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0209019.pdf - Improved harmonic approximation and the 2D...

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arXiv:hep-th/0209019v12 Sep 2002Improved harmonic approximation and the 2D Ising model atTnegationslash=Tcandhnegationslash= 0An´ıbal Iucci and Carlos M. Na´onInstituto de F´ısica La Plata. Departamento de F´ısica, Facultad de Ciencias Exactas,Universidad Nacional de La Plata. CC 67, 1900 La Plata, Argentina.Consejo Nacional de Investigaciones Cient´ıficas y T´ecnicas, Argentina.AbstractWe propose a new method to determine the unknown parameter associated to a self-consistentharmonic approximation. We check the validity of our technique in the context of the sine-Gordonmodel.As a non trivial application we consider the scaling regime of the 2D Ising model awayfrom the critical point and in the presence of a magnetic fieldh.We derive an expression thatrelates the approximate correlation lengthξ,TTcandh.*Electronic address: [email protected]; [email protected]1
The so called “self consistent harmonic approximation” (SCHA) is a non-perturbativetechnique that has been extensively employed in Statistical Mechanics [1][2] and CondensedMatter physics [3][4][5][6] applications. Roughly speaking it amounts to replacing an exactactionStrueby a trial actionStrialthat makes the problem tractable. UsuallyStrialis just aquadratic action that depends on certain unknown parameter Ω that must be determinedthrough some criterion such as the minimization of the free energy of the system.Thisapproximation is intimately related the the “gaussian effective potential” [7][8] in QuantumField Theories (QFT’s), a variational approximation to the effective potential which usesa gaussian wave functional depending on some mass parameter as the trial ground state.It also relies on a minimization principle often called “principle of minimal sensitivity” [9]to determine the additional parameter. In this work we point out that in two-dimensionalproblems there is an alternative way to obtain the quantity Ω.This method is based onConformal Field Theory (CFT) [10]. Moreover, we shall show that our method yields im-proved results with respect to the predictions of standard SCHA in the sine-Gordon (SG)model and allows us to give a new description of the off-critical 2D Ising model (2DIM). Inthe former we exploit the existence of exact results [11] [12] to check the consistency of ourproposal by obtaining a qualitatively good answer for the soliton mass. We then apply thesame idea to the 2D Ising model atTnegationslash=Tcandhnegationslash= 0, a non-integrable model in whichvery few quantitative results are known [13] [14]. We use the fermionic representation of the2DIM. Since the standard SCHA is restricted to bosonic models, the new procedure is alsoan extension of the gaussian approximation to fermionic 2D theories. Our main result is analgebraic equation which allows to get the behavior of the correlation length as function ofTTcandh.

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Quantum Field Theory, Statistical Mechanics, J Phys, Conformal field theory, Z Phys, Strial

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