Abstract
Elementary excitations in condensed matter capture the complex manybody dynamics of interacting basic entities in a simple quasiparticle picture. In magnetic systems the most established quasiparticles are magnons, collective excitations that reside in ordered spin structures, and spinons, their fractional counterparts that emerge in disordered, yet correlated spin states. Here we report on the discovery of elementary excitation inherent to spinstripe order that represents a bound state of two phason quasiparticles, resulting in a wigglinglike motion of the magnetic moments. We observe these excitations, which we dub “wigglons”, in the frustrated zigzag spin1/2 chain compound βTeVO_{4}, where they give rise to unusual lowfrequency spin dynamics in the spinstripe phase. This result provides insights into the stripe physics of stronglycorrelated electron systems.
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 modulationphase 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 twodimensional antiferromagnets^{7} and magnon bound states in ferromagnetic spin1/2 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 chargestripe order in hightemperature superconductors^{9,10,11,12,13,14,15} and promote enigmatic charge fluctuatingstripe (nematic) states.^{14,15,16}
Here we study elementary excitations of the spinstripe phase in the frustrated spin1/2 chain compound βTeVO_{4},^{17,18,19,20,21,22,23} which contains localized V^{4+} (S = 1/2) magnetic moments.^{19,22} This intriguing order involves two superimposed orthogonal incommensurate amplitudemodulated magnetic components with slightly different modulation periods (Fig. 1a), corresponding to two magnetic order parameters, that result in a nanometerscale spinstripe modulation.^{19} We show that in the lowfrequency (megahertz) range this, otherwise longrangeordered state, is in fact dynamical due to the presence of a lowenergy 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 spinstripe 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.
The peculiar spinstripe order in βTeVO_{4} evolves from a spindensitywave (SDW) phase, which develops below T_{N1} = 4.65K and is characterized by a single collinear incommensurate amplitudemodulated magnetic component (Fig. 1a). On cooling, a second superimposed incommensurate amplitudemodulated component with a different modulation period and orthogonal polarization emerges (Fig. 1a) at T_{N2} = 3.28K and the spinstripe order is formed. Finally, at T_{N3} = 2.28K the modulation periods of the two incommensurate amplitudemodulated components become equal and a vectorchiral (VC) phase (Fig. 1a) is established. For frustrated spin1/2 chains with ferromagnetic nearestneighbor and antiferromagnetic nextnearestneighbor exchange interactions, which in βTeVO_{4} amount to J_{1} ≈ −38K and J_{2} ≈ −0.8 J_{1}, respectively,^{19} the SDW and VC phases are predicted theoretically,^{24,25} while the intermediate spinstripe 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 localprobe muonspinrelaxation (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, B_{stat}, the corresponding 1/3 of the total muon polarization is constant, resulting in the socalled “1/3tail” at late times in the μSR signal.^{26} The remaining muon polarization precesses with the angular frequency γ_{μ}B_{stat} (γ_{μ} = 2π × 135.5 MHz/T), leading to oscillations in the time dependence of the μSR signal around the “1/3tail”.^{26} (See Supplementary information for details on μSR analysis, GinzburgLandau modeling, dielectric and highresolution inelasticneutronscattering 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 > T_{N1}, 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 T_{N1} (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 B_{stat}, a hallmark of the incommensurate amplitudemodulated magnetic order. Clearly, in βTeVO_{4}, this static damping is sufficiently strong that the μSR signal beyond ~1 μs can be attributed solely to the “1/3tail”. The latter notably decays (Fig. 2b), which proves that the local magnetic field is still fluctuating, as expected for incommensurate amplitudemodulated magnetic structures.^{32} Remarkably, below T_{N2}, in the spinstripe phase, the “1/3tail” is dramatically suppressed and the oscillation is lost (Fig. 2c), revealing a significant enhancement of localfield fluctuations. This indicates that the system enters an intriguing state that is completely dynamical on the μSR timescale. Finally, below T_{N3} the slowlyrelaxing “1/3tail” and the oscillation reappear (Fig. 2d), corroborating the establishment of a quasistatic VC state with almost fully developed magnetic moments.
To quantitatively account for the μSR signal we model the μSR polarization over the whole temperature range as a product of the two factors
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 B_{stat} and a width Δ. Since oscillations of the μSR polarization are almost completely damped already after the first visible minimum, the parameters B_{stat} and Δ must be comparable. Indeed, the best agreement with experiment was achieved for Δ/B_{stat} = 1.25(1) (Figs 2a–d), which was kept fixed for all temperatures. The second factor in Eq. (1) is the stretchedexponential function that describes the decay of the “1/3tail” due to additional local magneticfield 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. B_{stat} (Fig. 2e) grows from zero at T_{N1} to 9(1) mT at T_{N2}, which is a value of dipolar fields typical encountered by muons in spin1/2 systems.^{26} In the spinstripe phase, B_{stat} slightly decreases, while below T_{N3} 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 T_{N2}, i.e., in the spinstripe phase. Below T_{N3}, 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 dynamics^{34} of the disordered part of the magnetic moments in amplitudemodulated 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\(\gamma _\mu ^2B_{{\mathrm{dyn}}}^2\)/ν_{dyn},^{26} where B_{dyn} is the size of the fluctuating field and ν_{dyn} is the corresponding frequency. Considering that in amplitudemodulated magnetic structures B_{dyn} is comparable to B_{stat}, we can estimate that ν_{dyn} 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 fluctuatingfield distributions in the incommensurate amplitudemodulated magnetic structures.^{32}
While the μSR response in the SDW and VC phases is within expectations, the spinstripe phase shows a surprising enhancement of λ (Fig. 2f) and α (Fig. 2g) that reflects the severe decay of the “1/3tail” in this phase (Fig. 2c). This clearly demonstrates the appearance of an additional relaxation channel that is related to the spinstripe 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 spinstripe phase the static fields derived for the SDW phase fluctuate with a single correlation time 1/ν_{stripe}, where ν_{stripe} is the fluctuating frequency, and numerically calculate the resulting muon polarization function in a selfconsistent manner. Indeed, the resulting muon polarization function P_{stripe}(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 spinstripe phase (see Supplementary information). The derived temperature dependence of ν_{stripe} exhibits a continuous increase from 0.5(5) MHz at T_{N2} to 7.3(5) MHz at T_{N3} (Fig. 1b).
To further investigate the relation between the spinstripe order witnessed previously by neutron diffraction^{19,22} 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 magneticmoment 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 σ_{k} = M_{k}/μ_{B}, where M_{k} denotes the sublattice magnetization component associated with k and μ_{B} is the Bohr magneton.^{22} This allows for the comparison of ν_{stripe}(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 lowestorder 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 lowenergy excitations found in this state.
Discussion
To put the observed excitations into context in terms of frustrated quantumspin chains, we plot a schematic temperature–frequency diagram of excitations in βTeVO_{4} 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 ν_{stripe} fluctuations, which emerge at T_{N2} and reach a maximum of 7.3(5) MHz close to the T_{N3} transition, after which they disappear. The persistent spin dynamics, which can be significantly enhanced in frustrated spin1/2 systems due to quantum effects, exhibits even faster fluctuations at 0.1–1 GHz. The collective magnon excitations, determined by the main exchange interactions J_{i} (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 highresolution inelasticneutronscattering measurements.) The diagram is completed by spinon excitations, which form a continuum extending up to ~πJ_{i}σ^{2},^{35} in this case up to ~3 THz.^{23}
Next, we try to identify the physical mechanism responsible for the unusual spinstripe excitations. Among numerous experimental and theoretical studies of spin chains, considering different J_{1}/J_{2} ratios in an external magnetic field^{24,25,36} as well as in the presence of magnetic anisotropy^{37,38,39} and interchain interactions,^{40,41,42} there appears to be no record of multik magnetic structures that would resemble the spinstripe order observed in βTeVO_{4}, nor its associated excitations. Moreover, if persistent spin dynamics or any other lowenergy excitation inherent to the SDW (or VC) phase, e.g., phasons,^{5,38} were also primarily responsible for the spin relaxation in the spinstripe phase, there should be no significant difference between the SDW (or VC) and spinstripe dynamics. Contrary to this, spin dynamics in the spinstripe phase is completely different from the other ordered phases, as evidenced by the drastically enhanced muonspin relaxation rate (Fig. 2). Further comparison with dynamical processes in spin systems that do develop multik 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 midfrequency (MHz) range, as observed in βTeVO_{4}. Such dynamics, therefore, seems to originate from the peculiarities of the spinstripe phase, which are also responsible for the remarkable coincidence of the ν_{stripe}(T) and F_{4}(T) dependences (Fig. 1b).
To explain the sequence of the magnetic transitions as well as to clarify the existence of the spinstripe 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
where A_{i}, B_{i} (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 σ_{k} and σ_{k+Δk} that emerge at T_{N1} and T_{N2}, respectively. The fifth term represents exchange anisotropy that is responsible for a different magnetic wave vector for the σ_{k+Δk} component, i.e., favoring Δq = Δk/Δk_{0} ≠ 0, where \({\mathrm{\Delta }}k_0 \equiv {\mathrm{\Delta }}k(T_{N2})\) represents the discrepancy between the native magnetic wave vectors for the σ_{k+Δk} and σ_{k} components. The F_{4} 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(σ_{k}, σ_{k+Δk}) accounts for the size limitation of the V^{4+} 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, σ_{k} and σ_{k+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 magneticmoment components through the C term. On cooling, however, the ordered magnetic moments increase, causing the F_{4} term to prevail and thus to stabilize the VC phase with Δq = 0 below T_{N3}. Finally, the derived parameters allow us to calculate the temperature dependence of the F_{4} term. Comparison of the derived temperature dependence F_{4}(T) with experimentally determined ν_{stripe}(T), i.e., assuming that hν_{stripe} = cF_{4}, where h is the Planck constant, we obtain a very good agreement for c = 0.7 (Fig. 1b), corroborating the connection between the F_{4} term and the ν_{stripe} dynamics.
Having established the intimate relation between lowfrequency excitations and the spinstripe phase, the open question that remains concerns the microscopic nature of the spinstripe excitation mode. In contrast to ordinary magnon and spinon modes, these excitations arise from a fourthorder freeenergy term that couples magnetic components with different modulation periods (Fig. 1a). Higherorder 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 amplitudemodulated magnetic components. The latter may either be two phasons, i.e., linearly dispersing zerofrequency Goldstone modes that change the phase of the modulation,^{5,51} two amplitudons, i.e., highfrequency modes that change the amplitude of the modulation,^{5} or a combination of the two. Given that ν_{stripe} is very small compared to the exchange interactions, the spinstripe excitation is most likely a twophason bound mode that has minimal energy at a certain wave vector k_{stripe} (Fig. 4a). Since k_{stripe}, 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 wigglinglike motion of the magnetic moments (see simulations in Supplementary Videos 1 and 2). Hence, we dubbed this type of spinstripe excitations “wigglons”. Moreover, the corresponding amplitude variation is reminiscent of the longitudinal (amplitude) Higgs mode in the Ca_{2}RuO_{4} antiferromagnet, which has also been found to decay into a pair of Goldstone modes.^{7} Finally, we point out that “wigglon” dynamics in the spinstripe phase of βTeVO_{4} might share similarities with fluctuatingchargestripe phases,^{13,14} where the nematic response is ascribed to fast stripe dynamics.^{16}
Our results reveal an intriguing spinonly manifestation of fluctuatingstripe physics, which has, so far, been studied exclusively in the context of nematic phases in hightemperature superconductors. We show that βTeVO_{4} displays an intriguing spinstripe order, which, due to a slow wiggling motion of the magnetic moments, appears static on the neutronscattering 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 spinstripe phase that hosts an extraordinary type of excitation, driven by the fourthorder coupling term in the magnetic free energy. Our discovery draws attention to other frustrated spin1/2 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 nuclearmagneticresonance measurements that are highly sensitive to spin dynamics in the relevant MHz range even in a sizable applied magnetic field.
Methods
Sample description
The singlecrystal samples were grown from TeO_{2} and VO_{2} powders by chemical vapor transport reaction, using twozone furnace and TeCl_{4} as a transport agent, as explained in ref. ^{19}. Powder samples were obtained by grinding singlecrystal 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 SurfaceMuon instrument (GPS) at the Paul Scherrer Institute (PSI), Switzerland. The measurements at PSI were performed in zerofield, 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 mm^{3} singlecrystal on the tripleaxisspectrometer 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.
Data availability
The data that support the findings of this study will be available via https://doi.org/10.15128/r1h989r325b or from the corresponding author upon reasonable request.
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Acknowledgements
We thank T. Lancaster for fruitful discussions and valuable comments. We are grateful for the provision of beam time at the Science and Technology Facilities Council (STFC) ISIS Facility, Rutherford Appleton Laboratory, UK, and SμS, Paul Scherrer Institut, Switzerland. This work has been funded by the Slovenian Research Agency (project J19145 and program No. P10125), the Swiss National Science Foundation (project SCOPES IZ73Z0_152734/1) and the Croatian Science Foundation (project IP2013111011). This research project has been supported by the European Commission under the 7th Framework Programme through the “Research Infrastructures” action of the “Capacities’ Programme” NMI3II Grant number 283883, Contract No. 283883NMI3II. M.G. is grateful to EPSRC (UK) for financial support (grant No. EP/N024028/1). We are grateful to M. Enderle for local support at Institut LaueLangevin, Grenoble, France.
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M.P., A.Z. and D.A. designed and supervised the project. The samples were synthesized by H.B. The μSR experiments were performed by A.Z., M.G., H.L. and F.C. and analyzed by M.P., M.G. and A.Z. The neutron diffraction experiments were performed by M.P., O.Z. and J.S.W. and analyzed by M.P. The dielectric experiments were performed by T.I. and D.R.G. All authors contributed to the interpretation of the data and to the writing of the manuscript.
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Pregelj, M., Zorko, A., Gomilšek, M. et al. Elementary excitation in the spinstripe phase in quantum chains. npj Quantum Mater. 4, 22 (2019). https://doi.org/10.1038/s4153501901605
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