Similarity renormalization group (SRG) flow equations can be used to unitarily soften nuclear Hamiltonians by decoupling high-energy intermediate state contributions to low-energy observables while maintaining the natural hierarchy of many-body forces. Analogous flow equations can be used to consistently evolve operators so that observables are unchanged if no approximations are made. The question in practice is whether the advantages of a softer Hamiltonian and less correlated wave functions might be offset by complications in approximating and applying other operators.Here we examine the properties of SRG-evolved operators, focusing in this article on applications to the deuteron but leading toward methods for few-body systems. We find the advantageous features generally carry over to other operators with additional simplifications in some cases from factorization of the unitary transformation operator.

Using ab initio lattice methods, we calculate the finite temperature thermodynamics of homogeneous two-dimensional spin-1/2 fermions with attractive short-range interactions. We present results for the density, pressure, compressibility, and quantum anomaly (i.e. Tan's contact) for a wide range of temperatures and coupling strengths, focusing on the unpolarized case. Within our statistical and systematic uncertainties, our prediction for the density equation of state differs quantitatively from the prediction by Luttinger-Ward theory in the strongly coupled region of parameter space, but otherwise agrees well with it. We also compare our calculations with the second-and third-order virial expansion, with which they are in excellent agreement in the low-fugacity regime.

We present finite-temperature, lattice Monte Carlo calculations of the particle number density, compressibility, pressure, and Tan's contact of an unpolarized system of short-range, attractively interacting spin-1/2 fermions in one spatial dimension, i.e., the Gaudin-Yang model. In addition, we compute the second-order virial coefficients for the pressure and the contact, both of which are in excellent agreement with the lattice results in the low-fugacity regime. Our calculations yield universal predictions for ultracold atomic systems with broad resonances in highly constrained traps. We cover a wide range of couplings and temperatures and find results that support the existence of a strong-coupling regime in which the thermodynamics of the system is markedly different from the noninteracting case. We compare and contrast our results with identical systems in higher dimensions.

By choosing appropriate generators for the Similarity Renormalization Group (SRG) flow equations, different patterns of decoupling in a Hamiltonian can be achieved. Sharp and smooth block-diagonal forms of phase-shift equivalent nucleon-nucleon potentials in momentum space are generated as examples and compared to analogous low-momentum interactions ("V low k "). Here s is a flow parameter and the flow operator G s specifies the type of SRG [4]. Decoupling between low-energy and highenergy matrix elements is naturally achieved in a momentum basis by choosing a momentum-diagonal flow operator such as the kinetic energy T rel or the diagonal of H s ; either drives the Hamiltonian toward band-diagonal form. This decoupling leads to dramatically improved variational convergence in fewbody nuclear systems compared to unevolved phenomenological or chiral effective field theory (EFT) potentials [5,6]. Renormalization Group (RG) methods that evolve NN interactions with a sharp or smooth cutoff in relative momentum, known generically as V low k , rely on the invariance of the two-nucleon T matrix [7,8]. These approaches achieve a block-diagonal form characterized by a cutoff (see left plots in Figs. 1 and 2). As implemented in Refs. [7,8], the high-momentum matrix elements are defined to be zero, but this is not required.Block-diagonal decoupling of the sharp V low k form can be generated using SRG flow equations by choosing a blockdiagonal flow operator (see Refs. [9,10] for details),with projection operators P and Q = 1 − P . In a partial-wave momentum representation, P and Q are step functions defined by a sharp cutoff on relative momenta. This choice for G s , which means that η s is nonzero only where G s is zero, suppresses off-diagonal matrix elements such that the Hamiltonian approaches a block-diagonal form as s increases. If one considers a measure of the off-diagonal coupling of the Hamiltonian,then its derivative is easily evaluated by applying the SRG equation, Eq. (1):Thus, the off-diagonal QH s P block will decrease in general as s increases [9,10]. The right plots in Figs. 1 and 2 result from evolving the N 3 LO potential from Ref.[11] using the block-diagonal G s of Eq. (2) with = 2 fm −1 until λ ≡ 1/s 1/4 = 0.5 fm −1 . The parameter λ is a quantitative measure of the progress toward block diagonalization made by the SRG evolution. The agreement between V low k and SRG potentials for momenta below is striking. A similar degree of universality is found in the other partial waves. Deriving an explicit connection between these approaches is the topic of an ongoing investigation.The evolution with λ of two representative partial waves ( 3 S 1 and 1 P 1 ) are shown in Figs. 3 and 4. The evolution of the "off-diagonal" matrix elements (meaning those outside the P H s P and QH s Q blocks) can be roughly understood from the dominance of the kinetic energy on the diagonal. Let the indices p and q run over indices of the momentum states in the P and Q spaces, respectively. To good approximation we can replace P H ...

The choice of generator in the Similarity Renormalization Group (SRG) flow equation determines the evolution pattern of the Hamiltonian. The kinetic energy has been used in the generator for most prior applications to nuclear interactions with other options largely unexplored. Here we show how variations of this standard choice can allow the evolution to proceed more efficiently without losing its advantages.