## Periodic Reporting for period 3 - TENSORNETSIM (Accurate simulations of strongly correlated systems with tensor network methods)

Reporting period: 2019-09-01 to 2021-02-28

One of the key challenges in physics is understanding the emergent phenomena in systems of strongly interacting quantum particles, called strongly correlated quantum many-body systems. These systems can give rise to very remarkable phenomena, such as high-temperature superconductivity where an electrical current can flow without resistance at much higher temperatures than in conventional superconductors, or so-called quantum spin liquids which exhibit a new type of order called topological order and which host new types of quasi-particle excitations. These systems can often be described by relatively simple models on a lattice, but their study is nevertheless extremely challenging because conventional numerical simulations often fail in these cases. A famous example is the two-dimensional Hubbard model, a simple model of interacting electrons on a square lattice which is believed to capture the physics of high-temperature superconductivity. Despite an enormous effort during the last 50 years the Hubbard model has still not been conclusively solved, and the mechanism leading to high-temperature superconductivity remains to be uncovered. Thus, in order to make progress in our understanding of quantum many-body systems - which also forms the basis for future groundbreaking technological applications - it is essential to develop new numerical techniques for the accurate simulation of strongly correlated systems which are beyond today’s state-of-the-art.

This project focuses on the development and application of so-called two-dimensional tensor network methods to simulate challenging open problems in quantum many-body physics. A tensor network is an ansatz for quantum many-body states in which the accuracy can be systematically controlled by the so-called bond dimension.

On the methods development side, the main goals of this project include:

- the development of new techniques to optimize a tensor network ansatz and methods to extrapolate data to the exact infinite bond dimension limit

- the development of tensor network methods at finite temperature

- the development of algorithms to compute excitation spectra based on tensor networks

- tools to characterize and classify topological states of matter using tensor networks

- new types of tensor networks and algorithms to perform realistic simulations of high-temperature superconductors

- extensions of tensor network methods to quantum systems in 3 dimensions

- parallelization of tensor network codes

On the application side the main goals include:

- Simulations of the two-dimensional Hubbard model and multi-band Hubbard models to get new insights into high-temperature superconductivity, at zero and at finite temperature.

- Simulations of the SU(N) Hubbard and SU(N) Heisenberg models to predict new phases of matter which can be realized in experiments on ultra-cold alkaline earth atoms in optical lattices

- Simulations of realistic frustrated spin systems in two and three dimensions at zero and finite temperature for a quantitative understanding of frustrated materials

This project focuses on the development and application of so-called two-dimensional tensor network methods to simulate challenging open problems in quantum many-body physics. A tensor network is an ansatz for quantum many-body states in which the accuracy can be systematically controlled by the so-called bond dimension.

On the methods development side, the main goals of this project include:

- the development of new techniques to optimize a tensor network ansatz and methods to extrapolate data to the exact infinite bond dimension limit

- the development of tensor network methods at finite temperature

- the development of algorithms to compute excitation spectra based on tensor networks

- tools to characterize and classify topological states of matter using tensor networks

- new types of tensor networks and algorithms to perform realistic simulations of high-temperature superconductors

- extensions of tensor network methods to quantum systems in 3 dimensions

- parallelization of tensor network codes

On the application side the main goals include:

- Simulations of the two-dimensional Hubbard model and multi-band Hubbard models to get new insights into high-temperature superconductivity, at zero and at finite temperature.

- Simulations of the SU(N) Hubbard and SU(N) Heisenberg models to predict new phases of matter which can be realized in experiments on ultra-cold alkaline earth atoms in optical lattices

- Simulations of realistic frustrated spin systems in two and three dimensions at zero and finite temperature for a quantitative understanding of frustrated materials

We have already made significant progress in achieving the goals of this project, on the methodological side as well as on the application side. The main achievements in the methods development include:

- the development of new algorithms to optimize the tensors by performing an energy minimization which provides more accurate results than previous techniques.

- the development of an extrapolation technique based on a so-called truncation error which yields accurate estimates of ground state energies in the infinite bond dimension limit.

- a breakthrough in the study of 2D quantum critical systems based on a so-called finite correlation length scaling, enabling the accurate determination of critical couplings and universal critical exponents

- the development and testing of tensor network methods to compute properties at finite temperature

- the implementation and testing of new schemes to compute excitations with tensor networks

- testing and benchmarking tensor network approaches for 3D quantum systems

On the application side of this project the main achievements so far include:

- a major breakthrough in simulating the 2D Hubbard model where were able to obtain, for the first time, a conclusive answer regarding the nature of the ground state at a particularly challenging point in the phase diagram (U/t=8, 1/8 doping), namely that the ground state is a stripe state and not a uniform state.

- extension of these simulations to an extended 2D Hubbard model with an additional next-nearest neighbor hopping which yields the same stripe period as observed in experiments

- simulations of the SU(6) Heisenberg model on the honeycomb lattice which revealed a plaquette state

- pioneering tensor network simulations of an SU(N) Hubbard model, for the case N=3 on the honeycomb lattice, which clearly demonstrates that these models have become within reach of state-of-the-art tensor network simulations

- discovery of an unexpected Haldane phase in S=1 spin systems

- new insights into the physics of the frustrated material SrCu2(BO3)2 described by the Shastry-Sutherland model, including a new explanation for the new anomalies in the magnetization process observed in experiments, and new theory for the system under pressure described by a deformed Shastry-Sutherland model

- the development of new algorithms to optimize the tensors by performing an energy minimization which provides more accurate results than previous techniques.

- the development of an extrapolation technique based on a so-called truncation error which yields accurate estimates of ground state energies in the infinite bond dimension limit.

- a breakthrough in the study of 2D quantum critical systems based on a so-called finite correlation length scaling, enabling the accurate determination of critical couplings and universal critical exponents

- the development and testing of tensor network methods to compute properties at finite temperature

- the implementation and testing of new schemes to compute excitations with tensor networks

- testing and benchmarking tensor network approaches for 3D quantum systems

On the application side of this project the main achievements so far include:

- a major breakthrough in simulating the 2D Hubbard model where were able to obtain, for the first time, a conclusive answer regarding the nature of the ground state at a particularly challenging point in the phase diagram (U/t=8, 1/8 doping), namely that the ground state is a stripe state and not a uniform state.

- extension of these simulations to an extended 2D Hubbard model with an additional next-nearest neighbor hopping which yields the same stripe period as observed in experiments

- simulations of the SU(6) Heisenberg model on the honeycomb lattice which revealed a plaquette state

- pioneering tensor network simulations of an SU(N) Hubbard model, for the case N=3 on the honeycomb lattice, which clearly demonstrates that these models have become within reach of state-of-the-art tensor network simulations

- discovery of an unexpected Haldane phase in S=1 spin systems

- new insights into the physics of the frustrated material SrCu2(BO3)2 described by the Shastry-Sutherland model, including a new explanation for the new anomalies in the magnetization process observed in experiments, and new theory for the system under pressure described by a deformed Shastry-Sutherland model

Our simulations of the 2D Hubbard model can be seen as a major milestone in computational physics and demonstrate that even a very challenging model like the 2D Hubbard model has become within reach of modern computational methods. Our work sets a new state-of-the-art in the simulation of ground states of strongly correlated electron systems. Similarly, we expect highly accurate results also for multi-band Hubbard models to gain new insights into high-temperature superconductivity in the near future until the end of this project.

Our pioneering work on the SU(N) Hubbard models clearly show that these challenging systems can now be accurately studied by 2D tensor network simulations, offering a unique opportunity to predict new phases of matter which can potentially be realized in future quantum simulators with alkaline-earth atoms in optical lattices.

Our studies of frustrated spin systems further establish 2D tensor network as a reliable and powerful tool to study these systems in order to understand the physics of frustrated materials. First results at finite temperature also demonstrate that tensor networks can help to shed new light onto challenging problems at finite temperature where quantum Monte Carlo fails due to the negative sign problem. Also our first benchmarks in 3D look promising, and we expect to be able to address challenging problems in 3D in the second part of this project.

With the rapid progress on the methodological side so far we expect to have established and fully developed all methodological goals until the end of the project.

Our pioneering work on the SU(N) Hubbard models clearly show that these challenging systems can now be accurately studied by 2D tensor network simulations, offering a unique opportunity to predict new phases of matter which can potentially be realized in future quantum simulators with alkaline-earth atoms in optical lattices.

Our studies of frustrated spin systems further establish 2D tensor network as a reliable and powerful tool to study these systems in order to understand the physics of frustrated materials. First results at finite temperature also demonstrate that tensor networks can help to shed new light onto challenging problems at finite temperature where quantum Monte Carlo fails due to the negative sign problem. Also our first benchmarks in 3D look promising, and we expect to be able to address challenging problems in 3D in the second part of this project.

With the rapid progress on the methodological side so far we expect to have established and fully developed all methodological goals until the end of the project.