Exploring Hamiltonian Truncation in $\bf{d=2+1}$

Abstract

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.

Introduction to Lightcone Conformal Truncation: QFT Dynamics from CFT Data

“…If there are divergences in the theory, the lack of vacuum bubbles can often help reduce complexities associated with renormalization. This is especially true for Hamiltonian truncation methods, where truncation can introduce state-dependent sensitivities to the cutoff that are more challenging to address (e.g., see [40], as well as [18] for a proposed solution). In addition, the lack of vacuum bubbles also turns off matrix elements in the Hamiltonian where particles are produced from the vacuum, again simplifying the calculation.…”

“…In principle, one could construct all the primaries by writing out the action of K on the space of all monomials of a fixed scaling dimension and solving for the kernel of K. However, it is simpler to construct primary operators recursively by harnessing a result obtained by Penedones in [29]. 18 This result states that, given two holomorphic primary operators A and B in a generalized free theory, there is exactly one composite primary operator constructible using A and B for each non-negative integer . This composite operator is the double-trace operator…”

“…In addition, there have been crucial recent advancements that have greatly enhanced the scope and predictive power of truncation methods. These developments include the systematic incorporation of renormalization effects from high energy states allowing for high-precision studies of various models [11][12][13], progress in numerical diagonalization methods for large matrices [14,15], and formulations of truncation in d > 2 dimensions [16][17][18]. Indeed, we find ourselves in a time of rapid development for truncation methods in QFT.…”

“…Technically, the wavefunctions are monomial symmetric polynomials, but we will use "monomials" for short 18. See also earlier work by Mikhailov[56].…”

Computing the renormalization group flow of two-dimensional $ϕ^4$ theory with tensor networks

“…First, in (2+1)d, the λφ 4 coupling term is less relevant, which makes the method a priori converge slower. Second, it is no longer the case that the continuum Hamiltonian can be defined straightforwardly with normal ordering, and additional counter terms are needed in (2+1)d, requiring a non-trivial regularization (see however the recent advance [56]). Finally, the number of momenta to consider for a fixed energy cutoff is squared from (1+1)d to (2+1)d.…”