Ising spins coupled to a four-dimensional discrete Regge skeleton

Bittner, Elmar; Janke, Wolfhard; Markum, H.

doi:10.1103/physrevd.66.024008

Cited by 20 publications

(21 citation statements)

References 26 publications

(37 reference statements)

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“…These results state that if the leading scaling behaviour of the susceptibility differs from that of mean field theory only by multiplicative logarithmic corrections then the theory must be trivial. The perturbative arguments above demonstrate that the logarithmic corrections in the susceptibility are intimately linked to those of the Lee-Yang zeroes and that these are, in fact, independent of N. Thus independent numerical confirmation of their existence is strong evidence for the triviality of the theory for all N. Such non-perturbative evidence was provided in [16,19,22] (see also [12,13,15,18,20]). …”

Section: Partition Function Zeroes and Thermodynamic Functionsmentioning

confidence: 82%

“…Triviality of the O(4) φ 4 -theory in four dimensions is therefore an especially important issue. There is an abundance of analytical [1,8,9,10,11] and numerical [12,13,14,15,16,17,18,19,20,21,22] evidence in favour of triviality, mostly for the single-component theory. However, since a complete rigorous proof is still lacking, various ideas aimed at recovering a non-trivial continuum theory are also being explored [23].…”

Section: Introductionmentioning

confidence: 99%

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Finite size scaling for O(N) $phi;4-theory at the upper critical dimension

KENNA¹

2004

Nuclear Physics B

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A finite size scaling theory for the partition function zeroes and thermodynamic functions of O(N ) φ 4 -theory in four dimensions is derived from renormalization group methods. The leading scaling behaviour is mean-field like with multiplicative logarithmic corrections which are linked to the triviality of the theory. These logarithmic corrections are independent of N for odd thermodynamic quantities and associated zeroes and are N dependent for the even ones. Thus a numerical study of finite size scaling in the Ising model serves as a non-perturbative test of triviality of φ 4 4 -theories for all N .

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Section: Partition Function Zeroes and Thermodynamic Functionsmentioning

confidence: 82%

Section: Introductionmentioning

confidence: 99%

Finite size scaling for O(N) $phi;4-theory at the upper critical dimension

KENNA¹

2004

Nuclear Physics B

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“…On top of that, note that there are a number of quantum gravity approaches which rather work with quite simple (vertex) amplitudes [13][14][15][16][17][73][74][75][76] (we have in mind here the work of [16,17] in particular), in which the question of the large-scale limit can be addressed. The models proposed in our work here can be seen as small spin, cut-off versions of the full theory.…”

Section: Nonlocal Spin Foamsmentioning

confidence: 99%

“…The models proposed here can also be seen as candidates for an emergent gravity scenario. A related proposal is the truncated Regge models in [73][74][75][76], in which the continuous length variables of the Regge calculus are replaced by values in Z , with = 2 for "the Ising quantum gravity".…”

Section: Introductionmentioning

confidence: 99%

Spin Foam Models with Finite Groups

2013

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Spin foam models, loop quantum gravity, and group field theory are discussed as quantum gravity candidate theories and usually involve a continuous Lie group. We advocate here to consider quantum gravity-inspired models with finite groups, firstly as a test bed for the full theory and secondly as a class of new lattice theories possibly featuring an analogue diffeomorphism symmetry. To make these notes accessible to readers outside the quantum gravity community, we provide an introduction to some essential concepts in the loop quantum gravity, spin foam, and group field theory approach and point out the many connections to the lattice field theory and the condensed-matter systems.

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“…If this can be substantiated by further investigations and also for the Regge theory with continuous link lengths, the existence of a continuum limit at negative bare gravitational coupling is questionable. This is of major concern for a continuum theory of quantum gravity with matter fields [11]. …”

Section: Resultsmentioning

confidence: 99%

On the continuum limit of the discrete Regge model in 4d

Bittner¹,

Janke

Markum³

2003

Nuclear Physics B - Proceedings Supplements

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The Regge Calculus approximates a continuous manifold by a simplicial lattice, keeping the connectivities of the underlying lattice fixed and taking the edge lengths as degrees of freedom. The Discrete Regge model employed in this work limits the choice of the link lengths to a finite number. This makes the computational evaluation of the path integral much faster. A main concern in lattice field theories is the existence of a continuum limit which requires the existence of a continuous phase transition. The recently conjectured second-order transition of the four-dimensional Regge skeleton at negative gravity coupling could be such a candidate. We examine this regime with Monte Carlo simulations and critically discuss its behavior. REGGE QUANTUM GRAVITYOne promising method to quantize the theory of gravitation employs the Euclidean path integralwhere the partition function describes a fluctuating space-time manifold. In the Regge approach the (quadratic) link lengths q l represent the dynamical degrees of freedom [1], deforming continuously a simplicial lattice with fixed incidence matrix, whereas in the somehow complementary approach of dynamical triangulation the incidence matrix is fluctuating with constant link lengths. Any of these two approaches is plagued with various problems; for the Regge approach see [2,3]. In "conventional" Regge theory the ReggeEinstein action including a cosmological term [4],is used. The first sum runs over all products of triangle area A t times corresponding deficit angle δ t weighted by the bare gravitational coupling β. * Poster presented by E. B. and supported by Hochschuljubiläumsstiftung der Stadt Wien. † W. J. acknowledges partial support by the EC IHP Network grant HPRN-CT-1999-00161: "EUROGRID".The second sum extends over the volumes V s of the 4-simplices of the lattice and allows together with the cosmological constant λ to set an overall scale in the action. The Discrete Regge model was invented in an attempt to reformulate (1) as the partition function of a spin system [5,6]. It is defined by restricting the squared link lengths to take on only two valueswhere b l = 1, 2, 3, and 4 for edges, face diagonals, body diagonals and the hyperbody diagonal of a hypercube, respectively, is chosen to allow for fluctuations around flat space. The Euclidean triangle inequalities are fulfilled automatically as long as ǫ is smaller than a maximum value ǫ max . The measure D[q] in the quantum gravity path integral is taken to unity for all possible link configurations. Numerical simulations of the Z 2 system become extremely efficient by implementing look-up tables and a heat-bath algorithm. In this work typically 200 000 − 500 000 iterations have been generated. Calculations have been performed with the parameter ǫ = 0.0875 and the cosmological constant λ = 0 because (3) already fixes the average lattice volume.

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Ising spins coupled to a four-dimensional discrete Regge skeleton

Cited by 20 publications

References 26 publications

Finite size scaling for O(N) $phi;4-theory at the upper critical dimension

Finite size scaling for O(N) $phi;4-theory at the upper critical dimension

Spin Foam Models with Finite Groups

On the continuum limit of the discrete Regge model in 4d

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