Dynamical Quantum Phase Transitions: A Geometric Picture

Abstract

The Loschmidt echo (LE) is a purely quantum-mechanical quantity whose determination for large quantum many-body systems requires an exceptionally precise knowledge of all eigenstates and eigenenergies. One might therefore be tempted to dismiss the applicability of any approximations to the underlying time evolution as hopeless. However, using the fully connected transverse-field Ising model (FC-TFIM) as an example, we show that this indeed is not the case, and that a simple semiclassical approximation to systems well described by mean-field theory (MFT) is in fact in good quantitative agreement with the exact quantum-mechanical calculation. Beyond the potential to capture the entire dynamical phase diagram of these models, the method presented here also allows for an intuitive geometric interpretation of the fidelity return rate at any temperature, thereby connecting the order parameter dynamics and the Loschmidt echo in a common framework. Videos of the post-quench dynamics provided in the supplemental material visualize this new point of view.Equilibrium phase transitions are remarkable phenomena that have been under thorough experimental and theoretical investigation for decades. Over time, a number of advanced techniques such as scaling theory [1-3] and the renormalization group method [4-9] have been developed for the determination of the universal properties close to a critical point. One might ask whether an indepth study of dynamical critical phenomena far from equilibrium is possible along the lines established in the equilibrium framework. With the advent of modern ultracold atom [10-13] and ion-trap [14-16] experiments, this originally purely academic question has become accessible in laboratories as well.Dynamical quantum phase transitions (DPTs) occur in the dynamics of a quantum system after quenching a set of control parameters {Γ} of its Hamiltonian:Recently, the study of DPTs has focused on two largely independent concepts [17]. The first one, , resembles equilibrium Landau theory: A system undergoes a dynamical phase transition if the long-time limit of the order parameter is finite for one set {Γ i , Γ f }, whereas it vanishes for different final parameters {Γ f }. Furthermore, DPT-I also entails criticality in the transient dynamics of the order parameter and two-point correlators before reaching the steady state, giving rise to effects such as dynamic scaling and aging, which have been investigated theoretically [30][31][32] and also observed experimentally [33].The second concept, DPT-II, generalizes the nonanalytic behavior of the free energy at a phase transition in the thermodynamic limit (TL) to the out-ofequilibrium case. To this end, the LE has been introduced as a dynamical analog of a free energy per particle [34]. DPT-II has been extensively studied both theoretically [34][35][36][37][38][39][40][41][42][43] and in experiments [44,45]. As we aim to calculate dynamical phase transitions at finite preparation temperatures, we define the distance covered in Hilbert space between the pr...

Dynamical Topological Quantum Phase Transitions in Nonintegrable Models

“…Recently, direct experimental observation of DQPTs has been reported, where a transverse-field Ising model was realized with trapped ions [5,6]. For the better understanding of DQPTs several integrable models have been considered [4,[7][8][9][10][11], where the time evolution can be solved exactly. It has been revealed that, like equilibrium phase transitions, DQPTs also affect other observables.…”

Dynamical Phase Transitions in Topological Insulators

“…It is therefore interesting to generalise the concept of dynamical phase transitions to finite temperatures and density matrices. 29,[35][36][37][38][39] There is no unique way in which one may want to make such a figure 5(a). The solid dashed line is the entanglement entropy for cutting two dimers.…”

Out-of-equilibrium phase diagram of long-range superconductors

“…Indeed, it was then firmly established that the dynamical critical point, which separates different phases of DQPT, is in general distinct from the quantum equilibrium critical point and is strongly dependent on the initial condition [18,21,22]. Additionally, this dynamical critical point was shown to coincide with that of the type of dynamical phase transition based on a local order parameter [18,[20][21][22][23][24][25]. Also recently, the theory of DQPT has been extended to Floquet systems [38][39][40] and models with many-body localized phases [41,42].…”

“…Among the prominent thrusts of this research effort lies the phenomenon of dynamical phase transitions [8,9], which fall into two major categories when considering sudden quenches of a quantum many-body system. The first is characterized by a local order parameter, whose long-time behavior determines the dynamical phase of the steady state [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26]. In principle, this type of dynamical phase transition separates a ferromagnetic from a paramagnetic steady state in the wake of a quench, and this has been recently observed experimentally in trapped-ion setups [27].…”