Abstract Chapter 25 covers the Truncated Hilbert Space Approach provides a very efficient numerical algorithm to study many properties of a perturbed conformal field theory defined on a finite geometry, typically an infinite cylinder of radius R. These include the masses of the various excitations, their number below threshold, the presence of false vacua and resonances, on-shell three-particle coupling, etc. The implementation of this approach does not depend on the integrability of the off-critical model and therefore it is a very useful tool to extend study to the entire scaling region around the critical point. This chapter discusses the basis of such an algorithm and provides some interesting applications thereof.
Abstract Chapter 25 covers the Truncated Hilbert Space Approach provides a very efficient numerical algorithm to study many properties of a perturbed conformal field theory defined on a finite geometry, typically an infinite cylinder of radius R. These include the masses of the various excitations, their number below threshold, the presence of false vacua and resonances, on-shell three-particle coupling, etc. The implementation of this approach does not depend on the integrability of the off-critical model and therefore it is a very useful tool to extend study to the entire scaling region around the critical point. This chapter discusses the basis of such an algorithm and provides some interesting applications thereof.
We apply the massive analogue of the truncated conformal space approach to study the two dimensional φ4 theory in finite volume. We focus on the broken phase and determine the finite size spectrum of the model numerically. We interpret the results in terms of the Bethe- Yang spectrum, from which we extract the infinite volume masses and scattering matrices for various couplings. We compare these results against semiclassical analysis and perturbation theory. We also analyze the critical point of the model and confirm that it is in the Ising universality class.
<!--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-->Recent advances in Hamiltonian truncation have demonstrated it to be a useful tool for obtaining nonperturbative information about quantum field theories, even in the strongly-coupled regime. One particularly exciting feature of this technique is that it can be formulated in Minkowski rather than Euclidean space, making it easier to calculate real-time quantities. Truncation is particularly suited to obtaining spectral information (particle masses, operator spectral densities, etc.), but these are not the only observables of interest, especially when trying to make contact with high-energy experiments. I will explain how to use approximate knowledge of a QFT’s energy eigenstates to compute amplitudes for particle scattering, which can in turn be used to compute cross sections and decay rates. I will demonstrate this technique with an example of lightcone conformal truncation for the O(N) model in 2+1d at strong coupling, but it is worth emphasizing that the method can work with any numerical technique that outputs information about approximate Hamiltonian eigenstates. Along with the amplitude in the physical region, I will show that it is possible to analytically continue to the entire complex Mandelstam plane, and in fact that numerical convergence can be even better away from the physical region.
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.
In this thesis two rather different topics are addressed: • the application of the Truncated Conformal Spectrum Approach (TCSA) to pertur- bations of SU(2)k Wess-Zumino-Witten (WZW) models, and to Landau-Ginzburg (LG) theories; • the analysis of the role of quantum statistics in the steady flow patterns of 2d driven ideal Fermi gases in a non-integrable geometry. A common methodological question unifies the two topics: how to simulate numerically quantum systems with infinite spaces of states. This problem is typically faced either introducing a lattice in the real space or directly in the continuum, and the two addressed topics can be seen as examples of these two cases. The TCSA is aimed to tackle perturbed conformal field theories directly in the continuum. Instead, to study the ideal Fermi gas we introduce a lattice which naturally captures the features of the chosen non-integrable geometry. Chapter 1: This part of the thesis is mainly methodological and it is focused on the application of the TCSA to c = 1 conformal field theories. In particular, our interest is concerned with perturbations of WZW models with SU(2)k symmetry, which capture low energy properties of k/2-Heisenberg spin-chains. The new problem faced is how to apply the method in the presence of marginal perturbations, where UV divergences result in non-universal contributions in the simulations. A generic framework to cure this pathology is proposed, and tested in the case of the current-current deformation of SU(2)1. Among c = 1 theories, Landau-Ginzburg theories are of great interest, since their classical structure of the vacua captures the massive phases of deformation of minimal models. The most important methodological problem we faced here is that LG theories have an uncountable space of states. We show that the TCSA applies also in this case, once the target space of the LG field is properly compactified. The TCSA is thus used to very an existing conjecture on the number of stable neutral bound states. Chapter 2: Interacting quantum gases, such as the electron liquid, are known to admit a hydrodynamic description and, as for classical interacting fluids, they develop turbulence at sufficiently strong driving. On the other hand, ideal classical fluids cannot develop any turbulent flow. We ask whether this property extends to ideal quantum gases. Indeed, Fermi gases satisfy the exclusion principle that could play the role of an effective interaction, resulting in non-trivial flows. We are going to describe an ideal Fermi gas which flows form a narrow channel into a wider region. A lattice which approximates the geometry is introduced, and the system is tackled within two approaches: the micro-canonical formalism and the Lindblad equation. In the first case the system is closed, while in the second one it is coupled to external baths. Despite this difference, the observed physics turns out to be the same far from the boundaries. We show that, in specific regimes, quantum statistics induces non-trivial patterns in the vorticity, which generates local magnetic moments of a measurable intensity.