Emergent $\mathcal{D}_{8}^{(1)}$ spectrum and topological soliton excitation in CoNb₂O₆

Enhanced quantum fluctuations near QCPs can give rise to rich emergent phenomena [1, 2, 3, 4], including enhanced symmetry, deconfined fractional excitations, and emergent integrability. As a prototypical model for studying quantum criticality, the TFIC, which can be realised in a number of q1D quantum magnets, continues to be a research hotspot [1, 2, 5, 6, 7, 8, 9, 10, 11, 12, 13]. In q1D magnets, the interchain couplings, though weak, are relevant perturbation that causes 3D magnetic ordering of the system. As a consequence, the genuine 1D QCP is usually hidden inside the 3D ordered phase as illustrated in Fig. 1(a). This makes the critical behaviour even more complex and intriguing [13, 14]. The significance of the 1D quantum criticality in a TFIC is manifested by its spin excitation spectrum, and in the 3D magnetic ordered phase, the interchain interaction can be treated as an effective weak longitudinal field $\tilde{h}$ that confines the quasiparticles at critical into gapped bound states [8, 15] with quantum $E_{8}$ integrability, whose spectrum and scattering matrix are organised by the $E_{8}$ Lie algebra [16, 2].

The above scenario for the emergent $E_{8}$ integrability was long predicted theoretically [16] and the $E_{8}$ spectrum was initially proposed to be realised in the q1D Ising magnet CoNb₂O₆ [2]. Recently, the full $E_{8}$ spectrum and well defined dispersive $E_{8}$ quasiparticles have been observed in another q1D Ising magnet, BaCo₂V₂O₈, under transverse magnetic field [8, 17, 18, 15]. As for CoNb₂O₆, however, results based on recent measurements are still controversial.

CoNb₂O₆ is a renowned q1D Ising magnet with zigzag ferromagnetic (FM) chains along the crystalline $c$ axis forming a frustrated isosceles triangular lattice in the $a$-$b$ plane as depicted in Fig. 1(c) and (d). The FM intrachain coupling $J$ is much stronger than the antiferromagnetic (AFM) interchain ones, $J_{i}$ and $J_{i}^{\prime}$  [19, 20, 21, 22]. Experiments suggest that a 1D QCP in the TFIC universality class at $H_{c}^{\rm{1D}}\simeq 5.0-5.3$ T [2, 11, 12, 13, 22, 23, 10] is hidden in the 3D AFM ordered phase not far from the 3D QCP (at $H_{c}^{\rm{3D}}\simeq 5.5$ T [2]). The seminal inelastic neutron scattering (INS) measurement [2] provides a first evidence of $E_{8}$ spectrum: The energy ratio of two lowest peaks identified is close to the golden ratio, which is the exact mass ratio of the two lightest $E_{8}$ particles, when the system is tuned approaching $H_{c}^{\rm{1D}}$. Recent THz spectroscopy measurements with much higher energy resolution, however, revealed numerous additional satellite excitation modes that surpass the $E_{8}$ description [23, 24]. Besides, recent INS results indicate that the low-energy spectrum is influenced by the domain wall (DW) interaction associated with the reduced lattice symmetry of CoNb₂O₆ [9]. These results raise crucial questions about the adequacy of the Ising universality and the emergent $E_{8}$ spectrum.

In this article, we reexamine the spin excitations of CoNb₂O₆  near $H_{c}^{\rm{1D}}$ by performing iTEBD calculation and field theoretical analysis. We demonstrate that the low-energy physics in the vicinity of $H_{c}^{\rm{1D}}$ is controlled by the TFIC universality and the DW interaction is irrelevant. By suitably incorporating interchain spin fluctuations enhanced by spin frustration and the proximity to a 3D QCP, we find a remarkable result that the spectrum of CoNb₂O₆  is not consistent with the $E_{8}$ algebra but can be described by the $\mathcal{D}_{8}^{(1)}$ Lie algebra associated with the Ising${}_{h}^{2}$ integrable model. The satellite peaks incompatible with the $E_{8}$ algebra in the THz measurements are re-identified as the $\mathcal{D}_{8}^{(1)}$ soliton and/or breather excitations. The topological single soliton excitation, usually forbidden in other systems, is found to be robust over a finite field range. We propose that the $\mathcal{D}_{8}^{(1)}$ spectrum can be realised in a broad class of q1D quantum magnets. These results expand the realm of emergent phenomena in quantum magnetic systems.

We consider the following Hamiltonian for CoNb₂O₆,

where $\mathcal{H}_{\text{chain}}$ and $\mathcal{H}_{\text{ic}}$ include the intra- and inter-chain interactions, respectively. The intrachain Hamiltonian proposed to precisely describe the magnetic properties of CoNb₂O₆  reads as [9]

where the first two terms form a 1D XXZ model in the FM Ising limit, the third term refers to an AFM interaction between the next-nearest neighbouring (n.n.n.) spins, the fourth term is the so called DW interaction associated with the two DW bound state excitations, and the last term comes from the applied transverse magnetic field (along the crystalline $b$ axis). The AFM and DW terms originate from the zigzag geometry of the chain. Following Ref. 9, we take $J=2.7607$ meV, $\varepsilon=0.239$, $\lambda_{af}=0.1507$, $\lambda_{dw}=0.1647$, and $g=3.100$, which were shown to accurately describe the low-energy excitations of CoNb₂O₆. We note that more refined parameters for high-energy and strong-field excitations were recently proposed [22, 25].

It is believed that the dominant interchain interaction is also of Ising-type, so we consider the following Hamiltonian

where chain labels $\alpha$, $\beta$ run over n.n. chains. Given the weak coupling $J_{i}$, $\mathcal{H}_{\text{chain}}$ is usually treated at a chain mean-field level when the system is inside the 3D ordered phase, $\mathcal{H}_{\text{ic}}\approx-\tilde{h}\sum_{j}S^{z}_{j,\alpha}$, where $\tilde{h}=-J_{i}\sum_{\beta}\langle S^{z}_{j,\beta}\rangle$ is an effective longitudinal field acting on a single chain from its neighbours.

However, the frustrated interchain alignment causes cancellation of the effective fields from neighbouring chains and the interchain fluctuations, which are further enhanced by the proximity to the 3D QCP, must be treated in a more proper way. We then adopt a cluster mean-field approximation. Taking into account the two-sublattice nature of the 3D AFM order, we pick up a minimal unit consisting of two n.n. chains. The interchain couplings between these two chains are treated exactly, while the couplings to other chains are considered at the chain mean-field level as described above. The Hamiltonian of the minimal model then reads as

where $m$ is the chain index, and $\tilde{h}$ refers to the effective longitudinal field introduced by the chain mean-field approximation. Note that the minimal model recovers to two decoupled Ising chains when $J_{i}=0$, and an $E_{8}$ integrability is expected in this case. We will show in the following that the system crosses over to a novel quantum integrable class exhibiting the $\mathcal{D}_{8}^{(1)}$ mass spectrum in the $J_{i}/\tilde{h}\rightarrow\infty$ limit [Fig. 1(b)].

By comparing the experimental and calculated spectra and computing the critical exponents in the minimal model, we first show that the non-Ising terms, including the DM interaction, are irrelevant to the TFIC universality of the 1D QCP [see Supplemental Materials (SM) [26]]. Then we analyse the emergent integrabilities of the minimal model in Eq. (4). Without $\tilde{h}$, each chain at $H_{c}^{\rm{1D}}$ is described by a $(1+1)$D conformal field theory (CFT) with central charge $c=1/2$. Either a small $\tilde{h}$ field or a small coupling $J_{i}$ can push the ladder system away from criticality into a field-induced or intrinsic ordered phase, respectively. The former one has the system emerged the $E_{8}$ integrability emerges [27], while the later one drives the system approaching the Ising${}_{h}^{2}$ integrability. In the $\tilde{h}=0$ limit, the low-energy physics is described by the following Ising${}_{h}^{2}$ field theory

where $\mathcal{A}_{c=1/2}^{(\alpha)}$ denotes the $c=1/2$ CFT action for the critical chain $\alpha$, $\sigma^{(\alpha)}(x)$ and $\lambda$ refer to the correspondences of $S^{z}_{j,\alpha}$ and interchain coupling in the continuous limit, respectively. In the scaling limit [$a\;({\text{lattice spacing}}),\lambda\rightarrow 0$ with finite $\lambda/a$], the theory exhibits emergent Ising${}^{2}_{h}$ integrability characterised by the $\mathcal{D}_{8}^{(1)}$ Lie algebra [28]. In the most general case where both $\tilde{h}$ and $J_{i}$ are present, the mass spectrum of the minimal model in Eq. (4) crosses over from $E_{8}$$\times$$E_{8}$ to $\mathcal{D}_{8}^{(1)}$, as illustrated in Fig. 1(d).

As for CoNb₂O₆, the ordered moment at $H_{c}^{\rm{1D}}$ should be tiny since $H_{c}^{\rm{1D}}$ is very close to $H_{c}^{\rm{3D}}$. This, together with the frustrated alignment of chains, suppresses the effective field $\tilde{h}$. We therefore expect $J_{i}\gg\tilde{h}$ in CoNb₂O₆, so that its excitation spectrum near $H_{c}^{\rm{1D}}$ is characterised by the $\mathcal{D}_{8}^{(1)}$ algebra. Note that this is different from the case of BaCo₂V₂O₈, whose 1D QCP is located deep inside the 3D ordered phase [8]. There, the chain mean-field theory is a good approximation such that the excitation spectrum is well described by the $E_{8}$ algebra.

The integrable Ising${}_{h}^{2}$ model in Eq. (5) possesses excitations associated with the $\mathcal{D}_{8}^{(1)}$ algebra, which is characterised by a total of 8 types of particles, including one soliton ($S_{+1}$) and one antisoliton ($S_{-1}$), each with mass $m_{\text{s}}$, as well as 6 breathers $B_{n}$ with masses $m_{n}=2m_{\text{s}}\sin(n\pi/14)$, $(n=1,\dots,6)$  [28], which are referred to $\mathcal{D}_{8}^{(1)}$ particles in the following. We expect that peaks corresponding to these quasiparticles appear in the excitation spectrum of the minimal model at $\tilde{h}=0$ within the energy-momentum range where the Ising universality dominates. To check this, we numerically calculate the excitation spectrum at $k=0$ of an Ising ladder with $J_{i}=0.36J$ at $H_{c}^{\rm{1D}}$. As shown in Fig. 2(a), we identify a series of peaks with energies precisely corresponding to masses of $\mathcal{D}_{8}^{(1)}$ particles. Besides the single particle peaks, we also identify edges of several multiparticle excitation continua. Interestingly, the most prominent peak in the spectrum at about $0.74$ meV originates from the unusual topological single (anti)soliton. Also note that we cannot resolve any spectral signature corresponding to single $B_{1}$, $B_{3}$, or $B_{5}$ breather. This verifies a recent theoretical prediction that these odd-parity breathers cannot be excited from the ground state because of symmetry restriction [29]. The agreement between the numerical and analytical results indicates that the $\mathcal{D}_{8}^{(1)}$ physics is robust even for a sizeable interchain coupling.

We then compare our result with the spectrum obtained in a recent high-resolution THz measurement [23] for CoNb₂O₆  near $H_{c}^{\rm{1D}}$ (in Fig. 2(c)). Surprisingly, most peaks in the experiment can be assigned to single or multiple $\mathcal{D}_{8}^{(1)}$ particles, all the way up to about $2$ meV. To understand the two peaks (labelled by arrows) not captured by the Ising ladder model, we further calculate the spectrum of the minimal model with the non-Ising interactions and at $\tilde{h}=0$. As shown in Fig. 2(b), peaks consistent with the experiment emerge at arrow positions. We find these peaks are indeed associated with details of the microscopic model, instead of the $\mathcal{D}_{8}^{(1)}$ algebraic structure. In fact, the peak labelled by the red arrow appears at both $k=0$ and $k=\pi/2$, indicating it is a zone-folding peak caused by the n.n.n. AFM interaction $J_{\rm{AF}}$. The peak at the blue arrow, however, is attributed to the confining effect from the DW interaction, which gives rise to additional bound states out of the $2m_{1}$ continuum of the $\mathcal{D}_{8}^{(1)}$ spectrum (see Fig. 2(a)).

We further calculate the spectrum of a single chain under an effective field $\tilde{h}$, which is characterised by the $E_{8}$ algebra. As shown in Fig. 2, many features (satellite peaks) of the experimental spectrum are missing in the $E_{8}$ model but can be well captured by the $\mathcal{D}_{8}^{(1)}$ one. In Fig. 3(a), we plot the field dependence of several characteristic energies extracted from peaks in experimental spectra presented in Refs. 23 and 24. The energy ratios best fit to the $\mathcal{D}_{8}^{(1)}$ mass spectrum at about $5$ T. (Putative) $H_{c}^{\rm{1D}}$ determined in this way is very close to the value ($\sim 5.1-5.3$ T) determined from a previous NMR measurement [13]. Note that by assuming the $E_{8}$ structure, the same data gave $H_{c}^{\rm{1D}}\simeq 4.75$ T [23] which deviated much to the NMR value. All these results unambiguously justify the emergent $\mathcal{D}_{8}^{(1)}$ mass spectrum at low energies in CoNb₂O₆.

As illustrated in Fig. 2, Fig. 3(a), and Ref. 23, with the transverse field increasing near $H_{c}^{\rm{1D}}$, one signature of the spectrum is the splitting of $E_{8}$-like peaks where the $\mathcal{D}_{8}^{(1)}$  particles arise. Note that the mass ratio of $(m_{1}+m_{s})/m_{2}\approx 1.665$ in $\mathcal{D}_{8}^{(1)}$ and that of $m_{2}/m_{1}\approx 1.618$ in $E_{8}$  are very close. Taking into account the field dependence of peak positions, it is inadequate to assert any emergent integrability from calculating mass ratios of few low-energy peaks. An observation of a full spectrum is crucial.

From Fig. 3(a), we find that the mass ratio $m_{s}/m_{2}$ varies little over a finite field range near $H_{c}^{\rm{1D}}$, whereas the mass ratios of other $\mathcal{D}_{8}^{(1)}$ particles show much stronger field dependence. This is because the single (anti)soliton, corresponding to a single domain wall, is a topological excitation that is robust against local perturbations. A single soliton is usually forbidden. Here it is inherent to the $\mathcal{D}_{8}^{(1)}$  algebra and stabilised as confined by the interchain coupling.

In Fig. 3(b) we show the calculated spectrum of the minimal model in the full BZ using the same parameters as in Fig. 2(b). Clear dispersive bands corresponding to well defined $\mathcal{D}_{8}^{(1)}$ quasiparticles show up in low energies. The bands span about 30% of the BZ and follow the predicted relativistic dispersion of $\mathcal{D}_{8}^{(1)}$ particle $\omega=\sqrt{m_{n}^{2}c^{4}+k^{2}c^{2}}$, where $m_{n}$ refers to the mass of the $n$-th quasiparticle, and $c$ is a characteristic velocity. Moreover, the masses follow scaling relation $m_{n}\sim J_{i}^{4/7}$ [Fig. 3(c)] as predicted from the Ising${}_{h}^{2}$ integrable theory (SM [26]). This is to be contrast to the $E_{8}$ model in which $m_{n}\sim J_{i}^{8/15}$. Interestingly, the dispersion and scaling relation apply to both the topological (anti)soliton excitation and the odd-parity breathers (which are invisible in the excitation spectrum) [28, 29]. The unusual topological properties and symmetric restrictions make these particles potentially useful resources in quantum information. All these features deserve to be explored by spectral measurements such as INS and NMR.

Even though we do not perform fine tuning of model parameters, our minimal model describes the spectrum of CoNb₂O₆  quantitatively well. This suggests that our model, being a cluster mean-field approximation, captures the correct emergent low-energy physics of the system. We can understand this as follows: Fixing the transverse field to $H_{c}^{\rm{1D}}$, the interchain coupling is definitely relevant when going from decoupled chains to a ladder. It drives the system from critical to inside the ordered phase and causes emergent $\mathcal{D}_{8}^{(1)}$ mass spectrum. We can further extend the cluster by including more chains, until it covers the full 3D system, where the approximation becomes exact. During this procedure, the system becomes non-integrable. But we expect that the interchain coupling only causes less relevant perturbation to the $\mathcal{D}_{8}^{(1)}$ mass spectrum. This is indeed verified by our numerical calculation on 4 coupled Ising chains (Fig. S5 of the SM [26]). Interestingly, we find that the spin frustration plays a crucial role in stabilising the $\mathcal{D}_{8}^{(1)}$ spectrum. In the frustrated case, the spectrum resembles the $\mathcal{D}_{8}^{(1)}$ of two decoupled ladders, whereas the spectrum turns to $E_{8}$ in the unfrustrated case.

Note that although we show the robustness of the $\mathcal{D}_{8}^{(1)}$ mass spectrum for CoNb₂O₆, the experimental spectral intensity may crucially depend on both the microscopic details of interactions and experimental setup, and is generically non-universal. It is worth further noting that the emergent $\mathcal{D}_{8}^{(1)}$ mass spectrum is not limited to CoNb₂O₆, but can be applied to a large class of quantum magnets. It obviously shows up in spectra of weakly coupled quantum Ising ladders with rung interaction much weaker than that along the ladder direction. Moreover, we expect our argument for CoNb₂O₆  can be well applied to other weakly coupled Ising chains when $H_{c}^{\rm{1D}}$ is close to $H_{c}^{\rm{3D}}$. It would also be interesting to experimentally test the crossover from $E_{8}$ to $\mathcal{D}_{8}^{(1)}$ physics by tuning the distance of $H_{c}^{\rm{1D}}$ to $H_{c}^{\rm{3D}}$ via either rotating the field direction or applying a pressure. Last but not least, the physics should also be realised in specifically designed Rydberg atom systems.

In conclusion, we show that the spectrum of CoNb₂O₆  near its 1D QCP is described by the emergent $\mathcal{D}_{8}^{(1)}$ algebra. We show the emergent $\mathcal{D}_{8}^{(1)}$ mass spectrum contains breather and exotic single-soliton excitations. Our results advance the study on quantum integrability and topological excitations in quantum magnets.

We thank Linhao Li for helpful discussions. This work is supported by the National Key R&D Program of China (Grant No. 2023YFA1406500), the National Natural Science Foundation of China (Grant Nos. 12334008, 12274288, and 12174441), the Innovation Program for Quantum Science and Technology Grant No. 2021ZD0301900, and Natural Science Foundation of Shanghai with Grant No. 20ZR1428400.