<!--HTML-->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.
<!--HTML-->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.
We discuss the errors introduced by level truncation in the study of boundary renormalization group (RG) flows using the truncated conformal space approach (TCSA). We show that the TCSA results can have the qualitative form of a sequence of RG flows between different conformal boundary conditions. In the case of a perturbation by the field ϕ (13) , we propose a RG equation for the coupling constant which predicts a fixed point at a finite value of the TCSA coupling constant and we compare the predictions with data obtained using thermodynamic Bethe ansatz equations.
Abstract The Truncated conformal space approach (TCSA) is a numerical technique for finding finite size spectrum of Hamiltonians in quantum field theory described as perturbations of conformal field theories. The truncation errors of the method have been systematically studied near the UV fixed point (when the characteristic energy related to the coupling is less than the truncation cutoff) where a good theoretical understanding has been achieved. However numerically the method demonstrated a good agreement with other methods for much larger values of the coupling when the RG flow approaches a new fixed point in the infrared. In the present paper we investigate this regime for a number of boundary RG flows testing the leading exponent and truncation errors. We also study the flows beyond the first fixed point which have been observed numerically but yet lack a theoretical understanding. We show that while in some models such flows approximate reversed physical RG flows, in other models the spectrum approaches a stable regime that does not correspond to any local boundary condition. Furthermore we find that in general the flows beyond the first fixed point are very sensitive to modifications of the truncation scheme.
Truncated Conformal Space Approach (TCSA) is a highly efficient method to compute spectra, operator matrix elements and time evolution in quantum field theories defined as relevant perturbations of 1+1-dimensional conformal field theories. However, similarly to other exact diagonalisation methods, TCSA is ridden with the “curse of dimensionality”: the dimension of the Hilbert space increases exponentially with the (square root of the) truncation level, limiting its precision by the available memory resources. Here we describe an algorithm which exploits the chiral factorisation property of conformal field theory with periodic boundary conditions to achieve a substantial improvement in the truncation level. The Chirally Factorised TCSA (CFTCSA) algorithm presented here works with inputs describing the necessary CFT data in a specified format. It makes possible much more precise calculations with given computing resources and extends the reach of the method to problems requiring large Hilbert space dimensions. In fact, it has already been used in a number of recent works ranging from determination of form factors, through studying confinement of topological excitations to non-equilibrium dynamics. Besides the description of the algorithm, a MATLAB implementation of the algorithm is also provided as an ancillary file package, supplemented with example codes computing spectra, matrix elements and time evolution, and with CFT data for three different quantum field theories. We also give a detailed how-to guide for constructing the required CFT data for Virasoro minimal models with central charge c<1, and for the massless free boson with c=1.
Truncated Conformal Space Approach (TCSA) is a highly efficient method to compute spectra, operator matrix elements and time evolution in quantum field theories defined as relevant perturbations of 1+1-dimensional conformal field theories. However, similarly to other exact diagonalisation methods, TCSA is ridden with the “curse of dimensionality”: the dimension of the Hilbert space increases exponentially with the (square root of the) truncation level, limiting its precision by the available memory resources. Here we describe an algorithm which exploits the chiral factorisation property of conformal field theory with periodic boundary conditions to achieve a substantial improvement in the truncation level. The Chirally Factorised TCSA (CFTCSA) algorithm presented here works with inputs describing the necessary CFT data in a specified format. It makes possible much more precise calculations with given computing resources and extends the reach of the method to problems requiring large Hilbert space dimensions. In fact, it has already been used in a number of recent works ranging from determination of form factors, through studying confinement of topological excitations to non-equilibrium dynamics. Besides the description of the algorithm, a MATLAB implementation of the algorithm is also provided as an ancillary file package, supplemented with example codes computing spectra, matrix elements and time evolution, and with CFT data for three different quantum field theories. We also give a detailed how-to guide for constructing the required CFT data for Virasoro minimal models with central charge c<1, and for the massless free boson with c=1.
This thesis investigates two aspects of Conformal Field Theories (CFTs) in D dimensions. Its first part is devoted to conformal blocks, special functions that arise in the partial wave expansion of CFT four-point functions. We prove that these conformal blocks admit an expansion in terms of polar coordinates and show that the expansion coefficients are determined by recursion relations. Conformal blocks are naturally defined on the complex plane: we study their restriction to the real line, and show that they obey a fourth-order differential equation there. This ODE can be used to efficiently compute conformal blocks and their derivatives in general D. Several applications to the conformal bootstrap program are mentioned.The second half of this thesis investigates RG flows that are defined by perturbing a CFT by a number of relevant operators. We study such flows using the Truncated Conformal Space Approach (TCSA) of Yurov and Zamolodchikov, a numerical method that allows for controlled computations in strongly coupled QFTs. Two different RG flows are considered: the free scalar field deformed by a mass term, and phi^4 theory. The former is used as a benchmark, in order to compare numerical TCSA results to exact predictions. TCSA results for phi^4 theory display spontaneous Z_2 symmetry breaking at strong coupling: we study the spectrum of this theory both in the Z_2-broken and preserved phase, and we compare the critical exponents governing the phase transition to known values. In a separate chapter, we show how truncation errors can be reduced by adding suitable counterterms to the bare TCSA action, following earlier work in two dimensions.
We developed the theory of finite volume form factors in the presence of integrable defects. These finite volume form factors are expressed in terms of the infinite volume form factors and the finite volume density of states and incorporate all polynomial corrections in the inverse of the volume. We tested our results, in the defect Lee–Yang model, against numerical data obtained by truncated conformal space approach (TCSA), which we improved by renormalization group methods adopted to the defect case. To perform these checks we determined the infinite volume defect form factors in the Lee–Yang model exactly, including their vacuum expectation values. We used these data to calculate the two point functions, which we compared, at short distance, to defect CFT. We also derived explicit expressions for the exact finite volume one point functions, which we checked numerically. In all of these comparisons excellent agreement was found.