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arXiv:2312.09221 · SciPost Physics

Hamiltonian truncation crafted for UV-divergent QFTs

Olivier Delouche, Joan Elias Miró, James Ingoldby

Published Apr 19, 2024• 9 citations
arXivHTMLPDF

Abstract

We develop the theory of Hamiltonian Truncation (HT) to systematically study RG flows that require the renormalization of coupling constants. This is a necessary step towards making HT a fully general method for QFT calculations. We apply this theory to a number of QFTs defined as relevant deformations of d=1+1 CFTs. We investigated three examples of increasing complexity: The deformed Ising, Tricritical-Ising, and non-unitary minimal model M(3,7). The first two examples provide a crosscheck of our methodologies against well established characteristics of these theories. The M(3,7) CFT deformed by its \mathbb{Z}_2 ℤ 2 -even operators shows an intricate phase diagram that we clarify. At a boundary of this phase diagram we show that this theory flows, in the IR, to the M(3,5) CFT.

DOI
10.21468/scipostphys.16.4.105

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We develop the theory of Hamiltonian Truncation (HT) to systematically study RG flows that require the renormalization of coupling constants. This is a necessary step towards making HT a fully general method for QFT calculations. We apply this theory to a number of QFTs defined as relevant deformations of d=1+1 CFTs. We investigated three examples of increasing complexity: The deformed Ising, Tricritical-Ising, and non-unitary minimal model M(3,7). The first two examples provide a crosscheck of our methodologies against well established characteristics of these theories. The M(3,7) CFT deformed by its \mathbb{Z}_2 ℤ 2 -even operators shows an intricate phase diagram that we clarify. At a boundary of this phase diagram we show that this theory flows, in the IR, to the M(3,5) CFT.
A bstract Hamiltonian Truncation (HT) methods provide a powerful numerical approach for investigating strongly coupled QFTs. In this work, we develop HT techniques to analyse a specific Renormalization Group (RG) flow recently proposed in refs. [1–3]. These studies put forward Ginzburg-Landau descriptions for the conformal minimal models M (3, 10) and M (3, 8), as well as the RG-flow connecting them. Specifically, the RG-flow is defined by deforming the M (3, 10) with the relevant primary operator ϕ 1 , 7 (whose indices denote its position in the Kac table), yielding M (3, 10) + ϕ 1 , 7 . From the perspective of HT, realising such an RG-flow presents significant challenges, as the ϕ 1 , 7 deformation requires renormalizing the UV theory up to third order in the coupling constant of the deformation. In this study, we carry out the necessary calculations to formulate HT for this theory and numerically investigate the spectrum of M (3, 10) + ϕ 1 , 7 in the large coupling regime, finding strong evidence in favor of the proposed flow.
<!--HTML-->Hamiltonian Truncation (HT) is a numerical approach for calculating observables in a Quantum Field Theory non-perturbatively. This approach can be applied to theories constructed by deforming a conformal field theory with a relevant operator of scaling dimension ∆. In this talk I will review the HT techniques and emphasise few key open problems. I will also discuss the recent efforts to extend these ideas to higher dimensions (d > 2) and for UV divergent relevant operators (d/2 <= ∆ < d).
Abstract Hamiltonian Truncation (HT) is a numerical approach for calculating observables in a Quantum Field Theory non-perturbatively. This approach can be applied to theories constructed by deforming a conformal field theory with a relevant operator of scaling dimension ∆. UV divergences arise when ∆ is larger than half of the spacetime dimension d. These divergences can be regulated by HT or by using a more conventional local regulator. In this work we show that extra UV divergences appear when using HT rather than a local regulator for ∆ ≥ d/2 + 1/4, revealing a striking breakdown of locality. Our claim is based on the analysis of conformal perturbation theory up to fourth order. As an example we compute the Casimir energy of d = 2 Minimal Models perturbed by operators whose dimensions take values on either side of the threshold d/2 + 1/4.
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.
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, 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.
For QFTs in AdS the boundary correlation functions remain conformal even if the bulk theory has a scale. This allows one to constrain RG flows with numerical conformal bootstrap methods. We apply this idea to flows between two-dimensional CFTs, focusing on deformations of the tricritical and ordinary Ising model. We provide non-perturbative constraints for the boundary correlation functions of these flows and compare them with conformal perturbation theory in the vicinity of the fixed points. We also reproduce a completely general constraint on the sign of the TTˉT\bar TTTˉ deformation in two dimensions.