Detection of Quantum Phase Transitions with a Lee-Yang Formalism and Many-Body Algorithms

Phase transitions in many-body quantum systems arise from the collective behaviour of many degrees of freedom. For many interacting quantum many-body systems, it remains challenging to determine their phase diagram due to the exponential growth in the Hilbert space size and the difficulty of existing numerical techniques to tackle generic interacting quantum many-body systems. In this thesis, we combine state-of-the-art numerical techniques with an approach to symmetrybreaking phase transitions inspired by the works of Lee and Yang on the zeros of partition functions in the complex plane. Specifically, we study the behaviour and distribution of zeros of the momentgenerating function of the relevant order parameters to locate the phase transitions. We refer to these zeros as Lee-Yang zeros. Our approach involves using tensor networks and neural quantum states as variational states to describe the ground states or finite temperature density matrices of the systems studied. While tensor networks allow for a direct evaluation of the moment-generating function, and therefore a direct determination of the position of these Lee-Yang zeros, this is not possible for neural quantum states. Therefore, we also present a method that uses high cumulants of the order parameter combined with knowledge about the symmetries of the Lee-Yang zeros to estimate their locations. By extrapolating the distance from the origin to these zeros to the thermodynamic limit, the presence of a phase transition can be determined. Using this Lee-Yang formalism, we map out the phase diagram of the transverse field Ising model and a fermionic chain, thereby showing its practical applicability to specific quantum many-body models. Also using neural quantum states and tensor networks, we determine the phase diagram of a tetramerized antiferromagnetic spin-1/2 J₁-J₂ Heisenberg model on the square lattice. Without relying on our Lee-Yang formalism, we are able to trace out the phase diagram using conventional means, such as studying the susceptibility, spin structure factor and the many-body gap. This model, which has been recently realized in experiment, exhibits an intriguing competition between conventional magnetically ordered phases and a higher-order symmetry protected topological phase. By mapping out its phase diagram, we contribute to guiding experiments towards the parameter regimes of interest.