Hamiltonian Truncation Effective Theory

Journal Article · Thu Aug 04 00:00:00 EDT 2022 · SciPost Physics

DOI:https://doi.org/10.21468/SciPostPhys.13.2.011· OSTI ID:1879775

Cohen, Timothy ^[1]; Farnsworth, Kara ^[2]; Houtz, Rachel ^[3]; Luty, Markus ^[4]

---

1. University of Oregon
2. Case Western Reserve University
3. Durham University
4. University of California, Davis

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 $H_0$ with eigenvalues below some energy cutoff E_\text{max} $E_{\text{max}}$ . 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} $E_{\text{max}}$ . The effective Hamiltonian can be computed by matching a transition amplitude to the full theory, and gives corrections order by order as an expansion in powers of 1/E_\text{max} $1/E_{\text{max}}$ . The effective Hamiltonian is non-local, with the non-locality controlled in an expansion in powers of H_0/E_\text{max} $H_0/E_{\text{max}}$ . The effective Hamiltonian is also non-Hermitian, and we discuss whether this is a necessary feature or an artifact of our definition. We apply our formalism to 2D \lambda\phi^4 $\lambda {\varphi }^4$ theory, and compute the the leading 1/E_\text{max}^2 $1/E_{\text{max}}^2$ corrections to the effective Hamiltonian. We show that these corrections nontrivially satisfy the crucial property of separation of scales. Numerical diagonalization of the effective Hamiltonian gives residual errors of order 1/E_\text{max}^3 $1/E_{\text{max}}^3$ , as expected by our power counting. We also present the power counting for 3D \lambda \phi^4 $\lambda {\varphi }^4$ theory and perform calculations that demonstrate the separation of scales in this theory.

• Sponsoring Organization: USDOE
• Grant/Contract Number: SC0009999; SC0011640
• OSTI ID: 1879775
• Alternate ID(s): OSTI ID: 1905570
• Journal Information: SciPost Physics, Journal Name: SciPost Physics Journal Issue: 2 Vol. 13; ISSN 2542-4653
• Publisher: Stichting SciPost
• Country of Publication: Netherlands
• Language: English