Constructing an S-matrix from the truncated conformal approach data

Dermot Mc Ateer

Máté Lencsés

Yury Adamov, Robert Konik, Alexei M. Tsvelik

G. Takács, G.M.T. Watts

D.X. Horváth, K. Hódsági, G. Takács

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.

Horváth, Dávid, Hódsági, Kristóf, Takács, Gábor

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.