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arXiv:2107.10285 Β· Journal of High Energy Physics

Form factors and spectral densities from Lightcone Conformal Truncation

Hongbin Chen, A. Liam Fitzpatrick, Denis Karateev

Published Apr 20, 2022β€’ 11 citations
arXivHTMLPDF

Abstract

Abstract We use the method of Lightcone Conformal Truncation (LCT) to obtain form factors and spectral densities of local operators π’ͺ in Ο•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 π’ͺ 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 π’ͺ 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 π’ͺ is the trace Θ of the stress tensor.

DOI
10.1007/jhep04(2022)109

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Abstract We use the method of Lightcone Conformal Truncation (LCT) to obtain form factors and spectral densities of local operators π’ͺ in Ο•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 π’ͺ 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 π’ͺ 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 π’ͺ is the trace Θ of the stress tensor.
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.
A bstract We use Lightcone Conformal Truncation (LCT)β€”a version of Hamiltonian truncation β€” to study the nonperturbative, real-time dynamics of Ο• 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 Ο• 4 -theory in lightcone quantization. We then use LCT with this counterterm prescription to study Ο• 4 -theory, focusing on the β„€ 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.
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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 Οƒ 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 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 βˆ† of the deforming cosine/cosinh potential is in the range βˆ† ≀ 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 (βˆ† < 0), which relates βˆ† ↔ 4/βˆ†. As the dimension approaches the upper limit βˆ† = 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 βˆ† > 1.
We study 1+1 dimensional Ο• 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, C \mathcal{C} C . We use these states to express the Hamiltonian of the full interacting theory in lightcone quantization. After truncating to states with C≀Cmax⁑ \mathcal{C}\le {\mathcal{C}}_{\max } C≀Cmax​ , 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 equivalent to real-time, infinite-volume correlation functions. These spectral densities, which include the Zamolodchikov C-function along the full RG flow, are calculable at any value of the coupling. Near criticality, our numerical results reproduce correlation functions in the 2D Ising model.