Unconventional spin-intertwined charge density wave in magnetic phases of kagome metal GdTi_3Bi_4

Domain: physics
Published: 2026-02-12
Editorial outcome: minor_revision

Abstract
The symmetry breaking density waves and their interplay with spin and charge are crucial for understanding the correlated electronic states in quantum materials. However, symmetry-breaking density waves involving simultaneous charge and spin modulations remain largely unexplored. Here, we report the unconventional spin-intertwined charge density wave in magnetic kagome metal GdTi3Bi4 by scanning tunneling spectroscopy. We observe a 3Q charge density wave that is incommensurate with the crystalline lattice in both periodicity and orientation and breaks all mirror and rotation symmetries. The CDW undergoes incommensurate-commensurate transitions in a magnetic field and 3Q-1Q transitions as a function of applied magnetic field and temperature. Remarkably, the quantum and thermal melting of the CDW exhibits a phase structure closely resembling the bulk magnetization phase diagram of GdTi3Bi4, which provides evidence for the observation and manipulation of spin-intertwined CDW as a bulk property. Our findings reveal the microscopic textures of the intricate magnetic phases with unconventional charge-spin coupling, and offer insights into a broad class of multi-components density wave formation in kagome and other correlated quantum materials.

Introduction
Electronic density waves, characterized by periodic spatial modulations of charge1,2, spin3,4 or Cooper pair density5,6 that break lattice translational symmetry1, have become widespread phenomena of correlated states in quantum materials. Recently, unconventional density waves have been reported to exhibit additional symmetry breaking that gives rise to novel electronic phenomena such as smectic orders with simultaneous rotational symmetry breaking7–10, chiral density waves that break all mirror symmetries11–14, and non-magnetic current density waves that break time-reversal symmetry15. Typically, a density wave involves periodic modulations in a single quantum degree of freedom, such as spin or charge. Consequently, a fundamental motivation is to explore the simultaneous modulation of both charge and spin, which is dubbed a spin-intertwined charge density wave (SCDW). The charge modulation of SCDW can be detected via scanning probe microscope16, while its spin content allows the tunability by an external magnetic field17. The SCDW has been proposed theoretically in the study of density waves on frustrated lattices18. However, up to date, the SCDW in the frustrated-lattice materials has rarely been observed.

Magnetic kagome metals have recently emerged as an ideal platform for investigating the interplay between the spin and charge degrees of freedom. Electronic density waves, which play a crucial role in the emergence of unconventional superconductivity6,19,20, have been proposed for the magnetic kagome metals21–23. An intriguing example is that in the magnetic kagome metal FeGe, a charge density wave (CDW) emerges at temperatures below the magnetic transition, but the ordered magnetic moment exhibits a sudden change at the CDW transition, indicating a strong coupling between the magnetism and CDW24,25. Recently, the RETi3Bi4 (RE=rare earth metals) family exhibits unique fractional magnetization plateaus and topological band structure26–30, making it a compelling candidate for studying the coupling between magnetic order and density wave formation, including the possibility of realizing unconventional charge-spin coupled states.

Here, we investigate GdTi3Bi4 using low-temperature scanning tunneling microscopy/spectroscopy (STM/STS) equipped with external magnetic field capability. We observe a three-component (3Q) CDW at the Gd-terminated surface in the magnetic ground state. By comparing the CDW and crystalline vectors, we show that the CDW is incommensurate with the crystalline lattice in both periodicity and orientation. In addition, the CDW is highly unconventional as it breaks all mirror and rotation symmetries, in addition to breaking translation symmetry. Remarkably, the CDW is highly manipulable by external magnetic fields, demonstrating its spin-charge coupled nature and providing evidence of spin-intertwined CDW. We further study the quantum and thermal melting phase diagram of the CDW in magnetic fields as a function of temperature, which reflects similar trends to the magnetization phase diagram of bulk GdTi3Bi4, providing compelling evidence for an intertwined spin-charge ordered state.

Results
GdTi3Bi4 has a layered orthorhombic lattice structure in the space group Fmmm (Fig. 1a). The prototype structure consists of four slightly distorted Ti kagome-lattice layers per unit cell, which is larger than that of kagome superconductor CsV3Sb531. Another interesting structural motif of interest is the formation of Gd-Gd zigzag chains along a direction (Fig. 1a). The weak interlayer interactions between adjacent Gd/Bi layers (light green shade in Fig. 1a) result in the Gd terminated surface (Fig. 1b) after cleaving the GdTi3Bi4 crystal32. The GdTi3Bi4 crystal exhibits quasi-two-dimensional energy band. The Fermi surface contains three pairs of bands (Fig. 1c): triangular bands from Ti 3d orbitals (blue), quasi-1D band from Gd 5d orbitals (purple), and an inner band from Bi 6p orbitals (orange). The hybridization of the Gd-Gd zigzag chain with Bi 6p orbital band results in the elongated shape of the inner pocket. The uniaxial direction of the ellipse is perpendicular to quasi-1D band.

The GdTi3Bi4 exhibits an antiferromagnetic ground state below a Néel temperature of TN ~14 K, as demonstrated by magnetization measurements at a temperature range from 2 K to 120 K (Fig. 1d). In addition, GdTi3Bi4 shows a specific 1/3 plateau state with the magnetic field along the c-axis between Hc1 and Hc2. When the field is higher than Hc2, it turns to fully polarized ferromagnetic states (Fig. 1e).

We first study the electronic states of GdTi3Bi4 at atomic scale. The STM image of Gd terminated surfaces show flat terrace with randomly distributed defects (Fig. 1f). The height difference between two adjacent terraces is about 1.3 nm, consistent with half of the lattice constant of 1.3 nm along c direction (Fig. S1). The typical atomically-resolved STM images at relatively large sample bias, e.g., 320 mV, show hexagonal lattice with Gd atoms as Gd layer is slightly higher than the Bi atomic layer (Figs. 1f and S2). When approaching the tip closer to the sample surface by using small sample bias e.g., −10 mV, the STM images resolve both Gd and Bi atoms (Fig. 1g, for details see Fig. S2). The local density of states (LDOS) at Gd terminated surface display a V-shape metallic feature around the Fermi level (EF) with a broad conductance peak at 270 mV, as evidenced by the spatially averaged dI/dV spectra (Fig. 1h). The broad peak is spatially localized at the Gd atoms, as evidenced by the spatial distribution of dI/dV spectra (Fig. S3). In addition, the broad peak gradually split with increasing external magnetic field perpendicular to the sample surface (Fig. 1h, for details see Fig. S4). Thus, we attribute the broad peak to a localized magnetic state from Gd atoms.

In addition to the local magnetic state at 270 meV, there are some electronic states at low energies in the dI/dV spectra of Gd surface. The spatial distribution of LDOS around EF e.g., 20 meV (Fig. 2a), shows the real-space LDOS oscillation around the defects originated from quasiparticle interferences (QPI). These QPI signals result in an elongated circular pattern centered at Г point in the corresponding Fourier transform (FT) image of dI/dV map (Fig. 2b). The elongated circular QPI is originated from the intra-band scattering of ellipse band. The long axis of such QPI pattern corresponds to the uniaxial direction of the ellipse band from Bi band. By comparing the chain direction with the calculated Fermi surface, we attribute the long axis of elongated circular QPI pattern (Q3B in Fig. 2b) to the zigzag chain direction (a direction).

The FT cut along one direction (orange dashed line in Fig. 2b) in the q space shows a dispersing parabolic curve (white dashed dotted curve in Fig. 2c), corresponding to the expending elongated circular QPI patterns which grows outward with increasing energy (details see Fig. S5). The dispersive parabolic band is interrupted by the localized magnetic state at an energy of 270 meV. Such interruption is attributed to the hybridization between itinerant electrons and local moments from Gd atoms33,34.

We then focus on the itinerant electronic states near EF. In the FT of both STM topography (Fig. 2d, e) and low-energy dI/dV maps (Fig. 2f, g), there are three pairs of additional peaks around the circular QPI pattern (For details, see Fig. S6). To understand the origin of these extra peaks in the FT, we collect the dI/dV maps at various energies, i.e., g(r, E) (r is the real space location while E is the energies). The g(r, E) maps in large energies (e.g., −280 meV, Fig. 2f) show super-modulations propagating along three directions, forming a superlattice over the pristine structural lattice of Gd atoms (Fig. 2h). These additional modulation gives rise to the additional peak structures seen in the FTs, termed as qi (i=1, 2, 3) vectors (Fig. 2g).

To distinguish signals between dispersive QPI and non-dispersive CDW orders, we study the energy dependence of the qi vectors associated with the peaks in the FT. To do this, we obtain line cuts from FTs of the LDOS maps along the three directions in momentum space and plot this as a function of energy (Fig. 2i). Contrary to energy-dispersive features such as QPIs, we find that the magnitudes of these three q vectors show no energy dependence (details see Fig. S7). This indicates that the observed modulations arise from a CDW order in this material. We label the CDW vectors qi by the blue (q1), red (q2), and green (q3) squares. The periodicities of the CDW in three directions are close to but not equal to three times of the lattice constant a0 (the vector length: |q1CDW| = 0.31|Q1B|, |q2CDW| = 0.37|Q2B| and |q3CDW| = 0.40|Q3B|). The 3Q CDW is incommensurate with the lattice periodicity. In addition, the three CDW vectors do not align with crystalline lattice directions precisely (cyan lines in Fig. 2g). This leads to distorted hexagonal unit cell of the CDW (Fig. 3a) that breaks rotation and all mirror symmetries, indicating that the CDW is chiral11. We stress that the intensities of the CDW components, which are reflected in the heights of the CDW peaks in the FT images, are generally different in different directions and difficult to characterize systematically in our measurements. Thus, we will not comment specifically on the rotation symmetry and rotation symmetry breaking reflected by the CDW intensity in the conductance maps. Instead, we focus on the symmetry properties of the CDW defined by the wave vectors qi, i.e., the CDW peak locations in the FT images. To reveal the spatial configuration of the CDW, we filter the FT images by passing qiCDW and QiB spots and produce the real space image by the inverse FT (Fig. 3b), which highlights that the CDW is incommensurate with both the periodicity and orientation of the underlying lattice.

The incommensurate CDW with chirality is tunable by the external magnetic field. We study the evolution of three CDW vectors with the external magnetic field perpendicular to the sample surface (Bz) at 0.4 K. In the magnetic ground state (B < μHc1), the 3Q CDW remain approximately unchanged (Fig. 3c). In contrast, increasing the field to the 1/3 plateau state (μHc1 < B < μHc2), the three CDW vectors jump quickly to align with the crystalline directions. Independent with the magnetic field, the three qiCDW vectors satisfy the triangle relationship q1CDW + q2CDW + q3CDW = 0, (Fig. S9). Further increasing B, the q3CDW vector first disappears as the CDW gradually melts and all the CDW peaks become invisible in the FM state when B > μHc2 (Fig. 3c), indicating the complete quantum melting of the CDW in the fully polarized ferromagnetic phase. It suggests the strong coupling of quantum mechanical ground state of CDW to the magnetic ordering.

We note the 3Q CDW order in the 1/3 plateau magnetization phase, e.g., at 2.25 T, shows restored rotation and all mirror symmetries in terms of both vector orientation and magnitude, forming a nearly commensurate 3a0 × 3a0 CDW pattern (Fig. 3d). In addition, the inverse FT image obtained by filtering out qiCDW and QiB spots reveals real-space 3a0 × 3a0 modulations that are nearly commensurate with the underlying crystalline lattice (Fig. 3e).

To demonstrate the robustness and repeatability of the field-induced switching of 3Q CDW between ground state and 1/3 plateau state, we systematically study the CDW under the sequence of magnetic field of (−2.25 T) - 0 T - (2.25 T) (Fig. 3f, g). We quantitively study the variation of 3Q CDW with external magnetic field by defining the relative angle and relative length. The relative angle is the angle difference between the CDW vectors and corresponding Bragg vectors. The positive (negative) value means the CDW vector is on the right (left) of the closest Bragg vector. The three relative angles transform from finite values to zero when Bz increase to μHc1 (Fig. 3f). The values of the three relative lengths (|qiCDW|/|QiB|) deviate from 1/3 at 0 T and approach to 1/3 when Bz increases to μHc1 (Fig. 3g). Reversing the filed direction, we observe similar transitions in both relative angle and length with weak hysteretic behaviors (Fig. S10). Therefore, the incommensurate CDW in the AFM ground state evolves into a nearly commensurate 3a0 × 3a0 CDW in the 1/3 plateau state.

Having studied the quantum melting at low temperature by the magnetic field35, we turn to investigate the thermal melting36 of the 3Q CDW by increasing the temperature in zero magnetic field. We measure and compare the FTs of dI/dV maps at various temperatures over an identical surface region (Fig. 4a). When the temperature is increased from 0.4 K, the three CDW peaks remain stable at 2.0 K. However, in the temperature range from 2.2 K to 2.6 K, we observe that the intensities of two CDW peaks (q2 and q3) diminish sequentially (see method and Fig. S12). At 4.2 K, only a unidirectional 1Q-CDW order described by the single vector q1 is observed (Fig. 4a). Further increasing the temperature, the intensity of q1 diminishes and disappear at approximately the Néel temperature (TN ~ 14 K). The closeness of the melting temperature of 1Q-CDW TCDW to the Néel temperature TN indicates that the thermal melting of CDW is accompanied by that of the magnetic order, suggesting that the CDW is intertwined with magnetism thermodynamically.

It is instructive to investigate the magnetic field dependence of the 1Q-CDW phase. When the field is increased to above 1 T, the 1Q CDW transitions into another nearly commensurate 3Q CDW state (Fig. 4b, c). The latter is then gradually suppressed when the magnetic field is increased to 3.4 T. (Fig. S14). The field-induced melting at 4.2 K is smaller than the value at 0.4 K, indicating the importance of the underlying thermal fluctuations at reducing the ordered magnetic moment.

The phase diagram for the observed cascades of density wave in the plane of temperature and magnetic field is constructed using the STM data taken at the phase locations marked by the red symbols (Fig. 4d, for details see Fig. S10, S13 and S14). For comparison, we also present the magnetic phase diagram determined recently by bulk magnetization and magnetic force microscopy measurements (Fig. 4e, details see method part and Fig. S16, S17). The two phase diagrams (Fig. 4d, 4e) show a remarkable similarity in the overall structure of phase regions with approximate alignment of the phase boundaries. The magnetic ground state (AFM-I in Fig. 4e) has AFM order with the magnetic moments in a–c plane without b component37. The corresponding ground state observed by STM (orange region in Fig. 4d) is an incommensurate 3Q charge density wave. Increasing the magnetic field, the magnetic ground state undergoes a hysteretic transition to the phase with 1/3 magnetization plateau and magnetic moments still confined to the a–c plane (1/3-plateau-I in Fig. 4e). Intriguingly, this field-induced magnetic transition is captured in the density wave phase diagram by the field induced, weakly hysteretic incommensurate to nearly commensurate 3Q CDW transition in Fig. 4d (and Figs. 3f, g and S10). Similarly, increasing the temperature from the magnetic ground state drives a transition from AFM-I to the phase AFM-II characterized by a bifurcation in magnetic anisotropy that introduces a b-axis component to the magnetic moments37 (Fig. 4e). This magnetic transition corresponds to the incommensurate 3Q CDW to an incommensurate 1Q CDW transition observed by STM (light yellow in Fig. 4d). Driving the latter by increasing magnetic field into the nearly commensurate 3Q CDW in Fig. 4d accompanies the field-induced transition from AFM-II to the 1/3 plateau-II phase (Fig. 4e).

Methods
Single crystal growth of GdTi3Bi4
The flux method is utilized to synthesize the high–quality single crystals. The starting elements of Gd:Ti:Bi in the molar ratio of 1.2:1:20 are mixed in an alumina crucible and sealed in a quartz ampule. The ampule is placed in a furnace and heated to 1000 °C at a rate of 1 °C/min. After maintaining it at 1000 °C for 16 h, the ampule is slowly cooled down to 520 °C in 2000 min. The hexagonal shaped single crystals are obtained after centrifuging in order to remove the excess flux32.

Scanning tunneling microscopy/spectroscopy
The samples used in the STM/S experiments were cleaved at low temperature (80 K) and immediately transferred to an STM chamber and cooled down to 4.2 K. Experiments were performed in an ultra-high vacuum (1×10-10 mbar) ultra-low temperature STM system equipped with external magnetic field perpendicular to the sample surface. The lowest base temperature is 0.4 K with an electronic temperature of 650 mK (calibrated using a standard superconductor, Nb crystal). The magnetic field up to 11 T was applied using the zero-field cooling technique. All the scanning parameters (setpoint voltage Vset and tunneling current Iset) of the STM topographic images are listed in the figure captions. The dI/dV spectra were acquired by a standard lock-in amplifier at a modulation frequency of 973.1 Hz; the modulation bias (Vmod) is listed in the figure captions. Non-magnetic tungsten tips were fabricated via electrochemical etching and calibrated on a clean Au(111) surface prepared by repeated cycles of sputtering with argon ions and annealing at 500 °C. To remove the effects of small piezoelectric and thermal drifts during the acquisition of dI/dV maps, we apply the Lawler-Fujita drift-correction algorithm40 in angle and relative length (|qiCDW|/|QiB|) calculation in Fig. 3, which corrects the atomic Bragg peaks in the FTs of dI/dV maps to a single pixel to enhance calculational accuracy. The Fourier analyses in Figs. 2 and 4b, c are direct Fast Fourier transforms of corresponding real space signals. Figure 3b, e are inverse Fourier transforms of the filtered Fourier transforms. The methodological details are listed in the caption of Fig. S8.

Magnetic force microscopy
The MFM experiments were captured by a commercial magnetic force microscope (attoAFM, attocube) using a commercial magnetic tip (MFMR, Nanoword) based on a closed-cycle He cryostat (Bluefors) and dilution refrigerator system (Bluefors), combined with a vector magnet (9-3-1, Bluefors). The scanning probe system was operated at the resonance frequency of the magnetic tip, approximately 70 kHz. The MFM images were taken in constant height mode with the scanning plane nominally ~250 nm above the sample surface. The MFM signal, i.e., the resonance frequency shift (Δf), is proportional to the out-of-plane stray field gradient. Darker (brighter) magnetic contrast indicated the more attractive (repulsive) interaction between the magnetic tip and the sample stray field. GdTi3Bi4 single crystals were cleaved under ambient conditions to expose fresh surfaces before conducting MFM measurements.

Critical temperature of 3Q-1Q transition of 3Q CDW
To quantitatively evaluate the melting of the CDW phase, we extract FT line cuts from the center of the FT through each of the three CDW peaks as a function of temperature from 1.0 K to 2.6 K (details see Fig. S12). By plotting the peak intensities as a function of temperature, we find that the q1 peak keeps unchanged while the other two peaks get progressively suppressed with increasing temperature (Fig. S12). The critical temperature for q3 is 2.5 K, which is higher than that of q2 (2.2 K). The distinct critical temperature for the suppression of three CDW peaks further demonstrate the independent order parameters of the 3Q CDW orders. The critical temperature around 2.2 K corresponds to the magnetic state transition in the MFM measurements (Figs. 4e and S16 and S17), further supporting the strong correlation between the CDW and magnetic orders.

Magnetic transition at the critical temperature of ~2 K
We investigate the magnetic transition below 2 K by performing magnetic force microscopy (MFM) measurements of GdTi3Bi4 single crystals at temperatures. At 0.4 K, we observe the formation and evolution of uniformly stripe domains along the a-axis direction during the magnetic transition process due to the quasi-1D magnetic structure of GdTi3Bi4, leading to a pronounced in-plane magnetic anisotropy (Fig. S16). During the increasing magnetic field procedure, stripe domains emerge at 1.79 T and vanish completely at 1.99 T, indicating a magnetic transition within a range of 0.20 T. However, during the decreasing magnetic field procedure, the stripe domains reappear between 1.53 T and 1.73 T, suggesting a magnetic field hysteresis of 0.24 T (Fig. S16). In contrast, as the temperature increases to 4 K, the stripe domains no longer align strictly along the a-axis and exhibits some deviation (Fig. S17). Meanwhile, the hysteresis becomes negligible, and the magnetic transition window narrows. The switching between the presence and absence of hysteresis behavior, occurring at temperatures below TN, indicates a transition in the microscopic spin structure41–44. Therefore, the difference in the orientation of stripe domains and the disappearance of hysteresis suggest the presence of two distinct magnetic structures, with a phase transition occurring between 0.4 K and 4.0 K. The precise temperature-dependent measurements reveal the vanishing of hysteresis around 2.5 K, indicating the manifestation of a spin phase transition, while the second magnetic transition supports the same inference. The spin structure is expected to show more pronounced characteristics at the atomic scale, and potential coupling with electronic states may also be observed. Accordingly, we plot H-T phase diagram of GdTi3Bi4 by extracting the magnetic transition window and field hysteresis at various temperatures (Fig. 4e).

Repeatable observation of CDW in GdTi3Bi4
Except for the data shown in the main Figures, we have verified the existence of the CDW exhibiting the same periodicity and orientation with the crystalline lattice across different samples and tips from three different growth batches (Fig. S15).

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "This manuscript reports low-temperature STM/STS studies on GdTi3Bi4, claiming the discovery of a charge-spin density wave (CSDW). The authors observe an incommensurate, field- and temperature-tunable CDW and argue that its evolution closely tracks bulk magnetic phase transitions, implying a charge-spin coupled nature.\nThe experiments are technically impressive, demonstrating challenging STM/STS measurements under magnetic fields with detailed spatial mapping. The observation of a field-tunable CDW in a kagome metal is interesting, and the suggestion of chirality adds further intrigue and relevance for the correlated electron materials community.\nHowever, the central claim of observing a CSDW is not convincingly supported. STM measures charge density modulations; spin modulation is inferred indirectly from field dependence and phase diagram similarities. This is insufficient to establish the presence of spin density waves at the same wavevector as the CDW, which is the defining criterion for a CSDW.\nMajor comments:\n1. Lack of direct evidence for spin modulation and comparison to existing literature\nThe identification of a CSDW in this manuscript is based solely on the field dependence of the CDW and its similarity to magnetic phase diagrams. While these observations suggest coupling between CDW and magnetism, they do not prove that the CDW carries spin density modulation at the same wavevector, which is a defining feature of a CSDW.\nA recent study by Hu et al. (Nat. Commun. 2022) demonstrated coexisting SDW and CDW on Cr(001) via direct spin-polarized STM measurements, unambiguously distinguishing spin and charge modulations. In contrast, the present manuscript lacks direct spin-sensitive data, relying solely on field-dependent CDW behavior to infer a CSDW. This represents a significant difference in evidence strength. While tunability of CDW by magnetic field suggests coupling, it does not imply intrinsic spin modulation without direct experimental confirmation.\nRecommendation: The authors should either provide direct spin-sensitive measurements (e.g. spin-polarized STM, neutron diffraction at q_CDW) to support the CSDW claim, or significantly tone down their conclusions, presenting the results as an observation of an unconventional CDW strongly coupled to magnetism.\n2. Lack of microscopic mechanism discussion\nThe manuscript does not discuss the microscopic origin of such a CSDW in GdTi3Bi4. Is it driven by Fermi surface nesting combined with exchange splitting, or another instability typical of kagome systems?\nRecommendation: Include a theoretical discussion outlining possible mechanisms for CSDW formation in this material to strengthen the interpretation.\n3. Alternative interpretations not addressed\nThe possibility that the observed CDW is purely charge in nature but stabilized or modified by magnetic ordering remains viable and is not ruled out. This is particularly important given that STM data alone cannot resolve spin components, and the field dependence of the CDW may arise from indirect coupling to magnetic order rather than intrinsic spin modulation.\nRecommendation: Discuss alternative explanations explicitly and justify why the CSDW interpretation is preferred over a scenario where a conventional CDW is stabilized by underlying magnetic order.\nMinor comments:\n1. Consistency and proofreading: The manuscript should be thoroughly proofread. For example, it states that magnetization data were collected up to 100 K, but Figure 1(d) shows data up to 120 K. Additionally, Figure 1(d) indicates 0.1 T in the axis label, while the caption states 100 Oe. Such inconsistencies must be corrected to ensure clarity and precision.\n2.Fourier analysis clarity: The manuscript includes Fourier transforms of STM topographs to identify CDW wavevectors, but lacks methodological details (e.g. window size, filtering, background subtraction). A brief clarification of the analysis procedure would improve transparency and reproducibility.\n3. References: Check for duplicate entries (e.g. Agterberg et al. Annu Rev Condens Matter Phys 2020).\n4. Terminology consistency: The manuscript uses terms like CDW, chiral CDW, and CSDW interchangeably. These should be clearly defined upon first use and applied consistently, as the distinction is critical to the interpretation of the data.", "reviewer_2": "This article presents beautiful STM data as a function of magnetic field and temperature in a Kagome metal. The data convincingly demonstrate a coupling between magnetic order and the charge density wave (CDW), a relationship that had been suspected but is now clearly proven. The experiments were executed with great care, and the interpretation is highly compelling. In conclusion, this is a strong piece of work that deserves publication in its current form and will undoubtedly attract significant interest from a broad scientific community.\nMinor comments:\n1.Surface-bulk correspondence:\nRegarding the cleavage of the material, what guarantees that the surface electronic structure reflects that of the bulk? In particular, the Gd-Gd zigzag chains are broken at the surface—is that correct? One might then expect that the charge-spin density waves observed at the surface differ from those in the bulk. For example, NbSe₃ exhibits different electronic orders at the surface compared to the bulk. How do the authors ensure that their measurements are representative of the bulk properties? Is this conclusion based on the excellent agreement between the deduced temperature–magnetic field phase diagram and previous bulk measurements (e.g., Figs. 2d–e)?\n2.Attribution of spectral features:\nRelated to the above point, the splitting observed in the dI/dV spectra (Fig. 1h) is attributed to the Gd zigzag chains. However, if these chains are broken at the surface, how can they contribute to the observed spectral features? Could the authors clarify this apparent contradiction?\n3.Figure 2b, clarity of QPI features:\nIn Fig. 2b, the quasiparticle interference (QPI) patterns are not particularly clear. Could the authors consider removing the dotted purple guide-for-the-eye in half of q-space, so the reader can compare the data with and without the overlay?", "reviewer_3": "Han et al study the magnetic kagome metal GdTi3Bi4. The material shows antiferromagnetic order at zero field, and the moments can be polarized with a relatively small field of a few tesla. Authors show that spectra change with magnetic field (figure 1). Authors also claim that the material has a CDW seen in STM, and that the wavelength and symmetry of the CDW changes from the antiferromagnetic to the polarized state. This is an interesting finding - often, magnetism can cause only weak changes to the electronic structure, and this material appears to be a case where the coupling to electronic structure is strong. So, it is a good contribution to the literature and should be published if the authors can make their claims solid. I have some comments and questions to help the authors improve their manuscript:\n1. There is no discussion of the magnetic order in the antiferromagnetic state. Does its wavelength have any relationship to the CDW seen in experiment? Without this, it is difficult to understand the importance/relevance of the main results.\n2. Relevant to fig 1: “Thus, we attribute the broad peak to a localized magnetic state from Gd zigzag chains” Can the authors show more magnetic field dependence? It is not clear whether the spectral changes are abrupt, like the magnetization, or whether the changes are gradual.\n3. Fig 2 b, e and g are central to the manuscript. However, their presentation is not so good - with many squares and circles drawn, it is difficult for the reader to independently assess the data. For example, in figure 2b, the authors put squares around where they think the CDW wavevectors are. But, looking at the Fourier transform data, there are many other pixels which have the same intensity as the ones called CDW, so it is not clear what the basis for the authors choice is. The authors need to present the data better to make the claim clear to the readers.\n4. It is useful to understand whether the CDW patters are truly incommensurate, or whether they are commensurate but with phase slips. The areas shown in figure 3 are too small to judge. Could the authors investigate this more carefully?\n5. There is a magnetic phase diagram mentioned from MFM measurements, but there is no data presented for this in the main text. I suggest that the authors substantiate their phase diagram with some data in the main text that allows the reader to understand where the phase diagram comes from."}, "author_response": "Point-by-point responses to the comments from the Reviewers\nWe thank the reviewers for the constructive criticisms, comments and suggestions. As an overall\nresponse, we toned down our claim that the CDW we observed are spin-intertwined CDW instead\nof spin-charge density wave and we discussed a few possible mechanisms including the SDW\norigin and CDW indirectly coupled with magnetic order. We have added microscopic mechanism\ndiscussion based on Ginzburg-Landau theory. The calculated phase diagram matches well with our\nexperiment results. Besides, with the helpful advice raised by Reviewers, we have improved the\ndata presentation, added more information to make the manuscript more accessible.\nBelow, we respond to the comments and criticisms of the reviewers point-by-point. The comments\nfrom the reviewers are included in black italic typeface. Our responses are in regular blue typeface.\nThe changes to the text in the revised manuscript are described or quoted in red.\nResponse to the Reviewer #1\nThis manuscript reports low-temperature STM/STS studies on GdTi3Bi4, claiming the discovery of a\ncharge-spin density wave (CSDW). The authors observe an incommensurate, field- and temperature-\ntunable CDW and argue that its evolution closely tracks bulk magnetic phase transitions, implying a\ncharge-spin coupled nature.\nThe experiments are technically impressive, demonstrating challenging STM/STS measurements under\nmagnetic fields with detailed spatial mapping. The observation of a field-tunable CDW in a kagome\nmetal is interesting, and the suggestion of chirality adds further intrigue and relevance for the correlated\nelectron materials community.\nHowever, the central claim of observing a CSDW is not convincingly supported. STM measures charge\ndensity modulations; spin modulation is inferred indirectly from field dependence and phase diagram\nsimilarities. This is insufficient to establish the presence of spin density waves at the same wavevector\nas the CDW, which is the defining criterion for a CSDW.\nResponse: We thank the Reviewer for the positive remarks of our work. We have revised our claim\nof CSDW to address the Reviewer’s main concerns of CSDW. We have discussed alternative\nmechanisms for the CDW, and we have checked carefully for the consistency and improved the\ndata presentation according to the Reviewer’s suggestions.\nMajor comments:\nComment 1: Lack of direct evidence for spin modulation and comparison to existing literature. The\nidentification of a CSDW in this manuscript is based solely on the field dependence of the CDW and its\nsimilarity to magnetic phase diagrams. While these observations suggest coupling between CDW and\nmagnetism, they do not prove that the CDW carries spin density modulation at the same wavevector,\nwhich is a defining feature of a CSDW.\nA recent study by Hu et al. (Nat. Commun. 2022) demonstrated coexisting SDW and CDW on Cr(001)\nvia direct spin-polarized STM measurements, unambiguously distinguishing spin and charge\nmodulations. In contrast, the present manuscript lacks direct spin-sensitive data, relying solely on field-\ndependent CDW behavior to infer a CSDW. This represents a significant difference in evidence strength.\nWhile tunability of CDW by magnetic field suggests coupling, it does not imply intrinsic spin modulation\nwithout direct experimental confirmation.\nRecommendation: The authors should either provide direct spin-sensitive measurements (e.g. spin-\npolarized STM, neutron diffraction at q_CDW) to support the CSDW claim, or significantly tone down\ntheir conclusions, presenting the results as an observation of an unconventional CDW strongly coupled\nto magnetism.\nResponse: We thank the Reviewer for the valuable comment regarding the need for direct evidence\nof spin modulation. In this work, we mainly report the observation of charge density wave which\nis strongly coupled to the magnetism in GdTi Bi . Based on this pronounced coupling, we initially\n3 4\ninterpreted the phenomenon as a manifestation of unconventional spin–charge density wave. We\nfully agree with the Reviewer that that our current data lack direct spin-sensitive measurements\n(such as spin-polarized STM or neutron diffraction) that could unambiguously identify a spin\ndensity modulation, as demonstrated in previous work (e.g., Hu et al., Nat. Commun. 2022).\nFollowing the Reviewer’s suggestion, we have toned down the conclusions and now describe the\nobserved state as a “spin-intertwined charge density wave” from an experimental perspective. This\nterminology more accurately reflects the present evidence—namely, that the CDW exhibits a strong\ncorrelation with the spin modulation, regardless of whether the spin modulation shares the same\nwavevector with the CDW or not. The novelty of our observation lies in this pronounced\nintertwining between spin and charge degrees of freedom, as manifested by the nearly identical\ncritical temperatures and magnetic field dependences of the CDW and magnetic transitions.\nAccordingly, we have revised the manuscript as follows:\nTitle: “Discovery of unconventional spin-intertwined charge density wave in magnetic phases of\nkagome metal GdTi Bi ”\n3 4\nAbstract:\n“… Here, we report the discovery of unconventional spin-intertwined charge density wave\n(CDW) … which provides strong evidence for the observation and manipulation of spin-\nintertwined CDW as a bulk property…”\nDiscussion part:\n“Indeed, a commensurate 1Q spin density wave (SDW) order has recently been detected by neutron\nscattering30 in a related magnetic kagome compound CeTi Bi . It is natural to speculate that the\n3 4\nSDW thus directly detectable by spin-polarized STM on the surface of GdTi Bi , which deserves\n3 4\nfurther studied in the future (Hu et al. Nat. Commun. 2022).”\nComment 2: Lack of microscopic mechanism discussion\nThe manuscript does not discuss the microscopic origin of such a CSDW in GdTi3Bi4. Is it driven by\nFermi surface nesting combined with exchange splitting, or another instability typical of kagome\nsystems?\nRecommendation: Include a theoretical discussion outlining possible mechanisms for CSDW formation\nin this material to strengthen the interpretation.\nResponse: We thank the Reviewer for the insightful suggestion. As addressed in our response to\nComment 1, we have revised our interpretation by toning down the claim of a spin–charge density\nwave, and now describe the observed state as a “spin-intertwined charge density wave” from an\nexperimental perspective, regardless of whether the spin modulation shares the same wavevector\nwith the CDW or not.\nFollowing the Reviewer’s recommendation, we have added a discussion on the possible\nmicroscopic mechanism underlying the spin-intertwined charge density wave. To\nphenomenologically understand the complex density-wave phase diagram observed in GdTi Bi ,\n3 4\nwe have constructed a Ginzburg-Landau (GL) model that incorporates both charge modulation and\nspin orders, as well as their coupling. We have added the corresponding results as Figure S18, S19\nand Table S1 in the Supplementary Information.\nIn this framework, the spin order is microscopically motivated by Fermi surface nesting, which\nhelps identify the most probable spin density wave (SDW) ordering vectors 𝒒 . Crucially, the\n𝑆𝐷𝑊\ninteraction term in our GL free energy imposes a strict momentum conservation condition:\n2𝒒 +𝒒 =𝑮, where 𝑮 is the reciprocal lattice constant. This provides a concrete theoretical\n𝑆𝐷𝑊,𝑖 𝑖\nlink, explaining how the nesting-driven SDW order can induce the experimentally observed CDW\nmodulations.\nWhile a full microscopic derivation for the CDW is challenging — given that the incommensurate\nnature of the primary SDW order would lead to highly complex band folding—our\nphenomenological approach provides a robust explanation. This framework, driven by nesting-\ninduced coupling and competition, successfully reproduces the key experimental observations,\nincluding the simultaneous onset of the CDW and SDW orders, as well as the overall topology of\nthe complex temperature-magnetic field phase diagram.\nWe have added the theoretical discussion in our manuscript as below:\n‘Therefore, we develop a Ginzburg-Landau (GL) model incorporating strongly coupled CDW and\nAFM order parameters (For details, see Section VI in the Supplementary information) to\nphenomenologically understand the observed phase diagram of CDW in GdTi Bi . The model’s\n3 4\ncrucial feature is the strong coupling between spin and charge, which is microscopically justified\nby the Fermi surface nesting driving the AFM order, and the momentum conservation that links the\nAFM and CDW ordering vectors. The model also includes mutual competition between different\nAFM components, a consequence of their overlapping nesting regions on the Fermi surface (Fig.\nS18). This theoretical framework, driven by nesting-induced coupling and competition,\nsuccessfully reproduces the key experimental observations: simultaneous onset of the CDW and\nAFM orders, and the overall topology of the complex temperature-magnetic field phase diagram\n(Fig. S19). It thus provides a concrete understanding of the intricate spin-charge interplay in\nGdTi Bi .’\n3 4\nWe have added the corresponding results as Figure S18, S19 and Table S1 in the Supplementary\nInformation as below:\nGinzburg-Landau analysis for the spin-intertwined charge density wave\nTo phenomenologically understand the complex density-wave phase diagram observed in\nGdTi Bi , we construct a Ginzburg-Landau (GL) model that incorporates both CDW and SDW\n3 4\norders, as well as their coupling. The total free energy 𝑓 of the system can be written as:\n𝑡𝑜𝑡\n𝑓 [{𝜙 },{𝒎 };𝑇] = 𝑓 [{𝜙 };𝑇]+𝑓 [{𝒎 };𝑇]+𝑓 [{𝜙 },{𝒎 }] (𝑆1)\n𝑡𝑜𝑡 𝑖 𝑖 𝐶𝐷𝑊 𝑖 𝑆𝐷𝑊 𝑖 𝑖𝑛𝑡 𝑖 𝑖\nwhere 𝑓 [{𝜙 };𝑇] and 𝑓 [{𝒎 };𝑇] denote the free energies associated with CDW and\n𝐶𝐷𝑊 𝑖 𝑆𝐷𝑊 𝑖\nSDW orders, respectively; 𝑓 [{𝜙 },{𝒎 }] represents their interaction; {𝜙 } (𝑖 = 1,2,3) are\n𝑖𝑛𝑡 𝑖 𝑖 𝑖\ncomplex scalar order parameters corresponding to the three CDW components observed in the\nexperiment; and {𝒎 } are complex vector SDW order parameters that couple to the CDW.\n𝑖\nWe start by examining the CDW part of the free energy 𝑓 [{𝜙 };𝑇] . The three\n𝐶𝐷𝑊 𝑖\ncomponents {𝜙 } (𝑖 = 1,2,3) correspond to density modulations at wavevectors 𝒒 described by\n𝑖 𝑖\n𝜌(𝑟) = 𝜌 +∑ (𝜙 𝑒𝑖𝒒𝑖⋅𝑟 +𝑐.𝑐.). Since experimentally the wavevectors 𝒒 satisfy the momentum\n0 𝑖 𝑖 𝑖\nconservation condition 𝒒 +𝒒 +𝒒 = 𝟎 , the free energy density for the CDW order\n1 2 3\n𝑓 [{𝜙 };𝑇], expanded up to quartic order, is given by:\n𝐶𝐷𝑊 𝑖\n3\n𝛽 𝜂\n𝑓 [{𝜙 };𝑇] = ∑[𝛼 (𝑇)|𝜙 |2+ 𝑖 |𝜙 |4]+∑𝛾 |𝜙 |2|𝜙 | 2 − (𝜙 𝜙 𝜙 +𝑐.𝑐.) (𝑆2)\n𝐶𝐷𝑊 𝑖 𝑖 𝑖 2 𝑖 𝑖𝑗 𝑖 𝑗 2 1 2 3\n𝑖=1 𝑖<𝑗\nHere, 𝛼 (𝑇) = 𝑎 (𝑇−𝑇 )/𝑇 is the temperature-dependent quadratic coefficient, which\n𝑖 𝑖 𝑐,𝑖 𝑐𝑖\nchanges sign at the bare CDW transition temperature 𝑇 for each component; the quartic\n𝑐,𝑖\ncoefficients 𝛽 > 0 ensures the stability of the transition; 𝛾 term describes the quartic interaction\n𝑖 𝑖𝑗\nbetween different components, and we take 𝛾 > 0 for mutual competition. The cubic coupling\n𝑖𝑗\nterm, permitted by momentum conservation, locks the relative phases of the three components and\ntypically favors a 3Q-CDW configuration; we take 𝜂 > 0 without loss of generality.\nWe now turn to the discussion of possible SDW orders. Despite the close match between\nthe CDW and magnetic phase diagrams observed experimentally, suggesting a strong coupling\nbetween them, the available MFM measurements are still insufficient to unambiguously identify\nthe specific SDW ordering pattern. Here, we theoretically identify the most probable SDW orders\nby a combination of Ginzburg-Landau constraints and Fermi surface properties. To capture the\ncoupling between CDW and SDW, we consider that the SDW order parameter 𝒎 , which is\n𝑖\ngenerally a vector and its lowest-order coupling to the CDW order 𝜙 to the free energy thus takes\n𝑖\nthe form (𝒎 ⋅𝒎 )𝜙 +𝑐.𝑐. . This coupling leads to a momentum conservation condition\n𝑖 𝑖 𝑖\n2𝒒 +𝒒 = 𝑮, where 𝑮 is a reciprocal lattice vector. Given the experimental observed CDW\n𝑆𝐷𝑊,𝑖 𝑖\nmomentum 𝒒 , we specifically consider the two SDW orders 𝒒 and 𝒒 , as illustrated in\n𝑖 𝑆𝐷𝑊,1 𝑆𝐷𝑊,2\nFig. S18, since these choices efficiently nest a large portion of the Gd Fermi surface and satisfies\nthe momentum conservation condition. Denoting the corresponding SDW order parameters as\n{𝒎 } (𝑖 = 1,2), the free energy for the SDW order 𝑓 [{𝒎 };𝑇], expanded up to quartic order, is\n𝑖 𝑆𝐷𝑊 𝑖\nwritten as:\n𝛽\n𝑓\n𝑆𝐷𝑊\n[{𝒎\n𝑖\n};𝑇] = ∑ [𝛼\n𝑀,𝑖\n(𝑇)|𝒎\n𝑖\n|2+ 𝑀\n2\n,𝑖 |𝒎\n𝑖\n|4]+𝛾\n𝑀\n|𝒎\n1\n|2|𝒎\n2\n|2 (𝑆3)\n𝑖=1,2\nwhere, similar to the CDW case, 𝛼 (𝑇) = 𝑎 (𝑇−𝑇 )/𝑇 changes sign at bare SDW\n𝑀,𝑖 𝑀,𝑖 𝑀,𝑖 𝑀𝑖\ntransition temperature 𝑇 ; 𝛽 > 0 ensures the stability; and 𝛾 > 0 introduces competition\n𝑀,𝑖 𝑀,𝑖 𝑀\nbetween the two SDW components, since their associated Fermi surfaces are nearly identical. As\ndiscussed earlier, the interaction term 𝑓 [{𝜙 },{𝒎 }] is given by:\n𝑖𝑛𝑡 𝑖 𝑖\n𝜆\n𝑖\n𝑓\n𝑖𝑛𝑡\n[{𝜙\n𝑖\n},{𝒎\n𝑖\n}]= − ∑\n2\n[(𝒎\n𝑖\n⋅𝒎\n𝑖\n)𝜙\n𝑖\n+𝑐.𝑐.] (𝑆4)\n𝑖=1,2\nThis coupling term is crucial for SDW order 𝒎 to induce a finite CDW order 𝜙 even above its\n𝑖 𝑖\nbare transition temperature 𝑇 . And without loss of generality, we take 𝜆 > 0.\n𝑐,𝑖 𝑖\nHaving defined the Ginzburg-Landau free energy, we numerically minimize it respect to\nthe amplitudes |𝜙 | and |𝒎 | to investigate the phase diagram. The parameters adopted in our\n𝑖 𝑖\nanalysis are listed in Tab. S1. Notably, all bare CDW transition temperatures 𝑇 are relatively low,\n𝑐,𝑖\ncorresponding to the emergence of the 3Q phase at ~2.5K; while the bare SDW transition\ntemperature 𝑇 are relatively high, consistent with the magnetic transition temperature ~14K. The\n𝑀,𝑖\ninterplay between CDW and SDW orders strongly modifies this bare behavior, resulting in phase\ndiagrams consistent with experimental observations.\nWe first focus on the zero magnetic field case. As shown in Fig. S19(a), at 𝑇 = 14K,\n𝑀,𝑖\ndue to the relatively strong repulsion term 𝛾 |𝒎 |2|𝒎 |2 between 𝒎 and 𝒎 , only one non-zero\n𝑀 1 2 1 2\nSDW order 𝒎 exists. The strong SDW-CDW coupling 𝑓 [{𝜙 },{𝒎 }] simultaneously drives the\n1 𝑖𝑛𝑡 𝑖 𝑖\nonset of CDW order 𝜙 (or 𝜙 ), well above the bare 𝑇 = 2.5K, resulting in a 1Q-CDW/SDW\n1 2 𝑐,1\nphase. As the temperature is lowered, the system undergoes a second-order transition into the 3Q-\nCDW/SDW phase at 𝑇 = 2.5 K, due to the sign change of the quadratic term 𝛼 (𝑇)|𝜙 |2 and the\n𝑖 𝑖\n𝜂\ntrilinear coupling (𝜙 𝜙 𝜙 +𝑐.𝑐.). This sequence is consistent with experimental observations\n1 2 3\n2\nat zero magnetic field.\nTo further simulate the effect of an external magnetic field, we treat the quartic SDW\ncoefficient 𝛽 as a tuning parameter, where a larger 𝛽 suppresses the SDW order, mimicking the\n𝑀 𝑀\neffect of an external magnetic field. Our numerical results, as shown in Fig. S19(b), reveal that\nincreasing 𝛽 drives a transition from the 1Q-CDW/SDW phase to a 3Q-CDW/SDW phase. The\n𝑀\nmechanism for this transition can be understood as follows: The applied field suppresses the overall\nmagnitude of the SDW order parameters |𝒎 |, which significantly weakens the competition term\n𝑖\n𝛾 |𝒎 |2|𝒎 |2. This reduction allows the coexistence both SDW order, 𝒎 and 𝒎 . Through the\n𝑀 1 2 1 2\nSDW-CDW coupling 𝑓 [{𝜙 },{𝒎 }] , these two SDW orders then induce their corresponding\n𝑖𝑛𝑡 𝑖 𝑖\nCDW components, 𝜙 and 𝜙 . Finally, the trilinear coupling among the CDW orders,\n1 2\n𝜂\n(𝜙 𝜙 𝜙 +𝑐.𝑐.), stabilizes the third component 𝜙 , leading to the emergence of the 3Q state\n1 2 3 3\n2\nobserved at higher fields which qualitatively explained the experiment.\n𝑇 𝑎 𝛽 𝛾 𝜂 𝑇 𝑎 𝛽 𝛾 𝜆\n𝑐,𝑖 𝑖 𝑖 𝑖𝑗 𝑀,𝑖 𝑀,𝑖 𝑀,𝑖 𝑀 𝑖\n2.5 1.0 1.0 0.5 1.0 14.0 1.0 0.5 1.5 0.5\nTable. S1. Parameters used in the Ginzburg-Landau free energy. Indices take all allowed values:\n𝑖,𝑗 = 1,2,3 for parameters in 𝑓 [{𝜙 };𝑇] ; 𝑖 = 1,2 for parameters in 𝑓 [{𝒎 };𝑇] and\n𝐶𝐷𝑊 𝑖 𝑆𝐷𝑊 𝑖\n𝑓 [{𝜙 },{𝒎 }]. The parameter 𝛽 is varied when considering the effect of magnetic field.\n𝑖𝑛𝑡 𝑖 𝑖 𝑀,𝑖\nFigure. S18. Illustration of the CDW and SDW ordering wavevectors. The experimentally\nobserved CDW wavevectors 𝒒 , 𝒒 , 𝒒 are illustrated in blue, red, green arrows respectively,\n1 2 3\nwhere the momentum conservation condition 𝒒 +𝒒 +𝒒 = 𝟎 is also explicitly checked at the\n1 2 3\nupper right portion of the figure. 𝑸 ,𝑸 ,𝑸 are three reciprocal wavevectors. Here, we consider\n1 2 3\ntwo probable SDW orders with wavevectors 𝒒 , 𝒒 , illustrated as dashed purple arrow.\n𝑆𝐷𝑊,1 𝑆𝐷𝑊,2\nThe momentum conservation conditions 2𝒒 +𝒒 = 𝑸 and 2𝒒 +𝒒 = 𝑸 , as well as\n𝑆𝐷𝑊,1 1 1 𝑆𝐷𝑊,2 2 2\nthe nesting property, are explicitly illustrated in the figure.\nFigure. S19. Phase diagrams obtained from the Ginzburg-Landau analysis. (a) Temperature\ndependence of the amplitude of the CDW and SDW order parameters, |𝜙 | and |𝒎 | , at zero\n𝑖 𝑖\nmagnetic field, showing the sequential onset of the 1Q-CDW/SDW and 3Q-CDW/SDW phases. (b)\nThe phase diagram as a function of temperature 𝑇 and quartic SDW coefficient 𝛽 , which mimics\n𝑀\nthe effect of an external magnetic field. Increase 𝛽 drives a transition from the 1Q-CDW/SDW\n𝑀\nphase to the 3Q-CDW/SDW phase. The black arrowed line presents the evolution of the spin-\nintertwined charge density wave under magnetic field in the experiments results at 0.4 K, where\nthe CDW remains 3q. The white arrowed line presents the evolution of the CDW under magnetic\nfield in the experiments results at 4 K, where a 1q – 3q transition is observed. The yellow arrowed\nline presents the temperature evolution of the CDW under zero field, where a 3q – 1q transition is\nobserved.\nComment 3: Alternative interpretations not addressed\nThe possibility that the observed CDW is purely charge in nature but stabilized or modified by magnetic\nordering remains viable and is not ruled out. This is particularly important given that STM data alone\ncannot resolve spin components, and the field dependence of the CDW may arise from indirect coupling\nto magnetic order rather than intrinsic spin modulation.\nRecommendation: Discuss alternative explanations explicitly and justify why the CSDW interpretation\nis preferred over a scenario where a conventional CDW is stabilized by underlying magnetic order.\nResponse: We sincerely thank the Reviewer for this valuable comment, which prompts us to\nexplicitly address the possibility that the observed CDW is charge in nature but stabilized or\nmodified by magnetic ordering.\nWe consider this scenario unlikely based on both experimental and theoretical evidence. First, the\ntemperature dependence of the CDW is fully consistent with the evolution of magnetic order. In\nsystems where CDWs are merely stabilized or modified by magnetic order (such as FeGe,\nNature\n609, 490–495 (2022)), the magnetic transition typically occurs at a higher temperature than the\nCDW transition. In contrast, in the present case of GdTi Bi , the critical temperature of the CDW\n3 4\nis nearly identical to the Néel temperature (T ~14 K), suggesting that the two orders develop\nN\ncooperatively rather than hierarchically.\nSecond, theoretical calculations show that the energy difference between the AFM-I and AFM-II\nstates in GdTi Bi at zero field is very small (Phys. Rev. Lett. 134, 226704 (2025)). If the CDW\n3 4\nwere simply stabilized by static AFM order, such a small magnetic energy scale would be\ninsufficient to drive the experimentally observed 3q-to-1q CDW transition at the AFM-I to AFM-\nII transition.\nTogether, these observations indicate that the CDW in GdTi Bi is not a conventional charge\n3 4\nmodulation stabilized by magnetism, but is instead intimately coupled to the spin degree of\nfreedom—consistent with the spin–intertwined charge density wave interpretation.\nAccordingly, we have added the following discussion to the revised manuscript:\n“The possibility that the CDW is purely charge in nature but stabilized or modified by magnetic\nordering is ruled out by the close correspondence between the CDW phase diagram and the\ncomplex magnetic behavior. In systems where magnetic order stabilizes the CDW (e.g., FeGe\n[Nature 609, 490–495 (2022)]), the magnetic transition typically occurs at a higher temperature\nthan the CDW transition. In contrast, in GdTi Bi , the CDW transition temperature is nearly\n3 4\nidentical to the Néel temperature (Tₙ ≈ 14 K), indicating a cooperative rather than hierarchical\nrelationship. Furthermore, theoretical calculations reveal that the energy difference between the\nAFM-I and AFM-II states at zero field is too small (Phys. Rev. Lett. 134, 226704 (2025)) to account\nfor the observed 3q-to-1q CDW transition if the CDW were merely stabilized by magnetism. These\nresults suggest that the CDW is intrinsically coupled to spin degrees of freedom, consistent with a\nspin–intertwined charge density wave.”\nMinor comments:\nComment 4: Consistency and proofreading: The manuscript should be thoroughly proofread. For\nexample, it states that magnetization data were collected up to 100 K, but Figure 1(d) shows data up to\ninconsistencies must be corrected to ensure clarity and precision.\nResponse: We thank the Reviewer for the careful reading of our manuscript. The label should be\n0.01 T, and the magnetization data were collected up to 120 K. We have checked the consistency\nand did careful proofreading throughout the paper.\nRevised Fig. 1d. Temperature dependence of magnetization under a fixed magnetic field of 0.01 T,\nshowing the antiferromagnetic transition occurs at T ~14 K.\nN\nWe have corrected the description of Fig. 1d in the revised manuscript as below:\n‘The GdTi Bi exhibits an antiferromagnetic ground state below a Néel temperature of T ~14 K,\n3 4 N\nas demonstrated by magnetization measurements at a temperature range from 2 K to 100 K 120 K\n(Fig. 1d).’\nComment 5: Fourier analysis clarity: The manuscript includes Fourier transforms of STM topographs\nto identify CDW wavevectors, but lacks methodological details (e.g. window size, filtering, background\nsubtraction). A brief clarification of the analysis procedure would improve transparency and\nreproducibility.\nResponse: We have taken the Reviewer’s suggestions and added a brief clarification of the analysis\nprocedure:\n(1) The Fourier analyses in Figure 2, Fig. 4b and Fig. 4c are direct Fast Fourier transforms of\ncorresponding real space signals without further processing.\n(2) Fig. 3b and Fig. 3e are inverse Fourier transforms of the filtered Fourier transforms. The\nmethodological details are listed in the caption of Fig. S8.\n(3) Fig 3c and Fig. 4a are Zoom-in FTs of dI/dV data to highlight the CDW q vectors, and we have\nadded statement in the figure caption.\nAccordingly, we have added methodological details for the Fourier analysis in the Method part as\nbelow:\n“The Fourier analyses in Figure 2, Fig. 4b and Fig. 4c are direct Fast Fourier transforms of\ncorresponding real space signals. Fig 3b and Fig 3e are inverse Fourier transforms of the filtered\nFourier transforms. The methodological details are listed in the caption of Fig. S8.”\nWe have added statements in the caption of Fig. 3c and Fig. 4a as below:\n“Fig. 3c: ‘c, Zoom-in FTs of dI/dV maps …’\nFig. 4a: ‘a, Zoom-in FTs of dI/dV maps at a temperature of 0.4 K, 2.0 K, 4.2 K, 8.0 K and 11.0 K,\nrespectively…”\nRevised Figure S8. The unprocessed data of Fig. 3b, e and methodological details. a, Large-\nsized dI/dV map at 0 T, -180 mV. b, Zoom-in of the data in the red square of (a), which is the\nunprocessed data of Fig. 3b. c, Filtered Fourier transform of (a) by 2D Gaussian function, the\nfiltering window size σ = 5 pixels, leaving only Q and Q . d, Inverse Fourier transform of (c),\nB CDW\nshowing generally uniform CDW in this region. e, Large-sized dI/dV map at 2.25 T, -180 mV. f,\nZoom-in of the data in the red square of (e), which is the unprocessed data of Fig. 3e. g, Filtered\nFourier transform of (e) by 2D Gaussian function, the filtering window size σ = 5 pixels, leaving\nonly Q and Q . h, Inverse Fourier transform of (g), showing generally uniform CDW in this\nB CDW\nregion.\nComment 6: References: Check for duplicate entries (e.g. Agterberg et al. Annu Rev Condens Matter\nPhys 2020).\nResponse: We thank the reviewer for careful reading of the manuscript. We have checked for\nduplicates across the manuscript and corrected the typos.\nComment 7: Terminology consistency: The manuscript uses terms like CDW, chiral CDW, and CSDW\ninterchangeably. These should be clearly defined upon first use and applied consistently, as the\ndistinction is critical to the interpretation of the data.\nResponse: We have carefully checked throughout the manuscript and confirmed that all the terms\nare defined upon first use. Since we now interpret the unconventional density wave we observed\nhere as spin-intertwined CDW, we now use term CDW in the results of our experiments, and\ndiscussed its most possible origin as spin-intertwined charge density wave (SCDW).\nResponse to the Reviewer #2\nThis article presents beautiful STM data as a function of magnetic field and temperature in a Kagome\nmetal. The data convincingly demonstrate a coupling between magnetic order and the charge density\nwave (CDW), a relationship that had been suspected but is now clearly proven. The experiments were\nexecuted with great care, and the interpretation is highly compelling. In conclusion, this is a strong\npiece of work that deserves publication in its current form and will undoubtedly attract significant\ninterest from a broad scientific community.\nResponse: We thank the Reviewer for the nice summary and recommendation for publication. We\nhave adopted the Reviewer’s constructive suggestions and provide experimental details requested\nby the Reviewer as follows.\nMinor comments:\nComment 1: Surface-bulk correspondence:\nRegarding the cleavage of the material, what guarantees that the surface electronic structure reflects\nthat of the bulk? In particular, the Gd-Gd zigzag chains are broken at the surface—is that correct? One\nmight then expect that the charge-spin density waves observed at the surface differ from those in the\nbulk. For example, NbSe₃ exhibits different electronic orders at the surface compared to the bulk. How\ndo the authors ensure that their measurements are representative of the bulk properties? Is this\nconclusion based on the excellent agreement between the deduced temperature–magnetic field phase\ndiagram and previous bulk measurements (e.g., Figs. 2d–e)?\nResponse: We appreciate the Reviewer’s insightful comment. Indeed, the Gd–Gd zigzag chains\nare broken at the surface. We agree that, in general, surface cleavage can possibly lead to atomic\nreconstruction and possibly distinct surface magnetic order compared to the bulk, as reported in\nother magnetic metals (Phys. Rev. B 106, 035124 (2022)). However, in GdTi Bi , although the Gd–\n3 4\nGd zigzag chains are broken at the surface, our STM data reveal that the atomic arrangement at the\nsurface remains consistent with the pristine bulk crystal structure, showing no evidence of Gd\nreconstruction. Moreover, the strong interlayer magnetic coupling between the topmost and\nunderlying layers ensures that the surface electronic and magnetic properties could reflect those of\nthe bulk.\nTherefore, we believe that the density wave observed at the surface originates from the bulk\nmagnetism. This conclusion is further supported by the excellent agreement between our deduced\ntemperature–magnetic field phase diagram and previous bulk measurements (Figs. 4d, e).\nTo clarify this point, we have added the following discussion to the revised manuscript:\n“The consistence between CDW and MFM diagram demonstrates that the CDW observed by STM\non the surface is a bulk property that reveals the microscopic textures of the intricated magnetic\nstructure in bulk GdTi Bi crystals. Notably, Although the Gd–Gd zigzag chains are disrupted upon\n3 4\ncleavage, the surface atomic structure remains essentially identical to that of the bulk, with no\nindication of Gd reconstruction (Phys. Rev. B 106, 035124 (2022)). Considering the strong\ninterlayer magnetic coupling, the observed spin-intertwined CDW is reasonably attributed to the\nbulk state.”\nComment 2: Attribution of spectral features:\nRelated to the above point, the splitting observed in the dI/dV spectra (Fig. 1h) is attributed to the Gd\nzigzag chains. However, if these chains are broken at the surface, how can they contribute to the\nobserved spectral features? Could the authors clarify this apparent contradiction?\nResponse: We agree with the Reviewer that we cannot directly attribute the observed spectral peak\nat Gd surface to the Gd zigzag chain because the zigzag chains are broken at the surface. Our main\nclaim is that there is a local moment around the Gd atom at the surface as the spatial distribution of\nthe peak is localized at Gd atoms and these peak split with increasing magnetic field.\nAccordingly, we revised the manuscript as follows:\n“Thus, we attribute the broad peak to a localized magnetic state from Gd zigzag chain atoms.”\nComment 3: Figure 2b, clarity of QPI features:\nIn Fig. 2b, the quasiparticle interference (QPI) patterns are not particularly clear. Could the authors\nconsider removing the dotted purple guide-for-the-eye in half of q-space, so the reader can compare the\ndata with and without the overlay?\nResponse: We sincerely appreciate the Reviewer’s insightful suggestion regarding the clarity of\nthe QPI features in Fig. 2b. Following this recommendation, we have removed the dotted purple\nguide-for-the-eye in half of the q-space, allowing for a direct comparison between the raw QPI data\nand the overlaid region. This modification significantly improves the visual clarity of the figure. In\naddition, we have removed the colored squares in Fig. 2b, as suggested by Reviewer #3 in his/her\nComment 3.\nRevised Fig. 2b. FT of Fig. 2a without purple-dotted overlay, showing the elongated circular QPI\npatterns (marked by the purple arrow) around impurities.\nResponse to the Reviewer #3\nHan et al study the magnetic kagome metal GdTi3Bi4. The material shows antiferromagnetic order at\nzero field, and the moments can be polarized with a relatively small field of a few tesla. Authors show\nthat spectra change with magnetic field (figure 1). Authors also claim that the material has a CDW seen\nin STM, and that the wavelength and symmetry of the CDW changes from the antiferromagnetic to the\npolarized state. This is an interesting finding - often, magnetism can cause only weak changes to the\nelectronic structure, and this material appears to be a case where the coupling to electronic structure is\nstrong. So, it is a good contribution to the literature and should be published if the authors can make\ntheir claims solid.\nResponse: We thank the Reviewer for the helpful comments and questions on improving the\nmanuscript. We have adopted the Reviewer’s constructive suggestions and corrected claims that\nmight be confusing to make them solid.\nI have some comments and questions to help the authors improve their manuscript:\nComment 1: There is no discussion of the magnetic order in the antiferromagnetic state. Does its\nwavelength have any relationship to the CDW seen in experiment? Without this, it is difficult to\nunderstand the importance/relevance of the main results.\nResponse: We thank the Reviewer for this insightful question, which prompts us to clarify the\nrelationship between the CDW and the underlying magnetic order.\nThe magnetic structure of GdTi Bi is indeed not yet fully established, as direct experimental probes\n3 4\nsuch as neutron scattering are lacking. However, recent studies using Magnetic Force Microscopy\n(Phys. Rev. Lett. 134, 226704, 2025) and ARPES (Phys. Rev. X 15, 031012 (2025)) in the ReTe\n3\nfamily proposed a possible magnetic structure with a period of approximately 3×𝑎 in real space\n0\nwithin the ab plane. This periodicity closely matches the CDW wave vector observed in our STM\nmeasurements. Such correspondence between the magnetic and charge modulations strongly\nsuggests that the CDW is intrinsically coupled to the spin degrees of freedom, consistent with a\nspin–intertwined charge density wave scenario.\nIn order to make the relevance of the main results, we have revised our manuscript as below:\n“We next turn to discuss the possible origin of observed CDW. The observation of the cascades of CDW\nclosely tracking the magnetic phases and transitions supports that the translation symmetry breaking\ninvolves both charge and spin in the primary modulations, giving rise to the possible formation of novel\nspin-intertwined CDW. To date, the magnetic structure of GdTi Bi is not yet fully established, as direct\n3 4\nexperimental probes such as neutron scattering are lacking. However, recent studies using Magnetic\nForce Microscopy [Phys. Rev. Lett. 134, 226704, 2025] proposed a possible magnetic structure with a\nperiod of approximately 3×a in real space within the ab plane. This periodicity closely matches the\nComment 2: Relevant to fig 1: “Thus, we attribute the broad peak to a localized magnetic state from\nGd zigzag chains” Can the authors show more magnetic field dependence? It is not clear whether the\nspectral changes are abrupt, like the magnetization, or whether the changes are gradual.\nResponse: Yes, we can show more magnetic field dependence in the new Fig. S4 below, which\nexhibits the gradual changes with magnetic field.\nWe have revised our manuscript as below:\n‘In addition, the broad peak gradually split with increasing external magnetic field perpendicular to the\nsample surface (Fig. 1h, for details see Fig. S4).’\nFigure S4. Evolution of the localized states on Gd atoms under magnetic field. a, dI/dV spectra\nof the localized state under different magnetic field with the intensity color map filled between each\nspectrum, showing a gradual evolution of the peak splitting around 270 mV. The spectra are\nnormalized and shifted for clarity. For all dI/dV data, V = 600 mV, I = 800 pA, V = 2.5 mV.\nset t mod\nComment 3: Fig 2 b, e and g are central to the manuscript. However, their presentation is not so good\n- with many squares and circles drawn, it is difficult for the reader to independently assess the data. For\nexample, in figure 2b, the authors put squares around where they think the CDW wavevectors are. But,\nlooking at the Fourier transform data, there are many other pixels which have the same intensity as the\nones called CDW, so it is not clear what the basis for the authors choice is. The authors need to present\nthe data better to make the claim clear to the readers.\nResponse: We thank the Reviewer for the insightful suggestion. We have removed the colored\nsquare and purple dashed overlay in Fig. 2b according to Comment 3 of Reviewer#2 to make a\nbetter presentation of the QPI patterns. But still, the CDW signals are observable in Fig. 2b even\nwith QPI patterns around, which is clearly shown in the line profiles along three Г-q directions\nCDW\nin the new Fig. S6 below. The CDW signals are well-established in high energy dI/dV map and\ncorresponding Fourier transform as shown in Fig. 2f, g, where the QPI intensity is much weaker.\nAnd the line profiles of CDW signals at different energy are already shown in Fig. 2i. Since the\nSTM topography is the integration of dI/dV signals from Fermi level to the setting voltage, the\nintegration of QPI signals makes the CDW signals relatively weak in Fig. 2e. But still, they are\nobservable in the line profiles in Fig. S6d.\nRevised Fig. 2b. FT of Fig. 2a without purple-dotted overlay, showing the elongated circular QPI\npatterns (marked by the purple arrow) around impurities.\nFigure S6. CDW signal inside the QPI pattern. a, The same Fourier transform as Fig. 2b in the\nmain text. b, Line profiles along three red dashed lines in (a), showing the q vectors of the CDW\ninside the QPI patterns are observable (marked by the gray arrowed line). The signal near q = 0 is\nmanually set to zero for clarity. c, The same Fourier transform as Fig. 2d in the main text. d, Line\nprofiles along three red dashed lines in (c), showing the q vectors of the CDW inside the QPI\npatterns (marked by the gray arrowed line). The signal near q = 0 is manually set to zero for clarity.\nWe revised our manuscript as below:\n‘(outlined by blue, green and red squares in Fig. 2b, For details, see Fig. S6).’\nComment 4:. It is useful to understand whether the CDW patters are truly incommensurate, or whether\nthey are commensurate but with phase slips. The areas shown in figure 3 are too small to judge.\nCould the authors investigate this more carefully?\nResponse: We have adopted the Reviewer’s advice and added further analysis of the large-sized\ndI/dV data of Fig. 3b and 3e in the new Fig. S8. It is now clear that in the large-sized region, the\nCDW patterns are truly incommensurate (Fig. S8c and Fig. 3a).\nRevised Figure S8. The unprocessed data of Fig. 3b, e and methodological details. a, Large-\nsized dI/dV map at 0 T, -180 mV. b, Zoom-in of the data in the red square of (a), which is the\nunprocessed data of Fig. 3b. c, Filtered Fourier transform of (a) by 2D Gaussian function, the\nfiltering window size σ = 5 pixels, leaving only Q and Q . d, Inverse Fourier transform of (c),\nB CDW\nshowing generally uniform CDW in this region. e, Large-sized dI/dV map at 2.25 T, -180 mV. f,\nZoom-in of the data in the red square of (e), which is the unprocessed data of Fig. 3e. g, Filtered\nFourier transform of (e) by 2D Gaussian function, the filtering window size σ = 5 pixels, leaving\nonly Q and Q . h, Inverse Fourier transform of (g), showing generally uniform CDW in this\nB CDW\nregion.\nComment 5: There is a magnetic phase diagram mentioned from MFM measurements, but there is no\ndata presented for this in the main text. I suggest that the authors substantiate their phase diagram with\nsome data in the main text that allows the reader to understand where the phase diagram comes from.\nResponse: We thank the Reviewer for the helpful advice. We have put some data in the magnetic\nphase diagram in the revised Fig. 4e as below.\nRevised Figure 4e. The data shown in the four phases are MFM data measured in the same region.\nThe domain contrast between data comes from magnetization difference between the four magnetic\nphases (for details, see Fig. S16 and S17).\nWe have revised the caption of Fig. 4e as below:\n‘The magnetic phase diagram based on the MFM measurements (e). DW: density wave; AFM:\nantiferromagnetic state; FM: ferromagnetic state; PM: paramagnetic state. The difference between\nphase I and phase II is that the b component of the magnetic moment is absent in phase I but present\nin phase II. The data shown in the four phases are MFM data measured in the same region. The\ndomain contrast between data comes from magnetization difference between the four magnetic\nphases (for details, see Fig. S16 and S17).’\nPoint-by-point response to the comments from the reviewers\nResponse to Reviewer 1#\nThe authors have addressed all of the comments and have substantially improved the\nmanuscript. I have no further concerns, and I recommend the paper for publication.\nResponse: We thank the Reviewer for positive remarks and recommendation.\nResponse to Reviewer 2#\nI am fully satisfied with the responses from the authors. The manuscript can be\npublished as it is.\nResponse: We thank the Reviewer for positive remarks and recommendation.\nResponse to Reviewer 3#\nThe revisions to the manuscript are good and helped clarify my main questions from\nthe previous round. While there could always be further work done, I believe the\nmanuscript in its current form is a good addition to the literature and is suitable for\npublication.\nResponse: We thank the Reviewer for positive remarks and recommendation."}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors have addressed all of the comments and have substantially improved the manuscript. I have no further concerns, and I recommend the paper for publication.", "reviewer_2": "I am fully satisfied with the responses from the authors. The manuscript can be published as it is.", "reviewer_3": "The revisions to the manuscript are good and helped clarify my main questions from the previous round. While there could always be further work done, I believe the manuscript in its current form is a good addition to the literature and is suitable for publication."}, "author_response": ""}]

Decision letter