© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

Physical review. B, Condensed matter

New percolation structures in dilute antiferromagnetic Potts models

Henry Phillip Fried, Michael Schick C

Published Mar 1, 1990• 8 citations

Abstract

We introduce a new class of site-percolation models and study them by Monte Carlo methods. These percolation models are defined as the zero-temperature limit of a quench-diluted spin model whose symmetries are those of the antiferromagnetic three-state Potts model on a triangular lattice. Many of these percolation models exhibit a sequence of two percolation transitions. A new phase, excluded in the absence of dilution, is found at intermediate spin concentrations. This phase, which we call the chiral phase, exhibits the symmetries of only a subgroup of the full symmetry group of the Potts model. The critical properties of the transitions found in these percolation models have been studied. The results indicate that, for this entire class of percolation models, the observed transitions are all contained in the universality class of the geometric percolation (q=1 Potts) model. From this result we conclude that the breaking of symmetry, which is relevant for thermally driven transitions, is not relevant for percolation transitions.

DOI: 10.1103/physrevb.41.4389

Related papers

Nonlocal percolation in an antiferromagnetic Potts model

Henry Phillip Fried, Michael Schick C

We devise an algorithm to determine the critical site-occupation probability, ${x}_{c}$, at which antiferromagnetic order in the three-state Potts model first propagates on a triangular lattice. We find ${x}_{c}=0.735$, larger than the value of 0.5 necessary to percolate ferromagnetic order. Even though the means to propagate the order in this problem can be nonlocal as well as local, our results for critical exponents indicate that the percolation transition is in the same universality class as that of ferromagnetic percolation.

Physical review. B, Condensed matter•Jul 1, 1988•15 citations

Three-state Potts model on the centered triangular lattice

Zhe Fu, Wenan Guo, Henk W. J. Blöte

We study phase transitions of the Potts model on the centered-triangular lattice with two types of couplings, namely, K between neighboring triangular sites, and J between the centered and the triangular sites. Results are obtained by means of a finite-size analysis based on numerical transfer-matrix calculations and Monte Carlo simulations. Our investigation covers the whole (K,J) phase diagram, but we find that most of the interesting physics applies to the antiferromagnetic case K<0, where the model is geometrically frustrated. In particular, we find that there are, for all finite J, two transitions when K is varied. Their critical properties are explored. In the limits J→±∞ we find algebraic phases with infinite-order transitions to the ferromagnetic phase.

arXiv:1909.04979•Physical review. E•Jan 14, 2020•5 citations

Quantum Percolation in Two-Dimensional Antiferromagnets

Rong Yu, Tommaso Roscilde, Stephan Haas

The interplay of geometric randomness and strong quantum fluctuations is an exciting topic in quantum many-body physics, leading to the emergence of novel quantum phases in strongly correlated electron systems. Recent investigations have focused on the case of homogeneous site and bond dilution in the quantum antiferromagnet on the square lattice, reporting a classical geometric percolation transition between magnetic order and disorder. In this study we show how inhomogeneous bond dilution leads to percolative quantum phase transitions, which we have studied extensively by quantum Monte Carlo simulations. Quantum percolation introduces a new class of two-dimensional spin liquids, characterized by an infinite percolating network with vanishing antiferromagnetic order parameter.

arXiv:cond-mat/0410041•Physical Review Letters•May 20, 2005•42 citations

Quantum Phase Transition of the Randomly Diluted Heisenberg Antiferromagnet on a Square Lattice

Kiyoshi Katō, Synge Todo, Kenji Harada +3 more

Ground-state magnetic properties of the diluted Heisenberg antiferromagnet on a square lattice are investigated by means of the quantum Monte Carlo method with the continuous-time loop algorithm. It is found that the critical concentration of magnetic sites is independent of the spin size S, and equal to the two-dimensional percolation threshold. However, the existence of quantum fluctuations makes the critical exponents deviate from those of the classical percolation transition. Furthermore, we found that the transition is not universal, i.e., the critical exponents significantly depend on S.

arXiv:cond-mat/9905379•Physical Review Letters•May 1, 2000•68 citations

QUANTUM MONTE CARLO STUDY OF SITE-DILUTED HEISENBERG ANTIFERROMAGNET ON A SQUARE LATTICE

Synge Todo, Kiyoshi Katō, Hajime Takayama +3 more

Ground-state phase transition of site-diluted Heisenberg antiferromagnets on a square lattice is studied. By using the continuous-time loop algorithm, we perform large-scale quantum Monte Carlo simulation on large systems at quite low temperatures. It is found that the critical concentration of magnetic sites is independent of the spin size S, and equal to the classical percolation threshold. However, the existence of quantum fluctuations makes the critical exponents deviate from those of the classical percolation transition. It is found that the transition is not universal, i.e., the critical exponents depend on the spin size S.

International Journal of Modern Physics C•Dec 1, 1999•1 citation

Quantum Phase Transition of Two-Dimensional Diluted Heisenberg Antiferromagnet

Synge Todo, Chitoshi Yasuda, Kiyoshi Katō +4 more

Quantum phase transition of site-diluted and bond-diluted Heisenberg antiferromagnets on square lattices is studied. By using the novel continuous-time loop algorithm, we perform quantum Monte Carlo simulations on quite larger lattices at extremely lower temperatures than the previous numerical studies. It is found that the antiferromagnetic long-range order at T = 0 persists so long as a cluster of magnetic sites percolates, that is, the critical concentration is equal to the classical percolation threshold, in both of the site-diluted and bond-diluted cases. Furthermore, we find that some critical exponents, such as the magnetization exponent β, are non-classical and strongly depend on the spin size S. On the other hand, we show that the correlation-length exponent ν is universal and is equal to the classical value (ν=4/3).

Progress of Theoretical Physics Supplement•Jan 1, 2000•2 citations

We introduce a new class of site-percolation models and study them by Monte Carlo methods. These percolation models are defined as the zero-temperature limit of a quench-diluted spin model whose symmetries are those of the antiferromagnetic three-state Potts model on a triangular lattice. Many of these percolation models exhibit a sequence of two percolation transitions. A new phase, excluded in the absence of dilution, is found at intermediate spin concentrations. This phase, which we call the chiral phase, exhibits the symmetries of only a subgroup of the full symmetry group of the Potts model. The critical properties of the transitions found in these percolation models have been studied. The results indicate that, for this entire class of percolation models, the observed transitions are all contained in the universality class of the geometric percolation (q=1 Potts) model. From this result we conclude that the breaking of symmetry, which is relevant for thermally driven transitions, is not relevant for percolation transitions.

We devise an algorithm to determine the critical site-occupation probability,

{x}_{c}

, at which antiferromagnetic order in the three-state Potts model first propagates on a triangular lattice. We find

{x}_{c}=0.735

, larger than the value of 0.5 necessary to percolate ferromagnetic order. Even though the means to propagate the order in this problem can be nonlocal as well as local, our results for critical exponents indicate that the percolation transition is in the same universality class as that of ferromagnetic percolation.

We study phase transitions of the Potts model on the centered-triangular lattice with two types of couplings, namely, K between neighboring triangular sites, and J between the centered and the triangular sites. Results are obtained by means of a finite-size analysis based on numerical transfer-matrix calculations and Monte Carlo simulations. Our investigation covers the whole (K,J) phase diagram, but we find that most of the interesting physics applies to the antiferromagnetic case K<0, where the model is geometrically frustrated. In particular, we find that there are, for all finite J, two transitions when K is varied. Their critical properties are explored. In the limits J→±∞ we find algebraic phases with infinite-order transitions to the ferromagnetic phase.

The interplay of geometric randomness and strong quantum fluctuations is an exciting topic in quantum many-body physics, leading to the emergence of novel quantum phases in strongly correlated electron systems. Recent investigations have focused on the case of homogeneous site and bond dilution in the quantum antiferromagnet on the square lattice, reporting a classical geometric percolation transition between magnetic order and disorder. In this study we show how inhomogeneous bond dilution leads to percolative quantum phase transitions, which we have studied extensively by quantum Monte Carlo simulations. Quantum percolation introduces a new class of two-dimensional spin liquids, characterized by an infinite percolating network with vanishing antiferromagnetic order parameter.

Ground-state magnetic properties of the diluted Heisenberg antiferromagnet on a square lattice are investigated by means of the quantum Monte Carlo method with the continuous-time loop algorithm. It is found that the critical concentration of magnetic sites is independent of the spin size S, and equal to the two-dimensional percolation threshold. However, the existence of quantum fluctuations makes the critical exponents deviate from those of the classical percolation transition. Furthermore, we found that the transition is not universal, i.e., the critical exponents significantly depend on S.

Ground-state phase transition of site-diluted Heisenberg antiferromagnets on a square lattice is studied. By using the continuous-time loop algorithm, we perform large-scale quantum Monte Carlo simulation on large systems at quite low temperatures. It is found that the critical concentration of magnetic sites is independent of the spin size S, and equal to the classical percolation threshold. However, the existence of quantum fluctuations makes the critical exponents deviate from those of the classical percolation transition. It is found that the transition is not universal, i.e., the critical exponents depend on the spin size S.

Quantum phase transition of site-diluted and bond-diluted Heisenberg antiferromagnets on square lattices is studied. By using the novel continuous-time loop algorithm, we perform quantum Monte Carlo simulations on quite larger lattices at extremely lower temperatures than the previous numerical studies. It is found that the antiferromagnetic long-range order at T = 0 persists so long as a cluster of magnetic sites percolates, that is, the critical concentration is equal to the classical percolation threshold, in both of the site-diluted and bond-diluted cases. Furthermore, we find that some critical exponents, such as the magnetization exponent β, are non-classical and strongly depend on the spin size S. On the other hand, we show that the correlation-length exponent ν is universal and is equal to the classical value (ν=4/3).