We initiate the application of Hamiltonian Truncation methods to solve strongly coupled QFTs in $d=2+1$. By analysing perturbation theory with a Hamiltonian Truncation regulator, we pinpoint the challenges of such an approach and propose a way that these can be addressed. This enables us to formulate Hamiltonian Truncation theory for $ϕ^4$ in $d=2+1$, and to study its spectrum at weak and strong coupling. The results obtained agree well with the predictions of a weak/strong self-duality possessed by the theory. The $ϕ^4$ interaction is a strongly relevant UV divergent perturbation, and represents a case study of a more general scenario. Thus, the approach developed should be applicable to many other QFTs of interest.
Hamiltonian Truncation Methods are a useful numerical tool to study strongly coupled QFTs. In this work we present a new method to compute the exact corrections, at any order, in the Hamiltonian Truncation approach presented by Rychkov et al. in Refs. [1-3]. The method is general but as an example we calculate the exact $g^2$ and some of the $g^3$ contributions for the $ϕ^4$ theory in two dimensions. The coefficients of the local expansion calculated in Ref. [1] are shown to be given by phase space integrals. In addition we find new approximations to speed up the numerical calculations and imp…
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 Phi^4 theory in the phase where the Z_2-symmetry is spontaneously broken, complementing our earlier study of the Z_2-invariant phase and of the critical point. We also check numerically the weak/strong duality of this theory discussed long ago by Chang.
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 $H_0$ with eigenvalues below some energy cutoff $E_\text{max}$. 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 $E_\text{max}$. The effective Hamiltonian c…
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 \emph{state-dependent}, and UV sensitivity cannot be canceled with standard local operator counterterms. 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. W…
We study Hamiltonian truncation in boosted frames. We consider the thermal and magnetic field deformations of the 2d Ising model using TCSA at finite momentum. We find that even with moderate momenta, the spectrum and time-dependent correlation functions become significantly less dependent on the volume of the system. This allows for a more reliable determination of infinite volume observables.
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 $Δ$. UV divergences arise when $Δ$ 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 $Δ\geq d/2+1/4$, revealing a striking breakdown of localit…
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 re…
Strongly-coupled Quantum Field Theories (QFTs) are ubiquitous in high energy physics and many-body physics, yet our ability to do precise computations in such systems remains limited. Hamiltonian Truncation is a method for doing nonperturbative computations of real-time evolution in strongly coupled QFT in the continuum limit, and works by numerically solving the Schrodinger equation in a truncated subspace of the full Hilbert space. Recent advances in understanding this method have opened the door to progress in a range of applications, from gauge theories in $d\ge 2$ dimensions to relativist…
Hamiltonian Truncation (a.k.a. Truncated Spectrum Approach) is a numerical technique for solving strongly coupled QFTs, 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 n…
We combine the methods of Hamiltonian Truncation and the recently proposed generalisation of the S-matrix bootstrap that includes local operators to determine the two-particle scattering amplitude and the two-particle form factor of the stress tensor at $s>0$ in the 2d $ϕ^4$ theory. We use the form factor of the stress tensor at $s\le 0$ and its spectral density computed using Lightcone Conformal Truncation (LCT), and inject them into the generalized S-matrix bootstrap set-up. The obtained results for the scattering amplitude and the form factor are fully reliable only in the elastic regime. W…
Higher-order structure of Hamiltonian truncation effective theory
Università degli Studi di Pavia（帕维亚大学）；Dipartimento di Fisica “A. Volta”, Università degli Studi di Pavia（帕维亚大学A.沃尔塔物理系）；Istituto Nazionale di Fisica Nucleare, Sezione di Pavia（意大利国家核物理研究所帕维亚分部）
We defend the Fock-space Hamiltonian truncation method, which allows 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 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…
We implement the Rayleigh-Ritz method in supersymmetric quantum mechanics with flat directions, and extract the S-matrix and metastable resonances. The effectiveness of the method is demonstrated in two strongly coupled systems: an N=1 toy supermembrane model, and an N=4 model with a U(1) gauge multiplet and a charged chiral multiplet.
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 …
We consider Lorentzian CFT Wightman functions in momentum space. In particular, we derive a set of reference formulas for computing two- and three-point functions, restricting our attention to three-point functions where the middle operator (corresponding to a Hamiltonian density) carries zero spatial momentum, but otherwise allowing operators to have arbitrary spin. A direct application of our formulas is the computation of Hamiltonian matrix elements within the framework of conformal truncation, a recently proposed method for numerically studying strongly-coupled QFTs in real time and infini…
In contrast to lattice systems where powerful numerical techniques such as matrix product state based methods are available to study the non-equilibrium dynamics, the non-equilibrium behaviour of continuum systems is much harder to simulate. We demonstrate here that Hamiltonian truncation methods can be efficiently applied to this problem, by studying the quantum quench dynamics of the 1+1 dimensional Ising field theory using a truncated free fermionic space approach. After benchmarking the method with integrable quenches corresponding to changing the mass in a free Majorana fermion field theo…
Toward Hamiltonian simulations of Maxwell-Chern-Simons theory: constant modes and gauge field truncation
University of Bonn（波恩大学）；Helmholtz Institute for Radiation and Nuclear Physics (HISKP)（赫尔姆霍兹辐射与核物理研究所 (HISKP)）；Transdisciplinary Research Area “Building Blocks of Matter and Fundamental Interactions” (TRA Matter)（跨学科研究领域“物质基本组成与基本相互作用”(TRA Matter)）；University of Perugia（佩鲁贾大学）；INFN（意大利国家核物理研究所）；Dipartimento di Fisica e Geologia（物理与地质学系）；NiPS Laboratory（NiPS实验室）；Northeastern University - London（伦敦东北大学）；Khoury College of Computer Sciences, Northeastern University（东北大学库里计算机学院）；The Cyprus Institute（塞浦路斯研究所）；Deutsches Elektronen-Synchrotron DESY（德国电子同步加速器中心 (DESY)）
Entanglement is the fundamental difference between classical and quantum systems and has become one of the guiding principles in the exploration of high- and low-energy physics. The calculation of entanglement entropies in interacting quantum field theories, however, remains challenging. Here, we present the first method for the explicit computation of reduced density matrices of interacting quantum field theories using truncated Hamiltonian methods. The method is based on constructing an isomorphism between the Hilbert space of the full system and the tensor product of Hilbert spaces of sub-i…
We describe a nonperturbative method to compute the partition function and correlation functions for scalar QFTs set on the $d$-dimensional sphere $S^d$. The method relies on a Hamiltonian picture, where the theory is quantized on $S^{d-1}$ and states evolve in time by means of a time-dependent Hamiltonian. Crucially, the Hilbert space on $S^{d-1}$ is truncated to a finite set of states below a cutoff. Throughout this work we focus on the $ϕ^2$ and $iϕ^3$ flows in three dimensions. In the first part of this paper we analyze the cutoff-dependence of various observables, computing both divergent…
We both review and augment the lightcone conformal truncation (LCT) method. LCT is a Hamiltonian truncation method for calculating dynamical quantities in QFT in infinite volume. This document is a self-contained, pedagogical introduction and "how-to" manual for LCT. We focus on 2D QFTs which have UV descriptions as free CFTs containing scalars, fermions, and gauge fields, providing a rich starting arena for LCT applications. Along our way, we develop several new techniques and innovations that greatly enhance the efficiency and applicability of LCT. These include the development of CFT radial…
We apply a method of group averaging to states and operators appearing in (truncations of) the Spin(9) x SU(N) invariant matrix models. We find that there is an exact correspondence between the standard supersymmetric Hamiltonian and the Spin(9) average of a relatively simple lower-dimensional model.
Probing weak chaos in $\mathcal N=4$ super Yang-Mills and long-range spin chains
Stockholm University（斯德哥尔摩大学）；Yukawa Institute for Theoretical Physics, Kyoto University（京都大学汤川理论物理研究所）；University of Warsaw（华沙大学）；Weizmann Institute of Science（魏茨曼科学研究所）；MIT（麻省理工学院）；Julius-Maximilians-Universität Würzburg（维尔茨堡大学）；CUNEF Universidad（CUNEF大学）
