Fermion representation of the three-dimensional Ising model

Kavalov, Al. R.; Sedrakyan, A.

doi:10.1016/0550-3213(87)90338-5

Cited by 39 publications

(46 citation statements)

References 15 publications

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“…Random networks were recast as lattice models in Ref. [73] in connection with the string representation of the 3D Ising model, and further studied in Refs. [74][75][76][77][78].…”

Section: 69mentioning

confidence: 99%

Geometrically disordered network models, quenched quantum gravity, and critical behavior at quantum Hall plateau transitions

et al. 2017

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Recent results for the critical exponent of the localization length at the integer quantum Hall transition differ considerably between experimental (νexp ≈ 2.38) and numerical (νCC ≈ 2.6) values obtained in simulations of the Chalker-Coddington (CC) network model. The difference is at least partially due to effects of the electron-electron interaction present in experiments. Here we propose a mechanism that changes the value of ν even within the single-particle picture. We revisit the arguments leading to the CC model and consider more general networks with structural disorder. Numerical simulations of the new model lead to the value ν ≈ 2.37. We argue that in a continuum limit the structurally disordered model maps to free Dirac fermions coupled to various random potentials (similar to the CC model) but also to quenched two-dimensional quantum gravity. This explains the possible reason for the considerable difference between critical exponents for the CC model and the structurally disordered model. We extend our results to network models in other symmetry classes.

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“…Random networks were recast as lattice models in Ref. [73] in connection with the string representation of the 3D Ising model, and further studied in Refs. [74][75][76][77][78].…”

Section: 69mentioning

confidence: 99%

Geometrically disordered network models, quenched quantum gravity, and critical behavior at quantum Hall plateau transitions

et al. 2017

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“…The aim of this article is the formulation of the lattice model in 2+1 dimensional space and investigation of its critical states, which,as it seems to us, is lying in the basis of such physical problems as 3D-Ising model (3DIM) [1,2,3] and edge excitations, responsible for the plato transitions in the Hall effect [4,6,7].…”

Section: Introductionmentioning

confidence: 99%

“…In the presented model we have used the essential ingredients of the model for so-called sign-factor, formulated in the string representation if the 3DIM [3], namely, the chiral fermions, which are hopping along the arrows on the Manhattan lattice (ML) ( Fig.1) in the presence of some external SU(2) gauge field.…”

Section: Introductionmentioning

confidence: 99%

Edge excitations of an incompressible fermionic liquid in a staggered magnetic field

Sedrakyan¹

1999

Nuclear Physics B

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The model of lattice fermions in 2+1 dimensional space is formulated, the critical states of which are lying in the basis of such physical problems, as 3D Ising Model(3DIM) and the edge excitations in the Hall effect. The action for this exitations coincides with the action of so called sign-factor model in 3DIM at one values of its parameters, and represent a model for the edge excitations, which are responsible for the plato transitions in the Hall effect, at other values. The model can be formulated also as a loop gas models in 2D, but unlikely the O(n) models, where the loop fugacity is real, here we have directed (clochwise and conterclochwise) loops and phase factors e ±2π p q i for them. The line of phase transitions in the parametric space will be found and corresponding continuum limits of this models will be constructed. It appears, that besides the ordinary critical line, which separates the dense and diluted phases of the models(like in ordinary O(n) models), there is a line, which corresponds to the full covering of the space by curves. The N = 2 twisted superconformal models with SU(2)/U(1) coset model coupling constant k = q p − 2 describes this states.1

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“…In our picture, this amounts to considering [4] a sort of double scaling limit (DSL) [5] at the level of the reduced EQG-partition function on the Riemann surface Σ g ∼ ∂N 3 . Indeed, the DSL is the usual way of formulating the genus expansion, i.e.…”

Section: The Euclidean 3d-qg Partition Function and The Alexander-conmentioning

confidence: 99%

“…One may then study the unknown correlation functions of 3D-Euclidean quantum gravity, in a way similar to the 3D-Ising model, [3] directly in the fermionic formulation, which is Gaussian, rather than in the (non-linear) Chern-Simons gauge description ⋄ . In particular, in the case M 3 is a hyperbolic three-manifold N 3 , ∂N 3 = ∅, we shall show that Z EQG (N 3 ) is related to the reduced partition function of the 3D-Ising model on the lattice Λ which is embeddable in ∂N 3 * .…”

Section: Introductionmentioning

confidence: 99%

2+1 Dimensional Quantum Gravity as a Gaussian Fermionic System and the 3D-Ising Model

Bonacina¹,

Martellini²,

Rasetti³

1995

Series on Knots and Everything

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We show that 2+1-dimensional Euclidean quantum gravity is equivalent, under some mild topological assumptions, to a Gaussian fermionic system. In particular, for manifolds topologically equivalent to Σ g × IR with Σ g a closed and oriented Riemann surface of genus g, the corresponding 2+1-dimensional Euclidean quantum gravity may be related to the 3D-lattice Ising model before its thermodynamic limit.

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Fermion representation of the three-dimensional Ising model

Cited by 39 publications

References 15 publications

Geometrically disordered network models, quenched quantum gravity, and critical behavior at quantum Hall plateau transitions

Geometrically disordered network models, quenched quantum gravity, and critical behavior at quantum Hall plateau transitions

Edge excitations of an incompressible fermionic liquid in a staggered magnetic field

2+1 Dimensional Quantum Gravity as a Gaussian Fermionic System and the 3D-Ising Model

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