[Paper Review] Possibility of Deconfined Criticality in SU(N) Heisenberg Models at Small N

Kenji Harada, Takafumi Suzuki+6 more|Kyoto University Research Information Repository (Kyoto University)|Jul 1, 2013

Physics of Superconductivity and Magnetism7 citations

TL;DR

This study investigates deconfined quantum criticality in SU(N) Heisenberg models with four- and six-body interactions on square and honeycomb lattices using large-scale quantum Monte Carlo simulations. Despite apparent universal scaling behavior up to L=256, systematic increases in critical exponents with system size suggest a possible first-order transition, challenging the universality of deconfined criticality in SU(2) and SU(3) models.

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($N$) Néel 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 $y \equiv 1/ν$ as the system size increases. We also confirm the relevance of a squared valence-bond solid field $Ψ^2$ for the SU(3) model.

Motivation & Objective

• To test the validity of the deconfined criticality (DCP) scenario in SU(N) Heisenberg models with small N.
• To examine whether the Néel-to-valence-bond solid (VBS) quantum phase transition is truly continuous or potentially first-order.
• To assess the relevance of the squared VBS order parameter Ψ² in the SU(3) model under the DCP framework.
• To explore the role of Berry phase and lattice symmetry in stabilizing or destabilizing the DCP fixed point.
• To determine whether the observed scaling behavior is consistent with the DCP scenario or indicative of a nearby first-order transition.

Proposed method

• Quantum Monte Carlo simulations were performed on SU(2), SU(3), and SU(4) Heisenberg models with four-body (square lattice) and six-body (honeycomb lattice) interactions.
• The J–Q model Hamiltonian was used, with J and Q terms involving projectors onto SU(N) singlet states: H = -J∑P_ij - Q∑P_ijP_kl for nearest-parallel bonds.
• World-line Monte Carlo with loop updates was employed to simulate the quantum many-body systems at zero temperature.
• Finite-size scaling (FSS) was applied to critical exponents, including 1/ν (y), using observables such as magnetization and VBS order parameters.
• Block-averaged order parameters were used to define Ψ̄², and two-point correlation functions C_Ψ²(R) were computed to estimate scaling dimensions.
• The relevance of Ψ² was assessed via the scaling dimension 2x_Ψ² ≈ 4.0, implying y_Ψ² ≈ 1.0 > 0, indicating relevance at the critical point.

Experimental results

Research questions

• RQ1 Is the Néel-to-VBS transition in SU(2) and SU(3) J–Q models truly continuous, or does it exhibit signs of a first-order transition as system size increases?
• RQ2 Does the squared VBS order parameter Ψ² remain relevant in the SU(3) model, implying that the DCP fixed point cannot be realized in systems with Z₂ lattice symmetry?
• RQ3 To what extent do finite-size scaling results support the DCP scenario, given the observed trend in critical exponents?
• RQ4 How does the critical behavior of the SU(4) model compare to SU(2) and SU(3) in terms of scaling and potential first-order character?
• RQ5 Can the apparent universal scaling behavior observed in small to intermediate systems be explained by proximity to a DCP fixed point, even if the true transition is first-order?

Key findings

• Finite-size scaling analysis shows consistent critical exponents across different lattices and observables up to system sizes L=256 (square) and L=96 (honeycomb), supporting apparent universality.
• For SU(2) and SU(3), the estimated critical exponent y ≡ 1/ν systematically increases with system size, indicating a trend toward a first-order transition.
• The scaling dimension of the squared VBS order parameter Ψ² is estimated as 2x_Ψ² ≈ 4.0, implying y_Ψ² ≈ 1.0 > 0, confirming that Ψ² is a relevant perturbation in the SU(3) model.
• The relevance of Ψ² in SU(3) implies that the DCP scenario cannot describe the Néel-VBS transition in models with only Z₂ lattice rotational symmetry.
• The SU(4) model shows no such systematic trend in y, suggesting a possible distinction in critical behavior from SU(2) and SU(3), though further study is needed.
• Despite good finite-size scaling and agreement with 1/N expansion predictions, the persistent increase in y for SU(2) and SU(3) keeps open the possibility of a first-order transition.