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arXiv:1307.0501 · Physical Review B

Possibility of deconfined criticality in SU( N ) Heisenberg models at small N

Kenji Harada, Takafumi J. Suzuki, Tsuyoshi Okubo, Haruhiko Matsuo, Jie Lou, Hiroshi Watanabe, Synge Todo, Naoki Kawashima

Published Dec 13, 2013• 116 citations
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Abstract

To examine the validity of the scenario of the deconfined critical phenomena, we carry out a quantum Monte Carlo simulation for the SU($N$) generalization of the Heisenberg model with four-body and six-body interactions. The quantum phase transition between the SU($N$) N\'eel and valence-bond solid phases is characterized for $N=2$, 3, and 4 on the square and honeycomb lattices. While finite-size scaling analysis works well up to the maximum lattice size ($L=256$) 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.

DOI
10.1103/physrevb.88.220408

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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≡1/\ensuremathνy\ensuremath{\equiv}1/\ensuremath{\nu}y\ensuremath≡1/\ensuremathν as the system size increases. We also confirm the relevance of a squared valence-bond solid field \ensuremathΨ2{\ensuremath{\Psi}}^{2}\ensuremathΨ2 for the SU(3) model.
We investigate thermal phase transitions to a valence-bond solid phase in SU(N)\mathrm{SU}(N)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 XYXYXY model with a Z4{Z}_{4}Z4​ 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 Z4{Z}_{4}Z4​ 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 Z3{Z}_{3}Z3​ field remains relevant in the SU(3) and SU(4) cases.
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.
We study a spin-1/2 SU(2) model on the honeycomb lattice with nearest-neighbor antiferromagnetic exchange J that favors Néel order and competing six-spin interactions Q that favor a valence-bond-solid (VBS) state in which the bond energies order at the "columnar" wave vector K=(2π/3,-2π/3). We present quantum Monte Carlo evidence for a direct continuous quantum phase transition between Néel and VBS states, with exponents and logarithmic violations of scaling consistent with those at analogous deconfined critical points on the square lattice. Although this strongly suggests a description in terms of deconfined criticality, the measured threefold anisotropy of the phase of the VBS order parameter shows unusual near-marginal behavior at the critical point.
Using ground-state projector quantum Monte Carlo simulations in the valence-bond basis, it is demonstrated that nonfrustrating four-spin interactions can destroy the Néel order of the two-dimensional S=1/2 Heisenberg antiferromagnet and drive it into a valence-bond solid (VBS) phase. Results for spin and dimer correlations are consistent with a single continuous transition, and all data exhibit finite-size scaling with a single set of exponents, z=1, nu=0.78+/-0.03, and eta=0.26+/-0.03. The unusually large eta and an emergent U(1) symmetry, detected using VBS order parameter histograms, provide strong evidence for a deconfined quantum critical point.
A quantum phase transition is typically induced by tuning an external parameter that appears as a coupling constant in the Hamiltonian. Another route is to vary the global symmetry of the system, generalizing, e.g., SU(2) to SU(N)\text{SU}(N)SU(N). In that case, however, the discrete nature of the control parameter prevents one from identifying and characterizing the transition. We show how this limitation can be overcome for the SU(N)\text{SU}(N)SU(N) Heisenberg model with the help of a singlet projector algorithm that can treat NNN continuously. On the square lattice, we find a direct, continuous phase transition between N\'eel-ordered and crystalline bond-ordered phases at Nc=4.57(5){N}_{\text{c}}=4.57(5)Nc​=4.57(5) with critical exponents z=1z=1z=1 and $\ensuremath{\beta}/\ensuremath{\nu}=0.81(3)$.
We generalize the SU(N=2) S=1/2 square-lattice quantum magnet with nearest-neighbor antiferromagnetic coupling (J(1)) and next-nearest-neighbor ferromagnetic coupling (J(2)) to arbitrary N. For all N>4, the ground state has valence-bond-solid order for J(2)=0 and Néel order for J(2)/J(1)≫1, allowing us access to the transition between these types of states for large N. Using quantum Monte Carlo simulations, we show that both order parameters vanish at a single quantum-critical point, whose universal exponents for large enough N (here up to N=12) approach the values obtained in a 1/N expansion of the noncompact CP(N-1) field theory. These results lend strong support to the deconfined quantum-criticality theory of the Néel-valence-bond-solid transition.