New Algorithm for the Ising Problem: Partition Function for Finite Lattice Graphs

Galluccio, Anna; Loebl, Martin; Vondrák, Jan

doi:10.1103/physrevlett.84.5924

Cited by 66 publications

(105 citation statements)

References 18 publications

Supporting

Mentioning

102

Contrasting

Unclassified

Order By: Relevance

“…We consider a L × L lattice with given quenched couplings J ij , and we determine F (T ) and F DW (T ) via the exact computation of the partition function Z [17,18,19]. We can then average over different samples to obtain (up to some statistical errors) the correlation functions C O (L, T, ∆T ).…”

mentioning

confidence: 99%

See 1 more Smart Citation

Temperature chaos in two-dimensional Ising spin glasses with binary couplings: a further case for universality

Lukic¹,

Marinari²,

Martin³

et al. 2006

J. Stat. Mech.

View full text Add to dashboard Cite

We study temperature chaos in a two-dimensional Ising spin glass with random quenched bimodal couplings, by an exact computation of the partition functions on large systems. We study two temperature correlators from the total free energy and from the domain wall free energy: in the second case we detect a chaotic behavior. We determine and discuss the chaos exponent and the fractal dimension of the domain walls.PACS numbers: 75.10. Nr, 05.50.+q, 75.40.Gb, 75.40.Mg Introduction -A characteristic feature of spin glasses is the presence of chaos under small changes in the quenched couplings, in the temperature, or in the magnetic field [1,2,3,4,5,6,7,8,9]. With the expression temperature chaos we refer to the fragility of the equilibrium states of a disordered system under small temperature changes. Let us consider two typical equilibrium configurations of such a system under the same realization of the quenched disorder: the first configuration is in equilibrium at temperature T , while the second one is in equilibrium at temperature T ′ = T + ∆T . One says that there is temperature chaos if for arbitrarily small (but non-zero) values of ∆T , the typical overlap of two configurations at T and T ′ goes to zero when the system size diverges. The spatial distance ℓ(T, ∆T ) over which such overlaps decay is called the chaos length, and, as we will discuss better in the following, it scales as ℓ ∼ ∆T

show abstract

mentioning

confidence: 99%

“…Our study is based on exact partition function computations for large but finite size systems, for many realizations of the quenched disordered couplings [17,18]. Availability of the full density of states of a large system is a very powerful tool and it allows us to investigate the chaotic behavior of the free energy for the ±J model.…”

mentioning

confidence: 99%

Temperature chaos in two-dimensional Ising spin glasses with binary couplings: a further case for universality

Lukic¹,

Marinari²,

Martin³

et al. 2006

J. Stat. Mech.

View full text Add to dashboard Cite

show abstract

“…An exact calculation of the energy-entropy correlation for L = 50, or possibly L = 60, could be performed using the method of Galluccio, Loebl and Vondrák. 21 It should be noted, however, that it is not really necessary to calculate S 0 exactly. It would be more than sufficient to have an approximate calculation of S 0 which was accurate to one part in 10 4 .…”

Section: Discussionmentioning

confidence: 99%

Finite-Size Scaling in the Energy-Entropy Plane for the 2D ± Ising Spin Glass

Fisch¹

2006

J Stat Phys

View full text Add to dashboard Cite

For L × L square lattices with L ≤ 20 the 2D Ising spin glass with +1 and -1 bonds is found to have a strong correlation between the energy and the entropy of its ground states. A fit to the data gives the result that each additional broken bond in the ground state of a particular sample of random bonds increases the ground state degeneracy by approximately a factor of 10/3. For x = 0.5 (where x is the fraction of negative bonds), over this range of L, the characteristic entropy defined by the energy-entropy correlation scales with size as L 1.78(2) . Anomalous scaling is not found for the characteristic energy, which essentially scales as L 2 . When x = 0.25, a crossover to L 2 scaling of the entropy is seen near L = 12. The results found here suggest a natural mechanism for the unusual behavior of the low temperature specific heat of this model, and illustrate the dangers of extrapolating from small L.

show abstract

“…While our algorithm has superior speed to that of Ref. [5] only near the percolation threshold, we believe there are advantages to a transparent algorithm which operates directly in the spin representation. In addition, our method is highly parallelizable, and can execute in as little as O L time if a sufficient number of nodes are available.…”

mentioning

confidence: 99%

“…The algorithm requires O L 3 time to compute the partition function Z T of an L L square lattice at a single temperature T; for bond-diluted problems near the percolation threshold it requires only O L 2 ln L time, which is the fastest method to our knowledge in this case [2]. In comparison, the fermion network method takes O L 4 time [3], spin-basis transfer matrix methods [4] take O 2 L time, and the Pfaffian method with nested dissection takes O L 3 time [5,6]. While our algorithm has superior speed to that of Ref.…”

mentioning

confidence: 99%

Efficient Algorithm for Random-Bond Ising Models in 2D

Loh

Carlson

2006

Phys. Rev. Lett.

View full text Add to dashboard Cite

We present an efficient algorithm for calculating the properties of Ising models in two dimensions, directly in the spin basis, without the need for mapping to fermion or dimer models. The algorithm gives numerically exact results for the partition function and correlation functions at a single temperature on any planar network of N Ising spins in O(N 3/2 ) time or less. The method can handle continuous or discrete bond disorder and is especially efficient in the case of bond or site dilution, where it executes in O(L 2 lnL) time near the percolation threshold. We demonstrate its feasibility on the ferromagnetic Ising model and the ±J random-bond Ising model (RBIM) and discuss the regime of applicability in cases of full frustration such as the Ising antiferromagnet on a triangular lattice.

show abstract

New Algorithm for the Ising Problem: Partition Function for Finite Lattice Graphs

Cited by 66 publications

References 18 publications

Temperature chaos in two-dimensional Ising spin glasses with binary couplings: a further case for universality

Temperature chaos in two-dimensional Ising spin glasses with binary couplings: a further case for universality

Finite-Size Scaling in the Energy-Entropy Plane for the 2D ± Ising Spin Glass

Efficient Algorithm for Random-Bond Ising Models in 2D

Contact Info

Product

Resources

About