[Paper Review] Monte Carlo simulation of quantum Potts model

Chengxiang Ding, Yangcheng Wang+2 more|arXiv (Cornell University)|Feb 9, 2017

Quantum and electron transport phenomena3 citations

TL;DR

This study employs stochastic series expansion (SSE) Monte Carlo simulations with a cluster algorithm generalized from the quantum Ising model to investigate the three-state quantum Potts model. It reveals that the one-dimensional ferromagnetic case belongs to the 2D classical Potts universality class, the two-dimensional ferromagnetic case undergoes a first-order transition, and the antiferromagnetic case exhibits a continuous transition in the 3D classical XY universality class due to emergent O(2) symmetry, despite Z6 order in the ground state.

ABSTRACT

Using Monte Carlo simulations in the frame of stochastic series expansion (SSE), we study the three-state quantum Potts model. The cluster algorithm we used is a direct generalization of that for the quantum Ising model. The simulations include the one dimensional and two dimensional ferromagnetic three-state quantum Potts model and the two dimensional antiferromagnetic three-state quantum Potts model. Our results show that the phase transition of the one dimensional ferromagnetic quantum Potts model belongs to the same universality class of the two dimensional classical Potts model, the two dimensional ferromagnetic quantum Potts model undergoes a first order transition, which is also in analogy to its classical correspondence. The phase transition of the antiferromagnetic quantum Potts model is continuous, whose universality class belongs to the three-dimensional classical XY model, owing to an `emergent' O(2) symmetry at the critical point, although its ordered phase breaks the Z_6 symmetry.

Motivation & Objective

• To investigate the phase transitions and critical phenomena of the three-state quantum Potts model using finite-temperature Monte Carlo simulations.
• To determine the universality class of the phase transition in the one-dimensional and two-dimensional ferromagnetic and antiferromagnetic cases.
• To explore the emergence of continuous symmetry in the antiferromagnetic quantum Potts model despite explicit Z6 symmetry breaking in the ordered phase.
• To validate the universality of quantum phase transitions by comparing with classical counterparts in d+1 dimensions.

Proposed method

• The stochastic series expansion (SSE) method is used to represent the partition function as a Taylor series expansion in inverse temperature β.
• The Hamiltonian is decomposed into elementary lattice operators H_{t,a}, enabling the construction of SSE configurations with states |α_p⟩ and operator sequences.
• A cluster algorithm is implemented as a direct generalization of the quantum Ising model’s cluster update, enabling efficient sampling of the SSE configuration space.
• Observables such as energy, magnetization, and Binder ratios are measured to identify phase transitions and estimate critical points.
• Finite-size scaling is applied to the Binder ratio and order parameter to extract critical exponents and universality class.
• Histograms of the staggered magnetization are analyzed to detect emergent O(2) symmetry at the critical point.

Experimental results

Research questions

• RQ1 Does the one-dimensional ferromagnetic quantum Potts model exhibit a phase transition in the same universality class as the two-dimensional classical Potts model?
• RQ2 What is the nature of the phase transition in the two-dimensional ferromagnetic quantum Potts model—continuous or first-order?
• RQ3 Does the two-dimensional antiferromagnetic quantum Potts model undergo a continuous phase transition, and if so, to which universality class does it belong?
• RQ4 What is the origin of the observed universality class in the antiferromagnetic case, and how does emergent O(2) symmetry arise despite explicit Z6 symmetry breaking?
• RQ5 Can the critical exponents of the antiferromagnetic model be consistently fitted to the 3D classical XY model universality class?

Key findings

• The one-dimensional ferromagnetic quantum Potts model undergoes a continuous phase transition in the same universality class as the two-dimensional classical Potts model.
• The two-dimensional ferromagnetic quantum Potts model exhibits a first-order phase transition, consistent with its classical three-dimensional counterpart.
• The two-dimensional antiferromagnetic quantum Potts model undergoes a continuous phase transition with critical exponents y_t = 1.48(1) and y_h = 2.48(1), matching the 3D classical XY universality class.
• Despite the ordered phase breaking Z6 symmetry, the critical point exhibits emergent O(2) symmetry, confirmed by histograms of the staggered magnetization showing circular symmetry.
• The critical point of the antiferromagnetic model is estimated at g_c = 1.7173(3), with the Binder ratio Q_c = 0.72(1) consistent with the 3D XY model.
• The results for the antiferromagnetic case are analogous to the classical mixed Potts model on a simple cubic lattice, supporting universality in the quantum-classical correspondence.