The one-dimensional transverse Ising model is a paradigmatic example of quantum criticality. In spin-orbit coupled systems, however, effective Ising interaction emerges from the interplay between the Kitaev ( K ) and Heisenberg ( J ) exchanges, and is accompanied by an additional bond-dependent interaction (Γ), leading to complex magnetic orders beyond the pure Ising limit. We first explore how the generic J - K -Γ model with four-fold screw symmetry in a spin-orbit-entangled chain manifests via sublattice order, and then test whether field-driven transitions retain Ising universality. We find sublattice-dependent magnetic order below the critical field and a distinct sublattice pattern persisting above it. Despite these complex magnetic ordered structures, the transition remains in the Ising universality class with central charge c = 1/2. Our work provides a route to the microscopic Hamiltonian and emergent Ising criticality while allowing microscopic physics of the sublattice orders to manifest at low and high fields. Application to Ising materials such as chains of Co 2+ ions is also discussed.
The one-dimensional transverse Ising model is a paradigmatic example of quantum criticality. In spin-orbit coupled systems, however, effective Ising interaction emerges from the interplay between the Kitaev ( K ) and Heisenberg ( J ) exchanges, and is accompanied by an additional bond-dependent interaction (Γ), leading to complex magnetic orders beyond the pure Ising limit. We first explore how the generic J - K -Γ model with four-fold screw symmetry in a spin-orbit-entangled chain manifests via sublattice order, and then test whether field-driven transitions retain Ising universality. We find sublattice-dependent magnetic order below the critical field and a distinct sublattice pattern persisting above it. Despite these complex magnetic ordered structures, the transition remains in the Ising universality class with central charge c = 1/2. Our work provides a route to the microscopic Hamiltonian and emergent Ising criticality while allowing microscopic physics of the sublattice orders to manifest at low and high fields. Application to Ising materials such as chains of Co 2+ ions is also discussed.
The one-dimensional transverse Ising model is a paradigmatic example of quantum criticality. In spin-orbit coupled systems, however, effective Ising interactions arise alongside bond-dependent couplings such as Kitaev (K) and Γ terms in addition to the Heisenberg (J) interaction, leading to complex magnetic orders beyond the pure Ising limit. We first explore how the generic J−K−Γ model with four-fold screw symmetry in a spin-orbit-entangled pseudospin-1/2 chain manifests via sublattice order, and then test whether field-driven transitions retain Ising universality. We find sublattice-dependent magnetic order below the critical field and a distinct sublattice pattern persisting above it. Despite these complex magnetic order structures, the transition remains in the Ising universality class with central charge c=1/2. Our work provides a route to the microscopic Hamiltonian and emergent Ising criticality while allowing microscopic physics of the sublattice orders to manifest at low and high fields. Application to antiferromagnetic Ising materials such as BaCo2V2O8 is also discussed.
For many years, CoNb_{2}O_{6} has served as an exemplar of the one-dimensional Ising model. However, recent experimental and theoretical analyses challenge its applicability to this material. Prior to that, a tailored spin model for 3d^{7} systems such as Co^{2+}, known as the JKΓ model, has emerged, featuring Heisenberg (J), Kitaev (K), and Gamma (Γ) interactions. While these interactions are permitted by the symmetry of the system, their role in CoNb_{2}O_{6} remains enigmatic. We present a microscopic theory based on spin-orbit entangled J_{eff}=1/2 states, aimed at elucidating the roles of Kitaev and Gamma interactions in shaping Ising anisotropy. Leveraging strong coupling theory, we identify a dominant ferromagnetic Kitaev interaction. Furthermore, by comparing dynamical structure factors obtained via exact diagonalization with those from inelastic neutron scattering experiments, we find an antiferromagnetic Γ interaction, which dictates the Ising axis and explains the mechanism behind moment pinning. Our theory provides a microscopic origin for Ising behavior in spin-orbit coupled one-dimensional chains and posits CoNb_{2}O_{6} as a rare Kitaev chain.
Motivated by recent experiments on quasi-one-dimensional vanadium oxides, we study quantum phase transitions in a one-dimensional spin-orbital model describing a Haldane chain and a classical Ising chain locally coupled by the relativistic spin-orbit interaction. By employing a field-theoretical approach, we analyze the topology of the ground-state phase diagram and identify the nature of the phase transitions. In the strong coupling limit, a long-range N\'eel order of entangled spin and orbital angular momenta appears in the ground state. We find that, depending on the relative scales of the spin and orbital gaps, the linear chain follows two distinct routes to reach the N\'eel state. First, when the orbital exchange is the dominating energy scale, a two-stage ordering takes place in which the magnetic transition is followed by melting of the orbital Ising order; both transitions belong to the two-dimensional Ising universality class. In the opposite limit, the low-energy orbital modes undergo a continuous reordering transition which represents a line of Gaussian critical points. On this line the orbital degrees of freedom form a Tomonaga--Luttinger liquid. We argue that the emergence of the Gaussian criticality results from merging of the two Ising transitions in the strong hybridization region where the characteristic spin and orbital energy scales become comparable. Finally, we show that, due to the spin-orbit coupling, an external magnetic field acting on the spins can induce an orbital Ising transition.
Quantum criticality in one-dimensional quantum systems is characterized by emergent conformal field theories (CFTs), whose central charge counts independent gapless degrees of freedom. Establishing a microscopic connection between this universal conformal structure and the momentum-space topology of the underlying quasiparticle spectrum remains challenging. Here, we uncover a direct correspondence between Fermi-point topology, conformal criticality, and quantum entanglement in an extended quantum spin chain with competing cluster interactions, exchange anisotropy, and a transverse magnetic field. We show that interaction- and field-driven Lifshitz transitions generate conformal critical phases with effective central charges ceff=1/2, 1, 3/2, 2, and 3, including a multicritical point where Ising and Luttinger-liquid sectors coexist. Importantly, the central charge is not determined simply by the number of lattice gap closings or Fermi points, but by the number and conformal content of independent low-energy continuum sectors after accounting for lattice symmetries, reciprocal-lattice identifications, and mode equivalences. Thus, Lifshitz transitions may leave the conformal anomaly unchanged or modify the central charge depending on whether spectral reconstruction generates new independent continuum sectors. Real- and momentum-space entanglement spectra provide complementary microscopic signatures, revealing both the conformal content and momentum-space organization of critical modes. Our results establish a microscopic framework linking Fermi-point topology to emergent CFTs and show how interaction-driven spectral reconstruction can generate higher-central-charge criticality and unconventional multicritical behavior.
The dynamical structure factor of the transverse field Ising model (TFIM) shows universal power-law divergence at its quantum critical point, signatures of which have been arguably observed in inelastic neutron scattering studies of quantum spin chain materials, for example CoNb 2 O 6 . However, it has recently been suggested that its microscopic description is better captured in terms of a twisted Kitaev spin chain (TKSC) with bond-anisotropic couplings. Here, we present exact results for the dynamical structure factor of the TKSC across its quantum critical point, analyzing both the universal low-frequency response and the non-universal high-energy features. In addition, we explore extensions of the model including broken glide symmetry as well as the case of random, and incommensurate magnetic fields. Notably, in the latter case, the fermionic excitations exhibit a localization-delocalization transition, which is manifest in the dynamical response as a distinct signature at finite frequency. We discuss the relevance of these features for the observation of quantum critical response in experiments.
Motivated by intense research on two-dimensional spin-21 Kitaev materials, Kitaev spin chains and ladders, though geometrically limited, have been studied for their numerical simplicity and insights into extended Kitaev models. The phase diagrams under the magnetic field were also explored for these quasi-one-dimensional models. For an isotropic Kitaev chain, it was found that a magnetic field polarizes the ground state except along the symmetric field angle, where the chain is found to remain gapless up to a critical field strength where it enters an intriguing soliton phase before reaching the polarized state at higher field strengths. Here we study an anisotropic Kitaev chain under a magnetic field using the density matrix renormalization group technique, where the ground state has a macroscopic degeneracy with a finite gap in the absence of the magnetic field. When the field is mainly aligned parallel to the strong bond, four-site and large unit-cell ordered phases arise. In a certain angle of the field, another ordered phase characterized by a uniform chirality with six-site periodicity emerges. We employ a perturbation theory to understand such field-induced ordered phases. The effective model uncovers the presence of transverse Ising and Dzyaloshinskii-Moriya interactions between unit cells, as well as further-neighbor Ising interaction induced by the magnetic field, which collectively explain the mechanisms behind these ordered states. Open questions and challenges are also discussed.