arXiv:2205.13047 · Physical Review Research

Critical behavior of the three-state random-field Potts model in three dimensions

Manoj Kumar, Varsha Banerjee, Sanjay Puri, Martin Weigel

Published Nov 30, 2022• 5 citations

Abstract

A quasi-exact ground-state algorithm is used for a systematic study of the random-field Potts model, a generalization of the celebrated random-field Ising model. The application of accurate finite-size scaling techniques reveals a universality class for the model with three states.

DOI: 10.1103/physrevresearch.4.l042041

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arXiv:2205.13047 · Physical Review Research

Critical behavior of the three-state random-field Potts model in three dimensions

Manoj Kumar, Varsha Banerjee, Sanjay Puri, Martin Weigel

Published Nov 30, 2022• 5 citations

arXiv HTML PDF

Abstract

DOI: 10.1103/physrevresearch.4.l042041

Related papers

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The self-dual random-bond eight-state Potts model is studied numerically through large-scale Monte Carlo simulations using the Swendsen-Wang cluster flipping algorithm. We compute bulk and surface order parameters and susceptibilities and deduce the corresponding critical exponents at the random fixed point using standard finite-size scaling techniques. The scaling laws are suitably satisfied. We can conclusively rule out that the model belongs to the 2D Ising model universality class. The dimensions of the relevant bulk and surface scaling fields are found to be ${y}_{h}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.849$ , ${y}_{t}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.977$ , and ${y}_{{h}_{s}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.54$ , in contrast to their Ising values $15/8$ , $1$ , and $1/2$ , respectively.The self-dual random-bond eight-state Potts model is studied numerically through large-scale Monte Carlo simulations using the Swendsen-Wang cluster flipping algorithm. We compute bulk and surface order parameters and susceptibilities and deduce the corresponding critical exponents at the random fixed point using standard finite-size scaling techniques. The scaling laws are suitably satisfied. We can conclusively rule out that the model belongs to the 2D Ising model universality class. The dimensions of the relevant bulk and surface scaling fields are found to be ${y}_{h}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.849$, ${y}_{t}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.977$, and ${y}_{{h}_{s}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.54$, in contrast to their Ising values $15/8$, $1$, and $1/2$, respectively.

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The disorder-driven phase transition of the random-field Ising model is observed using exact ground-state computer simulations for hypercubic lattices in

d=5,6,7

dimensions. Finite-size scaling analyses are used to calculate the critical point and the critical exponents of the specific heat, magnetization, susceptibility, and of the correlation length. For dimensions

d=6,7

which are larger or equal to the assumed upper critical dimension,

{d}_{u}=6

, mean-field behavior is found, i.e., $\ensuremath{\alpha}=0, \ensuremath{\beta}=1/2, \ensuremath{\gamma}=1, \ensuremath{\nu}=1/2$. For the analysis of the numerical data, it appears to be necessary to include recently proposed corrections to scaling at and beyond the upper critical dimension.

{y}_{h}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.849

{y}_{t}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.977

, and

{y}_{{h}_{s}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.54

, in contrast to their Ising values

15/8

1

, and

1/2

, respectively.

The random field Ising model in three dimensions with Gaussian random fields is studied at zero temperature for system sizes up to

{60}^{3}.

For each realization of the normalized random fields, the strength of the random field, $\ensuremath{\Delta}$ and a uniform external, H is adjusted to find the finite-size critical point. The finite-size critical point is identified as the point in the $H\ensuremath{-}\ensuremath{\Delta}$ plane where three degenerate ground states have the largest discontinuities in the magnetization. The discontinuities in the magnetization and bond energy between these ground states are used to calculate the magnetization and specific heat critical exponents and both exponents are found to be near zero.