We formulate the next-to-leading-order nuclear effective field theory without pions in the two-nucleon sector on a spatial lattice and investigate nonperturbative renormalization group flows in the strong-coupling region by diagonalizing the Hamiltonian numerically. The cutoff (proportional to the inverse of the lattice constant) dependence of the coupling constants is obtained by changing the lattice constant with the binding energy and the asymptotic normalization constant for the ground state being fixed. We argue that the critical line can be obtained by looking at the finite-size dependence of the ground-state energy. We determine the relevant operator and locate the nontrivial fixed point, as well as the physical flow line corresponding to the deuteron in the two-dimensional plane of dimensionless coupling constants. It turns out that the location of the nontrivial fixed point is very close to the one obtained by the corresponding analytic calculation, but the relevant operator is quite different.
Abstract These lectures are a pedagogical—not comprehensive—introduction to the applications of effective field theory (EFT) in the context of nuclear and atomic physics. A common feature of these applications is the interplay between non-perturbative physics (needed at leading order to produce non-relativistic bound states and resonances) and controlled perturbative corrections (crucial for predictive power). The essential ideas are illustrated with the simplest nuclear EFT, pionless EFT, which contains only contact interactions and, with minor changes, can be adapted to certain atomic systems. This EFT exploits the two-body unitarity limit, where renormalization leads to discrete scale invariance in systems of three and more bodies. Remarkably complex structures then arise from very simple leading-order interactions. It briefly describes some of the challenges and rewards of including long-range forces—pion exchange in chiral EFT for nuclear systems or Van der Waals forces between atoms.
We consider two nonperturbative methods originally used to derive shell model effective interactions in nuclei. These methods have been applied to the two nucleon sector to obtain an energy independent effective interaction V_{low k}, which preserves the low momentum half-on-shell T matrix and the deuteron pole, with a sharp cutoff imposed on all intermediate state momenta. We show that V_{low k} scales with the cutoff precisely as one expects from renormalization group arguments. This result is a step towards reformulating traditional model space many-body calculations in the language of effective field theories and the renormalization group. The numerical scaling properties of V_{low k} are observed to be in excellent agreement with our exact renormalization group equation.
This thesis work has investigated the Renormalization Group theory for the nucleonnucleon interaction.Conventional nuclear many-body calculations have the following sources of non-perturbative physics: 1. a strongly repulsive short-range interaction, 2. a tensor force, e.g. from pion exchange, which is highly singular at short-distances, 3. the presence of low-energy bound states or nearly bound states (in the S waves).The RG approach exploits the insensitivity of low-energy processes to the details of the high-energy physics.Using any of the high-precision potentials as input, the highmomentum intermediate states in the Lippmann-Schwinger equation for the T matrix in a particular partial-wave are cut-off at Λ.The details of the physics beyond this limit of resolution are integrated out and included in the potential by requiring that the half-off shell T matrix elements be independent of the cut-off Λ.This requirement leads to a low-momentum potential "V low k ", which is energy independent.The choice of the regulator which cuts off the high momentum intermediate states is investigated.Sharp cut-offs, though straight forward, lead to convergence issues in few-body calculations that are eliminated using smooth regulators.The construction of low-momentum potentials using a smooth regulator is explored in detail.In the course of this study, a three-step process to calculate V low k requiring the cut-off independence of the fully-off shell T matrix elements has been established and this yields better numerical stability than the energy-independent RG.
The current paradigm to describe the nuclear interaction is within the frame of Chiral Effective Field Theory (ₓEFT) which organizes contributions to observables in a serie of decreasing importance. It happens that the leading contribution already requires to solve exactly the Schrödinger equation with a particular Hamiltonian. The same requirement is at play in pionless EFT which considers only nucleonic degrees of freedom. Such calculations are numerically intractable for A-body observables with A >> 10. One must design an additional expansion and truncation for many-body observables. In this thesis, non-perturbative approximations on the basis self-consistent Green’s function (SCGF) and on many-body perturbation theory (MBPT) are considered together with a pionless EFT. The goal of the present thesis is to investigate, in such framework, the renormalization invariance of many-body observables computed in A-body sectors with A >> 10. Hopefully the lessons learnt can be extended to ₓEFT. Analysis of numerical calculations realized with a state-of-the-art SCGF code reveals a critical numerical approximation leading to renormalization dependent observables. A necessary fix is proposed and must be implemented before any calculations based on SCGF and EFT in the future. This emphasizes the criticality of numerical approximations for any calculation within a pionless EFT. At the same time, renormalization invariance of observables computed within MBPT is studied formally, opening the path to formulate the renormalization of a wide range of many-body truncation schemes in the future.
The stability or lack thereof of nonrelativistic fermionic systems to interactions is studied within the renormalization-group (RG) framework, in close analogy with the study of critical phenomena using ${\ensuremath{\varphi}}^{4}$ scalar field theory. A brief introduction to ${\ensuremath{\varphi}}^{4}$ theory in four dimensions and the path-integral formulation for fermions is given before turning to the problem at hand. As for the latter, the following procedure is used. First, the modes on either side of the Fermi surface within a cutoff $\ensuremath{\Lambda}$ are chosen for study, in analogy with the modes near the origin in ${\ensuremath{\varphi}}^{4}$ theory, and a path integral is written to describe them. Next, an RG transformation that eliminates a part of these modes, but preserves the action of the noninteracting system, is identified. Finally the possible perturbations of this free-field fixed point are classified as relevant, irrelevant or marginal. A
d=1 warmup calculation involving a system of fermions shows how, in contrast to mean-field theory, which predicts a charge-density wave for arbitrarily weak repulsion, and superconductivity for arbitrarily weak attraction, the renormalization-group approach correctly yields a scale-invariant system (Luttinger liquid) by taking into account both instabilities. Application of the renormalization group in
d=2and3, for rotationally invariant Fermi surfaces, automatically leads to Landau's Fermi-liquid theory, which appears as a fixed point characterized by an effective mass and a Landau function
F, with the only relevant perturbations being of the superconducting (BCS) type. The functional flow equations for the BCS couplings are derived and separated into an infinite number of flows, one for each angular momentum. It is shown that similar results hold for rotationally noninvariant (but time-reversal-invariant) Fermi surfaces also, with obvious loss of rotational invariance in the parametrization of the fixed-point interactions. A study of a nested Fermi surface shows an additional relevant flow leading to charge-density-wave formation. It is pointed out that, for small $\frac{\ensuremath{\Lambda}}{{K}_{F}}$, a
N1 expansion emerges, with $N=\frac{{K}_{F}}{\ensuremath{\Lambda}}$, which explains why one is able to solve the narrow-cutoff theory. The search for non-Fermi liquids in
d=2 using the RG is discussed. Bringing a variety of phenomena (Landau theory, charge-density waves, BCS instability, nesting, etc.) under the one unifying principle of the RG not only allows us to better understand and unify them, but also paves the way for generalizations and extensions. The article is pedagogical in nature and is expected to be accessible to any serious graduate student. On the other hand, its survey of the vast literature is mostly limited to the RG approach.