials is the role of quantum geometry in governing
their physical responses [8], mainly encoded by Berry curvature and the Berry curvature dipole
(BCD) [9–11]. Berry curvature acts as an effective valley contrasting magnetic field in momentum
space[12], influencing the anomalous Hall [13] and valleyHall [14–16] effects in systems wheretime-
reversal or inversion symmetry is broken. In contrast, the BCD, which captures the asymmetric
distribution of Berry curvature, generates the nonlinear Hall effect (NLHE) and dominates the Hall
transport in time-reversal invariant systems with broken crystalline rotation symmetry. Together,
they offer a comprehensive framework to understand and predict the transport behavior of 2D
materials. Strain introduces an additional degree of freedom in this framework to manipulate both
BerrycurvatureandBCD.Bymodifyinglatticesymmetriesandelectronicwavefunctions,straincan
modify Berry curvature distribution in momentum space and induce asymmetries that significantly
alter the BCD [17].

Existing studies have explored how static strain affects Berry curvature and the BCD in 2D
materials. In TMDCs, strain has been shown to enhance valley-contrasting phenomena, such as
the valley Hall effect [18–20]. Moiré systems such as twisted bilayer graphene exhibit strain-tunable
Berry curvature distributions, which can modify band topology by altering Chern numbers, lead-
ing to nontrivial topological transport phenomena [21–23]. Experimental and theoretical studies
have demonstrated that strain-induced symmetry breaking enhances the BCD, giving rise to a pro-
nounced nonlinear Hall effect in materials like WTe and MoTe [24, 25]. Recent studies on twisted
doublebilayergraphene(TDBG)havedemonstratedthattheBCDcansensetopologicaltransitions
in moiré superlattices, underscoring the role of strain in tuning Berry curvature distribution and
topological properties [26, 27]. Although these findings highlight the sensitivity of Berry curvature
and BCD to strain, they predominantly address static strain, which is challenging to control due to
the inherent unpredictability of stacking-induced strain.

In this work, we introduce adiabatic oscillatory strain as a tool to dynamically modulate Berry
curvature and BCD in 2D materials, providing direct access to their temporal evolution [28, 29].
Figure 1a illustrates a representative untilted parabolic band with a symmetric Berry curvature
distribution over the Fermi surface–corresponding to a vanishing BCD. Dynamic strain perturbs
the symmetry in two distinct ways–band tilting (Figure 1b) and valley separation (Figure 1c), each
leaving a unique fingerprint in transport. The oscillatory strain periodically tilts the electronic
bands, distorting the Berry curvature distribution over the Fermi surface, leading to BCD modu-
lation as schematically shown in figure 1b. Also, the strain modulates the separation between the
K and K valleys as shown in figure 1c. Microscopically, this modifies the hopping amplitudes in
the hexagonal lattice, inducing a time-dependent gauge field [30, 31] that leads to pseudo-electric
field [32–35](See Supplementary Information section IV).

Thiscapabilityallowsforreal-timecontrolofBerrycurvatureandhelpsdisentangleothernonlin-
eareffectsfromquantumgeometriccontributions. Toexplorestrain-inducedmodulationofquantum
geometry, wefocusonTDBGandBernalBilayergraphene(BLG)fortheircomplementaryfeatures.
TDBG is a highly tunable system where the twist angle between two BLG sheets creates flat elec-
tronicbands[36,37]. Thetunabilityofitsbandstructureviaperpendicularelectricfieldandcarrier
density, combined with Berry curvature hotspots near the band edges [38–40], makes it ideal for
studying strain-driven Berry curvature dynamics. BLG, in contrast, serves as a simpler and well-
understood reference system, where a perpendicular electric field breaks inversion symmetry and
opens a bandgap at the charge neutrality point (CNP) [2]. To enable controlled strain application
in these systems, we fabricate our devices on trampoline-like, freely suspended silicon nitride mem-
branes. This platform allows mechanical coupling without substrate interference and is particularly
well suited for delicate moiré heterostructures. Through experiments on TDBG and BLG devices,
we show that strain oscillations modify the nonlinear Hall signal–arising from the time-dependent
BCD–at specific mixed frequencies, directly revealing the dynamic response of quantum geometry.
Alongside, we report direct observation of pseudo-electric field coupling to the Berry curvature,
resulting in an external field-free anomalous Hall voltage. Our findings, using versatile 2D systems,
introduce a novel approach for dynamic tuning of quantum geometric properties of materials of any
dimensionality beyond static perturbations.

Results
The TDBG device, consisting of two bilayer graphene sheets stacked with a twist angle of 1.1 , and
the BLG device are fabricated on 300 nm thick and 300 µm 300 µm silicon nitride (SiN) mem-
branes to facilitate the strain application [41, 42] (see Supplementary Figure S4 and Supplementary
Information section VII and VIII). Finite element analysis COMSOL simulation is used for place-
ment of the device stack on the SiN membrane for optimal strain application (see Supplementary
Information section IX). The devices are configured in a dual-gated geometry, enabling independent
control of the perpendicular electric field D/ϵ and carrier density n (see Supplementary Informa-
tion section X.1 for details). Time-dependent strain δu (t) and DC strain u are applied to the
device (see Figure 2a) using a Razorbill CS130 strain cell, and all measurements are performed at
1.5 K (further details on measurements are provided in Supplementary Information section X).

A sinusoidal strain of frequency ω , δu (t) sin(ω t) is applied to the devices via the strain
cell. ω is kept low to ensure adiabatic evolution of the electronic wavefunction. In addition to the
strain modulation, an AC current Iωc at frequency ω is applied to the device and the resulting Hall
responseVω ismeasuredatmixedfrequencies,ω = ω αω (α = 0,1,2). Phase-sensitivedetection
of the Hall response is performed by separately analyzing the real and imaginary components of
the AC signal, depending on the frequency combination. In the following sections, we present the
experimental results and analyze the corresponding Hall responses.

Figure 2b depicts the honeycomb lattice under uniaxial strain. The unstrained lattice is shown
in gray, while the strained lattice under tension and compression is represented by brown and
green, respectively. Strain in graphene can tilt the band structure [43], altering the Berry curvature
distribution over the Fermi surface, as schematically depicted. Consequently, the valley-specific
Berry curvature and BCD are modified. The theoretically calculated variation of Berry curvature
andBCDwithstrainwithinKT-valleyfor1.1 TDBGisshowninFigure2cand2d,respectively. The
Berry curvature and BCD are calculated at the maxima of the valence band (see Supplementary
Information section I), considering uniaxial strain along the zig-zag direction of graphene. Both
theoretical plots emphasize the sensitivity of Berry curvature and BCD to strain in TDBG.

Figure2epresentsacolormapofthelongitudinalresistanceR asafunctionofthefillingfactor
(ν, bottom x-axis), carrier density (n, top x-axis), and perpendicular electric field (D/ϵ , y-axis)
at 1.5 K under zero strain. At D/ϵ = 0, the two flat bands in TDBG remain gapless; however,
as a perpendicular electric field is applied, a bandgap opens between the two flat bands, which is
observed as a prominent resistance peak at ν = 0. In addition, well-defined resistance peaks appear
at ν = 4, corresponding to the full filling of the moiré flat bands; this measurement highlights the
tunability of the band structure of TDBG by a perpendicular electric field, consistent with prior
studies on TDBG [37, 38, 40, 44].

The longitudinal resistance (R ) changes notably with applied DC strain. Figure 2f shows R
as a function of the DC piezo voltage (Vdc ), which controls the applied strain. The resistance,
measured at a fixed carrier density (n = 3.46 1012 cm 2) slightly above full filling (ν = 4) at
1.5 K, responds to both tensile (Vdc > 0) and compressive (Vdc < 0) strain. In 2D materials,
strain modifies interatomic distances and breaks lattice symmetries, leading to significant changes
in carrier mobility [45], band structure, and Berry curvature distribution [46]. The observed strain
dependence of R thus confirms effective strain application and underscores its impact on the
electronic properties of the device.

Berry Curvature Dipole Modulation

A key outcome of strain modulation in TDBG is its profound impact on the BCD (Λ) and its role
in non-linear Hall transport [26, 47]. In 2D devices such as TDBG and BLG, a pre-existing static
strain (u ) often arises due to fabrication-induced imperfections or can be deliberately applied
via a DC voltage to the strain cell. This static strain breaks the C symmetry of the lattice,
leading to an asymmetric Berry curvature distribution in momentum (k) space. Such symmetry
breakingisresponsibleforanon-zeroBCD,whichgeneratesnonlinearHallvoltages(Supplementary
Information section XI), in presence of an in-plane electric field E = xˆE sin(ω t),
j(2) = e3τ ˆz (Λ E ) E .
2(1 iω τ) × ·

In addition to the static strain, when an oscillatory strain δu (t) sin(ω t), experimentally
controlled by an AC piezoelectric voltage, V = V sin(ω t) is applied to TDBG, the non-linear
Hallcurrentexhibitsastrain-dependentvariation,astheappliedmechanicaldeformationintroduces
adynamiccomponenttotheBCD,givenbyΛ(u +δu ) = Λ +δΛ cos(ω t)+.... Thismanifests
asmixingoffrequenciesbetweentheappliedin-planeelectricfieldandtheoscillatorystrain, leading
to a non-linear Hall response that displays sidebands at combined frequencies of the strain (ω )
and current modulation (ω ),
j(2) 0 0 0 0 cos(2ω t)+ 0 m cos(ω t) 0 m [cos((ω +2ω )t)+cos((ω 2ω )t)].
y ∝ 2 − 2 c 2 m − 4 m c m − c

Here,
δΛ = V ω δt
m m m
∂Λ ∂u
∂u ∂V
is the change of BCD due to strain over a small time interval δt, and Λ = Λ(u ) is the static BCD.
∂Λ captures the dependence of the BCD on strain, determined by the band structure, while
∂u u0
∂u dependsontheexperimentalsetupandtheeffectivecouplingbetweenpiezovoltageandstrain
∂V
(see Supplementary Information section III for detailed calculation).

The time-independent component of the BCD (Λ ) generates a DC rectification and a conven-
tional nonlinear Hall voltage at 2ω . The oscillatory component of the BCD (δΛ ) contributes a
rectification at ω and produces sidebands at ω 2ω . Since these frequency components arise
from dynamic strain-induced BCD oscillations, they exhibitphase differences relativeto theapplied
strain and current. Accurate detection of these nonlinear Hall components requires phase-sensitive
measurements. The nonlinear Hall signals appear 90 out of phase with the applied alternating
current, requiring measurement of its imaginary component to capture contributions at 2ω ,
ω , ω +2ω , and ω 2ω (for details see Supplementary Information section III).

Figure 3a and 3b present the nonlinear Hall signals at the frequencies ω +2ω (see Supplemen-
taryInformationsectionXIIforω 2ω sig nal)andω ,respectively,asfunctionsofcarrierdensity
(n) or filling factor (ν) and the perpendicular electric field (D/ϵ ) at 1.5 K under zero magnetic
field. These measurements are performed using an alternating current of 98 nA and a sinusoidal
strain generated by a 10 V peak-to-peak AC piezo excitation. Both the Hall signals at ω and
ω +2ω frequencies exhibit antisymmetric behavior under reversal of the perpendicular electric
field (D/ϵ ) near the CNP. For positive D, the Hall signals are negative for ν < 0 and become
positive when D is flipped to negative values (see Supplementary Information section XIV for data
from another TDBG device). This behavior is consistent with the fact that the Berry curvature
and BCD reverse sign with the direction of the electric field for ABAB TDBG [48].

To confirm that oscillating strain underlies the nonlinear Hall signals, we systematically study
their dependence on parameters characterizing the strain, the amplitude and frequency of the strain
modulation. Figure3cand3dillustratethescalingofthenonlinearHallsignalsatω +2ω andω ,
respectively; theamplitudeofthestrainmodulation, representedbytheACpiezoexcitationvoltage
(Vωm ) at ν = 0.037 and D/ϵ = 0.354 V/nm at 1.5 K. Both Vωm+2ωc and Vωm signals vary
piezo − 0 xy xy
linearly with the strain amplitude. Additionally, as the frequency determines the time derivative
of the BCD, we examined the frequency dependence of the Hall signals. Figure 3e and 3f illustrate
the linear dependence for both Vωm+2ωc and Vωm on ω . These results are in agreement with
xy xy m
equation (2) and (3), and offer compelling experimental evidence for the dynamic modulation of
the BCD under strain. Figure 3g and 3h further depict the scaling of the nonlinear Hall signals at
ω +2ω and ω with the square of the applied current Iωc, at ν = 0.007, D/ϵ = 0.354 V/nm.
m c m xx 0
The quadratic current dependence confirms that the nonlinear response stems from a second-order
process with the in-plane electric field. Notably, Vωm remains finite even at zero current, suggesting
xy
contributions beyond simple BCD modulation.

Coupling of Pseudo-Electric Field with Static Berry Curvature

As outlined earlier, oscillatory strain can also generate a pseudo-electric field via a time-dependent
gauge potential. Unlikerealelectricfields, thepseudo-electricfield E isvalley-dependentandcan
be written as E = χE (t) [32–34], where χ = 1 denotes the valley chirality. In the presence
ωm m
±
of nonzero Berry curvature Ω, the semiclassical equation of motion yields an anomalous velocity
e(Ω χE ), that is even under time reversal symmetry. This chirality-dependent coupling drives
m
×
carriers in both valleys in the same direction (Figure 7a), resulting in an anomalous Hall effect at
frequency ω solely dependent on the pseudo-electric field (see row 2 of Table 5 in Supplementary
m
Information section V), a notable aspect of our work. While earlier studies have explored couplings
between pseudo-electric and pseudo-magnetic fields[33], we emphasize that the effect reported here
originates from Berry curvature. This mechanism is distinct from pseudo-magnetic fields, which
instead arise from spatial strain gradients.

This mechanism also accounts for the deviation from the expected ratio between the signals
Vωm and Vωm+2ωc observed in Figure 3(a–d). Specifically, the pseudo-electric field contributes
xy xy
independently of the strain-induced modulation of the BCD, leading to a finite value of Vωm even
xy
at Iωc = 0(Figure 7b). This offset reflects the intrinsic Hall voltage originating completely from the
xx
pseudo-electric field. Figure 3i shows the BCD-modulated component of the nonlinear Hall voltage
∆Vωm = Vωm Vωm(Iωc = 0). As suggested by equation (2), this term is expected to be twice the
xy xy − xy xx
magnitude of Vωm+2ωc with opposite sign. Accordingly, we scale ∆Vωm by a factor of 1/2, and
xy xy −
observe that 1∆Vωm accurately overlays with Vωm+2ωc in Figure 3i, in excellent agreement with
−2 xy xy
theoretical expectations.

These observations not only validate the dynamic modulation of the BCD under strain, but
also suggest the presence of additional nonlinear transport phenomena beyond the BCD framework.
In particular, the finite offset in Vωm points toward an intriguing microscopic mechanism–namely,
xy
the emergence of strain-induced pseudo-electric field. If such a pseudo-field is indeed present, it is
expected to give rise to Hall-like signals at mixed frequencies ω ω as a consequence (see row 6,7
m c
±
of table 5 in Supplementary Information section V). This motivates our following investigation of
the Vωm ωc response in detail.
xy ±

Coupling of Pseudo-Electric Field and Band Velocity

Having established the presence of a finite Vωm offset, indicative of a pseudo-electric field, we now
xy
examine its dynamical consequences–specifically, the appearance of transverse voltages at mixed
frequencies ω ω . These components are not captured within the conventional BCD framework
m ± c
but emerge naturally from the interplay between a strain-induced pseudo-electric field and the
applied current. Time-dependent strain gives rise to a gauge potential χA [30, 31, 49], whose
ωm
temporal derivative defines the pseudo-electric field, E (t) = χ∂ A (t).
ωm
−
t ωm

Within the Boltzmann transport formalism (Supplementary Section V), this pseudo-electric
field, when combined with an in-plane AC current at frequency ω , leads to a second-order non-
c
linear response at ω ω . Notably, the corresponding current arises not from the anomalous
m c
±
velocity, which is Berry curvature dependent, but from the band velocity (v = ∂ε) term, yielding
b ∂k
a transverse voltage that is Hall-like in appearance but distinct in origin,
(eτ)2
jωm+ωc eχf ωm,ωcv eχf ωc,ωmv ; f ωj,ωk = (E )(E f ).
∝ − 2 b − 2 b 2 4[1+i(ω +ω )τ](1+iω τ) j ·∇ k ·∇ 0
j k k

Here, f
ωj,ωk
denotes the second-order correction to the electron distribution function driven by the
pseudo-electric and in-plane electric fields. For strain driven by a piezoelectric voltage, the resulting
pseudo-electric field can be expressed as (see Supplementary Information Section VI),
∂A β d
E = χ
ωm
= χ U V ω cos(ω t)
ωm
∂t − 2a dV
m m m
where depends on symmetrized strain tensor (Supplementary Information equation (45)). This
U
pseudo-electric field gives rise to a transverse Hall-like response (see Supplementary Information
equation (67) for detailed expression),
j y ωm ± ωc ∝ ± χv by V m ω m E 0 sin[(ω m ± ω c )t].
Both the dynamic strain-induced anomalous Hall component Vωm(Iωc = 0) and the band velocity-
xy xx
driven Vωm ωc arise due to the the pseudo-electric field. These signals vanish identically in a
xy ±
time-reversal symmetric system when subjected to a real in-plane electric field (see Table 2 and 4
of Supplementary Information).

Figure 4a illustrates the transverse response of TDBG at the mixed frequency ω + ω (see
m c
Supplementary Information section XIII for ω ω signal) as a function of the perpendicular
m c
−
electric field (D/ϵ ) and filling factor (ν) at 1.5K. Unlike the signals shown in Figure 3a and 3b,
0
which arise from Berry curvature and BCD contributions, the ω +ω response exhibits a distinctly
m c
different behavior. Specifically, the sign of Vωm+ωc remains positive for fillings just below full-band
xy
occupation (ν ⪅ 4,0,+4) and negative for fillings just above (ν ⪆ 4,0,+4), independent of
− −
D/ϵ , indicative of a band velocity–driven mechanism, an aspect that we discuss later in this work.
0

InFigure4b, thetransversevoltageatω +ω isplottedagainstthethepeak-to-peakpiezoelec-
m c
tric excitation voltage (Vωm ), which represents the amplitude of the pseudo electric field,
piezo
E , at a filling ν = 0.037 near the CNP and D/ϵ = 0.354 V/nm (See Figure 8 for scaling with
m 0
ω ). The observed behavior confirms the linear dependency of Vωm+ωc on E . Figure 4c further
m xy m
demonstrates the linear scaling of the Hall voltage with the amplitude of the applied current (Iωc)
xx
at ν = 0.037 and D/ϵ = 0.354 V/nm, in agreement with equation (5). To further improve our
0
understanding, next we probe a non-flatband system, Bernal BLG.

Measurements on a Bernal Bilayer Graphene

BLG is a simpler system where similar strain-induced physics is expected. Strain in BLG in combi-
nation with the Berry curvature hotspots near the band edges [15, 16] induce a finite BCD giving
risetononlinearHallvoltagenearCNP[50]. Whenanoscillatorystrainisintroduced, BLGexhibits
mixed-frequency nonlinear Hall responses analogous to those in TDBG. Figure 5a presents the Hall
voltage at frequency ω +2ω , associated with the dynamic modulation of the BCD in BLG. These
m c
measurements, performed at 1.5 K using an alternating current of 94 nA (17 Hz) and a 10 V peak-
to-peak AC piezo excitation at 177 Hz, reveal a pronounced signal across the CNP, consistent with
the formation of Berry curvature hotspots near the band edges. Similar to ABAB-stacked TDBG,
AB-stacked BLG also exhibits a sign reversal of both Berry curvature and BCD with the direction
of the applied displacement field (see Supplementary Information section II). Accordingly, the sign
of the ω +2ω response reverses with field polarity, consistent with its BCD origin.
m c

Thetransversevoltageatω +ω , arisingfromtheinterplaybetweenthestrain-inducedpseudo-
m c
electric field and the applied AC current is shown in Figure 5b. Unlike the BCD-driven response in
Figure 5a, this signal does not reverse sign with the perpendicular electric field, consistent with a
band velocity–dominated mechanism. As schematically illustrated in Figure 5c, the band velocity
v = ∂ε/∂k switches sign across the CNP and does not depend on field direction. This behavior
b
corroboratesthepseudo-electricfieldoriginoftheω +ω response. Figure5dand5efurtherconfirm
m c
the linear scaling of Vωm+2ωc and Vωm+ωc with the amplitude of the AC piezo excitation voltage in
xy xy
alignment with the theoretical formulation (equation 3 and 5). Additional characteristics, such as
the linear scaling of Vωm+ωc, quadratic scaling of Vωm+2ωc with AC current, and the dependence of
xy xy
the signals on piezo excitation frequency, are provided in the Figure 9 and 10 (see Supplementary
Figure S13, S14 and S15 for nonlinear Hall responses at other frequency components).

Methods
Device fabrication

The 1.1 twisted double bilayer graphene and Bernal bilayer graphene devices are fabricated using a
dry-transfer process and integrated onto 300 nm thick, 300 µm×300 µm trampoline-like suspended
silicon nitride membranes to enable controlled strain application. To ensure an optimal strain
distribution, Finite Element Analysis (COMSOL) was employed to determine the placement of the
heterostructure stack on the membrane. The devices were configured in a dual-gated geometry,
allowing independent control of the carrier density (n) and the perpendicular electric displacement
field (D/ϵ ).
0

Measurements

Low-temperature transport measurements are performed at 1.5 K using a closed-cycle Oxford Tes-
latron cryostat. An AC excitation current is sourced from a Zurich Instruments UHF (ZLI) lock-in
amplifier and applied to the device through a 10 MΩ ballast resistor to maintain a constant cur-
rent. Charge carrier density (n) and the perpendicular electric field (D/ϵ ) are tuned by applying
0
independent DC voltages to the top and bottom gates. External strain is controlled via a Razorbill
CS130straincellconsistingoftwoindependentpiezoelectricstacks. AconstantDCvoltage(V2)ap-
ply static strain, while a sinusoidal AC voltage (V1) at 177 Hz induced dynamic strain modulation.
The nonlinear Hall response is decomposed into linear combinations of the modulation frequencies
using the Amplitude Modulation and Demodulation feature of the ZLI, enabling the simultaneous
detection of signals at ω , ω ω , ω 2ω . Additional details are provided in Supplementary
m m c m c
± ±
Information section VII-X.

Theory calculation of pseudo-electric field driven transport

To understand the physical mechanisms underlying the observed transport phenomena, we per-
formed a semiclassical calculation using the Boltzmann transport equation (BTE) in the relaxation
time approximation. We model the system as a two-dimensional material with valley specific Berry
Curvature subjected to a superposition of two distinct driving fields, a standard externally applied
AC electric field and a time dependent strain-induced pseudo-electric field.

The total electric field (t) is expressed as:
E (t) = Re[E eiωct]+Re[χE eiωmt]
c m

Here, E is the amplitude of the real external electric field oscillating at frequency ω , and E
c c m
denotesthepseudo-electricfieldgeneratedbythetime-dependentstrainatfrequencyω . Thefactor
m
χ = 1 represents the valley index (K and K ). The defining characteristic of the pseudo-electric
± ′
field is its chirality; unlike the real field E which acts uniformly on all carriers, the pseudo-field
c
χE couples to the two valleys with opposite signs, arising from the gauge field associated with the
m
dynamical lattice deformation.

The carrier dynamics are governed by the BTE,
τ∂ f eτ f = f f,
t k 0
− E ·∇ −
where f is the equilibrium Fermi-Dirac distribution. The total current density includes contribu-
0
tions from both the band velocity v and the anomalous velocity v = e(Ω ), where Ω is
b anom
− ×E
the Berry curvature.

Strain-induced anomalous Hall effect at ω
m

A distinct and significant feature emerging from the Boltzmann calculation is the existence of an
anomalous Hall current at the strain frequency ω , even in the complete absence of an external real
m
electric field (E = 0). Expanding the distribution function and velocity terms, we isolate a current
c
component proportional to,
j e2χ f (k)[Ω (k) E ]d2k+contributions from higher moments of Berry Curvature
ωm
∝
0 0
×
m
BZ

This term represents an external electric field-free anomalous Hall effect. Physically, it arises from
the anomalous velocity induced by the pseudo-electric field acting on the equilibrium carrier distri-
bution (f ).
0

In a time-reversal symmetric system, the intrinsic Berry curvature Ω is odd under valley ex-
0
change (Ω = Ω ). Typically, this leads to the cancellation of the anomalous Hall effect for
K K
− ′
real electric fields. However, the pseudo-electric field possesses a valley-dependent sign (χ). Conse-
quently, the transverse deflection for a given valley, driven by Ω (χE ), acquires the same sign
m
∼ ×
for both K and K valleys (since the product χΩ is even). This results in a constructive addition of
′
transverse currents from both valleys, generating a net macroscopic anomalous Hall voltage driven
solely by the mechanical strain rate, without any charge current flow driven by a real bias.

Frequency mixing and Berry curvature modulation

When the real external field E is applied alongside the pseudo-field, additional nonlinear currents
c
arise due to frequency mixing. We expanded the BTE solution to second order in the fields (f ) and
2
accounted for the time-dependent modulation of the Berry curvature, Ω(t) Ω +Re[δΩ eiωmt],
0 m
≈
caused by the oscillating strain.

WhileBerrycurvaturemodulation(δΩ )contributestoharmonicssuchas2ω andω 2ω ,our
m m c m c
±
analysis identifies that there are another contribution to the mixed-frequency signal at ω = ω ω
m c
±
originates from the band velocity acting on the second-order distribution function induced by the
interplay of real and pseudo-fields. The resulting nonlinear current is derived as,
e3τ2 ∂f ∂E
0 m
j = χ (v )E (E ) + f sin[(ω ω )t]
ωm
±
ωc
± 2
b 0
·∇
k
∂k x ∂k x ·∇
k 0
m
±
c

This expression reveals that the mixing signal scales linearly with the applied field E = xˆE ,
c 0
pseudo-electric field E , and the strain frequency ω . Detailed derivations of all current har-
m m
monics, including the symmetry constraints and the full table of surviving terms, are provided in
Supplementary Information section V.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The authors have addressed all of my concerns.\nThe manuscript has substantially been improved.\nSo I recommend it for publication in Nature Commun. now.", "reviewer_2": "After carefully reviewing the authors’ rebuttal and the additional material provided, I am not convinced that the manuscript meets the conceptual novelty threshold required for publication in Nature Communications. While the responses are extensive, technically competent, and clearly written, they do not resolve the central issue: the claimed advance of “dynamic modulation of quantum geometry” does not constitute a qualitatively new physical regime, but rather a reinterpretation and repackaging of already established strain-driven transport phenomena.\nThe manuscript repeatedly emphasizes the term dynamic to distinguish the present work from prior studies employing surface acoustic waves (SAWs) or other time-dependent strain fields. However, from a physical standpoint, SAWs already generate explicitly time-periodic strain, leading to time-dependent band structures, gauge fields, and transport responses. Within the semiclassical Boltzmann framework used here, such time dependence necessarily implies a modulation of Berry curvature and related quantities. Consequently, the introduction of the label “dynamic quantum geometry” does not, by itself, define a new regime. The experiments are performed in the explicitly adiabatic limit (ωτ ≪ 1), where the electronic system follows the instantaneous band structure. In this limit, the response is more appropriately described as quasi-static modulation rather than genuinely dynamical quantum geometry involving non-adiabatic effects, geometric phase accumulation over a cycle, or Floquet band formation.\nThe authors argue that their main novelty lies in separating three contributions in the frequency domain—Berry curvature dipole modulation, coupling of a pseudo-electric field to Berry curvature, and pseudo-electric-field–band-velocity coupling. While this frequency-resolved analysis is carefully executed and pedagogically useful, it does not establish fundamentally new physics. Frequency mixing is a generic consequence of nonlinear transport under time-dependent perturbations and has already been observed and discussed extensively in SAW-based experiments. The present manuscript does not demonstrate that the observed mixed-frequency components are unique fingerprints of Berry-curvature modulation, nor that they cannot be reproduced within existing strain–gauge-field frameworks.\nThe comparison with Zhao et al. (PRL 128, 256601, 2022) is particularly unconvincing. The rebuttal repeatedly emphasizes that Zhao et al. studied monolayer graphene, which lacks Berry curvature at finite doping, whereas the present work focuses on TDBG and BLG. This distinction is too narrow to support a strong novelty claim. Zhao et al. already demonstrated time-dependent strain-induced synthetic electric fields, frequency-mixed Hall responses, and external-field-free transverse voltages. The present manuscript does not show that the presence of Berry curvature enables a qualitatively new observable or control mechanism; rather, it offers an alternative interpretative framework for closely related experimental signatures. Differences in terminology or band-structure context do not, on their own, constitute a Nature-level advance.\nMore broadly, the rebuttal relies heavily on interpretational arguments—assigning certain signals to “quantum geometry” rather than to pseudo-electromagnetic fields—without demonstrating access to a new control parameter, symmetry class, or physical limit inaccessible to prior work. Interpretation alone, however elegant, is insufficient justification for publication in Nature Communications. A genuinely compelling case would require, for example, non-adiabatic effects, topological pumping over a strain cycle, or a geometric invariant that can only arise from time-dependent Berry curvature in a way fundamentally distinct from SAW physics.\nI emphasize that this assessment does not question the technical quality of the experiments or the care with which the data are analyzed. The manuscript is well written, the measurements are competently performed, and the frequency-domain bookkeeping is clear and instructive. Nevertheless, the overall contribution represents an incremental extension and recontextualization of established strain-driven nonlinear Hall physics rather than a decisive conceptual breakthrough. For these reasons, I do not recommend publication in Nature Communications. The work would be more appropriately placed in a specialized condensed-matter journal, where its technical merits and detailed analysis can be appreciated without overstating its conceptual scope.", "reviewer_3": "The revised manuscript by S. Layek et al. has thoughtfully addressed all issues raised by three reviewers. The updated manuscript and supplementary information included new illustrations and experimental results that better clarified several key points. This manuscript is now well written, which contains convincing experimental and theoretical evidence that demonstrates unusual dynamic strain-induced Hall response due to coupling between pseudo-electric field and quantum geometry in twisted double bilayer graphene (TDBG) with two different twist angles and in Bernal bilayer graphene (BLG). Additionally, the underlying concepts have the potential of being generalized to systems of any dimensionality if they possess Berry curvature. Therefore, the revised manuscript is now suitable for publication in Nature Communications."}, "author_response": "Point-by-point response to reviewer comments\nWe thank all the reviewers for their careful reviews and comments.\nThe reviewers’ comments are in blue-colored italic fonts, and our responses are in black fonts.\nReferee #1 (Remarks to the Author):\nThe authors have addressed all of my concerns.\nThe manuscript has substantially been improved.\nSo I recommend it for publication in Nature Commun. now.\nResponse:\nWe thank the Reviewer for his/her positive evaluation and recommendation for publication.\nReferee #2 (Remarks to the Author):\nAfter carefully reviewing the authors’ rebuttal and the additional material provided, I am not convinced\nthat the manuscript meets the conceptual novelty threshold required for publication in Nature\nCommunications. While the responses are extensive, technically competent, and clearly written, they do\nnot resolve the central issue: the claimed advance of “dynamic modulation of quantum geometry” does\nnot constitute a qualitatively new physical regime, but rather a reinterpretation and repackaging of\nalready established strain-driven transport phenomena.\nThe manuscript repeatedly emphasizes the term dynamic to distinguish the present work from prior\nstudies employing surface acoustic waves (SAWs) or other time-dependent strain fields. However, from a\nphysical standpoint, SAWs already generate explicitly time-periodic strain, leading to time-dependent\nband structures, gauge fields, and transport responses. Within the semiclassical Boltzmann framework\nused here, such time dependence necessarily implies a modulation of Berry curvature and related\nquantities. Consequently, the introduction of the label “dynamic quantum geometry” does not, by itself,\ndefine a new regime. The experiments are performed in the explicitly adiabatic limit (ωτ ≪ 1), where the\nelectronic system follows the instantaneous band structure. In this limit, the response is more\nappropriately described as quasi-static modulation rather than genuinely dynamical quantum geometry\ninvolving non-adiabatic effects, geometric phase accumulation over a cycle, or Floquet band formation.\nThe authors argue that their main novelty lies in separating three contributions in the frequency\ndomain—Berry curvature dipole modulation, coupling of a pseudo-electric field to Berry curvature, and\npseudo-electric-field–band-velocity coupling. While this frequency-resolved analysis is carefully executed\nand pedagogically useful, it does not establish fundamentally new physics. Frequency mixing is a generic\nconsequence of nonlinear transport under time-dependent perturbations and has already been observed\nand discussed extensively in SAW-based experiments. The present manuscript does not demonstrate\nthat the observed mixed-frequency components are unique fingerprints of Berry-curvature modulation,\nnor that they cannot be reproduced within existing strain–gauge-field frameworks.\nThe comparison with Zhao et al. (PRL 128, 256601, 2022) is particularly unconvincing. The rebuttal\nrepeatedly emphasizes that Zhao et al. studied monolayer graphene, which lacks Berry curvature at finite\ndoping, whereas the present work focuses on TDBG and BLG. This distinction is too narrow to support a\n1\nstrong novelty claim. Zhao et al. already demonstrated time-dependent strain-induced synthetic electric\nfields, frequency-mixed Hall responses, and external-field-free transverse voltages. The present\nmanuscript does not show that the presence of Berry curvature enables a qualitatively new observable or\ncontrol mechanism; rather, it offers an alternative interpretative framework for closely related\nexperimental signatures. Differences in terminology or band-structure context do not, on their own,\nconstitute a Nature-level advance.\nMore broadly, the rebuttal relies heavily on interpretational arguments—assigning certain signals to\n“quantum geometry” rather than to pseudo-electromagnetic fields—without demonstrating access to a\nnew control parameter, symmetry class, or physical limit inaccessible to prior work. Interpretation alone,\nhowever elegant, is insufficient justification for publication in Nature Communications. A genuinely\ncompelling case would require, for example, non-adiabatic effects, topological pumping over a strain\ncycle, or a geometric invariant that can only arise from time-dependent Berry curvature in a way\nfundamentally distinct from SAW physics.\nI emphasize that this assessment does not question the technical quality of the experiments or the care\nwith which the data are analyzed. The manuscript is well written, the measurements are competently\nperformed, and the frequency-domain bookkeeping is clear and instructive. Nevertheless, the overall\ncontribution represents an incremental extension and recontextualization of established strain-driven\nnonlinear Hall physics rather than a decisive conceptual breakthrough. For these reasons, I do not\nrecommend publication in Nature Communications. The work would be more appropriately placed in a\nspecialized condensed-matter journal, where its technical merits and detailed analysis can be\nappreciated without overstating its conceptual scope.\nResponse: We note the reviewer’s comments. However, we respectfully disagree with the opinion of the\nreviewer.\nReferee #3 (Remarks to the Author):\nThe revised manuscript by S. Layek et al. has thoughtfully addressed all issues raised by three\nreviewers. The updated manuscript and supplementary information included new illustrations and\nexperimental results that better clarified several key points. This manuscript is now well written, which\ncontains convincing experimental and theoretical evidence that demonstrates unusual dynamic\nstrain-induced Hall response due to coupling between pseudo-electric field and quantum geometry in\ntwisted double bilayer graphene (TDBG) with two different twist angles and in Bernal bilayer graphene\n(BLG). Additionally, the underlying concepts have the potential of being generalized to systems of any\ndimensionality if they possess Berry curvature. Therefore, the revised manuscript is now suitable for\npublication in Nature Communications.\nResponse:\nWe thank the Reviewer for his/her positive endorsement and feedback regarding the improved clarity of\nthe manuscript.\n2"}]

Decision letter