In equilibrium, phase transition happened when the free energy density develops non-analyticity at thermodynamic limit. For out of equilibrium quantum systems, the dynamics are governed by Schrodinger equation. The Loschmidt echo can be formally interpreted as partition function. The rate function of the Loschmidt echo thus plays a similar role to the free energy density. The dynamical quantum phase transition(DQPT) happens at the critical time when this quantity develops non-analyticity. The notion of phase transition is then generalized to a dynamical quantum system. The study of related phenomena is still at its infancy and most understanding is built from exact solvable models or low dimensional systems. It will be nice if we can find more examples that is beyond above mentioned situations. The idea of quantum simulation has bee developed and applied on condensed matter related system for decades. Recently, the concept has been pushed forward to simulate gauge theories. One of the obstacles in quantum simulator is the local Hilbert space is usually of finite dimensional. Quantum Link Model(QLM) thus plays a special role that it provides the gauge structure using finite dimensional Hilbert space. Besides, recent quantum simulation experiments demonstrate the possibility of engineering gauge invariant Hamiltonian and coherent many-body real time evolution. In this project, we explore the quench dynamics of gauge theories through the lens of dynamical quantum phase transitions. Using time-evolving block decimation (TEBD) and Lanczos-based exact diagonalization, we explore (1+1) d and (2+1) d QLMs. The advantage of using TEBD is it is almost exact in (1+1) d at short time. For (2+1) d, naive full spectrum ED will hit the wall of enumerating of states and it can only reach 4 by 4 (32 spin) system in a reasonable time. Using Lanczos-based method has two primary advantages. First, we perform exact time-evolution within the Krylov subspace where the information of the sparse many-body Hamiltonian is compacted in a dense effective matrix which is easy to solve. Second, the Krylov space is generated at run time which reduces the requirement of memory and allows us to go to larger system such as 4 by 6(48 spins) and 6 by 6(72 spins). Also, the Krylov space will stay in the same gauge sector automatically as the Hamiltonian is gauge invariant. This property helps us bypass the enumeration of gauge invariant states in a neat way. The problem thus becomes manageable. Using the code I developed, we found DQPT in both (1+1) d and (2+1) d U(1) QLMs. We also found the corresponding order parameters and describe their physical meaning.

[1] Yi-Ping Huang, Debasish Banerjee, and Markus Heyl, Phys. Rev. Lett. 122, 250401 (2019)