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arXiv:1505.06273 · Physical Review B

Thermal phase transition of generalized Heisenberg models for SU(N) spins on square and honeycomb lattices

Takafumi J. Suzuki, Kenji Harada, Haruhiko Matsuo, Synge Todo, Naoki Kawashima

Published Mar 16, 2015• 13 citations
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Abstract

We investigate thermal phase transitions to a valence-bond solid phase in $\mathrm{SU}(N)$ Heisenberg models with four- or six-body interactions on a square or honeycomb lattice, respectively. In both cases, a thermal phase transition occurs that is accompanied by rotational symmetry breaking of the lattice. We perform quantum Monte Carlo calculations in order to clarify the critical properties of the models. The estimated critical exponents indicate that the universality classes of the square- and honeycomb-lattice cases are identical to those of the classical $XY$ model with a ${Z}_{4}$ symmetry-breaking field and the three-state Potts model, respectively. In the square-lattice case, the thermal exponent, $\ensuremath{\nu}$, monotonically increases as the system approaches the quantum critical point, while the values of the critical exponents, $\ensuremath{\eta}$ and $\ensuremath{\gamma}/\ensuremath{\nu}$, remain constant. From a finite-size scaling analysis, we find that the system exhibits weak universality, because the ${Z}_{4}$ symmetry-breaking field is always marginal. In contrast, $\ensuremath{\nu}$ in the honeycomb-lattice case exhibits a constant value, even in the vicinity of the quantum critical point, because the ${Z}_{3}$ field remains relevant in the SU(3) and SU(4) cases.

DOI
10.1103/physrevb.91.094414

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We study thermal transitions of the generalized SU(N) Heisenberg models with four-body interactions on a square lattice and with six-body interactions on a honeycomb lattice. In both models for the N=3 and 4 cases, a singlet-dimer state is stabilized at a very low temperature, where a rotational symmetry of lattice is broken spontaneously. We discuss the universality class of thermal transition to the singlet dimer phases, performing quantum Monte Carlo calculations. From the finite-size scaling analysis, we find that the criticality for the square lattice case is well explained by the 2D weak Ising universality, while the 2D three-state Potts universality is observed in the honeycomb lattice case.
To examine the validity of the scenario of the deconfined critical phenomena, we carry out a quantum Monte Carlo simulation for the SU(NNN) generalization of the Heisenberg model with four-body and six-body interactions. The quantum phase transition between the SU(NNN) N\'eel and valence-bond solid phases is characterized for N=2N=2N=2, 3, and 4 on the square and honeycomb lattices. While finite-size scaling analysis works well up to the maximum lattice size (L=256L=256L=256) and indicates the continuous nature of the phase transition, a clear systematic change towards the first-order transition is observed in the estimates of the critical exponent $y\ensuremath{\equiv}1/\ensuremath{\nu}$ as the system size increases. We also confirm the relevance of a squared valence-bond solid field ${\ensuremath{\Psi}}^{2}$ for the SU(3) model.
The scaling of the transition temperature into an ordered phase close to a quantum critical point as well as the order parameter fluctuations inside the quantum critical region provide valuable information about universal properties of the underlying quantum critical point. Here, we employ quantum Monte Carlo simulations to examine these relations in detail for two-dimensional quantum systems that exhibit a finite-temperature Ising-transition line in the vicinity of a quantum critical point that belongs to the universality class of either (i) the three-dimensional Ising model for the case of the quantum Ising model in a transverse magnetic field on the square lattice or (ii) the chiral Ising transition for the case of a half-filled system of spinless fermions on the honeycomb lattice with nearest-neighbor repulsion. While the first case allows large-scale simulations to assess the scaling predictions to a high precision in terms of the known values for the critical exponents at the quantum critical point, for the later case, we extract values of the critical exponents $\ensuremath{\nu}$ and $\ensuremath{\eta}$, related to the order parameter fluctuations, which we discuss in relation to other recent estimates from ground-state quantum Monte Carlo calculations as well as analytical approaches.
We study the S=1/2S=1/2S=1/2 Heisenberg (JJJ) model on the two-dimensional (2D) square lattice in the presence of additional higher-order spin interactions (QQQ) which lead to a valence-bond-solid (VBS) ground state. Using quantum Monte Carlo simulations, we analyze the thermal VBS transition. We find continuously varying exponents, with the correlation-length exponent $\ensuremath{\nu}$ close to the Ising value for large Q/JQ/JQ/J and diverging when Q/JQ/JQ/J approaches the quantum-critical point (the critical temperature ${T}_{c}\ensuremath{\rightarrow}0$). We identify the transition with a class of conformal field theories with charge c=1c=1c=1 and critical exponents varying between those of the 2D Ising model and the Kosterlitz-Thouless (KT) fixed point. We find explicit evidence for KT physics by studying the emergence of U(1)U(1)U(1) symmetry of the order parameter at T=TcT={T}_{c}T=Tc​ when ${T}_{c}\ensuremath{\rightarrow}0$.
In order to gain a deeper understanding of the quantum criticality in the explicitly staggered dimerized Heisenberg models, we study a generalized staggered dimer model named the J 0 — J 1 — J 2 model, which corresponds to the staggered J — J ′ model on a square lattice and a honeycomb lattice when J 1 / J 0 equals 1 and 0, respectively. Using the quantum Monte Carlo method, we investigate all the quantum critical points of these models with J 1 / J 0 changing from 0 to 1 as a function of coupling ratio α = J 2 / J 0 . We extract all the critical values of the coupling ratio α c for these models, and we also obtain the critical exponents ν, β/ν, and η using different finite-size scaling ansätz,. All these exponents are not consistent with the three-dimensional Heisenberg universality class, indicating some unconventional quantum critical points in these models.
We determine the finite-temperature phase diagram and critical behavior of the classical square-lattice Heisenberg-compass model using large-scale Monte Carlo simulations and finite-size scaling. Six symmetry distinct ordered phases are identified. The four phases that simultaneously break the spin-lattice C4C_4C4​ and in-plane spin-inversion symmetries undergo continuous transitions in the Ashkin-Teller universality class, with the associated critical lines terminating at four-state Potts points, beyond which the transitions become first order. In contrast, the two zzz-polarized phases display conventional two-dimensional Ising criticality. Our results reveal how the interplay between Heisenberg exchange and compass anisotropy organizes these distinct critical regimes, thereby completing the characterization of the model's thermal phase transitions.