Precise determination of critical exponents and equation of state by field theory methods

Zinn-Justin, Jean

doi:10.1016/s0370-1573(00)00126-5

Cited by 139 publications

(185 citation statements)

References 66 publications

Supporting

Mentioning

166

Contrasting

Unclassified

Order By: Relevance

“…5) by a short dashed line. The PT results for ν(N) agree within a percent with those from other field theoretical methods [33]. The N dependence of ω(N) is reproduced within 5%.…”

Section: Comparisonsupporting

confidence: 79%

“…This qualitative difference between exact and PT flows is linked to their fundamental inequivalence, even to this order of the approximation [29]. Still, the large effective radius of convergence explains why the derivative expansion for the PT flow (33) should have even better convergence properties. While this explains the convergence behaviour found in Ref.…”

Section: Discussionmentioning

confidence: 99%

“…For N = 1 and 2, the boundary values stem from experimental data. The bounds on field theoretical methods, reviewed in [33], are much tighter. They agree surprisingly well with the findings of [29] (within less than 1% for ν and around 5% for ω, for all N = 0, · · · , 4), and are represented by the short-dashed line in Figs.…”

Section: Comparisonmentioning

confidence: 99%

See 2 more Smart Citations

Derivative expansion and renormalisation group flows

Litim

2001

J. High Energy Phys.

117

137

View full text Add to dashboard Cite

We study the convergence of the derivative expansion for flow equations. The convergence strongly depends on the choice for the infrared regularisation. Based on the structure of the flow, we explain why optimised regulators lead to better physical predictions. This is applied to O(N )-symmetric real scalar field theories in 3d, where critical exponents are computed for all N . In comparison to the sharp cut-off regulator, an optimised flow improves the leading order result up to 10%. An analogous reasoning is employed for a proper time renormalisation group. We compare our results with those obtained by other methods.

show abstract

“…5) by a short dashed line. The PT results for ν(N) agree within a percent with those from other field theoretical methods [33]. The N dependence of ω(N) is reproduced within 5%.…”

Section: Comparisonsupporting

confidence: 79%

Section: Discussionmentioning

confidence: 99%

See 1 more Smart Citation

Derivative expansion and renormalisation group flows

Litim

2001

J. High Energy Phys.

117

137

View full text Add to dashboard Cite

show abstract

“…∆ 0 = 3 − 1/ν and ∆ 1 = 1 2 + η 2 ) in the literature including the MC simulations [28], resummed perturbation theories (MS and MZM) [38] as well as conformal bootstrap analysis [4,10]. We also find an estimate of higher charged operators that are summarized in table I.…”

Section: Appendix B: Summary Of Scaling Dimensions In the Literaturementioning

confidence: 67%

Necessary Condition for Emergent Symmetry from the Conformal Bootstrap

Nakayama

Ohtsuki

2016

Phys. Rev. Lett.

View full text Add to dashboard Cite

We use the conformal bootstrap program to derive necessary conditions for emergent symmetry enhancement from discrete symmetry (e.g. Zn) to continuous symmetry (e.g. U (1)) under the renormalization group flow. In three dimensions, in order for Z2 symmetry to be enhanced to U (1) symmetry, the conformal bootstrap program predicts that the scaling dimension of the order parameter field at the infrared conformal fixed point must satisfy ∆1 > 1.08. We also obtain the similar conditions for Z3 symmetry with ∆1 > 0.580 and Z4 symmetry with ∆1 > 0.504 from the simultaneous conformal bootstrap analysis of multiple four-point functions. Our necessary conditions impose severe constraints on many controversial physics such as the chiral phase transition in QCD, the deconfinement criticality in Néel-VBS transitions and anisotropic deformations in critical O(n) models. In some cases, we find that the conformal bootstrap program dashes hopes of emergent symmetry enhancement proposed in the literature.

show abstract

“…Unfortunately, it is well known that the ǫ expansion is often an asymptotic series [7], sometime providing a better estimate of the exponents at first order than at second order. For all that one knows the third loop-order may bring the exponent down, maybe even below the first loop-order.…”

mentioning

confidence: 99%

Self-affine roughness of a crack front in heterogeneous media

2007

View full text Add to dashboard Cite

The long-ranged elastic model, which is believed to describe the evolution of a self-affine rough crack-front, is analyzed to linear and non-linear orders. It is shown that the nonlinear terms, while important in changing the front dynamics, are not changing the scaling exponent which characterizes the roughness of the front. The scaling exponent thus predicted by the model is much smaller than the one observed experimentally. The inevitable conclusion is that the gap between the results of experiments and the model that is supposed to describe them is too large, and some new physics has to be invoked for another model.The self-affine roughness of a crack-front propagating under a tensile load in a randomly heterogeneous system is a well studied issue, both experimentally and theoretically. Experimentally one measures the position h(x, t) of the crack front, where h is the position of the front as a function of the span-wise coordinate x at time t, and finds that this is a self-affine function whose roughness is characterized by a scaling exponent ζ (defined below in Eq. (4)) in the range of 0.5-0.65 [1,2]. Theoretically there appears to be a consensus that the appropriate model for such dynamical roughening is a long-ranged elastic string close to its depinning threshold. This model is defined by the equation of motion for a front h(x, t), which is allowed to move only forward due to the irreversibility of the fracture process [3] ∂h(x, t) ∂tHere and bellow the integral is meant in the Cauchy Principal Value sense. The RHS of Eq. (1) is the difference between the local driving force (below referred to as G, related physically to the energy release rate driving the crack [4]), and Γ(x, h) which is a random quenched noise (representing the random material fracture energy [4]). G (0) is the control parameter that represents the energy release rate of a straight front. The integral term stands for the long ranged restoring forces stemming from bulk elastic degrees of freedom. The correspondence between the theoretical model and the experimental findings remained however unclear, since the best numerical studies of the resulting self-affine graph h(x) of this model came up with a roughness exponent ζ = 0.388 ± 0.002 [5], clearly outside the range of error of the experimental measurements. This apparent difficulty led to a number of interesting studies, insisting that the model is basically right, and that the result concerning the scaling exponent is not final. Thus, for example, in [6] the authors analyzed Eq.(1) using a functional renormalization group. They have calculated the scaling exponent ζ to one-and two-loop orders in ǫ, where ǫ = 2 − d. To one-loop order the result is ζ = ǫ/3, predicting ζ = 1/3 at d = 1, deviating considerably from the best numerical estimate.To two-loop order the prediction increases to about 0.466, leading to a statement that the model probably describes properly the experimental findings. Unfortunately, it is well known that the ǫ expansion is often an asymptotic series [7], sometime ...

show abstract

Precise determination of critical exponents and equation of state by field theory methods

Cited by 139 publications

References 66 publications

Derivative expansion and renormalisation group flows

Derivative expansion and renormalisation group flows

Necessary Condition for Emergent Symmetry from the Conformal Bootstrap

Self-affine roughness of a crack front in heterogeneous media

Contact Info

Product

Resources

About