We both review and augment the lightcone conformal truncation (LCT) method. LCT is a Hamiltonian truncation method for calculating dynamical quantities in QFT in infinite volume. This document is a self-contained, pedagogical introduction and "how-to" manual for LCT. We focus on 2D QFTs which have UV descriptions as free CFTs containing scalars, fermions, and gauge fields, providing a rich starting arena for LCT applications. Along our way, we develop several new techniques and innovations that greatly enhance the efficiency and applicability of LCT. These include the development of CFT radial quantization methods for computing Hamiltonian matrix elements and a new SUSY-inspired way of avoiding state-dependent counterterms and maintaining chiral symmetry. We walk readers through the construction of their own basic LCT code, sufficient for small truncation cutoffs. We also provide a more sophisticated and comprehensive set of Mathematica packages and demonstrations that can be used to study a variety of 2D models. We guide the reader through these packages with several examples and illustrate how to obtain QFT observables, such as spectral densities and the Zamolodchikov $C$-function. Specific models considered are finite $N_c$ QCD, scalar $ϕ^4$ theory, and Yukawa theory.
We use the method of Lightcone Conformal Truncation (LCT) to obtain form factors and spectral densities of local operators $\mathcal{O}$ in $ϕ^4$ theory in two dimensions. We show how to use the Hamiltonian eigenstates from LCT to obtain form factors that are matrix elements of a local operator $\mathcal{O}$ between single-particle bra and ket states, and we develop methods that significantly reduce errors resulting from the finite truncation of the Hilbert space. We extrapolate these form factors as a function of momentum to the regime where, by crossing symmetry, they are form factors of $\m…
We study the chiral limit of 2d QCD with a single quark flavor at finite $N_c$ using LCT. By modifying the LCT basis according to the quark mass in a manner motivated by 't Hooft's analysis, we are able to restore convergence for quark masses much smaller than the QCD strong coupling scale. For such small quark masses, the IR of the theory is expected to be well described by the Sine-Gordon model. We verify that LCT numerics are able to capture in detail the spectrum and correlation functions of the Sine-Gordon model. This opens up the possibility for studying deformations of various integrabl…
A recent paper by Gross and Erler (hep-th/0406199) showed that by making a certain well-defined, unitary transformation on the mode basis for the open bosonic string--one that identifies the lightcone component of position with the string midpoint--it is possible to render the action for cubic string field theory local in lightcone time. In this basis, then, cubic string field theory possesses a well-defined initial value formulation and a conserved Hamiltonian. With this new understanding it seems natural to study time dependent solutions representing the the decay of an unstable D-branes. In…
We use Lightcone Conformal Truncation (LCT) -- a version of Hamiltonian truncation -- to study the nonperturbative, real-time dynamics of $ϕ^4$-theory in 2+1 dimensions. This theory has UV divergences that need to be regulated. We review how, in a Hamiltonian framework with a total energy cutoff, renormalization is necessarily \emph{state-dependent}, and UV sensitivity cannot be canceled with standard local operator counterterms. To overcome this problem, we present a prescription for constructing the appropriate state-dependent counterterms for (2+1)d $ϕ^4$-theory in lightcone quantization. W…
Strongly-coupled Quantum Field Theories (QFTs) are ubiquitous in high energy physics and many-body physics, yet our ability to do precise computations in such systems remains limited. Hamiltonian Truncation is a method for doing nonperturbative computations of real-time evolution in strongly coupled QFT in the continuum limit, and works by numerically solving the Schrodinger equation in a truncated subspace of the full Hilbert space. Recent advances in understanding this method have opened the door to progress in a range of applications, from gauge theories in $d\ge 2$ dimensions to relativist…
We initiate the application of Hamiltonian Truncation methods to solve strongly coupled QFTs in $d=2+1$. By analysing perturbation theory with a Hamiltonian Truncation regulator, we pinpoint the challenges of such an approach and propose a way that these can be addressed. This enables us to formulate Hamiltonian Truncation theory for $ϕ^4$ in $d=2+1$, and to study its spectrum at weak and strong coupling. The results obtained agree well with the predictions of a weak/strong self-duality possessed by the theory. The $ϕ^4$ interaction is a strongly relevant UV divergent perturbation, and represe…
Quantum Field Theories (QFTs) in Anti-de Sitter (AdS) spacetime are often strongly coupled when the radius of AdS is large, and few methods are available to study them. In this work, we develop a Hamiltonian truncation method to compute the energy spectrum of QFTs in two-dimensional AdS. The infinite volume of constant timeslices of AdS leads to divergences in the energy levels. We propose a simple prescription to obtain finite physical energies and test it with numerical diagonalization in several models: the free massive scalar field, $ϕ^4$ theory, Lee-Yang and Ising field theory. Along the …
We combine the methods of Hamiltonian Truncation and the recently proposed generalisation of the S-matrix bootstrap that includes local operators to determine the two-particle scattering amplitude and the two-particle form factor of the stress tensor at $s>0$ in the 2d $ϕ^4$ theory. We use the form factor of the stress tensor at $s\le 0$ and its spectral density computed using Lightcone Conformal Truncation (LCT), and inject them into the generalized S-matrix bootstrap set-up. The obtained results for the scattering amplitude and the form factor are fully reliable only in the elastic regime. W…
We set up and analyze the lightcone Hamiltonian for an abelian Chern-Simons field coupled to $N_f$ fermions in the limit of large $N_f$ using conformal truncation, i.e. with a truncated space of states corresponding to primary operators with dimension below a maximum cutoff $Δ_{\rm max}$. In both the Chern-Simons theory, and in the $O(N)$ model at infinite $N$, we compute the current spectral functions analytically as a function of $Δ_{\rm max}$ and reproduce previous results in the limit that the truncation $Δ_{\rm max}$ is taken to $\infty$. Along the way, we determine how to preserve gauge …
We present a prescription for an effective lightcone (LC) Hamiltonian that includes the effects of zero modes, focusing on the case of Conformal Field Theories (CFTs) deformed by relevant operators. We show how the prescription resolves a number of issues with LC quantization, including i) the apparent non-renormalization of the vacuum, ii) discrepancies in critical values of bare parameters in equal-time vs LC quantization, and iii) an inconsistency at large N in CFTs with simple AdS duals. We describe how LC quantization can drastically simplify Hamiltonian truncation methods applied to some…
Hamiltonian Truncation Methods are a useful numerical tool to study strongly coupled QFTs. In this work we present a new method to compute the exact corrections, at any order, in the Hamiltonian Truncation approach presented by Rychkov et al. in Refs. [1-3]. The method is general but as an example we calculate the exact $g^2$ and some of the $g^3$ contributions for the $ϕ^4$ theory in two dimensions. The coefficients of the local expansion calculated in Ref. [1] are shown to be given by phase space integrals. In addition we find new approximations to speed up the numerical calculations and imp…
Hamiltonian Truncation (a.k.a. Truncated Spectrum Approach) is an efficient numerical technique to solve strongly coupled QFTs in d=2 spacetime dimensions. Further theoretical developments are needed to increase its accuracy and the range of applicability. With this goal in mind, here we present a new variant of Hamiltonian Truncation which exhibits smaller dependence on the UV cutoff than other existing implementations, and yields more accurate spectra. The key idea for achieving this consists in integrating out exactly a certain class of high energy states, which corresponds to performing re…
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, $\mathcal{C}$. The relevant deformation is then considered using lightcone quantization, with the resulting Hamiltonian expressed in terms of this CFT basis. Truncating to states with $\mathcal{C} \leq \mathcal{C}_{\max}$, one can numerically find the resulting spectrum, as well as other dynamical quantities, such as spectral densities of operators. This method re…
Hamiltonian Truncation (a.k.a. Truncated Spectrum Approach) is a numerical technique for solving strongly coupled QFTs, in which the full Hilbert space is truncated to a finite-dimensional low-energy subspace. The accuracy of the method is limited only by the available computational resources. The renormalization program improves the accuracy by carefully integrating out the high-energy states, instead of truncating them away. In this paper we develop the most accurate ever variant of Hamiltonian Truncation, which implements renormalization at the cubic order in the interaction strength. The n…
With the purpose of holographically describing flows from a large family of four dimensional ${\cal N}=1$ and ${\cal N}=2$ conformal field theories, we discuss truncations of seven dimensional supergravity to five dimensions. We write explicitly the reduced gauged supergravity and find BPS equations for simple configurations. Lifting these flows to eleven dimensions or Massive IIA supergravity, we present string duals to RG flows from strongly coupled conformal theories when deformed by marginal and/or relevant operators. We further discuss observables common to infinite families of ${\cal N}=…
We consider Lorentzian CFT Wightman functions in momentum space. In particular, we derive a set of reference formulas for computing two- and three-point functions, restricting our attention to three-point functions where the middle operator (corresponding to a Hamiltonian density) carries zero spatial momentum, but otherwise allowing operators to have arbitrary spin. A direct application of our formulas is the computation of Hamiltonian matrix elements within the framework of conformal truncation, a recently proposed method for numerically studying strongly-coupled QFTs in real time and infini…
We use Lightcone Conformal Truncation to analyze the RG flow of the two-dimensional supersymmetric Gross-Neveu-Yukawa theory, i.e. the theory of a real scalar superfield with a $\mathbb{Z}_2$-symmetric cubic superpotential. The theory depends on a single dimensionless coupling $\bar{g}$, and is expected to have a critical point at a tuned value $\bar{g}_*$ where it flows in the IR to the Tricritical Ising Model (TIM); the theory spontaneously breaks the $\mathbb{Z}_2$ symmetry on one side of this phase transition, and breaks SUSY on the other side. We calculate the spectrum of energies as a fu…
We prove that any D-dimensional theory comprising gravity, an antisymmetric n-index field strength and a dilaton can be consistently reduced on S^n in a truncation in which just $n$ scalar fields and the metric are retained in (D-n)-dimensions, provided only that the strength of the couping of the dilaton to the field strength is appropriately chosen. A consistent reduction can then be performed for n\le 5; with D being arbitrary when n\le 3, whilst D\le 11 for n=4 and D\le 10 for n=5. (Or, by Hodge dualisation, $n$ can be replaced by (D-n) in these conditions.) We obtain the lower dimensional…
We describe a nonperturbative method to compute the partition function and correlation functions for scalar QFTs set on the $d$-dimensional sphere $S^d$. The method relies on a Hamiltonian picture, where the theory is quantized on $S^{d-1}$ and states evolve in time by means of a time-dependent Hamiltonian. Crucially, the Hilbert space on $S^{d-1}$ is truncated to a finite set of states below a cutoff. Throughout this work we focus on the $ϕ^2$ and $iϕ^3$ flows in three dimensions. In the first part of this paper we analyze the cutoff-dependence of various observables, computing both divergent…
We study 1+1 dimensional $ϕ^4$ theory using the recently proposed method of conformal truncation. Starting in the UV CFT of free field theory, we construct a complete basis of states with definite conformal Casimir, $\mathcal{C}$. We use these states to express the Hamiltonian of the full interacting theory in lightcone quantization. After truncating to states with $\mathcal{C} \leq \mathcal{C}_{\max}$, we numerically diagonalize the Hamiltonian at strong coupling and study the resulting IR dynamics. We compute non-perturbative spectral densities of several local operators, which are equivalen…
