Singular renormalization group transformations and first order phase transitions (II). Monte Carlo renormalization group results

Decker, Karsten M.; Hasenfratz, Anna; Hasenfratz, P.

doi:10.1016/0550-3213(88)90225-8

Cited by 31 publications

(22 citation statements)

References 24 publications

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“…Indeed the results, always for lattices with periodic boundary conditions and with runs of O(10 4 ) length, have usually depended on which method was used for studying the phase transition. Most studies using the Monte-Carlo renormalization group transformation [6][7][8][9] have claimed a second order phase transition, but sometimes with very different critical exponents, whereas most works based on finite size scaling theory [10][11][12] have concluded to a first-order transition, but again with a critical exponent not consistent with the first-order prediction. These discrepancies led the subject into an expectant state.…”

Section: Introductionmentioning

confidence: 98%

High-statistics finite size scaling analysis of U(1) lattice gauge theory with a Wilson action

Klaus

Roiesnel

1998

Phys. Rev. D

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We describe the results of a systematic high-statistics Monte-Carlo study of finite-size effects at the phase transition of compact U(1) lattice gauge theory with Wilson action on a hypercubic lattice with periodic boundary conditions. We find unambiguously that the critical exponent ν is lattice-size dependent for volumes ranging from 4 4 to 12 4 . Asymptotic scaling formulas yield values decreasing from ν(L ≥ 4) ≈ 0.33 to ν(L ≥ 9) ≈ 0.29. Our statistics are sufficient to allow the study of different phenomenological scenarios for the corrections to asymptotic scaling. We find evidence that corrections to a first-order transition with ν = 0.25 provide the most accurate description of the data. However the corrections do not follow always the expected first-order pattern of a series expansion in the inverse lattice volume V −1 . Reaching the asymptotic regime will require lattice sizes greater than L = 12. Our conclusions are supported by the study of many cumulants which all yield consistent results after proper interpretation.

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Section: Introductionmentioning

confidence: 98%

High-statistics finite size scaling analysis of U(1) lattice gauge theory with a Wilson action

Klaus

Roiesnel

1998

Phys. Rev. D

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“…This behaviour appears very similar to the one in the U(1) lattice gauge theory [18][19][20][21][22], (cf. the discussion in sect.…”

Section: /?_<24mentioning

confidence: 55%

“…1). As suggested by Hasenfratz et al [16,21] it might be associated with a singular behaviour of the BST (first observed for the three-state Potts model [35]). In U(1) the Coulomb phase -like the Higgs phase in the adjoint Higgs model -has a massless photon.…”

Section: /?_<24mentioning

confidence: 97%

“…To the authors knowledge up to now no DFP has been observed within the context of gauge theories. On the other hand, in U(1) gauge theories [17][18][19][20][21][22] there is evidence for a hybrid type of FP, positioned on the first order part of the PT surface, with a flow away from the singular surface on the hot side and a flow towards a FP within the surface on the cold side. Similar discontinuous flows are observed in the four-dimensional Z(2) gauge theory [23].…”

Section: Introductionmentioning

confidence: 99%

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The renormalization flow in the adjoint SU(2) lattice Higgs model

Baier¹,

Lang²,

Reusch³

1988

Nuclear Physics B

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Applying the Monte Carlo renormalization group method we investigate the flow of coupling constants for the 84 and 164 SU(2) lattice Higgs model with triplet Higgs fields. The couplings of the renormalized actions are determined using Schwinger-Dyson equations. From the flow we find new evidence for the existence of a tricritical point at finite values of the inverse gauge coupling fl but no indication for a discontinuity fixed point.

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“…On the one hand, the existence of several phases raises the possibility that the RG map may be discontinuous or multi-valued 173,197,305] on the rst-order transition surface, as the numerical evidence reported by several groups 33,233,72,163] seems to suggest. On the other hand, Gri ths and Pearce 172,173,171] have pointed out some \peculiarities" of the commonly used discrete-spin position-space RG transformations (decimation, majority rule, etc.…”

Section: General Introductionmentioning

confidence: 99%

Regularity properties and pathologies of position-space renormalization-group transformations

Enter

Fernández

Sokal

1991

Nuclear Physics B - Proceedings Supplements

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We reconsider the conceptual foundations of the renormalization-group (RG) formalism, and prove some rigorous theorems on the regularity properties and possible pathologies of the RG map. Regarding regularity, we show that the RG map, de ned on a suitable space of interactions (= formal Hamiltonians), is always single-valued and Lipschitz continuous on its domain of de nition. This rules out a recently proposed scenario for the RG description of rst-order phase transitions. On the pathological side, we make rigorous some arguments of Grifths, Pearce and Israel, and prove in several cases that the renormalized measure is not a Gibbs measure for any reasonable interaction. This means that the RG map is ill-de ned, and that the conventional RG description of rst-order phase transitions is not universally valid. For decimation or Kadano transformations applied to the Ising model in dimension d 3, these pathologies occur in a full neighborhood f > 0 ; jhj < ( )g of the low-temperature part of the rstorder phase-transition surface. For block-averaging transformations applied to the Ising model in dimension d 2, the pathologies occur at low temperatures for arbitrary magnetic-eld strength. Pathologies may also occur in the critical region for Ising models in dimension d 4. We discuss in detail the distinction between Gibbsian and non-Gibbsian measures and the possible occurrence of the latter in other situations, and give a rather complete catalogue of the known examples. Finally, we discuss the heuristic and numerical evidence on RG pathologies in the light of our rigorous theorems.

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Singular renormalization group transformations and first order phase transitions (II). Monte Carlo renormalization group results

Cited by 31 publications

References 24 publications

High-statistics finite size scaling analysis of U(1) lattice gauge theory with a Wilson action

High-statistics finite size scaling analysis of U(1) lattice gauge theory with a Wilson action

The renormalization flow in the adjoint SU(2) lattice Higgs model

Regularity properties and pathologies of position-space renormalization-group transformations

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