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

Abstract

A boundary transfer matrix formulation allows to calculate the Loschmidt echo for onedimensional quantum systems in the thermodynamic limit. We show that nonanalyticities 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 nonintegrable cases-where the Loschmidt echo does not show nonanalyticities although the quench leads across an equilibrium phase transition, and examples where nonanalyticities 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.

Quench echo and work statistics in integrable quantum field theories

“…One can begin to understand this in terms of the corresponding Fisher zeroes of the lattice partition function: if one does a quench originating from one phase and arriving in a different phase, one may expect (not always) a dynamical phase transition in the time evolution, which is governed by the Fisher zeroes [78]. With a reversed logic, when quenching inside the same phase, as in our case, a dynamical phase transition is not expected (although in some cases they were observed also without crossing a phase transition line, see [80,81]), thus neither are Fisher zeroes, i.e. branch points.…”

“…This non-analytic behaviour was later shown to be robust under the inclusion of non-integrable interactions, irrelevant or relevant in the RG sense, using the tDMRG numerical method [79]. However more recently it was also shown that such nonanalyticities are caused more generally by a crossing of eigenvalues in the spectrum of the transfer matrix [80] and that they may not appear even if a critical line is crossed, thus indicating that the presence of zeros in Z(z) and non-analyticities in the free energy and L(t) are not a characteristic feature of dynamical phase transitions [81]. Further progress in the analytical calculation of the above quantity for integrable spin chains was made using Algebraic Bethe Ansatz techniques [82,83].…”

Tuning the presence of dynamical phase transitions in a generalizedXYspin chain

“…The initial observation by Heyl et al 4 that DPTs are associated with the sudden quenches across the QCP has been verified in several studies [7][8][9][10]16 . However, subsequently it has been shown that DPTs are not necessarily connected with the passage through the equilibrium QCP and may occur following a sudden quench even within the same phase (i.e., not crossing the QCP) for both integrable 11 as well as non-integrable models 12 .…”

Overlap distributions for quantum quenches in the anisotropic Heisenberg chain

“…The behavior of the Loschmidt echo after a quantum quench has been studied in a large body of recent literature, see, e.g., [18,22,25,34,. In particular, the Loschmidt echo is central to the study of so-called dynamical phase transitions [67][68][69][70][71][72][73][74][75][76]. By expanding |Ψ 0 in terms of the eigenstates |n of the final Hamiltonian H f , we find…”

“…In the current era of non-equilibrium physics, it has been widely used as a model system to explore and exemplify non-equilibrium phenomena, including studies of the Loschmidt echo (e.g. [22,46,[64][65][66][67][68]), studies of quenches of the anisotropy parameter ∆ (e.g. [99,100]), and of quenches starting from a Néel state (e.g.…”