Derivatives and inequalities for order parameters in the Ising spin glass

Identities and inequalities are proved for the order parameters, correlation functions and their derivatives of the Ising spin glass. The results serve as additional evidence that the ferromagnetic phase is composed of two regions, one with strong ferromagnetic ordering and the other with the effects of disorder dominant. The Nishimori line marks a crossover between these two regions.Comment: 10 pages; 3 figures; new inequalities added, title slightly change

Multicritical Points of Potts Spin Glasses on the Triangular Lattice

We predict the locations of several multicritical points of the Potts spin glass model on the triangular lattice. In particular, continuous multicritical lines, which consist of multicritical points, are obtained for two types of two-state Potts (i.e., Ising) spin glasses with two- and three-body interactions on the triangular lattice. These results provide us with numerous examples to further verify the validity of the conjecture, which has succeeded in deriving highly precise locations of multicritical points for several spin glass models. The technique, called the direct triangular duality, a variant of the ordinary duality transformation, directly relates the triangular lattice with its dual triangular lattice in conjunction with the replica method.Comment: 18 pages, 2, figure

Location of the Multicritical Point for the Ising Spin Glass on the Triangular and Hexagonal Lattices

A conjecture is given for the exact location of the multicritical point in the phase diagram of the +/- J Ising model on the triangular lattice. The result p_c=0.8358058 agrees well with a recent numerical estimate. From this value, it is possible to derive a comparable conjecture for the exact location of the multicritical point for the hexagonal lattice, p_c=0.9327041, again in excellent agreement with a numerical study. The method is a variant of duality transformation to relate the triangular lattice directly with its dual triangular lattice without recourse to the hexagonal lattice, in conjunction with the replica method.Comment: 9 pages, 1 figure; Minor corrections in notatio

Exact location of the multicritical point for finite-dimensional spin glasses: A conjecture

We present a conjecture on the exact location of the multicritical point in the phase diagram of spin glass models in finite dimensions. By generalizing our previous work, we combine duality and gauge symmetry for replicated random systems to derive formulas which make it possible to understand all the relevant available numerical results in a unified way. The method applies to non-self-dual lattices as well as to self dual cases, in the former case of which we derive a relation for a pair of values of multicritical points for mutually dual lattices. The examples include the +-J and Gaussian Ising spin glasses on the square, hexagonal and triangular lattices, the Potts and Z_q models with chiral randomness on these lattices, and the three-dimensional +-J Ising spin glass and the random plaquette gauge model.Comment: 27 pages, 3 figure

Symmetry, complexity and multicritical point of the two-dimensional spin glass

We analyze models of spin glasses on the two-dimensional square lattice by exploiting symmetry arguments. The replicated partition functions of the Ising and related spin glasses are shown to have many remarkable symmetry properties as functions of the edge Boltzmann factors. It is shown that the applications of homogeneous and Hadamard inverses to the edge Boltzmann matrix indicate reduced complexities when the elements of the matrix satisfy certain conditions, suggesting that the system has special simplicities under such conditions. Using these duality and symmetry arguments we present a conjecture on the exact location of the multicritical point in the phase diagram.Comment: 32 pages, 6 figures; a few typos corrected. To be published in J. Phys.

Duality and Multicritical Point of Two-Dimensional Spin Glasses

Determination of the precise location of the multicritical point and phase boundary is a target of active current research in the theory of spin glasses. In this short note we develop a duality argument to predict the location of the multicritical point and the shape of the phase boundary in models of spin glasses on the square lattice.Comment: 4 pages, 1 figure; Reference updated, definition of \tilde{V} added; to be published in J. Phys. Soc. Jp

Typical performance of low-density parity-check codes over general symmetric channels

Typical performance of low-density parity-check (LDPC) codes over a general binary-input output-symmetric memoryless channel is investigated using methods of statistical mechanics. Theoretical framework for dealing with general symmetric channels is provided, based on which Gallager and MacKay-Neal codes are studied as examples of LDPC codes. It has been shown that the basic properties of these codes known for particular channels, including the property to potentially saturate Shannon's limit, hold for general symmetric channels. The binary-input additive-white-Gaussian-noise channel and the binary-input Laplace channel are considered as specific channel noise models.Comment: 10 pages, 4 figures, RevTeX4; an error in reference correcte

Tracing the Evolution of Physics on the Backbone of Citation Networks

Many innovations are inspired by past ideas in a non-trivial way. Tracing these origins and identifying scientific branches is crucial for research inspirations. In this paper, we use citation relations to identify the descendant chart, i.e. the family tree of research papers. Unlike other spanning trees which focus on cost or distance minimization, we make use of the nature of citations and identify the most important parent for each publication, leading to a tree-like backbone of the citation network. Measures are introduced to validate the backbone as the descendant chart. We show that citation backbones can well characterize the hierarchical and fractal structure of scientific development, and lead to accurate classification of fields and sub-fields.Comment: 6 pages, 5 figure

Duality in finite-dimensional spin glasses

We present an analysis leading to a conjecture on the exact location of the multicritical point in the phase diagram of spin glasses in finite dimensions. The conjecture, in satisfactory agreement with a number of numerical results, was previously derived using an ansatz emerging from duality and the replica method. In the present paper we carefully examine the ansatz and reduce it to a hypothesis on analyticity of a function appearing in the duality relation. Thus the problem is now clearer than before from a mathematical point of view: The ansatz, somewhat arbitrarily introduced previously, has now been shown to be closely related to the analyticity of a well-defined function.Comment: 12 pages, 3 figures; A reference added; to appear in J. Stat. Phy

Gauge Theory for Quantum Spin Glasses

The gauge theory for random spin systems is extended to quantum spin glasses to derive a number of exact and/or rigorous results. The transverse Ising model and the quantum gauge glass are shown to be gauge invariant. For these models, an identity is proved that the expectation value of the gauge invariant operator in the ferromagnetic limit is equal to the one in the classical equilibrium state on the Nishimori line. As a result, a set of inequalities for the correlation function are proved, which restrict the location of the ordered phase. It is also proved that there is no long-range order in the two-dimensional quantum gauge glass in the ground state. The phase diagram for the quantum XY Mattis model is determined.Comment: 15 pages, 2 figure