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Hamiltonian Truncation Effective Theory
Timothy Cohen, Kara Farnsworth, Rachel Houtz, Markus A. Luty
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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 H₀ with eigenvalues below some energy cutoff Eₘₐₓ. 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 Eₘₐₓ. 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/Eₘₐₓ. The effective Hamiltonian is non-local, with the non-locality controlled in an expansion in powers of H₀/Eₘₐₓ. 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 λϕ⁴ theory, and compute the the leading 1/Eₘₐₓ² corrections to the effective Hamiltonian. We show that these corrections non-trivially satisfy the crucial property of separation of scales. Numerical diagonalization of the effective Hamiltonian gives residual errors of order 1/Eₘₐₓ³, as expected by our power counting. We also present the power counting for 3D λϕ⁴ theory and perform calculations that demonstrate the separation of scales in this theory.
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