We study the statistical properties of an ensemble of disordered 1D spatial spin-chains (SSCs) of certain length in the external field. On nodes of spin-chain lattice the recurrent equations and corresponding inequal-ity conditions are obtained for calculation of local minimum of a classical Hamiltonian. Using these equa-tions for simulation of a model of 1D spin-glass an original high-performance parallel algorithm is developed. Distributions of different parameters of unperturbed spin-glass are calculated. It is analytically proved and shown by numerical calculations that the distribution of the spin-spin interaction constant in the Heisenberg nearest-neighboring Hamiltonian model as opposed to the widely used Gauss-Edwards-Anderson distribu-tion satisfies the Lévy alpha-stable distribution law which does not have variance. We have studied critical properties of spin-glass depending on the external field amplitude and have shown that even at weak external fields in the system strong frustrations arise. It is shown that frustrations have a fractal character, they are self-similar and do not disappear at decreasing of calculations area scale. After averaging over the fractal structures the mean values of polarizations of the spin-glass on the scales of external field's space-time peri-ods are obtained. Similarly, Edwards-Anderson’s ordering parameter depending on the external field ampli-tude is calculated. It is shown that the mean values of polarizations and the ordering parameter depending on the external field demonstrate phase transitions of first-order.
We study the statistical properties of an ensemble of disordered 1D spatial spin-chains (SSCs) of certain length in the external field. On nodes of spin-chain lattice the recurrent equations and corresponding inequal-ity conditions are obtained for calculation of local minimum of a classical Hamiltonian. Using these equa-tions for simulation of a model of 1D spin-glass an original high-performance parallel algorithm is developed. Distributions of different parameters of unperturbed spin-glass are calculated. It is analytically proved and shown by numerical calculations that the distribution of the spin-spin interaction constant in the Heisenberg nearest-neighboring Hamiltonian model as opposed to the widely used Gauss-Edwards-Anderson distribu-tion satisfies the Lévy alpha-stable distribution law which does not have variance. We have studied critical properties of spin-glass depending on the external field amplitude and have shown that even at weak external fields in the system strong frustrations arise. It is shown that frustrations have a fractal character, they are self-similar and do not disappear at decreasing of calculations area scale. After averaging over the fractal structures the mean values of polarizations of the spin-glass on the scales of external field's space-time peri-ods are obtained. Similarly, Edwards-Anderson’s ordering parameter depending on the external field ampli-tude is calculated. It is shown that the mean values of polarizations and the ordering parameter depending on the external field demonstrate phase transitions of first-order.
We study statistical properties of 3D classical spin glass layer of certain width and infinite length. The 3D spin glass is represented as an ensemble of disordered 1D spatial spin-chains (SSC) where interactions are random between spin-chains (nonideal ensemble of 1D SSCs). It is proved that at the limit of Birkhoff's ergodic hypothesis performance 3D spin glasses can be generated by Hamiltonian of disordered 1D SSC with random environment. Disordered 1D SSC is defined on a regular lattice where one randomly oriented spin is put on each node of lattice. Also it is supposed that each spin randomly interacts with six nearest-neighboring spins (two spins on lattice and four in the environment). The recurrent transcendental equations are obtained on the nodes of spin-chain lattice. These equations combined with the Silvester conditions allow step by step construct spin-chain in the ground state of energy where all spins are in minimal energy of classical Hamiltonian. On the basis of these equations an original high-performance parallel algorithm is developed for 3D spin glasses simulation. Distributions of different parameters of unperturbed spin glass are calculated. In particular, it is analytically proved and by numerical calculations shown that the distribution of spin-spin interaction constant in Heisenberg nearest-neighboring Hamiltonian model as opposed to widely used Gauss-Edwards- Anderson distribution satisfies Lévy alpha-stable distribution law which does not have variance. A new formula is proposed for construction of partition function in kind of one-dimensional integral on energy distribution of 1D SSCs
We have investigated the statistical properties of an ensemble of disordered 1D spatial spin chains (SSCs) of finite length, placed in an external field, with consideration of relaxation effects. The short-range interaction complex-classical Hamiltonian was first used for solving this problem. A system of recurrent equations is obtained on the nodes of the spin-chain lattice. An efficient mathematical algorithm is developed on the basis of these equations with consideration of advanced Sylvester conditions which allows one to step by step construct a huge number of stable spin chains in parallel. The distribution functions of different parameters of spin glass system are constructed from first principles by analyzing the calculation results of the 1D SSCs ensemble. It is shown that the behaviors of different distributions parameters are quite different even at weak external fields. The ensemble energy and constants of spin-spin interactions are being changed smoothly depending on the external field in the limit of statistical equilibrium, while some of them such as the mean value of polarizations of the ensemble and parameters of its orderings are frustrated. We have also studied some critical properties of the ensemble such as catastrophes in the Clausius-Mossotti equation depending on the value of the external field. We have shown that the generalized complex-classical approach excludes these catastrophes, which allows one to organize continuous parallel computing on the whole region of values of the external field including critical points. A new representation of the partition function is suggested based on these investigations. Being opposite to the usual definition, it is a complex function and its derivatives are everywhere defined, including at critical points.
We study the classical 1D Heisenberg spin glasses in the framework of nearest-neighboring model. Based on the Hamilton equations we obtained the system of recurrence equations which allows to perform node-by-node calculations of a spin-chain. It is shown that calculations from the first principles of classical mechanics lead to ℕℙ hard problem, that however in the limit of the statistical equilibrium can be calculated by ℙ algorithm. For the partition function of the ensemble a new representation is offered in the form of one-dimensional integral of spin-chains’ energy distribution.
For the spin-glass chain in an external field h, a nonzero weight at the origin of the bond distribution $\ensuremath{\rho}(J)$ is known to induce a nonanalytical magnetization at zero temperature; for $\ensuremath{\rho}(J)\ensuremath{\sim}A{\ensuremath{\mid}J\ensuremath{\mid}}^{\ensuremath{\mu}\ensuremath{-}1}$ near $J\ensuremath{\rightarrow}0$, the magnetization follows the Chen-Ma scaling $M\ensuremath{\sim}{h}^{\ensuremath{\mu}∕(2+\ensuremath{\mu})}$. In this paper, we numerically revisit this model to obtain detailed statistical information on the ground-state configuration and on the low-energy two-level excitations that govern the low-temperature properties. The ground state consists of long unfrustrated intervals separated by weak frustrated bonds. We accordingly compute the strength distribution of these frustrated bonds, as well as the length and magnetization distributions of the unfrustrated intervals. We find that the low-energy excitations are of two types: (i) one frustrated bond of the ground state may have two positions that are nearly degenerate in energy and (ii) two neighboring frustrated bonds of the ground state may be annihilated or created with nearly zero energy cost. For each excitation type, we compute its probability density as a function of its length. Moreover, we show that the contributions of these excitations to various observables (specific heat, Edwards-Anderson order parameter, susceptibility) are in full agreement with the direct transfer matrix evaluations at low temperature. Finally, following the special bimodal case $\ifmmode\pm\else\textpm\fi{}J$, where a Ma-Dasgupta, RG procedure has been previously used to compute explicitly the above observables, we discuss the possibility of an extended RG procedure. We find that the ground state can be seen as the result of a hierarchical ``fragmentation'' procedure that we describe.
Abstract. Numerical results for the local field distributions of a family of Ising spinglass models are presented. In particular, the Edwards-Anderson model in dimensions two, three, and four is considered, as well as spin glasses with long-range powerlaw-modulated interactions that interpolate between a nearest-neighbour Edwards-Anderson system in one dimension and the infinite-range Sherrington-Kirkpatrick model. Remarkably, the local field distributions only depend weakly on the range of the interactions and the dimensionality, and show strong similarities except for near zero local field. PACS numbers: 75.50.Lk, 75.40.Mg, 05.50.+qLocal field distributions in spin glasses 2 1.
The main subjects of this dissertation are spin glass applications in other disciplines and spin glass algorithms. Spin glasses are magnetic systems with disorder and frustration, and the essential physics of spin glasses lies not in the details of their microscopic interactions but rather in the competition between quenched ferromagnetic and antiferromagnetic interactions. Concepts that arose in the study of spin glasses have led to applications in areas as diverse as computer science, biology, and finance, as well as a variety of others.\n\nIn the first part of this dissertation I study the equilibrium and non-equilibrium properties of Boolean decision problems with competing interactions on scale-free networks in an external bias (a magnetic field). First, I perform finite-temperature Monte Carlo simulations in a field to test the robustness of the spin-glass phase and I show that the system has a spin-glass phase in a field, i.e., it exhibits a de Almeida–Thouless line. Then I study avalanche distributions when the system is driven by a field at zero temperature to test whether the system displays self-organized criticality. The numerical results suggest that avalanches (damage) can spread across the entire system with nonzero probability when the decay exponent of the interaction degree is less than or equal to 2, i.e., that Boolean decision problems on scale-free networks with competing interactions can be fragile when the system is not in thermal equilibrium.\n\nIn the second part of this dissertation I discuss the best-case performance of quantum annealers on native spin-glass benchmarks, i.e., how chaos can affect success probabilities. We perform classical parallel-tempering Monte Carlo simulations of the archetypal benchmark problem, an Ising spin glass, on the native chip topology. Using realistic uncorrelated noise models for the D-Wave Two quantum annealer, I study the best-case resilience, or the probability that the ground-state configuration is not affected by random fields and random-bond fluctuations found on the chip. We compute the upper-bound success probabilities for different instance classes based on these simple error models, and I present strategies for developing robust and hard benchmark instances.\n\nIn the third part of this dissertation I present a cluster algorithm for Ising spin glasses that works in any space dimension and speeds up thermalization by several orders of magnitude at temperatures where thermalization is typically difficult. Our isoenergetic cluster moves are based on the Houdayer cluster algorithm for two-dimensional spin glasses and lead to a speedup over conventional state-of-the-art methods that increases with the system size. We illustrate the benefits (improved thermalization and achievement of more equiprobable sampling of ground states) of the isoenergetic cluster moves in two and three space dimensions, as well as in the nonplanar Chimera topology found in the D-Wave quantum annealing machine.\n\nFinally, I study the thermodynamic properties of the two-dimensional Edwards-Anderson Ising spin-glass model on a square lattice using the tensor renormalization group method based on a higher-order singular-value decomposition. Our estimates of the partition function without a high precision data type lead to negative values at very low temperatures, thus illustrating that the method can not be applied to frustrated magnetic systems.