Higher-Order Structure of Hamiltonian Truncation Effective Theory

Abstract

We study the Hamiltonian truncation for the two-dimensional $λφ^4$ theory within the framework of Hamiltonian truncation effective theory, where truncation artifacts are mitigated through a systematic inclusion of corrective terms organized in inverse powers of the ultraviolet energy cut-off $E_{\rm max}$. Building on the leading-order matching program, we develop two complementary extensions. First, we derive compact all-order expressions for the local matching corrections to the mass and quartic coupling by resumming infinite classes of diagrams sharing fixed topologies within the local approximation. Second, we extend the non-local sector by computing the next-to-next-to-local corrections contributing at $\mathcal{O}(E_{\rm max}^{-4})$, following a continuum-first matching procedure, in which the effective corrections are computed in infinite volume and the spatial direction is subsequently re-compactified to obtain a separable Hilbert-space basis on which the truncated operator construction is implemented. Our results show that an increasingly rich operator basis is necessary to describe the theory beyond leading order.

Figures (4)

• Figure 1: First non-local corrections $\xi$, $\alpha_1^{(2)}$, $\alpha_2^{(2)}$, and $\beta_2^{(2)}$, computed at finite (continuous line) and infinite (dotted line) volume as a function of the energy cut-off $E_{\max}$, for $2\pi R = 10$, $\lambda/4\pi = 1$ and $m=1$, in the range $E_{\max}\in[8,25]$. The upper panel shows the corrections $\alpha_1^{(2)}$ and $\alpha_2^{(2)}$, while the lower panel shows $\xi$ and $\beta_2^{(2)}$. In both panels, the lower portion of each plot shows the corresponding percent error with respect to the continuum value. For the discrete corrections, the sums over momenta are evaluated up to a finite maximum momentum $K_{\mathrm{UV}}=1000$.
• Figure 2: First energy gap $\Delta E_1$ as a function of the truncation scale $E_{\max}$, at fixed parameters $m=1$, $\lambda/(4\pi)=1$, and $2\pi R=10$. The curve (dot-dashed lines with star markers) labeled "Raw" corresponds to the bare truncation $H_{\mathrm{raw}}=PHP$. The curve (dashed lines with circular markers) labeled "LO" includes the leading local counterterms. The curve (solid lines with triangular markers) labeled "Resummed" includes the resummed local corrections implemented via Eqs. \ref{['eq:delta_lambda_resum']} and \ref{['eq:delta_m_resum']}.
• Figure 3: First energy gap $\Delta E_1$ as a function of the cut-off $E_{\max}$ for two circumferences, $L=10$ and $L=15$, at fixed parameters $m=1$ and $\lambda/(4\pi)=1$. The upper panel shows the LO implementation (including corrections up to $\mathcal{O}(E_{\max}^{-2})$), while the lower panel shows the NLO implementation (including corrections up to $\mathcal{O}(E_{\max}^{-3})$). In each panel, the lower sub-plot reports the percent deviation of the discrete result from the corresponding continuous evaluation. We observe that, in the infinite-dimensional case, the dependence of the spectrum on the value of the circumference originates from the operatorial sector rather than from the coefficients.
• Figure 4: Dependence of the energy gaps on the cut-off $E_{\max}$, comparing different HT and HTET prescriptions: Raw (stars with dash-dotted lines), LO (circles with dashed lines), NLO (squares with dotted lines), and NNLO (triangles with solid lines). Clockwise from the top left: $\Delta E_1$ with $\lambda/(4\pi)=1$, $\Delta E_5$ with $\lambda/(4\pi)=1$, $\Delta E_5$ with $\lambda/(4\pi)=3$, and $\Delta E_1$ with $\lambda/(4\pi)=3$.