We study the phase diagram of isotropic spin-1 models in the vicinity of the Uimin-Lai-Sutherland (ULS) model. This is done with the help of a level-one SU(3) Wess-Zumino-Witten model with certain marginal perturbations. We find that the renormalization-group flow has infrared stable and unstable trajectories divided by a critical line on which the ULS model is located. The infrared unstable trajectory produced by a marginally relevant perturbation generates an exponential mass gap for the Haldane phase, and thus the universality class of the transition from the massless phase to the Haldane phase at the ULS point is identified with the Berezinski\ifmmode \breve{}\else \u{}\fi{}\ifmmode \imath \else \i \fi{}-Kosterlitz-Thouless type. Our results support recent numerical studies by F\'ath and S\'olyom. In the massless phase, we calculate logarithmic finite-size corrections of the energy for the SU(\ensuremath{\nu})-symmetric and asymmetric models in the massless phase.
We study the phase diagram of isotropic spin-1 models in the vicinity of the Uimin-Lai-Sutherland (ULS) model. This is done with the help of a level-one SU(3) Wess-Zumino-Witten model with certain marginal perturbations. We find that the renormalization-group flow has infrared stable and unstable trajectories divided by a critical line on which the ULS model is located. The infrared unstable trajectory produced by a marginally relevant perturbation generates an exponential mass gap for the Haldane phase, and thus the universality class of the transition from the massless phase to the Haldane phase at the ULS point is identified with the Berezinski\ifmmode \breve{}\else \u{}\fi{}\ifmmode \imath \else \i \fi{}-Kosterlitz-Thouless type. Our results support recent numerical studies by F'ath and S'olyom. In the massless phase, we calculate logarithmic finite-size corrections of the energy for the SU(\ensuremath{\nu})-symmetric and asymmetric models in the massless phase.
We investigate a phase transition in an SU(3) symmetric spin-1 chain. To study this transition, we numerically diagonalize an SU(3) symmetric Hamiltonian combining the Uimin-Lai-Sutherland (ULS) model Hamiltonian with the Hamiltonian of the exact trimer ground state (Z3 symmetry breaking). Our numerical results are discussed on the basis of the conformal field theory (CFT) and the renormalization group (RG). We then show the phase transition between a trimer liquid phase (massless, no long-range order) and a trimer phase (spontaneously Z3 symmetry breaking). Next, we find the central charge c=2 in the massless phase, and c<2 in the Z3 symmetry broken phase. From these results, we identify the universality class of the critical trimer liquid phase, as well as the phase boundary between these two phases.
We study critical phenomena of the SU(3) symmetric spin-1 chains when adding the SU(3) asymmetric term. To investigate such system, we numerically diagonalize the Dimer-Trimer (DT) model Hamiltonian around the SU(3) symmetric point, named the pure trimer (PT) point. We analyze our numerical results with the conformal field theory (CFT). First of all, we discover soft modes at the wave number q = 0 and q = 2π/3 for the PT point, and then the system is critical. Secondly, we find that the system at the PT point belongs to the CFT with the central charge c = 2 and the scaling dimension x = 2/3. Finally, by investigating the eigenvalues of the Hamiltonian in the vicinity of the PT point, we find that there is a phase transition at the PT point from a massive phase to a massless phase. From these numerical results, the phase transition at the PT point belongs to the Berezinskii-Kosterlitz-Thouless (BKT)-like universality class that is explained by the level-1 SU(3) Wess-Zumino-Witten (SU(3) 1 WZW) model.
We investigate spin chains with bilinear-biquadratic (BLBQ) spin interactions as a function of an applied magnetic field h. At the Uimin-Lai-Sutherland (ULS) critical point we find a gapless to gapless transition revealed by the dynamical structure factor $S(q,\ensuremath{\omega})$ as a function of h. At h=0, the envelope of the lowest-energy excitations goes soft at two points, ${q}_{1}=2\ensuremath{\pi}/3$ and ${q}_{2}=4\ensuremath{\pi}/3$, dubbed the phase A. With increasing field, the spectral peaks at each of the gapless points bifurcate, making in total four soft modes, and combine to form a new set of excitations that soften at a single point $q=\ensuremath{\pi}$ at ${h}_{c1}\ensuremath{\approx}0.94$. Beyond hc1 the system enters another gapless B phase until the transition at hc2=4 to the fully polarized phase. We compare the ULS model results with those for the Affleck-Kennedy-Lieb-Tasaki model as a representative of the gapped Haldane phase. We explain the mechanism of the gapless to gapless transition in the ULS model using its conserved charges and a spinon band picture. We also discuss the universality of central charges of the BLBQ family of models subjected to a magnetic field.
We study critical phenomena of an SU(3) symmetric spin-1 chain when adding an SU(3) asymmetric term. To investigate such phenomena, we numerically diagonalize the dimer–trimer (DT) model Hamiltonian around the SU(3) symmetric point, named the pure trimer (PT) point. We analyze our numerical results on the basis of the conformal field theory (CFT). First of all, we discover that soft modes appear at the wave number q = 0 and ±2π/3 for the PT point, and then the system is critical. Secondly, we find that the system at the PT point can be described by the CFT with the central charge c = 2 and the scaling dimension x = 2/3. Finally, by investigating the eigenvalues of the Hamiltonian in the vicinity of the PT point, we find that there is a phase transition at the PT point from a massive phase to a massless phase. From these numerical results, the phase transition at the PT point belongs to the Berezinskii–Kosterlitz–Thouless (BKT)- like universality class that is explained by the level-1 SU(3) Wess–Zumino–Witten [SU(3)1 WZW] model.
The possible ground states and the phase transitions of a spin-1 chain with bilinear-biquadratic exchange are studied numerically. In order to see better the real asymptotic behavior of the infinite system and to determine the necessary degeneracies, a twisted boundary condition is used that reduces the finite-size effects. This calculation confirms that a second-order Ising-type phase transition occurs at the Takhtajan-Babujian point from the dimerized phase to the singlet Haldane phase. The transition at the Lai-Sutherland point, to a massless phase with three soft modes, is found to be of Kosterlitz-Thouless type.
We investigate critical phenomena of a spin-1 chain in the vicinity of the SU(3) symmetric critical point, which we already specified in a previous study [Mashiko and Nomura, Phys. Rev. B 104, 155405 (2021)]. We numerically diagonalize a Hamiltonian combining the bilinear-biquadratic Hamiltonian with the trimer Hamiltonian. We then discuss the numerical results based on the conformal field theory and the renormalization group. As a result, we first verify that the critical point found in our previous study is the tricritical point among the Haldane phase, the trimer phase, and the the trimer-liquid (TL) phase. Second, with regard to the TL-trimer transition and the TL-Haldane transition, we find that the critical phenomena around this tricritical point belong to the Berezinskii-Kosterlitz-Thouless-like universality class. Third, we find the boundary between the Haldane phase and the trimer phase, which is illustrated by the massive self-dual sine-Gordon model.