We show that bound states moving in a finite periodic volume have an energy correction which is topological in origin and universal in character. The topological volume corrections contain information about the number and mass of the constituents of the bound states. These results have broad applications to lattice calculations involving nucleons, nuclei, hadronic molecules, and cold atoms. We illustrate and verify the analytical results with several numerical lattice calculations.

The unitarity limit describes interacting particles where the range of the interaction is zero and the scattering length is infinite. We present precision benchmark calculations for two-component fermions at unitarity using three different ab initio methods: Hamiltonian lattice formalism using iterated eigenvector methods, Euclidean lattice formalism with auxiliary-field projection Monte Carlo, and continuum diffusion Monte Carlo with fixed and released nodes. We have calculated the ground state energy of the unpolarized four-particle system in a periodic cube as a dimensionless fraction of the ground state energy for the non-interacting system. We obtain values 0.211(2) and 0.210(2) using two different Hamiltonian lattice representations, 0.206(9) using Euclidean lattice, and an upper bound of 0.212(2) from fixed-node diffusion Monte Carlo. Released-node calculations starting from the fixed-node result yield a decrease of less than 0.002 over a propagation of 0.4E −1 F in Euclidean time, where E F is the Fermi energy. We find good agreement among all three ab initio methods.

We present continuum and lattice calculations for elastic scattering between a fermion and a bound dimer in the shallow binding limit. For the continuum calculation we use the SkorniakovTer-Martirosian (STM) integral equation to determine the scattering length and effective range parameter to high precision. For the lattice calculation we use the finite-volume method of Lüscher. We take into account topological finite-volume corrections to the dimer binding energy which depend on the momentum of the dimer. After subtracting these effects, we find from the lattice calculation κa f d = 1.174(9) and κr f d = −0.029(13). These results agree well with the continuum values κa f d = 1.17907(1) and κr f d = −0.0383(3) obtained from the STM equation. We discuss applications to cold atomic Fermi gases, deuteron-neutron scattering in the spin-quartet channel, and lattice calculations of scattering for nuclei and hadronic molecules at finite volume.

We investigate the attractive Fermi polaron problem in two dimensions using non-perturbative Monte Carlo simulations. We introduce a new Monte Carlo algorithm called the impurity lattice Monte Carlo method. This algorithm samples the path integral in a computationally efficient manner and has only small sign oscillations for systems with a single impurity. As a benchmark of the method, we calculate the universal polaron energy in three dimensions in the scale-invariant unitarity limit and find agreement with published results. We then present the first fully non-perturbative calculations of the polaron energy in two dimensions and density correlations between the impurity and majority particles in the limit of zero range interactions. We find evidence for a smooth crossover transition from fermionic quasiparticle to molecular state as a function of interaction strength.PACS numbers: 67.85. Lm, 02.70.Ss, One of the most interesting and fundamental problems in quantum many-body physics is the polaron problem, where a mobile impurity interacts with a bath of particles. With the advent of trapped ultracold atomic gases, the polaron problem can now be realized for both bosonic and fermionic baths, and also in the universal limit where the range of the particle interactions are negligible [1]. In a fermionic medium, the impurity can undergo a transition and change its quantum statistics by binding fermions from the surrounding Fermi gas [2,3]. The impurity is dressed by fluctuations of the Fermi sea forming a quasiparticle or polaron state. But with increasing particle interaction strength, molecules will form by capturing one or even two particles from the Fermi sea, and this behavior has been shown to depend on the mass ratio of the two components of the Fermi gas for the 3D case [2][3][4][5][6][7][8][9][10]. In 1D, the exact analytical solution for equal masses shows that the polaron-molecule transition is a smooth crossover [11,12].In 2D the Fermi polaron properties have been studied using different theoretical and experimental approaches, and these have predicted various scenarios for the existence or absence of a polaron-molecule transition [13][14][15][16][17][18][19][20][21]. The Fermi polaron system has been studied using diagrammatic Monte Carlo (diag MC) [20,21]. The diag MC method uses a worm algorithm to stochastically sample Feynman diagrams to high orders in the coupling constant. In this work we introduce a non-perturbative ab initio approach called impurity lattice Monte Carlo (ILMC) [22] to investigate highlyimbalanced Fermi gases. Unlike diag MC, impurity lattice Monte Carlo directly samples the path integral and so is a fully non-perturbative calculation. We present calculations for the energy of the 2D polaron and density correlations between the impurity and majority particles as a function of interaction strength. Our results show evidence for a smooth crossover from polaron to molecule.Impurity lattice Monte Carlo method. The impurity lattice Monte Carlo method is a hybrid of two Monte Carlo al...