/Subtype /Link /W 1 >> << /D (cite.erzan) /Parent 5 0 R /D (cite.Cassi) >> 25 0 obj >> endobj /Kids [11 0 R 12 0 R] /C [0 1 0] A variation of the scale loosely corresponds to changing the magnifying power of an appropriate microscope for viewing the system. /MediaBox [0 0 612 792] << endobj endobj 30 0 obj /Producer (pdfTeX-1.40.12) /Border [0 0 1] /Next 34 0 R /Subtype /Link /C [0 1 0] /H /I (3.20), the result is called the Gaussian theory: Z = & detK (1 2 π)N ’ 1/2 # Dφ(x)exp * − #, rφ2(x)−Dφ(x)∇2φ(x)-dx +, (3.24) The absence of … /S /GoTo /Limits [(cite.BA) (equation.2.3)] /D (cite.Berlin) /D (cite.Chung) 16 0 obj /Rect [185.05 510.505 192.024 518.917] The Renormalization Group in Momentum Space 39 3.2 The Gaussian Model If only the terms quadratic in φ(k) are retained in Eq. /Rect [266.988 394.136 273.962 402.549] /S /S << /H /I /A << /A << /Type /Page /Rect [257.026 394.136 264 402.549] /C [1 0 0] /Rect [168.611 127.028 180.566 135.441] >> Gaussian Model: the "free" theory << /Keywords () 27 0 obj /Parent 11 0 R /Subtype /Link /Border [0 0 1] /Type /Annot /BS << endobj >> /Count 8 >> >> Simplified renormalization group transformation 106 12.2. /Author () Thes 4 model 101 subspaces) 159 4.1. /H /I Renormalization of the Gaussian Model The renormalization group analysis of the ˚4theory can be performed exactly above the upper critical dimension d uc, where the Gaussian version of the theory is sucient to describe the critical behavior. /A << /H /I /C [0 1 0] /Subtype /Link /Kids [56 0 R 57 0 R] /BS << /Rect [173.25 510.505 180.223 518.917] << /H /I /Title (I Introduction) endobj << >> /Kids [44 0 R 45 0 R 46 0 R 47 0 R 48 0 R 49 0 R] /Type /Annot << /A << /D (cite.Goldenfeld) /S /GoTo /S /GoTo We compute the specific heat and magnetic field exponents. /Subtype /Link /Subtype /Link /Type /Annot >> /S /GoTo >> Thec-expansionand a non-trivialfixed point 107 12.3. << << /Count 14 /S /GoTo << /A << Multiple fixed points, domains, and 4.3. >> /H /I >> /S /S 10 0 obj /S /GoTo /A << >> /Border [0 0 1] /BS << 26 0 R 27 0 R 28 0 R 29 0 R] 1 0 obj endobj /BS << /Filter /FlateDecode /S /S /A << 3 0 obj << /S /S /Border [0 0 1] /OpenAction 3 0 R /PTEX.Fullbanner (This is pdfTeX, Version 3.1415926-2.3-1.40.12 \(TeX Live 2011\) kpathsea version 6.0.1) /A << The critical behaviour of the Gaussian model on dierent non-spatial networks is studied by means of the spectral renormalization group we have recently proposed. /Limits [(equation.2.4) (section*.13)] stream /S /S endobj /Creator (LaTeX with hyperref package) << /S /GoTo >> /Rect [541.971 125.037 548.945 136.992] endobj /BS << /C [0 1 0] >> 19 0 obj 17 0 obj /Kids [43 0 R] II, we define the spectral renormalization group for the Gaussian model on a generic network. /Kids [50 0 R 51 0 R 52 0 R 53 0 R 54 0 R 55 0 R] 24 0 obj /Subtype /Link endobj >> << the couplings of a system are running couplings, running with the scale. /C [0 1 0] /Subtype /Link /S /S /Type /Pages /W 1 In Sec. /Type /Annot /H /I /S /S >> From Gaussian models to Renormalization Group We have studied Ising type models (but the study can be easily extended to ferromagnetic models with O(N) symmetry) with short range interac- tions and determined the behaviour of the thermodynamic functions near a continuous phase transition, within the framework of the quasi-Gaussian or mean field approximations. << 15 0 obj /W 1 /H /I /D (cite.Wilson2) /Subtype /Link The renormalization group (RG) then describes how these parameters are changed when the system is considered at a dierent scale, i.e. /Type /Annot >> /W 1 /Subject () /D (cite.Berlin) >> /W 1 /S /S /BS << /A << /C [0 1 0] /Names 2 0 R endobj >> >> >> /S /S 12 0 obj /BS << /Kids [13 0 R 14 0 R 15 0 R] /Type /Annot /BS << endobj /Contents [30 0 R 31 0 R] endobj /H /I << /Subtype /Link /Border [0 0 1] /Parent 4 0 R 21 0 obj << /A << /BS << /Subtype /Link /Title () /Limits [(section*.14) (table.1)] << /Parent 5 0 R >> endobj /D (cite.Hattori) /W 1 endobj /Type /Annot /Outlines 4 0 R /Rect [142.285 476.133 149.259 484.546] /H /I >> endobj /Border [0 0 1] I. GAUSSIAN MODEL We will take free massless scalar field in D-dimension as the first example, which represents the gapless Gaussian fluctuations in the ordered side in Landau’s theory for continuous phase transition. /Type /Annot >> /H /I /Type /Outlines >> /Rect [171.968 556.333 178.942 564.746] 29 0 obj /Prev 36 0 R /Subtype /Link /BS << >> /S /S /W 1 /Type /Annot << /C [0 1 0] In Sec. >> >> >> /Parent 4 0 R /A 33 0 R >> /D (cite.diamond) 4 0 obj >> /D [8 0 R /Fit] >> /Last 10 0 R /Pages 5 0 R /D (equation.2.2) >> endobj << >> >> /Count 7 /Rect [138.157 172.857 150.112 181.269] 7 0 obj %���� >> /W 1 >> /A << 5 0 obj Linearized equations and calculation. /S /GoTo /S /GoTo /Border [0 0 1] /Rect [350.887 179.682 357.861 188.095] /Border [0 0 1] >> 8 0 obj 14 0 obj /Rect [152.804 476.133 159.778 484.546] 23 0 obj /W 1 /S /S /S /S /C [0 1 0] /Kids [37 0 R 8 0 R 38 0 R 39 0 R 40 0 R 41 0 R 42 0 R] 6 0 obj << >> >> 9 0 obj endobj << /Rect [442.371 332.293 449.345 340.706] << /D (cite.Chung) /BS << /Trapped /False /Type /Annot