We discuss ultracold Fermi gases in two dimensions, which could be realized in a strongly confining one-dimensional optical lattice. We obtain the temperature versus effective interaction phase diagram for an s-wave superfluid and show that, below a certain critical temperature Tc, spontaneous vortex-antivortex pairs appear for all coupling strengths. In addition, we show that the evolution from weak-to-strong coupling is smooth, and that the system forms a square vortex-antivortex lattice at a lower critical temperature TM.

We discuss the possibility of a quantum phase transition in ultra-cold spin-polarized Fermi gases which exhibit a p-wave Feshbach resonance. We show that when fermionic atoms form a condensate that can be externally tuned between the BCS and BEC limits, the zero temperature compressibility and the spin susceptibility of the fermionic gas are non-analytic functions of the two-body bound state energy. This non-analyticity is due to a massive rearrangement of the momentum distribution in the ground state of the system. Furthermore, we show that the low temperature superfluid density is also nonanalytic, and exhibits a dramatic change in behavior when the critical value of the bound state energy is crossed.

The BCS to BEC evolution has been recently the focus of studies in superconductors and cold atomic gases. For a d-wave system, we show that a Lifshitz transition occurs at a critical particle density which separates two topologically distinct phases according to their quasiparticle excitation energies: a BCS-like gapless superconductor in the higher density limit and a BEC-like fully gapped superconductor in the lower density limit. This transition is second order according to Ehrenfest's classification, but it occurs without a change in the symmetry of the order parameter, and thus can not be classified under Landau's scheme. To illustrate the nature of the transition, we compute the compressibility and the superfluid density as functions of particle density.

Recent progress in scientific machine learning (SciML) has opened up the possibility of training novel neural network architectures that solve complex partial differential equations (PDEs). Several (nearly data free) approaches have been recently reported that successfully solve PDEs, with examples including deep feed forward networks, generative networks, and deep encoder-decoder networks. However, practical adoption of these approaches is limited by the difficulty in training these models, especially to make predictions at large output resolutions (≥ 1024 × 1024).Here we report on a software framework for data parallel distributed deep learning that resolves the twin challenges of training these large SciML models training in reasonable time as well as distributing the storage requirements. Our framework provides several out of the box functionality including (a) loss integrity independent of number of processes, (b) synchronized batch normalization, and (c) distributed higher-order optimization methods.We show excellent scalability of this framework on both cloud as well as HPC clusters, and report on the interplay between bandwidth, network topology and bare metal vs cloud. We deploy this approach to train generative models of sizes hitherto not possible, showing that neural PDE solvers can be viably trained for practical applications. We also demonstrate that distributed higher-order optimization methods are 2-3× faster than stochastic gradient-based methods and provide minimal convergence drift with higher batch-size.

We study the branched polymer growth model (BPGM) introduced by Lucena et al. [Phys. Rev. Lett. 72, 230 (1994)] in two dimensions. First the BPGM was simulated in very large lattices with concentrations of impurities q=0 and q=0.2. The scaling of the mass in chemical space gives accurate estimates of the critical branching probabilities b(c) and of the chemical dimensions Dc at criticality, improving previous results. Estimates of the fractal dimension D(F) at criticality are consistent with a universal value along the critical line. Our results for q=0 suggest small deviations of Dc and D(F) from the percolation values. We also simulated the BPGM in finite lattices of lengths between L=32 and L=512 for the same concentrations q. Using finite-size scaling techniques, we confirm the previous estimates of D(F) and the universality along the critical line, and obtain the correlation exponent nu=1.43+/-0.06. It proves that the BPGM is not in the same universality class of percolation in two dimensions. Finally, we simulate random walks on the critical polymers grown in very large lattices with q=0 and q=0.2, and obtain the random walk dimension Dw and the spectral dimension Ds. Dw is larger and Ds is smaller than the corresponding values in critical percolation clusters, due to the lower connectivity of the polymers. The scaling relation Ds=2D(F)/Dw is not satisfied, as observed in other tree-like structures.