High-precision calculations in strongly coupled quantum field theory with next-to-leading-order renormalized Hamiltonian Truncation

Abstract

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. Keywords: Field Theories in Lower Dimensions, Nonperturbative EffectsArXiv ePrint: 1706.06121Open Access, c The Authors. Article funded by SCOAP 3 .https://doi.org/10.1007/JHEP10 (2017)213 JHEP10 (2017)213 Introduction. Many interesting strongly interacting Quantum Field Theories (QFTs) are not amenable to analytical treatment. Such theories are often studied via Lattice Monte Carlo (LMC) numerical simulations, starting from the discretized Euclidean action. However, LMC has some drawbacks, for example it cannot easily compute real-time observables, it is rather computationally expensive, and it cannot directly describe renormalization group (RG) flows starting from interacting fixed points. Therefore, it is worth exploring other numerical approaches to strongly interacting QFTs. One promising alternative is provided by the Hamiltonian methods, which look for the eigenstates of the quantum Hamiltonian. These methods use various finite-dimensional approximations to the full infinite-dimensional QFT Hilbert space. Notable examples are the methods using Matrix Product States [1, 2] and more general Tensor Networks [3] such as MERA [4] or PEPS [5]. In this paper we will be concerned with another representative of this group of methods -Hamiltonian Truncation (HT), also known as the Truncated Spectrum (or Space) Approach, which is a direct generalization of the variational Rayleigh-Ritz (RR) method from quantum mechanics. This method goes back to the seminal work of Yurov and Al. Zamolodchikov [6,7] and has since been applied in many contexts. See [8] for a recent extensive review and the bibliography.The idea of HT is simple. The QFT Hamiltonian operator H is split as H 0 + V where H 0 is an exactly solvable Hamiltonian whose eigenstates form the basis of the Hilbert space. One quantizes at surfaces of constant time and works in finite volume so that the spectrum is discrete. 1 The Hilbert space is then truncated to the low-lying eigenvectors of H 0 . The matrix of H in this truncated Hilbert space is diagonalized exactly on a computer, to find the low-energy spectrum of interacting eigenstates.As was understood early on [16], the numerical convergence of the HT...

λϕ4 theory — Part II. the broken phase beyond NNNN(NNNN)LO

“…It also features a simple but non trivial duality symmetry, Chang duality [4]. For these reasons the φ 4 theory is an ideal laboratory to possibly test new methods, or improve on old ones, for studying quantum field theories at strong coupling, such as lattice simulations [5][6][7][8], hamiltonian truncations [9][10][11][12][13][14][15][16][17][18] or resummation of the perturbative series [19][20][21][22][23]1]. In the context of resummations, Figure 1: Phase structure of the φ 4 theory according to Chang duality, as a function of the couplings g and g. Points in the same vertical line connecting the black and red lines correspond to different descriptions of the same theory.…”

Simple non-perturbative resummation schemes beyond mean-field: case study for scalar ϕ4 theory in 1+1 dimensions

“…While recent works such as Refs. [4,5,7,8] consistently find g c 2.8, the critical coupling in the R3 approximation is approximately g c 2.2. Curiously, both the vacuum energy and the critical exponent c 2 = 0.12 (5) in the R3 approximation are numerically close to the corresponding results from precision calculations.…”

“…1, where also comparison to recent precision calculations of the same quantities from Ref. [7,8] are shown. As can be seen from Fig.…”

“…1, R2 results for the mass gap and vacuum energy are qualitatively similar to those obtained from Refs. [7,8], but show clear quantitative deviations. Moreover, the R2 scheme predicts a phase transition close to g c 1.89 with a critical exponent c 1 that is not consistent with the exact value c 1 = 1.…”

Simple non-perturbative resummation schemes beyond mean-field II: Thermodynamics of scalar ϕ4 theory in 1 + 1 dimensions at arbitrary coupling

“…2 To avoid confusion, re-instating powers of λ, the zero temperature contribution − 35,36] when requiring the cosmological constant to vanish at λ = 0. Thus, the renormalization condition adopted in this work differs from these references.…”