scholarly journals POTENTIAL FLOW OF THE RENORMALIZATION GROUP IN A SIMPLE TWO-COMPONENT MODEL

1993 ◽  
Vol 08 (32) ◽  
pp. 3103-3110 ◽  
Author(s):  
BRIAN P. DOLAN
Keyword(s):  
Renormalization Group ◽  
Scalar Field ◽  
Simple Model ◽  
Correlation Functions ◽  
Potential Flow ◽  
Dimensional Space ◽  
Component Model ◽  
Two Dimensional ◽  
Two Component ◽  
Point Correlation

The renormalization group (RG) flow on the space of couplings of a simple model with two couplings is examined. The model considered is that of a single component scalar field with φ4 self interaction coupled, via Yukawa coupling, to a fermion in flat four-dimensional space. The RG flow on the two-dimensional space of couplings, [Formula: see text], is shown to be derivable from a potential to sixth order in the couplings, which requires two-loop calculations of the β-functions. The identification of a potential requires the introduction of a metric on [Formula: see text] and it is shown that the metric defined by Zamolodchikov, in terms of two-point correlation functions of composite operators, gives potential flow to this order.

scholarly journals Geometrical four-point functions in the two-dimensional critical Q-state Potts model: the interchiral conformal bootstrap

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Yifei He ◽  
Jesper Lykke Jacobsen ◽  
Hubert Saleur
Keyword(s):  
Potts Model ◽  
Correlation Functions ◽  
Liouville Theory ◽  
Analytical Determination ◽  
Conformal Weight ◽  
Two Dimensional ◽  
Conformal Blocks ◽  
Conformal Bootstrap ◽  
Point Correlation

Abstract Based on the spectrum identified in our earlier work [1], we numerically solve the bootstrap to determine four-point correlation functions of the geometrical connectivities in the Q-state Potts model. Crucial in our approach is the existence of “interchiral conformal blocks”, which arise from the degeneracy of fields with conformal weight hr,1, with r ∈ ℕ*, and are related to the underlying presence of the “interchiral algebra” introduced in [2]. We also find evidence for the existence of “renormalized” recursions, replacing those that follow from the degeneracy of the field $$ {\Phi}_{12}^D $$ Φ 12 D in Liouville theory, and obtain the first few such recursions in closed form. This hints at the possibility of the full analytical determination of correlation functions in this model.

scholarly journals A bulk localized state and new holographic renormalization group flow in 3D spin-3 gravity

2018 ◽  
Vol 33 (12) ◽  
pp. 1850061 ◽  
Author(s):  
Ryuichi Nakayama ◽  
Tomotaka Suzuki
Keyword(s):  
Renormalization Group ◽  
Scalar Field ◽  
Central Charge ◽  
Dimensional Space ◽  
Space Time ◽  
Internal Space ◽  
Conformal Weight ◽  
Weight Changes ◽  
Localized State ◽  
Bulk Space

We construct a localized state of a scalar field in 3D spin-3 gravity. 3D spin-3 gravity is thought to be holographically dual to [Formula: see text]-extended CFT on a boundary at infinity. It is known that while [Formula: see text] algebra is a nonlinear algebra, in the limit of large central charge [Formula: see text] a linear finite-dimensional subalgebra generated by [Formula: see text] [Formula: see text] and [Formula: see text] [Formula: see text] is singled out. The localized state is constructed in terms of these generators. To write down an equation of motion for a scalar field which is satisfied by this localized state, it is necessary to introduce new variables for an internal space [Formula: see text], [Formula: see text], [Formula: see text], in addition to ordinary coordinates [Formula: see text] and [Formula: see text]. The higher-dimensional space, which combines the bulk space–time with the “internal space,” which is an analog of superspace in supersymmetric theory, is introduced. The “physical bulk space–time” is a 3D hypersurface with constant [Formula: see text], [Formula: see text] and [Formula: see text] embedded in this space. We will work in Poincaré coordinates of AdS space and consider [Formula: see text]-quasi-primary operators [Formula: see text] with a conformal weight [Formula: see text] in the boundary and study two and three point functions of [Formula: see text]-quasi-primary operators transformed as [Formula: see text]. Here, [Formula: see text] and [Formula: see text] are [Formula: see text] generators in the hyperbolic basis for Poincaré coordinates. It is shown that in the [Formula: see text] limit, the conformal weight changes to a new value [Formula: see text]. This may be regarded as a Renormalization Group (RG) flow. It is argued that this RG flow will be triggered by terms [Formula: see text] added to the action.

An exactly solvable two component classical Coulomb system

1984 ◽  
Vol 26 (2) ◽  
pp. 119-128 ◽  
Author(s):  
Peter J. Forrester
Keyword(s):  
Coulomb Potential ◽  
Correlation Functions ◽  
Asymptotic Form ◽  
Coulomb System ◽  
Two Dimensional ◽  
Coupling Constant ◽  
Sum Rule ◽  
Exactly Solvable ◽  
Two Component ◽  
Special Value

AbstractA two component classical Coulomb system is considered, in which particles of charge +q and + 2q are constrained to lie on a circle and interact via the two-dimensional Coulomb potential. At a special value of the coupling constant the correlation functions are calculated exactly and the asymptotic form of the truncated charge-charge correlation is found to obey Jancovici's sum rule.

Fourth-order bounds on the effective bulk and shear moduli of random dispersions of penetrable spheres

1995 ◽  
Vol 450 (1940) ◽  
pp. 651-665 ◽  
Keyword(s):  
Fourth Order ◽  
Correlation Functions ◽  
Structural Parameters ◽  
Numerical Calculations ◽  
Shear Moduli ◽  
Two Component ◽  
Point Correlation ◽  
Order Bounds ◽  
Penetrable Spheres

The structural parameters entering the fourth-order bounds of Milton & Phan-Thien ( Proc . R . Soc . Lond . A 380, 305 (1982)) on the effective bulk and shear moduli of two-component materials are expressed in terms of integrals over three- and four-point correlation functions to facilitate numerical calculations on systems for which the correlation functions are known. These integral expressions are then used to numerically calculate the structural parameters of random dispersions of penetrable spheres.

scholarly journals Line and surface defects for the free scalar field

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Edoardo Lauria ◽  
Pedro Liendo ◽  
Balt C. van Rees ◽  
Xiang Zhao
Keyword(s):  
Scalar Field ◽  
Correlation Functions ◽  
Surface Defects ◽  
Intermediate Result ◽  
Conformal Theory ◽  
Free Theory ◽  
Free Spectrum ◽  
Point Correlation ◽  
Free Scalar ◽  
Almost All

Abstract For a single free scalar field in d ≥ 2 dimensions, almost all the unitary conformal defects must be ‘trivial’ in the sense that they cannot hold interesting dynamics. The only possible exceptions are monodromy defects in d ≥ 4 and co-dimension three defects in d ≥ 5. As an intermediate result we show that the n-point correlation functions of a conformal theory with a generalized free spectrum must be those of the generalized free theory.

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