Finite-size Analysis of O(N) Nonlinear σ Model on Semi-compact Spaces
نویسنده
چکیده
Fisher’s phenomenological renormalization method is used to calculate the mass gap and the correlation length of the O(N) nonlinear σ model on a semi-compact space S1 ×R2. This shows that the ultraviolet momentum cut-off does not conflict with the infrared cut-off along the S1 direction. The mass gap on S2 × R1 is also discussed. ∗e-mail address: [email protected] 1 The O(N) nonlinear σ (NLσ) model on a semi-compact space S × R has been intensively studied recently in the context of two-dimensional quantum spin systems[1] and in researching the critical phenomena in three dimensions[2][3]. The calculation of the partition functions and correlation length (or mass gap) is done with the ultraviolet (UV) momentum cut-off in the R direction and usually without any UV momentum cut-off along the S direction being imposed[1][4]. This procedure is quite natural in the view of the imaginary time path integral method, which discritizes only the spatial coordinates and leaves the temporal one continuous. However, this scheme might conflict with the infrared (IR) cut-off along the S direction because of its unrenormalizability. In this paper, we consider the gap equation on the lattice and make use of Fisher’s phenomenological renormalization (PR) method[5][6] to clarify the above point. The Pr method is a renormalization method in the real space based on a hypothesis of the pseudoscale invariance for large but finite systems. In this scheme, we have the advantage that the UV cut-off does not appear explicitly. In terms of the PR method, the correlation length can be calculated exactly on S × S × R or S × S × S[5]. We apply the PR method to the O(N) NLσ model on S1×R2 and S2×R1. These PR procedures reproduce the mass-gap obtained previously in the cut-off regularization[1]. Therefore, we consider that the UV cut-off regularization does not conflict with the IR cut-off along the S or S direction. Of course, it is widely believed that the physical quantities can be calculated independently of the regularization in renormalizable systems. Our results are consistent with this intuition despite its unrenormalizablilty in the ordinary meaning. We consider the O(N) NLσ model on a semi-compact space S ×R2 with the radius of S, L. The partition function is given by Z = ∫
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