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Dynamical quantum phase transitions and the Loschmidt echo: A transfer matrix approach

A boundary transfer matrix formulation allows to calculate the Loschmidt echo for one-dimensional quantum systems in the thermodynamic limit. We show that non-analyticities in the Loschmidt echo and zeros for the Loschmidt amplitude in the complex plane (Fisher zeros) are caused by a crossing of eigenvalues in the spectrum of the transfer matrix. Using a density-matrix renormalization group algorithm applied to these transfer matrices we numerically investigate the Loschmidt echo and the Fisher zeros for quantum quenches in the XXZ model with a uniform and a staggered magnetic field. We give examples---both in the integrable and the non-integrable case---where the Loschmidt echo does not show non-analyticities although the quench leads across an equilibrium phase transition, and examples where non-analyticities appear for quenches within the same phase. For a quench to the free fermion point, we analytically show that the Fisher zeros sensitively depend on the initial state and can lie exactly on the real axis already for finite system size. Furthermore, we use bosonization to analyze our numerical results for quenches within the Luttinger liquid phase.