The Effective Hamiltonian

Jun-Hang Ren, Ming-Yong Ye, Xiu‐Min Lin

Effective Hamiltonian method is widely used in quantum information. We introduce a method to calculate effective Hamiltonians and give two examples in quantum information to demonstrate the method. We also give a relation between the effective Hamiltonian in the Shrödinger picture and the corresponding effective Hamiltonian in the interaction picture. Finally, we present a relation between our effective Hamiltonian method and the James–Jerke method which is currently used by many authors to calculate effective Hamiltonians in quantum information science.

H. Matsumoto

General properties of an effective Hamiltonian in thermo field dynamics, which includes reservoir effects through coarse graining, are investigated. The necessary and sufficient conditions for the effective Hamiltonian to describe the reservoir effect are obtained. The final canonical distribution is discussed with relation to each given interaction. The result gives a quick and easy guide to construct an effective Hamiltonian and to identify the final equilibrium state.

Timothy Cohen A, Kara Farnsworth, R. Houtz +1 more

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 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 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 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 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 λ ϕ 4 theory, and compute the the leading 1/E_\text{max}^2 1 / E 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 max 3 , as expected by our power counting. We also present the power counting for 3D \lambda \phi^4 λ ϕ 4 theory and perform calculations that demonstrate the separation of scales in this theory.