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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 [1,2] 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., Comment: Lectures prepared for the ECT* Doctoral Training Program, "Computational Nuclear Physics", April 13 - May 22, 2015. Submitted to Lect. Notes Phys., "An advanced course in computational nuclear physics: Bridging the scales from quarks to neutron stars", Morten Hjorth-Jensen, Maria Paola Lombardo, Ubirajara van Kolck, Editors. Code to be included online with published version

Probability distributions for correlation functions of particles interacting via random-valued fields are discussed as a novel tool for determining the spectrum of a theory. In particular, this method is used to determine the energies of universal N-body clusters tied to Efimov trimers, for even N, by investigating the distribution of a correlation function of two particles at unitarity. Using numerical evidence that this distribution is log-normal, an analytical prediction for the N-dependence of the N-body binding energies is made., Comment: 6 pages, 3 figures. Invited contribution to the 21st International Conference on Few-Body Problems in Physics (FB21)

Neutron matter at densities of the order of the nuclear saturation density is believed to have neutrons paired in the 3P2 channel. We study the low lying modes of this phase and find two massless modes (angulons), resulting from the spontaneous breaking of rotational symmetry as well as three other, gapped modes. We compute their masses at arbitrary temperatures., Comment: 8 pages, 2 figures

We present results for the ground-state mass shifts of octet baryons due to the presence of a medium of pions or kaons from a lattice QCD calculation performed at a single value of the quark mass, corresponding to a pion mass of $m_\pi$ ~ 390 MeV, and a spatial volume V ~ (4fm)^3. We use a canonical approach in which correlators are formed using a single baryon propagator and a fixed number of meson propagators, up to n=9. From the ground-state energies we calculate two- and three-body interaction parameters. We also extract combinations of low-energy constants by comparing our results to tree level chiral perturbation theory at non-zero isospin/kaon chemical potential., Comment: 7 pages, 3 figures, Proceedings of the 31st International Symposium on Lattice Field Theory (Lattice 2013), July 29 - August 3, 2013, Mainz, Germany

We present a lattice QCD calculation of the ground-state energy shifts of various baryons in a medium of pions or kaons at a single value of the quark mass corresponding to a pion mass of m_\pi~390 MeV and a kaon mass of m_K~540 MeV, and in a spatial volume V (4fm)^3. All systems are created using a canonical formalism in which quark propagators are contracted into correlation functions of fixed isospin/kaon density. We study four different systems, \Sigma^+(\pi^+)^n, \Xi^0(\pi^+)^n, p(K^+)^n, and n(K^+)^n, for up to n=11 mesons. From the ground-state energy shifts we extract two- and three-body scattering parameters, as well as linear combinations of low-energy constants appearing in tree-level chiral perturbation theory., Comment: published version

The condensation of neutrons into a 3P2 superfluid phase occurs at densities relevant for the interior of neutron stars. The triplet pairing breaks rotational symmetry spontaneously and leads to the existence of gapless modes (angulons) that are relevant for many transport coefficients and to the star's cooling properties. We derive the leading terms of the low energy effective field theory, including the leading coupling to electroweak currents, valid for a variety of possible 3P2 phases., Comment: Extended discussions, minor changes. Published version

Due to the presence of light pions in the theory, lattice QCD at finite densities suffers from issues with noise in both grand canonical and canonical formulations. We study two different formulations of the Nambu-Jona-Lasinio model reduced to 2+1 dimensions at large N, where N is the number of flavors. At finite chemical potential one formulation has a severe sign problem and a fermion correlator which displays a broad probability distribution with small mean. In the other we find no sign problem and a distribution amenable to the cumulant expansion techniques developed in earlier work., Comment: 6 pages, proceedings for the "Extreme QCD" workshop, Washington DC, August 2012

Lattice field theory methods, usually associated with non-perturbative studies of quantum chromodynamics, are becoming increasingly common in the calculation of ground-state and thermal properties of strongly interacting non-relativistic few- and many-body systems, blurring the interfaces between condensed matter, atomic and low-energy nuclear physics. While some of these techniques have been in use in the area of condensed matter physics for a long time, others, such as hybrid Monte Carlo and improved effective actions, have only recently found their way across areas. With this topical review, we aim to provide a modest overview and a status update on a few notable recent developments. For the sake of brevity we focus on zero-temperature, non-relativistic problems. After a short introduction, we lay out some general considerations and proceed to discuss sampling algorithms, observables, and systematic effects. We show selected results on ground- and excited-state properties of fermions in the limit of unitarity. The appendix contains details on group theory on the lattice., Comment: 64 pages, 32 figures; topical review for J. Phys. G; replaced with published version

The Nambu-Jona-Lasinio model reduced to 2+1 dimensions has two different path integral formulations: at finite chemical potential one formulation has a severe sign problem similar to that found in QCD, while the other does not. At large N, where N is the number of flavors, one can compute the probability distributions of fermion correlators analytically in both formulations. In the former case one finds a broad distribution with small mean; in the latter one finds a heavy tailed positive distribution amenable to the cumulant expansion techniques developed in earlier work. We speculate on the implications of this model for QCD., Comment: 16 pages, 5 figures; Published version with minor changes from the original

We perform lattice Monte Carlo simulations for up to 66 unitary fermions in a finite box using a highly improved lattice action for nonrelativistic spin 1/2 fermions. We obtain a value of $0.366^{+0.016}_{-0.011}$ for the Bertsch parameter, defined as the energy of the unitary Fermi gas measured in units of the free gas energy in the thermodynamic limit. In addition, for up to four unitary fermions, we compute the spectrum of the lattice theory by exact diagonalization of the transfer matrix projected onto irreducible representations of the octahedral group for small to moderate size lattices, providing an independent check of our few-body simulation results. We compare our exact numerical and simulation results for the spectrum to benchmark studies of other research groups, as well as perform an extended analysis of our lattice action improvement scheme, including an analysis of the errors associated with higher partial waves and finite temporal discretization., Comment: Significant revisions from previous version. Included data at a larger volume and performed an infinite volume extrapolation of the Bertsch parameter. Published version

The ground state energies of universal N-body clusters tied to Efimov trimers, for N even, are shown to be encapsulated in the statistical distribution of two particles interacting with a background auxiliary field at large Euclidean time when the interaction is tuned to the unitary point. Numerical evidence that this distribution is log-normal is presented, allowing one to predict the ground-state energies of the N-body system., Comment: Extended discussion of results; published version

We show how sign problems in simulations of many-body systems can manifest themselves in the form of heavy-tailed correlator distributions, similar to what is seen in electron propagation through disordered media. We propose an alternative statistical approach for extracting ground state energies in such systems, illustrating the method with a toy model and with lattice data for unitary fermions and for baryons in QCD., Comment: Plenary talk presented at the XXIX International Symposium on Lattice Field Theory (Lattice 2011), Squaw Valley, Lake Tahoe, California, July 10-16, 2011

A recently developed lattice method for large numbers of strongly interacting nonrelativistic fermions exhibits a heavy tail in the distributions of correlators for large Euclidean time {\tau} and large number of fermions N, which only allows the measurement of ground state energies for a limited number of fermions using standard techniques. In such cases, it is suggested that measuring the log of the correlator is more efficient, and a cumulant expansion of this quantity can be exactly related to the correlation function. The cumulant expansion technique allows us to determine the ground state energies of up to 66 unpolarized unitary fermions on lattices as large as 72$\times$14^3, and up to 70 unpolarized unitary fermions trapped in a harmonic potential on lattices as large as 72$\times$64^3. We have also improved our lattice action with a Galilean invariant form for the four-fermion interaction, which results in predictive volume scaling of the lowest energy of three fermions in a periodic box and in good agreement of our results for N \leq 6 trapped unitary fermions with those from other benchmark calculations., Comment: 7 pages, 6 figures, Presented at 29th International Symposium on Lattice Field Theory (Lattice2011), Squaw Valley, Lake Tahoe, CA, USA, 10-16 July 2011

We present a new lattice Monte Carlo approach developed for studying large numbers of strongly interacting nonrelativistic fermions, and apply it to a dilute gas of unitary fermions confined to a harmonic trap. Our lattice action is highly improved, with sources of discretization and finite volume errors systematically removed; we are able to demonstrate the expected volume scaling of energy levels of two and three untrapped fermions, and to reproduce the high precision calculations published previously for the ground state energies for N = 3 unitary fermions in a box (to within our 0.3% uncertainty), and for N = 3, . . ., 6 unitary fermions in a harmonic trap (to within our ~ 1% uncertainty). We use this action to determine the ground state energies of up to 70 unpolarized fermions trapped in a harmonic potential on a lattice as large as 64^3 x 72; our approach avoids the use of importance sampling or calculation of a fermion determinant and employs a novel statistical method for estimating observables, allowing us to generate ensembles as large as 10^8 while requiring only relatively modest computational resources., Comment: 48 pages, 19 figures, published version, some text revised, typos corrected, title changed in journal; previously "Unitary fermions on the lattice I: in a harmonic trap"

We show how sign problems in simulations of many-body systems can manifest themselves in the form of heavy-tailed correlator distributions, similar to what is seen in electron propagation through disordered media. We propose an alternative statistical approach for extracting ground state energies in such systems, illustrating the method with a toy model and with lattice data for unitary fermions., Comment: 4 pages, 4 figures

A fundamental constant in systems of unitary fermions is the so-called Bertsch parameter, the ratio of the ground state energy for spin paired unitary fermions to that for free fermions at the same density. I discuss how we computed this parameter as well as the pairing gap using a recently developed lattice construction for unitary fermions, by measuring correlation functions for up to 38 fermions in a finite box. Our calculation illustrates interesting issues facing the study of many-body states on the lattice, which may eventually be confronted in QCD calculations as well., Comment: 7 pages, 6 figures, The XXVIII International Symposium on Lattice Field Theory, Lattice2010, June 14-19, 2010, Villasimius, Italy

A novel lattice approach is presented for studying systems comprising a large number of interacting nonrelativistic fermions. The construction is ideally suited for numerical study of fermions near unitarity--a strongly coupled regime corresponding to the two-particle s-wave scattering phase shift delta_0 = pi/2. Such systems may be achieved experimentally with trapped atoms, and provide a starting point for an effective field theory description of nuclear physics. We discuss the construction of our lattice theory, which allows us to study systems of up to (but by no means limited to) 38 fermions with high accuracy and modest computational resources, and offer an overview of several applications of the technique. A more detailed discussion of applications and simulation results will be described in companion proceedings by A. N. N. and J-W. L., Comment: 7 pages, 4 figures, Presented at the XXVIII International Symposium on Lattice Field Theory, June 14-19, 2010, Villasimius, Sardinia Italy

We present a lattice study of up to N=20 unitary fermions confined to a harmonic trap. Our preliminary results show better than 1% agreement with high precision solutions to the many-body Schrodinger equation for up to N=6. We are able to make predictions for larger N which were inaccessible by the Hamiltonian approach due to computational limitations. Harmonic traps are used experimentally to study cold atoms tuned to a Feshbach resonance. We show that they also provide certain benefits to numerical studies of many-body correlators on the lattice. In particular, we anticipate that the methods described here could be used for studying nuclear physics., Comment: 7 pages, 5 figures, presented at the XXVIII International Symposium on Lattice Field Theory (Lattice 2010), Villasimius, Italy, June 14-19 2010

The study of nuclear physics using lattice QCD is hindered by an exponentially large signal-to-noise problem which is conventionally alleviated by raising the quark masses to unphysically high values. We propose a novel form of partial quenching for calculations involving nucleons in which the sea quark masses are taken to be smaller than the valence quark masses. It is shown that lowering the sea quark masses toward their physical values actually improves signal-to-noise. An optimized approach to the physical point in the (m_s, m_v) plane is proposed, with a full analysis of the cost benefit. Improvements in computing time of ~ 10^(2(A-1)), where A is the number of nucleons in the system, are shown to be possible., Comment: 5 pages, 2 figures. Typo corrected, references updated, parts of text revised