Hamiltonian Truncation with larger dimensions

Abstract

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 ∆. UV divergences arise when ∆ 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 ∆ ≥ d/2 + 1/4, revealing a striking breakdown of locality. Our claim is based on the analysis of conformal perturbation theory up to fourth order. As an example we compute the Casimir energy of d = 2 Minimal Models perturbed by operators whose dimensions take values on either side of the threshold d/2 + 1/4.

Effective Hamiltonians and Counterterms for Hamiltonian Truncation

“…In ref. [20], we showed that the Hamiltonian Truncation UV regulator gives results that are inconsistent with those derived using a local regularisation (such as a short distance cutoff), implying that non-local counterterms are needed to implement renormalisation in Hamiltonian Truncation.…”

“…In (3.5) we have set R = 1, in the rest of the paper we measure all dimensionful quantities in units of R. Similar expressions can be derived for the excited states. Further details can be found in [20].…”

“…Furthermore, the use of the ∆ T -cutoff to regularise the theory introduces UV divergences at higher orders than would have been present had a local regulator been used: in ref. [20] it was shown that for ∆ ∈ [d/2 + 1/4, 3d/4), the fourth order correction to the Casimir energy is finite if a local regulator is employed but diverges if the ∆ T cutoff is used.…”

“…Here we perform a check using perturbation theory that the effective Hamiltonian constructed following the procedure outlined in section 3 contains the extra interactions required to cancel the UV divergences found in ref. [20] which were inconsistent with local regularisation.…”

“…See for example refs. [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25], as well as the reviews [26,27]. A particularly important focus for research in this area has been on the development of effective Hamiltonians for improving the accuracy of Hamiltonian Truncation estimates [4,[28][29][30][31][32][33][34].…”