Renormalisation-group calculations of finite systems: order parameter and specific heat for epitaxial ordering

Berker, A. Nihat; Östlund, Stellan

doi:10.1088/0022-3719/12/22/035

Cited by 428 publications

(298 citation statements)

References 45 publications

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“…The Migdal-Kadanoff approximation amounts to replace the hypercubic lattice by a hierarchical fractal diamond lattice [18][19][20] which is constructed recursively as follows. Two boundary spins S A and S B are linked by K branches each branch containing two links and a middle spin S i , so that the corresponding energy reads…”

Section: Exactness For the Migdal-kadanoff Approximationmentioning

confidence: 99%

One-dimensional Ising spin-glass with power-law interaction: real-space renormalization at zero temperature

Monthus¹

2014

J. Stat. Mech.

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For the one-dimensional long-ranged Ising spin-glass with random couplings decaying with the distance r as J(r) ∼ r −σ and distributed with the Lévy symmetric stable distribution of index 1 < µ ≤ 2 (including the usual Gaussian case µ = 2), we consider the region σ > 1/µ where the energy is extensive. We study two real space renormalization procedures at zero temperature, namely a simple box decimation that leads to explicit calculations, and a strong disorder decimation that can be studied numerically on large sizes. The droplet exponent governing the scaling of the renormalized couplings JL ∝ L θµ(σ) is found to be θµ(σ) = 2 µ −σ whenever the long-ranged couplings are relevant θµ(σ) ≥ −1. For the statistics of the ground state energy E GS L over disordered samples, we obtain that the droplet exponent θµ(σ) governs the leading correction to extensivity of the averaged value E GS L ≃ Le0 + L θµ (σ) e1. The characteristic scale of the fluctuations around this average is of order L 1 µ , and the rescaledGaussian distributed for µ = 2, or displays the negative power-law tail in 1/(−u) 1+µ for u → −∞ in the Lévy case 1 < µ < 2. Finally we apply the zero-temperature renormalization procedure to the related Dyson hierarchical spin-glass model where the same droplet exponent appears.

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Section: Exactness For the Migdal-kadanoff Approximationmentioning

confidence: 99%

One-dimensional Ising spin-glass with power-law interaction: real-space renormalization at zero temperature

Monthus¹

2014

J. Stat. Mech.

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“…Also the behavior of magnetic models on random complex networks has been investigated [5,6]. AN's have the appealing advantages of geometrical sets defined through an exact inflation rule, where renormalization techniques, leading to exact results can be applied [7][8][9]. In fact, this type of techniques has been considered for different hierarchical structures to study critical phenomena [10][11][12][13][14].…”

Section: Introductionmentioning

confidence: 99%

q-state Potts model on the Apollonian network

2010

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The q-state Potts model is studied on the Apollonian network with Monte Carlo simulations and the Transfer Matrix method. The spontaneous magnetization, correlation length, entropy, and specific heat are analyzed as a function of temperature for different number of states, q. Different scaling functions in temperature and q are proposed. A quantitative agreement is found between results from both methods. No critical behavior is observed in the thermodynamic limit for any number of states.

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“…A van der Waals loop [13] develops in the stress-strain dependence signaling a weak phase transition that replaces the continuous Potts phase transition. While in the renormalization-group calculations [14,15] that are exact [16] on hierarchical lattices [17,18] the Potts transitions are always continuous, in the Monte Carlo simulations we can see both continuous transitions (for small q) and discontinuous transitions (for large q). This allows us to explore the influence of the order of the Potts transition on the mechanical properties of the solid, such as stress dependence on temperature and interatomic distance.…”

Section: Introductionmentioning

confidence: 90%

“…The recursion equations (16)- (18) represent the exact solutions [16] for the diamond hierarchical lattice [17,18]. of those quantities is scaled by the total number of lattice bonds.…”

Section: Renormalization Groupmentioning

confidence: 99%

Equation of state from the Potts-percolation model of a solid

Kaufman

Diep

2011

Phys. Rev. E

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We expand the Potts-percolation model of a solid to include stress and strain.Neighboring atoms are connected by bonds. We set the energy of a bond to be given by the Lennard-Jones potential. If the energy is larger than a threshold the bond is more likely to fail, while if the energy is lower than the threshold the bond is more likely to be alive. In two dimensions we compute the equation of state: stress as function of inter-atomic distance and temperature by using renormalization group and Monte Carlo simulations. The phase diagram, the equation of state, and the isothermal modulus are determined. When the Potts heat capacity is divergent the continuous transition is replaced by a weak first-order transition through the van der Waals loop mechanism. When the Potts transition is first order the stress exhibits a large discontinuity as function of the inter-atomic distance.

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Renormalisation-group calculations of finite systems: order parameter and specific heat for epitaxial ordering

Cited by 428 publications

References 45 publications

One-dimensional Ising spin-glass with power-law interaction: real-space renormalization at zero temperature

One-dimensional Ising spin-glass with power-law interaction: real-space renormalization at zero temperature

q-state Potts model on the Apollonian network

Equation of state from the Potts-percolation model of a solid

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