## Final Report Summary - FDIAGMC (Quantum Monte Carlo simulation of Feynman diagrams)

The main highlights of this project can be ordered in three different categories: (i) Diagrammatic Monte Carlo simulations, which deal with novel algorithmic developments and results for the Fermi-polaron system, (ii) novel results for strongly interacting bosonic lattice systems, with the computation of response functions in strongly-interacting regimes, fundamental new insight in strong disorder and many-body localized phases, and the development of new approximate methods for bosons subject to synthetic gauge fields, and (iii) the intricacies of (super)solid Helium-4 in continuum space. Below we elaborate these topics.

Diagrammatic Monte Carlo —

The main goal of the project was to develop new quantum Monte Carlo methods for strongly interacting systems. These are based on a sampling of Feynman diagrams and go under the name of diagrammatic Monte Carlo simulations, with several pros and cons: Compared to path integral Monte Carlo methods, the infamous sign problem is not leading to an exponential scaling in the system size or inverse temperature, but instead in the expansion order. The hope is hence that the answer can be obtained, either converged or through extrapolation, in low expansion orders. However, most perturbative expansions in physics are asymptotic, and this may lead to serious difficulties. Over the past years we learned that the method needs further refinements for competing instabilities, but could be applied successfully when there is a small parameter (which may be emergent). This has led to the following main results:

- P. Kroiss and L. Pollet, Phys Rev B 90, 104510 (2014) and Phys Rev B 91, 144507 (2015): In these works we applied the diagrammatic Monte Carlo method for the Fermi-polaron problem in the quasi-2d regime and in 3d for mass imbalance, respectively. The main result is that we could numerically prove that a variational approach in the 2particle-hole channel remains quantitatively accurate for all cases considered, an idea that appeared in the literature before in a number of different contexts but numerical results have not been analyzed along these lines before. This allowed us to obtain a rather precise phase diagram by combining different methods, as well as compute all thermodynamic properties of interest.

- L. Pollet, M. N. Kiselev, N. V. Prokof’ev, and B. V. Svistunov, Grassmannization of classical models, New J Phys Vol 18, Iss 11,p 113025 (2016): In order to overcome Dyson’s collapse argument, we developed the Grassmannization technique for classical bond models, giving us a Feynman’s diagrammatic technique based on Grassmann free propagators and Wick’s theorem. The method is general and can form the basis for plaquette-type interactions and even gauge fields. As an application, we derived the Feynman series (to 14th order) for the spin-spin correlation function of the 2D classical Ising model.

Interacting bosonic systems —

We have also studied a large number of bosonic systems with path-integral Monte Carlo simulations with worm-type updates in different contexts.

We list the main results:

- In Phys. Rev. A 89, 023605 (2014) we studied the 1d-2d dimensional crossover for the Bose-Hubbard model. It turns out that the location of the tip of the lobe is determined by the 1d physics and follows a Kosterlitz-Thouless behavior.

- In Phys Rev A 88, 033628 (2013) we studied mixtures of bosonic particles with SU(2) symmetry. For large occupation numbers the system phase separates, which can also be understood analytically.

- In Phys. Rev. Lett. 112, 030402 (2013) we studied the universal properties of the Higgs particle in (2+1)d for models with U(1) symmetry. This showed that the paradigm of quantum simulation also applies to response functions. We saw that the first peak shifts to lower frequency when tuning closer to the quantum critical point. Very close to the critical point, the response is dominated by the critical exponents of the XY model in 3 dimensions as well as a universal scaling function, which we determined.

- In Phys Rev B 93, 094404 (2016) the classical mutual information of the Sherington-Kirkpatrick model was analyzed and compared with an extension of Parisi’s famous solution.

- In Phys Rev B 94, 195119 (2016) we developed bosonic selfenergy functional theory. This allows to study bosonic systems with very few parameters that are determined variationally on an impurity problem, which is then embedded into the full system. The accuracy of the method is very high for systems where we can check the answer, but it can also straightforwardly be applied to systems where conventional quantum Monte Carlo simulations fail due to the sign problem. With a simplified version of this method, we could find interaction-induced symmetry protected topological phases in the bosonic Harper-Hofstader model (arXiv:1609.03760).

- Phys. Rev. B 89, 054204 (2014) and Comptes rendus Physique 14, 712-724 (2013) are works focusing on bosonic systems on a lattice subject to diagonal disorder. We contributed to developing a strong-disorder theory for bosons in 1d (they have a Kosterlitz-Thouless theory with a non-universal value of the Luttinger parameter at the transition. In contrast to previous works, our theory is asymptotically exact. We also wrote a review article.

- In Phys Rev Lett 117, 120402 (2016) we suggested to study many-body localized phases with Monte Carlo simulations by shifting the spectrum of the Hamiltonian through so-called localized-bits, which we could approximate through an exact diagonalization procedure. It may pave the way to study many-body localized phases in higher dimension. Many-body localization is currently a very hot topic in condensed matter physics. It deals with generic, interacting systems that do not forget their initial conditions, even after an infinitely long time evolution. Very few methods other than full diagonalization can be used to study these systems.

Supersolidity and Helium-4 —

We also performed a few studies relevant for superfluid and supersolid Helium-4 using path integral Monte Carlo simulations in the continuum:

In Phys Rev B 90, 184508 (2014) we explained the role of quantum kinks and jogs in dislocations in solid Helium-4, which may lead to the syringe effect and quantum plasticity. The ground state of such a solid may well be much richer than previously thought and be considered a quantum liquid of such dislocation excitations. In Phys Rev B 87, 094514 (2013) we studied the phase diagram of Helium-4 adsorbed on graphene, which was found to be very similar to one of Helium-4 adsorbed on graphite. In Phys Rev Lett 113, 045301 (2014) we studied the phase diagram of Helium-4 confined to a long cylinder: Unusually strong hysteresis was found, which was explained by the topology ( a disclination) of the cylinder. Finally, we obtained with the same method the ground state phase diagram of hypothetical particles interacting via gaussian interactions, showing the usual superfluid and crystalline phases, but no supersolid phase (Phys Rev B 93, 174520 (2016)).

contact:

Prof Dr Lode Pollet

Department of Physics

Arnold Sommerfeld Center

Theresienstrasse 37

80333 Munich

Germany

E-mail: Lode.Pollet@lmu.de

website: http://www.theorie.physik.uni-muenchen.de/lsschollwoeck/pollet_group/index.html

Diagrammatic Monte Carlo —

The main goal of the project was to develop new quantum Monte Carlo methods for strongly interacting systems. These are based on a sampling of Feynman diagrams and go under the name of diagrammatic Monte Carlo simulations, with several pros and cons: Compared to path integral Monte Carlo methods, the infamous sign problem is not leading to an exponential scaling in the system size or inverse temperature, but instead in the expansion order. The hope is hence that the answer can be obtained, either converged or through extrapolation, in low expansion orders. However, most perturbative expansions in physics are asymptotic, and this may lead to serious difficulties. Over the past years we learned that the method needs further refinements for competing instabilities, but could be applied successfully when there is a small parameter (which may be emergent). This has led to the following main results:

- P. Kroiss and L. Pollet, Phys Rev B 90, 104510 (2014) and Phys Rev B 91, 144507 (2015): In these works we applied the diagrammatic Monte Carlo method for the Fermi-polaron problem in the quasi-2d regime and in 3d for mass imbalance, respectively. The main result is that we could numerically prove that a variational approach in the 2particle-hole channel remains quantitatively accurate for all cases considered, an idea that appeared in the literature before in a number of different contexts but numerical results have not been analyzed along these lines before. This allowed us to obtain a rather precise phase diagram by combining different methods, as well as compute all thermodynamic properties of interest.

- L. Pollet, M. N. Kiselev, N. V. Prokof’ev, and B. V. Svistunov, Grassmannization of classical models, New J Phys Vol 18, Iss 11,p 113025 (2016): In order to overcome Dyson’s collapse argument, we developed the Grassmannization technique for classical bond models, giving us a Feynman’s diagrammatic technique based on Grassmann free propagators and Wick’s theorem. The method is general and can form the basis for plaquette-type interactions and even gauge fields. As an application, we derived the Feynman series (to 14th order) for the spin-spin correlation function of the 2D classical Ising model.

Interacting bosonic systems —

We have also studied a large number of bosonic systems with path-integral Monte Carlo simulations with worm-type updates in different contexts.

We list the main results:

- In Phys. Rev. A 89, 023605 (2014) we studied the 1d-2d dimensional crossover for the Bose-Hubbard model. It turns out that the location of the tip of the lobe is determined by the 1d physics and follows a Kosterlitz-Thouless behavior.

- In Phys Rev A 88, 033628 (2013) we studied mixtures of bosonic particles with SU(2) symmetry. For large occupation numbers the system phase separates, which can also be understood analytically.

- In Phys. Rev. Lett. 112, 030402 (2013) we studied the universal properties of the Higgs particle in (2+1)d for models with U(1) symmetry. This showed that the paradigm of quantum simulation also applies to response functions. We saw that the first peak shifts to lower frequency when tuning closer to the quantum critical point. Very close to the critical point, the response is dominated by the critical exponents of the XY model in 3 dimensions as well as a universal scaling function, which we determined.

- In Phys Rev B 93, 094404 (2016) the classical mutual information of the Sherington-Kirkpatrick model was analyzed and compared with an extension of Parisi’s famous solution.

- In Phys Rev B 94, 195119 (2016) we developed bosonic selfenergy functional theory. This allows to study bosonic systems with very few parameters that are determined variationally on an impurity problem, which is then embedded into the full system. The accuracy of the method is very high for systems where we can check the answer, but it can also straightforwardly be applied to systems where conventional quantum Monte Carlo simulations fail due to the sign problem. With a simplified version of this method, we could find interaction-induced symmetry protected topological phases in the bosonic Harper-Hofstader model (arXiv:1609.03760).

- Phys. Rev. B 89, 054204 (2014) and Comptes rendus Physique 14, 712-724 (2013) are works focusing on bosonic systems on a lattice subject to diagonal disorder. We contributed to developing a strong-disorder theory for bosons in 1d (they have a Kosterlitz-Thouless theory with a non-universal value of the Luttinger parameter at the transition. In contrast to previous works, our theory is asymptotically exact. We also wrote a review article.

- In Phys Rev Lett 117, 120402 (2016) we suggested to study many-body localized phases with Monte Carlo simulations by shifting the spectrum of the Hamiltonian through so-called localized-bits, which we could approximate through an exact diagonalization procedure. It may pave the way to study many-body localized phases in higher dimension. Many-body localization is currently a very hot topic in condensed matter physics. It deals with generic, interacting systems that do not forget their initial conditions, even after an infinitely long time evolution. Very few methods other than full diagonalization can be used to study these systems.

Supersolidity and Helium-4 —

We also performed a few studies relevant for superfluid and supersolid Helium-4 using path integral Monte Carlo simulations in the continuum:

In Phys Rev B 90, 184508 (2014) we explained the role of quantum kinks and jogs in dislocations in solid Helium-4, which may lead to the syringe effect and quantum plasticity. The ground state of such a solid may well be much richer than previously thought and be considered a quantum liquid of such dislocation excitations. In Phys Rev B 87, 094514 (2013) we studied the phase diagram of Helium-4 adsorbed on graphene, which was found to be very similar to one of Helium-4 adsorbed on graphite. In Phys Rev Lett 113, 045301 (2014) we studied the phase diagram of Helium-4 confined to a long cylinder: Unusually strong hysteresis was found, which was explained by the topology ( a disclination) of the cylinder. Finally, we obtained with the same method the ground state phase diagram of hypothetical particles interacting via gaussian interactions, showing the usual superfluid and crystalline phases, but no supersolid phase (Phys Rev B 93, 174520 (2016)).

contact:

Prof Dr Lode Pollet

Department of Physics

Arnold Sommerfeld Center

Theresienstrasse 37

80333 Munich

Germany

E-mail: Lode.Pollet@lmu.de

website: http://www.theorie.physik.uni-muenchen.de/lsschollwoeck/pollet_group/index.html