[Paper Review] Dynamical Phase Transitions as Properties of the Stationary State: Analytic Results after Quantum Quenches in the Spin-1/2 XXZ Chain

Maurizio Fagotti|arXiv (Cornell University)|Aug 1, 2013

Quantum many-body systems13 citations

TL;DR

This paper analytically investigates the dynamical free energy density after quantum quenches in the spin-1/2 XXZ chain, showing that non-analyticities (dynamical phase transitions) arise from the absence of a gap in the excitation spectrum of the generalized Gibbs ensemble (GGE). It establishes that these singularities are not exclusive to quenches across critical points but are instead tied to the gapless nature of the GGE's generalized Hamiltonian, providing a unified framework linking non-equilibrium dynamics to stationary-state properties.

ABSTRACT

The (Loschmidt) overlap between the state at different times after a quantum quench is attracting increasing interest, as it was recently shown that in the thermodynamic limit its logarithm per unit of length has a non-analytic behavior if a Hamiltonian parameter is quenched across a critical point. This phenomenon was called a "dynamical phase transition" in analogy with the behavior of the canonical partition function at an equilibrium phase transition. We distinguish between local and nonlocal contributions to the aforementioned quantity and derive an analytic expression for the time evolution of the local part after quantum quenches in the XXZ spin-1/2 chain. The state that describes the stationary properties of (local) observables can be represented by a Gibbs ensemble of a generalized Hamiltonian; we reveal a deep connection between the appearance of singularities and the excitation energies of the generalized Hamiltonian.

Motivation & Objective

• To understand the origin of non-analytic behavior in the dynamical free energy density after quantum quenches in integrable systems.
• To distinguish between local and nonlocal contributions to the Loschmidt overlap and isolate the bulk (local) part of the dynamical free energy.
• To investigate whether dynamical phase transitions are exclusively linked to quenches across critical points or can occur in gapped phases.
• To establish a connection between singularities in the time evolution and the excitation spectrum of the generalized Hamiltonian in the GGE.

Proposed method

• Derives a system of integral equations for the bulk part of the dynamical free energy density after a quantum quench in the XXZ chain.
• Uses the formalism of the generalized Gibbs ensemble (GGE) to describe the stationary state of local observables post-quench.
• Analyzes the dressed energy of the generalized Hamiltonian to identify conditions under which non-analyticities emerge.
• Applies an effective ansatz involving rapidly oscillating phases to approximate the large-time asymptotics of the Loschmidt amplitude.
• Performs a qualitative analysis linking singularities in the dynamical free energy to the absence of a gap in the excitation spectrum of the GGE Hamiltonian.
• Validates results numerically for quenches from infinite to finite anisotropy, observing non-analytic behavior even within the gapped phase.

Experimental results

Research questions

• RQ1 What determines the appearance of non-analyticities in the dynamical free energy density after a quantum quench in the XXZ chain?
• RQ2 Are dynamical phase transitions exclusively associated with quenches across critical points, or can they occur in gapped phases?
• RQ3 How is the non-analytic behavior of the Loschmidt overlap related to the excitation spectrum of the generalized Gibbs ensemble?
• RQ4 Can the bulk (local) contribution to the dynamical free energy be analytically described using integral equations derived from the GGE formalism?
• RQ5 What role does the gaplessness of the generalized Hamiltonian's excitation spectrum play in the emergence of singularities in the time evolution?

Key findings

• Non-analytic behavior in the bulk dynamical free energy density arises when the generalized Hamiltonian of the GGE has a gapless excitation spectrum, not only when quenching across a critical point.
• The paper identifies the absence of a gap in the excitation spectrum of the generalized Hamiltonian as the key condition for the appearance of singularities in the time evolution of the Loschmidt overlap.
• Numerical results show that non-analyticities persist even in the gapped phase of the XXZ model, indicating that criticality is not a necessary condition for dynamical phase transitions.
• The large-time asymptotics of the Loschmidt amplitude are well-approximated by an effective ansatz involving a rapidly oscillating phase and the dressed energy of the GGE.
• Singularities at late times are approximately periodic, with times of non-analyticity given by t*(xc) = (2/sinh η) * (2π(n + 1/2))/d(xc), where d(xc) is the derivative of the dressed energy and x_c is a zero of the dressed energy.
• The analysis reveals that the dynamical free energy density's singularities are intrinsic properties of the stationary state described by the GGE, not just transient features of the time evolution.