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Recurrence relations in one-dimensional Ising models.

C M Silva da Conceição1, R N P Maia2

  • 1Universidade Federal Fluminense, RHS/RCN, 28895-532 Rio das Ostras, Rio de Janeiro, Brazil.

Physical Review. E
|January 20, 2018
PubMed
Summary
This summary is machine-generated.

Researchers derived the exact partition function for the 1D Ising model using algebraic operators. This method reveals a recursive formula for the partition function, potentially applicable to higher dimensions.

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Area of Science:

  • Statistical Mechanics
  • Condensed Matter Physics

Background:

  • The Ising model is a fundamental tool for studying magnetism and phase transitions.
  • Calculating the partition function, especially for finite-size systems and nonhomogeneous interactions, is computationally challenging.

Purpose of the Study:

  • To derive the exact finite-size partition function for the one-dimensional (1D) nonhomogeneous Ising model.
  • To explore potential recursive properties of the partition function and its relation to generalized Lucas polynomials.
  • To investigate the impact of quenched disorder on magnetic susceptibility.

Main Methods:

  • Utilizing algebraic operator approaches to compute the partition function.
  • Employing matrix representation of linear second-order recurrence relations with nonconstant coefficients.
  • Analyzing the relationship between the partition function and generalized Lucas polynomials for homogeneous cases.
  • Investigating quenched averaged magnetic susceptibility under disorder.

Main Results:

  • An exact finite-size partition function for the 1D nonhomogeneous Ising model was successfully derived.
  • A recursive formula for the partition function was established for the homogeneous model, linked to generalized Lucas polynomials.
  • The study suggests potential for recurrence relations in higher-dimensional Ising models.
  • Quenched disorder introduces nontrivial behavior in magnetic susceptibility due to altered ferromagnetic concentration probabilities.

Conclusions:

  • The algebraic operator method provides an effective means to calculate the exact partition function for finite-size 1D Ising models.
  • The discovered recursive property highlights a significant characteristic of the Ising model's partition function.
  • The findings offer insights into the influence of disorder on magnetic properties, with implications for statistical physics models.