## Final Report Summary - QUSIMGAS (Quantum Simulation of Many-Body Physics in Ultracold Gases)

The goal of the project QUSIMGAS was to develop novel methods for studying quantum systems in order to reach lower temperatures, strongly correlated regimes, and exploring exotic phases of matter. Ideally, such methods can first be benchmarked against experiments in the field of "quantum simulation" and, second, be applied to other systems and parameter regimes.

For fermionic systems, diagrammatic Monte Carlo methods, first introduced by Prokof'ev and Svistunov, were explored. The bare series was used to study the Hubbard model with explicitly broken SU(2) spin symmetry, which can be considered a spin-nematic deformation of the Fermi surface. Instead of a previously proposed Bose metal we found for attractive interactions a p-wave pairing between species of the same type and incommensurate density waves. For repulsive interactions and at half filling, a Z2 finite temperature transition to an Ising antiferromagnet could be proven. We also studied for attractive interactions the case with spin imbalance but with SU(2) spin symmetry restored. In this case, we could show that for polarizations about 1/4 there is a leading instability of the Fermi liquid at finite temperature towards a FFLO state, which is an s-wave superconductor with non-zero center of mass momentum.

The Hubbard model was also studied with lattice-determinant Monte Carlo methods to study the dimensional crossover shedding new light on certain organic superconductors.

We also contributed to the method development by developing a new diagrammatic technique called Grassmannization relevant for studying classical bond models. In order to improve the convergence properties of the diagrammatic series, as well as find a way to circumvent dealing with high-dimensional objects, we extended the homotopy analysis method to the case of field theories. It provides a way to deal with the Dyson-Schwinger formulation of field theories in a controllable fashion, and was benchmarked against the prototypical phi^4 theory.

To strengthen the paradigm of quantum simulation, we performed conventional path integral Monte Carlo simulations of the one-dimensional Hubbard model in collaboration with the experimental teams led by I. Bloch and C. Gross at the Max-Planck Institute for Quantum Optics in Garching and the Ludwig-Maximilian-University in Munich, Germany. We focused on the thermometry and the magnetic properties and saw specific instances of spin-charge separation.

Impurity systems, which generally consist of a single quantum mechanical object coupled to an environment, could be studied successfully with diagrammatic Monte Carlo techniques. For the Fermi-polaron problem, in which an itinerant fermion is coupled to a bath of other, non-interacting fermions, we computed the ground-state properties such as the polaron-to-molecule transition, the ground state energy and quasi-particle residue in case the mass of the impurity differs from the one of the majority component in three dimensions. We also analyzed this problem in a quasi-two-dimensional setup. An important technical insight was that grouping terms in the perturbative Feynman expansion along the number of hole propagators showed a fast convergence and allows to make a connection with the path-integral formalism and variational wavefunctions.

For bosonic systems, we developed novel tools based on exact diagonalization, momentum cluster perturbation theory, and symmetry breaking. This allowed us to compute the phase diagram of the Harper-Hofstadter-Mott model for bosons with infinitely strong local repulsive interactions and subject to a magnetic field which has a quarter flux per plaquette of the square lattice.

The accuracy of the method seems to be better than five percent in the most difficult regimes, and there is room for improvement in future research.

The quasi-one-dimensional phase diagram is even under better control as a finite size analysis could be performed. Its phase diagram displays more (analogs of) gapped phases compared to the two-dimensional case.

Of direct interest to cold gas experiments were the studies of certain response functions of the Bose-Hubbard model near the tip of the Mott lobe, where a U(1) symmetry emerges. We computed the response to an amplitude modulation of the laser constructing the optical lattice, and determined hereby the universal scaling function of the massive Goldstone (sometimes referred to as 'Higgs' mode) and determined realistic conditions under which this can be observed in experiment with existing technology. We also computed the universal scaling function of the optical conductivity, which would correspond to a phase modulation. The improvement in accuracy by one to two orders of magnitude allowed furthermore a high-precision test of the predictions of holographic theories for these quantities.

In conclusion, despite the substantial progress for fermionic systems and the additional control we have obtained, dealing with competing instabilities remains extremely challenging. Impurity systems such as the physics of polarons or quantum mechanical devices with few degrees of freedom coupled to external baths, can, with some effort, be studied rather accurately. The challenge is shifting towards real-time dynamics and out-of-equilibrium phenomena. For non-positive bosonic systems, the progress made in this project shows that, by a suitable combination of techniques, it may become feasible to study the phase diagrams of interacting bosonic systems subject to artificial magnetic fields in the near future. The interplay between superfluidity, localization through interactions and non-trivial band structures may lead to rich physics reminiscent of the quantum Hall effect and give rise to exotic phases of matter.

For fermionic systems, diagrammatic Monte Carlo methods, first introduced by Prokof'ev and Svistunov, were explored. The bare series was used to study the Hubbard model with explicitly broken SU(2) spin symmetry, which can be considered a spin-nematic deformation of the Fermi surface. Instead of a previously proposed Bose metal we found for attractive interactions a p-wave pairing between species of the same type and incommensurate density waves. For repulsive interactions and at half filling, a Z2 finite temperature transition to an Ising antiferromagnet could be proven. We also studied for attractive interactions the case with spin imbalance but with SU(2) spin symmetry restored. In this case, we could show that for polarizations about 1/4 there is a leading instability of the Fermi liquid at finite temperature towards a FFLO state, which is an s-wave superconductor with non-zero center of mass momentum.

The Hubbard model was also studied with lattice-determinant Monte Carlo methods to study the dimensional crossover shedding new light on certain organic superconductors.

We also contributed to the method development by developing a new diagrammatic technique called Grassmannization relevant for studying classical bond models. In order to improve the convergence properties of the diagrammatic series, as well as find a way to circumvent dealing with high-dimensional objects, we extended the homotopy analysis method to the case of field theories. It provides a way to deal with the Dyson-Schwinger formulation of field theories in a controllable fashion, and was benchmarked against the prototypical phi^4 theory.

To strengthen the paradigm of quantum simulation, we performed conventional path integral Monte Carlo simulations of the one-dimensional Hubbard model in collaboration with the experimental teams led by I. Bloch and C. Gross at the Max-Planck Institute for Quantum Optics in Garching and the Ludwig-Maximilian-University in Munich, Germany. We focused on the thermometry and the magnetic properties and saw specific instances of spin-charge separation.

Impurity systems, which generally consist of a single quantum mechanical object coupled to an environment, could be studied successfully with diagrammatic Monte Carlo techniques. For the Fermi-polaron problem, in which an itinerant fermion is coupled to a bath of other, non-interacting fermions, we computed the ground-state properties such as the polaron-to-molecule transition, the ground state energy and quasi-particle residue in case the mass of the impurity differs from the one of the majority component in three dimensions. We also analyzed this problem in a quasi-two-dimensional setup. An important technical insight was that grouping terms in the perturbative Feynman expansion along the number of hole propagators showed a fast convergence and allows to make a connection with the path-integral formalism and variational wavefunctions.

For bosonic systems, we developed novel tools based on exact diagonalization, momentum cluster perturbation theory, and symmetry breaking. This allowed us to compute the phase diagram of the Harper-Hofstadter-Mott model for bosons with infinitely strong local repulsive interactions and subject to a magnetic field which has a quarter flux per plaquette of the square lattice.

The accuracy of the method seems to be better than five percent in the most difficult regimes, and there is room for improvement in future research.

The quasi-one-dimensional phase diagram is even under better control as a finite size analysis could be performed. Its phase diagram displays more (analogs of) gapped phases compared to the two-dimensional case.

Of direct interest to cold gas experiments were the studies of certain response functions of the Bose-Hubbard model near the tip of the Mott lobe, where a U(1) symmetry emerges. We computed the response to an amplitude modulation of the laser constructing the optical lattice, and determined hereby the universal scaling function of the massive Goldstone (sometimes referred to as 'Higgs' mode) and determined realistic conditions under which this can be observed in experiment with existing technology. We also computed the universal scaling function of the optical conductivity, which would correspond to a phase modulation. The improvement in accuracy by one to two orders of magnitude allowed furthermore a high-precision test of the predictions of holographic theories for these quantities.

In conclusion, despite the substantial progress for fermionic systems and the additional control we have obtained, dealing with competing instabilities remains extremely challenging. Impurity systems such as the physics of polarons or quantum mechanical devices with few degrees of freedom coupled to external baths, can, with some effort, be studied rather accurately. The challenge is shifting towards real-time dynamics and out-of-equilibrium phenomena. For non-positive bosonic systems, the progress made in this project shows that, by a suitable combination of techniques, it may become feasible to study the phase diagrams of interacting bosonic systems subject to artificial magnetic fields in the near future. The interplay between superfluidity, localization through interactions and non-trivial band structures may lead to rich physics reminiscent of the quantum Hall effect and give rise to exotic phases of matter.