This project focuses on ultra-cold atoms and quantum crystals where collective behavior is governed by laws of quantum mechanics. Understanding these systems is crucial for theoretical modeling, condensed matter physics, and materials science because of the prospects for discovering new states of matter. One of such states --- supersolidity of Helium-4 --- remains one of the biggest puzzles in the modern low-temperature physics. One finds interacting quantum systems across all fields of physics, quantum chemistry, and materials science and there is urgent need for universal unbiased first-principles methods to deal with them in their full complexity. This project is aimed at developing such methods, with the particular focus on: (i) Studying collective phenomena in disordered, multi-component, and other non-trivial cold-atomic ensembles in optical lattices and in continuous space, including interacting fermions in the crossover regime with physics intermediate between that of conventional superconductors and bosonic superlfuids; (ii) Understanding the microscopic picture behind and novel phenomena associated with the supersolidity in Helium-4; (iii) Advancing Monte Carlo techniques and algorithms as a universal tool for solving quantum-statistical problems - diagrammatic Monte Carlo for fermions and Worm Algorithm for bosons.
An unbiased theoretical description of collective quantum phenomena is of vital interdisciplinary importance for a number of applied and fundamental areas, such as quantum computing and high-energy physics. High-end computing methods and techniques often find applications outside the physics community. Simulations of complex models with multiple constraints, randomness, and a variable number of continuous parameters are typical in polymer science, neural networks, computer science, behavioral, social and economics studies. The algorithms developed in the project provide an example of how some of the difficulties may be circumvented. An integral part of the project is the training of graduate students and post-doctoral associate in advanced numeric techniques, quantum statistics, topical problems of atomic and solid state physics, network administration, and parallel supercomputing. This project includes: (i) developing tools for visualizing quantum statistical phenomena in terms of Feynman's paths (worldlines) and diagrams; (ii) maintaining an interactive web site popularizing, teaching and disseminating new algorithms and codes; (iii) upgrading and administrating major shared computational facilities at both Universities; (iv) developing and teaching a multi-institutional graduate tele-course on advanced numeric methods; (v) writing a book on superfluid states of matter and developing on its basis a graduate course; (vi) promoting higher standards in science education at schools; (vii) organizing a workshop on supersolidity.
Goals of the project: (i) studying collective phenomena in strongly correlated systems of cold atoms, (ii) understanding the microscopic picture behind and novel phenomena associated with the supersolidity in He-4, (iii) advancing Diagrammatic Monte Carlo as a universal tool for quantum-statistical problems. Key outcomes: 1. Bold Diagrammatic Monte Carlo for Resonant Fermions. The model of resonant fermions in the regime of BCS-BEC crossover is an accurate model for a two-species ultra-cold fermionic gas under the conditions of Feshbach resonance; it is also directly relevant to the equation of state of neutron matter. The BCS-BEC crossover is one of the hottest topics of modern experiments with ultra-cold atoms. While being fundamentally interesting on its own, the system of resonant fermions serves as a perfect testing ground for first-principles theoretical methods. We have developed Bold Diagrammatic Monte Carlo (BDMC) algorithm for resonant fermions. In a collaborative effort with the group of Martin Zwierlein (MIT) we have cross-validated the theory and experiment down to the critical temperature. The experimental and theoretical results for the equation of state (at unitarity) agree within a few percent. This result establishes BDMC as a controllable theoretical tool for strongly-correlated fermions. 2. Universal fermionization. Dyson's collapse and/or constraints on the Hilbert space is the major obstacle for using Feynman's diagrams in combination with diagrammatic Monte Carlo as a universal first-principles tool for solving quantum-statistical problems for strongly correlated bosons and spins, as well as fermions with projected Hamiltonians. We have found a universal way of solving the problem by generalizing the Popov-Fedotov fermionization trick (originally proposed for spin-1/2 lattice systems). We have established a general way of mapping the above-mentioned systems onto the system of multi-component fermions, for which there exists Feynman's diagrammatic technique with finite convergence radius. 3. Role of Bose Statistics in Crystallization and Quantum Jamming. By first principles simulations of dipolar bosons and bulk condensed helium-4 (and also by general qualitative analysis), we have shown that indistinguishability of particles is a major factor destabilizing crystalline order in Bose systems. Contrary to conventional wisdom, zero-point motion alone cannot prevent helium-4 crystallization at near zero pressure. Secondly, Bose statistics leads to quantum jamming at finite temperature, dramatically enhancing the metastability of superfluid glasses. 4. Higgs Mode. We obtained solid evidence for the existence of a well-defined Higgs amplitude mode in two-dimensional relativistic field theories based on analytically continued results from quantum Monte Carlo simulations of the Bose-Hubbard model in the vicinity of the superfluid-Mott insulator quantum critical point, featuring emergent particle-hole symmetry and Lorentz-invariance. 5. Bold Diagrammatic Monte Carlo for Frustrated Spin Systems. Using fermionic representation of spin degrees of freedom within the Popov-Fedotov approach, we developed algorithm for Monte Carlo sampling of skeleton Feynman diagrams for Heisenberg-type models. Our scheme works without modifications for any dimension of space, lattice geometry, and interaction range, i.e., it is suitable for dealing with frustrated magnetic systems at finite temperature. As a practical application, we computed uniform magnetic susceptibility of the antiferromagnetic Heisenberg model on the triangular lattice and compared our results with the best available high-temperature expansions. We also produced results for the momentum dependence of the static magnetic susceptibility throughout the Brillouin zone. 6. Deconfined Criticality Flow in the Heisenberg Model with Ring-Exchange Interactions. We performed a direct comparison of quantum phase transition points in the J-Q model—the test bed of the deconfined critical point theory—and the SU(2)-symmetric discrete noncompact CP1 representation of the deconfined critical action. We found that the flows of two systems coincide in a broad region of linear system sizes (10 Broader impacts: Unbiased theoretical access to collective quantum phenomena is of vital interdisciplinary importance for a number of applied and fundamental areas, such as ab initio modeling of q-bits for quantum computing, verifying challenging predictions and searching for new discoveries in the theory of quantum phase transitions and high-energy physics. High-end computing methods and techniques may find applications outside the physics community. Simulations of complex models with multiple constraints, randomness, and a variable number of continuous parameters are typical in polymer science, behavioral and economics studies. Worm algorithm and Diagrammatic Monte Carlo provide an example of how some of the difficulties encountered in such studies may be circumvented. An integral part of the project was training of 4 graduate students (including one female) and two post-doctoral associates in advanced numeric techniques, in quantum statistics, topical problems of atomic and solid state physics, network administration, and parallel/super computing. The list of broader impacts of the project also includes: (i) developing tools for visualizing quantum statistical phenomena in terms of Feynman's paths (worldlines) and diagrams, (ii) maintaining an interactive web site popularizing, teaching and disseminating new algorithms and codes, (iii) upgrading and administrating major shared computational facilities.
