Hamiltonian Truncation Effective Theory

Abstract

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 $H_0$ with eigenvalues below some energy cutoff $E_{\text{max}}$. 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 $E_{\text{max}}$. 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 $1/E_{\text{max}}$. The effective Hamiltonian is non-local, with the non-locality controlled in an expansion in powers of $H_0/E_{\text{max}}$. 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 $\lambda {\varphi }^4$ theory, and compute the the leading $1/E_{\text{max}}^2$ 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 $1/E_{\text{max}}^3$, as expected by our power counting. We also present the power counting for 3D $\lambda {\varphi }^4$ theory and perform calculations that demonstrate the separation of scales in this theory.

51 pages, 9 figures, v2: Clarifications and additional discussion added in response to referee reports. Conclusions unchanged

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Cited by in corpus (9)

• Effective Hamiltonians and Counterterms for Hamiltonian Truncation
• Towards a nonperturbative construction of the $S$-matrix
• Enhancing Quantum Field Theory Simulations on NISQ Devices with Hamiltonian Truncation
• Hamiltonian Truncation Crafted for UV-divergent QFTs
• Testing the RG-flow $M(3,10)+{\varphi }_{1,7}\to M(3,8)$ with Hamiltonian Truncation
• Quantum phase transitions in a bidimensional $O(N)\times {\mathbb{Z}}_2$ scalar field model
• Renormalized dual basis for scalable simulations of 2+1D compact quantum electrodynamics
• Hamiltonian formulation of the $1+1$-dimensional ${\varphi }^4$ theory in a momentum-space Daubechies wavelet basis
• Higher-order structure of Hamiltonian truncation effective theory