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arXiv:2212.07266 · Journal of High Energy Physics

Effective Hamiltonians and Counterterms for Hamiltonian Truncation

Joan Elias Miró, James Ingoldby

Published Jul 6, 2023• 15 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1007/jhep07(2023)052

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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.
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.
Hamiltonian truncation is a non-perturbative numerical method for calculating observables of a quantum field theory. The starting point for this method is to truncate the interacting Hamiltonian to a finite-dimensional space of states spanned by the eigenvectors of the free Hamiltonian H_0 H 0 with eigenvalues below some energy cutoff E_\text{max} E max . In this work, we show how to treat Hamiltonian truncation systematically using effective field theory methodology. We define the finite-dimensional effective Hamiltonian by integrating out the states above E_\text{max} E max . The effective Hamiltonian can be computed by matching a transition amplitude to the full theory, and gives corrections order by order as an expansion in powers of 1/E_\text{max} 1 / E max . The effective Hamiltonian is non-local, with the non-locality controlled in an expansion in powers of H_0/E_\text{max} H 0 / E max . The effective Hamiltonian is also non-Hermitian, and we discuss whether this is a necessary feature or an artifact of our definition. We apply our formalism to 2D \lambda\phi^4 λ ϕ 4 theory, and compute the the leading 1/E_\text{max}^2 1 / E max 2 corrections to the effective Hamiltonian. We show that these corrections nontrivially satisfy the crucial property of separation of scales. Numerical diagonalization of the effective Hamiltonian gives residual errors of order 1/E_\text{max}^3 1 / E max 3 , as expected by our power counting. We also present the power counting for 3D \lambda \phi^4 λ ϕ 4 theory and perform calculations that demonstrate the separation of scales in this theory.
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.
We defend the Fock-space Hamiltonian truncation method, which allows us to calculate numerically the spectrum of strongly coupled quantum field theories, by putting them in a finite volume and imposing a UV cutoff. The accuracy of the method is improved via an analytic renormalization procedure inspired by the usual effective field theory. As an application, we study the two-dimensional ${\ensuremath{\phi}}^{4}$ theory for a wide range of couplings. The theory exhibits a quantum phase transition between the symmetry-preserving and symmetry-breaking phases. We extract quantitative predictions for the spectrum and the critical coupling and make contact with previous results from the literature. Future directions to further improve the accuracy of the method and enlarge its scope of applications are outlined.