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arXiv:0803.3114 · Journal of Physics A Mathematical and Theoretical

Critical intermediate phase and phase transitions in a triangular-lattice three-spin interaction model: Level-spectroscopy approach

Hiromi Otsuka, K. Nomura

Published Aug 7, 2008• 3 citations
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Abstract

We investigate infinite-order phase transitions like the Berezinskii-Kosterlitz-Thouless transition observed in a triangular-lattice three-spin interaction model. Based on a field theoretical description and the operator-production-expansion technique, we perform the renormalization-group analysis, and then clarify properties of marginal operators near the phase transition points. The results are utilized to establish criteria to determine the transition points and some universal relations among excitation levels to characterize the transitions. We verify these predictions via the numerical analysis on eigenvalue structures of the transfer matrix. Also, we discuss an enhancement of symmetry at the end points of a critical intermediate phase in connection with a transition observed in the ground state of the bilinear-biquadratic spin-1 chain.

DOI
10.1088/1751-8113/41/37/375001

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We investigate infinite-order phase transitions like the Berezinskii-Kosterlitz-Thouless transition observed in a triangular-lattice three-spin interaction model. Based on a field theoretical description and the operator-production-expansion technique, we perform the renormalization-group analysis, and then clarify properties of marginal operators near the phase transition points. The results are utilized to establish criteria to determine the transition points and some universal relations among excitation levels to characterize the transitions. We verify these predictions via the numerical analysis on eigenvalue structures of the transfer matrix. Also, we discuss an enhancement of symmetry at the end points of a critical intermediate phase in connection with a transition observed in the ground state of the bilinear-biquadratic spin-1 chain.
We study a Berezinskii-Kosterlitz-Thouless-like phase transition in a triangular-lattice three-spin interaction model. The vector sine-Gordon field theory describes defects involved, and enables us to perform the renormalization-group analysis of the transition. Based on it, we discuss properties of the spin-spin correlation function, and also analyze the helicity modulus for the present model. To provide the data to support our analysis, we perform the Monte-Carlo simulations to calculate them. Further, we exhibit snapshots of some typical defect conflgurations, which are helpful to illustrate the defect-mediated phase transition.
The spin-1 Ising model on the square lattice with nearest-neighbor ferromagnetic exchange interactions [both bilinear (JJJ) and biquadratic (KKK) and crystal-field interaction ($\ensuremath{\Delta}$) is studied via a renormalization-group transformation in position space. The phase diagram in JJJ, KKK, $\ensuremath{\Delta}$ space is found to have one surface of critical phase transitions and two surfaces of first-order phase transitions. These surfaces are variously bounded by an ordinary tricritical line, an isolated critical line, and a line of critical end points. These three lines joint at a special tricritical point corresponding to the transition of the three-state Potts model. The over-all phase diagram is qualitatively similar to that obtained with the mean-field approximation, except in the vicinity of the Potts transition where a four-phase coexistence line in mean-field theory shrinks into a special tricritical point in renormalization-group theory. Symmetry considerations guide the construction of our truncated renormalization-group transformation. The global connectivity and local exponents of the thirteen separate fixed points underlying this quite complicated structure are determined. Local analysis with respect to magnetic field (HHH) and another odd interaction (LLL) is performed. A one-adjusted-parameter version of our transformation yields remarkably quantitative results, predicting the Potts transition temperature, for example, within 0.3% of the exact value.
Quantum Ising model on a triangular lattice hosts a finite temperature Berezinskii-Kosterlitz-Thouless (BKT) phase with emergent U(1) symmetry, and it will transit into an up-up-down (UUD) phase with C3{C}_{3}C3​ symmetry breaking upon an infinitesimal external field along the longitudinal direction, but the overall phase diagram spanned by the axes of external field and temperature remains opaque due to the lack of systematic investigations with controlled methodologies. By means of quantum Monte Carlo at finite temperature and ground state density matrix renormalization group simulations, we map out the phase diagram of triangular quantum Ising model. Starting from the upper BKT temperature at zero field, we obtain the phase boundary between the UUD and paramagnetic phases with its 2D q=3q=3q=3 Potts universality at weak field and weakly first order transition at strong field. Originated from the lower BKT temperature at zero field, we analyze the low temperature phase boundary between the clock phase and the UUD phase with Ising symmetry breaking at weak fields and the quantum phase transition between the UUD and fully polarized phases at strong fields. The accurate many-body numerical results are consistent with our field theoretical analysis. The experimental relevance towards the BKT magnet TmMgGaO4{\mathrm{TmMgGaO}}_{4}TmMgGaO4​ and programmable quantum simulators are also discussed.
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.
The field-induced Berezinskii-Kosterlitz-Thouless (BKT) transition in the ground state of the triangular antiferromagnetic Ising model is studied by the level-spectroscopy method. We analyze dimensions of operators around the BKT line, and estimate the BKT point H(c) approximately equal to 0.5229 +/-0.001, which is followed by the level-consistency check to demonstrate the accuracy of our estimate. Further we investigate the anisotropic case to clarify the stability of the field-induced string-density plateau against an incommensurate liquid state by the density-matrix renormalization-group method.
We discuss an effective field theory of a triangular-lattice three-spin interaction model defined by the Zp{\mathbb Z}_pZp​ variables. Based on the symmetry properties and the ideal-state graph concept, we show that the vector dual sine-Gordon model describes the long-distance properties for p≥5p\ge5p≥5; we then compare its predictions with the previous argument. To provide the evidences, we numerically analyze the eigenvalue structure of the transfer matrix for p=6p=6p=6, and we check the criticality with the central charge c=2c=2c=2 of the intermediate phase and the quantization condition of the vector charges.