Concavity of Magnetization of an Ising Ferromagnet in a Positive External Field
01 Mar 1970-Journal of Mathematical Physics (American Institute of Physics)-Vol. 11, Iss: 3, pp 790-795
TL;DR: In this paper, an inequality for correlation functions in an Ising model with purely ferromagnetic interactions between pairs of spins is established and used to show that the magnetization in such a model is a concave function of external field H for H > 0.
Abstract: An inequality for correlation functions in an Ising model with purely ferromagnetic interactions between pairs of spins is established and used to show that the magnetization in such a model is a concave function of external field H for H > 0. The concavity of magnetization, which holds not only for spin‐½ but also for arbitrary‐spin Ising ferromagnets, provides a basis for certain thermodynamic inequalities near the ferromagnetic critical point, including one involving the ``high temperature'' indices α and γ.
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TL;DR: In this paper, the equality of two critical points -the percolation threshold pH and the point pτ where the cluster size distribution ceases to decay exponentially -is proven for all translation invariant independent per-colation models on homogeneous d-dimensional lattices.
Abstract: The equality of two critical points - the percolation threshold pH and the point pτ where the cluster size distribution ceases to decay exponentially - is proven for all translation invariant independent percolation models on homogeneous d-dimensional lattices (d^ 1). The analysis is based on a pair of new nonlinear partial differential inequalities for an order parameter M(β, h\ which for h = Q reduces to the percolation density P^ - at the bond density p = l—e~β in the single parameter case. These are: (1) M^hdM/dh + M2 + βMdM/dβ, and (2) dM/dβ^\J\MdM/dh. Inequality (1) is intriguing in that its derivation provides yet another hint of a "φ3 structure" in percolation models. Moreover, through the elimination of one of its derivatives, (1) yields a pair of ordinary differential inequalities which provide information on the critical exponents β and δ. One of these resembles an Ising model inequality of Frόhlich and Sokal and yields the mean field bound (5^2, and the other implies the result of Chayes and Chayes that β^ί. An inequality identical to (2) is known for Ising models, where it provides the basis for Newman's universal relation /?((5 —1)^1 and for certain extrapolation principles, which are now made applicable also to independent percolation. These results apply to both finite and long range models, with or without orientation, and extend to periodic and weakly inhomogeneous systems.
457 citations
Cites methods from "Concavity of Magnetization of an Is..."
...14) is identical to one obeyed by the magnetization in ferromagnetic Ising spin systems, for which it follows from the GriffithsHurst-Sherman inequality [10]....
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TL;DR: In this article, it was shown that the φ4 Euclidean field theory with a lattice cut-off is inevitably free in the single phase regime in ind>4 dimensions, and that the critical behavior in Ising models is in exact agreement with the mean-field approximation in high dimensions, but not in the low dimensiond=2.
Abstract: We provide here the details of the proof, announced in [1], that ind>4 dimensions the (even) φ4 Euclidean field theory, with a lattice cut-off, is inevitably free in the continuum limit (in the single phase regime). The analysis is nonperturbative, and is based on a representation of the field variables (or spins in Ising systems) as source/sink creation operators in a system of random currents — which may be viewed as the mediators of correlations. In this dual representation, the onset of long-range-order is attributed to percolation in an ensemble of sourceless currents, and the physical interaction in the φ4 field — and other aspects of the critical behavior in Ising models — are directly related to the intersection properties of long current clusters. An insight into the criticality of the dimensiond=4 is derived from an analogy (foreseen by K. Symanzik) with the intersection properties of paths of Brownian motion. Other results include the proof that in certain respect, the critical behavior in Ising models is in exact agreement with the mean-field approximation in high dimensionsd>4, but not in the low dimensiond=2 — for which we establish the “universality” of hyperscaling.
393 citations
23 Nov 2017
TL;DR: In this paper, the authors give a friendly, rigorous introduction to fundamental concepts in equilibrium statistical mechanics, covering a selection of specific models, including the Curie-Weiss and Ising models, the Gaussian free field, O(n) models, and models with Kac interactions.
Abstract: This motivating textbook gives a friendly, rigorous introduction to fundamental concepts in equilibrium statistical mechanics, covering a selection of specific models, including the Curie–Weiss and Ising models, the Gaussian free field, O(n) models, and models with Kac interactions. Using classical concepts such as Gibbs measures, pressure, free energy, and entropy, the book exposes the main features of the classical description of large systems in equilibrium, in particular the central problem of phase transitions. It treats such important topics as the Peierls argument, the Dobrushin uniqueness, Mermin–Wagner and Lee–Yang theorems, and develops from scratch such workhorses as correlation inequalities, the cluster expansion, Pirogov–Sinai Theory, and reflection positivity. Written as a self-contained course for advanced undergraduate or beginning graduate students, the detailed explanations, large collection of exercises (with solutions), and appendix of mathematical results and concepts also make it a handy reference for researchers in related areas.
383 citations
Book•
01 Jan 1992
TL;DR: In this article, the authors used random walk representations as a tool to derive correlation inequalities for critical-exponent theory and the consequences of these inequalities for the theory of critical phenomena and quantum field theory.
Abstract: The subject of this book is equilibrium statistical mechanics, in particular the theory of critical phenomena, and quantum field theory. The central theme is the use of random-walk representations as a tool to derive correlation inequalities. The consequences of these inequalities for critical-exponent theory are expounded in detail. The book contains some previously unpublished results. It addresses both the researcher and the graduate student in modern statistical mechanics and quantum field theory.
350 citations
Cites background from "Concavity of Magnetization of an Is..."
...14) is the Lebowitz inequality [279, 362, 491, 162, 92, 97, 5], which has had numerous applications in quantum field theory and statistical mechanics....
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...The only general result available for the truncated three-point function is the GHS inequality [279, 491, 162] 〈φx; φy; φz〉 ≤ 0 , (12....
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TL;DR: In this article, a unified approach to many recently established correlation inequalities was presented, and a simple proof of the mass gap for the λ(φ4)2 quantum field model was obtained.
Abstract: Ferromagnetic lattice spin systems can be expressed as gases of random walks interacting via a soft core repulsion. By using a mixed spinrandom walk representation we present a unified approach to many recently established correlation inequalities. As an application of these inequalities we obtain a simple proof of the mass gap for the λ(φ4)2 quantum field model. We also establish new upper bounds on critical temperatures.
316 citations
Cites background from "Concavity of Magnetization of an Is..."
...2) is, of course, the well-known Lebowitz inequality [25-29, 1,3]....
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References
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TL;DR: The theory of critical phenomena in systems at equilibrium is reviewed at an introductory level with special emphasis on the values of the critical point exponents α, β, γ,..., and their interrelations as mentioned in this paper.
Abstract: The theory of critical phenomena in systems at equilibrium is reviewed at an introductory level with special emphasis on the values of the critical point exponents α, β, γ,..., and their interrelations. The experimental observations are surveyed and the analogies between different physical systems - fluids, magnets, superfluids, binary alloys, etc. - are developed phenomenologically. An exact theoretical basis for the analogies follows from the equivalence between classical and quantal `lattice gases' and the Ising and Heisenberg-Ising magnetic models. General rigorous inequalities for critical exponents at and below Tc are derived. The nature and validity of the `classical' (phenomenological and mean field) theories are discussed, their predictions being contrasted with the exact results for plane Ising models, which are summarized concisely. Pade approximant and ratio techniques applied to appropriate series expansions lead to precise critical-point estimates for the three-dimensional Heisenberg and Ising models (tables of data are presented). With this background a critique is presented of recent theoretical ideas: namely, the `droplet' picture of the critical point and the `homogeneity' and `scaling' hypotheses. These lead to a `law of corresponding states' near a critical point and to relations between the various exponents which suggest that perhaps only two or three exponents might be algebraically independent for any system.
1,792 citations
Book•
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01 Jan 1962
TL;DR: In this article, the axiom of choice of choice is used to define connectedness path problems in directed graphs and cyclic graphs, as well as Galois correspondences of connectedness paths.
Abstract: Fundamental concepts Connectedness Path problems Trees Leaves and lobes The axiom of choice Matching theorems Directed graphs Acyclic graphs Partial order Binary relations and Galois correspondences Connecting paths Dominating sets, covering sets and independent sets Chromatic graphs Groups and graphs Bibliography List of concepts Index of names.
1,732 citations
TL;DR: In this paper, an inequality relating binary correlation functions for an Ising model with purely ferromagnetic interactions is derived by elementary arguments and used to show that such a ferromagnet cannot exhibit a spontaneous magnetization at temperatures above the mean-field approximation to the Curie or critical point.
Abstract: An inequality relating binary correlation functions for an Ising model with purely ferromagnetic interactions is derived by elementary arguments and used to show that such a ferromagnet cannot exhibit a spontaneous magnetization at temperatures above the mean-field approximation to the Curie or “critical” point. (As a consequence, the corresponding “lattice gas” cannot undergo a first order phase transition in density (condensation) above this temperature.) The mean-field susceptibility in zero magnetic field at high temperatures is shown to be an upper bound for the exact result.
445 citations
289 citations
TL;DR: In this article, it was shown that Griffiths' result widens the conclusion due to Griffiths by weakening the hypothesis and widening the conclusion of Griffiths's result.
Abstract: Let N = (1, 2, ⋯, n). For each subset A of N, let JA ≥ 0. For eachi∈N, let σi ± 1. For each subset A of N, define σA=∏i∈A σi. Let the Hamiltonian be − ΣACN JA σA. Then for each A, B⊂N, 〈σA〉≥0 and 〈σAσB〉−〈σA〉〈σB〉≥0. This weakens the hypothesis and widens the conclusion of a result due to Griffiths.
251 citations