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arXiv:2112.09049 · Journal of High Energy Physics

Hamiltonian Truncation with larger dimensions

Joan Elias Miró, James Ingoldby

Published May 23, 2022• 9 citations
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Abstract

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.

DOI
10.1007/jhep05(2022)151

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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.
<!--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).
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 We outline a procedure for applying Hamiltonian Truncation to Quantum Field Theories (QFTs) that have UV divergences. To do this, we derive a novel representation of an Effective Hamiltonian which makes manifest some of its important properties (e.g. the non-perturbative matching of the spectra between the UV theory and the theory described by the Effective Hamiltonian). We check the consistency of our procedure using Conformal Perturbation Theory. Finally we comment on how the Effective Hamiltonian, which incorporates non-local interactions, describes a local QFT.
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 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.
A bstract 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 way, we discuss spontaneous symmetry breaking in AdS and derive a compact formula for perturbation theory in quantum mechanics at arbitrary order. Our results suggest that all conformal boundary conditions for a given theory are connected via bulk renormalization group flows in AdS.