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arXiv:2104.10689 · Journal of High Energy Physics

Hamiltonian truncation in Anti-de Sitter spacetime

Matthijs Hogervorst, Marco Meineri, João Penedones, Kamran Salehi Vaziri

Published Aug 16, 2021• 22 citations
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Abstract

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.

DOI
10.1007/jhep08(2021)063

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We study interacting scalar quantum field theories in anti-de Sitter (AdS) space using a tensor-network approach based on matrix product states. We develop methods for extracting low-lying boundary spectra from global-AdS energy levels, probing bulk criticality through finite-size scaling in the AdS radius, and identifying conformal boundary conditions. Applying these methods to scalar ϕ4ϕ^4ϕ4 theory in AdS2\mathrm{AdS}_2AdS2​, we compute the lowest non-trivial Z2\mathbb{Z}_2Z2​-odd and even boundary scaling dimensions non-perturbatively. We then locate the bulk Z2\mathbb{Z}_2Z2​ symmetry-breaking transition and extract critical exponents and central charge consistent with the 2D Ising universality class. At the critical point, we determine the low-lying boundary spectrum and use it to identify the Z2\mathbb{Z}_2Z2​-preserving free/ordinary Ising conformal boundary condition, thereby characterising both the bulk and boundary universality classes.
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.
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.
We compute entanglement entropy for free massive scalar fields in anti-de Sitter (AdS) space. The entangling surface is a minimal surface whose boundary is a sphere at the boundary of AdS. The entropy can be evaluated from the thermal free energy of the fields on a topological black hole by using the replica method. In odd-dimensional AdS, exact expressions of the Rényi entropy S n are obtained for arbitrary n. We also evaluate 1-loop corrections coming from the scalar fields to holographic entanglement entropy. Applying the results, we compute the leading difference of entanglement entropy between two holographic CFTs related by a renormalization group flow triggered by a double trace deformation. The difference is proportional to the shift of a central charge under the flow.
The goal of these notes is to introduce, in a very elementary way, the idea of the anti de-Sitter/Conformal Field Theory (AdS/CFT) correspondence to condensed matter physicists. This theory relates a gravity theory in a (d+1)- dimensional anti-de Sitter spacetime to a strongly-coupled d- dimensional quantum field theory living on its boundary. The AdS/CFT correspondence can be used to study finite temperature real time processes, such as response functions and dynamics far from equilibrium in quantum critical points in condensed matter systems. Computation of these quantities is reduced to solving classical gravitational equations in one higher dimension than the original theory.