A path-integral representation for the d-dimensional q-state quantum Potts model has been obtained starting from the microscopic Hamiltonian. The terms in the functional Hamiltonian, which are relevant in the renormalization-group sense, are given explicitly up to a quartic term in the order-parameter field. The critical properties of the model at zero temperature near five dimensions are analyzed within the field-theoretic renormalization-group approach with the help of the minimal subtraction method. In particular, some critical exponents to first order in \ensuremath{\epsilon}=5d are presented, and the occurrence of a d\ensuremath{\rightarrow}d+1 dimensional crossover is pointed out.A path-integral representation for the d-dimensional q-state quantum Potts model has been obtained starting from the microscopic Hamiltonian. The terms in the functional Hamiltonian, which are relevant in the renormalization-group sense, are given explicitly up to a quartic term in the order-parameter field. The critical properties of the model at zero temperature near five dimensions are analyzed within the field-theoretic renormalization-group approach with the help of the minimal subtraction method. In particular, some critical exponents to first order in \ensuremath{\epsilon}=5d are presented, and the occurrence of a d\ensuremath{\rightarrow}d+1 dimensional crossover is pointed out.

Physical review. B, Condensed matterAug 1, 19904 citations

Potts-model formulation of the random resistor network

Amanda Harris, T. C. Lubensky

The randomly diluted resistor network is formulated in terms of an n-replicated s-state Potts model with a spin-spin coupling constant J in the limit when first n, then s, and finally 1/J go to zero. This limit is discussed and to leading order in 1/J the generalized susceptibility is shown to reproduce the results of the accompanying paper where the resistor network is treated using the xy model. This Potts Hamiltonian is converted into a field theory by the usual Hubbard-Stratonovich transformation and thereby a renormalization-group treatment is developed to obtain the corrections to the critical exponents to first order in \ensuremath{\epsilon}=6-d, where d is the spatial dimensionality. The recursion relations are shown to be the same as for the xy model. Their detailed analysis (given in the accompanying paper) gives the resistance crossover exponent as ${\ensuremath{\varphi}}_{1}$=1+\ensuremath{\epsilon}/42, and determines the critical exponent, t for the conductivity of the randomly diluted resistor network at concentrations, p, just above the percolation threshold: t=(d-2)\ensuremath{\nu}+${\ensuremath{\varphi}}_{1}$, where \ensuremath{\nu} is the critical exponent for the correlation length at the percolation threshold. These results correct previously accepted results giving \ensuremath{\varphi}=1 to all orders in \ensuremath{\epsilon}. The new result for ${\ensuremath{\varphi}}_{1}$ removes the paradox associated with the numerical result that t>1 for d=2, and also shows that the Alexander-Orbach conjecture, while numerically quite accurate, is not exact, since it disagrees with the \ensuremath{\epsilon} expansion.The randomly diluted resistor network is formulated in terms of an n-replicated s-state Potts model with a spin-spin coupling constant J in the limit when first n, then s, and finally 1/J go to zero. This limit is discussed and to leading order in 1/J the generalized susceptibility is shown to reproduce the results of the accompanying paper where the resistor network is treated using the xy model. This Potts Hamiltonian is converted into a field theory by the usual Hubbard-Stratonovich transformation and thereby a renormalization-group treatment is developed to obtain the corrections to the critical exponents to first order in \ensuremath{\epsilon}=6-d, where d is the spatial dimensionality. The recursion relations are shown to be the same as for the xy model. Their detailed analysis (given in the accompanying paper) gives the resistance crossover exponent as ${\ensuremath{\varphi}}_{1}$=1+\ensuremath{\epsilon}/42, and determines the critical exponent, t for the conductivity of the randomly diluted resistor network at concentrations, p, just above the percolation threshold: t=(d-2)\ensuremath{\nu}+${\ensuremath{\varphi}}_{1}$, where \ensuremath{\nu} is the critical exponent for the correlation length at the percolation threshold. These results correct previously accepted results giving \ensuremath{\varphi}=1 to all orders in \ensuremath{\epsilon}. The new result for ${\ensuremath{\varphi}}_{1}$ removes the paradox associated with the numerical result that t>1 for d=2, and also shows that the Alexander-Orbach conjecture, while numerically quite accurate, is not exact, since it disagrees with the \ensuremath{\epsilon} expansion.

Physical review. B, Condensed matterMay 1, 198715 citations

Finite size scaling and critical point exponents of the potts model

M. P. Nightingale, Henk W. J. Blöte

Physica A Statistical Mechanics and its ApplicationsNov 1, 198054 citations

THE DYNAMIC CRITICAL EXPONENT OF THE Q=3 AND Q=4 STATE POTTS-MODEL

Lucilla DE ARCANGELIS, Naeem Jan

Jan 1, 19860 citations

Bethe $M$ -layer construction for the percolation problem

Maria Chiara Angelini, Saverio Palazzi, Tommaso Rizzo +1 more

One way to perform field theory computations for the bond percolation problem is through the Kasteleyn and Fortuin mapping to the n+1 n + 1 states Potts model in the limit of n \to 0 n → 0 . In this paper, we show that it is possible to recover the \epsilon ϵ -expansion for critical exponents in finite dimension directly using the M M -layer expansion, without the need to perform any analytical continuation. Moreover, we also show explicitly that the critical exponents for site and bond percolation are the same. This computation provides a reference for applications of the M M -layer method to systems where the underlying field theory is unknown or disputed.One way to perform field theory computations for the bond percolation problem is through the Kasteleyn and Fortuin mapping to the n+1 n + 1 states Potts model in the limit of n \to 0 n → 0 . In this paper, we show that it is possible to recover the \epsilon ϵ -expansion for critical exponents in finite dimension directly using the M M -layer expansion, without the need to perform any analytical continuation. Moreover, we also show explicitly that the critical exponents for site and bond percolation are the same. This computation provides a reference for applications of the M M -layer method to systems where the underlying field theory is unknown or disputed.

arXiv:2407.00338SciPost PhysicsJan 23, 20255 citations