Quantum spin dynamics of quasi-one-dimensional Heisenberg-Ising magnets in a transverse field: confined spinons, E 8 spectrum, and quantum phase transitions

We report on high-resolution terahertz spectroscopic studies of quantum spin dynamics in the quasi-one-dimensional Ising-like ferromagnet CoNb2O6 and antiferromagnet BaCo2V2O8 as a function of an applied transverse magnetic field. In the ordered phases stabilized by inter-chain couplings, we reveal characteristics for confined spinon excitations, E 8 dynamical spectrum, and field-induced quantum phase transitions. The connections between these characteristic dynamical features are found in the field-dependent evolution of the excitation spectra.


Introduction
The one-dimensional (1D) quantum spin models are a subject of a constant stream of theoretical studies [1][2][3][4][5][6]. Despite their apparent simplicity, the spin-chain models exhibit a rich spectrum of fascinating physics, such as quantum phase transitions [2,[7][8][9], fractionalization [6,[10][11][12], complex bound states [13,14], and quantum information [4]. It is possible to achieve an exact understanding of the physics, since the 1D models are analytically or numerically solvable. Invented over a century ago, the spin-1/2 Ising chain is a celebrated 1D model [15][16][17]. This model exhibits a long-range order only at zero temperature and a spin excitation gap in zero magnetic field. In an applied transverse field, the spin gap is reduced and finally closed at a critical field B c,1D ⊥ (see figure 1). A quantum phase transition occurs at the critical field, above which the long-range order is suppressed and the system enters the paramagnetic phase again with a finite spin gap.
An experimental realization and investigation of the Ising spin-chain model in a solid-state material is not straightforward. In a real material, it is not natural to have an infinite Ising anisotropy, but rather a finite one in addition to a Heisenberg isotropic exchange interaction. At the same time, interchain couplings are not exactly zero, but can be much smaller in comparison with intrachain interactions. In these so-called quasi-1D spin-chain materials, the interchain couplings can be treated as a small perturbation or are even negligible under certain conditions. In presence of the interchain couplings, the quasi-1D spin materials exhibit various interesting phenomena, such as a three-dimensional (3D) quantum phase transition [8,[18][19][20], confined spinon excitations [10,11,[21][22][23][24], or the characteristic E 8 dynamical spectrum at a hidden 1D quantum critical point [10,[25][26][27]. As illustrated in figure 1, a 3D long-range order can be stabilized at finite temperature (T C for ferromagnet and T N for antiferromagnet). In an applied transverse magnetic field, the ordering temperature reduces continuously. A 3D quantum phase transition occurs at a critical field B c,3D before the long-range order is completely suppressed.
At zero field, the spin dynamics of a Heisenberg-Ising (or XXZ) chain is characterized by fractionalized spinon excitations (or domain-wall excitations). Below the ordering temperature, the interchain couplings provide a confining potential to the spinon-pairs, because the interchain interactions are frustrated for the spins within the two spinons (or domain walls). The energy of the system increases linearly with the distance between the two spinons, thus the dynamics of a spinon-pair can be effectively described by the 1D Schrödinger equation [10,11] − ℏ 2 µ where 2E 0 is the excitation energy threshold, λ|x| is the linear confining potential due to interchain coupling with λ = 2B ∥ ⟨S z ⟩/c, andc is the lattice constant. The solution features states with eigenenergies Schematic phase diagram of spin-1/2 one-dimensional and quasi-onedimensional Ising-like magnets in an applied transverse magnetic field. Without interchain couplings a 1D Ising-like ferromagnet (FM) or antiferromagnet (AFM) exhibits a 1D long-range order only at zero temperature. With interchain couplings a 3D longrange order is formed at finite temperature below T C or T N for ferromagnet or antiferromagnet, respectively. The 1D and 3D long-range orders can be suppressed by an applied transverse field B c,1D ⊥ or B c,3D ⊥ , respectively. Inset: a zigzag ferromagnetic spin chain of CoNb 2 O 6 and a screw antiferromagnetic spin chain of BaCo 2 V 2 O 8 based on magnetic Co 2+ ions [25,27].
which is a linear function of ζ i , with ζ i being the negative zeros of the Airy function, Ai(−ζ i ) = 0. In general, one may treat small interchain couplings as an effective longitudinal magnetic field, and model a quasi-1D spin material by an effective 1D model Hamiltonian where S i denotes the spin on site i, J > 0 is the nearest-neighbor exchange interaction for a ferromagnetic spin chain, and ⟨i, j⟩ indicates the summation over all pairs of nearest neighbors. B ⊥ denotes an applied transverse field. The effective fields due to the interchain coupling are longitudinal and denoted by B ∥ . The corresponding Hamiltonian for an antiferromagnetic chain is found by performing a transformation S i → (−1) i S i , such that with a staggered effective field (−1) i B ∥ .
Due to the effective fields B ∥ , the systems exhibit a spin gap at B c,1D ⊥ = J/2, where otherwise a 1D transverse field Ising quantum critical point should occur. Around this perturbed quantum critical point, an integrable field theory was found, whose dynamical spectrum is described as corresponding to an E 8 Lie algebra [7,[28][29][30]. The dynamic spectrum is characterized by excitations of eight single quasiparticles and also continua of their multiple excitations. The energies of the eight quasiparticles are uniquely defined by their ratios. For instance, the ratio of the first two, m 2 /m 1 ≈ 1.618, is the golden ratio.
Previously, experimental studies suggested that the quasi-1D Ising-like ferromagnet CoNb 2 O 6 [10,20,21,27,31,32] and antiferromagnet BaCo 2 V 2 O 8 [9,18,22,25,26,33] could be very good realizations of the Hamiltonians in equations (3) and (4), respectively. In this work, we perform terahertz (THz) absorption spectroscopy in an applied transverse magnetic field, and precisely track the evolution of the spin dynamics in these two compounds. Starting from zero-field confined spinons, the spin dynamics evolves through a characteristic E 8 dynamics, and finally exhibits a gap closing-reopening behavior, crossing the orderdisorder phase boundary. These features are found to be sharply dependent on the THz radiation polarization relative to the spin chain direction, which appeals for further theoretical studies.

Magneto-THz spectroscopy of BaCo 2 V 2 O 8
The spin-chain antiferromagnet BaCo 2 V 2 O 8 orders below T N = 5.5 K. Whereas the compound crystallizes in a tetragonal structure above T N , a small orthorhombic distortion was found below T N [18,19]. The BaCo 2 V 2 O 8 single crystals were measured down to 2.7 K in a liquidhelium-bath cryostat, equipped with a superconducting magnet for applying fields up to 17 T. The intensity of THz radiation transmitted through the sample was measured with a bolometer, which was kept at 300 mK by a vacuum-insulated He-3 cooling circuit. The transmission spectra were recorded using a Martin-Puplett interferometer [34]. To study polarization dependence and field dependence of the spin excitations, we measured a BaCo 2 V 2 O 8 single crystal cut along the crystallographic ac plane both in Faraday and Voigt configurations, i.e. with the THz radiation travelling parallel and perpendicular to the applied magnetic field, respectively. In the following, we denote the in-plane tetragonal a axis by a, while the other equivalent tetragonal axis perpendicular to the sample surface will be denoted by b. The sample has a surface area of about 4 × 4 mm 2 and a thickness of 0.76 mm. We controlled the incident THz polarization using a rotating polarizer in front of the sample.
The absorption coefficient due to magnetic excitations is found by comparing the transmission spectra below T N with a reference spectrum above T N , for which we chose the zerofield transmission spectrum at 10 K. Since BaCo 2 V 2 O 8 undergoes a phase transition at T N , the absorption peaks associated with the low-temperature phase are not present at 10 K. The absorption coefficient α is then equivalent to the differential absorbance where I(B, T) is the transmitted intensity in a given field B at a given temperature T, and d denotes the sample thickness. For this insulator with relatively weak absorption peaks of magnetic excitations, we assume the reflection coefficient to be constant in the given temperature range.

Magneto-THz spectroscopy of CoNb 2 O 6
The spin-chain ferromagnet CoNb 2 O 6 crystallizes in an orthorhombic structure with a longrange magnetic order formed below T C = 2.85 K. A single crystal with a thickness of 0.5 mm and a diameter of about 3 mm was measured down to 250 mK in a He-3/He-4 dilution refrigerator with a 12 T superconducting magnet. For the detection of the THz radiation, a bolometer was cooled to 400 mK by a separate He-3 cooling circuit. A Martin-Puplett interferometer was utilized for the THz transmission experiment. The measurements were performed in Faraday configuration with an ac-cut sample, where the unpolarized light propagates along the b axis parallel to the applied magnetic field. With this experimental setup it was not feasible to perform measurements for Voigt configuration or with different THz polarizations.
Since warming up the sample above its ordering temperature in the dilution refrigerator is not attainable, we find the absorption coefficient by first calculating the differential absorbance at different magnetic field values using the zero-field spectrum at 250 mK as the reference. The differential absorbance is where d is the sample thickness, and a constant reflection coefficient is once again assumed for this insulator. This, of course, introduces negative peaks at those frequencies where the absorption is present only in zero field. We then eliminate negative absorption and recover the zero-field absorption spectrum by subtracting a baseline from the differential absorbance, which gives us the absorption coefficient α. The subtracted baseline is obtained statistically from the median absorption values of all spectra measured at different strengths of the applied magnetic field. In addition, we measured the same CoNb 2 O 6 sample above T C at 4 K by using a liquidhelium-bath cryostat. The sample was measured in the same configuration as in the dilution refrigerator, and the same procedure for calculating the absorption coefficient was utilized. In the liquid-helium-bath cryostat the rotating polarizer in front of the sample allowed us to measure the spectra with two orthogonal incident light polarizations.

Absorption spectra in BaCo 2 V 2 O 8
BaCo 2 V 2 O 8 crystallizes in a tetragonal structure with the spin chains running along the c axis. Figure 2 displays the absorption spectra of BaCo 2 V 2 O 8 , measured at 2.7 K for various applied magnetic fields up to 17 T. The magnetic fields are applied perpendicular to the Ising axis (the c axis), with directions denoted by B ∥ a or B ∥ b. These two orientations are equivalent because of the tetragonal crystalline structure. The spectra exhibit very clear polarization dependence. In zero field, a series of absorption peaks are only present when the THz oscillating magnetic field B ω is oriented along the a axis (B ω ∥ a, see figures 2(a) and (c)), while they completely disappear when B ω is along the c axis (B ω ∥ c, see figures 2(b) and (d)). In an applied magnetic field, the spectra of B ω ∥ c are practically identical with the applied field directions B ∥ a (figure 2(b)) and B ∥ b ( figure 2(d)). This also indicates that the a and b axes are equivalent in this material, as expected for the tetragonal structure. A single absorption peak is observed in the finite magnetic fields, which shifts to higher energy for higher fields and becomes increasingly stronger.
In contrast, the spectra of B ω ∥ a are sensitive to the orientation of the applied magnetic field. As shown in figures 2(a) and (c), the spectra for B ∥ B ω exhibit richer features than that of B ⊥ B ω , even though the THz polarization relative to the crystal axes is the same. For B ∥ B ω ∥ a ( figure 2(a)), the zero-field modes split in finite fields. In particular, the lowest-lying mode softens with increasing field until 9.5 T, above which it hardens again. This is a signature of a field-induced quantum phase transition, which is consistent with the observed phase boundary at B c,3D ⊥ = 9.5 T between a low-field 3D ordered phase and high-field disordered phase [18], see figure 1. For B ⊥ B ω ∥ a (figure 2(c)), all the observed zero-field modes harden at higher magnetic fields. Above B c,3D ⊥ = 9.5 T, only the lowest-lying mode remains visible while the higher-energy modes disappear in the disordered phase. As we will discuss below,  these higher-energy modes are the confined spinon excitations, which are present in the 3D ordered phase due to the interchain couplings.

Absorption spectra in CoNb 2 O 6
The absorption spectra of CoNb 2 O 6 above T C and below T C are presented in figures 3 and 4, respectively, for an applied transverse magnetic field along the crystallographic b axis. Above T C , the spectra with two orthogonal polarizations are shown in figure 3(a) for B ω ∥ a and in figure 3(b) for B ω ∥ c. In zero field, the B ω ∥ a spectrum exhibits two modes at 7.5 and 16 cm −1 , while the spectrum of B ω ∥ c is rather featureless. The energy of the 7.5 cm −1 mode stays almost constant in the applied magnetic field until around 2 T, and then rapidly increases. In contrast, the 16 cm −1 mode splits into a softening and a hardening mode. Interestingly, the hardening branch is absent with B ω ∥ c, while the overall spectra above 2 T look very similar for the two polarizations. Above approximately 5 T, the spectra are mainly characterized by a rather broad absorption band, which shifts from 5 cm −1 to 17 cm −1 at 15 T. This band seems to be composed of more than one mode, which overlap with each other and can hardly be distinguished. Figure 4 presents the spectra measured at 250 mK below T C with unpolarized THz radiation for the transverse fields up to 12 T. In contrast to the spectra above T C , the spectra of the magnetically ordered phase exhibit much richer features (see also figures 9(a) and (b)). There are more absorption peaks resolved, they are sharper and can be clearly followed in the applied magnetic fields. At higher fields above 5 T, the spectra are similar to those measured at 4 K, see figure 3.

Confined spinons in the zero-field 3D ordered phases
The dynamics in a 1D quantum spin-1/2 magnet is characterized by fractionalized spinon excitations with a quantum number S = 1/2. A single spin-flip creates a pair of spinons, which could be viewed as two domain-walls. For a Heisenberg-Ising chain the spinon-pair excitations form a continuum with a spin-excitation gap. For a quasi-1D system below the ordering temperature, the spinons are confined in a linear potential due to the interchain couplings. The eigenenergies of the confined spinon excitations should follow the equation (2), featuring a linear dependence on ζ i . The intercept 2E 0 reflects the spin gap, which is due to the Ising anisotropy, while the slope λ 2/3 ( ℏ 2 µ ) 1/3 contains the λ-term reflecting the strength of the confinement. The zero field spectra of BaCo 2 V 2 O 8 below T N and of CoNb 2 O 6 below T C were more concisely discussed previously in [25,27], respectively, while they are represented here in figures 5 and 6 for further quantitative comparison. In the case of BaCo 2 V 2 O 8 , a series of five absorption peaks is observed at E 1 = 13.4, E 2 = 20.6, E 3 = 25.5, E 4 = 29.6, and E 5 = 33.1 cm −1 ( figure 5(a)). These frequencies are presented as a function of ζ i for i = 1, 2, . . . , 5 in figure 5(b), which exhibits clearly a linear dependence. A fit to the equation (2) provides 2E 0 = 5.73 cm −1 (∼0.71 meV) and λ 2/3 ( ℏ 2 µ ) 1/3 = 3.51 cm −1 (∼0.44 meV). In the zero-field spectrum of CoNb 2 O 6 at 250 mK, one can observe six modes at relatively lower frequencies, E 1 = 9.6, E 2 = 10.7, E 3 = 11.6, E 4 = 12.6, E 5 = 13.4, and E 6 = 14.1 cm −1 ( figure 6(a)). The dependence of these frequencies on ζ i , as shown in figure 6(b), follows also the linear function in equation (2), for the fit parameters 2E 0 = 7.97 cm −1 (∼0.99 meV) and λ 2/3 ( ℏ 2 µ ) 1/3 = 0.68 cm −1 (∼0.084 meV).
The excellent linear fits show that the Schrödinger equation in equation (1) figure 7, also with that of another spin-chain antiferromagnet SrCo 2 V 2 O 8 [11]. While the threshold energy in CoNb 2 O 6 is the highest, the confining effect is least evident. In contrast, in BaCo 2 V 2 O 8 it requires the lowest energy to excite a spinon-pair, which experiences the largest confining among the three systems.
It is interesting to note that the confined spinons in BaCo 2 V 2 O 8 are absent in the spectrum if the polarized THz magnetic field is parallel to the Ising easy axis, i.e. B ω ∥ c (see figures 2(b) and (d)), but are observed with B ω perpendicular to the easy axis (see figures 2(a) and (c)).  (2)). The linear fit parameters correspond to the linear confinement potential and the threshold energy for creating a spinon pair. This can be understood as selection rule of magnetic-dipole excitations. In the 3D ordered phase, the spins are aligned antiferromagnetically along the c axis. Therefore, the coupling of the polarized THz magnetic field B ω with the spin component S z will not trigger a spinflip. A spin-flip excitation will occur when the THz magnetic field B ω is along the transverse orientations, i.e. B ω ∥ a or B ω ∥ b.

Spin dynamics in the transverse magnetic fields: E 8 characteristics and quantum phase transitions
The field dependence of the observed spin excitations in BaCo 2 V 2 O 8 is represented in figure 8, where the resonance frequencies of the spin excitations are plotted as a function of the applied magnetic field. The spectra exhibit very clear contrast for different polarizations and for different orientations of the applied magnetic field, as compared in figures 8(a) and (b). While there are no excitations resolved in zero field for B ω ∥ c, a single mode is observed in a finite magnetic field above 2 T, which shifts to higher energy as the field is increased. The observation of this mode reflects that, in the applied transverse fields, the spins are tilted away from the c axis (the easy axis), which allows a spin-flip excitation through the coupling with the THz magnetic field. The field dependence of this mode is essentially identical to the one with the transverse fields applied along the a or the b axis, as a consequence of the pseudo-tetragonal symmetry.
While a quantitative understanding of the observed field-dependent behavior requires a detailed numerical calculation of the dynamical spectra, we highlight a few interesting observations. There is a clear contrast between two applied field directions B ∥ a and B ∥ b with the polarization B ω ∥ a. For B ω ⊥ B, the absorption modes corresponding to the confined spinons harden monotonically in the applied magnetic field, whereas for B ω ∥ B these modes split in finite field and exhibit complex field-dependent evolutions (see figures 2(a) and 8(a)).
For B ω ∥ B ∥ a, the lowest-lying mode softens continuously until about 9.5 T and then hardens monotonically in higher fields. This indicates a field-induced quantum phase transition at B c,3D ⊥ , above which the 3D order is suppressed as revealed by other experiments [9,18]. At the same time, the higher-energy confined-spinon modes observed with B ∥ b disappear above B c,3D ⊥ as well. The lowest spinon-pair mode splits into two, with the lower-lying one exactly overlapping with the branch of B ω ∥ c (see figure 8(b)).
In a different representation, figure 8(c), we plot the energy ratios of the higher-energy modes with respect to the lowest-lying mode m 1 for B ω ∥ B ∥ a. We note that, between 4 and 6 T, a small satellite peak can be observed very close to the lowest-lying mode (see figure 2(a)) ⊥ . The ratios approach theoretically predicted values for the E 8 excitations (marked with dashed lines) at 5 T. In (c) the frequency of m 1 is taken as the average of the two low-lying peaks, which split from a single peak due to anisotropy (see figure 2(a)).
confined spinons no yes no yes E 8 excitations no yes no no due to small magnetic anisotropy in the 3D ordered phase [18,35]. Thus, in figure 8(c) the value of m 1 is taken as the average of the two low-lying peaks, which split from a single peak due to anisotropy. With increasing field, the ratios increase monotonically above 3 T. At 5 T, these ratios reach the expected ratios of the so-called E 8 particles (dashed lines) which was predicted for the perturbed 1D transverse-field Ising quantum critical point [7], corresponding to the Hamiltonians in equations (3) and (4) for ferromagnetic and antiferromagnetic chains, respectively. We can see that apart from the single-particle channel, the two-particle excitations m 1 + m 2 are also clearly observed. The observations for B ω ∥ B ∥ a were reported with a very good agreement with theoretical results [25,30] and are also consistent with recent reports of inelastic neutron scattering [26]. The observed evolution of the spin dynamics from the confined spinons in zero field to the E 8 excitations at a perturbed 1D transverse-field Ising quantum critical point, which is also polarization dependent, is not straightforward to understand. The corresponding selection rules are summarized in table 1. Both the confined spinons and E 8 excitations are simultaneously allowed in only one configuration, B ω ∥ B ∥ a. The observation of the E 8 excitations points to a realization of the phase diagram as illustrated in figure 1, i.e. the 1D quantum critical point is hidden in the 3D ordered phase, B c,1D ⊥ . In the 3D ordered phase, the interchain couplings provide the required perturbative longitudinal field for the emergence of the E 8 excitations.

CoNb 2 O 6 .
A similar phase diagram is also realized in the ferromagnetic chain CoNb 2 O 6 . As shown in figures 4 and 9(a), the lower-lying confined-spinon modes harden while the higher-lying ones soften in the applied transverse field. Above 2 T, these modes exhibit different field dependencies. The overall field dependence of the spin excitation spectra is not the same as in BaCo 2 V 2 O 8 . This shows that the observed field-dependent behavior is not universal but material-dependent, and a quantitative description, therefore, requires a material-or model-specific numerical analysis. Nonetheless, we can see that, above 2 T, the lowest-energy mode softens continuously until a critical field B c,3D ⊥ ≈ 5.3 T for CoNb 2 O 6 , which is followed by hardening in higher fields (see also figure 4). This corresponds to the suppression of the 3D ordered phase by the applied magnetic field, which is consistent with previous reports [20]. Approaching the critical field from below, the energy ratios with respect to the lowest-lying mode exhibit a monotonic increase. At B c,1D ⊥ = 4.75 T, these ratios simultaneously reach the expected ratios of the E 8 excitations up to the energy of the sixth particle m 6 (figure 9(c)) [27]. This not only confirms the observation of the E 8 excitations beyond the energy of 2m 1 [36], but also demonstrates the material-independent universality of the E 8 dynamics characteristic of the perturbed 1D transverse-field Ising quantum critical point.

Summary
We have performed THz spectroscopy of quantum spin dynamics in the quasi-one-dimensional spin-1/2 Ising-like antiferromagnet BaCo 2 V 2 O 8 and ferromagnet CoNb 2 O 6 in applied transverse magnetic fields. Due to interchain couplings, 3D ordering phases are formed below T N = 5.5 K for BaCo 2 V 2 O 8 and below T C = 2.85 K for CoNb 2 O 6 . In zero field, the spin dynamics is characterized by confined-spinon (domain-wall) excitations. With increasing magnetic field, the systems evolve in material-dependent complex ways through a regime where a characteristic E 8 dynamical spectrum is observed (around 5 T for BaCo 2 V 2 O 8 and 4.75 T for CoNb 2 O 6 ), and finally enter the field-induced disordered phase above the 3D quantum critical field B c,3D ⊥ (9.5 T for BaCo 2 V 2 O 8 and 5.3 T for CoNb 2 O 6 ). Moreover, we determine the polarization dependence of the confined-spinon excitations and the E 8 excitations in BaCo 2 V 2 O 8 . While the detected confined spinons reflect material-dependent magnetic properties, the observed E 8 dynamical spectrum exhibits universal characteristics of a perturbed 1D transverse-field Ising quantum critical point. Beyond the critical regime, a quantitative understanding of the field-dependent spin dynamics remains a subject for further theoretical studies.

Data availability statement
The data that support the findings of this study are available upon reasonable request from the authors.