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arXiv:hep-th/0403178 · Journal of Statistical Mechanics Theory and Experiment

Renormalization group limit cycles and field theories for ellipticS-matrices

André LeClair, Germán Sierra

Published Aug 17, 2004• 21 citations
arXivHTMLPDF

Abstract

The renormalization group (RG) for maximally anisotropic su (2) current interactions in two dimensions is shown to be cyclic at one-loop level. The fermionized version of the model exhibits spin–charge separation of the four-fermion interactions and has symmetry. It is proposed that the S -matrices for these theories are the elliptic Zamolodchikov and Mussardo–Penati S -matrices. The S -matrix parameters are related to Lagrangian parameters by matching the period of the RG. All models exhibit two characteristic signatures of an RG limit cycle: periodicity of the S -matrix as a function of energy and the existence of an infinite number of resonance poles satisfying Russian doll scaling.

DOI
10.1088/1742-5468/2004/08/p08004

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The renormalization group (RG) for maximally anisotropic su (2) current interactions in two dimensions is shown to be cyclic at one-loop level. The fermionized version of the model exhibits spin–charge separation of the four-fermion interactions and has symmetry. It is proposed that the S -matrices for these theories are the elliptic Zamolodchikov and Mussardo–Penati S -matrices. The S -matrix parameters are related to Lagrangian parameters by matching the period of the RG. All models exhibit two characteristic signatures of an RG limit cycle: periodicity of the S -matrix as a function of energy and the existence of an infinite number of resonance poles satisfying Russian doll scaling.
RG Limit Cycles in BKT Flows ≡\equiv≡ Periodic Real-Time Dynamics in the Current-Current Perturbed SU(2)1SU(2)_1SU(2)1​ WZW Model
We establish an exact correspondence between renormalization-group (RG) evolution and real-time quantum dynamics in the anisotropic current-current perturbed SU(2)1SU(2)_1SU(2)1​ Wess-Zumino-Witten (WZW) model. Using its integrable fermionic representation, we construct the exact time-dependent many-body wavefunction by means of the generalized Bethe ansatz and show that periodic boundary conditions lead to quantum Knizhnik-Zamolodchikov equations associated with the XXZ trigonometric RRR-matrix, which corresponds to the quantum affine algebra Uq(sl2^)\mathcal{U}_q(\widehat{\mathfrak{sl}_2})Uq​(sl2​​) evaluated in its spin-1/21/21/2 evaluation representation. Consistency of these equations constrains the temporal evolution of the longitudinal and transverse interaction strengths. In the universal regime, these integrability conditions are exactly equivalent to the Berezinskii-Kosterlitz-Thouless RG flow equations upon identifying physical time with the logarithmic RG scale. Consequently, the RG limit cycles of the anisotropic current-current perturbed SU(2)1SU(2)_1SU(2)1​ WZW model are realized as periodic real-time evolution of its interaction strengths. Our results provide an exact dynamical realization of RG limit cycles and establish a direct connection between cyclic RG flows, quantum integrability, and periodically driven interacting quantum field theories.
Focusing on Bethe-Ansatz integrable models, robust to both time-reversal symmetry breaking and disorder, we consider the Russian Doll Model (RDM) for finite system sizes and energy levels. Suggested as a time-reversal-symmetry breaking deformation of Richardson’s model, the well-known and simplest model of superconductivity, RDM revealed an unusual cyclic renormalization group (RG) over the system size N N , where the energy levels repeat themselves, shifted by one after a finite period in \ln N ln N , supplemented by a hierarchy of superconducting condensates, with the superconducting gaps following the so-called Efimov (exponential) scaling. The equidistant single-particle spectrum of RDM made the above Efimov scaling and cyclic RG to be asymptotically exact in the wideband limit of the diagonal potential. Here, we generalize this observation in various respects. We find that, beyond the wideband limit, when the entire spectrum is considered, the periodicity of the spectrum is not constant, but appears to be energy-dependent. Moreover, we resolve the apparent paradox of shift in the spectrum by a single level after the RG period, despite the disappearance of a finite fraction of energy levels. We also analyze the effects of disorder in the diagonal potential on the above periodicity and show that it survives only for high energies beyond the energy interval of the disorder amplitude. Our analytic analysis is supported with exact diagonalization.
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We apply field theoretical renormalization group (RG) methods to describe the Tomonaga–Luttinger model as an important test ground to deal with spin–charge separation effects in higher spatial dimensions. We calculate the RG equations for the renormalized forward couplings g 2 R and g 4 R up to two-loop order and demonstrate that they do not flow in the vicinities of the Fermi points (FPs). We solve the Callan–Symanzik equation for G ( a ) R in the vicinities of the FPs. We calculate the related spectral function and the momentum distribution function at p = k F , p 0 = ω. We compute the renormalized one-particle irreducible function Γ (2) (+) R ( p , p 0 = 0; Λ) and show that it carries important spin–charge separation effects in agreement with well-known results. Finally, we discuss the implementation of the RG scheme taking into account the important simplifications produced by the Ward identities.