Wegner-Houghton equation in low dimensions

Carmona, J. M.; Polónyi, János; Tarancón, A.

doi:10.1103/physrevd.61.085018

Cited by 4 publications

(4 citation statements)

References 45 publications

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“…Increasing the number N of the encountered Fourier-modes of the blocked potential does not alter this general behaviour. This behaviour is qualitatively different from that for the polynomial case when all dimensionful coupling constants (with even indices) tend to a non-vanishing constant for k → 0 [26,39].…”

Section: E Effective Potentialcontrasting

confidence: 70%

“…There were no appreciable changes in the numerical results by increasing p further. The numerics have been tested by reproducing the results of [39] for the polynomial case. Below the critical scale k c where the spinodal instability occurs for the periodic case, the RG flow was determined by the help of the tree-level blocking relation (25).…”

Section: E Effective Potentialmentioning

confidence: 99%

“…The upper and lower plots show, respectively the scale-dependence above and below the scale kc of the spinodal instability which occurs at kc = 0.01413 for N = 10. The solution for the corresponding polynomial case is also indicated (solid line) [26,39]. (37)] in the (J, y)-plane.…”

Section: Figmentioning

confidence: 99%

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Renormalization-group analysis of the generalized sine-Gordon model and of the Coulomb gas for d>~3 dimensions

2004

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Renormalization-group (RG) flow equations have been derived for the generalized sine-Gordon model (GSGM) and the Coulomb gas (CG) in d ≥ 3 of dimensions by means of Wegner's and Houghton's, and by way of the real-space RG approaches. The UV scaling laws determined by the leading-order terms of the flow equations are in qualitative agreement for all dimensions d ≥ 3, independent of the dimensionality, and in sharp contrast to the special case d = 2. For the 4-dimensional GSGM it is demonstrated explicitly (by numerical calculations), that the blocked potential tends to a constant effective potential in the infrared (IR) limit, satisfying the requirements of periodicity and convexity. The comparison of the RG flows for the three-dimensional GSGM, the CG, and the vortex-loop gas reveals a significant dependence on the renormalization schemes and the approximations used.PACS numbers: 11.10.Hi, 11.10.Kk I. INTRODUCTIONThe renormalization of a two-dimensional one-component scalar field theory with a periodic self-interaction, and the two-dimensional generalized sine-Gordon model (GSGM) have been investigated for d = 2 dimensions in great detail over the last three decades [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]. Here, we only mention some of the main results:• The blocked potential for the two-dimensional GSGM tends to a constant effective potential in the IR limit, independent of the field [12,13,14].• Following the procedure proposed in [16], one can identically rewrite the partition function of the d-dimensional GSGM in the form of a d-dimensional static Coulomb gas, which we call here the equivalent Coulomb gas (ECG). For d = 2 it is well-known that the GSGM and the ECG, as well as the XY model describing classical planar spins belong to the same universality class [10]. The XY model has a dual theory in the sense of [17] that can be reformulated as a gas of topological excitations: the two-dimensional vortex gas (VG) for d = 2, and the three-dimensional vortex-loop gas (VLG) for d = 3 (see [10] and Refs. therein). For dimensions d = 2, it was shown that the ECG and the VG can be transformed into one another by an appropriate duality transformation [3,7,8,15] that inverts the temperature. Two-dimensional generalized models are well known where both the ECG and the VG are included as particular limiting cases [3,7,8,15] and are self-dual under the duality transformation.While the relations of the various models for two dimensions are well-established, those of the corresponding models for d = 3 are not completely settled (see e.g. [10]). In particular, it is not proven that the VLG and the threedimensional XY-model belong to the same class of universality. Therefore, the investigation of the renormalization of the GSGM and the ECG for d ≥ 3 dimensions is essential for a further clarification of this issue. The purpose of the present work is threefold:1. to investigate the renormalization of the GSGM for dimensions d ≥ 2 by the Wegner-Houghton renormalizationgroup (WH-RG) method [18] and to demonstrate numerically tha...

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Section: E Effective Potentialcontrasting

confidence: 70%

Section: E Effective Potentialmentioning

confidence: 99%

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Renormalization-group analysis of the generalized sine-Gordon model and of the Coulomb gas for d>~3 dimensions

2004

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“…There is a remarkable difference in the behavior of the theory with a periodic potential and that with the corresponding polynomial potential [29]. Namely, that all the dimensionful coupling constants g n (k) obtained for the periodic potential tend to zero in contrary to those of the polynomial potential which remain finite as k → 0.…”

Section: Numerical Solutionmentioning

confidence: 96%

Renormalization of periodic potentials

2001

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The renormalization of the periodic potential is investigated in the framework of the Euclidean onecomponent scalar field theory by means of the differential RG approach. Some known results about the sine-Gordon model are recovered in an extremely simple manner. There are two phases: an ordered one with asymptotical freedom and a disordered one where the model is nonrenormalizable and trivial. The order parameter of the periodicity, the winding number, indicates spontaneous symmetry breaking in the ordered phase where the fundamental group symmetry is broken and the solitons acquire dynamical stability. It is argued that the periodicity and the convexity are such strong constraints on the effective potential that it always becomes flat. This flattening is reproduced by integrating out the RG equation.

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Functional Callan–Symanzik equation for QED

Alexandre¹,

Polónyi²,

Sailer³

2002

Physics Letters B

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Wegner-Houghton equation in low dimensions

Cited by 4 publications

References 45 publications

Renormalization-group analysis of the generalized sine-Gordon model and of the Coulomb gas for d>~3 dimensions

Renormalization-group analysis of the generalized sine-Gordon model and of the Coulomb gas for d>~3 dimensions

Renormalization of periodic potentials

Functional Callan–Symanzik equation for QED

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Wegner-Houghton equation in low dimensions

Cited by 4 publications

References 45 publications

Renormalization-group analysis of the generalized sine-Gordon model and of the Coulomb gas for d&gt;~3 dimensions

Renormalization-group analysis of the generalized sine-Gordon model and of the Coulomb gas for d&gt;~3 dimensions

Renormalization of periodic potentials

Functional Callan–Symanzik equation for QED

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Renormalization-group analysis of the generalized sine-Gordon model and of the Coulomb gas for d>~3 dimensions

Renormalization-group analysis of the generalized sine-Gordon model and of the Coulomb gas for d>~3 dimensions