Quantum field theories require a cutoff to regulate divergences that result from local interactions, and yet physical results can not depend on the value of this cutoff. The renormalization group employs a transformation that changes the cutoff to isolate hamiltonians that produce cutoff-independent eigenvalues. The similarity renormalization group is based on similarity transformations that regulate off-diagonal matrix elements, forcing the hamiltonian towards a band-diagonal form as the cutoff is lowered. This avoids pathologies that plagued tradition transformations acting on hamiltonians, making it possible to produce a well-behaved perturbative approximation of renormalized hamiltonians in asymptotically free theories. We employ a simple two-dimensional delta function example to illustrate this new renormalization technique.
A general nonperturbation problem, described by the Hamiltonian
H, is considered. Approximate creation operators ${{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$ are defined as operators which satisfy the Hamiltonian commutator equations $[\mathcal{H},{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}]={\ensuremath{\omega}}_{i}{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}+{{R}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$, where ${\ensuremath{\omega}}_{i}$ are the creation energies, and the remainder operators ${{R}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$ are small in the sense that statistical averages involving the ${{R}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$ are small. A zeroth approximation is to neglect the ${{R}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$. In this case, a straight-forward derivation provides general relations between statistical averages such as $〈{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{\Omega}〉$ and $〈{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{\Omega}\ifmmode\pm\else\textpm\fi{}\ensuremath{\Omega}{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}〉$, where $\ensuremath{\Omega}$ is any operator. The usual boson and fermion occupation numbers are a trivial result of these zeroth-order relations. Corrections to the general zeroth-order relations arise from the ${{R}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$; such corrections are considered in the spirit of perturbation theory. An explicit first-order correction is obtained for the averages $〈{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}{\ensuremath{\theta}}_{i}〉$, and this correction is related to a self-consistent energy-renormalization procedure. The present method is similar to the method of thermodynamic Green's functions in that the derivations given here do not require evaluation of any state vectors or of the partition function. The general results are applied to the Heisenberg ferromagnet problem. At low temperatures, the zeroth-order relations give the Bloch spin-wave results, and an approximate evaluation of the first-order contribution gives the leading term in Dyson's
T4 correction to the spontaneous magnetization. For arbitrary temperatures, the zeroth-order relations give the Tyablikov equations for
〈Sz〉, and Callen's results are obtained from the first-order corrections. These examples illustrate the simplicity of calculating statistical averages of functions of the ${{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$, ${\ensuremath{\theta}}_{i}$ operators by the present method.
A general system composed of many weakly interacting bosons and/or fermions is considered. The object is to develop a procedure for renormalizing the single-particle creation operators, so as to remove the interactions to successively higher orders of perturbation. The basic idea is that creation operators ${{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$ must satisfy the Hamiltonian commutator equations $[\mathcal{H},{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}]={\ensuremath{\omega}}_{i}{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$, where
H is the Hamiltonian and ${\ensuremath{\omega}}_{i}$ are the particle energies. For the many-weakly-interacting-particle system, the zeroth-order boson and fermion creation operators satisfy these equations to zeroth order. It is shown that if the creation operators are renormalized so as to satisfy the Hamiltonian commutator equations to order
m, and also to satisfy the appropriate boson commutators and fermion anticommutators to order
m, then the problem is solved to order
m. In particular, the vectors formed by operating on the ground state with the renormalized creation operators, according to the usual boson and fermion occupation-number representation, are eigenvectors of
H and are orthonormal, all to order
m. A procedure is otained for finding the (
m+1)-order contributions to the particle creation energies, in terms of the
m-order operators. Explicit first-order calculations of these general results are provided for a system of bosons and a system of fermions, and these first-order results are shown to include similar results of Rayleigh-Schr\"odinger perturbation theory, the random-phase approximation, and the method of thermodynamic Green's functions. The problem of anharmonic lattice dynamics is studied in detail, and a method of undetermined coefficients is used to renormalize the phonon creation operators to first order. The phonon energies are calculated to second order, and this calculation shows that the interactions between renormalized phonons cannot be removed in second order. Statistical averages of the phonon energies give the energy shifts and lifetimes which have been calculated previously by various propagator techniques. In addition, the renormalized energies are used to calculate the temperature-dependent part of the Helmholtz free energy correct to second order. As a final example, electron-phonon interactions in a normal metal are studied. The electron and phonon creation operators are renormalized to first order, the particle energies are calculated to second order, and statistical averages of the particle energies recover the usual thermodynamic Green's-function results for energy shifts and lifetimes. These examples show the simplicity by which the renormalization procedure obtains a great amount of detailed information about the single-particle nature of a many-particle system.
We present two new methods for performing ab initio calculations of excited states for closed-shell systems within the in-medium similarity renormalization group (IMSRG) framework. Both are based on combining the IMSRG with simple many-body methods commonly used to target excited states, such as the Tamm-Dancoff approximation (TDA) and equations-of-motion (EOM) techniques. In the first approach, a two-step sequential IMSRG transformation is used to drive the Hamiltonian to a form where a simple TDA calculation (i.e., diagonalization in the space of
1p1h excitations) becomes exact for a subset of eigenvalues. In the second approach, EOM techniques are applied to the IMSRG ground-state-decoupled Hamiltonian to access excited states. We perform proof-of-principle calculations for parabolic quantum dots in two dimensions and the closed-shell nuclei
16O and
22O. We find that the TDA-IMSRG approach gives better accuracy than the EOM-IMSRG when calculations converge, but it is otherwise lacking the versatility and numerical stability of the latter. Our calculated spectra are in reasonable agreement with analogous EOM-coupled-cluster calculations. This work paves the way for more interesting applications of the EOM-IMSRG approach to calculations of consistently evolved observables such as electromagnetic strength functions and nuclear matrix elements, and extensions to nuclei within one or two nucleons of a closed shell by generalizing the EOM ladder operator to include particle-number nonconserving terms.
The conventional many-body perturbation theory suffers from three major ills: (i) some terms in the perturbation series may be unbounded; (ii) the series as a whole may not have a limit and (iii) despite convergence,the calculated limit may be false ('bogus' convergence). In renormalizable field theories, as the name indicates, problem (i) can be bypassed. Problems (ii) and (iii) remain. Here we demonstrate that these two problems can also be resolved, by choosing an appropriate Kohn-Sham Hamiltonian as the 'unperturbed' Hamiltonian. For this to be possible, some pertinent ground-state densities must be pure-state non-interacting v-representable. We illustrate our findings by means of two examples.
Dynamical Nuclear Field Theory which was modified in the previous paper so as to bring complete equivalence with Boson Expansion Theory (BET) is now extended to derive a BET under the influence of two-body correlations. For this purpose, the driving Hamiltonian of the vibrating Hartree field approximation is supplemented by the residual two-body interaction. Effective forms of Tamm-Dancoff fermion-pair and scattering operators are expanded perturbatively. Effects of three kinds of bubble diagrams can be incorporated by the renormalization procedure, but the contributions from unlinked diagrams remain uncancelled. Comparisons with Nuclear Field Theory are discussed.