Exact Calculation Of Partition Function Research Articles

Polynomial algorithm for exact calculation of partition function for binary spin model on planar graphs

In this paper we propose and realize an algorithm for exact calculation of partition function for planar graph models with binary variables. The complexity of the algorithm is O(N 2) Experiments show good agreement with Onsager’s analytical solution for the two-dimensional Ising model of infinite size.

Exactly solvable mixed-spin Ising-Heisenberg diamond chain with biquadratic interactions and single-ion anisotropy

An exactly solvable variant of mixed spin-(1/2,1) Ising-Heisenberg diamond chain is considered. Vertical spin-1 dimers are taken as quantum ones with Heisenberg bilinear and biquadratic interactions and with single-ion anisotropy, while all interactions between spin-1 and spin-1/2 residing on the intermediate sites are taken in the Ising form. The detailed analysis of the $T=0$ ground state phase diagram is presented. The phase diagrams have shown to be rather rich, demonstrating large variety of ground states: saturated one, three ferrimagnetic with magnetization equal to 3/5 and another four ferrimagnetic ground states with magnetization equal to 1/5. There are also two frustrated macroscopically degenerated ground states which could exist at zero magnetic filed. Solving the model exactly within classical transfer-matrix formalism we obtain an exact expressions for all thermodynamic function of the system. The thermodynamic properties of the model have been described exactly by exact calculation of partition function within the direct classical transfer-matrix formalism, the entries of transfer matrix, in their turn, contain the information about quantum states of vertical spin-1 XXZ dimer (eigenvalues of local hamiltonian for vertical link).

Exact Potts model partition functions on strips of the honeycomb lattice

We present exact calculations of the partition function of the $q$-state Potts model on (i) open, (ii) cyclic, and (iii) M\"obius strips of the honeycomb (brick) lattice of width $L_y=2$ and arbitrarily great length. In the infinite-length limit the thermodynamic properties are discussed. The continuous locus of singularities of the free energy is determined in the $q$ plane for fixed temperature and in the complex temperature plane for fixed $q$ values. We also give exact calculations of the zero-temperature partition function (chromatic polynomial) and $W(q)$, the exponent of the ground-state entropy, for the Potts antiferromagnet for honeycomb strips of type (iv) $L_y=3$, cyclic, (v) $L_y=3$, M\"obius, (vi) $L_y=4$, cylindrical, and (vii) $L_y=4$, open. In the infinite-length limit we calculate $W(q)$ and determine the continuous locus of points where it is nonanalytic. We show that our exact calculation of the entropy for the $L_y=4$ strip with cylindrical boundary conditions provides an extremely accurate approximation, to a few parts in $10^5$ for moderate $q$ values, to the entropy for the full 2D honeycomb lattice (where the latter is determined by Monte Carlo measurements since no exact analytic form is known).

EXACT PARTITION FUNCTION FOR THE POTTS MODEL WITH NEXT-NEAREST NEIGHBOR COUPLINGS ON ARBITRARY-LENGTH LADDERS

We present exact calculations of partition function Z of the q-state Potts model with next-nearest-neighbor spin–spin couplings, both for the ferromagnetic and antiferromagnetic case, for arbitrary temperature, on n-vertex ladders with free, cyclic, and Möbius longitudinal boundary conditions. The free energy is calculated exactly for the infinite-length limit of these strip graphs and the thermodynamics is discussed. Considering the full generalization to arbitrary complex q and temperature, we determine the singular locus ℬ in the corresponding [Formula: see text] space, arising as the accumulation set of partition function zeros as n → ∞.