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# Algorithms

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In order to tackle problems living at the forefront of scientific research, a dynamic environment focused on the development of methods designed to go beyond what is currently reachable is a must. In our group we pursue a wide variety of approaches incorporating the latest developments in the most promising numerical methods.

Dynamical mean field approximation/theory (DMFA/DMFT) and its cluster extensions are methods based on a physically motivated approximation of the self-energy suitable for models with local interaction (e.g. Hubbard model). Here a reduction of the problem’s Hilbert space is achieved via a self-consistent mapping onto a quantum impurity problem. DMFT is, in combination with other band structure methods (e.g. density functional theory), the state-of-the-art approach for the study of correlated materials.

The dynamical cluster approximation (DCA) is a cluster extension of DMFT which is exact in the limit of infinite cluster size and provides a systematic way of reducing the mean field effects. It is of particular use for study of models. We adapted it for multisite cell structures enabling the simulation of general (non-Bravais) lattices. Some our more recent efforts are targeted towards developing an efficient algorithm for the impurity problem based on strong-coupling continuous-time Monte Carlo.

Series expansions methods have been long successful in the history of condensed matter physics. They constitute a series of very powerful methods allowing us to access systems which are difficult, or impossible, to study using Monte-Carlo based approaches and complement the physical picture obtained by other methods. Most notably, they are applicable to frustrated spin systems as well as fermionic lattice models alike, granting access both to the study of the high-temperature regime as well as the ground-state, directly in the thermodynamic limit.

Here we are currently developing a generic framework to perform high-temperature and perturbative T=0 series expansions for various models. By incorporating state-of-the-art high-performance computing techniques in the framework we bring our methods on today's supercomputers trying to push them far beyond their previous applications.

Path integral Monte Carlo (PIMC) can provide essentially exact thermodynamic measurements for systems in equilibrium. Unlike many other methods which purport to do the same, PIMC is not based off ground state wave functions but off the semi-classical path integral picture of the many-body density matrix originally introduced by Feynman. This feature grants PIMC simple access to finite-temperature quantum-mechanical quantities while maintaining a classical analogue of a vibrating polymer as shown above.

From the definition of the Fermi density matrix, we quickly realize there is an issue with the alternating sign, i.e. the fermion sign problem. One possible route to circumventing the sign problem is the fixed-node approximation, which constrains the sign of the density matrix according to an ansatz. However, since the actual nodal structure of the density matrix is unknown, this method comes with an uncontrolled approximation. We are currently exploring ways around this difficultly, both by allowing variational improvement of the nodal structure, as well as improving the explicit sampling of permutation space.

We develop continuous-time quantum Monte Carlo (CTQMC) methods for the simulations of correlated fermions, bosons and quantum spins. Besides their efficiency and accuracy, CTQMC methods are more flexible and have broader application range compared to the traditional discrete-time approaches. CTQMC methods also provide better access to quantities of interest to contemporary condensed matter physics research, such as quantum information measures (entanglement entropy and fidelity susceptibility). We adopt these new algorithms to study exotic quantum phases and topological materials.

Optimization methods have a wide range of real-world applications across a plethora of scientific disciplines. The minimization of a cost function is one of the foundations of physics. However, solving this often complex optimization problem poses formidable numerical and theoretical challenges, many of which remain unsolved. Because the effort needed to solve most interesting optimization problems scales worse than polynomial in the size of the input, advances in algorithm development are of paramount importance over ever increasing hardware requirements. Most recently, a push towards using quantum instead of classical hardware has emerged. Although promising, there is yet no concrete indication that optimization algorithms exploiting quantum effects pose an advantage over classical heuristic and exact optimization algorithms. Here we are working to gain further understanding in this challenging area of research, developing novel classical algorithms and exploring the potential advantage of quantum effects in finding the solution more efficiently than the classical counterparts.

Recently arising as a new, entanglement-based, approach to the study of strongly correlated lattice models, tensor network states are aimed towards the efficient description of a particularly relevant zero-measure subspace, of an otherwise exponentially growing Hilbert space, corresponding to states obeying a so-called area law of entanglement entropy. Examples of states living in this subspace include all ground-states of 1D gapped local hamiltonians. More interestingly, it is by now well-accepted that the applicability of these methods spans well beyond this scenario and have been successfully used for the study of local 2D Hamiltonians. as well as their time evolution and systems at finite temperature.

Matrix Product States (MPS), arising naturally within the Density Matrix Renormalization Group (DMRG) framework, have become the golden standard for the simulation of 1D lattice systems of either spins, bosons or fermions. Projected Entangled-Pair States (PEPS) or their thermodynamic limit extension infinite PEPS (iPEPS), correspond to the generalization of MPS to 2D lattices and have, over the past decade, become one of the reference methods in the study of physically relevant models such as the t-J model, connected to the physics of the High-Tc cuprates. Here, we have developed a massively parallel DMRG code designed to tackle 2D simulations, where the problem scales exponentially in the system width, and where, for Hubbard-type systems, we have been able to perform simulations employing bond dimensions beyond 40.000. In the realm of PEPS we have successfully applied it to a rich variety of models including fermionic as well as frustrated spin models and we are currently exploring approaches towards achieving a reduced computational complexity.