scholarly journals Exact Solution of a One-dimensional Spin System

1970 ◽  
Vol 23 (5) ◽  
pp. 927 ◽  
Author(s):  
RW Gibberd
Keyword(s):  
Phase Transition ◽  
Exact Solution ◽  
Free Energy ◽  
Partition Function ◽  
Thermodynamic Limit ◽  
Spin System ◽  
Spin Systems ◽  
One Dimensional ◽  
Gibb’S Free Energy ◽  
Theory Of Superconductivity

The partition function and the Gibb's free energy are calculated exactly in the thermodynamic limit, using techniques which are well known in the theory of superconductivity. This calculation illustrates explicitly the similarity between the phase transition in superconductivity and the molecular field transitions in spin systems.

scholarly journals Contraction: A Unified Perspective of Correlation Decay and Zero-Freeness of 2-Spin Systems

2021 ◽  
Vol 185 (2) ◽  
Author(s):  
Shuai Shao ◽  
Yuxin Sun
Keyword(s):  
Partition Function ◽  
Approximation Algorithms ◽  
External Field ◽  
Spin System ◽  
Spin Systems ◽  
Interaction Parameters ◽  
Entire Family ◽  
Bounded Degree ◽  
Correlation Decay ◽  
Complex Valued

AbstractWe study the connection between the correlation decay property (more precisely, strong spatial mixing) and the zero-freeness of the partition function of 2-spin systems on graphs of bounded degree. We show that for 2-spin systems on an entire family of graphs of a given bounded degree, the contraction property that ensures correlation decay exists for certain real parameters implies the zero-freeness of the partition function and the existence of correlation decay for some corresponding complex neighborhoods. Based on this connection, we are able to extend any real parameter of which the 2-spin system on graphs of bounded degree exhibits correlation decay to its complex neighborhood where the partition function is zero-free and correlation decay still exists. We give new zero-free regions in which the edge interaction parameters and the uniform external field are all complex-valued, and we show the existence of correlation decay for such complex regions. As a consequence, we obtain approximation algorithms for computing the partition function of 2-spin systems on graphs of bounded degree for these complex parameter settings.

3-D SPIN GLASSES AND ISING MODELS NEW METHODS ON NEW ARCHITECTURES

1993 ◽  
Vol 04 (01) ◽  
pp. 217-221
Author(s):  
GYAN BHANOT
Keyword(s):  
Phase Transition ◽  
Free Energy ◽  
Ising Model ◽  
Thermodynamic Limit ◽  
Spin Glasses ◽  
Weak Coupling ◽  
Ising Models ◽  
Parallel Architectures ◽  
Strong Indication ◽  
New Methods

I describe work on 3-d Spin Glasses and the 3-d Ising Model done in collaboration with Michael Creutz at BNL and Jan Lacki at IAS Princeton. We have developed novel techniques to study these systems that make use of parallel architectures. For 3-d spin glasses, our results give strong indication that there is no phase transition in the thermodynamic limit whereas for the Ising model, we are able to extend the weak coupling expansion of the average free energy to 50 excited bonds.

scholarly journals Fast Algorithms for General Spin Systems on Bipartite Expanders

2021 ◽  
Vol 13 (4) ◽  
pp. 1-18
Author(s):  
Andreas Galanis ◽  
Leslie Ann Goldberg ◽  
James Stewart
Keyword(s):  
Partition Function ◽  
Spin System ◽  
Weighted Graph ◽  
Arbitrary Spin ◽  
Spin Assignment ◽  
Spin Systems ◽  
Regular Graphs ◽  
Canonical Class ◽  
Finite Set ◽  
Polymer Models

A spin system is a framework in which the vertices of a graph are assigned spins from a finite set. The interactions between neighbouring spins give rise to weights, so a spin assignment can also be viewed as a weighted graph homomorphism. The problem of approximating the partition function (the aggregate weight of spin assignments) or of sampling from the resulting probability distribution is typically intractable for general graphs. In this work, we consider arbitrary spin systems on bipartite expander Δ-regular graphs, including the canonical class of bipartite random Δ-regular graphs. We develop fast approximate sampling and counting algorithms for general spin systems whenever the degree and the spectral gap of the graph are sufficiently large. Roughly, this guarantees that the spin system is in the so-called low-temperature regime. Our approach generalises the techniques of Jenssen et al. and Chen et al. by showing that typical configurations on bipartite expanders correspond to “bicliques” of the spin system; then, using suitable polymer models, we show how to sample such configurations and approximate the partition function in Õ( n 2 ) time, where n is the size of the graph.

scholarly journals Approximation of the n-vicinity method

2019 ◽  
Vol 487 (3) ◽  
pp. 246-251
Author(s):  
B. V. Kryzhanovsky
Keyword(s):  
Free Energy ◽  
Small Parameter ◽  
Spin System ◽  
Nearest Neighbors ◽  
Spin Systems

We determined a small parameter that determines the possibility of using the n-vicinity 06_method to calculate the free energy of a spin system, and found the types of spin systems for which this method is applicable. It is shown that this method is applicable for the analysis of spin systems, where the number of nearest neighbors is greater than 16/3.

Phase transition to blob-hole coherent structure in the Hasegawa–Mima model for plasmas

2021 ◽  
Vol 87 (6) ◽  
Author(s):  
Chjan C. Lim
Keyword(s):  
Phase Transition ◽  
Free Energy ◽  
Critical Temperature ◽  
Partition Function ◽  
Open System ◽  
Ground State ◽  
Spherical Model ◽  
Toroidal Plasmas ◽  
Fixed Energy ◽  
System A

An equilibrium statistical mechanics theory for the Hasegawa–Mima equations of toroidal plasmas, with canonical constraint on energy and microcanonical constraint on potential enstrophy, is solved exactly as a spherical model. The use of a canonical energy constraint instead of a fixed-energy microcanonical approach is justified by the preference for viewing real plasmas as an open system. A significant consequence of the results obtained from the partition function, free energy and critical temperature, is the condensation into a ground state exhibiting a blob-hole-like structure observed in real plasmas.

Grain size effect on magnetic and phase transition features in one-dimensional S = 1/2 Heisenberg spin chain molecular crystals

2015 ◽  
Vol 39 (7) ◽  
pp. 5395-5401 ◽  
Author(s):  
Guo-Jun Yuan ◽  
Yun-Xia Sui ◽  
Jian-Lan Liu ◽  
Xiao-Ming Ren
Keyword(s):  
Phase Transition ◽  
Grain Size ◽  
Spin Chain ◽  
Molecular Crystals ◽  
Spin Systems ◽  
Grain Size Effect ◽  
Heisenberg Spin ◽  
One Dimensional ◽  
Heisenberg Spin Chain ◽  
Molecular Spin

Magnetic and thermal behaviors and the phase transition nature are strongly influenced by grain size in one-dimensional S = 1/2 molecular spin systems.

scholarly journals Geometric phases and criticality in spin systems

2006 ◽  
Vol 364 (1849) ◽  
pp. 3463-3476 ◽  
Author(s):  
Jiannis K Pachos ◽  
Angelo C.M Carollo
Keyword(s):  
Phase Transition ◽  
Quantum Phase Transition ◽  
Spin System ◽  
Arbitrary Spin ◽  
Spin Systems ◽  
Quantum Phase ◽  
Geometric Phases ◽  
Interacting Spin Systems ◽  
The Way

A general formalism of the relation between geometric phases produced by circularly evolving interacting spin systems and their criticality behaviour is presented. This opens up the way for the use of geometric phases as a tool to probe regions of criticality without having to undergo a quantum phase transition. As a concrete example, a spin-1/2 chain with XY interactions is considered and the corresponding geometric phases are analysed. Finally, a generalization of these results to the case of an arbitrary spin system is presented.

Untersuchungen zur Quantenkondensation

1974 ◽  
Vol 29 (10) ◽  
pp. 1387-1393
Author(s):  
P. L. Lin
Keyword(s):  
Magnetic Field ◽  
Phase Transition ◽  
External Magnetic Field ◽  
Thermodynamic Limit ◽  
Order Phase Transition ◽  
Rotating System ◽  
One Dimensional ◽  
Charged System ◽  
Three Space ◽  
Second Order Phase Transition

Abstract It has recently been proved that quantum condensation can possibly occur only when the thermodynamic limit is formed with respect to all three space dimensions. Following this idea, it is shown that a rotating system is practically one-dimensional and therefore does not permit quantum condensation. The same is true for a charged system in an external magnetic field. However, an exact proof is given only for a second order phase transition.

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