Dynamical phase transitions in the collisionless pre-thermal states of isolated quantum systems: theory and experiments

Abstract

We overview the concept of dynamical phase transitions in isolated quantum systems quenched out of equilibrium. We focus on non-equilibrium transitions characterized by an order parameter, which features qualitatively distinct temporal behaviour on the two sides of a certain dynamical critical point. Dynamical phase transitions are currently mostly understood as long-lived prethermal phenomena in a regime where inelastic collisions are incapable to thermalize the system. The latter enables the dynamics to substain phases that explicitly break detailed balance and therefore cannot be encompassed by traditional thermodynamics. Our presentation covers both cold atoms as well as condensed matter systems. We revisit a broad plethora of platforms exhibiting pre-thermal DPTs, which become theoretically tractable in a certain limit, such as for a large number of particles, large number of order parameter components, or large spatial dimension. The systems we explore include, among others, quantum magnets with collective interactions, φ 4 quantum field theories, and Fermi-Hubbard models. A section dedicated to experimental explorations of DPTs in condensed matter and AMO systems connects this large variety of theoretical models.

Theory of dynamical phase transitions in quantum systems with symmetry-breaking eigenstates

“…The first one, DPT-I, occurs when the dynamics of observables qualitatively changes at a critical value of a control parameter [6][7][8][9][10][11]. DPTs-I are characterized by a dynamical order parameter, given by the long-time average of an observable which changes from a nonzero value to zero at the critical value of the control parameter (for a recent review, see [12]). The second one, DPT-II, happens when the dynamics becomes non-analytic at particular critical times [13][14][15][16][17][18][19]; hence, it is a purely non-equilibrium phenomenon which cannot be described by a dynamical order parameter.…”

Dynamical and excited-state quantum phase transitions in collective systems

“…On the one hand, for quenches with average energy below and above the ESQPT of Fig. 4(a,c), that is < or > , respectively, the time when the quantum dynamics deviates from the classical expectation follows a power-law behavior of the form SC ∼ √ , as expected [19,39]; however, this time is much smaller for the quench ending at the ESQPT critical energy, revealing a logarithmic law instead, SC ∼ log 10 . Such a logarithmic scaling essentially precludes a realistic description of the quantum dynamics by means of the classical limit; for a macroscopic system with = 10 24 atoms, the quantum evolution would follow the semiclassical curve only up to ≈ log 10 10 24 = 24 s, which is negligible compared to ≈ √ 10 24 = 10 12 s = 10 6 s as obtained for quenches ending below or above .…”

“…This operator plays an important role in DPTs-I as it can be used to define an order parameter [19] [i.e., ( ) = ˆ ( ) in Fig. 1].…”

“…In the quantum domain, the same phenomenon appears in closed quantum systems evolving under unitary dynamics. A typical scenario occurs in quantum systems with long-range or infinite-range interactions which undergo a quantum phase transition (QPT) separating two ground-state phases [19]: one phase where a discrete Z 2 symmetry is broken, and another where the same symmetry is restored. Under such circumstances, a quench from a broken-symmetry ground state may lead to two different dynamical behaviors: one in which oscillations around a broken-symmetry effective state are observed, and another one in which the same kind of oscillations occur around a symmetric state.…”

Dynamical quantum phase transitions in spin-$S$ $\mathrm{U}(1)$ quantum link models

“…The field of far-from-equilibrium quantum many-body physics currently finds itself in a remarkable era of active quantum-simulation efforts seeking to realize evermore * jad.halimeh@physik.lmu.de exotic phenomena with no true counterpart in equilibrium [1][2][3][4][5]. Naturally, such efforts align with the ultimate quest for possible dynamical quantum universality classes, for which various concepts of dynamical phase transitions have been proposed [6][7][8][9][10].…”