Introduction

The concept of elementary excitations provides an elegant description of dynamical processes in condensed matter.1 Its use is widespread and represents the theoretical foundation for our understanding of vibrational motions of atoms in crystals as phonons,1 the excitations of the valence electrons in metals as plasmons,2 the bound states of an electron and an electron hole in semiconductors as excitons,3 etc. In magnetic systems, this approach inspired the spinon picture of fractional excitations in spin liquids,4 the phason description of the modulation-phase oscillations in amplitude modulated structures,5 and the magnon picture of collective spin excitations in ordered states.6 The latter led to further intriguing discoveries, including longitudinal Higgs modes in two-dimensional antiferromagnets7 and magnon bound states in ferromagnetic spin-1/2 chains.8 Yet, for systems where several order parameters interact, the elementary excitations remain mysterious. A prominent example are the elusive excitations that cause the melting of charge-stripe order in high-temperature superconductors9,10,11,12,13,14,15 and promote enigmatic charge fluctuating-stripe (nematic) states.14,15,16

Here we study elementary excitations of the spin-stripe phase in the frustrated spin-1/2 chain compound β-TeVO4,17,18,19,20,21,22,23 which contains localized V4+ (S = 1/2) magnetic moments.19,22 This intriguing order involves two superimposed orthogonal incommensurate amplitude-modulated magnetic components with slightly different modulation periods (Fig. 1a), corresponding to two magnetic order parameters, that result in a nanometer-scale spin-stripe modulation.19 We show that in the low-frequency (megahertz) range this, otherwise long-range-ordered state, is in fact dynamical due to the presence of a low-energy excitation mode that results from the binding of two phasons from the two orthogonal magnetic components. This type of elementary excitation, which we dub a “wigglon”, is inherent to the spin-stripe order (Fig. 1) and provides insights into the dynamics of stripe phases that may be found when there are two or more order parameters coupled together.

Fig. 1
figure 1

Magnetic structures and magnetic phase diagram. a Magnetic structure models corresponding to the spin-density-wave (SDW), spin-stripe and vector-chiral (VC) phases in β-TeVO4. Note the difference in modulation periods of the two ordered components in the spin-stripe phase, d and d′, associated with the magnetic wave vectors k and k + Δk, respectively. b The corresponding excitations extend over a wide frequency range, from low-energy dielectric dynamics up to high-energy spinon excitations, i.e., from ~105 Hz up to ~1012 Hz. The derived frequency of the magnetic-order dynamics νstripe in the spin-stripe phase corresponds well to the product of intensities of the two neutron reflections, associated with the fourth-order coupling term F4, and to the phenomenological model (solid line). The error bars represent an uncertainty of 1 s.d.

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The peculiar spin-stripe order in β-TeVO4 evolves from a spin-density-wave (SDW) phase, which develops below TN1 = 4.65K and is characterized by a single collinear incommensurate amplitude-modulated magnetic component (Fig. 1a). On cooling, a second superimposed incommensurate amplitude-modulated component with a different modulation period and orthogonal polarization emerges (Fig. 1a) at TN2 = 3.28K and the spin-stripe order is formed. Finally, at TN3 = 2.28K the modulation periods of the two incommensurate amplitude-modulated components become equal and a vector-chiral (VC) phase (Fig. 1a) is established. For frustrated spin-1/2 chains with ferromagnetic nearest-neighbor and antiferromagnetic next-nearest-neighbor exchange interactions, which in β-TeVO4 amount to J1 ≈ −38K and J2 ≈ −0.8 J1, respectively,19 the SDW and VC phases are predicted theoretically,24,25 while the intermediate spin-stripe phase is not. The formation of the latter has been associated with exchange anisotropies and interchain interactions,19,22 but still awaits a comprehensive explanation.

Results

We explore the spin dynamics in these phases by employing the local-probe muon-spin-relaxation (μSR) technique, which is extremely sensitive to internal magnetic fields and can distinguish between fluctuating and static magnetism in a broad frequency range (from ~100 kHz to ~100 GHz).26 We used a powder sample obtained by grinding single crystals (see Methods) to ensure that on average 1/3 of the muon polarization was parallel to the local magnetic field. In the case of static local fields, Bstat, the corresponding 1/3 of the total muon polarization is constant, resulting in the so-called “1/3-tail” at late times in the μSR signal.26 The remaining muon polarization precesses with the angular frequency γμBstat (γμ = 2π × 135.5 MHz/T), leading to oscillations in the time dependence of the μSR signal around the “1/3-tail”.26 (See Supplementary information for details on μSR analysis, Ginzburg-Landau modeling, dielectric and high-resolution inelastic-neutron-scattering measurements, which also includes refs. 27,28,29,30,31) The only way for the muon polarization to relax below 1/3 at late times is thus provided by dynamical local fields.

At T = 4.7K > TN1, the measured μSR polarization decays monotonically (Fig. 2a), as expected in the paramagnetic state where fluctuations of the local magnetic fields are fast compared to the muon lifetime.26 The muon relaxation curve changes dramatically at TN1 (Fig. 2b), where the polarization at early times suddenly drops, reflecting the establishment of static internal fields in the SDW phase. The corresponding oscillations are severely damped, i.e., only the first oscillation at t < 1 μs can be clearly resolved (Fig. 2b), which indicates a wide distribution of Bstat, a hallmark of the incommensurate amplitude-modulated magnetic order. Clearly, in β-TeVO4, this static damping is sufficiently strong that the μSR signal beyond ~1 μs can be attributed solely to the “1/3-tail”. The latter notably decays (Fig. 2b), which proves that the local magnetic field is still fluctuating, as expected for incommensurate amplitude-modulated magnetic structures.32 Remarkably, below TN2, in the spin-stripe phase, the “1/3-tail” is dramatically suppressed and the oscillation is lost (Fig. 2c), revealing a significant enhancement of local-field fluctuations. This indicates that the system enters an intriguing state that is completely dynamical on the μSR timescale. Finally, below TN3 the slowly-relaxing “1/3-tail” and the oscillation reappear (Fig. 2d), corroborating the establishment of a quasi-static VC state with almost fully developed magnetic moments.

Fig. 2
figure 2

μSR results. a–d The μSR polarization data. Lines are fits to the model Eq. (1). The arrows indicate the first minimum due to oscillations caused by static local magnetic fields at 4.4 and 1.6K. e The derived mean value of the static local magnetic fields at the muon stopping site. f The derived relaxation rate associated with the local magnetic-field fluctuations. The line is a fit to a simple linear model λ ∝ a+bT. The apparent gap in the model at 1K is just a gap in the horizontal scale. g The derived stretching exponent, reflecting the distribution of the local magnetic-field fluctuations. The error bars represent an uncertainty of 1 s.d.

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To quantitatively account for the μSR signal we model the μSR polarization over the whole temperature range as a product of the two factors

$$\begin{array}{*{20}{l}} {P(t) = \left[ {\frac{1}{3} + \frac{2}{3}{\kern 1pt} {\mathrm{cos}}(\gamma _\mu B_{{\mathrm{stat}}}t)e^{ - (\gamma _\mu {\mathrm{\Delta }}t)^2/2}} \right]e^{ - (\lambda t)^\alpha }.} \hfill \end{array}$$
(1)

The exponential in the first factor in Eq. (1) accounts for the muon relaxation due to a Gaussian distribution of static magnetic fields with a mean value Bstat and a width Δ. Since oscillations of the μSR polarization are almost completely damped already after the first visible minimum, the parameters Bstat and Δ must be comparable. Indeed, the best agreement with experiment was achieved for Δ/Bstat = 1.25(1) (Figs 2a–d), which was kept fixed for all temperatures. The second factor in Eq. (1) is the stretched-exponential function that describes the decay of the “1/3-tail” due to additional local magnetic-field fluctuations. Here, λ is the mean relaxation rate while α is the stretching exponent accounting for a distribution of relaxation rates.33

The results of our fits of the μSR data to Eq. (1) are summarized in Fig. 2. Bstat (Fig. 2e) grows from zero at TN1 to 9(1) mT at TN2, which is a value of dipolar fields typical encountered by muons in spin-1/2 systems.26 In the spin-stripe phase, Bstat slightly decreases, while below TN3 it starts growing again and reaches a 15(1) mT plateau at the lowest temperatures. On the contrary, the relaxation rate λ does not change significantly throughout the SDW phase, but it escalates by more than an order of magnitude below TN2, i.e., in the spin-stripe phase. Below TN3, however, it reduces and resumes following the same linear temperature dependence (solid lines in Fig. 2f) as in the SDW phase. This is a characteristic of the persistent spin dynamics34 of the disordered part of the magnetic moments in amplitude-modulated magnetic structures.32 As these fluctuations are fast compared to the muon precession, i.e., they do not suppress the minimum in the μSR signal described by the first factor in Eq. (1), one can assume that λ = 2\(\gamma _\mu ^2B_{{\mathrm{dyn}}}^2\)/νdyn,26 where Bdyn is the size of the fluctuating field and νdyn is the corresponding frequency. Considering that in amplitude-modulated magnetic structures Bdyn is comparable to Bstat, we can estimate that νdyn ranges between 0.1 and 1 GHz (Fig. 1b). Finally, the stretching exponent α in the SDW and VC phase (Fig. 2g) amounts to 0.43(5) and 0.25(2), respectively, as expected for broad fluctuating-field distributions in the incommensurate amplitude-modulated magnetic structures.32

While the μSR response in the SDW and VC phases is within expectations, the spin-stripe phase shows a surprising enhancement of λ (Fig. 2f) and α (Fig. 2g) that reflects the severe decay of the “1/3-tail” in this phase (Fig. 2c). This clearly demonstrates the appearance of an additional relaxation channel that is related to the spin-stripe order only. Moreover, the increase of λ is accompanied with the loss of the oscillation in the μSR signal, which indicates that the corresponding fluctuations are associated with the ordered part of the magnetic moments. To account for these experimental findings we introduce the dynamics of the magnetic order into our minimal model of Eq. (1) via the strong collision approach.26 Namely, we assume that in the spin-stripe phase the static fields derived for the SDW phase fluctuate with a single correlation time 1/νstripe, where νstripe is the fluctuating frequency, and numerically calculate the resulting muon polarization function in a self-consistent manner. Indeed, the resulting muon polarization function Pstripe(t), (see Supplementary information, which also includes refs. 27,28,29,30,31) with all other parameters fixed to the values derived for the SDW phase, explains the response of the μSR signal throughout the spin-stripe phase (see Supplementary information). The derived temperature dependence of νstripe exhibits a continuous increase from 0.5(5) MHz at TN2 to 7.3(5) MHz at TN3 (Fig. 1b).

To further investigate the relation between the spin-stripe order witnessed previously by neutron diffraction19,22 and the stripe dynamics observed by μSR, we performed additional neutron diffraction measurements (see Methods). We measured the temperature dependence of the strongest magnetic reflection and its satellites (Fig. 3a), the latter being associated with the orthogonal b magnetic-moment component (Fig. 1a)22 that emerges at slightly different wave vectors, shifted by ±Δk from the main magnetic wave vector k.19 The intensity of an individual magnetic reflection, scales with the square of the corresponding order parameter σk = Mk/μB, where Mk denotes the sublattice magnetization component associated with k and μB is the Bohr magneton.22 This allows for the comparison of νstripe(T) with the temperature evolution of \(F_4 \sim \sigma _{ - {\mathbf{k}}}^2\sigma _{{\mathbf{k}} + {\bf{\Delta }}{\mathbf{k}}}\sigma _{{\mathbf{k}} - {\bf{\Delta }}{\mathbf{k}}}\), i.e., the lowest-order term in the magnetic free energy that couples all magnetic components with different modulation periods (k, k+Δk, and k−Δk).22 We find a very good correspondence (Fig. 1b), which confirms a direct link between the formation of spin stripes and the remarkable low-energy excitations found in this state.

Fig. 3
figure 3

Neutron diffraction results. a The temperature dependence of the strongest magnetic reflection and the accompanying satellite reflections. The temperature dependences of b the order parameters σk and σk+Δk as well as c Δq derived from the neutron diffraction experiment (symbols) and calculated using a phenomenological model (solid lines; see text). The error bars represent an uncertainty of 1 s.d.

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Discussion

To put the observed excitations into context in terms of frustrated quantum-spin chains, we plot a schematic temperature–frequency diagram of excitations in β-TeVO4 in Fig. 1b. At the lowest energies, we find dielectric dynamics, which peaks at ~0.4 MHz and is most pronounced in the multiferroic VC phase (see Supplementary information for complementary dielectric measurements). These are followed by strong νstripe fluctuations, which emerge at TN2 and reach a maximum of 7.3(5) MHz close to the TN3 transition, after which they disappear. The persistent spin dynamics, which can be significantly enhanced in frustrated spin-1/2 systems due to quantum effects, exhibits even faster fluctuations at 0.1–1 GHz. The collective magnon excitations, determined by the main exchange interactions Ji (i = 1,2), develop in the VC phase above the gap, most likely induced by spin–orbit coupling, which is also responsible for exchange anisotropy, i.e., at frequencies of 0.1–1 THz.23 (See Supplementary information for details on high-resolution inelastic-neutron-scattering measurements.) The diagram is completed by spinon excitations, which form a continuum extending up to ~πJiσ2,35 in this case up to ~3 THz.23

Next, we try to identify the physical mechanism responsible for the unusual spin-stripe excitations. Among numerous experimental and theoretical studies of spin chains, considering different J1/J2 ratios in an external magnetic field24,25,36 as well as in the presence of magnetic anisotropy37,38,39 and interchain interactions,40,41,42 there appears to be no record of multi-k magnetic structures that would resemble the spin-stripe order observed in β-TeVO4, nor its associated excitations. Moreover, if persistent spin dynamics or any other low-energy excitation inherent to the SDW (or VC) phase, e.g., phasons,5,38 were also primarily responsible for the spin relaxation in the spin-stripe phase, there should be no significant difference between the SDW (or VC) and spin-stripe dynamics. Contrary to this, spin dynamics in the spin-stripe phase is completely different from the other ordered phases, as evidenced by the drastically enhanced muon-spin relaxation rate (Fig. 2). Further comparison with dynamical processes in spin systems that do develop multi-k magnetic structures, e.g., skyrmion phases,43 does not reveal any similarity either. Namely, in contrast to our case, in such systems spin dynamics are typically driven either by very slow domain fluctuations in the range between 1 Hz to 1 kHz,44 or stem from much faster collective breathing, magnon or even electromagnon excitations in the GHz range.45,46,47 Finally, dynamics in alternating patterns of spin and charge stripes in oxides was found between 1 and 100 GHz.48 To summarize, there exists no report of electron spin dynamics in the mid-frequency (MHz) range, as observed in β-TeVO4. Such dynamics, therefore, seems to originate from the peculiarities of the spin-stripe phase, which are also responsible for the remarkable coincidence of the νstripe(T) and F4(T) dependences (Fig. 1b).

To explain the sequence of the magnetic transitions as well as to clarify the existence of the spin-stripe phase and its corresponding excitations we undertake a phenomenological approach based on the classical Ginzburg–Landau theory of phase transitions. We construct the following expression for the magnetic free energy

$$\begin{array}{*{20}{l}} F \hfill & = \hfill & {A_1\left( {T/T_{N1} - 1} \right)\sigma _{\mathbf{k}}^2 + A_2\left( {T/T_{N2} - 1} \right)\sigma _{{\mathbf{k}} + {\bf{\Delta }}{\mathbf{k}}}^2} \hfill \\ {} \hfill & {} \hfill & { +\, B_1\sigma _{\mathbf{k}}^4 + B_2\sigma _{{\mathbf{k}} + {\bf{\Delta }}{\mathbf{k}}}^4 + C\left( {1 - {\mathrm{\Delta }}q} \right)^2\sigma _{{\mathbf{k}} + {\bf{\Delta }}{\mathbf{k}}}^2 + F_4,} \hfill \end{array}$$
(2)
$$F_4 = \left[ {D + Ef(\sigma _{\mathbf{k}},\sigma _{{\mathbf{k}} + {\bf{\Delta }}{\mathbf{k}}}){\mathrm{\Delta }}q^2} \right]\sigma _{ - {\mathbf{k}}}^2\sigma _{{\mathbf{k}} + {\mathrm{\Delta }}{\mathbf{k}}}\sigma _{{\mathbf{k}} - {\mathrm{\Delta }}{\mathbf{k}}},$$
(3)

where Ai, Bi (i = 1, 2), C, D, and E are scaling constants. The first four terms in Eq. (2) describe the evolution of two independent magnetic order parameters σk and σk+Δk that emerge at TN1 and TN2, respectively. The fifth term represents exchange anisotropy that is responsible for a different magnetic wave vector for the σk+Δk component, i.e., favoring Δq = Δk/Δk0 ≠ 0, where \({\mathrm{\Delta }}k_0 \equiv {\mathrm{\Delta }}k(T_{N2})\) represents the discrepancy between the native magnetic wave vectors for the σk+Δk and σk components. The F4 term is associated with the coupling between the two order parameters and favors fully developed magnetic moments, i.e., it acts against the discrepancy between the two modulation periods (Δq → 0). The function f(σk, σk+Δk) accounts for the size limitation of the V4+ S = 1/2 magnetic moments and thus smoothly changes from 0 to 1, when \(\sqrt {\sigma _{\mathbf{k}}^2 + \sigma _{{\mathbf{k}} + {\mathbf{\Delta k}}}^2}\) exceeds the limiting value (see Supplementary information). Considering \(\sigma _{{\mathbf{k}} - {\mathbf{\Delta k}}} \approx \sqrt {0.2} \sigma _{{\mathbf{k}} + {\mathbf{\Delta k}}}\) (Fig. 3 and ref. 19), the minimization of Eq. (2) with respect to Δq, σk and σk+Δk (see Supplementary information) returns the corresponding temperature dependences (Fig. 3b, c) that almost perfectly describe the observed behavior. In particular, we find that in the vicinity of the paramagnetic phase, where ordered magnetic moments are still small, a sizable exchange anisotropy can impose different modulations for different magnetic-moment components through the C term. On cooling, however, the ordered magnetic moments increase, causing the F4 term to prevail and thus to stabilize the VC phase with Δq = 0 below TN3. Finally, the derived parameters allow us to calculate the temperature dependence of the F4 term. Comparison of the derived temperature dependence F4(T) with experimentally determined νstripe(T), i.e., assuming that hνstripe = c|F4|, where h is the Planck constant, we obtain a very good agreement for c = 0.7 (Fig. 1b), corroborating the connection between the F4 term and the νstripe dynamics.

Having established the intimate relation between low-frequency excitations and the spin-stripe phase, the open question that remains concerns the microscopic nature of the spin-stripe excitation mode. In contrast to ordinary magnon and spinon modes, these excitations arise from a fourth-order free-energy term that couples magnetic components with different modulation periods (Fig. 1a). Higher-order terms in the free energy impose an interaction between the basic elementary excitations.8,49,50 The observed excitations are thus most likely bound states of two elementary excitations of the incommensurate amplitude-modulated magnetic components. The latter may either be two phasons, i.e., linearly dispersing zero-frequency Goldstone modes that change the phase of the modulation,5,51 two amplitudons, i.e., high-frequency modes that change the amplitude of the modulation,5 or a combination of the two. Given that νstripe is very small compared to the exchange interactions, the spin-stripe excitation is most likely a two-phason bound mode that has minimal energy at a certain wave vector kstripe (Fig. 4a). Since kstripe, in principle, differs from both k and k ± Δk, the bound mode imposes additional expansion and contraction of the modulation periods on top of the phase changes induced by individual phasons (Fig. 4b). Consequently, positions, sizes and orientations of maxima in the magnetic structure exhibit completely different time dependences than for individual phason (Fig. 4c), resulting in a wiggling-like motion of the magnetic moments (see simulations in Supplementary Videos 1 and 2). Hence, we dubbed this type of spin-stripe excitations “wigglons”. Moreover, the corresponding amplitude variation is reminiscent of the longitudinal (amplitude) Higgs mode in the Ca2RuO4 antiferromagnet, which has also been found to decay into a pair of Goldstone modes.7 Finally, we point out that “wigglon” dynamics in the spin-stripe phase of β-TeVO4 might share similarities with fluctuating-charge-stripe phases,13,14 where the nematic response is ascribed to fast stripe dynamics.16

Fig. 4
figure 4

Wigglon excitation. a Schematic dispersion relations for individual gapless phasons corresponding to σk and σk+Δk (dotted lines), as well as for the gaped two-phason bound wigglon excitation (solid line). b Schematic representation of the magnetic order in the spin-stripe phase, decomposed into two orthogonal magnetic components (green and violet arrows) and summed together (magenta arrows), for the two decoupled phason modes (left) and for the two-phason bound wigglon excitation (right). Consecutive figures represent the time evolution, while large arrows denote shifts of the central maximum (see simulations in Supplementary Videos 1 and 2). Note the uniform shifts for decoupled phasons and nonuniform extensions/contractions induced by a wigglon. c The position, amplitude and orientation of the marked maximum in the magnetic structure (thick black and magneta arrows in b) as a function of time for decoupled phason excitations (left) and a wigglon excitation (right)

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Our results reveal an intriguing spin-only manifestation of fluctuating-stripe physics, which has, so far, been studied exclusively in the context of nematic phases in high-temperature superconductors. We show that β-TeVO4 displays an intriguing spin-stripe order, which, due to a slow wiggling motion of the magnetic moments, appears static on the neutron-scattering timescale,19 i.e., at ν > 10 GHz, while it is in fact dynamical at MHz frequencies. The phenomenon is driven by sizable exchange anisotropy, which prevails in a finite temperature range where it stabilizes the dynamical spin-stripe phase that hosts an extraordinary type of excitation, driven by the fourth-order coupling term in the magnetic free energy. Our discovery draws attention to other frustrated spin-1/2 chain compounds with complicated and unresolved magnetic phase diagrams,52,53 where similar effects may be anticipated to play a role. Finally, more details of the wigglon excitation, such as their dependence on the applied magnetic field, should be explored by complementary nuclear-magnetic-resonance measurements that are highly sensitive to spin dynamics in the relevant MHz range even in a sizable applied magnetic field.

Methods

Sample description

The single-crystal samples were grown from TeO2 and VO2 powders by chemical vapor transport reaction, using two-zone furnace and TeCl4 as a transport agent, as explained in ref. 19. Powder samples were obtained by grinding single-crystal samples.

μSR experiments

The experiments were performed on the MuSR instrument at the STFC ISIS facility, Rutherford Appleton Laboratory, United Kingdom, and on the General Purpose Surface-Muon instrument (GPS) at the Paul Scherrer Institute (PSI), Switzerland. The measurements at PSI were performed in zero-field, while at ISIS a small longitudinal field of 4 mT was applied to decouple the muon relaxation due to nuclear magnetism at long times. The dead time at the GPS instrument is ~0.01 μs, whereas its is ~0.1 μs at the MuSR instrument. For details on background subtraction (see Supplementary information, which also includes refs. 27,28,29,30,31).

Neutron diffraction

Neutron diffraction measurements were performed on a 2 × 3 × 4 mm3 single-crystal on the triple-axis-spectrometer TASP at PSI. To assure the maximal neutron flux the wavelength of 3.19 Å was chosen for the experiment. An analyzer was used to reduce the background, while the standard ILL orange cryostat was used for cooling.