Quantum Monte Carlo study of strongly correlated electrons

Understanding strongly correlated electrons is an important long-term goal, not only for uncoveringfundamental physics behind, but also for their emergence of lots of novel states which have potentialapplications in quantum control and quantum computations. Meanwhile, the strongly correlated electrons areusually extremely hard problems, and it is generally impossible to understand them unbiasedly. QuantumMonte Carlo is a typical unbiased numeric method, which does not depend on any perturbation, and it can helpus to exactly understand the strongly correlated electrons, so that it is widely used in high energy andcondensed matter physics. However, quantum Monte Carlo usually suffers from the notorious sign problem. Inthis paper, we introduce general ideas to design sign problem free models and discuss the sign bound theory weproposed recently. In the sign bound theory, we build a direct connection between the average sign and theground state properties of the system. We find usually the average sign has the conventional exponential decaywith system size increasing, leading to exponential complexity; but for some cases it can have algebraic decay,so that quantum Monte Carlo simulation still has polynomial complexity. By designing sign problem free oralgebraic sign behaved strongly correlated electron models, we can approach to several long outstandingproblems, such as the itinerant quantum criticality, the competition between unconventional superconductivityand magnetism, as well as the recently found correlated phases and phase transitions in moire quantum matter