Abstract
We address the computation of the Loschmidt echo in interacting integrable spin chains after a quantum quench. We focus on the massless regime of the XXZ spin-1/2 chain and present exact results for the dynamical free energy (Loschmidt echo per site) for a special class of integrable initial states. For the first time we are able to observe and describe points of non-analyticities using exact methods, by following the Loschmidt echo up to large real times. The dynamical free energy is computed as the leading eigenvalue of an appropriate Quantum Transfer Matrix, and the non-analyticities arise from the level crossings of this matrix. Our exact results are expressed in terms of “excited-state” thermodynamic Bethe ansatz equations, whose solutions involve non-trivial Riemann surfaces. By evaluating our formulas, we provide explicit numerical results for the quench from the Néel state, and we determine the first few non-analytic points.
Highlights
Despite their long history, in the past decade the theory of integrable models has witnessed a series of unexpected developments
A simple but physically interesting protocol which has proven to be within the reach of integrability has been the one of quantum quenches [2]: a system is prepared in some well-defined state | 0 and left to evolve unitarily with some Hamiltonian H
We addressed the computation of the Loschmidt echo in the XXZ spin-1/2 chain, for a special class of integrable initial states [90]
Summary
In the past decade the theory of integrable models has witnessed a series of unexpected developments. Despite the Quench Action approach provides a formal representation for the time evolution of local observables, it is usually overwhelmingly complicated to evaluate, and so far this task was carried out only in the case of interaction quenches in one-dimensional Bose gases [45] Motivated by this problem, an analytic computation of the so-called Loschmidt echo in the XXZ Heisenberg chain was initiated in [53,54]. A promising analytical approach to its computation was proposed in [53], which is based on the so called Quantum Transfer Matrix (QTM) formalism [87,88,89] This method can be applied quite generally to an infinite family of initial integrable states, which have been introduced and studied in [90]. The most technical part of our work is consigned to the appendix
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