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Exact partition function of the Potts model on the Sierpinski gasket and the Hanoi lattice

Pedro D. Alvarez

Abstract

We present an analytic study of the Potts model partition function on the Sierpinski and Hanoi lattices, which are self-similar lattices of triangular shape with non integer Hausdorff dimension. Both lattices are examples of non-trivial thermodynamics in less than two dimensions, where mean field theory does not apply. We used and explain a method based on ideas of graph theory and renormalization group theory to derive exact equations for appropriate variables that are similar to the restricted partition functions. We benchmark our method with Metropolis Monte Carlo simulations. The analysis of fixed points reveals information of location of the Fisher zeros and we provide a conjecture about the location of zeros in terms of the boundary of the basins of attraction.

Exact partition function of the Potts model on the Sierpinski gasket and the Hanoi lattice

Abstract

We present an analytic study of the Potts model partition function on the Sierpinski and Hanoi lattices, which are self-similar lattices of triangular shape with non integer Hausdorff dimension. Both lattices are examples of non-trivial thermodynamics in less than two dimensions, where mean field theory does not apply. We used and explain a method based on ideas of graph theory and renormalization group theory to derive exact equations for appropriate variables that are similar to the restricted partition functions. We benchmark our method with Metropolis Monte Carlo simulations. The analysis of fixed points reveals information of location of the Fisher zeros and we provide a conjecture about the location of zeros in terms of the boundary of the basins of attraction.
Paper Structure (11 sections, 48 equations, 13 figures, 6 tables)

This paper contains 11 sections, 48 equations, 13 figures, 6 tables.

Table of Contents

  1. The Sierpinski gasket
  2. Dichromatic Polynomial and the Partition function
  3. Connectivity coefficients
  4. Dynamical equations
  5. Dynamical equations on Hanoi graph
  6. Thermodynamic functions
  7. Fixed points
  8. Fixed points on the Hanoi lattice
  9. Changing the lattice by changing the initial data
  10. Locci of the zeros of the partition function
  11. Conclusions

Figures (13)

  • Figure 1: Sierpinski iteration procedure. $T(1)$ is a lattice with $V=3$ and $E=3$, $T(2)$ is a lattice with $V=6$ and $E=9$, $T(3)$ is a lattice with $V=15$ and $E=27$ and so on. In general, the number of vertices at step $n$ is $V_n=(3^n+3)/2$ while the number of edges is $E_n=3^n$. The number of internal (bulk) sites of $T(n)$ is $(3^n-3)/2$ and the number of external sites is always $3$.
  • Figure 2: Hanoi iteration procedure. $T(1)$ is a lattice with $V=3$ and $E=3$, $T(2)$ is a lattice with $V=9$ and $E=12$, $T(3)$ is a lattice with $V=27$ and $E=39$ and so on. In general, the number of vertices at step $n$ is $V_n=3^n$ while the number of edges is $E_n=(3/2) (3^n - 1)$.
  • Figure 3: Diagrammatic illustration of the base used in eq. (\ref{['connectiondefinition1nosymm']}). The double line means that such pair of vertex are idenfied, according to the second term in (\ref{['dp1']}).
  • Figure 4: Sierpinski composition of three graphs $A \wedge B \wedge C$. The products summarized in Table \ref{['connpattable']}are understood according to this diagram.
  • Figure 5: "Hanoi" composition of three graphs $A \wedge B \wedge C$. The products summarized in Table \ref{['connpattablehanoi']} are understood according to this diagram.
  • ...and 8 more figures