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arXiv:1412.3460 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Hamiltonian truncation study of the φ 4 theory in two dimensions

Slava Rychkov, Lorenzo G. Vitale

Published Apr 8, 2015• 149 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevd.91.085011

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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 \ensuremathϕ4{\ensuremath{\phi}}^{4}\ensuremathϕ4 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.
Hamiltonian Truncation (a.k.a. Truncated Spectrum Approach) is an efficient numerical technique to solve strongly coupled QFTs in d = 2 spacetime dimensions. Further theoretical developments are needed to increase its accuracy and the range of applicability. With this goal in mind, here we present a new variant of Hamiltonian Truncation which exhibits smaller dependence on the UV cutoff than other existing implementations, and yields more accurate spectra. The key idea for achieving this consists in integrating out exactly a certain class of high energy states, which corresponds to performing renormalization at the cubic order in the interaction strength. We test the new method on the strongly coupled two-dimensional quartic scalar theory. Our work will also be useful for the future goal of extending Hamiltonian Truncation to higher dimensions d ≥ 3.
Hamiltonian truncation (also known as ``truncated spectrum approach'') is a numerical technique for solving strongly coupled quantum field theories, in which the full Hilbert space is truncated to a finite-dimensional low-energy subspace. The accuracy of the method is limited only by the available computational resources. The renormalization program improves the accuracy by carefully integrating out the high-energy states, instead of truncating them away. In this paper, we develop the most accurate ever variant of Hamiltonian Truncation, which implements renormalization at the cubic order in the interaction strength. The novel idea is to interpret the renormalization procedure as a result of integrating out exactly a certain class of high-energy ``tail states.'' We demonstrate the power of the method with high-accuracy computations in the strongly coupled two-dimensional quartic scalar theory and benchmark it against other existing approaches. Our work will also be useful for the future goal of extending Hamiltonian truncation to higher spacetime dimensions.
<!--HTML-->Hamiltonian truncation is a non-perturbative numerical method for calculating observables of a quantum field theory by truncating the Hilbert space to states with energy below a maximum energy cutoff. In this talk I will present an effective field theory approach to Hamiltonian truncation, which provides a systematic way of improving the calculations without increasing the energy cutoff. I will demonstrate this with numerical results for the two dimensional phi^4 theory, and talk about future applications of this method to more complicated theories.
The Fock-space Hamiltonian truncation method is developed further, paying particular attention to the treatment of the scalar field zero mode. This is applied to the two-dimensional ${\ensuremath{\phi}}^{4}$ theory in the phase where the Z2{\mathbb{Z}}_{2}Z2​-symmetry is spontaneously broken, complementing our earlier study of the Z2{\mathbb{Z}}_{2}Z2​-invariant phase and of the critical point. We also check numerically the weak/strong duality of this theory discussed long ago by Chang.
We study the spectrum of two-dimensional Yukawa theory using Hamiltonian truncation, with a focus on bound states. We extend the framework of Hamiltonian truncation effective theory to include fermions and logarithmic UV divergences. With these corrections, the energy levels converge as a function of the cutoff with the scaling predicted by power counting. We identify a strategy to follow the evolution of fermion-antifermion bound states from weak to strong coupling, and investigate its binding energy, finding good agreement with finite-volume perturbation theory at weak coupling. Finally, we explore the heavy-scalar regime toward the massive Thirring/Sine-Gordon limit, recovering the expected weak-coupling behavior and outlining how the calculation can be extended toward this limit.
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.