In solid-state physics, unique phenomena emerge when a long-range order is suppressed by an external parameter towards lower temperatures where quantum fluctuations dominate. This suppression often reveals a ‘dome’-like new phase at the bottom of a conical cross-over region and a significant change of the critical dynamics1. Famous examples for this are superconductivity2,3, heavy fermion systems4,5, and quantum spin liquids6, all of which are accompanied by changes of the excitation spectra. To investigate the influence of quantum fluctuations on the critical dynamics, systems with low dimensionality are a good starting point as they ideally do not order even at T → 0 K. Experimentally, strictly 1D systems are, of course, not available, but systems where the magnetic ions are located on structures with lower dimensionality, e.g., spin-chain compounds, have been shown to undergo magnetic field-driven phase transitions at very low temperatures. As these quantum phase transitions are driven by quantum fluctuations and not by temperature unusual dynamics are expected to emerge in their proximity. The influence of quantum fluctuations is even stronger when the systems have almost isotropic spin S = 1/2 in so-called Heisenberg spin chains. One way to introduce vector spin chirality in such a spin chain is frustration due to competing nearest and next-nearest neighbor interactions, J1 and J2, in the presence of a small easy-plane anisotropy7,8. Apart from being chiral, these phases can be magnetoelectric multiferroic9,10 as has been observed in many compounds11,12. The coupling between magnetization M and polarization P has potential applications for data storage13,14 and, most important for this work, it allows to observe the dynamics of the magnetic system via measurements of the polarization dynamics15,16. In particular, in the case of magnetoelectric multiferroics an electromagnon17,18, i.e., collective magnetic excitations that also carry electric polarization, can be observed. The corresponding fluctuation dynamics show critical slowing down at the multiferroic phase transition19 that can be tuned towards T → 0 K in an external magnetic field. Close to this quantum critical point a change in the fluctuation dynamics can be observed via dielectric measurements20. It has also been shown, that quantum critical behavior can be observed above the critical point that terminates a first-order phase transition21.

Here, we report experimental evidence for a chiral quantum spin-liquid state in LiCuVO4. This evidence is based on the analysis of the T-, H-, and ν-dependence of the complex permittivity ε* of LiCuVO4 close to the multiferroic phase transition down to 0.025 K with frequencies in the GHz range. Our results show classical critical slowing down at the multiferroic phase transition with a critical dynamical exponent νξz ≈ 1.3 above 1.0 K in agreement with mean-field predictions15. For an external magnetic field, the transition is shifted towards low temperatures. In close proximity to the multiferroic phase transition, we observe both quantum critical slowing down of the fluctuations as well as the emergence of a nearly gapless excitation that is in agreement with the chiral soliton proposed for LiCuVO4 in8 based on quantum spin-liquid theory. From direct measurements, we find an energy gap of ESE ≈ 14.1 μeV for this excitation.

Results and discussions

LiCuVO4 has a distorted inverse spinel-type structure22, see Fig. 1a. Here, a low-dimensional spin system is realized by Cu2+ ions with S = 1/2 that form 1D chains along the b axis separated in a direction by VO4 tetrahedra and by Li along the c direction. Competing ferromagnetic J1 < 0 and antiferromagnetic J2 > 0 interactions23 lead to frustration along the spin chains and cycloidal Néel ordering of the spins within the ab plane is observed below TN = 2.4 K due to weak interchain coupling, as sketched in Fig. 1b. In this phase, LiCuVO4 is multiferroic with an electric polarization Pk × (Si × Si+1) parallel to the a axis. By applying a magnetic field along the c axis the 3D order can be weakened and TN is continuously suppressed down to 0 K at μ0Hc ≈ 7.4 T16,24,25, raising the question if a quantum phase transition occurs.

Fig. 1: Introduction of LiCuVO4.
figure 1

a A fraction of the crystal structure of LiCuVO4 (data from ref. 23), the ordered magnetic moments of Cu in the multiferroic phase are indicated. b A sketch of the temperature (T) vs magnetic field (μ0H) phase diagram. The pre-factor of H is the vacuum permeability μ0 = 4π 10−7 Vs(Am)−1. ce The temperature-dependent thermal expansion coefficients αi for all three crystallographic axes change sign once the external magnetic field crosses the critical magnetic field μ0Hc 7.4 T, and f an analogous sign change occurs in the magnetic field-dependent measurements of the magnetic Grüneisen parameter ΓH = 1/T ∂T/∂(μ0H).

A characteristic signature of pressure-dependent quantum phase transitions26,27, that is observed in various quasi-1D quantum spin materials28,29,30,31,32,33, is a sign change of the Grüneisen ratio Γ = αi/Cp (with the uniaxial thermal expansion coefficient αi and the specific heat Cp > 0) close to Hc. Below Hc, the transition into the multiferroic phase causes sharp anomalies in αi = 1/L ∂L/∂T (i denotes the orthorhombic axes a, b, c), as is shown for 7.0 T (red) in Fig. 1c–e. Upon increasing the magnetic field Hc above Hc, these anomalies vanish and αi changes sign for each axis i. Note that the sign change of αa occurs above 7.5 T, which may arise either from a stabilization of the multiferroic phase for uniaxial pressure along a and/or from a slight misorientation of the magnetic field. Experimentally, magnetic field-dependent measurements are more reliable when the phase boundary is very steep close to Hc, see Fig. 1b. Therefore, we measured the magnetic field-dependent Grüneisen parameter ΓH = 1/T ∂T/∂(μ0H), which is obtained from the ratio of the magnetocaloric effect and Cp32,33, that confirms the sign change (Fig. 1f).

Based on coexisting phases neutron scattering experiments suggest that the field-driven phase transition is of first-order at very low temperatures34,35. However, there are no indications of hysteresis effects in any of our measured macroscopic quantities. In combination with the observed quantum critical behavior, we propose that the first-order transition at very low temperatures changes at a critical point into a second-order phase transition. This could be a Lifshitz point36 which separates the low-field cycloidal phase with constant wave vector q from the spin-modulated phase with a varying q(H)34,35. In this case, the quantum critical behavior also ends at the critical point and not at T=0 K as discussed e.g., for the critical end point in Sr3Ru2O721.

When increasing the magnetic field above Hc at low temperatures the system enters a collinear spin-modulated phase with short-range order35,37 and vector-chiral correlations were reported to exist already for T > TN38. Above 41 T, LiCuVO4 enters a spin-nematic phase and a saturated phase above 44 T39, both of which are well beyond the magnetic field range used in this work.

The multiferroic phase transition can not only be observed in thermodynamic quantities, but also in dielectric spectroscopy. As this technique also measures a background of regular phonon contributions we subtracted measurements away from the phase transition to isolate the additional multiferroic contribution Δε* close to the phase transition.

Thermally activated fluctuations

On approaching the multiferroic phase transition, thermally activated chiral fluctuations appear already above TN whose size and lifetime diverge for T → TN. In this regime dispersion is observed in \({{\Delta }}\varepsilon ^{\prime}\), see Fig. 2a, as the high frequencies break away from the quasi-static response that is given by the envelope of \({{\Delta }}\varepsilon ^{\prime}\) (dashed line). This behavior is accompanied by corresponding contributions in the dielectric loss Δε at temperatures above TN shown in Fig. 2b. These characteristic features of a phonon mode’s critical slowing down are expected in both ferroelectric as well as multiferroic materials15. As a well-founded expectation for these results already exists we can use the fluctuations to both demonstrate the viability of the background correction as well as introduce the classical effects that overlay the quantum fluctuation effects discussed later on. To quantify the dynamics above the phase transition we utilize critical dynamical scaling that relates the relaxation time τξz to the correlation length ξ via the dynamical exponent z. Close to the phase transition the correlation length ξ diverges, \(\xi (T)\propto | T/{T}_{{{{{{{{\rm{N}}}}}}}}}-1{| }^{-{\nu }_{\xi ,T}}\), with the positive correlation length exponent νξ. Therefore, we look at the permittivity in the frequency domain where ε*(ν) is described by a Debye relaxation

$${\varepsilon }_{{{{{{{{\rm{EM}}}}}}}}}^{* }(\nu )=\frac{{{\Delta }}{\varepsilon }_{{{{{{{{\rm{s}}}}}}}}}}{1-i2\pi \nu \tau }.$$

with the step height Δεs in \(\varepsilon ^{\prime}\) and the relaxation time τ. In Fig. 2e the complex permittivity is shown for different temperatures as both, \(\varepsilon ^{\prime} (\nu )\) (circles) and ε(ν) (triangles) with fits of the Debye relaxation (lines). Here, the arrows mark the position νp = 1/2πτ of the maxima in the dielectric loss, and their shift towards lower frequencies upon cooling demonstrates the critical slowing down close to the phase transition. Additional information on the dynamics can be gained from the height of the step Δεs. According to the Lyddane–Sachs–Teller relation40 the mode’s eigenfrequency ωEM is connected to the critical increase in the static permittivity by \({{\Delta }}{\varepsilon }_{{{{{{{{\rm{s}}}}}}}}}\propto {\omega }_{{{{{{{{\rm{EM}}}}}}}}}^{-2}\). As relaxation is characterized by high damping \({\omega }_{{{{{{{{\rm{EM}}}}}}}}}^{2}\approx {{{\Gamma }}}_{{{{{{{{\rm{EM}}}}}}}}}/\tau\) holds and Δεs is therefore also proportional to τ. In Fig. 2g we show that a critical exponent νξ,Tz ≈ 1.3 simultaneously describes νp = 1/2πτ and 1/Δεs when choosing the position of the maximum for the lowest frequency as TN. This result is in agreement with the expectation of νξ,Tz ≈ 1.272 for materials of the 3D-Ising universality class15.

Fig. 2: T- and H-dependent critical dynamics of thermally activated fluctuations.
figure 2

Real and imaginary part of the complex permittivity in the frequency range of 56 MHz–5.6 GHz from temperature (T) dependent measurements in zero magnetic fields are shown in a, b and magnetic field (H) dependent measurements at 1.0 K are shown in c, d. Contributions of faster polarization effects have been subtracted. In e, f the measured permittivity spectra (circles for \({{\Delta }}\varepsilon ^{\prime} (\nu )\) and triangles for Δε(ν)) are shown with the fitted Debye relaxation as solid lines for the T − and H − dependent measurements respectively. Here, the spectra are offset for enhanced visibility and the peak positions in the dielectric loss are marked by arrows. The T − (in g) and H − dependence (in h) of the parameters νp and 1/Δεs can be described with a critical dynamical exponent of 1.3. The error bars in g, h represent the root mean squared error obtained for the fits of the Debye relaxations.

At the steep phase boundary close to Hc we performed measurements of Δε(ν, H) at constant T. Compared to μ0H = 0 T the feature at T = 1.0 K is slightly smaller as the spins are canted into the direction of the magnetic field and, thus, reducing Pk × (Si × Si+1), see Fig. 2c, d. The observed dynamics, on the other hand, are almost identical and also show critical slowing down, as seen in the spectra in Fig. 2f. Evaluating the critical exponent from νp and Δεs, in this case its magnetic field dependence \(\tau (H)\propto | H/{H}_{{{{{{{{\rm{c}}}}}}}}}-1{| }^{-{\nu }_{\xi ,H}z}\) with Hc again from the maximum at the lowest frequency, yields νξ,Hz ≈ 1.3, also in agreement with the expected mean-field behavior15.

Dynamics from quantum fluctuations

Below 1.0 K we observe a fundamental change in the fluctuation dynamics in both, the permittivity \({{\Delta }}\varepsilon ^{\prime}\) shown in Fig. 3a and the dielectric loss Δε in Fig. 3b. In \({{\Delta }}\varepsilon ^{\prime}\) the dispersive peak from temperature-activated fluctuations evolves into an up to ≈1 GHz dispersionless maximum at Hc. In the dielectric loss, a corresponding peak is only visible in a narrow frequency range between 2 and 3 GHz. This missing low-frequency contributions in Δε indicate, that excitation with low damping is observed at lowest temperatures. While the observed absolute value of the permittivity depends on H and vanishes outside of 6.5 T H 8 T, within the experimental accuracy the observed excitation shows the same frequency dependence for all magnetic fields within this magnetic field range. Therefore, we only discuss the data close to Hc.

Fig. 3: Low-temperature dielectric measurements.
figure 3

The magnetic field-dependent complex permittivity Δε* in the vicinity of Hc for different frequencies and temperatures. In a the evolution of \({{\Delta }}\varepsilon ^{\prime}\) from a dispersive feature at a higher temperature into a peak with no dispersion up to approximately 1 GHz (green) with a steep drop above this frequency with cooling is shown. The dielectric loss Δε in b also changes from a broad, dispersive maximum at higher temperatures observed for all frequencies to a, in the frequency domain, a narrow peak located roughly between 2 and 3 GHz at low temperature.

To examine the change in the fluctuation dynamics in more detail we again look at the data in the frequency domain. The excitation observed at the lowest temperatures is taken into account by adding a Lorentz oscillator with the plasma frequency ωp, the undamped eigenfrequency ω0, and the damping Γ to eq. (1):

$${\varepsilon }^{* }(\nu )={\varepsilon }_{{{{{{{{\rm{EM}}}}}}}}}(\nu )+\frac{{\omega }_{{{{{{{{\rm{p}}}}}}}}}^{2}}{{\omega }_{0}^{2}-{(2\pi \nu )}^{2}+i2\pi \nu {{\Gamma }}}.$$

The real and imaginary part of ε*(ν) are shown for several temperatures and constant magnetic field μ0H = 7.4 T, i.e., close to Hc, in Fig. 4a, b. The results of the fits are shown as solid lines in both panels and are in good agreement with the data, the two individual processes are shown as gray (relaxation) and blue (excitation) dashed lines. A single relaxation process is sufficient to describe the data at 1.0 K, both processes are needed in an intermediate temperature regime until only the excitation remains below 0.2 K.

Fig. 4: Dynamics at low temperatures.
figure 4

Measured spectra (points) of the complex permittivity Δε*(ν) at H ≈ Hc with fits of Equation (2) (lines) are shown in a for the real part, \({{\Delta }}\varepsilon ^{\prime}\), and the imaginary part, the dielectric loss Δε, in b as a function of the frequency ν. The fitted peak frequencies of the maxima in Δε are shown in c for both the relaxation (gray squares) and the emergent excitation (blue circles) on the left scale with the corresponding damping factor for the excitation (triangles) on the right scale. The error bars in c correspond to the root mean squared error that was obtained when fitting the spectra with Equation (2).

For the relaxation, the shift towards lower frequencies on cooling again indicates critical slowing down. In contrast to the previously discussed case of thermally activated fluctuations, these measurements were taken within the quantum critical regime and a different critical scaling is observed. The temperature dependence of the peak frequencies νp(T), shown as gray dots in Fig. 4c, follows a critical exponent of νξ,Tz ≈ 2 (solid line) which was also observed for multiferroic Ba2CoGe2O720. Different from this material, the temperature of the critical point Tcp ≈ 0.37 K has to be taken into account in LiCuVO4 when discussing the slowing down of the quantum fluctuations.

In contrast to the relaxation, the excitation does not show critical slowing down upon cooling, but instead its maximum shifts to higher frequencies, as shown as blue dots in Fig. 4c. This apparent “speeding up”41 can be explained by a reduced damping Γ with decreasing T, plotted as triangles in the figure, while the eigenfrequency ω0 is independent of temperature. As this excitation is observed with dielectric spectroscopy the underlying mechanism has to include electric polarization. Therefore, the type of excitation, namely the stimulation via an electric field, demonstrates that we are not dealing with simple spin dipolar fluctuations but with magnetoelectric, vector-chiral fluctuations. This also excludes the bimagnon excitation proposed in LiCuVO435 because the bimagnon excitation should stem from a collinear spin arrangement and this would not carry an electric dipole moment. Thus, our experimental results show the existence of chiral solitons and thereby support the occurrence of a chiral spin-liquid state close to Hc as has been predicted in ref. 8.

Chiral solitons

In a theoretical work by Furukawa et al.8 it has been shown that quantum fluctuations in frustrated S = 1/2 systems can lead to the emergence of a spin-liquid state in which the spins dimerize. The characteristic excitations of this state are chiral solitons that are proposed to work as follows8: starting from a uniform spin chain in Fig. 5a the suppression of long-range order leads to dimerization. To enhance visibility neighboring dimers in the uniform spin chain are highlighted with alternating dashes in green and blue that have almost inverted spin orientations. In the beginning, the vector handedness of the spiral and, therefore, its polarization is uniform, here clockwise. By flipping both spins in a dimer it changes its “type”, here from blue to green, see Fig. 5b. During this process, two defects in the as yet uniform chirality of the spin chain are created, that are marked in red, and cause a local flip of the electric dipole moment. By exchanging neighboring dimer pairs these chiral solitons can move through the spin chain, e.g. in opposite directions with effective q = 0, see Fig. 5c. While the model used by Furukawa et al. does not explicitly use an external magnetic field it has been shown7 that despite magnetic fields perpendicular to the easy plane the vector-chiral state remains for small magnetic fields and can also be present in the U(1) symmetric case.

Fig. 5: Chiral solitons.
figure 5

a The pristine spin chain with uniform clockwise vector handedness. Dimers are marked by colored backgrounds to differentiate between the alternating species that make up the uniform spin chain. b By flipping both spins in a dimer it transforms into the other species and two chiral solitons are created. This process is slightly gapped due to the presence of interchain coupling in LiCuVO4. c By exchanging two neighboring dimers the soliton can move through the spin chain without creating additional changes in its handedness. Based on ref. 8.

Due to the change in the chirality, the soliton’s electric dipole moments locally point against the ferroelectric background of the uniform spin chain and they are thus visible by dielectric spectroscopy. For LiCuVO4 these excitations have been predicted to be slightly gapped due to interchain coupling, which is in perfect agreement with our results. We find that the gap of the soliton excitation is indeed very small, ω0/2π ≈ 3.4 GHz which corresponds to a gap energy of ESE = 14.1 μeV.

The chiral solitons are not only observed close to the first-order phase transition where the 3D order is weakened. Above the critical point, the chiral solitons can be observed as long as the lifetime of the quantum fluctuations is larger than the inverse peak frequency of the excitation, τEM > 1/ω0. At higher temperatures, the lifetime of the fluctuations decreases and the damping increases, such that the chiral solitons become too broad to be observed.


Our measurements of the complex permittivity close to the multiferroic phase transition in LiCuVO4 demonstrate the changing polarization dynamics in the presence of increasing quantum fluctuations. At temperatures above 1.0 K we see the mean-field behavior expected for the critical dynamics at a multiferroic phase transition: critical slowing down for both, the temperature and magnetic field-dependent measurements. When TN is suppressed further the critical slowing down of thermal fluctuations is insufficient to describe the dynamics. Instead, thermal and quantum fluctuations compete until the latter condense into a state with a characteristic chiral soliton excitation that we assume to coexist with the multiferroic and collinear phases. Close to Hc, the long-range order of the multiferroic phase is weakened and, at the same time, the short-range modulated collinear phase is not yet fully established. The coexistence of both later phases has been shown from neutron data35, giving further credit to the possible coexistence of a third phase in the same magnetic field range. From the measurements shown in Fig. 3, we can estimate that this state occurs only for magnetic fields ranging from approximately 6.5 to 8.0 T. Here we measure a contribution to the permittivity spectra that, despite being located in the GHz regime, does not show overdamped relaxational but excitational character with comparably low damping. This low-energy excitation with a tiny gap of ESE ≈ 14.1 μeV is consistent with the predicted chiral soliton excitation of a quantum spin-liquid state in LiCuVO48. To determine the correlation length and to investigate the possible quantum spin-liquid behavior further neutron scattering experiments would be helpful.


Single crystals of LiCuVO4 were grown from LiVO3 flux in the temperature range between 910 and 853 K with an applied cooling rate of 0.1 K/h.

Dielectric spectroscopy

To observe the fluctuation dynamics at the edge of the multiferroic phase we mounted a LiCuVO4 crystal on a microstrip sample holder for dielectric spectroscopy measurements in the GHz range. The measurements were done with different network analyzers (Agilent PNA-X and Rohde & Schwarz ZVB4 and ZNB8) in two cryostats, a top-loading dilution refrigerator (Oxford Instruments KELVINOX) and a Quantum Design PPMS.

Thermal expansion

Thermal expansion was measured in a home-built capacitance dilatometer with an ac capacitance bridge (Andeen–Hagerling AH2550). The uniaxial thermal expansion coefficient was obtained numerically via αi = 1/L∂ΔL/∂T. The magnetic Grüneisen parameter ΓH was measured using a home-built calorimeter with a technique explained in ref. 32.