Critical behavior of the Ising model on the four-dimensional cubic lattice

Lundow, Per-Håkan; Markström, Klas

doi:10.1103/physreve.80.031104

Cited by 77 publications

(82 citation statements)

References 16 publications

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“…In the absence of theã μ (r) field, such a rank-2 theory undergoes a "rank-2 confinement/deconfinement" transition at some coupling strength λ c (assuming space-time dimensionality greater or equal to four). In the (3+1)D case, this transition has Ising-like critical exponents, which implies that the heat capacity peak should increase very slowly with the system size [10,11], and our results in Fig. 4 are in agreement with this.…”

Section: Explicit Demonstration Of Fractionalization Of Faraday supporting

confidence: 82%

Model of fractionalization of Faraday lines in compact electrodynamics

Geraedts

Motrunich

2014

Phys. Rev. B

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Motivated by ideas of fractionalization and intrinsic topological order in bosonic models with short-range interactions, we consider similar phenomena in formal lattice gauge theory models. Specifically, we show that a compact quantum electrodynamics (CQED) can have, besides the familiar Coulomb and confined phases, additional unusual confined phases where excitations are quantum lines carrying fractions of the elementary unit of electric field strength. We construct a model that has N -tupled monopole condensation and realizes 1/N fractionalization of the quantum Faraday lines. This phase has another excitation which is a Z N quantum surface in spatial dimensions five and higher, but can be viewed as a quantum line or a quantum particle in four or three spatial dimensions, respectively. These excitations have statistical interactions with the fractionalized Faraday lines; for example, in three spatial dimensions, the particle excitation picks up a Berry phase of e i2π/N when going around the fractionalized Faraday line excitation. We demonstrate the existence of this phase by Monte Carlo simulations in (3+1) space-time dimensions.

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Section: Explicit Demonstration Of Fractionalization Of Faraday supporting

confidence: 82%

Model of fractionalization of Faraday lines in compact electrodynamics

Geraedts

Motrunich

2014

Phys. Rev. B

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“…For both cases the straight lines which fit to these data give T c (∞) = 6.6830(51), T c (∞) = 6.6840(55), T c (∞) = 6.6845(56) and T c (∞) = 6.6788(65), T c (∞) = 6.6798(69), T c (∞) = 6.6802(70) without and with logarithmic factors for 7, 14, and 21 independent simulations, respectively (Table 2). T c (∞) = 6.6802(70) is consistent with the series expansion results of T c (∞) = 6.6817 (15) [37], T c (∞) = 6.6802(2) [38], the dynamic Monte Carlo result of T c (∞) = 6.6803(1) [38], the cluster Monte Carlo result of T c (∞) = 6.680(1) [11,12], the Creutz cellular automaton results of T c (∞) = 6.680, T c (∞) = 6.6802 (2), T c (∞) = 6.682, T c (∞) = 6.67 and the Monte Carlo using Metropolis and Wolff-cluster algorithm result of T c (∞) = 6.6802632 ± 5 × 10 −5 [39].…”

Section: Resultsmentioning

confidence: 98%

The Effect of the Number of Simulations on the Exponents Obtained by Finite-Size Scaling Relations of the Order Parameter and the Magnetic Susceptibility for the Four-Dimensional Ising Model on the Creutz Cellular Automaton

Merdan

Güzelsoy

2011

Int J Theor Phys

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The four-dimensional Ising model is simulated on the Creutz cellular automaton using finite-size lattices with linear dimension 4 ≤ L ≤ 8. The exponents in the finitesize scaling relations for the order parameter and the magnetic susceptibility at the finitelattice critical temperature are computed to be β = 0.49(7), β = 0.49(5), β = 0.50(1) and γ = 1.04(4), γ = 1.03(4), γ = 1.02(4) for 7, 14, and 21 independent simulations, respectively. As the number of independent simulations increases, the obtained results are consistent with the renormalization group predictions of β = 0.5 and γ = 1. The values for the critical temperature of the infinite lattice T c (∞) = 6.6788(65), T c (∞) = 6.6798(69), T c (∞) = 6.6802(70) are obtained from the straight-line fit of the magnetic susceptibility maxima using 4 ≤ L ≤ 8 for 7, 14, and 21 independent simulations, respectively. As the number of independent simulations increases, the obtained results are in very good agreement with the series expansion results of T c (∞) = 6.6817(15), T c (∞) = 6.6802(2), the dynamic Monte Carlo result of T c (∞) = 6.6803(1), the cluster Monte Carlo result of T c (∞) = 6.680(1) and the Monte Carlo using Metropolis and Wolff-cluster algorithm result of T c (∞) = 6.6802632 ± 5 × 10 −5 .

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“…The theory of abelian 2-form (and also higher nform) fields on a lattice had already been explored from varying perspectives in [1][2][3][4][5][6][7] and non-abelian lattice theories were also proposed in [8][9][10][11]. Later, rather similar ideas were pursued in [23][24][25]. In the lattice gerbe theories the gauge variables lived on the faces of the cubes of a hypercubic lattice and the Hamiltonian (i.e.…”

mentioning

confidence: 83%

“…Since d = 4 is the upper critical dimension of the Ising model the exponents will be mean field (though there is still some discussion about the nature of the specific heat divergence, if it exists, [25]). The M 4,3 model has been simulated in [6] to compare with the predictions from duality from the 4d Ising model and in [7] with a view to formulating a viable cluster algorithm.…”

mentioning

confidence: 99%

Z2lattice Gerbe theory

Johnston¹

2014

Phys. Rev. D

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Discretized formulations of 2-form abelian [1][2][3][4][5][6][7] and non-abelian [8][9][10][11] gauge fields on ddimensional hypercubic lattices have been discussed in the past by various authors and most recently in [12]. In this note we recall that the Hamiltonian of a Z2 variant of such theories is one of the family of generalized Ising models originally considered by Wegner [13]. For such "Z2 lattice gerbe theories" general arguments can be used to show that a phase transition for Wilson surfaces will occur for d > 3 between volume and area scaling behaviour. In 3d the model is equivalent under duality to an infinite coupling model and no transition is seen, whereas in 4d the model is dual to the 4d Ising model and displays a continuous transition. In 5d the Z2 lattice gerbe theory is self-dual in the presence of an external field and in 6d it is self-dual in zero external field. The study of the lattice discretization of 1-form gauge fields, lattice gauge theory, is now a mature subject. Higher form gauge fields on the lattice have received less attention, although 2-form gauge fields or gerbes have also proved to be of considerable interest. In string theory interest in higher form fields dates back to the work of Kalb and Ramond [14] and they have also arisen more recently in M-theory, particularly with regard to describing M5-branes. Abelian gerbes suffice to describe the six-dimensional low energy effective theory arising from a single M5-brane [15,16], but multiple M5-branes require a non-abelian theory. This has proved difficult to formulate in the continuum and motivated the authors of [12] to revisit both abelian and non-abelian gerbe theories on Euclidean hypercubic lattices in the manner of the Wilsonian formulation of lattice gauge theory [17][18][19][20][21][22]. The theory of abelian 2-form (and also higher nform) fields on a lattice had already been explored from varying perspectives in [1][2][3][4][5][6][7] and non-abelian lattice theories were also proposed in [8][9][10][11]. Later, rather similar ideas were pursued in [23][24][25]. In the lattice gerbe theories the gauge variables lived on the faces of the cubes of a hypercubic lattice and the Hamiltonian (i.e. action in field theoretic language) was given by the product of face variables round elementary cubes, which were called cubets in [12], summed over the lattice.We take a step further back here and consider what might be learnt from the simplest Hamiltonians that maintain the 2-form gauge invariance, which are constructed from Z 2 spins and which we call Z 2 lattice gerbe theories, borrowing the nomenclature of [12]. Many of the characteristic features of a lattice gauge theory are already discernible in the simplest Z 2 variant, so a reasonable expectation is that such a Z 2 lattice gerbe theory might offer a similar degree of insight into the behaviour of more general, including non-abelian, lattice gerbe theories.

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Critical behavior of the Ising model on the four-dimensional cubic lattice

Cited by 77 publications

References 16 publications

Model of fractionalization of Faraday lines in compact electrodynamics

Model of fractionalization of Faraday lines in compact electrodynamics

The Effect of the Number of Simulations on the Exponents Obtained by Finite-Size Scaling Relations of the Order Parameter and the Magnetic Susceptibility for the Four-Dimensional Ising Model on the Creutz Cellular Automaton

Z2lattice Gerbe theory

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