On properties of the Ising model for complex energy/temperature and magnetic field

Matveev, Victor; Shrock, Robert

doi:10.1088/1751-8113/41/13/135002

Cited by 23 publications

(31 citation statements)

References 44 publications

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“…The original analysis of the ferromagnetic Ising model created a huge impact and a vast literature extending results 1 on other models and/or non-equilibrium cases. For a very limited list of references see, e.g., [2][3][4][5][6] .…”

Section: Introduction and Motivation Disorder Line In The Classicmentioning

confidence: 99%

Infinite cascades of phase transitions in the classical Ising chain

Timonin

Chitov

2017

Phys. Rev. E

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We report the new exact results on one of the best studied models in statistical physics: the classical antiferromagnetic Ising chain in a magnetic field. We show that the model possesses an infinite cascade of thermal phase transitions (also known as "disorder lines" or geometric phase transitions). The phase transition is signalled by a change of asymptotic behavior of the nonlocal string-string correlation functions when their monotonous decay becomes modulated by incommensurate oscillations. The transitions occur for rarefied (m-periodic) strings with arbitrary odd m. We propose a duality transformation which maps the Ising chain onto the m-leg Ising tube with nearest-neighbor couplings along the legs and the plaquette four-spin interactions of adjacent legs. Then the m-string correlation functions of the Ising chain are mapped onto the two-point spin-spin correlation functions along the legs of the m-leg tube. We trace the origin of these cascades of phase transitions to the lines of the Lee-Yang zeros of the Ising chain in m-periodic complex magnetic field, allowing us to relate these zeros to the observable (and potentially measurable) quantities.

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Section: Introduction and Motivation Disorder Line In The Classicmentioning

confidence: 99%

Infinite cascades of phase transitions in the classical Ising chain

Timonin

Chitov

2017

Phys. Rev. E

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“…Let us denote βJ ≡ F and consider the pure imaginary magnetic field βh ≡ iθ/2. Shifting θ by 2π induces for each vertex the same factor e ±iπ = −1 which has no relevant effect on the partition function (2). The partition function is also invariant with respect to the transformation θ → −θ and therefore one can restrict oneself to θ ∈ [0, π].…”

Section: Recapitulation Of the 1d Casementioning

confidence: 99%

“…where the first sum goes over all nearest-neighbor pairs of lattice sites and the second sum over all lattice sites. The partition function is defined by (2).…”

Section: Mapping Onto a Symmetric Vertex Modelmentioning

confidence: 99%

“…This model was investigated mainly in its antiferromagnetic version, on two-dimensional (2D) square or honeycomb lattices. The first studies were oriented to the location of Yang-Lee zeros in the complex temperature-magnetic field plane [1][2][3]. Similarly as in the case of the real magnetic field [4][5][6], dividing the lattice into two interwoven sublattices A and B the antiferromagnet exhibits two possible phases: the disordered paramagnetic phase at high temperatures with equivalent sublattice magnetizations m A = m B and the symmetry-broken antiferromagnetic phase at low temperatures with m A = m B .…”

Section: Introductionmentioning

confidence: 99%

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Ising ferromagnets and antiferromagnets in an imaginary magnetic field

Krčmár,

Gendiar,

Šamaj

2021

Preprint

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We study classical Ising spin-1 2 models on the 2D square lattice with ferromagnetic or antiferromagnetic nearest-neighbor interactions, under the effect of a pure imaginary magnetic field. The complex Boltzmann weights of spin configurations cannot be interpreted as a probability distribution which prevents from application of standard statistical algorithms. In this work, the mapping of the Ising spin models under consideration onto symmetric vertex models leads to real (positive or negative) Boltzmann weights. This enables us to apply accurate numerical methods based on the renormalization of the density matrix, namely the corner transfer matrix renormalization group and the higher-order tensor renormalization group. For the 2D antiferromagnet, varying the imaginary magnetic field we calculate with a high accuracy the curve of critical points related to the symmetry breaking of magnetizations on the interwoven sublattices. The critical exponent β and the anomaly number c are shown to be constant along the critical line, equal to their values β = 1 8 and c = 1 2 for the 2D Ising in a zero magnetic field. As concerns the 2D ferromagnet, the free energy turns out to be well defined for any temperature. On the other hand, a "stability" border of first-order transition temperatures below which the thermodynamic limit of the magnetization per spin is ill-defined (it varies chaotically with the size of the system) is determined as a function of the imaginary magnetic field by using two different approaches.

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“…For T > T c the endpoint of the arc of zeros is called the Lee-Yang edge. The confining of partition function zeros to an arc in the complex z plane for real temperatures is to be contrasted with the zeros of the partition function in the complex plane u = e −4E/kB T for real H, where computer studies [72], on systems of size 4 up to 16 × 16 show that the zeroes for H = 0 are located on curves only for very special boundary conditions [73], and for H = 0 there are regions of the complex u-plane where, even for these special boundary conditions, the zeros do not lie on curves.…”

Section: Conformal Field Theorymentioning

confidence: 99%

The Importance of the Ising Model

McCoy

Maillard

2012

Progress of Theoretical Physics

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Understanding the relationship which integrable (solvable) models, all of which possess very special symmetry properties, have with the generic non-integrable models that are used to describe real experiments, which do not have the symmetry properties, is one of the most fundamental open questions in both statistical mechanics and quantum field theory. The importance of the two-dimensional Ising model in a magnetic field is that it is the simplest system where this relationship may be concretely studied. We here review the advances made in this study, and concentrate on the magnetic susceptibility which has revealed an unexpected natural boundary phenomenon. When this is combined with the Fermionic representations of conformal characters, it is suggested that the scaling theory, which smoothly connects the lattice with the correlation length scale, may be incomplete for H = 0.Subject Index: 010, 040 §1. IntroductionIt may be rightly said that the two-dimensional-Ising model for H = 0 is one of the most important systems studied in theoretical physics. It is the first statistical mechanical system which can be exactly solved which exhibits a phase transition. From the exact results for the free energy, 1) spontaneous magnetization, 2),3) and correlation functions 4)−8) a point of view has been developed, which embraces the concepts of scaling, universality and conformal field theory, that extends the exact results of the Ising model to more general situations. These concepts are widely used to analyze both experiments and models of critical phenomena. Furthermore the correlation functions provide very concrete realizations of the concepts of mass and wave function renormalization used to define Euclidean quantum field theories.However, starting with the work of Nickel 9),10) on the magnetic susceptibility new properties of the Ising model have been uncovered 11)−27) which go beyond what has been seen in the computations of the free energy, spontaneous magnetization and correlation functions. These new features need to be explored to see if there is relevant physics which is not incorporated in our current view of critical phenomena. In this article we will review these new phenomena and the relation they have with scaling theory and Euclidean quantum field theory.In §2 we define what will be meant by an Ising model. In §3 we review the known exact results for H = 0. In addition to the well known results for the free energy 1) and the magnetization 2),3) we will put particular emphasis on the magnetic susceptibility which has an expansion analogous 6),9),10) to a Feynman diagram expansion. These Ising model integrals share with Feynman diagram integrals the

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On properties of the Ising model for complex energy/temperature and magnetic field

Cited by 23 publications

References 44 publications

Infinite cascades of phase transitions in the classical Ising chain

Infinite cascades of phase transitions in the classical Ising chain

Ising ferromagnets and antiferromagnets in an imaginary magnetic field

The Importance of the Ising Model

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