Lattice Methods and Effective Field Theory

Abstract

Lattice field theory is a non-perturbative tool for studying properties of strongly interacting field theories, which is particularly amenable to numerical calculations and has quantifiable systematic errors. In these lectures we apply these techniques to nuclear Effective Field Theory (EFT), a non-relativistic theory for nuclei involving the nucleons as the basic degrees of freedom. The lattice formulation of Endres et al. (Phys Rev A 84:043644, 2011; Phys Rev A 87:023615, 2013) for so-called pionless EFT is discussed in detail, with portions of code included to aid the reader in code development. Systematic and statistical uncertainties of these methods are discussed at length, and extensions beyond pionless EFT are introduced in the final section.

Appendix

5.1.1 Compilation and Running the Code

This code requires the use of the FFTW library, which you may download and install from fftw.org. The script “create_lib.sh” should be run first from the head directory. Once this script is successful, you may go into the production directory, modify the script “create_binary.sh” to reflect your path to the FFTW library, and compile by running this script. The executable created is called “a.out”, which should be run without specifying any additional parameters in the command line. Input parameters are specified in the files included in the “arg” folder. The parameters for each file are described in the header “arg.h”. The codes can be downloaded from the link https://github.com/ManyBodyPhysics/LectureNotesPhysics/tree/master/Programs/Chapter5-programs/. Output is created in the folder “results”. The file gives a list of the values (real part listed first, imaginary second) of the two-particle correlation function calculated at different values of Euclidean time, on a set of auxiliary field configurations. The organization of the output is as follows:

$$\displaystyle{ \begin{array}{ccccccc} \mathrm{Re}[C(\phi _{1},\tau _{1})] & \mathrm{Im}[C(\phi _{1},\tau _{1})] & \mathrm{Re}[C(\phi _{1},\tau _{2})] & \mathrm{Im}[C(\phi _{1},\tau _{2})] &\cdots & \mathrm{Re}[C(\phi _{1},\tau _{N_{\tau }})] & \mathrm{Im}[C(\phi _{1},\tau _{N_{\tau }})] \\ \mathrm{Re}[C(\phi _{2},\tau _{1})] & \mathrm{Im}[C(\phi _{2},\tau _{1})] & \mathrm{Re}[C(\phi _{2},\tau _{2})] & \mathrm{Im}[C(\phi _{2},\tau _{2})] &\cdots & \mathrm{Re}[C(\phi _{2},\tau _{N_{\tau }})] & \mathrm{Im}[C(\phi _{2},\tau _{N_{\tau }})]\\ &&&\vdots&&& \\ \mathrm{Re}[C(\phi _{N_{\mbox{ cfg}}},\tau _{1})]&\mathrm{Im}[C(\phi _{N_{\mbox{ cfg}}},\tau _{1})]&\mathrm{Re}[C(\phi _{N_{\mbox{ cfg}}},\tau _{2})]&\mathrm{Im}[C(\phi _{N_{\mbox{ cfg}}},\tau _{2})]&\cdots &\mathrm{Re}[C(\phi _{N_{\mbox{ cfg}}},\tau _{N_{\tau }})]&\mathrm{Im}[C(\phi _{N_{\mbox{ cfg}}},\tau _{N_{\tau }})]\\ \end{array} }$$

where N _τ and N _cfg are the total number of time steps, specified in “do.arg”, and total number of configurations, specified in “evo.arg”, respectively. To calculate the correlation function at a given time, τ, average over all values: $C(\tau ) =\sum _{i}\left (\mathrm{Re}\left [C(\phi _{i},\tau )\right ] + i\ \mathrm{Im}\left [C(\phi _{i},\tau )\right ]\right )$.

5.1.2 Exercises

5.6. Set the first value in the file “interaction.arg” to a coupling of your choice, and the remaining couplings to 0. Use the long time behavior of the effective mass function, $\ln \frac{C(\tau )} {C(\tau +1)}\mathop{\longrightarrow }\limits_{\tau \rightarrow \infty }E_{0}$ (see Sect. 5.3), to determine the ground state energy for your choice of coupling, g. Compare this with what you expect from Eq. (5.64), using the relation $\lambda = e^{-E_{0}}$, as the number of lattice points is increased. You may test the improved interaction, Sect. 5.2.2.5, using coefficients calculated from your code developed in Prob. 4 by setting multiple couplings in the “interaction.arg” file. Be careful to set the dispersion relation in “kinetic.arg” to match the one used in setting up your transfer matrix for the tuning.

5.7. Add a harmonic potential by setting the parameters in potential.arg. The three numerical values correspond to the spring constant, κ, for the x, y, z-directions. Set the interaction coefficients to correspond to unitarity, then find the energies of two unitary fermions in a harmonic trap, exploring and removing finite volume and discretization effects by varying the parameters, $L,L_{0} = \left (\kappa M\right )^{-1/4}$, and performing extrapolations in these quantities if necessary. Compare your result to the expected value of 2ω, where $\omega = \sqrt{\kappa /M}$, and the mass M is set in the file “kinetic.arg”.

5.8. Construct sources for three fermions in an l = 0 and l = 1 state and find the lowest energies corresponding to each state at unitarity. Which l corresponds to the true ground state of this system?