Ground States of Two-Dimensional Ising Spin Glasses: Fast Algorithms, Recent Developments and a Ferromagnet-Spin Glass Mixture

Hartmann, Alexander K.

doi:10.1007/s10955-011-0272-1

Cited by 26 publications

(21 citation statements)

References 75 publications

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“…This part of the integral can be expressed in polar coordinates as ǫ 2 0 2π 0 ρ(r cos θ)ρ(r sin θ)rdrdθ. By substituting from (5) it can be seen easily that the angular part is convergent and the radial part is proportional to ǫ 2(1+µ) . Note that this is actually the probability of having two weak bonds that we discussed above for general w. Nonsingular part of the integral is of order ǫ.…”

Section: Analytical Results For Weak Magnetic Field: Chen-ma Scalingmentioning

confidence: 97%

Ground-state magnetization of the Ising spin glass: A recursive numerical method and Chen-Ma scaling

Sepehrinia

Chalangari

2018

Phys. Rev. B

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The ground-state properties of quasi-one-dimensional (Q1D) Ising spin glass are investigated using an exact numerical approach and analytical arguments. A set of coupled recursive equations for the ground-state energy are introduced and solved numerically. For various types of coupling distribution, we obtain accurate results for magnetization, particularly in the presence of a weak external magnetic field. We show that in the weak magnetic field limit, similar to the 1D model, magnetization exhibits a singular power-law behavior with divergent susceptibility. Remarkably, the spectrum of magnetic exponents is markedly different from that of the 1D system even in the case of two coupled chains. The magnetic exponent makes a crossover from being dependent on the distribution function to a constant value independent of distribution. We provide an analytic theory for these observations by extending the Chen-Ma argument to the Q1D case. We derive an analytical formula for the exponent which is in perfect agreement with the numerical results.

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Section: Analytical Results For Weak Magnetic Field: Chen-ma Scalingmentioning

confidence: 97%

Ground-state magnetization of the Ising spin glass: A recursive numerical method and Chen-Ma scaling

Sepehrinia

Chalangari

2018

Phys. Rev. B

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“…By using this approach, large planar spin glasses of up to

spins can be treated [ 14 ]. In general, the application of this and similar fast GS algorithms [ 16 ] have been successfully used to clarify the nature of GSs of two-dimensional spin glasses [ 13 , 14 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 ].…”

Section: Spin Glasses Ground States and Domain Wallsmentioning

confidence: 99%

From Spin Glasses to Negative-Weight Percolation

Hartmann

Melchert

Norrenbrock

2019

Entropy

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Spin glasses are prototypical random systems modelling magnetic alloys. One important way to investigate spin glass models is to study domain walls. For two dimensions, this can be algorithmically understood as the calculation of a shortest path, which allows for negative distances or weights. This led to the creation of the negative weight percolation (NWP) model, which is presented here along with all necessary basics from spin glasses, graph theory and corresponding algorithms. The algorithmic approach involves a mapping to the classical matching problem for graphs. In addition, a summary of results is given, which were obtained during the past decade. This includes the study of percolation transitions in dimension from d = 2 up to and beyond the upper critical dimension d u = 6 , also for random graphs. It is shown that NWP is in a different universality class than standard percolation. Furthermore, the question of whether NWP exhibits properties of Stochastic–Loewner Evolution is addressed and recent results for directed NWP are presented.

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“…In effect, frustration rules out a GS in which all the bonds are satisfied. As limiting cases one can identify the random bond Ising ferromagnet (FM) as µ → ∞ and the (Gaussian) 2D Edwards Anderson Ising spin glass (SG) [11][12][13] at µ = 0. A GS spin configuration σ GS is simply a minimizer of the energy function Eq.…”

Section: A 2d Random Bond Ising Modelmentioning

confidence: 99%

Information-theoretic approach to ground-state phase transitions for two- and three-dimensional frustrated spin systems

Melchert

Hartmann

2013

Phys. Rev. E

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The information-theoretic observables entropy (a measure of disorder), excess entropy (a measure of complexity), and multi-information are used to analyze ground-state spin configurations for disordered and frustrated model systems in two and three dimensions. For both model systems, ground-state spin configurations can be obtained in polynomial time via exact combinatorial optimization algorithms, which allowed us to study large systems with high numerical accuracy. Both model systems exhibit a continuous transition from an ordered to a disordered ground state as a model parameter is varied. By using the above information-theoretic observables it is possible to detect changes in the spatial structure of the ground states as the critical point is approached. It is further possible to quantify the scaling behavior of the information-theoretic observables in the vicinity of the critical point. For both model systems considered, the estimates of critical properties for the ground-state phase transitions are in good agreement with existing results reported in the literature.

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Ground States of Two-Dimensional Ising Spin Glasses: Fast Algorithms, Recent Developments and a Ferromagnet-Spin Glass Mixture

Cited by 26 publications

References 75 publications

Ground-state magnetization of the Ising spin glass: A recursive numerical method and Chen-Ma scaling

Ground-state magnetization of the Ising spin glass: A recursive numerical method and Chen-Ma scaling

From Spin Glasses to Negative-Weight Percolation

Information-theoretic approach to ground-state phase transitions for two- and three-dimensional frustrated spin systems

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