Observation of topological braiding and dynamical criticality in time reflection and refraction

Domain: neuroscience
Published: 2026-01-27
Editorial outcome: minor_revision

Abstract
Time reflection and refraction are among the most canonical phenomena emerging from the burgeoning field of time-varying materials. However, their connection with topology remains largely unexplored. Here, we experimentally report that temporal scattering processes exhibit two distinct topological characteristics: eigenstate topological braiding and dynamical topological phase transition. By developing approaches to implement Schrödinger dynamics and engineer temporal boundaries, various temporal scattering processes are implemented in circuit metamaterials. Through measurements of scattering coefficients, we observe diverse topological braiding and linking structures, whose linking numbers are governed by the difference in topological winding numbers before and after the temporal boundary. Furthermore, we reveal that these processes exhibit dynamical critical behavior and host dynamical topological phase transitions, evidenced by quantized jumps at critical time points in the measured dynamical topological invariants. Our study demonstrates fundamental links between temporal scattering and braiding and dynamical topology, opening avenues for exploring the topological aspects of time-varying and nonequilibrium phases of matter.

Introduction
Time-varying materials have recently emerged as a transformative platform for wave control1–5. An ultrafast and spatially uniform modulation of material properties creates temporal boundaries6. Unlike their counterparts under the space-time duality, namely spatially modulated systems, temporal boundaries or interfaces cause wave reflection and refraction in a fundamentally distinct manner by altering energy while conserving momentum. This distinctive mechanism unlocks wave-engineering capabilities, including broadband frequency translation7,8, nonreciprocal wave propagation9 and reconfigurable wavefront shaping10–15. In the past decade, time reflection and refraction have been observed in a wide range of systems, including water waves16, photonics17–20, acoustics21, microwaves22, mechanics23, and quantum matter waves24.

Recently, it has been discovered that abruptly changing parameters in a topological lattice model can provide us a discrete lattice platform to study both time reflection and time refraction24–26. Along this line, theoretical study further revealed a connection between topology and temporal scattering processes, predicting that the time reflection and refraction coefficients exhibit topological braiding structures that can be utilized to probe the differences in topological invariants across a temporal boundary27. Distinct from the topological braiding associated with eigenvalues in non-Hermitian systems28–40, this braiding arises from eigenstates and their dynamics in time-varying Hermitian systems. Despite these timely advances, the intrinsically time-varying nature of such systems poses fundamental challenges to conventional topological classifications and experimental detections41–50, leaving the connection between temporal scattering processes and topological physics largely unexplored from both theoretical and experimental perspectives.

In this work, we experimentally bridge this gap by exploring the dynamical topological aspects of time reflection and refraction in time-varying circuit metamaterials. A dual-circuit-lattice architecture is implemented to encode and simulate the Schrödinger dynamics of both the real and imaginary components of the wavefunction. Furthermore, a technology is introduced to create and control temporal boundaries for realizing temporal interfaces and manipulating temporal scatterings. By measuring the time reflection and refraction coefficients across various temporal scattering scenarios, we observe a rich variety of topological braiding and linking structures, arising from the intertwining of these coefficients throughout the Brillouin zone (see Fig. 1a, b). Beyond this geometric-topological link, we reveal the emergence of dynamical criticality and dynamical topological phase transitions inside the time reflection and refraction processes. Intriguingly, the dynamical topological phase transition cannot be described by conventional static topological invariants, but rather by dynamical topological invariants (see Fig. 1c), which are defined through the time evolution of Bloch states. Experimentally, we observe the dynamical criticality at critical time points through the nonanalytic behavior in the dynamics of rate functions associated with the temporal scattering coefficients. At precisely the critical time points, quantized jumps are detected in the measured dynamical topological invariants, providing direct evidence for the occurrence of dynamical topological phase transitions. Together, these results offer fresh insights into the dynamical topological aspect of temporal scattering in time-varying metamaterials.

Results
Topological braiding of time reflection and refraction
To be specific, we investigate time reflection and refraction in a long-range SSH lattice model, as illustrated in Fig. 1d. The corresponding model Hamiltonian in the momentum space is given by

H=(J1+J2cosk+J3cos2k)σx+(J2sink+J3sin2k)σy (1)

where J1 and J2 respectively represent the nearest-neighbor intra- and inter-cell coupling strengths, J3 denotes the long-range coupling strengths. σx,y are the Pauli matrices defined by the sublattices a and b. This model features a two-band dispersion relation, E±(k)=±\sqrt{J1^2+2J1(J2+J3)cosk+2J2J3cos2k}, where \Delta J^2=J1^2+J2^2+J3^2 and ± denote the upper and lower energy bands, respectively. Its phase diagram is illustrated in Fig. 1e in the space of lattice parameters in terms of topological winding number w. Due to the presence of long-range couplings, the system exhibits three distinct topological phases, characterized by w=0, 1, 2. To introduce a temporal boundary, we abruptly change the values of the couplings J at a specific time tb, thereby creating the boundary at the moment tb. Before the temporal boundary, the system is described by the initial Hamiltonian Hi with eigenstates |ψi±(k)⟩ and topological winding number wi. After the temporal boundary, the system evolves under a final Hamiltonian Hf, with eigenstates |ψf±(k)⟩ and topological winding number wf. Notably, one may initialize the system in the eigenstate |ψi±(k)⟩ and let it evolve under Hf, which lead to the same topological braiding discussed below. Nevertheless, the introduction of a temporal boundary enables us to establish a direct connection between temporal scattering dynamics and topological braiding, offering opportunities for manipulating wave scattering with topological protection in the time domain.

For the two-band energy dispersion at a given k, there exist two eigenstates propagating in opposite directions, due to their group velocities ∂E±(k)/∂k having opposite signs but the same amplitudes. To study the wave packet scattering at the temporal boundary, we suppose the system is initially prepared in the lower-band eigenstate at the moment tb (green dots in Figs. 3, 4a, g), i.e., |ψi-(k)⟩. For t>tb, the state evolves as

|ψ(k,t)⟩=e-iϕ-r-(k)e-iϕ- |ψf-(k)⟩+r+(k)e-iϕ+ |ψf+(k)⟩

where ϕ±=Ei-(k)tb+Ef±(k)(t-tb) is the dynamical phase, Ei-(k) and Ef±(k) are the eigenenergies of Hi and Hf, respectively (see Supplementary Note 1 for more details). As shown, the initial-state wave packet is scattered into two propagating wave packets, governed by the upper- and lower-band eigenstates of Hf (blue dots in Figs. 3, 4a, g), with group velocities having the same amplitude but opposite signs. The corresponding scattering coefficients are specifically defined as

r-(k)=⟨ψf-(k)|ψi-(k)⟩, (2)

r+(k)=⟨ψf+(k)|ψi-(k)⟩: (3)

Notably, the wave packet with the same sign of group velocity as the initial state is referred to as the time refraction wave, while the other one with the opposite sign is called the time reflection wave. Consequently, the |r±(k)|2 represent the probabilities of time refraction and reflection.

The topological braiding of r±(k) in the momentum space is investigated for four temporal scattering processes, labeled by (i)-(iv), as indicated by the arrows in Fig. 1e. The starting and ending points of the arrows represent the lattice parameters before and after the temporal boundary, each of which is associated with a well-defined topological winding number. We start with a long-range SSH lattice in which the influence of nearest-neighbor couplings dominates over the long-range couplings. For this initial case, the topological winding number of the lattice Hamiltonian is wi=0 before the boundary and wf=1 after. As shown in panel (i) of Fig. 1f, the real and imaginary parts for the time reflection and refraction coefficients r±(k) exhibit braiding in the Brillouin zone. Due to the periodic boundary conditions in the momentum space, the momenta at k=0 and k=2π are equivalent, the endpoints of each strand are identical and can be connected, which thus closes the braiding and forms a topological Hopf link (see the right inset for the braiding diagram). The topology of this link, characterized by the topological linking number v27, is determined by the topological winding numbers before and after the temporal boundary (see Supplementary Note 1 for more details), given by

v=wf-wi: (4)

Accordingly, we find the topological linking number v=1, indicating that a right-handed topological Hopf link emerges in the temporal scattering process (i).

In case (ii), we show that increasing the long-range couplings J3 can lead to more complex links. Specifically, J3 is increased to make wf=2 after the temporal boundary. As shown in panel (ii) of Fig. 1f, the resulting winding and braiding of r±(k) form a nontrivial topological Solomon link, with a topological linking number v=2. Notably, in general, introducing even longer-range couplings can further enrich the variety of topological links. Case (iii) highlights that the handedness of the links, i.e., the signs of the topological linking numbers, can also be controlled by tuning the values of wi,f. This case is specifically exhibited by evolving across a temporal boundary from wi=2 to wf=1. The corresponding braiding of r±(k) shown in panel (iii) of Fig. 1f is similar to that in case (i), but with opposite orientation, resulting in a left-handed topological Hopf link, with a topological linking number v=-1. Finally, in case (iv), we consider a temporal boundary with wi=wf=2. The winding of r+(k) and r-(k) do not braid with one another, resulting in a topologically trivial unlink, characterized by a topological linking number v=0.

Implementation of temporal dynamics in circuit metamaterials
We experimentally implement the temporal dynamics in circuit metamaterials. Significant recent advances in topolectrical circuits, realizing temporal dynamics governed by the Schrödinger equation remains challenging51–54. This challenge was recently addressed through a circuit design55; however, the preparation and measurement of complex-valued wavefunctions remain beyond reach. To overcome these limitations, we establish a circuit framework that employs two sets of circuit nodes to separately encode the real-time dynamics of the real and imaginary components of the wavefunction, as illustrated in Fig. 2a–c. This design enables the direct preparation and detection of complex-valued wavefunctions and their real-time dynamics. Specifically, odd circuit nodes in j(k) correspond to Re[ψan] (Im[ψan]), while even j(k) correspond to Re[ψbn] (Im[ψbn]). These nodes are interconnected via three types of a current-inversion negative impedance converters (INICs, see Supplementary Note 3 for more details), as indicated by the green, blue, and yellow arrows in Fig. 2b, which respectively implement the lattice couplings J1,2,3 and realize the long-range SSH model. The blue circuit nodes are grounded through both capacitor C0 and INICs, whereas the red circuit nodes are grounded through capacitor C0 and normal resistors. The corresponding circuit dynamics for the voltages are governed by Kirchhoff’s current law,

C0 dVj(t)/dt = Vj(t)/R0 + ∑k (Vk(t)-Vj(t))/Rjk, (5)

C0 dVk(t)/dt = Vk(t)/Rk + ∑j (Vj(t)-Vk(t))/Rjk, (6)

where Vj,k(t) represents the voltage at the nodes, and Rjk=±Rjk is the effective resistance at those nodes. By mapping these voltages into a vector, V(t)=[V1(t), V2(t), ..., V4n(t)]T, the above equation can be recast as the form of Schrödinger equation,

i dV(t)/dt = Hc V(t): (7)

Due to the separation of the real and imaginary components of the wavefunction, Hc = iH0 takes the antidiagonal form, where H is the long-range SSH model Hamiltonian H given in Eq. (1). The corresponding lattice couplings are given by J1=1/C0R0, J2=1/C0Rb and J3=1/C0Ry, where Rb=R1,2 and Ry=R3,4 denote the effective resistances of the blue and yellow INICs, respectively, as determined by the switch states within the INICs. For implementing the time evolution of an initial state |ψ(0)⟩=(ψ1, ψ2, ..., ψn)T, we excite the circuit with the initial voltage V(0)=[Re[ψ1], Im[ψ1], ..., Re[ψn], Im[ψn]]T. As, the circuit dynamics are governed by the Schrödinger equation, the time-evolved state is given by V(t)=e-iHtV(0). By assigning the encoding of the imaginary part of the wavefunction the value i, i.e., |ψ(t)⟩=(1, i)T I2n V(t)=e-iHt|ψ(0)⟩.

This procedure is benchmarked by measuring the time evolution of single-site excitation state |ψ0⟩=|0...,1,0,...,0⟩ under open boundary conditions, specifically with the excitation initialized at site b15. Such an initial state is experimentally prepared by exciting the circuit node j=30 using an external voltage source. Figure 2d specifically exhibits ψb15(t) and ψa17(t) under the dynamics e-iHt|ψ(0)⟩. Due to the chiral symmetry of the Hamiltonian and the fact that all coupling terms are real, the wavefunction satisfies ψ*(t)=ψ(-t), with ψan(t)=ψan(-t) and ψbn(t)=ψbn(-t) under the initial state considered above. As a result, the wavefunction at site b15 remains purely real throughout the evolution, while that at site a17 becomes purely imaginary. Measured results confirm this behavior, with both the measured real and imaginary parts of the wavefunctions showing excellent agreement with numerical simulations, thereby validating the reliability of our approach of implementing temporal dynamics in circuit metamaterials.

Furthermore, such dynamics allow us to probe the topological invariants of the long-range SSH lattice. To this end, we introduce the voltage mean chiral displacement as a measurable observable, which is defined as P(t)=1/n(ay a - by b). Experimentally, a voltage is injected at a bulk circuit node j=30, and subsequently the time evolution of such an initial state is monitored for probing the dynamics of the SSH lattice in three distinct topological phases shown in Fig. 1e, characterized by the topological winding number w=0, 1, and 2. As shown, the measured results agree well with theoretical predictions. The three curves clearly oscillate around distinct center values, qualitatively manifesting the signature of different topological winding numbers. Due to finite evolution time, the measured oscillation centers are 0.09, 0.59 and 1.18, respectively, agree well with the simulated values 0.01, 0.49, and 1.01, giving the measured topological winding numbers w=0.18, 1.18, and 2.36. Nevertheless, the clear distinction between the measured nontrivial and trivial topological winding numbers enables an unambiguous identification of the three topological phases.

Observation of topological braiding of time reflection and refraction
We now experimentally demonstrate the topological braiding of time reflection and refraction at the temporal boundaries. In our experiment, to implement temporal modulation, we introduce electrically controlled circuit switches S1,2,3,4 within the blue and yellow INICs and the red grounding nodes (see Fig. 2b). These switches abruptly reconfigure the resistances R at a specific time tb=20 ms, inducing sharp temporal variations in the coupling strengths J2,3 and thereby creating a temporal boundary at this time point (see Supplementary Note 4 for more details). Across the temporal boundary, the topological winding number changes from wi to wf, with their values determined by J2,3. Since our experimental detection is conducted in the real space, we need to perform a Fourier transformation to map the lower-energy Bloch state |ψi-(k)⟩ from momentum space to real space, and then apply a Gaussian modulation to localize the excitation. The Gaussian envelope ensures that the wave packet is well-localized in real space while simultaneously possesses a narrow momentum distribution, thereby enabling a clear visualization of reflection and refraction dynamics at the temporal boundary. Specifically, the wave packet in the n-th unit cell is therefore initialized as |ψi-(k)⟩n=(ψan, ψbn)T=1/\sqrt{4πσ2} e-(n-n0)2/2σ2 eikn |ψi-(k)⟩, where ψan and ψbn represent the amplitudes at sites an and bn, respectively, n0 denotes the central unit cell of the initial wave packet, σ is its width, and k is the wave packet momentum. The implemented long-range SSH lattice in this case is subject to periodic boundary conditions to enable access to momentum-space features. The dynamics of this wave packet across the temporal boundary is presented in Figs. 3 and 4 for the four temporal scattering processes in Fig. 1f. Here, σ=6.67, and n0 is chosen such that the wave packet remains centered within the lattice at tb.

Figure 3a presents the simulated and measured band structures before and after the temporal boundary (see “Method” for more details), across which the topological winding number changes from wi=0 to wf=1, corresponding to the case (i) in Fig. 1f. The green dot indicates the initial lower-band eigenstate at a specific momentum, while the blue dots represent the two scattered states after the temporal boundary, correspondingly leading to the time reflection and refraction wave packets. For k=8π/15, 2π/3, 4π/5 (see Supplementary Note 7 for the results of other momenta), the initial wave packet has a positive (negative) group velocity and thus propagates along the positive (negative) direction. After crossing the temporal boundary, it splits into two components: one that retains the original group velocity sign (time refraction) and one with the opposite sign (time reflection). In particular, the time refraction and reflection probabilities of the two components are almost equal (see Supplementary Note 2 for more details). These observations agree with the measured and simulated time reflection and refraction probabilities |r±(k)|2 in Fig. 3e. Notably, |r-|2=0 and |r+|2=1 at k=π, which reveals that a band inversion occurs at this point. This feature allows us to identify that there exists a topological phase transition between Hi and Hf. The real and imaginary parts of the time reflection and refraction coefficients, r±(k), are shown in detail in Fig. 3(f) (see “Method” for more details). As illustrated, r+(k) and r-(k) braid around each other to form a right-handed topological Hopf link, with a topological linking number v=1. Similarly, Fig. 3g–l exhibit the temporal scattering process within which wi=2 and wf=1, where a left-handed Hopf link emerges, characterized by a topological linking number v=-1. This result underscores the capability of tuning the handedness of link structure. All experimental measurements are in good agreement with theoretical simulations.

Figure 4 further demonstrates that the topological structures of the resulting links can become significantly richer or trivial through the varying of wi,f. Specifically, the band structures in Fig. 4a examine the case wi=0 and wf=2. The corresponding wave packet dynamics across the temporal boundary are shown in Fig. 4b–d for three representative momenta k=π/3, 7π/15, 8π/15. By experimentally scanning over all momenta for such dynamics, we measure out the time reflection and refraction coefficients r±(k) in the full momentum space. As shown in Fig. 4e, |r-|2=0 and |r+|2=1 at k≈0.58π and k≈1.42π, indicating the occurrence of band inversion at these points, signaling a topological phase transition from wi=0 and wf=2. The winding and braiding of r±(k) presented in Fig. 4f, forms a complex topological Solomon link, with a topological linking number v=2. In contrast, the links can also be trivial by setting wi=wf. Figure 4g–l experimentally study the case wi=wf=2. As shown, the characteristic features |r-|2=0 and |r+|2=1 do not appear, consistent with the absence of a topological phase transition here. The winding of r±(k) in this case does not braid but instead form a trivial unlink with a topological linking number v=0.

Observation of temporal boundary induced dynamical topological phase transitions
We further find that dynamical phase transitions can emerge in temporal scattering processes driven by explicitly time-dependent interactions in time-varying metamaterials. Our work parallels earlier studies where such transitions were primarily identified in the real-time dynamics of quantum many-body spin chains or systems exhibiting static topological phases57–61. Unlike conventional equilibrium phase transitions, dynamical criticality and dynamical phase transitions manifest as nonanalytic behavior in dynamical observables at specific critical times57–61, such as the real-time dynamics of Loschmidt amplitudes. Notably, in previously reported experiments62,63, dynamical phase transitions were indirectly detected from the measured time evolution of density distributions, since directly probing both the real and imaginary components of the wavefunction and its real-time dynamics remained challenging there. In our experiment, by contrast, the real-time dynamics are emulated using a dual-lattice circuit structure, which enables direct measurement of both the real and imaginary parts of the wavefunction at each lattice site. As shown below, this direct access provides a real-time view of topology-related dynamical phenomena, including the dynamics of Loschmidt amplitudes, rate functions, and dynamical topological invariants.

Specifically, the Loschmidt amplitude in the temporal scattering process is defined as G(t)=∑k G(k,t), which characterizes the overlap between the states before and after the topological temporal boundary, where the k-resolved Loschmidt amplitude G(k,t)=⟨ψi-(k)|ψ(k,t)⟩ captures the contribution from each momentum mode. Interestingly, G(k,t) is connected to the time reflection and refraction coefficients, G(k,t)=|r+(k)|2e-iϕ+ +|r-(k)|2e-iϕ- (8)

Furthermore, we define the rate function g(t)=-N-1log|G(t)|2, where N is the lattice size. This quantity serves as a dynamical counterpart to free energy in the framework of equilibrium phase transition. It turns out that, when the topological invariants differ across the temporal boundary, the time reflection and refraction coefficients would cross at the critical momentum kc (see in Supplementary Note 6 for more details), i.e., |r+(kc)|2 = |r-(kc)|2 = 0.5 (as manifested in Figs. 3 and 4 (e, k)), which renders G(kc,t)=0 at the critical times tc,m=tb + (Ef+(kc)-Ef-(kc))-1 π(2m+1), where m∈Z. This vanishing leads to a nonanalyticities in g(t), hallmarking the emergence of dynamical criticality and dynamical phase transition. Remarkably, such a nonequilibrium phase transition can be characterized by a dynamical topological invariant

vd(t)=1/2π ∫0π dk ∂k ϕG(k,t), (9)

where ϕG(k,t) represents the Pancharatnam geometrical phase accumulated during the temporal scattering process (see “Method” for more details). Notably, the nonanalyticities of g(t) at the critical times tc are accompanied by the jumps of vd(t) by 1, revealing a deep connection between dynamical criticality and topological invariant.

Figure 5a shows the measured and simulated rate functions g(t) for the temporal scattering case with v=1 (Fig. 3a). Both curves exhibit a peak at the critical time point tc≈23.4 ms, reflecting the underlying nonanalyticity and manifesting the dynamical criticality. The deviations between the measured and simulated results arise from practical experimental imperfections. Direct identification of the nonanalyticity remains challenging, as it theoretically requires a larger lattice size to access the critical momentum kc, and experimentally demands nearly perfect time evolution. However, as shown in Fig. 5b, the jump in the dynamical topological invariant vd(t) at tc provides a clear and robust signature of the nonanalyticity and allows unambiguous identification of the dynamical phase transition. In this case, the measured and simulated vd(t) show good agreement, owing to the topological protection that endows vd with robustness against imperfections. The insets illustrate the topological winding of the Pancharatnam geometric phase ϕG at two selected time points, corresponding to vd=0 and vd=1, respectively. As shown, the sign of vd is clearly manifested by the winding direction.

Figure 5c investigates the dynamical phase transition in the temporal scattering case v=2 (Fig. 4a), where the rate functions take nonanalyticities at two closely spaced critical time points t1c≈23.02 ms and t2c≈23.48 ms. As shown in Fig. 5d, although the peaks for g(t) at t2c are less pronounced due to the finite size effect, the dynamical topological invariant vd displays distinct jump at this critical time. In contrast to Fig. 5b, vd in this case firstly jumps to -1 at t1c, and then to 0 at t2c, confirming the change of v=2 (Fig. 3a), where the rate functions curve resemble that of the v=1 case, exhibiting a peak at a different critical time tc≈23.42 ms. For comparison, we also examine the trivial case v=0 (Fig. 4g) in Fig. 5g, in which the rate function evolves smoothly without singular features, and vd(t) remains zero throughout the evolution, indicating the absence of a dynamical topological phase transition. Taken together, the four cases demonstrate that there emerges dynamical criticality within the temporal scattering processes.

Methods
Sample fabrications and circuit temporal boundary
Our electric circuits, including the PCB composition, stack-up layout, internal layer and grounding settings, are designed through PAD program software (see Supplementary Note 5 for more details). To implement the ideal INIC realized long-range SSH model coupling, we employed the LT1363 operational amplifier, which features high speed and a high slew rate. The operational amplifiers were powered by ±5V supply voltages. Two chip resistors Ra=10 kΩ were used as auxiliary components in the positive and negative feedback loops of the operational amplifier.

To create a temporal boundary in the system, we employed high-speed CMOS analog switches (model COS12A12511) to abruptly change the system parameters. These switches, powered by ±5V supplies, exhibit a typical switching time of less than 70 ns, so the switching can be regarded as instantaneous compared to the system’s millisecond-scale (~ ms) dynamics. The analog switches are digitally triggered and can be programmed to activate at precise time points, enabling controlled manipulation of the circuit’s temporal boundary. To initialize the circuit before each experimental run, the grounding capacitance C0 of each node was momentarily connected to predefined voltage sources via CMOS analog switches to set the initial conditions. The voltage dynamics of all nodes are then monitored using a Rigol digital oscilloscope with multi-channel input and high temporal resolution. This setup enables time-resolved measurements of the node voltages, thereby allowing us to extract dynamical features.

Dynamical phase transitions and topological invariants
The formation of a dynamical phase transition after the temporal boundary is indicated by the emergence of Fisher zeros in the real-time dynamics of the Loschmidt amplitudes. These zeros occur at critical momenta and time points, kc and tc, where G(t)=∑k G(k,t)=0 (see the discussion around Eq. (6) for a detailed derivation), leading to nonanalyticities in the rate functions g(t). Such nonanalytic behavior serves as a clear hallmark of dynamical phase transitions and provides experimentally accessible signatures of dynamical criticality. Below, we further show that such a dynamical phase transition features a topological origin. The Loschmidt amplitude G(k,t) can be expressed in polar form as

G(k,t)=r(k,t)eiϕ(k,t), (10)

where r(k,t) denotes the amplitude and ϕ(k,t) is the total phase. The Pancharatnam geometric phase, which characterizes the geometric part of ϕ(k,t), is obtained by subtracting the dynamical phase,

ϕG(k,t)=ϕ(k,t)-ϕdyn(k,t): (11)

The dynamical phase ϕdyn is given by

ϕdyn(k,t)=-∫0tb dt ⟨ψ(k,t)|H|ψ(k,t)⟩-∫tb t dt ⟨ψ(k,t)|Hf|ψ(k,t)⟩,

which can be compactly written as

ϕdyn(k,t)=Ef+(t-tb)(|r-|2-|r+|2)+Ei+tb: (12)

At the high-symmetry momenta k=0 and k=π, one of the time-scattering coefficients (r+ or r-) vanishes, forcing the Pancharatnam geometric phase to be pinned at zero. Consequently, the momentum interval [0, π] can be topologically identified as a closed loop by connecting its endpoints. Within this structure, the dynamical topological invariant is defined as vd(t)=1/2π ∫0π dk ∂k ϕG(k,t), which quantifies the topological winding of the Pancharatnam geometric phase over [0, π]. The dynamical topological invariant exhibits quantized jumps at critical times tc,m, thereby providing a topological characterization of nonequilibrium criticality associated with dynamical phase transitions.

Experimental measurement methods
The procedures for measuring the energy band structures, the temporal scattering coefficients, the rate functions and dynamical topological invariants are outlined as follows. Based on the experimentally measured time-evolved wavefunctions, we directly extract the Loschmidt amplitudes G(k,t). Subsequently, a Fourier transformation is applied to the Loschmidt amplitudes, where the peak positions in the resulting spectra correspond to the energy eigenvalues E(k). Since the initial state is precisely the lower-energy eigenstate of Hi, the Fourier spectrum of G(k,t<tb) exhibits a single peak corresponding to Ei-(k). In contrast, performing the Fourier transform over the time window t>tb, the spectrum displays two peaks which yield Ef+(k) and Ef-(k). By scanning over k and repeating the above procedure, we reconstruct the energy band structures of both Hi and Hf, and obtain the rate function g(t). Using the measured band energies, we can further measure the dynamical phase ϕdyn(k,t). Subtracting this from the total phase of G(k,t) yields the Pancharatnam geometric phase ϕG(k,t), which allows us to detect the dynamical topological invariant vd(t). To extract the time reflection and refraction coefficients, we project the measured time-evolved state after the temporal boundary onto the eigenstates of the final Hamiltonian Hf, ⟨ψf±(k)|ψ(k,t)⟩=e-iEi-(k)tbe-iEf±(k)(t-tb)r±(k). By removing the measured dynamical phases, the coefficients r±(k) are directly obtained. Scanning over the momentum k enables us to reconstruct the momentum-resolved trajectories of r±(k), thereby clearly visualizing the braiding structure of time reflection and refraction in momentum space.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "This manuscript introduces a novel and exciting approach to implementing Schrödinger dynamics in circuit metamaterials.\nBased on this, the authors present compelling experimental evidence for topological braiding and dynamical phase\ntransitions in time reflection and refraction. The concept of topological braiding was proposed very recently and has never\nbefore been experimentally demonstrated, making this work a significant contribution to the field. The experimental work is\nclearly well-executed, and the results are promising. Before I make my final recommendation, I would encourage the authors\nto address a few points to further strengthen the manuscript.\n1. The theoretical work (Ref. 55) that proposes one of the core physics of this work\n(topological braiding in time reflection and refraction) should be well introduced in the introduction; In addition, the dynamical phase transition, another core\nphysics, has been theoretically and experimentally investigated in other physical systems [Ref. 56-59, and Phys. Rev. B\n100, 024310 (2019); Phys. Rev. Lett. 124, 043001 (2020); Phys. Rev. Lett. 126, 200602 (2021)]. The author should point out\nthe theoretical and experimental progress of the circuit experiment compared to previous research. It is necessary for people\nto understand the research background and accurately evaluate the contribution of this work.\n2. In the introduction, the authors state that ‘However, realizing Schrodinger dynamics in circuit metamaterials remains a\nsignificant experimental challenge’. It seems that topological circuits are also governed by the Schrodinger-like equation\n(Ref. [51-54]) and the temporal dynamics can be found in the work: Nat Commun 16, 198 (2025). In contrast to these works\nthat do not need additional nodes, the implementation in this work uses additional circuit nodes to separately encode the\nreal and imaginary parts of the wavefunctions, which is obviously more complex than others. Could the authors explain the\nadvantage of this new setup compared with the formers?\n3. On page 5, the authors state: ‘The initial state |ψi−(k)⟩ is prepared by making a Fourier transformation and modulating it\nwith a Gaussian envelope, such that…’. Do the authors mean that initial state is prepared by making a Fourier transformation\nto the momentum-space wavefunction? This should be clarified.\n4. In Fig. 4f, it seems that some experimental data (green dots) are lost around k=1.5pi, compared with the data in Fig. 4e.\n5. Fig. 1c is not mentioned in the main text.", "reviewer_2": "The authors experimentally demonstrate that temporal scattering processes exhibit two distinct topological features:\neigenstate topological braiding and dynamical topological phase transitions. Expanding on their earlier theoretical work\n(Ref. [50]), they implement various temporal scattering processes in circuit metamaterials and observe diverse topological\nbraiding and linking structures, establishing their utility as probes of bulk topology. Notably, the linking numbers of these\nbraiding structures are determined by the difference in topological winding numbers across the temporal boundary. Given\nthe novelty and soundness of the experimental results, I’d like to recommend this work for publication if the authors fully\naddress the following comments:\n1.Is the temporal boundary necessary? I mean, the authors could simply initialize the system H_f in the eigenstate\n|\\psi^{i}\\rangle of H_i and observe its evolution under H_f to determine r_{+} and r_{-} via Eq.(2). It is unclear why introducing\na topological system first, followed by a temporal boundary to scatter the wavefunction, is necessary for this analysis.\n2.Is necessary for H_f to have two eigenstates (i.e., two bands)? If H_f is instead a single-site lattice system (with only one\nband), forward and backward waves may still exist at the temporal boundary. However, would the resulting dynamics\nreproduce the key findings shown in Fig. 1?\n3.Since knots are a primary braiding structure, is it possible to have knot-like structures in this system? If so, could the\nauthors discuss their potential formations? I suspect they might emerge when H_i is chosen to be a gapless system, though I\nam uncertain.\n4.Dynamical topological phase transitions are really interesting, as they demonstrate a potential method to identify the\ntopological difference between H_i and H_f through temporal scattering. I have two questions: (1) The dynamical topological\ninvariant in Eq. (7) appears to take multiple values (e.g, |v_d|>1). Could the authors show some examples? (2) In Fig. 5(e,f),\nthe dynamics for the braiding case with v=−1 closely resemble those for the braiding with v=1. Why do these topologically\ndistinct braiding structures share nearly identical rate functions and dynamical topological invariants?\n5.Could this framework be generalized to four-strand systems (because of chiral symmetry) for realizing non-Abelian\nbraiding? For instance, by implementing a four-band Hamiltonian as H_f, one would obtain four eigenstates in Eq.(2).\nHowever, the connection between this non-Abelian braiding topology and the Abelian (integer-valued) topological invariants\n(e.g., winding number) remains unclear. This may be beyond the scope of the current work, but I would appreciate it if the\nauthors could give some discussions about how these distinct topological properties might be reconciled within their\ntheoretical framework.\nMinor comments:\n1.It looks Fig.1(c) does not appear to be referred in the main text. Please refer it in the proper place.\n2.The supplementary material is quite informative. I suggest the authors consider integrating some of its key parts into the\nMethods section of the main text. Currently, the Methods section only focuses on sample fabrication and temporal boundary\nrealizations. Important details, such as the formation of dynamic topological phase transition, the mathematical definition of\nthe Pancharatnam geometrical phase in Eq. (7), and measurement scheme, would be more appropriately placed there."}, "author_response": "Our reply: We sincerely thank the reviewer for evaluating our experimental method of\nimplementing Schrödinger dynamics in circuit metamaterials as a “novel and exciting\napproach”.\nOur reply: We greatly thank the reviewer for the careful evaluation of our manuscript\nand the accurate summary of our main results, particularly highlighting the compelling\nexperimental evidence for topological braiding and dynamical phase transitions in time\nreflection and refraction. We sincerely appreciate the reviewer for assessing our work\nas “a significant contribution to the field”. We are also grateful for the reviewer’s\npositive comments on the quality of our experimental work and the promising nature\nof the results.\nOur reply: We greatly thank the reviewer’s expertise and suggestions on helping us to\nfurther strengthen our manuscript. We have carefully revised the manuscript point-by-\npoint, and have provided additional clarifications, analyses, and discussions to further\nstrengthen the paper.\n1. The theoretical work (Ref. 55) that proposes one of the core physics of this work\n(topological braiding in time reflection and refraction) should be well introduced in the\nintroduction;\nOur reply: We greatly thank the reviewer for raising this point. In the second paragraph\nof the Introduction section, we have added further text to better introduce the theoretical\nwork on topological braiding in time reflection and refraction.\nIn addition, the dynamical phase transition, another core physics, has been theoretically\nand experimentally investigated in other physical systems [Ref. 56-59, and Phys. Rev.\nB 100, 024310 (2019); Phys. Rev. Lett. 124, 043001 (2020); Phys. Rev. Lett. 126,\n200602 (2021)]. The author should point out the theoretical and experimental progress\nof the circuit experiment compared to previous research. It is necessary for people to\nunderstand the research background and accurately evaluate the contribution of this\nwork.\nOur reply: We greatly thank the reviewer for raising this point.\nIn previous theoretical studies (Refs. 56-59 and Phys. Rev. Lett. 126, 200602\n(2021)), dynamical phase transitions were primarily identified in the real-time\ndynamics of quantum many-body spin chains or systems exhibiting static topological\nphases. In contrast, our work shows that dynamical phase transitions can also emerge\nin temporal scattering processes driven by explicitly time-dependent interactions in\ntime-varying metamaterials.\nIn previous experimental studies (Phys. Rev. B 100, 024310 (2019); Phys. Rev.\nLett. 124, 043001 (2020)), it was a challenge to directly measure both the real and\nimaginary components of the wavefunction and its real-time dynamics. Consequently,\ndynamical phase transitions were inferred indirectly from the time evolution of density\ndistributions. In our experiment, by contrast, the real-time dynamics are emulated using\na dual-lattice circuit structure (as discussed in response to the reviewer’s comment),\nwhich enables direct measurement of both the real and imaginary parts of the\nwavefunction at each lattice site. This direct access provides a real-time view of\ntopology-related dynamical phenomena, including the dynamics of Loschmidt\namplitudes, rate functions, and dynamical topological invariants.\nIn the revised manuscript, we have updated the discussion in the first paragraph of\npage 8 on dynamical topological phase transitions. We have added texts to explicitly\ncompare our theoretical framework and experimental achievements with previous\ntheoretical and experimental studies. We have also cited Phys. Rev. B 100, 024310\n(2019), Phys. Rev. Lett. 124, 043001 (2020), and Phys. Rev. Lett. 126, 200602 (2021)\nas the references [61-63].\n2. In the introduction, the authors state that ‘However, realizing Schrodinger dynamics\nin circuit metamaterials remains a significant experimental challenge’. It seems that\ntopological circuits are also governed by the Schrodinger-like equation (Ref. [51-54])\nand the temporal dynamics can be found in the work: Nat Commun 16, 198 (2025). In\ncontrast to these works that do not need additional nodes, the implementation in this\nwork uses additional circuit nodes to separately encode the real and imaginary parts of\nthe wavefunctions, which is obviously more complex than others. Could the authors\nexplain the advantage of this new setup compared with the formers?\nOur reply: We greatly thank the reviewer for raising this point. Below, we provide an\nexplanation to clarify the novelty of our experimental setup in comparison with the\nprevious works.\nIn these previous studies (Refs. [51–54]), the observed dynamics originated from\nexternally driven responses rather than intrinsic time evolution under a Hamiltonian.\nThe applied time-dependent voltage pulses 𝑉(𝑡) = 𝑒𝑥𝑝[−(𝑡−𝑡 )2/𝑡2]𝑠𝑖𝑛(2𝜋𝑓𝑡),\n0 𝑑 𝑐\ncorrespond to the Gaussian wave packets with a central frequency 𝑓 and a bandwidth\n𝑐\nof 1/𝑡 , acting as external sources that generate forced oscillations governed by\n𝑑\nKirchhoff’s laws. Consequently, the measured voltages represent the resonant\nresponses of driven modes rather than temporal evolution of a prepared state. This is\nsimilar to see the resonant response of putting photons on the edges of a topological\nphotonic lattice and observe its subsequent transport, which is not exactly\ncorresponding to the real-time dynamics governed by the Schrödinger equation.\nThe previous study in Nat. Commun. 16, 198 (2025) realized a circuit lattice\ncapable of emulating Schrödinger dynamics. However, it did not address the challenge\nof directly preparing and measuring complex-valued wavefunctions, including both\ntheir real and imaginary components. That work was limited to preparing and detecting\nonly real-valued wavefunctions (voltages), such as 𝑉(𝑥) = (±1,0,0,0,…,0).\nIn our study, we overcome this limitation by establishing a new framework that\nemploys two sets of circuits to explicitly and independently encode the real-time\ndynamics of the real and imaginary parts of the wavefunction. This circuit design not\nonly enables the implementation of the Schrödinger dynamics, but also allows direct\npreparation of arbitrary initial-state wavefunctions and real-time detection of their\nevolution, including both the real and imaginary components.\nIn the revised manuscript, we have added explanatory text in the first paragraph of\npage 4 to highlight these distinctions and to better convey the novelty of our work.\n3. On page 5, the authors state: ‘The initial state |ψi−(k)⟩ is prepared by making a\nFourier transformation and modulating it with a Gaussian envelope, such that…’. Do\nthe authors mean that initial state is prepared by making a Fourier transformation to the\nmomentum-space wavefunction? This should be clarified.\nOur reply: We greatly thank the reviewer for raising this point. Since our experimental\ndetection is conducted in the real space, we need to perform a Fourier transformation\nto convert the lower-energy Bloch state |𝜓𝑖(𝑘)⟩ (i.e., the initial state) from the\n−\nmomentum space to real space. After the transformation, the wavefunction for the 𝑛-\nth unit cell is expressed as |𝜓 (𝑘)⟩ = 1 𝑒𝑖𝑘𝑛|𝜓𝑖 (𝑘)⟩. As shown, both the initial and\n𝑖 𝑛 −\n√𝑁\nits temporally scattered states are superpositions of plane waves in real space, making\nit difficult to directly visualize the scattering dynamics of a localized wave packet, as\nillustrated in Fig. 1(a). To address this, we apply a Gaussian envelope to the real-space\nwavefunction: |𝜓 (𝑘)⟩ = 1 𝑒−(𝑛−𝑛0)2/2𝜎2 𝑒𝑖𝑘𝑛|𝜓𝑖(𝑘)⟩ , which confines the\n𝑖 𝑛 4 √𝜋𝜎2 −\nwavefunction around the central unit cell 𝑛 , ensures a well-localized wave packet and\n0\nfacilitates its direct experimental visualization. This modulation introduces a\nmomentum linewidth in the corresponding state in the momentum space, |𝜓⟩ =\nσ 2 𝑞2\n∫𝑑𝑞𝐶̃(𝑞)|𝜓 − 𝑖(𝑘+𝑞)⟩, where the amplitude 𝐶̃(𝑞) ∝ 𝑒 − 2 𝑒−i𝑞𝑛0 , 𝑘 denotes the\ncenter momentum and Δ𝑞 ≈ 1/σ represents the momentum linewidth. When 𝜎→∞,\nthe Gaussian modulation disappears, and the wave packet becomes a plane wave\ncorresponding to a single momentum eigenstate. In our experiment, we choose 𝜎=6.67\nto balance spatial localization with momentum resolution, ensures that the resulting\nwave packet is well-localized in real space while simultaneously possessing a narrow\nmomentum distribution, thereby enabling a clear observation of the reflection and\nrefraction of Gaussian wave packets with a central momentum k at the temporal\nboundary.\nIn the revised manuscript, we have revised the corresponding sentence on page 5\nto clarify this point.\n4. In Fig. 4f, it seems that some experimental data (green dots) are lost around k=1.5pi,\ncompared with the data in Fig. 4e.\nOur reply: We greatly thank the reviewer for raising this point. In the revised manuscript,\nthe incorrect Fig. 4(f) has now been replaced to fix this point.\n5. Fig. 1c is not mentioned in the main text.\nOur reply: We greatly thank the reviewer for pointing this out. Figure 1(c) has now been\nappropriately referenced in the last paragraph of the Introduction section in the revised\nmanuscript.\nOur reply: We greatly thank the reviewer for the careful reading and accurate summary\nof our experimental results. We also greatly appreciate the reviewer’s recognition of the\ntwo distinct topological features observed in our study, namely eigenstate topological\nbraiding and dynamical topological phase transitions, as well as the remark on how the\nlinking numbers are determined by the difference in topological winding numbers\nacross the temporal boundary.\nOur reply: We greatly thank the reviewer for the recognition of the novelty and\nsoundness of our results, as well as for the encouraging recommendation for publication.\nWe also greatly appreciate the reviewer’s insightful questions regarding our results on\ntopological braiding. As detailed below, we provide point-by-point responses\naddressing all of the reviewer’s comments.\n1.Is the temporal boundary necessary? I mean, the authors could simply initialize the\nsystem H_f in the eigenstate |\\psi^{i}\\rangle of H_i and observe its evolution under H_f\nto determine r_{+} and r_{-} via Eq.(2). It is unclear why introducing a topological\nsystem first, followed by a temporal boundary to scatter the wavefunction, is necessary\nfor this analysis.\nOur reply: We greatly thank the reviewer for this insightful comment. From a theoretical\nperspective, it is not necessary to introduce a real temporal boundary as pointed out by\nthe reviewer; one can simply initialize the system in the eigenstate of 𝐻 (|𝜓𝑖 (𝑘)⟩)\n𝑖 ±\nand let it evolve under 𝐻 , where 𝑡 = 0 effectively serves as the temporal boundary,\n𝑓\nleading to the same topological braiding physics. Although this approach is\nconceptually simpler, the realization of a real temporal boundary is meaningful from an\nexperimental perspective. This offers new opportunities for manipulating waves with\ntopological protection in the time domain. Thus, we introduce a real temporal boundary\nand explore the topological feature from genuine temporal scattering dynamics.\nIn the revised manuscript, we have added text below Eq. (1) to clarify this point.\n2.Is necessary for H_f to have two eigenstates (i.e., two bands)? If H_f is instead a\nsingle-site lattice system (with only one band), forward and backward waves may still\nexist at the temporal boundary. However, would the resulting dynamics reproduce the\nkey findings shown in Fig. 1?\nOur reply: We greatly thank the reviewer for raising these insightful questions. It is\nunnecessary to require 𝐻 having two eigenstates. The presence of two scattering\n𝑓\nchannels is strictly necessary to realize the non-trivial topological links shown in Fig.\n1. This necessity arises from the topological nature of the braiding itself. We further\nexplain this point by discussing two cases for a single-band 𝐻 .\n𝑓\nFirst, if both 𝐻 and 𝐻 are single-site lattice systems, there is only one available\n𝑖 𝑓\nscattering channel in 𝐻 for any given momentum 𝑘 . Consequently, the time-\n𝑓\nscattering process will only yield a single scattering wave, and the non-trivial\ntopological links cannot be formed because they require the linking of at least two\nindependent closed curves.\nSecond, if 𝐻 is a two-band system (with two sites per unit cell), the single-site\n𝑖\nlattice system 𝐻 can also be viewed as a two-band system where two band is gapless\n𝑓\nwith band-folding. Then, for any given momentum 𝑘 in 𝐻 , there are two available\n𝑖\nscattering channels in 𝐻 , yielding the time-refracted and time-reflected waves with\n𝑓\ntwo distinct scattering coefficients 𝑟 and 𝑟 . Notably, the braiding of 𝑟 and 𝑟\n+ − + −\nstill extract the difference in Zak phase across the temporal interface as the proof of\nbraiding of time refraction and time reflection is not specified to be valid only in gapped\nsystem. This is further explained from the presence of knot-like structures in the next\nresponse.\nIn the first paragraph of revised Supplementary Note 8, we have explicitly\nexplained the role of scattering channels and the conditions required for realizing non-\ntrivial temporal braiding.\n3.Since knots are a primary braiding structure, is it possible to have knot-like structures\nin this system? If so, could the authors discuss their potential formations? I suspect they\nmight emerge when H_i is chosen to be a gapless system, though I am uncertain.\nOur reply: We greatly thank the reviewer for this thought-provoking question. The\nknot-like structures are possible when 𝐻 or 𝐻 is chosen to be a gapless system, as\n𝑖 𝑓\npointed out by the reviewer. For example, 𝐻 :𝐽 /𝐽 = 1,𝐽 /𝐽 = 2; 𝐻 :𝐽 /𝐽 =\n𝑖 2 1 3 1 𝑓 2 1\n1,𝐽 /𝐽 = 0. The system undergoes the topological phase transition at the band gap\n3 1\nclosing. Correspondingly, the topological invariant is quantized to a half-integer\nvalue. This condition of a half-integer invariant induces a critical distortion in the\nbraiding of 𝑟 and 𝑟 , which transforms the link structure (two separate closed\n+ −\ncurves) into a single, self-entangled closed curve, i.e., a knot.\nIn Supplementary Note 8, we have added a detailed discussion on the emergence\nof topological knot structures, including their connection to band degeneracy and the\nassociated phase evolution.\n4.Dynamical topological phase transitions are really interesting, as they demonstrate a\npotential method to identify the topological difference between H_i and H_f through\ntemporal scattering. I have two questions:\nOur reply: We greatly thank the reviewer for the positive comment on our results of\ndynamical topological phase transitions.\n(1) The dynamical topological invariant in Eq. (7) appears to take multiple values (e.g,\n|v_d|>1). Could the authors show some examples?\nOur reply: We greatly thank the reviewer for raising this point. In Supplementary Note\n6, we have added numerical simulations showing that the dynamical topological\ninvariant can take multiple integer values as the evolution time increases. However, the\nevolution time cannot be extended too long due to experimental imperfections, which\nrestricts the observable invariant to |𝑣 | ≤ 1. This clarification has been included in\n𝑑\nSupplementary Note 6.\n(2) In Fig. 5(e,f), the dynamics for the braiding case with v=−1 closely resemble those\nfor the braiding with v=1. Why do these topologically distinct braiding structures share\nnearly identical rate functions and dynamical topological invariants?\nOur reply: We greatly thank the reviewer for raising this point. Indeed, the dynamics\nfor the cases with 𝑣 = −1 and 𝑣 = 1 are different. The similarity between the results\nin Fig. 5(e,f), including both the rate functions and the dynamical topological invariants,\nis due to the specific choice of system parameters. In our practical experiments, because\nthe available commercial capacitive and resistive components provide only limited\nparameter values, the parameters associated with 𝐻 result in an accidental situation\n𝑖\nwhere the difference between the outcomes for 𝑣 = −1 and 𝑣 = 1 becomes very\nsmall. Below, we clarify this point by analyzing the analytical expressions of the critical\ntime points in the rate functions and the characteristics of the dynamical topological\ninvariants.\nThe rate function manifests a nonanalytic behavior at the critical time points:\n(2𝑚+1)𝜋\n𝑡 = 𝑡 + , where 𝑘 denotes the critical momentum satisfying\n𝑐,𝑚 𝑏 𝑓 𝑓 𝑐\n𝐸\n+\n(𝑘𝑐)−𝐸\n−\n(𝑘𝑐)\n|𝑟 (𝑘 )|2 = |𝑟 (𝑘 )|2 = 0.5 . The values of 𝑡 are therefore determined by the\n+ 𝑐 − 𝑐 𝑐,𝑚\nenergy difference of 𝐻 at 𝑘 . In principle, different parameter configurations\n𝑓 𝑐\ngenerally lead to different 𝑘 values and different 𝑡 . However, due to the\n𝑐 𝑐,𝑚\ncoincidental choice of parameters in our case, the difference of the critical momenta\nbetween the case with 𝑣 = −1 and 𝑣 = 1 is only Δ𝑘 ≈ 0.006𝜋, resulting in nearly\n𝑐\nidentical critical times and rate functions.\nThe dynamical topological invariant 𝑣 (𝑡) is governed by the evolution of the\n𝑑\nPancharatnam geometric phase 𝜙 (𝑘,𝑡), but it cannot distinguish different types of\n𝐺\ntopological phase transitions. The direction of this phase winding depends on how the\ntwo amplitudes |𝑟 (𝑘)|2 and |𝑟 (𝑘)|2 interchange during evolution: when |𝑟 (𝑘)|2\n+ − +\nincreases while |𝑟 (𝑘)|2 decreases around the critical momentum 𝑘 , the invariant\n− 𝑐\nchanges by +1, whereas the reverse process leads to a change of −1. Thus, although\nthe two cases (𝑣 = 1 and 𝑣 = −1) possess opposite braiding chiralities in momentum\nspace, they yield the same dynamical topological invariant due to the similar 𝑘 -\ndependence of the amplitudes |𝑟 (𝑘)|2. Moreover, as mentioned above, because the\n±\ncritical times are nearly identical under our experimental conditions, the jumps for the\ntwo cases take place at nearly the same critical time points.\nTo further verify this point, we have included a comparison in which the rate\nfunctions and dynamical topological invariants exhibit pronounced differences under\nother parameter configurations, as shown in Fig. S5 of Supplementary Note 6.\n5.Could this framework be generalized to four-strand systems (because of chiral\nsymmetry) for realizing non-Abelian braiding? For instance, by implementing a four-\nband Hamiltonian as H_f, one would obtain four eigenstates in Eq.(2). However, the\nconnection between this non-Abelian braiding topology and the Abelian (integer-\nvalued) topological invariants (e.g., winding number) remains unclear. This may be\nbeyond the scope of the current work, but I would appreciate it if the authors could give\nsome discussions about how these distinct topological properties might be reconciled\nwithin their theoretical framework.\nOur reply: We greatly thank the reviewer for this insightful question. We emphasize\nthat the results in this work demonstrate the Abelian topology. However, the framework\ncan be straightforwardly applied to multi-band non-Abelian Hamiltonians. Then, there\nemerges multiple scattering channels [Eq. (2)], giving rise to an intricate multi-strand\nlink. In such cases, the braiding of the scattering coefficients is not simply connected\nto the topological invariants or the Berry phases associated with the bands, which are\nnot integer times of π for the non-Abelian topology. Therefore, the exact\ncorrespondence between them [Eq. (3)] remains unclear and deserves to be further\nexplored. Nevertheless, we point out that distinct topological phases before and after\nthe temporal boundary can be characterized through dynamical topological phase\ntransitions.\nTo be specific, in the temporal scattering, one can extract the temporal scattering\ncoefficients, which are directly related to the Loschmidt amplitude with Eq. (6) in the\nmain text. When the system evolves across different topological phases, the Loschmidt\namplitude exhibits nonanalytic points at critical times, indicating the occurrence of a\ndynamical topological phase transition. These singularities are accompanied by\nquantized jumps in the dynamical topological order parameters, thereby providing a\nclear dynamical signature for distinguishing the topological phases before and after the\ntemporal scattering.\nIn new Supplementary Note 9, we have included a brief discussion on the\nextension to non-Abelian topology and the corresponding dynamical topological phase\ntransitions.\nMinor comments:\n1.It looks Fig.1(c) does not appear to be referred in the main text. Please refer it in the\nproper place.\nOur reply: We greatly thank the reviewer for pointing this out. In the revised manuscript,\nFigure 1(c) has been appropriately cited in the final paragraph of the Introduction.\n2.The supplementary material is quite informative. I suggest the authors consider\nintegrating some of its key parts into the Methods section of the main text. Currently,\nthe Methods section only focuses on sample fabrication and temporal boundary\nrealizations. Important details, such as the formation of dynamic topological phase\ntransition, the mathematical definition of the Pancharatnam geometrical phase in Eq.\n(7), and measurement scheme, would be more appropriately placed there.\nOur reply: We greatly thank the reviewer for this constructive suggestion. In the revised\nmanuscript, we have incorporated important details, including the formation of the\ndynamic topological phase transition, the mathematical definition of the Pancharatnam\ngeometric phase in Eq. (7), and the measurement scheme, into the revised Method\nsection."}, {"round": 2, "reviewer_comments": {"reviewer_1": "I am satisfied with the authors' responses and the revisions made in the manuscript. I recommend its acceptance for publication in Nature Communications.", "reviewer_2": "The authors have made significant efforts to answer all my comments. I can now recommend its publication in Nature Communications."}, "author_response": "Our reply: We greatly appreciate the referee's recommendation for publication in\nNature Communications.\nOur reply: We greatly appreciate the referee's recommendation for publication in\nNature Communications."}]

Decision letter