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arXiv:2501.03329 · Physical review. B./Physical review. B

Field-induced ordered phases in anisotropic spin- 1 2 Kitaev chains

Mandev Bhullar, Haoting Xu, Hae‐Young Kee

Published Mar 31, 2025• 3 citations
arXivHTMLPDF

Abstract

Motivated by intense research on two-dimensional spin-$\frac{1}{2}$ 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.

DOI
10.1103/physrevb.111.104439

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Motivated by intense research on two-dimensional spin-12\frac{1}{2}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.
We study the interplay of electric and magnetic order in the one-dimensional Heisenberg spin-12\frac{1}{2}21​ XXZ chain with large Ising anisotropy in the presence of the Dzyaloshinskii-Moriya (DM) interaction and with longitudinal and transverse magnetic fields, interpreting the DM interaction as a coupling between the local electric polarization and an external electric field. We obtain the ground state phase diagram using the density matrix renormalization group method and compute various ground state quantities like the magnetization, staggered magnetization, electric polarization and spin correlation functions, etc. In the presence of both longitudinal and transverse magnetic fields, there are three different phases corresponding to a gapped N\'eel phase with antiferromagnetic (AF) order, gapped saturated phase, and a critical incommensurate gapless phase. The external electric field modifies the phase boundaries but does not lead to any new phases. Both external magnetic fields and electric fields can be used to tune between the phases. We also show that the transverse magnetic field induces a vector chiral order in the N\'eel phase (even in the absence of an electric field) which can be interpreted as an electric polarization in a direction parallel to the AF order.
Recently, there has been a renewed interest in properties of the higher-spin Kitaev models, especially their low-dimensional analogs with additional interactions. These quasi-one-dimensional systems exhibit rich phase diagrams with symmetry-protected topological phases, Luttinger liquids, hidden order, and higher-rank magnetism. However, the nature of the pure spin-SSS Kitaev chains is not yet fully understood. Earlier works found a unique ground state with short-ranged correlations for S=1S=1S=1 and an intriguing double-peak structure in the heat capacity associated with an entropy plateau. To understand the low-energy excitations and thermodynamics for general SSS, we study the anisotropic spin-SSS Kitaev chain. Starting from the dimerized limit, we derive an effective low-energy Hamiltonian at finite anisotropy. For half-integer spins we find a trivial effective model, reflecting a nonlocal symmetry protecting the degeneracy, while for integer SSS we find interactions among the flux degrees of freedom that select a unique ground state. The effective model for integer spins is used to predict the low-energy excitations and thermodynamics, and we make a comparison with the semiclassical limit through linear spin wave theory. Finally, we speculate on the nature of the isotropic limit.
We investigate the fate of the Kitaev spin liquid (KSL) under the influence of an external magnetic field h in the [001] direction and upon tuning bond anisotropy of the Kitaev coupling Kz keeping Kx=Ky=K . Guided by density matrix renormalization group, exact diagonalization, and with insights from parton mean-field theory, we uncover a field-induced gapless-to-gapless Lifshitz transition from the nodal KSL to an intermediate gapless phase. The intermediate phase sandwiched between hc1 and hc2 , which persists for a wide range of anisotropy Kz/K>0 , is composed of weakly coupled one-dimensional quantum critical chains. This intermediate phase is a dimensional crossover, which asymptotically leads to the one-dimensional quantum Ising criticality characterized by the (1+1)D conformal field theory as the field reaches the phase transition at hc2 . Beyond hc2 the system enters a partially polarized phase describable as effectively decoupled bosonic chains in which spin waves propagate along the one-dimensional zigzag direction. Our findings provide a comprehensive phase diagram and offer insights into the unusual physics of dimensional reduction generated by a uniform magnetic field in an otherwise two-dimensional quantum spin liquid. Published by the American Physical Society 2024
Abstract We study the spin-1/2 Kitaev chain with additional XX and symmetric off-diagonal Gamma interactions. By combining the Jordan–Wigner transformation with density matrix renormalization group (DMRG) numerical simulations, we obtain the exact solution of the model and map out the phase diagram containing six distinct phases. The four gapped phases display ferromagnetic and antiferromagnetic magnetic orders along the (1,1,0)- and (1,-1,0)-spin directions, whereas in the gapless phases, the low-energy spectrum consists of two branches of helical Majorana fermions with unequal velocities. The transition lines separating different phases include deconfined quantum critical lines with dynamical critical exponent z = 1 and quadratic critical lines with z = 2. Our work reveals the rich interplay among symmetry, magnetic order, and quantum criticality in the Kitaev–XX–Gamma chain.
We study the Kitaev-Ising model, where ferromagnetic Ising interactions are added to the Kitaev model on a lattice. This model has two phases which are characterized by topological and ferromagnetic order. Transitions between these two kinds of order are then studied on a quasi-one-dimensional system, on a ladder, and on a two-dimensional periodic lattice, a torus. By exactly mapping the quasi-one-dimensional case to an anisotropic XYXYXY chain we show that the transition occurs at zero $\ensuremath{\lambda}$, where $\ensuremath{\lambda}$ is the strength of the ferromagnetic coupling. In the two-dimensional case the model is mapped to a two-dimensional Ising model in transverse field, where it shows a transition at a finite value of $\ensuremath{\lambda}$. A mean-field treatment reveals the qualitative character of the transition and an approximate value for the transition point. Furthermore with perturbative calculation, we show that the expectation value of Wilson loops behaves as expected in the topological and ferromagnetic phases.
In this thesis, we deal with 1-dimensional anisotropic spin models more specifically with the chain and ladder geometries. These models are fundamental to the study of condensed matter theory and quantum magnetism because of their simplicity. Research into the chain model dates back to 1930s with Hans Bethe. One method to study these systems is through numerical analysis including renormalization group techniques. These allow for the diagonalization of the Hamiltonian without sacrificing computation to a large virtual Hilbert space. In this thesis, we present results for both the spin chain and spin ladder geometries in open boundary conditions. This work uses the density matrix renormalization group technique to calculate system energy. Initially we will present a study into the spin energy gap of the spin-\sfrac{1}{2} anisotropic (XXZ) Heisenberg chain in open boundary conditions (OBCs). The energy gap is shown to be reduced to half in the ground sector when compared to periodic boundary conditions (PBCs) due to edge effects. Secondly, a full phase diagram for the spin-\sfrac{1}{2} anisotropic Heisenberg ladder is presented which shows the emergence of a rich schematic with a variety of phases. Lastly, weak coupling limit maps are compared to quantum field theory phase transition predictions. This research was motivated by the lack of comprehensive phase diagram results for ladders and by real materials being investigated at the University of Kent.