Regge gravity on general triangulations

Beirl, W.; Markum, H.; Riedler, J.

doi:10.1016/0370-2693(94)01265-2

Cited by 13 publications

(9 citation statements)

References 23 publications

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“…The same authors studied the inclusion of the higher-derivative term (15) in [36]. On the regular lattice, their findings for R(k, a) confirmed those by Hamber, apart from the fact that they found stable expectation values even for positive R. Inserting irregular vertices pushes R to larger values and leads again to the appearance of an additional transition point.…”

Section: Non-hypercubic Latticessupporting

confidence: 59%

“…Simulations on lattices with irregular link geometry (still with T 4 -topology) have been performed by Beirl et al [31,36]. They were obtained by adding a few vertices of low coordination number to otherwise regular lattices.…”

Section: Non-hypercubic Latticesmentioning

confidence: 99%

“…One finds that the average curvature R increases from negative to positive values, even for k < k c , as a result of the formation of spikes. In [36], the averages l 2 were monitored separately at the regular and the inserted vertices.…”

Section: Non-hypercubic Latticesmentioning

confidence: 99%

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Discrete Approaches to Quantum Gravity in Four Dimensions

Loll

1998

Living Rev. Relativ.

179

175

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The construction of a consistent theory of quantum gravity is a problem in theoretical physics that has so far defied all attempts at resolution. One ansatz to try to obtain a non-trivial quantum theory proceeds via a discretization of space-time and the Einstein action. I review here three major areas of research: gauge-theoretic approaches, both in a path-integral and a Hamiltonian formulation; quantum Regge calculus; and the method of dynamical triangulations, confining attention to work that is strictly four-dimensional, strictly discrete, and strictly quantum in nature.

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Section: Non-hypercubic Latticessupporting

confidence: 59%

Section: Non-hypercubic Latticesmentioning

confidence: 99%

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Discrete Approaches to Quantum Gravity in Four Dimensions

Loll

1998

Living Rev. Relativ.

179

175

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“…͑A brief numerical study of a kagome Ising model with three-spin interactions can be found in Ref. 13.͒ In the same manner in which we proceeded in Sec. III A, we can integrate out the a-spins to form an effective model for the b-spins.…”

Section: ͑23͒mentioning

confidence: 90%

XXZand Ising spins on the triangular kagome lattice

et al. 2008

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The recently fabricated two-dimensional magnetic materials Cu 9 X 2 ͑cpa͒ 6 · xH 2 O ͑cpa= 2-carboxypentonic acid and X =F,Cl,Br͒ have copper sites which form a triangular kagome lattice ͑TKL͒, formed by introducing small triangles ͑"a-trimers"͒ inside of each kagome triangle ͑"b-trimer"͒. We show that in the limit where spins residing on b-trimers have Ising character, quantum fluctuations of XXZ spins residing on the a-trimers can be exactly accounted for in the absence of applied field. This is accomplished through a mapping to the kagome Ising model, for which exact analytic solutions exist. We derive the complete finite-temperature phase diagram for this XXZ-Ising model, including the residual zero-temperature entropies of the seven ground-state phases. Whereas the disordered ͑spin liquid͒ ground state of the pure Ising TKL model has macroscopic residual entropy ln 72= 4.2767. . . per unit cell, the introduction of transverse ͑quantum͒ couplings between neighboring a-spins reduces this entropy to 2.5258. . . per unit cell. In the presence of applied magnetic field, we map the TKL XXZ-Ising model to the kagome Ising model with three-spin interactions and derive the ground-state phase diagram. A small ͑or even infinitesimal͒ field leads to a new phase that corresponds to a nonintersecting loop gas on the kagome lattice, with entropy 1.4053. . . per unit cell and a mean magnetization for the b-spins of 0.12͑1͒ per site. In addition, we find that for moderate applied field, there is a critical spin liquid phase that maps to close-packed dimers on the honeycomb lattice, which survives even when the a-spins are in the Heisenberg limit.

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“…for a generic large surface of volume V 4 and no accidental symmetry factors there will be V 4 such manifolds for each closed manifold since the boundary can be placed at any of the V 4 four-simplexes. If we assume that the canonical partition function is given by (36), we get: (40) which has the obvious interpretation that the probability of having a branching to a minbu of volume V 4 per unit volume is V 2 4 (1 V 4 =N 4 ) 2 . As shown in 34,19] \minbu counting" is a very convenient tool for measuring if is not too negative, both in two-dimensional and four-dimensional simplicial quantum gravity.…”

Section: Scaling Relationsmentioning

confidence: 99%

Scaling in four-dimensional quantum gravity

Ambjørn¹,

Jurkiewicz²

1995

Nuclear Physics B

100

159

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We discuss scaling relations in four dimensional simplicial quantum gravity. Using numerical results obtained with a new algorithm called \baby universe surgery" we study the critical region of the theory. The position of the phase transition is given with high accuracy and some critical exponents are measured. Their values prove that the transition is continuous. We discuss the properties of two distinct phases of the theory. For large values of the bare gravitational coupling constant the internal Hausdor dimension is two (the elongated phase), and the continuum theory is that of so called branched polymers. For small values of the bare gravitational coupling constant the internal Hausdor dimension seems to be in nite (the crumpled phase). We conjecture that this phase corresponds to a theory of topological gravity. At the transition point the Hausdor dimension might be nite and larger than two. This transition point is a potential candidate for a non-perturbative theory of quantum gravity.

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Regge gravity on general triangulations

Cited by 13 publications

References 23 publications

Discrete Approaches to Quantum Gravity in Four Dimensions

Discrete Approaches to Quantum Gravity in Four Dimensions

XXZand Ising spins on the triangular kagome lattice

Scaling in four-dimensional quantum gravity

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