Space and space-time topologies in a type-II hyperbolic lattice

Domain: physics
Published: 2026-03-16
Editorial outcome: major_revision

Abstract
Recent breakthroughs in hyperbolic lattices have expanded the study of topological phases of matter from Euclidean to non-Euclidean spaces. However, prior work has mostly focused on spatial topological states at the single outer edge of type-I hyperbolic lattices. The dynamic transfer of hyperbolic topological states across multiple edges, as well as the emergence of spatiotemporal topological phenomena, remains largely unexplored. Here, we establish both spatial and spatiotemporal topologies in a newly discovered type-II hyperbolic lattice possessing outer and inner edges. Using electric circuits, we experimentally realize a type-II hyperbolic Chern insulator and directly observe degenerate chiral edge states (CESs) of opposite chirality at its outer and inner edges. Furthermore, by dynamically modulating the coupled counterpropagating CESs, we demonstrate arbitrary-proportion dynamic transfer of the CESs during modulation from an anti–parity-time–broken phase to a quasi-exceptional point. Finally, we propose a novel paradigm for constructing a (2+1)-dimensional hyperbolic space-time crystal that hosts an intertwined topology of spatial Chern and temporal winding numbers, yielding a unique space-time topological string state. Our work expands the frontier of hyperbolic topological physics, paving the way for the spatiotemporal dynamic manipulation of hyperbolic topological states.

Introduction
The discovery of hyperbolic lattice, a discretized regularization of non-Euclidean space with constant negative curvature, in circuit quantum electrodynamics [1] and electric circuits [2] has opened new avenues to extend topological physics from the Euclidean to non-Euclidean spaces [3-21]. However, to date, previous studies on hyperbolic topological physics have been predominantly restricted to conventional type-I hyperbolic lattices and exclusively focused on the spatial topological states associated with a single outer edge [3-21]. This limitation leaves topological phenomena involving multi-edge spatial interactions and dynamic transitions, such as Laughlin and Thouless pumping [22-25], Landau-Zener transition [26-30], and non-Hermitian phase transition [30-32], largely unexplored. These dynamic topological effects are pivotal for hyperbolic topological physics, as they not only enable dynamic manipulation of hyperbolic topological states but also serve as powerful methods to characterize hyperbolic topological invariants and explore higher-dimensional hyperbolic topological physics [33,34]. The recently discovered type-II hyperbolic lattice, which features both outer and inner edges, provides an ideal platform for addressing this gap [28,35,36]. Although the dynamic transfer of chiral edge states (CESs) in such a system has been theoretically predicted [28], its experimental realization has been impeded by stringent and intricately tailored coupling requirements. Consequently, developing a more experimentally feasible type-II hyperbolic topological model and using it to directly observe the dynamic topological phenomena are highly sought after.

Besides the spatial topology relying on energy gaps and spatial interfaces, recent advancements in photonic time [37-41] and space-time crystals [42-45] have unveiled fascinating temporal topological interface states [37,41,44,45] and space-time topological events [44,45] that originate from the momentum [37,38,41,44,45] and energy-momentum band gaps [44,45], respectively. These fascinating results motivate deeper exploration at the intersection of topological physics and spatiotemporal frameworks. However, time- and space-time topologies have thus far been primarily limited to Euclidean space-time with only one spatial and one temporal dimension ((1+1)-D) [37,38,41,44,45]. In contrast, higher-dimensional hyperbolic space-time and its accompanying novel spatiotemporal topological phenomena remain completely unexplored.

Here, we present a systematic exploration of the space and space-time topologies in a type-II hyperbolic lattice. By mapping the celebrated Haldane model onto a type-II hyperbolic lattice, we experimentally realize a type-II hyperbolic Chern insulator and directly observe degenerate outer and inner CESs with opposite chirality. Furthermore, by dynamically modulating the coupled counterpropagating outer and inner CESs, we experimentally demonstrate the dynamic transfer of CESs with arbitrary proportions, originating from the coalescence process of two eigenstates with opposite power flows from an anti–parity-time–broken (APT-B) phase to a quasi-exceptional point (q-EP). Notably, by precisely engineering the transfer proportions in a modulated type-II hyperbolic Chern insulator, we construct a (2+1)-dimensional ((2+1)-D) hyperbolic space-time crystal with an intertwined space Chern and time winding topology. This non-trivial space-time topology gives rise to a unique topological state that is temporally localized at the time interface and spatially localized along the lines on the outer and inner space boundaries, thus forming a novel space-time topological string.

Results
Space topology in a type-II hyperbolic Chern insulator

We begin with a one-sheet hyperboloid (upper panel of Fig. 1a), which can be continuously projected as a planar hyperbolic model, i.e., “Poincaré ring”, preserving the local constant negative curvature and global topology [28,35,36]. This Poincaré ring can be further discretized as a type-II hyperbolic lattice (lower panel of Fig. 1a) via regular polygon tessellation [28,35,36]. The unique geometric degree of freedom of a type-II hyperbolic lattice can be fully characterized by an extended Schläfli symbol {𝜌, p, q}, where p and q represent q copies of p-sided polygons meeting at each vertex, and 𝜌 represents the characteristic radius. We map the celebrated Haldane model [46] onto a type-II hyperbolic {0.584, 8, 3} lattice, as illustrated in Fig. 1b. In the light of hyperbolic band theory (HBT) [14-19], a hyperbolic lattice without spatial translational symmetry can be effectively described in a higher-dimensional momentum space utilizing a hyperbolic Bravais lattice. Following this insight, we select a {0.584, 8, 8} Bravais lattice (purple lines) for our type-II hyperbolic Haldane model (see details in Supplementary Note 1). The corresponding unit cell (yellow region) is an octagon containing 16 sites, where the onsite potential of subsite A (B) is +M (-M), the coupling magnitude between nearest-neighbor (NN) sites is t1, and the coupling magnitude (phase) between next-nearest-neighbor (NNN) sites is t2 (𝜙). This unit cell is equipped with a Bloch vector k = (k1, k2, k3, k4) under a noncommutative translation group Γ = 〈𝛾1, 𝛾2, 𝛾3, 𝛾4 : 𝛾1𝛾2𝛾1^{-1}𝛾2^{-1}𝛾3𝛾4𝛾3^{-1}𝛾4^{-1}〉, whose components form a four-dimensional (4D) Brillouin zone (BZ). We henceforth set the system parameters t1 = 1, t2 = 0.2, 𝜙 = 𝜋/2 and M = 0. Figure 1c presents the bulk density of states (DOS) as a function of energy for the unit cell in Fig. 1b, computed using HBT by sampling k over the 4D BZ. The yellow energy regions with vanished DOS indicate two energy gaps with non-zero Chern vectors C = (-1,1) (see details in Supplementary Note 2), revealing a nontrivial Chern topology.

To demonstrate the CESs associated with these Chern gaps, we construct a finite type-II hyperbolic Haldane model with open boundary conditions along the k2, k3 and k4 directions, and periodic boundary conditions along the k1 direction, as illustrated in Fig. 2a. Figure 2b displays the corresponding band structure calculated using HBT, showing the chiral dispersions of the outer (red lines) and inner (green lines) edge modes, spanning the Chern gaps I and II (yellow regions). Figures 2c and 2d respectively show the eigenmode profiles and point-source-excited wave function amplitude distributions of degenerate modes (labeled by “①” and “②” in Fig. 2b) at a mid-gap energy E = -1.178 (or E = 1.178), verifying a counterclockwise (CCW) or clockwise (CW) unidirectional propagation of outer (upper panel) or inner (lower panel) edge-dominated state.

To experimentally demonstrate the degenerate outer and inner CESs with opposite chirality, we implement the type-II hyperbolic Haldane model to a type-II hyperbolic Chern circuit based on the topological circuit platform [47,48], as shown in Fig. 2e. Four circuit nodes are connected as a loop via capacitors C0 to form an effective lattice site. Voltages on these four nodes are defined as Vi,1, Vi,2, Vi,3 and Vi,4, which can be used to construct a pair of pseudospins Vi,↑/↓ = (Vi,1 ± iVi,2 - Vi,3 ∓ iVi,4)/2 to realize the required site couplings. As shown in the inset of Fig. 2e, four pairs of adjacent nodes are directly linked via capacitors C1 to implement the real NN couplings, while four pairs of adjacent nodes are crossly linked via capacitors C2 to implement the complex NNN couplings, and each node is grounded by an inductor L. Additionally, all boundary nodes are grounded via additional capacitors to ensure the same resonance frequency as the bulk nodes. Based on the Kirchoff’s law, we can derive the electrical circuit eigenequation which is consistent with the eigenequation of the type-II hyperbolic Haldane model, identifying the tight-binding model (TBM) parameters in terms of circuit elements as: t1 = C1/C0, t2 = C2/C0, 𝜙 = 𝜋/2, E = f^2/f0^2 - 2 - (3C1 + 6C2)/C0, where f is the circuit frequency and f0 = (2𝜋√LC0)^{-1} (see details in Supplementary Note 3). Specifically, the electrical circuit element parameters are selected as C0 = 1.5nF, C1 = 1.5nF, C2 = 0.3nF, and L = 10μH.

We first experimentally measure the impedance spectra at a bulk node, an outer edge node, and an inner edge node (illustrated by grey, red and green dots) in type-II hyperbolic Haldane circuit, as shown in Fig. 2f. We identify two gaps (yellow regions) in frequency ranges of 0.45-0.496 MHz (gap I) and 0.554-0.62 MHz (gap II), within which the outer and inner edge nodes exhibit significant impedance responses while the bulk node shows negligible impedance responses. Additionally, we measure the spatial impedance distributions of the type-II hyperbolic Haldane circuit at frequency f1 = 0.47 MHz in gap I and f2 = 0.586 MHz in gap II, respectively, as shown in Fig. 2g and 2h. These edge-dominated profiles match well with the superposition of the two edge eigenmodes in Fig. 2c, confirming the coexistence of outer and inner edge states.

Next, we apply a chiral voltage source (cyan star) [Vi,1, Vi,2, Vi,3, Vi,4] = [V0, iV0, -V0, -iV0]sin(2𝜋ft) (V0 = 0.5V) at an outer or inner edge node to selectively excite the spin-up voltage pseudospin to demonstrate the unidirectionality of these hyperbolic edge states. Figure 2i shows the measured voltage amplitude distributions of the outer (upper panel) and inner (lower panel) edge states at frequency f2 in gap II. It can be seen that the outer (inner) edge voltage field is tightly confined to the outer (inner) edge and propagates unidirectionally in a CCW (CW) direction. Additionally, the robustness of these CESs against defects has also been experimentally examined (see details in Supplementary Note 4).

Arbitrary-proportion dynamic transfer of the coupled counterpropagating CESs

Next, we explore the arbitrary-proportion dynamic transfer of the coupled counterpropagating outer and inner CESs in a modulated type-II hyperbolic Chern insulator. We start with a two-level model and employ the outer and inner CESs in Fig. 2c as its orthonormal basis {|𝜓o⟩, |𝜓i⟩}, whose purely CCW or CW power flows are defined as 𝛽o = |⟨𝜓o|𝜓o⟩|^2 - |⟨𝜓o|𝜓i⟩|^2 = 1 and 𝛽i = |⟨𝜓i|𝜓o⟩|^2 - |⟨𝜓i|𝜓i⟩|^2 = -1, respectively. To manipulate the couplings between these bases, we introduce a radial coupling channel (blue region) consisting of four modulated couplings (short red lines) mt1 into type-II hyperbolic Haldane model, as illustrated in Fig. 3a. We adopt an outer or inner edge source to excite the coupled CESs |𝜓m,k⟩ (k = 1,2) (see Supplementary Note 5) and define the corresponding power flow as 𝛽k = |⟨𝜓m,k|𝜓o⟩|^2 - |⟨𝜓m,k|𝜓i⟩|^2. Figure 3b presents the 𝛽k as a function of the modulation strength m, showcasing that as m increases, 𝛽k simultaneously changes from 1 (red circles) or -1 (blue circles) to 0. This phenomenon of power flow evolution can be effectively modeled by a Hamiltonian HAPT = (𝛽o i𝜅; i𝜅 𝛽i) = (1 i𝜅; i𝜅 -1), where 𝜅 denotes the effective coupling strength between the outer and inner CESs. Note that its apparent non-Hermiticity of the purely imaginary coupling does not originate from any physical gain or loss. Instead, it is an pseudo-Hermiticity, i.e., 𝜎z HAPT 𝜎z^{-1} = HAPT† with 𝜎z = (1 0; 0 -1), which follows from power-flow-difference conservation, a unique form of energy-conservation inherent to contra-directional coupling [30-32] (see Supplementary Note 6). This Hamiltonian is also anti-parity-time symmetric (APT-S), i.e., {HAPT, P̂T̂} = 0, and gives eigenvalues:

𝛽 = ±√1-𝜅^2 (1)

Using this equation, we can retrieve the corresponding 𝜅 of 𝛽k and replot 𝛽k as a function of 𝜅, as shown by red and blue circles in Fig. 3c, exhibiting a good quantitative agreement with the analytic solutions (red and blue solid lines). Specifically, in the weakly coupled regime |𝜅| < 1 (blue region), 𝛽 appear as two real numbers (purple dashed line), supporting two oppositely propagating eigenstates that correspond to the APT-B phase. The defining feature of this phase is that an excitation applied to either edge couples only partially to the opposite edge through the coupling channel. As a result, the wave function amplitude remains predominantly localized on the initial edge, as exemplified by the two modes (labeled “①” and “②”) with m = 6 in the left insets of Fig. 3b. When the coupling approaches a critical value |𝜅| = 1 (red dashed line), 𝛽 nearly coalesces to zero, indicating the formation of two nearly coalesced non-propagating eigenstates that correspond to the q-EP (red star). The hallmark of this point is that an excitation on either edge couples completely to the opposite edge via the coupling channel. In other words, the two edges become fully connected regardless of the excitation position, leading to a wave function amplitude equally distributed between both edges, as exemplified by the two modes (labeled “③” and “④”) with m = 41 in the right insets of Fig. 3b. Consequently, based on the intrinsic coalescence process of two eigenstates carrying opposite power flows from an APT-B phase to a q-EP, a continuous, arbitrary-proportion dynamic transfer of the coupled outer and inner CESs can be achieved. Figure 3d displays the local DOS distributions of the modes for m = 6 (upper panel) and m = 41 (lower panel), which correspond to the superposition of two APT-B modes and two q-EP modes in Fig. 3b, respectively. The key difference between the two mode profiles is that, in the former case, the amplitude at the coupling channel is significantly weaker than that at the edges, whereas in the latter case, the amplitude at the coupling channel is comparable to that at the edges, indicating that the two edges are partially and fully coupled, respectively. Note that these features persist even in the presence of losses, although attenuation modifies the wave-function amplitude distributions. Figures 3e and 3f show the wave function amplitude distributions with propagation loss 𝜂 = 0.04, excited by an outer or inner edge source (cyan star) with energy E2 for m = 6 and m = 41, respectively. It can be seen that, in the APT-B phase (m = 6), a CCW (CW) CES at the outer (inner) edge couples only partially to the CW (CCW) CES at the inner (outer) edge via the coupling channel; the wave function amplitude remains predominantly localized at the outer (inner) edge despite loss-induced attenuation. By contrast, at the q-EP (m = 41), a CCW (CW) CES at the outer (inner) edge couples completely back to a CW (CCW) CES at the inner (outer) edge via the coupling channel; however, the wave-function amplitude is no longer equally distributed between the two edges owing to loss-induced attenuation.

To experimentally demonstrate the arbitrary-proportion dynamic transfer of the coupled outer and inner CESs, we construct a radial coupling channel by parallelling five additional capacitors Cm1 = C0 (Cm2 = 8C0) to each of the four NN couplings to enhance their effective TBM parameter as tm1 = (5Cm1 + C1)/C0 = 6 (tm2 = (5Cm2 + C1)/C0 = 41), as illustrated in the inset of Fig. 3g, corresponding to a modulation strength m = 6 (m = 41). Figures 3h and 3i display the measured spatial impedance distributions of the modulated type-II hyperbolic Haldane circuit, i.e., effective local DOS, at frequency f2 for m = 6 and m = 41, respectively, whose mode profiles match well with the local DOS profiles shown in Fig. 3d. Figures 3j and 3k shows the measured voltage amplitude distributions of the coupled CESs excited by an outer or inner edge source (cyan star) at frequency f2 for m = 6 and m = 41, respectively. For the weak manipulation (m = 6), the outer (inner) CES partially transfers into the inner (outer) CES through the coupling channel, corresponding to an APT-B phase, thereby achieving the partially dynamic transfer of the outer and inner CESs. While for the strong manipulation (m = 41), the outer (inner) CES completely turns into the inner (outer) CES through the coupling channel, corresponding to a q-EP, thereby achieving the fully dynamic transfer of the outer and inner CESs.

Hyperbolic space-time topological string

Finally, by precisely engineering the transfer proportions governing pulse dynamics in a double-channel-modulated type-II hyperbolic Chern insulator, we construct a discrete (2+1)-D hyperbolic space-time crystal. With additional non-Hermitian modulations, we further demonstrate a space-time topological string. As illustrated in Fig. 4a, we introduce two distinct radial coupling channels C1 and C2 and two edge modulators MS and ML into a {0.9984, 8, 3} type-II hyperbolic Haldane model. Specifically, the modulation strengths of C1 and C2 are set to m = 8.3805 and m = 3001, respectively, and MS and ML are set near C1, both of which consist of edge on-site gain/loss and a phase shifter. For a Gaussian pulse with a specific width and an energy centered at E2 = -1.178, the outer and inner edges are equivalent to identical edge loops (see details in Supplementary Note 7). Meanwhile, C1 and C2 effectively function as 50:50 and 0:100 beam splitters (BSs), respectively, with an output phase difference of 𝜋/2 (see details in Supplementary Note 8), and MS and ML effectively operate as pulse modulators that provide a composite modulation e𝜑 = e𝜑a+i𝜑p, where 𝜑a and 𝜑p denote the amplitude and phase modulation strengths (see details in Supplementary Notes 9 and 10), as illustrated in Fig. 4b. In this way, the outer and inner edge loops are partitioned into two closed loops with distinct path lengths, denoted as loop-S (the shorter loop, marked in purple) and loop-L (the longer loop, marked in red). The two closed loops are separated at C2 and connected through C1, and are individually modulated by MS and ML, respectively. By defining a loop unit length Δ, the lengths of loop-S and loop-L are set as LS = 19.5Δ and LL = 20.5Δ, respectively (see details in Supplementary Note 11). Interestingly, to simplify the analysis without considering non-Hermitian modulations, when a single pulse is injected into loop-L, it undergoes iterative splitting and recombination at C1, consequently evolving into two pulse trains (one in each loop). Figure 4c shows the snapshots of intensity distribution of pulse trains in the edge loops at the first four time-steps 𝕥 = 0T, 1T, 2T, 3T, where T = (LS + LL)/2 represents the average time period for a pulse with a unit velocity propagating along the loop-S and loop-L. The corresponding intensity functions of sub-pulses in the loop-L (loop-S) with respect to the loop length s ∈ [0, s0], s0 = 18Δ (i.e., pulse counting region), |𝕧𝕥(s)|^2 (|𝕦𝕥(s)|^2), are displayed in the upper (lower) panel of each row in Fig. 4d. Owing to the length difference between two loops, within a time interval of 𝕥 = t×T, the evolved sub-pulses in loop-S or loop-L (black dashed lines) acquire negative or positive positional shifts with respect to the initial pulse (red dashed lines), with the displacement quantified in units of Δs = Δ/2. Remarkably, these shifts essentially index the relative positions of each pulse along the pulse trains, which map to the discrete space coordinate x, and the count of time-step of the pulse train maps to the discrete time coordinate t. By extracting the local maxima of |𝕧𝕥(s)|^2 (|𝕦𝕥(s)|^2) and inserting zeros between adjacent maxima, |𝕧𝕥(s)|^2 (|𝕦𝕥(s)|^2) can be discretely reduced into |𝜐t_x|^2 (|u_x^t|^2) (t = 0,1,2,3), as displayed in the upper (lower) panel of each row in Fig. 4e. The corresponding spatiotemporal distributions of the total pulse intensity, defined as P(x,t) = |𝜐t_x|^2 + |u_x^t|^2, are demonstrated in Fig. 4f. Particularly, another equivalent coordinate description regarding time delay of this form of time-multiplexing is that: after t round-trips evolutions, the sub-pulses in loop-S or loop-L accumulate the total time delay t×T ± x×Δt = t×T ± x×Δs, where the positive and negative signs indicate the relative time delay and time advance, respectively, thereby yielding a (1+1)-D synthetic space-time lattice consist of a mesh of BS (50:50), as illustrated in Fig. 4g. Together with the discrete radial spatial distance 𝜌, the pulse trains effectively evolve in a discrete (2+1)-D hyperbolic space-time (𝜌,x,t). Notably, the space-time topology in the space-time (𝜌,x,t) is essentially the intertwining of a space topology defined in the (𝜌,x) sub-space-time, ensuring the formation of spatial topological edge states, and a time topology defined in the (x,t) sub-space-time, determining the emergence of temporal topological interface states.

As illustrated in Fig. 5a, the pulse dynamics in a synthetic lattice with non-Hermitian modulations is governed by the recursive evolution equations (see details in Supplementary Note 11):

u_x^{t+1} = 1/√2 (-u_x^t + i𝜐_x+1^t) e^{𝜑u(x,t+1)}

𝜐_x+2^{t+1} = 1/√2 (i u_x+1^t - 𝜐_x+1^t) e^{𝜑𝜐(x+2,t+1)} (2)

The modulation parameter 𝜑u(x,t) (𝜑𝜐(x,t)) setting for u_x^t (𝜐_x^t) is set as amplitude modulation only, i.e., 𝜑u = 𝜑ua (𝜑𝜐 = 𝜑𝜐a). We select a unit cell (green shaded region) that has a two-step periodicity in space, accompanied by a four-step periodicity in time with gain-loss modulation settings [𝜑u(x,t), 𝜑u(x+1,t+1), 𝜑u(x,t+2), 𝜑u(x+1,t+3)] = [g, 0, 0, -g] and 𝜑𝜐 = -𝜑u, where g is the gain/loss strength. For the Hermitian case, i.e., g = 0, by applying Floquet-Bloch ansatz to Eq. (2), we can obtain the system’s band structure, as shown in Fig. 5b, which is gapless in both energy and momentum. While for the non-Hermitian case, e.g., g = 0.3 (g = -0.3), two momentum bandgaps (purple and blue regions) open at momenta k = 0 and 𝜋 (see details in Supplementary Note 12), as shown in Fig. 5h (Fig. 5i). The topology of the momentum bandgap can be characterized by a temporal winding number:

𝜈t = 1/2𝜋 ∑_{s=1}^2 ∫_{-𝜋}^𝜋 i ⟨𝜓ks(k)| 𝜕/𝜕E |𝜓ks(k)⟩ dE (3)

where 𝜓ks(k) represents the eigenstates of two momentum bands. Specifically, for g = 0.3, 𝜈t = +1, while for g = -0.3, 𝜈t = -1 (see details in Supplementary Note 13).

For the entire (2+1)-D hyperbolic space-time (𝜌,x,t), in the Hermitian case (g = 0), without the time winding topology, the space Chern topology supports a space-time topological world sheet, where topological states spatially localized on the outer and inner boundary x-t surfaces of the space-time and remain temporally delocalized. To demonstrate this spatiotemporal topological state, we construct a finite discrete (2+1)-D hyperbolic space-time crystal which possesses a pure space Chern topology, as illustrated in Fig. 5c. The corresponding values of g over time are shown in Fig. 5d. We initially inject a spatially broad wave-packet with a momentum k = 0 from the loop-L at time 𝕥 = 0. The evolution of the intensity function of the sub-wave-packet in the loop-S (loop-L) |𝕦𝕥(s)|^2 (|𝕧𝕥(s)|^2) over time is displayed in the left (right) panel of Fig. 5e. Figure 5f further shows the spatiotemporal distribution of the total wave-packet intensity P(x,t). The total wave-packet intensity P(t) = ∑_x (|𝜐_x^t|^2 + |u_x^t|^2) keep almost unchanged over time, as shown in Fig. 5g. This spatiotemporal topological state is exponentially localized along the spatial axis 𝜌, and extended along both the spatial axis x and temporal axis t, signifying the forming of a space-time topological world sheet. In the non-Hermitian case (g ≠ 0), the time winding topology is intertwined with the space Chern topology, the resulting space-time topology of this (2+1)-D hyperbolic space-time is characterized by a hybrid topological invariant (C, 𝜈t). This invariant guarantees the emergence of a space-time topological string, i.e., a time-constant slice of the space-time topological world sheet, where a topological state is not only spatially localized on the outer and inner boundary x-t surfaces of the space-time but also temporally localized at a specific time t slice. To demonstrate this space-time topological string state, we construct a finite discrete (2+1)-D hyperbolic space-time crystal which consists of regions I and II possessing space-time topologies of (C, 1) and (C, -1), respectively, thereby forming a temporal interface, as illustrated in Fig. 5j. The corresponding values of g over time are shown in Fig. 5k, where the black dashed line represents the temporal interface. Figures 5l and 4m show |𝕦𝕥(s)|^2 and |𝕧𝕥(s)|^2 over time and P(x,t) across all (x,t) coordinates, respectively. It can be seen that the wave-packet is localized at the time-topological interface where 𝜈t flips its sign, and P(t) exhibits exponential growth and decay before and after the temporal interface, as shown in Fig. 5n. This spatiotemporal topological state is exponentially localized along both the spatial axis 𝜌 and temporal axis t, and extended along spatial axis x, signifying the forming of a space-time topological string. Additionally, the robustness of this space-time topological string against temporal disorders has also been examined (see details in Supplementary Note 14). Note that this paradigm for achieving a space-time topological string is universal, applying to other lattices with inner and outer CESs—including flat lattices. However, type-II hyperbolic lattices with a higher edge-to-bulk ratio hold a clear advantage, as they require far fewer total lattice sites to achieve equivalent performance.

Methods
Sample fabrications

The type-II hyperbolic Haldane circuit is fabricated by standard printed circuit board (PCB) technology, where the capacitances are implemented by ceramic capacitors with C0 = 1.5nF (1% tolerance), C1 = 1.5nF (1% tolerance), C2 = 0.3nF (1% tolerance), the inductances are implemented by power inductors with L = 10μH (5% tolerance), 170mΩ series resistance, and a minimal quality factor of 40 at 1 MHz.

Measurement details

The grounded impedances are measured by using a Zurich Instruments MFIA impedance analyzer (5MHz) in a two-terminal measurement configuration. The voltage-field profiles are measured using a Zurich Instruments MFIA impedance analyzer (5 MHz) in a four-terminal measurement configuration, where two terminals are connected to a four-way power divider, whose outputs are phase-shifted by 0, 𝜋/2, 𝜋, and 3𝜋/2, respectively, to serve as the chiral excitation source for the chiral edge states, while the remaining two terminals function as a lock-in amplifier to measure the voltage amplitudes.

Data availability

All the data supporting this study are available in the paper and Supplementary Information. Additional data related to this paper are available from the corresponding authors upon request.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "In this manuscript, the authors demonstrated topological phenomena in type-II hyperbolic lattices, which are obtained through the transfer of type-I (Poincare-disk) lattices to the Poincare ring structure. Notably, the authors examined both space and temporal degrees of freedom in examining topological features in hyperbolic lattices, by utilizing a RF circuit composed of 288 sites. The major impact of this work is to extend the accessible spatio-temporal degrees of freedom—which have been an emerging research topic in optics and microwaves represented by the concept of photonic time crystals and momentum gaps—into non-Euclidean geometry.\n\nThe manuscript is well written. The theoretical and experimental analysis from the spatial chiral edge state analysis to their dynamical transfer, and to (2+1)-D space-time crystals is very concrete. In terms of exploring novel material phases under the effective implementation of Hamiltonians, the authors’ proposal and demonstration explicitly shows a clear novelty and impact in the fields of both non-Euclidean wave physics and dynamical photonic crystals. Considering the listed novelties, the in-depth analysis, and well-organized presentation, the reviewer strongly suggests acceptance after the following minor revision.\n\n1. In the discussion of temporal topological invariants v_t, it would be helpful to include a comparison with the winding number or Zak phase, and related topological phenomena in a spatial-domain Su-Schrieffer-Heeger (SSH) lattice to clarify the operation principle of the authors’ work based on time-varying non-Hermitian systems. The reviewer also suggests the inclusion of Eq. (19) in Supplementary Information because the temporal topological invariant is less familiar with general readership compared with its spatial counterpart.\n\n2. In the equation for temporal lattices described in Fig. 4, the hopping between neighboring sites is tunable only for its amplitude, which operates as gain or loss. The reviewer wonders whether the phase of hopping components can be manipulated along the temporal axis, for example, by modifying the properties of round-trip pulses in Supp. Fig. 5.\n\n3. Despite fertile physics in hyperbolic lattices, one of the critical issues in implementing the lattices would be scalability because of the substantial increase of element density near the boundary. In this context, it is necessary to include further discussion for the realizable number of lattice sites over both spatial and temporal axes.", "reviewer_2": "The authors have experimentally realized a type-II hyperbolic topological insulator using an electrical circuit and observed its edge modes. They have also theoretically proposed a way to realize space-time topology in such a type-II hyperbolic lattice, where topological modes are localized both spatially and temporally.\n\nTheir study on hyperbolic lattices and spatiotemporal topology is very timely and interesting for experts. However, I am not sure that the present manuscript brings a high impact on a broad community that meets the criteria for publication in Nature Communications. This is because the first and second parts in the result look very similar to a previous study [23] and considering existing studies on topological hyperbolic lattices in electrical circuits, their experimental realizations are not surprising. If I understand correctly, the last part on space-time topology utilizing edge pulse dynamics is a novel proposal in this field and the most interesting part of this paper. However, some details are hard to follow, and the absence of its experimental realization diminishes the impact.\n\nTherefore, I think that the present manuscript is more suitable for publication in a specialized journal. If the authors greatly revise the manuscript and focus more on the last part, I may change the evaluation. Below, I raise some comments that the authors should address.\n\n1. In line 84, specifying the references to hyperbolic band theory will be helpful for non-experts.\n\n2. In line 163, why is the effective Hamiltonian non-Hermitian, while the whole dynamics is governed by a Hermitian Hamiltonian?\n\n3. In my understanding, the time delay $\\Delta t$ plays a crucial role in introducing an effective spatial coordinate x in Eq. (2). The authors should discuss its role in the main text.\n\n4. I do not exactly understand what is calculated in Fig. 4. I am curious about why there are no propagating waves but standing waves in panels g, n, and how to obtain the information in discrete coordinates in panels h, o. The authors should clarify the numerical methods in e.g. Methods section.\n\n5. I am not convinced with the validity of Eq. (2) in non-Hermitian cases because gain and loss can broaden the pulse width, and thus C1 channel may not work as a 50:50 beam splitter after a long time. Furthermore, gain and loss are introduced independently of the discrete spatial coordinate, it cannot be true due to the time delay. Please discuss the robustness against such a non-Hermitian effect.\n\n6. The temporal winding number is calculated based on the effective model in Eq. (2). However, one should be able to define a topological number from the Hamiltonian of the whole system. Please discuss this possibility.\n\n7. Experimental realization of time-space topology seems straightforward. Please specify any technical reason for the absence of such an experimental result. If there is no reason, the author should conduct the experiment.\n\n8. There is a typo in Supplementary Fig. 5b; the sign before $2\\Delta t$ is wrong.\n\n9. I am confused about the notation in Eq. (16) in Supplementary Note 8. E and k may be swapped.", "reviewer_3": "The manuscript entitled \"Space and space-time topologies in a type-II hyperbolic lattice\" presents a meaningful study that utilizes coupled chiral edge states (CESs) to construct an anti-parity-time (APT) symmetric Hamiltonian, which further serves as a dynamic coupling channel for the construction of space-time topological string. The idea is conceptually interesting and represents an original advancement in the field of hyperbolic topological systems. However, I have several concerns related to the APT phase and the realization of the space-time topological string:\n\n1. In the regime of κ < 1, the system resides in the anti-PT-broken (APT-B) phase with real eigenvalues. The eigenstates, as shown in Fig. 3b, are mainly localized on either the outer or inner boundary. Since the APT Hamiltonian is constructed based on the power-flow basis (β₀ = +1 for CCW and β₁ = −1 for CW), one would expect that edge excitation in this regime would result in energy remaining mostly localized on the initial boundary. However, Fig. 3d shows that energy propagates along both edges, which seems contradictory to the eigenstate distribution in Fig. 3b. Could the authors clarify why this dual-boundary behavior occurs in the APT-B regime?\n\nSimilarly, at κ = 1 (the exceptional point), the eigenstates coalesce and theoretically become identical regardless of excitation direction. This should yield identical distributions on both boundaries. Yet, Fig. 3e does not appear to show this distribution. Could the authors explain the reason behind this discrepancy?\n\n2. In the proposed space-time topological string, as known from pulse dynamics, wave packets typically broaden during propagation. Given the necessity of multiple round-trips to simulate the time dimension, how is pulse broadening suppressed or compensated?\n\n3. The function of C₁ and C₂ as effective directional beam splitters (50:50 and 0:100) is crucial to the string construction, yet the description remains somewhat abstract. Can the authors provide a simulation that explicitly demonstrates the splitting behavior of wave packets through C₁ and C₂? For example, showing a localized input before and after the coupling channel would help visualize their function as couplers.\n\n4. After constructing the effective (2+1)D synthetic model, it seems like the authors did not revisit the original hyperbolic Haldane circuit model to validate the approximation or show consistency. A direct simulation based on the full Haldane-type circuit model—especially using actual pulse propagation and modulation profiles—would provide valuable insight into the validity and applicability of the synthetic space-time model.\n\n5. In Fig. 1a, the lattice on hyperboloid is labeled as periodic. However, the current figure does not clearly illustrate this periodicity.\n\n6. The eigenstate distributions in Fig. 3g and Fig. 3h appear nearly identical, which seems inconsistent with the significantly different mode patterns shown in Fig. 3b for the same modulation parameters (m = 6 and m = 41). Could this be a labeling or figure placement error?\n\nThis work is a valuable contribution to the field, but my final recommendation will depend on the authors’ detailed responses and clarifications."}, "author_response": "We are grateful for the constructive comments on our Nature Communications manuscript (NCOMMS-25-42311) from all the reviewers. In the text below, each of the comments from each reviewer is quoted in italics and is followed by the corresponding detailed response. We also revised the manuscript and the Supplementary Information accordingly, and these updates are highlighted in blue in those files. In the text below, the references to these updates are highlighted in a similar way. [Response continues with detailed point-by-point rebuttals for Reviewers 1, 2, and 3]"}, {"round": 2, "reviewer_comments": {"reviewer_1": "In this resubmission, the authors have addressed all comments and suggestions raised by the reviewers. The expanded discussion of topological quantities in space and time—particularly Supplementary Note S15 together with Fig. 5(m) in the main text—provides a clearer and more instructive presentation of the work. In addition, Supplementary Note S11, which discusses the scalability of the space–time circuit, offers a useful guideline for practical implementation. The discussion in Supplementary Note S10 further expands the design freedom for space–time topology by introducing the implementation of gauge fields in the proposed system.\n\nThe reviewer believes that the revised manuscript pushes the boundaries of research on hyperbolic lattices and meets the high standards of Nature Communications. Therefore, the reviewer recommends acceptance of the manuscript in its current form.", "reviewer_2": "The authors have greatly revised the manuscript and added discussions on the space-time topology of the hyperbolic lattice. This revision and the authors’ detailed response helps me understanding their results much better. However, there are still several critical issues to be addressed. In particular, the anti-parity-time (APT) phase transition discussed in the second section of Results is not a genuine phase transition. I also think that their numerical methods are still hard to follow. Thus, the authors should address the following problems before deciding my final recommendation.\n\n1. Since the effective Hamiltonian H_{APT} right before Eq. (1) is just apparently non-Hermitian, the parameter should never go into the genuinely non-Hermitian regime, i.e., |\\kappa|>1. Therefore, in the proposed setup, the APT phase transition from APT-broken to APT-symmetric phases cannot be observed. In fact, Figure 3b clearly shows this; two eigenvalues do not coalesce at any finite m. Since the coalescence of eigenvalues (the exceptional point) correspond to the transition point, this numerical result indicates that the system is always in the APT-broken phase and thus the phase transition does not happen. Instead, the approximate coalescence of two eigenvectors is a feature of the exceptional point.\n\nI suggest that the authors remove “Anti-parity-time phase transition” from the title of the section and focus on the exceptional point. They should also discuss the relationship between the proposed phenomenon and other exceptional-point-induced phenomena, such as coherent perfect absorption (cf. Phys. Rev. A 82, 031801(R) (2010), Phys. Rev. Lett. 106, 093902 (2011)) and unidirectional invisibility (cf. Phys. Rev. Lett. 106, 213901 (2011)).\n\n2. The importance of the type-II hyperbolic lattice should be more highlighted in the context of the time-space topological string. If I understand correctly, the existence of the outer and inner edge states is crucial because we need two chiral edge states. This point is partially explained in Introduction, but it should also be mentioned in the section of Hyperbolic space-time topological string. Another possible criticism is that a cylindrical geometry can also realize two coupled chiral edge states. Thus, the advantage of a type-II hyperbolic lattice over a cylindrical geometry should be discussed.\n\n3. All the calculations should be able to be reproduced without demanding the codes to authors. However, I cannot follow how the authors calculate the pulse dynamics in Figs. 4 and 5. They should clarify the following things. Making separate sections for numerical methods in Methods or Supplementary Information is preferable.\n(a) Based on which equations do they calculate the pulse dynamics?\n(b) What is the exact definition of u_t(s) and v_t(s) in the captions of Figs. 4 and 5? How do they calculate them from the wavefunction?\n(c) While the original model is a lattice model, Figs. 4b, c look continuum intensity fields without bulk. How do they obtain these figures from the lattice model with bulk?\n(d) How do they introduce the gain-loss in Fig. 5? They have partially explained it in the previous reply as it is on-site gain and loss at a small number of sites, but they should clarify at which sites they introduce it.\n\n4. While the authors have replied that the width of a gaussian pulse is not altered by loss, it can be false if the loss rate depends on the wavelength. I have another concern that the local gain or loss can behave as an effective defect and thus the chiral edge pulse avoids it, which results in no amplification or attenuation of the edge pulse. (cf. Topological edge modes at gain-loss induced effective boundaries are discussed in, e.g., Science 365, 1163 (2019).) Please explain more details on how the authors have numerically confirmed that the width of the gaussian pulse is not altered during the amplification or attenuation by local gain or loss.", "reviewer_3": "I have read the responses to Reviewers's comments. The authors have addressed the concerns raised, and the explanations and revisions are satisfactory. I therefore recommend the manuscript for publication."}, "author_response": "We are grateful for the constructive comments from all the reviewers on our Nature Communications manuscript (NCOMMS-25-42311A). In the text below, each of the comments from each reviewer is quoted in italics and is followed by the corresponding detailed response. We also revised the manuscript and the Supplementary Information accordingly, and these updates are highlighted in blue in those files. In the text below, references to these updates are highlighted in the same way. [Response continues with detailed point-by-point rebuttals for Reviewers 1, 2, and 3]"}, {"round": 3, "reviewer_comments": {"reviewer_2": "The authors have addressed all the comments and resolved all my major concerns about their theoretical proposal. As I mentioned in my first comment, their results on space-time topology are very timely and provide a novel method for realizing temporal boundary modes, which should be of immediate interest to condensed matter physicists and metamaterial researchers. Thus, I recommend the publication of the manuscript."}, "author_response": "We are grateful for the constructive comments on our Nature Communications manuscript (NCOMMS-25-42311B) from all the reviewers. In the text below, each of the comments from each reviewer is quoted in italics and is followed by the corresponding detailed response. We also revised the manuscript and the Supplementary Information accordingly, and these updates are highlighted in blue in those files. In the text below, the references to these updates are highlighted in a similar way."}]

Decision letter