Nonperturbative dynamics of (2+1)d ϕ4-theory from Hamiltonian truncation

Abstract

We use Lightcone Conformal Truncation (LCT)—a version of Hamiltonian truncation — to study the nonperturbative, real-time dynamics of ϕ4-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 state-dependent, and UV sensitivity cannot be canceled with standard local operator counter-terms. To overcome this problem, we present a prescription for constructing the appropriate state-dependent counterterms for (2+1)d ϕ4-theory in lightcone quantization. We then use LCT with this counterterm prescription to study ϕ4-theory, focusing on the ℤ2 symmetry-preserving phase. Specifically, we compute the spectrum as a function of the coupling and demonstrate the closing of the mass gap at a (scheme-dependent) critical coupling. We also compute Lorentz-invariant two-point functions, both at generic strong coupling and near the critical point, where we demonstrate IR universality and the vanishing of the trace of the stress tensor.

Bootstrapping 2d ϕ4 theory with Hamiltonian truncation data

“…More ambitiously, our method in principle can be applied to higher dimensions, the main challenge being that it is difficult to obtain the input data. LCT has been applied to the φ 4 model in 3d, and the stress tensor spectral density was obtained in [68]. 19 One would have to generalize our treatment of form factors to 3d, but the basic idea would be the same.…”

“…for recent LCT and DLCQ applications to 2d QCD 19. Both lightcone and equal-time Hamiltonian truncation have seen important recent progress for φ 4 theory in d > 2[68][69][70][71]. One of the main challenges has been dealing with state-dependent counterterms for divergences.…”

Hamiltonian truncation in Anti-de Sitter spacetime

“…It is worth noticing that, due to the mentioned UV/IR connection, the manifestation of the problem in perturbation theory, i.e. the state-dependence of the contributions of intermediate states close to the cutoff, is analogous to the situation in flat space when authentic UV divergences are present [12,26]. Contrary to the latter, though, the infinite volume divergences we are concerned with all come from disconnected contributions.…”

“…(2.13), vanishes. 12 From the point of view of the boundary, the condition (2.14) corresponds to the fine tuning of relevant operators at leading order inλ. Therefore, it is not surprising that it may not be sufficient for the theory to be well defined at finiteλ.…”

“…See the recent work[9] for a lattice Monte Carlo approach to address the finite-coupling regime in AdS2 7. In lightcone conformal truncation, the need to work with a finite basis of states leads to an effective IR cutoff, see for instance the discussion in appendix F of[12].…”