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arXiv:2412.12255 · Journal of High Energy Physics

Toolkit for general 2d scalar potential in LCT

A. Liam Fitzpatrick, Emanuel Katz, Yuan Xin

Published Jun 19, 2025• 1 citation
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Abstract

A bstract We present efficient algorithms for obtaining the Hamiltonian in Lightcone Conformal Truncation (LCT) for a 2d scalar field with a generic potential. We apply this method to the sine-Gordon and sinh-Gordon models in 1 +1 d , and find precise agreement with integrability results when the scaling dimension ∆ of the deforming cosine/cosinh potential is in the range ∆ ≤ 1. The agreement provides additional evidence for a recent conjecture for how to compute the effective lightcone Hamiltonian in this class of models. In addition, to high precision, we provide the first direct confirmation for the conjectured self-duality of the sinh-Gordon model (∆ < 0), which relates ∆ ↔ 4/∆. As the dimension approaches the upper limit ∆ = 1 from below, we show analytically that the Hamiltonian matrix elements exactly reproduce those of a free Majorana fermion, demonstrating how bosonization is manifested in the LCT basis. We comment on the possible extension of the approach to ∆ > 1.

DOI
10.1007/jhep06(2025)193

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A bstract We present efficient algorithms for obtaining the Hamiltonian in Lightcone Conformal Truncation (LCT) for a 2d scalar field with a generic potential. We apply this method to the sine-Gordon and sinh-Gordon models in 1 +1 d , and find precise agreement with integrability results when the scaling dimension ∆ of the deforming cosine/cosinh potential is in the range ∆ ≤ 1. The agreement provides additional evidence for a recent conjecture for how to compute the effective lightcone Hamiltonian in this class of models. In addition, to high precision, we provide the first direct confirmation for the conjectured self-duality of the sinh-Gordon model (∆ < 0), which relates ∆ ↔ 4/∆. As the dimension approaches the upper limit ∆ = 1 from below, we show analytically that the Hamiltonian matrix elements exactly reproduce those of a free Majorana fermion, demonstrating how bosonization is manifested in the LCT basis. We comment on the possible extension of the approach to ∆ > 1.
Quantum Field Theory (QFT) is the language that describes a wide spectrum of physics. However, it is notoriously hard at strong coupling regime. We approach this problem in an old Quantum Mechanical method - keep a finite number of states and diagonalize the Hamiltonian as a finite-size matrix. To study a QFT, we take the Hamiltonian to be the Conformal Field Theory as the Ultraviolet fixed point of the theory's Renormalization Group Flow, deformed by a relevant operator. We use a recent framework known as the Lightcone Conformal Truncation (LCT), where we use conformal basis and lightcone quantization. As an application of the method, we study the two dimensional Supersymmetric (SUSY) Gross-Neveu-Yukawa Model. The model is expected to have a critical point in the universality class of tri-critical Ising model, a massive phase and a massless SUSY-breaking phase. We use the LCT to compute the spectrum and the spectral density of the theory at all couplings and map the entire phase diagram.
Abstract We use the method of Lightcone Conformal Truncation (LCT) to obtain form factors and spectral densities of local operators 𝒪 in ϕ4 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 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 𝒪 between the vacuum and a two-particle asymptotic scattering state. We also compute the momentum-space time-ordered two-point functions of local operators in LCT. These converge quickly at momenta away from branch cuts, allowing us to indirectly obtain the time-ordered correlator and the spectral density at the branch cuts. We focus on the case where the local operator 𝒪 is the trace Θ of the stress tensor.
Lightcone Hamiltonian for Ising field theory I: T<TcT < T_cT<Tc​
We study 2d Ising Field Theory (IFT) in the low-temperature phase in lightcone quantization, and show that integrating out zero modes generates a very compact form for the effective lightcone interaction that depends on the finite volume vacuum expectation value of the \sigma σ operator. This form is most naturally understood in a conformal basis for the lightcone Hilbert space. We further verify that this simple form reproduces to high accuracy results for the spectra, the c c -function, and the form-factors from integrability methods for the magnetic deformation of IFT. For generic non-integrable values of parameters we also compute the above observables and compare our numeric results to those of equal-time truncation. In particular, we report on new measurements of various bound-state form-factors as well as the stress-tensor spectral density. We find that the stress tensor spectral density provides additional evidence that certain resonances of IFT are surprisingly narrow, even at generic strong coupling. Explicit example code for constructing the effective Hamiltonian is included in an appendix.
A bstract 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 state-dependent , and UV sensitivity cannot be canceled with standard local operator counter-terms. 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. We then use LCT with this counterterm prescription to study ϕ 4 -theory, focusing on the ℤ 2 symmetry-preserving phase. Specifically, we compute the spectrum as a function of the coupling and demonstrate the closing of the mass gap at a (scheme-dependent) critical coupling. We also compute Lorentz-invariant two-point functions, both at generic strong coupling and near the critical point, where we demonstrate IR universality and the vanishing of the trace of the stress tensor.
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 C≤Cmax⁡ \mathcal{C}\le {\mathcal{C}}_{\max } C≤Cmax​ , 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.
A bstract We develop a novel approach to the Wilsonian renormalisation of Hamiltonians for 2-dimensional quantum field theories on the cylinder described in the UV by marginally relevant deformations of conformal field theories. To introduce a Wilsonian short-distance cutoff we make essential use of free field realisations of the full vertex operator algebra in the UV. Our method is intrinsically non-perturbative; we derive a Hamiltonian analogue of Polchinski’s equation describing the flows of all couplings. As a primary example of our general method, we apply it to the marginal anisotropic deformation of the su2 {\mathfrak{su}}_2 su2​ su 2 Wess-Zumino-Witten model at level 1, which is equivalent to the sine-Gordon model on the cylinder. In particular, we reproduce the standard renormalisation group flow of the sine-Gordon model near the Kosterlitz-Thouless point to second order in the couplings, a result usually derived using Lagrangian/path-integral methods.