A numerical approach to 2-D relativistic field theories is suggested. Considering a field theory model as an ultraviolet conformal field theory perturbed by a suitable relevant scalar operator one studies it in finite volume (on a circle). The perturbed Hamiltonian acts in the conformal field theory space of states and its matrix elements can be extracted from the conformal field theory. Truncation of the space at a reasonable level results in a finite dimensional problem for numerical analyses. The nonunitary field theory with the ultraviolet region controlled by the minimal conformal theory {ital M}(2/5) is studied in detail.A numerical approach to 2-D relativistic field theories is suggested. Considering a field theory model as an ultraviolet conformal field theory perturbed by a suitable relevant scalar operator one studies it in finite volume (on a circle). The perturbed Hamiltonian acts in the conformal field theory space of states and its matrix elements can be extracted from the conformal field theory. Truncation of the space at a reasonable level results in a finite dimensional problem for numerical analyses. The nonunitary field theory with the ultraviolet region controlled by the minimal conformal theory {ital M}(2/5) is studied in detail.

Aug 20, 199024 citations

Defect scaling Lee–Yang model from the perturbed DCFT point of view

Zoltán Bajnok, László Holló, G.M.T. Watts

We analyze the defect scaling Lee–Yang model from the perturbed defect conformal field theory (DCFT) point of view. First the defect Lee–Yang model is solved by calculating its structure constants from the sewing relations. Integrable defect perturbations are identified in conformal defect perturbation theory. Then pure defect flows connecting integrable conformal defects are described. We develop a defect truncated conformal space approach (DTCSA) to analyze the one parameter family of integrable massive perturbations in finite volume numerically. Fusing the integrable defect to an integrable boundary the relation between the IR and UV parameters can be derived from the boundary relations. We checked these results by comparing the spectrum for large volumes to the scattering theory.We analyze the defect scaling Lee–Yang model from the perturbed defect conformal field theory (DCFT) point of view. First the defect Lee–Yang model is solved by calculating its structure constants from the sewing relations. Integrable defect perturbations are identified in conformal defect perturbation theory. Then pure defect flows connecting integrable conformal defects are described. We develop a defect truncated conformal space approach (DTCSA) to analyze the one parameter family of integrable massive perturbations in finite volume numerically. Fusing the integrable defect to an integrable boundary the relation between the IR and UV parameters can be derived from the boundary relations. We checked these results by comparing the spectrum for large volumes to the scattering theory.

arXiv:1307.4536Nuclear Physics BJun 26, 201413 citations

TCSA for beginners

Gábor Takács

I give an introduction the Truncated Conformal Space Approach, which was originally invented by Yurov and Zamolodchikov in the framework of the scaling Lee-Yang model. Since then it has become a very powerful tool to study not only finite volume spectra, but also form factors, resonances, phase transitions and renormalization group flows in one-dimensional quantum field theories, both integrable and non-integrable. Its scope has been extended to theories with boundaries and impurities, and also generalized to settings beyond perturbed conformal field theories. In this talk, beyond explaining the basics of the approach I also give a brief overview of the recently introduced renormalization methods to deal with cut-off dependence, which led to a major improvement in precision and also widened the potential scope of applications.I give an introduction the Truncated Conformal Space Approach, which was originally invented by Yurov and Zamolodchikov in the framework of the scaling Lee-Yang model. Since then it has become a very powerful tool to study not only finite volume spectra, but also form factors, resonances, phase transitions and renormalization group flows in one-dimensional quantum field theories, both integrable and non-integrable. Its scope has been extended to theories with boundaries and impurities, and also generalized to settings beyond perturbed conformal field theories. In this talk, beyond explaining the basics of the approach I also give a brief overview of the recently introduced renormalization methods to deal with cut-off dependence, which led to a major improvement in precision and also widened the potential scope of applications.

Jan 1, 20140 citations

TRUNCATED-FERMIONIC-SPACE APPROACH TO THE CRITICAL 2D ISING MODEL WITH MAGNETIC FIELD

V.P. Yurov, Al.B. Zamolodchikov

The critical Ising model perturbed by the spin field (conjugated to the magnetic field) is studied numerically by the method of truncated fermionic space of states. The matrix elements of the perturbed Hamiltonian are found between the states of free Majorana fermions living on a finite-length circle. The Hamiltonian spectrum is studied numerically in the level-5 truncated space. Reasonable estimations are obtained for the vacuum energy, a part of the mass spectrum and the simplest scattering amplitude of the perturbed system.The critical Ising model perturbed by the spin field (conjugated to the magnetic field) is studied numerically by the method of truncated fermionic space of states. The matrix elements of the perturbed Hamiltonian are found between the states of free Majorana fermions living on a finite-length circle. The Hamiltonian spectrum is studied numerically in the level-5 truncated space. Reasonable estimations are obtained for the vacuum energy, a part of the mass spectrum and the simplest scattering amplitude of the perturbed system.

International Journal of Modern Physics AOct 20, 1991156 citations

A conformal truncation framework for infinite-volume dynamics

Emanuel Katz, Zuhair U. Khandker, Matthew T. Walters

We present a new framework for studying conformal field theories deformed by one or more relevant operators. The original CFT is described in infinite volume using a basis of states with definite momentum, P , and conformal Casimir, C. The relevant deformation is then considered using lightcone quantization, with the resulting Hamiltonian expressed in terms of this CFT basis. Truncating to states with $\mathcal{C}\le {\mathcal{C}}_{\max }$ , one can numerically find the resulting spectrum, as well as other dynamical quantities, such as spectral densities of operators. This method requires the introduction of an appropriate regulator, which can be chosen to preserve the conformal structure of the basis. We check this framework in three dimensions for various perturbative deformations of a free scalar CFT, and for the case of a free O(N ) CFT deformed by a mass term and a non-perturbative quartic interaction at large-N . In all cases, the truncation scheme correctly reproduces known analytic results. We also discuss a general procedure for generating a basis of Casimir eigenstates for a free CFT in any number of dimensions.We present a new framework for studying conformal field theories deformed by one or more relevant operators. The original CFT is described in infinite volume using a basis of states with definite momentum, P , and conformal Casimir, C. The relevant deformation is then considered using lightcone quantization, with the resulting Hamiltonian expressed in terms of this CFT basis. Truncating to states with $$ \mathcal{C}\le {\mathcal{C}}_{\max } $$ , one can numerically find the resulting spectrum, as well as other dynamical quantities, such as spectral densities of operators. This method requires the introduction of an appropriate regulator, which can be chosen to preserve the conformal structure of the basis. We check this framework in three dimensions for various perturbative deformations of a free scalar CFT, and for the case of a free O(N ) CFT deformed by a mass term and a non-perturbative quartic interaction at large-N . In all cases, the truncation scheme correctly reproduces known analytic results. We also discuss a general procedure for generating a basis of Casimir eigenstates for a free CFT in any number of dimensions.

arXiv:1604.01766Journal of High Energy PhysicsJul 1, 201652 citations

Quantum field theories in finite volume: Excited state energies

Vladimir V. Bazhanov, Sergei L. Lukyanov, Alexander B. Zamolodchikov

We develop a method of computing the excited state energies in Integrable Quantum Field Theories (IQFT) in finite geometry, with the spatial coordinate compactified on a circle of circumference R. The IQFT “commuting transfer matrices” introduced earlier [Commun. Math. Phys. 177 (1996) 381] for Conformal Field Theories (CFT) are generalized to non-conformal IQFT obtained by perturbing CFT with the operator Φ1,3. We study the models in which the fusion relations for these “transfer matrices” truncate and provide closed integral equations which generalize the equations of the thermodynamic Bethe ansatz to excited states. The explicit calculations are done for the first excited state in the “scaling Lee-Yang model”.We develop a method of computing the excited state energies in Integrable Quantum Field Theories (IQFT) in finite geometry, with the spatial coordinate compactified on a circle of circumference R. The IQFT “commuting transfer matrices” introduced earlier [Commun. Math. Phys. 177 (1996) 381] for Conformal Field Theories (CFT) are generalized to non-conformal IQFT obtained by perturbing CFT with the operator Φ1,3. We study the models in which the fusion relations for these “transfer matrices” truncate and provide closed integral equations which generalize the equations of the thermodynamic Bethe ansatz to excited states. The explicit calculations are done for the first excited state in the “scaling Lee-Yang model”.

arXiv:hep-th/9607099Nuclear Physics BApr 1, 1997216 citations

\mathcal{C}\le {\mathcal{C}}_{\max }

, one can numerically find the resulting spectrum, as well as other dynamical quantities, such as spectral densities of operators. This method requires the introduction of an appropriate regulator, which can be chosen to preserve the conformal structure of the basis. We check this framework in three dimensions for various perturbative deformations of a free scalar CFT, and for the case of a free O(N ) CFT deformed by a mass term and a non-perturbative quartic interaction at large-N . In all cases, the truncation scheme correctly reproduces known analytic results. We also discuss a general procedure for generating a basis of Casimir eigenstates for a free CFT in any number of dimensions.