Quantum critical excitations in Ising-like spin chains: A heat transport perspective

Abstract

Regarded as the closest realization of the quantum Ising model in a magnetic field, which features stable quasiparticles with an emergent E_8 symmetry, CoNb₂O₆ keeps being a source of inspiration by manifesting intricacies beyond this model. A notable example is the recent finding of an unexpected band of localized gapless fermionic excitations around the field-induced quantum critical point (QCP), whose origin remains unclear. Despite differences in many aspects compared to CoNb₂O₆, BaCo₂V₂O₈ in a magnetic field also exhibits the E_8 physics, while allowing for better scrutiny of the critical excitations associated with its well-separated one- and three-dimensional QCPs. Here we study the low-lying magnetic excitations within the antiferromagnetic order of BaCo₂V₂O₈ by performing ultralow-temperature heat transport measurements. The field-independent thermal conductivity at the lowest temperatures and the suppressed thermal conductivity with an increasing field at higher temperatures point to heat-carrying phonons scattered by magnetic excitations. For the magnetic excitations, importantly, we find no evidence for the emergence, around the one-dimensional QCP, of gapless fermionic excitations similar to those observed in CoNb₂O₆. Based on the contrasting cases of BaCo₂V₂O₈ and CoNb₂O₆, we propose that the frustrated alignment of chains and the proximity of two QCPs in CoNb₂O₆ may put this system in a unique place to host the gapless fermionic excitations.

Graphical abstract

Keywords

quantum criticality / magnetic excitations / ultralow-temperature thermal conductivity / Ising chain

1 Introduction

As an archetype for the study of phase transitions and critical phenomena, the Ising model and its variants have been extended beyond mathematics and physics to diverse fields of complex systems ranging from neuroscience to financial markets, all of which are characterized by interacting components leading to emergent collective behavior [1]. The quantum transverse-field Ising chain (TFIC) shares the same universality class with the textbook classical two-dimensional Ising model, whose exact solution was first obtained by Onsager decades ago [2]. When the TFIC is further perturbed by a longitudinal field, the exact solvability is lost. Fortunately, however, the scaling limit of the model around the quantum critical point (QCP) can still be described by an integrable field theory with an emergent E_8 exceptional Lie algebra symmetry [3, 4]. The most salient feature is the appearance of a series of discrete modes corresponding to eight stable quasiparticles, the E_8 particles, with particular ratios of their mass gaps predicted by the field theory [4−7].

The quasi-one-dimensional (quasi-1D) spin-1/2 ferromagnetic Ising chain system CoNb₂O₆ has long been exploited for the demonstration of the E_8 physics [8−16]. For perfect 1D chains (isolated chains), there is a quantum phase transition from a ferromagnetic to a paramagnetic state at a critical transverse field, i.e., 1D QCP [17]. The quasi-1D nature of real materials, on the other hand, brings about weak but nonzero interchain interactions that result in a 3D long-range magnetic order, which terminates at the 3D QCP [11, 18, 19]. Discrete modes corresponding to the E_8 particles are expected to be evident around the 1D QCP, which is confirmed for CoNb₂O₆ by neutron and terahertz spectroscopy [10, 12−15, 20]. Moreover, recent studies suggest that CoNb₂O₆ may exhibit complexities beyond the E_8 physics: instead of a model of longitudinal-field-perturbed TFIC, the frustration induced quantum motion of domain walls in CoNb₂O₆ was argued to be alternatively described by a model of twisted Kitaev chain, reminiscent of the honeycomb Kitaev spin liquid [21]; On another front, gapless fermionic (GLF) excitations were identified around the closely-spaced QCPs by a linear-in-temperature term in the specific heat [16], and were further found to be localized according to the lack of a corresponding contribution from these excitations to thermal conductivity [22]. The existence of the GLF excitations is not expected by existing theory. However, the proximity of the 1D and 3D QCPs in CoNb₂O₆ hinders a clear look at the origin of these excitations: are they more related to the 1D or the 3D QCP? Do the low-lying GLF excitations always appear in conjunction with the higher-energy E_8 particles?

A way out may lie in the study of BaCo₂V₂O₈, which is a quasi-1D spin-1/2 antiferromagnetic Heisenberg-Ising (XXZ) chain with Ising-like anisotropy attracting recent attention due to its various novel quantum phases and excitations. In longitudinal fields, the research focus has been on the emergence of an incommensurate spin density wave phase and string excitations [23−29]. In transverse fields, the 3D QCP has been identified to correspond to a topological transition between two phases characterized by dual solitonic excitations [30, 31]. For smaller transverse fields, the “1D QCP hidden inside 3D order” picture for CoNb₂O₆ applies also to BaCo₂V₂O₈ [32−34], despite the antiferromagnetic intrachain coupling in BaCo₂V₂O₈ compared to the ferromagnetic one in CoNb₂O₆. BaCo₂V₂O₈ crystallizes in a tetragonal structure, with the edge-sharing CoO₆ octahedra forming a screw chain along the c axis [Fig. 1(a)]. Due to the screw chain structure, the local easy axes of the magnetic Co²⁺ ions tilt slightly away from the crystallographic c axis, leading to an anisotropic g tensor [32, 35, 36]. The consequence is that when a uniform field is applied along, say, the tetragonal a axis, a staggered field perpendicular to both the Ising axis and the uniform field is generated, serving as the transverse field in the TFIC model. Inside the 3D order (with T_{\text{N}}\sim 5.5\text{K} at zero field), the finite interchain coupling further induces an effective staggered longitudinal field along the Ising axis. These characteristics combined provide all the prerequisites for the realization of a TFIC perturbed by a longitudinal field or, equivalently, a quantum Ising chain in both transverse and longitudinal fields.

Indeed, similar to CoNb₂O₆, the E_8 spectrum in BaCo₂V₂O₈ in transverse fields around its 1D QCP has been identified by several experimental probes, showing quantitative agreement with theory [34, 37, 38]. Reflecting such physics, the phase diagram of BaCo₂V₂O₈ is shown in Fig. 1(d), with the Néel temperatures obtained from the magnetic susceptibility data in Fig. 1(b). For comparison, the phase diagram of CoNb₂O₆ adapted from Ref. [16] is shown in Fig. 1(c). Considering the remarkable similarities between the two systems, the low-lying quantum critical excitations in BaCo₂V₂O₈ are also expected to be similar to those in CoNb₂O₆ [37, 38]. More importantly, compared to CoNb₂O₆, the critical field values for the QCPs in BaCo₂V₂O₈ are {\mu }_0{H}_c^{1\text{D}}\sim 4.7\pm 0.3\text{T} and {\mu }_0{H}_c^{3\text{D}}\sim 10.4\pm 0.1\text{T}, respectively [34], creating a magnetic field range in which the influence of the 1D and 3D QCPs on the quantum critical excitations can be clearly separated.

Ultralow-temperature thermal conductivity measurement has proven to be a powerful means of studying magnetic excitations in low-dimensional magnets [22, 39−60] — a case in point being that it confirmed the localized nature of the GLF excitations in CoNb₂O₆ as mentioned above [22]. Here, we utilize such measurements down to 50 mK to investigate the quantum critical excitations around the 1D QCP of BaCo₂V₂O₈, in a transverse field along the a axis up to 8 T, i.e., with the presence of the 3D order. At lower temperatures, the heat-carrying phonons in the boundary-scattering limit result in field-independent thermal conductivity \kappa. With increasing temperature, the scattering of phonons by magnetic excitations thermally excited across their gap comes into play, leading to an ever more suppressed \kappa with an increasing magnetic field. Crucially, the lack of a linear-in-temperature contribution to \kappa, together with the monotonic evolution of \kappa across the 1D critical field, clearly rules out the existence of GLF excitations around the 1D QCP. To account for the magnetic excitations that scatter phonons, one needs to invoke only the gapped spinon pair excitations, which span the whole 3D order and manifest as the E_8 particles around the 1D QCP. The contrasting nature of the quantum critical excitations in the two Ising-like spin chains, i.e., the presence of GLF excitations in CoNb₂O₆ and their absence in BaCo₂V₂O₈, suggests that the GLF excitations might only emerge in systems in which the 1D and 3D QCPs are in close proximity.

2 Methods

Single crystals of BaCo₂V₂O₈ were grown by the self-nucleation method [34]. The magnetic susceptibility was measured with a Quantum Design magnetic property measurement system. For thermal conductivity measurements, the samples were cut into a rectangular shape, with the longest dimension along the c axis. Four silver wires were attached to the sample with silver paint as the thermal contacts. The thermal conductivity was measured in a dilution refrigerator, using a standard four wire steady-state method with two RuO₂ chip thermometers, calibrated in situ against a reference RuO₂ thermometer. The experimental setup for thermal conductivity measurements is schematically shown in Fig. 1(a), with a heat current along the chain direction c and transverse magnetic fields along a. For a typical sample, the distance between the T^{+} and T^{-} probes is 1.26 mm, with a width and thickness of 0.29 and 0.25 mm, respectively. We exploit specific procedures to account for heat leaks, thermal contact resistance, and other factors that may affect the data quality at ultralow temperatures. With these procedures, the experimental error bar can be as low as \kappa /T=\pm 0.005\text{mW}\cdot {\text{K}}^{-2}\cdot {\text{cm}}^{-1}.

3 Results and discussion

3.1 Absence of GLF excitations in BaCo₂V₂O₈

The thermal conductivity \kappa (T) of BaCo₂V₂O₈ under various fields is shown in Fig. 2(d), with the \kappa (T) of CoNb₂O₆ extracted from Ref. [22] also shown in Fig. 2(a) for comparison. The most salient feature of the \kappa (T) of BaCo₂V₂O₈ is that there are two temperature regions: the applied field barely affects \kappa at lower temperatures, while it clearly suppresses \kappa at higher temperatures. This is better visualized in Figs. 2(b) and 2(e), where \mathrm{\Delta }\kappa (T)\equiv \kappa (H,T)-\kappa (0\text{T},T) is plotted. The \mathrm{\Delta }\kappa (T) of CoNb₂O₆ is clearly negative at finite temperatures, and the curves corresponding to different magnetic fields do not overlap, as a result of the gapless but localized nature of the GLF excitations therein; while for BaCo₂V₂O₈, \mathrm{\Delta }\kappa (T) is negligible in region I but negative in region II, and the curves corresponding to different magnetic fields diverge from each other only in region II. The choice of the zero-field data as the baseline in the definition of \mathrm{\Delta }\kappa (T) will be justified later. The normalized \kappa (H) curves of BaCo₂V₂O₈ displayed in Fig. 2(f) are almost flat at lower temperatures, while they show a monotonic decrease with the increasing field at higher temperatures, consistent with the behavior of \kappa (T) and in sharp contrast to the behavior of CoNb₂O₆ where the normalized \kappa (H) shows a minimum around the closely-spaced 1D and 3D QCPs at all temperatures [Fig. 2(c)].

In the temperature region I, the fact that \kappa (for simplicity, we omit the independent variable T or H in the following) is insensitive to the magnetic field already indicates that phonons are the only heat carriers and the magnetic degrees of freedom are barely relevant. Nevertheless, we perform a fitting procedure for further analysis. The thermal conductivity of an insulating magnet can generally be decomposed as \kappa ={\kappa }_{\text{m}}+{\kappa }_{\text{p}}, in which the two terms represent the contributions from magnetic excitations and phonons, respectively. If the GLF excitations observed in CoNb₂O₆ also existed in BaCo₂V₂O₈, they would be expected to give rise to a linear contribution to \kappa, i.e., {\kappa }_{\text{m}}\sim T [22]. At low temperatures, phonons can be scattered by the sample boundaries or by magnetic excitations. The latter scattering channel can be excluded by the field independence of \kappa in region I. Therefore, {\kappa }_p\sim T^{\alpha }, where \alpha is usually between 2 and 3 [22, 42, 55]. Combining these two terms, one can fit the data to \kappa /T=a+b{T}^{\alpha -1}, and tell whether itinerant GLF excitations contribute to the heat transport based on the value of the residual linear term {\kappa }_0/T\equiv {\kappa }_{\text{m}}/T(T\to 0)=a.

A typical fitting line for \kappa around the 1D QCP of BaCo₂V₂O₈ is shown in the inset of Fig. 3, while the {\kappa }_0/T values obtained at all fields are summarized in the main panel. The obtained \alpha is weakly dependent on the magnetic field, with a typical value between 2.3 and 2.4, consistent with boundary-scattering limited phonon heat transport. The negligible {\kappa }_0/T at all fields clearly rules out the existence of itinerant GLF excitations in BaCo₂V₂O₈. First hint of a negligible magnetic contribution to \kappa was given based on the lack of difference between the \kappa with a heat current along the spin chain and perpendicular to it [32, 61] . Here, we determine \kappa down to ultralow temperatures, which allows for an unambiguous extraction of the zero-temperature asymptotic behavior, with a focus on \kappa in fields near the 1D QCP. For comparison, we note that in a typical antiferromagnetic Heisenberg chain copper benzoate, the spinons, which are gapless and fermionic, contribute to a {\kappa }_0/T\sim2\text{mW}\cdot {\text{K}}^{-2}\cdot {\text{cm}}^{-1} [58], two orders of magnitude higher than the upper limit here.

Since there are no direct contributions from magnetic excitations, the field-induced suppression of \kappa in the temperature region II can only be attributed to a phonon-dominated thermal conductivity with the phonons being more strongly scattered as the field increases. Moreover, such scattering must be ineffective at lower temperatures to be in accord with the data in region I. A scenario that can reconcile these observations is that there is a set of gapped magnetic excitations scattering phonons. The gap can be thermally overcome in region II, and the gap size decreases with increasing field. Consequently, (i) a substantial number of magnetic excitations start to participate in the scattering of phonons in region II, leading to a downward trend of \kappa compared to a simple extrapolation of the phonon-dominated \kappa in region I to region II; (ii) more magnetic excitations participate in the scattering of phonons as the field increases, leading to a suppressed \kappa. Since in this scenario the gap size is largest at zero field, \kappa (0\text{T}) can be viewed as the closest to a pure boundary-limited-phonon-dominated \kappa. In this sense, the value of \mathrm{\Delta }\kappa can act as a measure of the scattering strength of phonons by magnetic excitations thermally excited across their gap.

The above analysis delineates the restrictions on the magnetic excitations set by our data. In the following, we discuss which set of magnetic excitations specifically meets the requirements. The existence of itinerant GLF excitations in BaCo₂V₂O₈ has been ruled out by the negligible {\kappa }_0/T at all fields shown in Fig. 3. What about the possibility of a band of localized GLF excitations, as in the case of CoNb₂O₆ [16, 22]? If such localized GLF excitations exist around the 1D QCP of BaCo₂V₂O₈, a dip structure centering around H_c^{1\text{D}} in Fig. 2(f), similar to the case of CoNb₂O₆ in Fig. 2(c), is expected, which is in sharp contrast to the smooth evolution observed here. Furthermore, since the GLF excitations are gapless, if present, they would scatter phonons at any finite temperature, resulting in a nonzero \mathrm{\Delta }\kappa for all temperatures measured. This is again inconsistent with our observations of a discernible \mathrm{\Delta }\kappa only in region II [Fig. 2(e)]. Therefore, we can safely rule out the existence of GLF excitations, itinerant or localized, around the 1D QCP of BaCo₂V₂O₈.

In the 3D long-range ordered state of BaCo₂V₂O₈, the established magnetic excitations are spinon pair excitations associated with the formation of quantized spinon bound states [30, 34, 38]. The magnetic excitation gap corresponds to the confining potential of spinons. The gap size of the lowest-lying mode is determined by terahertz spectroscopy to be \sim 20\text{K} at zero field and decreases by a factor of about two upon approaching the 3D QCP [38], while neutron scattering and electron spin resonance measurements point to a steeper decrease to \sim 2\text{K} around H_c^{3\text{D}} [30, 62]. In Fig. 2(e), the onset temperature for a nonzero \mathrm{\Delta }\kappa, i.e., the onset temperature for phonon scattering, does not vary strongly with the field, pointing to a slower decrease of the gap. The zero-field gap is sufficiently large so that there are barely any excited spinon pair excitations in the whole temperature range of our measurements; as the gap size decreases with increasing field, more spinon pair excitations are thermally excited in region II, leading to a decreased \kappa due to the enhanced scattering inflicted by these excitations on phonons. Since even for the largest field of 8 T, the gap size remains large compared to our measurement temperature, the decrease of \kappa with the field is only moderate.

The large gap size across the whole field range of our measurement explains the lack of a direct contribution from the magnetic excitations. On the other hand, the phonon thermal transport appears to be a more sensitive probe of the increasing density of the magnetic excitations with increasing field, although the gap may not show a strong decrease with field, and the density of the magnetic excitations remains low all along. In this aspect, we note a recent heat transport study on a ferromagnetic Ising spin α−CoV₂O₆, where the phonon thermal conductivity below 1 K decreases noticeably with the application of a transverse field, while the magnetic excitation gap remains large (\sim 35\text{K}) even for a 9 T field [63]. The cases of BaCo₂V₂O₈ and α−CoV₂O₆ show that the phonon thermal conductivity provides a surprisingly sensitive probe of magnetic degrees of freedom in Ising spin chain systems in a transverse field, whose microscopic origin calls for future studies but is beyond the scope of this work. We remark that although the picture of phonon-dominated heat transport was already invoked in previous studies [32, 61], the different scattering conditions in regions I and II, and the identification of specific magnetic excitations (i.e., the gapped spinon pair excitations) as the phonon scatterers were not possible until now.

3.2 Why do the GLF excitations emerge only in CoNb₂O₆?

Finally, we discuss the implications of our results on the quantum critical excitations in various spin chain systems. The absence of the GLF excitations around the 1D QCP of BaCo₂V₂O₈ immediately indicates that the GLF excitations observed in CoNb₂O₆ are not a feature common to spin chain systems that can be described, at least to some extent, by the longitudinal-field-perturbed TFIC model, and the origin of the GLF excitations in CoNb₂O₆ must lie beyond the E_8 physics.

Are the GLF excitations then purely a consequence of the 3D QCP? It is worth noting that spectroscopic studies all point to a finite magnetic excitation gap at the 3D QCP of BaCo₂V₂O₈ and its sister compound SrCo₂V₂O₈ [30, 38, 62, 64, 65]. One may envisage a scenario in which gapless excitations may elude spectroscopic probes either due to a mismatch with the energy or momentum window that the measurements are sensitive to, and manifest themselves only in ultralow-temperature specific heat and heat transport measurements that treat all portions of the Brillouin zone on an equal footing [22]. This scenario may explain why the GLF excitations in CoNb₂O₆ were not found until recently when the specific heat and heat transport measurements were performed [16, 22]. Such a possibility cannot be excluded, leaving room for the GLF excitations to be present around the 3D QCP of BaCo₂V₂O₈. Previous heat transport studies showed that as a function of the applied transverse field, \kappa exhibits a dip around H_c^{3\text{D}} down to \sim 0.3\text{K}, attributed to the scattering of phonons by quantum fluctuations [32, 61]. One cannot tell from the existing data whether there are GLF excitations around H_c^{3\text{D}}, since their effect on \kappa is indistinguishable from quantum fluctuations if the potential GLF excitations are localized and can thus affect \kappa only by scattering phonons. The picture is further complicated considering the case of SrCo₂V₂O₈: although still under debate [66], the emergence of gapless excitations associated with an exposed 1D QCP outside the 3D order (H_c^{1\text{D}}>H_c^{3\text{D}}) was concluded from nuclear magnetic resonance measurements [65].

Considering that (i) out of the various Ising-like spin chain systems, the existence of the GLF excitations is only conclusive in CoNb₂O₆ around its closely located 1D and 3D QCPs, and (ii) we have demonstrated through the study of BaCo₂V₂O₈ that the realization of E_8 physics does not guarantee the emergence of the GLF excitations, deviations from the longitudinal-field-perturbed TFIC model must be considered in the effort to understand the origin of such novel quantum critical excitations.

In CoNb₂O₆, BaCo₂V₂O₈, and SrCo₂V₂O₈, the magnetic Co²⁺ ion with a d^7 electron configuration exhibits a total orbital angular momentum L=3 and total spin S=3/2. The ground state manifold is further split by the local octahedral crystal field and spin-orbit coupling, resulting in an J_{\text{eff}}=1/2 [7]. This is the common starting point for the effective (pseudo)spin-1/2 XXZ Hamiltonian governing the low-energy physics of all three systems. On the other hand, such a setting is reminiscent of the Kitaev physics in honeycomb cobalt-based systems. In fact, the proposal of Kitaev physics realized in cobalt compounds has attracted substantial attention [67−70], with prominent examples being the possible realization of quantum spin liquid in BaCo₂(AsO₄)₂ and Na₂Co₂TeO₆ [56, 60, 71−74]. For the zigzag chain systems, the glide symmetry breaking and the resultant additional terms in the Hamiltonian of CoNb₂O₆ have been scrutinized recently, leading to the proposal of CoNb₂O₆ as a realization of the twisted Kitaev chain, also known as the generalized quantum spin compass chain [21, 75−77], in which the bond-dependent interactions depend sensitively on the geometry of electron orbitals and hopping paths [76].

In this sense, the emergence of the GLF excitations around the two closely-spaced QCPs in CoNb₂O₆ may not be a coincidence: the specific combination of microscopic parameters, including the level of distortion of the CoO₆ octahedra in the crystal structure, the exchange anisotropy, and the spin-orbit coupling strength, etc., may give rise to the adjacent QCPs and the GLF excitations around them in CoNb₂O₆ but not in related systems. In CoNb₂O₆, one chain is coupled to two other chains on a triangle with antiferromagnetic interchain interactions, leading to strong frustration and, hence, a relatively weak interchain interaction. In BaCo₂V₂O₈, one chain is antiferromagnetically coupled to three other chains on a square. The frustration between nearest-neighbor chains persists, but the interaction with the diagonal chain becomes effective, leading to stronger interchain coupling: \sim 0.1J in BaCo₂V₂O₈ [26, 31], as compared to \sim 0.01J in CoNb₂O₆ [75], where J is the coupling constant in the XXZ Hamiltonian. Note that the precise nature of the interchain coupling is still largely unknown due to the screw nature of the chains, and the estimations are only on a phenomenological level.

It has been recently argued that in CoNb₂O₆, certain spectral features incompatible with the E_8 description originate from excitations described by the D_8^{(1)} Lie algebra associated with the {\text{Ising}}_h^2 integrable model [78, 79]. Crucially, a key prerequisite for the realization of the D_8^{(1)} physics is the strong interchain spin fluctuations, whose enhancement can be attributed to the frustrated alignment of chains and the proximity of the 1D and 3D QCPs [78, 79]. In CoNb₂O₆, the frustrated interchain alignment causes the cancellation of an effective longitudinal field \tilde{h} from neighboring chains. In addition, the ordered moment is minute at the 1D QCP since the long-range order is already close to its demise at the 3D QCP, leading to further suppression of \tilde{h}. Therefore, CoNb₂O₆ is closer to the small \tilde{h} limit where the D_8^{(1)} physics is expected [78]. Though the D_8^{(1)} physics is invoked to address excitations with a higher energy scale than the GLF excitations, it highlights that closely spaced 1D and 3D QCPs indeed have a profound impact on the quantum critical excitations in quasi-1D magnets, consistent with our observations.

In our previous work on CoNb₂O₆, we proposed that the frustrated alignment of chains may be a key factor in the localized nature of the GLF excitations [22]. The absence of GLF excitations in BaCo₂V₂O₈ demonstrated in this work leads us to further speculate that frustration may be pivotal in determining whether GLF excitations exist in the first place.

4 Summary

We measured the thermal conductivity of the Ising-like spin chain BaCo₂V₂O₈ within its three-dimensional antiferromagnetic order and down to ultralow temperatures. The thermal conductivity is dominated by phonons, while the effect of magnetic degrees of freedom is evident only at higher temperatures. The magnetic excitations scatter phonons more severely as their gap size decreases with the increase of the applied transverse field, leading to a suppressed thermal conductivity. Attributing the magnetic excitations to the spinon pair excitations associated with the E_8 physics provides a scenario consistent with all the experimental observations. Combining the absence of a linear-in-temperature contribution, the smooth evolution of the thermal conductivity around the one-dimensional quantum critical point, and the negligible magnetic scattering of phonons at lower temperatures, the existence of gapless fermionic excitations can be safely ruled out. The contrasting cases of BaCo₂V₂O₈ and CoNb₂O₆ suggest that intertwined quantum critical points of different dimensions may play a fundamental role in the realization of the gapless fermionic quantum critical excitations.

References

[1] S. G. Brush , History of the Lenz−Ising model, Rev. Mod. Phys. 39(4), 883 (1967)

[2] L. Onsager , Crystal statistics. I. A two-dimensional model with an order−disorder transition, Phys. Rev. 65(3−4), 117 (1944)

[3] H. Shiba , Quantization of magnetic excitation continuum due to interchain coupling in nearly one-dimensional Ising-like antiferromagnets, Prog. Theor. Phys. 64(2), 466 (1980)

[4] A. B. Zamolodchikov , Integrals of motion and S-matrix of the (scaled) T=Tc Ising model with magnetic field, Int. J. Mod. Phys. A 04, 4235 (1989)

[5] E. Lieb , T. Schultz , and D. Mattis , Two soluble models of an antiferromagnetic chain, Ann. Phys. 16(3), 407 (1961)

[6] P. Pfeuty , The one-dimensional Ising model with a transverse field, Ann. Phys. 57(1), 79 (1970)

[7] J. Yang , X. Wang , and J. Wu , Magnetic excitations in the one-dimensional Heisenberg−Ising model with external fields and their experimental realizations, J. Phys. A Math. Theor. 56(1), 013001 (2023)

[8] S. Kobayashi , S. Mitsuda , M. Ishikawa , K. Miyatani , and K. Kohn , Three-dimensional magnetic ordering in the quasi-one-dimensional Ising magnet CoNb₂O₆ with partially released geometrical frustration, Phys. Rev. B 60(5), 3331 (1999)

[9] S. Lee , R. K. Kaul , and L. Balents , Interplay of quantum criticality and geometric frustration in columbite, Nat. Phys. 6(9), 702 (2010)

[10] R. Coldea , D. A. Tennant , E. M. Wheeler , E. Wawrzynska , D. Prabhakaran , M. Telling , K. Habicht , P. Smeibidl , and K. Kiefer , Quantum criticality in an Ising chain: Experimental evidence for emergent E8, Science 327(5962), 177 (2010)

[11] A. W. Kinross , M. Fu , T. J. Munsie , H. A. Dabkowska , G. M. Luke , S. Sachdev , and T. Imai , Evolution of quantum fluctuations near the quantum critical point of the transverse field Ising chain system CoNb₂O₆, Phys. Rev. X 4(3), 031008 (2014)

[12] I. Cabrera , J. D. Thompson , R. Coldea , D. Prabhakaran , R. I. Bewley , T. Guidi , J. A. Rodriguez-Rivera , and C. Stock , Excitations in the quantum paramagnetic phase of the quasi-one-dimensional Ising magnet CoNb₂O₆ in a transverse field: Geometric frustration and quantum renormalization effects, Phys. Rev. B 90(1), 014418 (2014)

[13] N. J. Robinson , F. H. L. Essler , I. Cabrera , and R. Coldea , Quasiparticle breakdown in the quasi-one-dimensional Ising ferromagnet CoNb₂O₆, Phys. Rev. B 90(17), 174406 (2014)

[14] C. M. Morris , R. Valdés Aguilar , A. Ghosh , S. M. Koohpayeh , J. Krizan , R. J. Cava , O. Tchernyshyov , T. M. Mc Queen , and N. P. Armitage , Hierarchy of bound states in the one-dimensional ferromagnetic Ising chain CoNb₂O₆ investigated by high-resolution time-domain terahertz spectroscopy, Phys. Rev. Lett. 112(13), 137403 (2014)

[15] K. Amelin , J. Engelmayer , J. Viirok , U. Nagel , T. Rõõm , T. Lorenz , and Z. Wang , Experimental observation of quantum many-body excitations of E8 symmetry in the Ising chain ferromagnet CoNb₂O₆, Phys. Rev. B 102(10), 104431 (2020)

[16] T. Liang , S. M. Koohpayeh , J. W. Krizan , T. M. McQueen , R. J. Cava , and N. P. Ong , Heat capacity peak at the quantum critical point of the transverse Ising magnet CoNb₂O₆, Nat. Commun. 6(1), 7611 (2015)

[17] S. Sachdev, Quantum Phase Transitions, 2nd Ed., Cambridge University Press, 2011

[18] W. Scharf , H. Weitzel , I. Yaeger , I. Maartense , and B. M. Wanklyn , Magnetic structures of CoNb₂O₆, J. Magn. Magn. Mater. 13(1−2), 121 (1979)

[19] T. Hanawa , K. Shinkawa , M. Ishikawa , K. Miyatani , K. Saito , and K. Kohn , Anisotropic specific heat of CoNb₂O₆ in magnetic fields, J. Phys. Soc. Jpn. 63(7), 2706 (1994)

[20] Y. Wan and N. P. Armitage , Resolving continua of fractional excitations by spinon echo in THz 2D coherent spectroscopy, Phys. Rev. Lett. 122(25), 257401 (2019)

[21] C. M. Morris , N. Desai , J. Viirok , D. Hüvonen , U. Nagel , T. Rõõm , J. W. Krizan , R. J. Cava , T. M. McQueen , S. M. Koohpayeh , R. K. Kaul , and N. P. Armitage , Duality and domain wall dynamics in a twisted Kitaev chain, Nat. Phys. 17(7), 832 (2021)

[22] Y. Xu , L. S. Wang , Y. Y. Huang , J. M. Ni , C. C. Zhao , Y. F. Dai , B. Y. Pan , X. C. Hong , P. Chauhan , S. M. Koohpayeh , N. P. Armitage , and S. Y. Li , Quantum critical magnetic excitations in spin-1/2 and spin-1 chain systems, Phys. Rev. X 12(2), 021020 (2022)

[23] S. Kimura , H. Yashiro , K. Okunishi , M. Hagiwara , Z. He , K. Kindo , T. Taniyama , and M. Itoh , Field-induced order−disorder transition in antiferromagnetic BaCo₂V₂O₈ driven by a softening of spinon excitation, Phys. Rev. Lett. 99(8), 087602 (2007)

[24] S. Kimura , T. Takeuchi , K. Okunishi , M. Hagiwara , Z. He , K. Kindo , T. Taniyama , and M. Itoh , Novel ordering of an S = 1/2 quasi-1D Ising-like antiferromagnet in magnetic field, Phys. Rev. Lett. 100(5), 057202 (2008)

[25] M. Klanjšek , M. Horvatić , S. Krämer , S. Mukhopadhyay , H. Mayaffre , C. Berthier , E. Canévet , B. Grenier , P. Lejay , and E. Orignac , Giant magnetic field dependence of the coupling between spin chains in BaCo₂V₂O₈, Phys. Rev. B 92(6), 060408 (2015)

[26] Q. Faure , S. Takayoshi , V. Simonet , B. Grenier , M. Månsson , J. S. White , G. S. Tucker , C. Rüegg , P. Lejay , T. Giamarchi , and S. Petit , Tomonaga-Luttinger liquid spin dynamics in the quasi-one-dimensional Ising-like antiferromagnet BaCo₂V₂O₈, Phys. Rev. Lett. 123(2), 027204 (2019)

[27] S. Takayoshi , Q. Faure , V. Simonet , B. Grenier , S. Petit , J. Ollivier , P. Lejay , and T. Giamarchi , Phase transitions and spin dynamics of the quasi-one dimensional Ising-like antiferromagnet BaCo₂V₂O₈ in a longitudinal magnetic field, Phys. Rev. Res. 5(2), 023205 (2023)

[28] Z. Wang , M. Schmidt , A. Loidl , J. Wu , H. Zou , W. Yang , C. Dong , Y. Kohama , K. Kindo , D. I. Gorbunov , S. Niesen , O. Breunig , J. Engelmayer , and T. Lorenz , Quantum critical dynamics of a Heisenberg−Ising chain in a longitudinal field: Many-body strings versus fractional excitations, Phys. Rev. Lett. 123(6), 067202 (2019)

[29] W. Yang , J. Wu , S. Xu , Z. Wang , and C. Wu , One-dimensional quantum spin dynamics of Bethe string states, Phys. Rev. B 100(18), 184406 (2019)

[30] Q. Faure , S. Takayoshi , S. Petit , V. Simonet , S. Raymond , L. P. Regnault , M. Boehm , J. S. White , M. Månsson , C. Rüegg , P. Lejay , B. Canals , T. Lorenz , S. C. Furuya , T. Giamarchi , and B. Grenier , Topological quantum phase transition in the Ising-like antiferromagnetic spin chain BaCo₂V₂O₈, Nat. Phys. 14(7), 716 (2018)

[31] Q. Faure , S. Takayoshi , B. Grenier , S. Petit , S. Raymond , M. Boehm , P. Lejay , T. Giamarchi , and V. Simonet , Solitonic excitations in the Ising anisotropic chain BaCo₂V₂O₈ under large transverse magnetic field, Phys. Rev. Res. 3(4), 043227 (2021)

[32] S. K. Niesen , G. Kolland , M. Seher , O. Breunig , M. Valldor , M. Braden , B. Grenier , and T. Lorenz , Magnetic phase diagrams, domain switching, and quantum phase transition of the quasi-one-dimensional Ising-like antiferromagnet BaCo₂V₂O₈, Phys. Rev. B 87(22), 224413 (2013)

[33] Z. Wang , T. Lorenz , D. I. Gorbunov , P. T. Cong , Y. Kohama , S. Niesen , O. Breunig , J. Engelmayer , A. Herman , J. Wu , K. Kindo , J. Wosnitza , S. Zherlitsyn , and A. Loidl , Quantum criticality of an Ising-like spin-1/2 antiferromagnetic chain in a transverse magnetic field, Phys. Rev. Lett. 120(20), 207205 (2018)

[34] H. Zou , Y. Cui , X. Wang , Z. Zhang , J. Yang , G. Xu , A. Okutani , M. Hagiwara , M. Matsuda , G. Wang , G. Mussardo , K. Hódsági , M. Kormos , Z. He , S. Kimura , R. Yu , W. Yu , J. Ma , and J. Wu , E8 spectra of quasi-one-dimensional antiferromagnet BaCo₂V₂O₈ under transverse field, Phys. Rev. Lett. 127(7), 077201 (2021)

[35] S. Kimura , H. Yashiro , M. Hagiwara , K. Okunishi , K. Kindo , Z. He , T. Taniyama , and M. Itoh , High field magnetism of the quasi one-dimensional anisotropic antiferromagnet BaCo₂V₂O₈, J. Phys. Conf. Ser. 51, 99 (2006)

[36] S. Kimura , K. Okunishi , M. Hagiwara , K. Kindo , Z. He , T. Taniyama , M. Itoh , K. Koyama , and K. Watanabe , Collapse of magnetic order of the quasi one-dimensional Ising-like antiferromagnet BaCo₂V₂O₈ in transverse fields, J. Phys. Soc. Jpn. 82(3), 033706 (2013)

[37] B. Grenier , S. Petit , V. Simonet , E. Canévet , L. P. Regnault , S. Raymond , B. Canals , C. Berthier , and P. Lejay , Longitudinal and transverse Zeeman ladders in the Ising-like chain antiferromagnet BaCo₂V₂O₈, Phys. Rev. Lett. 114(1), 017201 (2015)

[38] Z. Zhang , K. Amelin , X. Wang , H. Zou , J. Yang , U. Nagel , T. Rõõm , T. Dey , A. A. Nugroho , T. Lorenz , J. Wu , and Z. Wang , Observation of E8 particles in an Ising chain antiferromagnet, Phys. Rev. B 101(22), 220411 (2020)

[39] M. Yamashita , N. Nakata , Y. Kasahara , T. Sasaki , N. Yoneyama , N. Kobayashi , S. Fujimoto , T. Shibauchi , and Y. Matsuda , Thermal-transport measurements in a quantum spin-liquid state of the frustrated triangular magnet κ-(BEDT-TTF)₂Cu₂(CN)₃, Nat. Phys. 5(1), 44 (2009)

[40] M. Yamashita , N. Nakata , Y. Senshu , M. Nagata , H. M. Yamamoto , R. Kato , T. Shibauchi , and Y. Matsuda , Highly mobile gapless excitations in a two-dimensional candidate quantum spin liquid, Science 328(5983), 1246 (2010)

[41] Y. Tokiwa , T. Yamashita , M. Udagawa , S. Kittaka , T. Sakakibara , D. Terazawa , Y. Shimoyama , T. Terashima , Y. Yasui , T. Shibauchi , and Y. Matsuda , Possible observation of highly itinerant quantum magnetic monopoles in the frustrated pyrochlore Yb₂Ti₂O₇, Nat. Commun. 7(1), 10807 (2016)

[42] Y. Xu , J. Zhang , Y. S. Li , Y. J. Yu , X. C. Hong , Q. M. Zhang , and S. Y. Li , Absence of magnetic thermal conductivity in the quantum spin-liquid candidate YbMgGaO₄, Phys. Rev. Lett. 117(26), 267202 (2016)

[43] Y. J. Yu , Y. Xu , L. P. He , M. Kratochvilova , Y. Y. Huang , J. M. Ni , L. Wang , S. W. Cheong , J. G. Park , and S. Y. Li , Heat transport study of the spin liquid candidate 1T-TaS₂, Phys. Rev. B 96(8), 081111 (2017)

[44] I. A. Leahy , C. A. Pocs , P. E. Siegfried , D. Graf , S. H. Do , K. Y. Choi , B. Normand , and M. Lee , Anomalous thermal conductivity and magnetic torque response in the honeycomb magnet α-RuCl₃, Phys. Rev. Lett. 118(18), 187203 (2017)

[45] Y. J. Yu , Y. Xu , K. J. Ran , J. M. Ni , Y. Y. Huang , J. H. Wang , J. S. Wen , and S. Y. Li , Ultralow-temperature thermal conductivity of the Kitaev honeycomb magnet α-RuCl₃ across the field-induced phase transition, Phys. Rev. Lett. 120(6), 067202 (2018)

[46] J. M. Ni , Q. Y. Liu , Y. J. Yu , E. J. Cheng , Y. Y. Huang , Z. Y. Liu , X. J. Wang , Y. Sui , and S. Y. Li , Ultralow-temperature heat transport in the quantum spin liquid candidate Ca₁₀Cr₇O₂₈ with a bilayer Kagome lattice, Phys. Rev. B 97(10), 104413 (2018)

[47] J. M. Ni , B. L. Pan , B. Q. Song , Y. Y. Huang , J. Y. Zeng , Y. J. Yu , E. J. Cheng , L. S. Wang , D. Z. Dai , R. Kato , and S. Y. Li , Absence of magnetic thermal conductivity in the quantum spin liquid candidate EtMe₃Sb[Pd(dmit)₂]₂, Phys. Rev. Lett. 123(24), 247204 (2019)

[48] P. Bourgeois-Hope , F. Laliberté , E. Lefrançois , G. Grissonnanche , S. R. de Cotret , R. Gordon , S. Kitou , H. Sawa , H. Cui , R. Kato , L. Taillefer , and N. Doiron-Leyraud , Thermal conductivity of the quantum spin liquid candidate EtMe₃Sb[Pd(dmit)₂]₂: No evidence of mobile gapless excitations, Phys. Rev. X 9(4), 041051 (2019)

[49] R. Hentrich , A. U. B. Wolter , X. Zotos , W. Brenig , D. Nowak , A. Isaeva , T. Doert , A. Banerjee , P. Lampen-Kelley , D. G. Mandrus , S. E. Nagler , J. Sears , Y. J. Kim , B. Büchner , and C. Hess , Unusual phonon heat transport in α-RuCl₃: Strong spin-phonon scattering and field-induced spin gap, Phys. Rev. Lett. 120(11), 117204 (2018)

[50] N. Li , Q. Huang , X. Y. Yue , W. J. Chu , Q. Chen , E. S. Choi , X. Zhao , H. D. Zhou , and X. F. Sun , Possible itinerant excitations and quantum spin state transitions in the effective spin-1/2 triangular-lattice antiferromagnet Na₂BaCo(PO₄)₂, Nat. Commun. 11(1), 4216 (2020)

[51] H. Murayama , Y. Sato , T. Taniguchi , R. Kurihara , X. Z. Xing , W. Huang , S. Kasahara , Y. Kasahara , I. Kimchi , M. Yoshida , Y. Iwasa , Y. Mizukami , T. Shibauchi , M. Konczykowski , and Y. Matsuda , Effect of quenched disorder on the quantum spin liquid state of the triangular-lattice antiferromagnet 1T-TaS₂, Phys. Rev. Res. 2(1), 013099 (2020)

[52] B. L. Pan , J. M. Ni , L. P. He , Y. J. Yu , Y. Xu , and S. Y. Li , Specific heat and thermal conductivity of the triangular-lattice rare-earth material KBaYb(BO₃)₂ at ultralow temperature, Phys. Rev. B 103(10), 104412 (2021)

[53] J. M. Ni , Y. Y. Huang , E. J. Cheng , Y. J. Yu , B. L. Pan , Q. Li , L. M. Xu , Z. M. Tian , and S. Y. Li , Giant isotropic magneto-thermal conductivity of metallic spin liquid candidate Pr₂Ir₂O₇ with quantum criticality, Nat. Commun. 12(1), 307 (2021)

[54] X. Rao , G. Hussain , Q. Huang , W. J. Chu , N. Li , X. Zhao , Z. Dun , E. S. Choi , T. Asaba , L. Chen , L. Li , X. Y. Yue , N. N. Wang , J. G. Cheng , Y. H. Gao , Y. Shen , J. Zhao , G. Chen , H. D. Zhou , and X. F. Sun , Survival of itinerant excitations and quantum spin state transitions in YbMgGaO₄ with chemical disorder, Nat. Commun. 12(1), 4949 (2021)

[55] Y. Y. Huang , Y. Xu , L. Wang , C. C. Zhao , C. P. Tu , J. M. Ni , L. S. Wang , B. L. Pan , Y. Fu , Z. Hao , C. Liu , J. W. Mei , and S. Y. Li , Heat transport in Herbertsmithite: Can a quantum spin liquid survive disorder, Phys. Rev. Lett. 127(26), 267202 (2021)

[56] X. Hong , M. Gillig , R. Hentrich , W. Yao , V. Kocsis , A. R. Witte , T. Schreiner , D. Baumann , N. Pérez , A. U. B. Wolter , Y. Li , B. Büchner , and C. Hess , Strongly scattered phonon heat transport of the candidate Kitaev material Na₂Co₂TeO₆, Phys. Rev. B 104(14), 144426 (2021)

[57] X. Hong , M. Behnami , L. Yuan , B. Li , W. Brenig , B. Büchner , Y. Li , and C. Hess , Heat transport of the Kagome Heisenberg quantum spin liquid candidate YCu₃(OH)_6.5Br_2.5: Localized magnetic excitations and a putative spin gap, Phys. Rev. B 106(22), L220406 (2022)

[58] B. Y. Pan , Y. Xu , J. M. Ni , S. Y. Zhou , X. C. Hong , X. Qiu , and S. Y. Li , Unambiguous experimental verification of linear-in-temperature spinon thermal conductivity in an antiferromagnetic Heisenberg chain, Phys. Rev. Lett. 129(16), 167201 (2022)

[59] X. Hong , M. Gillig , A. R. N. Hanna , S. Chillal , A. T. M. N. Islam , B. Lake , B. Büchner , and C. Hess , Spinon heat transport in the three-dimensional quantum magnet PbCuTe₂O₆, Phys. Rev. Lett. 131(25), 256701 (2023)

[60] X. Hong , M. Gillig , W. Yao , L. Janssen , V. Kocsis , S. Gass , Y. Li , A. U. B. Wolter , B. Büchner , and C. Hess , Phonon thermal transport shaped by strong spin-phonon scattering in a Kitaev material Na₂Co₂TeO₆, npj Quantum Mater. 9, 18 (2024)

[61] Z. Y. Zhao , X. G. Liu , Z. Z. He , X. M. Wang , C. Fan , W. P. Ke , Q. J. Li , L. M. Chen , X. Zhao , and X. F. Sun , Heat transport of the quasi-one-dimensional Ising-like antiferromagnet BaCo₂V₂O₈ in longitudinal and transverse fields, Phys. Rev. B 85(13), 134412 (2012)

[62] A. Okutani , H. Onishi , S. Kimura , T. Takeuchi , T. Kida , M. Mori , A. Miyake , M. Tokunaga , K. Kindo , and M. Hagiwara , Spin excitations of the S = 1/2 one-dimensional Ising-like antiferromagnet BaCo₂V₂O₈ in transverse magnetic fields, J. Phys. Soc. Jpn. 90(4), 044704 (2021)

[63] Y. Zhao , Z. Ma , Z. He , H. Liao , Y. C. Wang , J. Wang , and Y. Li , Quantum annealing of a frustrated magnet, Nat. Commun. 15(1), 3495 (2024)

[64] Z. Wang , J. Wu , S. Xu , W. Yang , C. Wu , A. K. Bera , A. T. M. N. Islam , B. Lake , D. Kamenskyi , P. Gogoi , H. Engelkamp , N. Wang , J. Deisenhofer , and A. Loidl , From confined spinons to emergent fermions: Observation of elementary magnetic excitations in a transverse-field Ising chain, Phys. Rev. B 94(12), 125130 (2016)

[65] Y. Cui , H. Zou , N. Xi , Z. He , Y. X. Yang , L. Shu , G. H. Zhang , Z. Hu , T. Chen , R. Yu , J. Wu , and W. Yu , Quantum criticality of the Ising-like screw chain antiferromagnet SrCo₂V₂O₈ in a transverse magnetic field, Phys. Rev. Lett. 123(6), 067203 (2019)

[66] X. Zhang , Z. He , Y. Hao , Y. Shen , S. Shen , Y. Feng , G. Xu , N. P. Butch , G. J. Nilsen , W. Wang , and J. Zhao , Quantum phase transitions in a quasi-one-dimensional Ising quantum magnet in transverse fields, Phys. Rev. B 103(14), 144405 (2021)

[67] H. Liu and G. Khaliullin , Pseudospin exchange interactions in d7 cobalt compounds: Possible realization of the Kitaev model, Phys. Rev. B 97(1), 014407 (2018)

[68] R. Sano , Y. Kato , and Y. Motome , Kitaev−Heisenberg Hamiltonian for high-spin d7 Mott insulators, Phys. Rev. B 97(1), 014408 (2018)

[69] H. Liu , J. Chaloupka , and G. Khaliullin , Kitaev spin liquid in 3d transition metal compounds, Phys. Rev. Lett. 125(4), 047201 (2020)

[70] C. Kim , H. S. Kim , and J. G. Park , Spin−orbital entangled state and realization of Kitaev physics in 3d cobalt compounds: A progress report, J. Phys.: Condens. Matter 34(2), 023001 (2022)

[71] R. Zhong , M. Chung , T. Kong , L. T. Nguyen , S. Lei , and R. J. Cava , Field-induced spin-liquid-like state in a magnetic honeycomb lattice, Phys. Rev. B 98(22), 220407 (2018)

[72] G. Lin , J. Jeong , C. Kim , Y. Wang , Q. Huang , T. Masuda , S. Asai , S. Itoh , G. Günther , M. Russina , Z. Lu , J. Sheng , L. Wang , J. Wang , G. Wang , Q. Ren , C. Xi , W. Tong , L. Ling , Z. Liu , L. Wu , J. Mei , Z. Qu , H. Zhou , X. Wang , J. G. Park , Y. Wan , and J. Ma , Field-induced quantum spin disordered state in spin-1/2 honeycomb magnet Na₂Co₂TeO₆, Nat. Commun. 12(1), 5559 (2021)

[73] W. Yao , K. Iida , K. Kamazawa , and Y. Li , Excitations in the ordered and paramagnetic states of honeycomb magnet Na₂Co₂TeO₆, Phys. Rev. Lett. 129(14), 147202 (2022)

[74] X. Zhang , Y. Xu , T. Halloran , R. Zhong , C. Broholm , R. J. Cava , N. Drichko , and N. P. Armitage, , A magnetic continuum in the cobalt-based honeycomb magnet BaCo₂(AsO₄)₂, Nat. Mater. 22(1), 58 (2023)

[75] M. Fava , R. Coldea , and S. A. Parameswaran , Glide symmetry breaking and Ising criticality in the quasi-1D magnet CoNb₂O₆, Proc. Natl. Acad. Sci. USA 117(41), 25219 (2020)

[76] P. Laurell , G. Alvarez , and E. Dagotto , Spin dynamics of the generalized quantum spin compass chain, Phys. Rev. B 107(10), 104414 (2023)

[77] D. Churchill and H. Y. Kee , Transforming from Kitaev to disguised Ising chain: Application to CoNb₂O₆, Phys. Rev. Lett. 133(5), 056703 (2024)

[78] N. Xi , X. Wang , Y. Gao , Y. Jiang , R. Yu , and J. Wu , Emergent D8(1) spectrum and topological soliton excitation in CoNb₂O₆, Phys. Rev. B 112(24), L241104 (2025)

[79] Y. Gao , X. Wang , N. Xi , Y. Jiang , R. Yu , and J. Wu , Spin dynamics and dark particle in a weak-coupled quantum Ising ladder with D8(1)spectrum, Phys. Rev. B 111(20), L201117 (2025)

RIGHTS & PERMISSIONS

Higher Education Press