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arXiv:2112.15053 · Journal of the Physical Society of Japan

Possibility of a Topological Phase Transition in Two-dimensional RP3 Model

Tsuyoshi Okubo, Naoki Kawashima

Published Oct 3, 2023• 0 citations
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Abstract

We study by large-scale Monte Carlo simulation the $RP^3$ model, which can be regarded as an effective low-energy model of a triangular lattice Heisenberg antiferromagnet. $Z_2$ vortices appear as elementary excitations in the triangular lattice Heisenberg antiferromagnet. Such $Z_2$ vortices are ubiquitous in other frustrated Heisenberg spin systems that have noncollinear long-range orders. In this study, we investigate a possible topological phase transition driven by the binding--unbinding of $Z_2$ vortices. By extracting important degrees of freedom, we map a frustrated spin system to an effective $RP^3$ model. From large-scale Monte Carlo simulation, we obtain an order parameter and a correlation length of up to $L=16384$. Concerning the existence of a $Z_2$-vortex transition, by extrapolating the order parameter to the thermodynamics limit assuming the $Z_2$-vortex transition, we obtain a finite transition temperature as $T_v/\tilde{J} \simeq 0.25$. Our estimate of the correlation length at $T_v$ is much larger than $L=16384$, which is beyond the previous estimate obtained with the triangular lattice Heisenberg model.

DOI
10.7566/jpsj.92.114701

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We study by large-scale Monte Carlo simulation the RP3RP^3RP3 model, which can be regarded as an effective low-energy model of a triangular lattice Heisenberg antiferromagnet. Z2Z_2Z2​ vortices appear as elementary excitations in the triangular lattice Heisenberg antiferromagnet. Such Z2Z_2Z2​ vortices are ubiquitous in other frustrated Heisenberg spin systems that have noncollinear long-range orders. In this study, we investigate a possible topological phase transition driven by the binding--unbinding of Z2Z_2Z2​ vortices. By extracting important degrees of freedom, we map a frustrated spin system to an effective RP3RP^3RP3 model. From large-scale Monte Carlo simulation, we obtain an order parameter and a correlation length of up to L=16384L=16384L=16384. Concerning the existence of a Z2Z_2Z2​-vortex transition, by extrapolating the order parameter to the thermodynamics limit assuming the Z2Z_2Z2​-vortex transition, we obtain a finite transition temperature as Tv/J~≃0.25T_v/\tilde{J} \simeq 0.25Tv​/J~≃0.25. Our estimate of the correlation length at TvT_vTv​ is much larger than L=16384L=16384L=16384, which is beyond the previous estimate obtained with the triangular lattice Heisenberg model.
We investigate nonequilibrium properties of the frustrated Heisenberg antiferromagnets on the triangular lattice. Nonequilibrium critical relaxation of frustrated Heisenberg antiferromagnets shows a dynamic transition (or, at least, sharp crossover) near the temperature Tu=0.282J{T}_{\mathrm{u}}=0.282JTu​=0.282J of the equilibrium Z2{\mathbb{Z}}_{2}Z2​-vortex unbinding. We show that starting from the high-temperature initial state, due to the presence of Z2{\mathbb{Z}}_{2}Z2​ vortices in the system being considered, in a broad temperature range T<TuT<{T}_{\mathrm{u}}T<Tu​ the dynamic properties in the intermediate time range are similar to those of the two-dimensional XY model below the Berezinskii-Kosterlitz-Thouless transition. The interaction of Z2{\mathbb{Z}}_{2}Z2​ vortices with spin-wave degrees of freedom does not emerge until rather long times.
A numerical method to study defect-mediated phase transitions is proposed, which combines a standard Monte Carlo technique with a particular type of boundary conditions. The method is applied to the two-dimensional XY (plane rotator) model as well as to the two-dimensional SO(3) Heisenberg model which models a triangular-lattice Heisenberg antiferromagnet. Our results give a direct numerical support for the occurrence of a topological transition in these systems driven by the vortex binding-unbinding.
Frustrated quantum magnets may exhibit fascinating collective phenomena. The main goal of this dissertation is to provide conclusive evidence for the emergence of novel phases of matter like quantum spin liquids in local quantum spin models. We develop novel algorithms for large-scale Exact Diagonalization computations. Sublattice coding methods for efficient use of lattice symmetries in the procedure of diagonalizing the Hamiltonian matrix are proposed and suggest a randomized distributed memory parallelization strategy. Benchmarks of computations on various supercomputers with system size up to 50 spin-1/2 particles have been performed. Results concerning the emergence of a chiral spin liquid in a frustrated kagome Heisenberg antiferromagnet are presented. The stability and extent of this phase are discussed. In an extended Heisenberg model on the triangular lattice, we establish another chiral spin liquid phase. We discuss the special case of the Heisenberg J1J_1J1​-J2J_2J2​ model and present a scenario where the critical point of phase transition from the 120-degree Néel to a putative Z2\mathbf{Z}_2Z2​ spin liquid is described by a Dirac spin liquid. A generalization of the SU(2) Heisenberg model with SU(N) degrees of freedom on the triangular lattice with an additional ring-exchange term is discussed. We present our contribution to the project and the final results that suggest a series of chiral spin liquid phases in an extended parameter range. Finally, we present preliminary data from a Quantum Monte Carlo study of an SU(N) version of the J-Q model on a square lattice for N=2,...,10, and multi-column representations. We establish the phase boundary between the Néel ordered phase and the disordered phases. The disordered phase in the four-column representation is expected to be a two-dimensional analog of the Haldane phase for the spin-1 Heisenberg chain.
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Using quantum Monte Carlo simulations, we map out the phase diagram of Hamiltonians interpolating between trivial and nontrivial bosonic symmetry-protected topological phases, protected by Z2{\mathbb{Z}}_{2}Z2​ and Z23{\mathbb{Z}}_{2}^{3}Z23​ symmetries, in two dimensions. In all cases, we find that the trivial and the topological phases are separated by an intermediate phase in which the protecting symmetry is spontaneously broken. Depending on the model, we identify a variety of magnetic orders on the triangular lattice, including ferromagnetism, $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ order, and stripe orders (both commensurate and incommensurate). Critical properties are determined through a finite-size scaling analysis. Possible scenarios regarding the nature of the phase transitions are discussed.