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Numerical Renormalization Group for Continuum One-Dimensional Systems

Robert Konik, Yury Adamov

We present a renormalization group (RG) procedure which works naturally on a wide class of interacting one-dimension models based on perturbed (possibly strongly) continuum conformal and integrable models. This procedure integrates Wilson's numerical renormalization group with Zamolodchikov's truncated conformal spectrum approach. The key to the method is that such theories provide a set of completely understood eigenstates for which matrix elements can be exactly computed. In this procedure the RG flow of physical observables can be studied both numerically and analytically. To demonstrate the approach, we study the spectrum of a pair of coupled quantum Ising chains and correlation functions in a single quantum Ising chain in the presence of a magnetic field.We present a renormalization group (RG) procedure which works naturally on a wide class of interacting one-dimension models based on perturbed (possibly strongly) continuum conformal and integrable models. This procedure integrates Wilson's numerical renormalization group with Zamolodchikov's truncated conformal spectrum approach. The key to the method is that such theories provide a set of completely understood eigenstates for which matrix elements can be exactly computed. In this procedure the RG flow of physical observables can be studied both numerically and analytically. To demonstrate the approach, we study the spectrum of a pair of coupled quantum Ising chains and correlation functions in a single quantum Ising chain in the presence of a magnetic field.

arXiv:cond-mat/0701605Physical Review LettersApr 6, 200784 citations

Theoretical Aspects of Mainly Low Dimensional Magnetic Systems

Hans C. Fogedby

Springer eBooksJan 1, 198029 citations

A study of truncation effects in boundary flows of the Ising model on a strip

Gábor Zsolt Tóth

We investigate the idea that the effect of the truncation applied in the truncated conformal space approach method to the spectrum coincides with the effect of a suitable changing of the coefficients of the terms in the Hamiltonian operator. The investigation is done in the case of the critical Ising model on a strip with an external magnetic field on one of the boundaries. A detailed quantum field theoretical description of this model is also given, and we propose a description as a perturbation of the infinite coupling limit. The investigation is also carried out for a truncation method which preserves the solvability of the model. The results of perturbative and numerical calculations presented support the above idea and show that the qualitative behaviour of the truncated spectrum as a function of the coupling constant depends on the truncation method.We investigate the idea that the effect of the truncation applied in the truncated conformal space approach method to the spectrum coincides with the effect of a suitable changing of the coefficients of the terms in the Hamiltonian operator. The investigation is done in the case of the critical Ising model on a strip with an external magnetic field on one of the boundaries. A detailed quantum field theoretical description of this model is also given, and we propose a description as a perturbation of the infinite coupling limit. The investigation is also carried out for a truncation method which preserves the solvability of the model. The results of perturbative and numerical calculations presented support the above idea and show that the qualitative behaviour of the truncated spectrum as a function of the coupling constant depends on the truncation method.

arXiv:hep-th/0612256Journal of Statistical Mechanics Theory and ExperimentApr 16, 200715 citations

The magnetic transition in finite systems

W. F. Hall, Roger E. De Wames, T. Wolfram

Physics Letters AMay 1, 19711 citation

Magnetism in one-dimensional disordered systems

George Theodorou, Morrel H. Cohen

Springer eBooksApr 6, 20081 citation

The renormalisation group for the truncated conformal space approach on the cylinder

Philip Giokas, G.M.T. Watts

In this paper we continue the study of the truncated conformal space approach to perturbed conformal field theories, this time applied to bulk perturbations and focusing on the leading truncation-dependent corrections to the spectrum. We find expressions for the leading terms in the ground state energy divergence, the coupling constant renormalisation and the energy rescaling. We apply these methods to problems treated in two seminal papers and show how these RG improvements greatly increase the predictive power of the TCSA approach. One important outcome is that the TCSA spectrum of excitations is predicted not to converge for perturbations of conformal weight greater than 3/4, but the ratios of excitation energies should converge.In this paper we continue the study of the truncated conformal space approach to perturbed conformal field theories, this time applied to bulk perturbations and focusing on the leading truncation-dependent corrections to the spectrum. We find expressions for the leading terms in the ground state energy divergence, the coupling constant renormalisation and the energy rescaling. We apply these methods to problems treated in two seminal papers and show how these RG improvements greatly increase the predictive power of the TCSA approach. One important outcome is that the TCSA spectrum of excitations is predicted not to converge for perturbations of conformal weight greater than 3/4, but the ratios of excitation energies should converge.

arXiv:1106.2448Jun 13, 201157 citations

Coupled one-dimensional magnetic systems: the truncated conformal spectrum approach

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