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Journal of Magnetism and Magnetic Materials

Kosterlitz–Thouless and topological Gaussian criticalities in a mixed-spin Heisenberg model with modulated single-ion anisotropy

João Chakrian, Maria S. S. Pereira, M. L. Lyra, Luan M. Veríssimo

Published Aug 7, 2026• 0 citations
Publisher

Abstract

The interplay between competing interactions in low-dimensional quantum many-body systems gives rise to a wide variety of quantum phases and critical phenomena. Motivated by this scenario, we investigate the quantum phase diagram of the mixed-spin 1 2 , 1 antiferromagnetic Heisenberg chain with modulated single-ion anisotropy by means of the density matrix renormalization group (DMRG) method. The resulting phase diagram comprises four distinct quantum phases: three gapped phases, namely the 1 / 3 magnetization plateau, the trivial nonmagnetic (TnM) phase, and the topological valence-bond-solid (TVBS) phase, together with a gapless Luttinger liquid (LL) phase. The different phases are characterized through the analysis of magnetization, spin correlation functions, nonlocal order parameters, and entanglement-based observables. The TVBS phase is identified by the presence of exponentially localized edge states, finite values of the nonlocal string order parameter and twist operator, and the characteristic degeneracy pattern of the entanglement spectrum. The boundaries of the LL phase are determined from the transverse spin correlation function and the corresponding Luttinger parameter, revealing two Kosterlitz–Thouless transition lines characterized by the critical value K c = 2.0 , in agreement with finite-size scaling analyses of the spin gap. In addition, we identify a Gaussian critical line associated with the topological quantum phase transition separating the TVBS and TnM phases. By employing the tangential finite-size scaling approach, we accurately determine this critical line by extracting the correlation-length critical exponent ν and the conformal field theory central charge c . Our results demonstrate that the competition between exchange interactions and modulated single-ion anisotropy gives rise to a rich interplay between topology, quantum criticality, and magnetic ordering, highlighting the fundamental role of these interactions in determining the ground-state properties of low-dimensional quantum spin systems.

DOI
10.1016/j.jmmm.2026.174464

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The interplay between competing interactions in low-dimensional quantum many-body systems gives rise to a wide variety of quantum phases and critical phenomena. Motivated by this scenario, we investigate the quantum phase diagram of the mixed-spin 1 2 , 1 antiferromagnetic Heisenberg chain with modulated single-ion anisotropy by means of the density matrix renormalization group (DMRG) method. The resulting phase diagram comprises four distinct quantum phases: three gapped phases, namely the 1 / 3 magnetization plateau, the trivial nonmagnetic (TnM) phase, and the topological valence-bond-solid (TVBS) phase, together with a gapless Luttinger liquid (LL) phase. The different phases are characterized through the analysis of magnetization, spin correlation functions, nonlocal order parameters, and entanglement-based observables. The TVBS phase is identified by the presence of exponentially localized edge states, finite values of the nonlocal string order parameter and twist operator, and the characteristic degeneracy pattern of the entanglement spectrum. The boundaries of the LL phase are determined from the transverse spin correlation function and the corresponding Luttinger parameter, revealing two Kosterlitz–Thouless transition lines characterized by the critical value K c = 2.0 , in agreement with finite-size scaling analyses of the spin gap. In addition, we identify a Gaussian critical line associated with the topological quantum phase transition separating the TVBS and TnM phases. By employing the tangential finite-size scaling approach, we accurately determine this critical line by extracting the correlation-length critical exponent ν and the conformal field theory central charge c . Our results demonstrate that the competition between exchange interactions and modulated single-ion anisotropy gives rise to a rich interplay between topology, quantum criticality, and magnetic ordering, highlighting the fundamental role of these interactions in determining the ground-state properties of low-dimensional quantum spin systems.
We investigate the phase diagram of a quantum mixed-spin (1/2,1/2,1,1)(1/2,1/2,1,1)(1/2,1/2,1,1) Heisenberg tetramer chain featuring bond-alternating antiferromagnetic nearest-neighbor exchange interactions and single-ion anisotropy. This model exhibits a rich phase structure comprising three gapped phases: the antiferromagnetic, topological valence-bond-solid (TVBS), and the trivial nonmagnetic phases (TnM). We identify a multicritical Wess-Zumino-Witten SU(2)1SU{(2)}_{1}SU(2)1​ point where two Ising critical lines merge, forming a higher-symmetry point that realizes conformal embedding on the lattice. Beyond this multicritical point, a Gaussian critical line emerges, determining the boundary between the TVBS and the TnM phases. Utilizing the tangential finite-size scaling method, we demonstrate the precision with which critical points and correlation length critical exponents can be extracted along this Gaussian line. Our results reveal rich critical phenomena induced by the interplay of bond-alternating exchange and anisotropic single-ion interactions in quantum many-body systems.
= 1 andS=3/2dimers is studied in detail as a function of two considered exchange interactions ascribed to similar and dissimilar spin pairs. At zero magnetic field, the ferrimagnetic mixed spin-(1, 1, 3/2, 3/2) Heisenberg tetramer chain displays, depending on a mutual interplay between two considered exchange interactions, three distinct gapped valence-bond-solid phases separated by gap-closing quantum critical points. Using density-matrix renormalization group calculations we construct the full ground-state phase diagram as a function of the interaction ratio and magnetic field, which exhibits besides three gapped valence-bond-solid phases special Kosterlitz-Thouless and topological quantum critical points. A tangential finite-size scaling analysis is employed to obtain precise estimates of the zero-field valence-bond-solid transitions and unveil their common logarithmic correction to a power-law scaling of the correlation length.
We investigate the ground-state phase diagram of the spin-1 Kitaev–Heisenberg (KH) chain in the presence of uniaxial single-ion anisotropy D z using the density-matrix renormalization-group method. The relative strength of the Heisenberg and Kitaev interactions is parametrized by a coupling angle ϕ , defined by J = cos ϕ and K = sin ϕ . By combining energy-based diagnostics with finite-size extrapolations of order parameters and correlation functions, we obtain a comprehensive phase diagram as a function of ϕ and D z . The phase diagram contains ferromagnetic and Néel ordered phases, collinear states with four-site spin modulation, magnetically disordered or critical regimes, two Kitaev-dominated spin-liquid-like regions, and a topological Haldane phase near the Heisenberg limit. We find that the Kitaev-dominated disordered regions acquire finite parameter widths in the spin-1 model, whereas the Haldane phase is fragile against Kitaev-type anisotropy, especially for D z < 0 . Positive D z generally suppresses magnetic order and expands nonmagnetic regimes, while negative D z enhances Ising-like magnetic order. At D z = 0 , we also identify an exactly solvable point at tan ϕ = − 2 ( ϕ / π ≃ 1.6476 ) , which gives rise to a first-order transition between the Néel- z state and a four-site collinear state. We further contrast these findings with the spin- 1 / 2 KH chain and with the spin-1 honeycomb KH model, highlighting the distinct roles of single-ion anisotropy and dimensionality in Kitaev-type magnets.
Density-matrix renormalization group calculations are used to determine the ground-state phase diagram of the mixed spin-(1/2, 5/2, 1/2) Heisenberg chain whose backbone consists of regularly alternating s=1/2s=1/2s=1/2 and S=5/2S=5/2S=5/2 spins, the latter of which are coupled to additional s=1/2s=1/2s=1/2 spins providing lateral branching. The proposed magnetic structure aims to describe the main characteristics of the heterotrimetallic coordination polymer $[{\mathrm{Cu}}^{\mathrm{II}}{\mathrm{Mn}}^{\mathrm{II}}({\mathrm{L}}^{1})][{\mathrm{Fe}}^{\mathrm{III}}(\mathrm{bpb}){(\mathrm{CN})}_{2}]\phantom{\rule{4pt}{0ex}}\ifmmode\cdot\else\textperiodcentered\fi{}\phantom{\rule{4pt}{0ex}}{\mathrm{ClO}}_{4}\phantom{\rule{4pt}{0ex}}\ifmmode\cdot\else\textperiodcentered\fi{}\phantom{\rule{4pt}{0ex}}{\mathrm{H}}_{2}\mathrm{O}$. The full ground-state phase diagram in the parameter space exchange anisotropy versus magnetic field unveils special critical points, at which the intermediate magnetization plateaus emergent at 3/7 and 5/7 of the saturation magnetization vanish. A detailed description of the zero-temperature magnetization process and pair entanglement entropy is accomplished with the aim to distinguish the main coupling regimes. A scaling analysis is performed to accurately locate Kosterlitz-Thouless and Gaussian critical points. In particular, we introduce a new finite-size scaling protocol to properly extract the critical parameters from the size dependence of the magnetic-field range corresponding to the magnetization plateau in the close vicinity of the Gaussian point at which two gapped phases meet.
Quantum many-body edge and extended magnon excitations from the 1/3 plateau of the anisotropic Heisenberg model on an open AB2{AB}_{2}AB2​ chain in a magnetic field hhh are unveiled using the density matrix renormalization group and exact diagonalization. By tuning both the anisotropy and hhh in the rich phase diagram, the edge states penetrate in the bulk, whose gap closes in a symmetry-protected topological Kosterlitz-Thouless transition. Also, we witness the squeezed chain effect, the breaking of the edge state degeneracy, and a topological change of the excitations from gapped magnons with quadratic long-wavelength dispersion to a linear spinon dispersion in the Luttinger liquid gapless phase as the anisotropy $\ensuremath{\lambda}$ approaches the critical point from the $\ensuremath{\lambda}>0$ side of the phase diagram.
We investigate the ground-state phase diagram of a frustrated spin-1/2 Heisenberg ladder in the transverse magnetic field with an anisotropic inter-rung exchange coupling ααα. This bond-anisotropic parameter continuously interpolates among the Ising-like limit α=0α=0α=0, the isotropic point α=1α=1α=1, and the XY-dominated regime α≫1α\gg 1α≫1. Using density-matrix renormalization group calculations within a matrix-product-state framework, we analyze bipartite entanglement, magnetization, and spin correlations to characterize the emergent quantum phases. We identify six distinct ground-state phases, including rung-singlet, Haldane-like, Tomonaga-Luttinger liquid, canted Ising-ordered, XY-polarized, and ordered ferromagnetic states. While magnetization and local correlations provide a first insight into phase classifications, the finite-size scaling of the entanglement measures offers a more sensitive and unified diagnostics of both gapped and gapless regimes. In the gapless regime, we extract the central charge c≃1c \simeq 1c≃1, confirming Tomonaga-Luttinger liquid behavior, while at the transition between the canted Ising and ferromagnetic phases, we observe c≃1/2c \simeq 1/2c≃1/2, consistent with the Ising universality. Finally, we find that both the Haldane-like behavior and the extended critical Tomonaga-Luttinger liquid regime are strongly confined to the vicinity of the isotropic point α=1α= 1α=1.