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arXiv:1712.09907 · Physical review. B./Physical review. B

Out-of-equilibrium dynamics driven by localized time-dependent perturbations at quantum phase transitions

Andrea Pelissetto C, Davide Rossini, Ettore Vicari

Published Mar 14, 2018• 29 citations
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Abstract

We investigate the quantum dynamics of many-body systems subject to local (i.e., restricted to a limited space region) time-dependent perturbations. If the system crosses a quantum phase transition, an off-equilibrium behavior is observed, even for a very slow driving. We show that, close to the transition, time-dependent quantities obey scaling laws. In first-order transitions, the scaling behavior is universal, and some scaling functions can be computed exactly. For continuous transitions, the scaling laws are controlled by the standard critical exponents and by the renormalization-group dimension of the perturbation at the transition. Our protocol can be implemented in existing relatively small quantum simulators, paving the way for a quantitative probe of the universal off-equilibrium scaling behavior, without the need to manipulate systems close to the thermodynamic limit.

DOI
10.1103/physrevb.97.094414

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We investigate the quantum dynamics of many-body systems subject to local (i.e., restricted to a limited space region) time-dependent perturbations. If the system crosses a quantum phase transition, an off-equilibrium behavior is observed, even for a very slow driving. We show that, close to the transition, time-dependent quantities obey scaling laws. In first-order transitions, the scaling behavior is universal, and some scaling functions can be computed exactly. For continuous transitions, the scaling laws are controlled by the standard critical exponents and by the renormalization-group dimension of the perturbation at the transition. Our protocol can be implemented in existing relatively small quantum simulators, paving the way for a quantitative probe of the universal off-equilibrium scaling behavior, without the need to manipulate systems close to the thermodynamic limit.
We investigate the emergence of universal dynamical scaling in quantum critical spin systems adiabatically driven out of equilibrium, with emphasis on quench dynamics which involves non-isolated critical points ( i.e. , critical regions) and cannot be a priori described through standard scaling arguments nor time-dependent perturbative approaches. Comparing to the case of an isolated quantum critical point, we find that non-equilibrium scaling behavior of a large class of physical observables may still be explained in terms of equilibrium critical exponents. However, the latter are in general non-trivially path-dependent, and detailed knowledge about the time-dependent excitation process becomes essential. In particular, we show how multiple level crossings within a gapless phase may completely suppress excitation depending on the control path. Our results typify non-ergodic scaling in continuous finite-order quantum phase transitions.
We investigate the nonequilibrium dynamics of quantum spin chains during a round-trip protocol that slowly drives the system across a quantum first-order transition. Out-of-equilibrium scaling behaviors \`a la Kibble-Zurek for the single-passage protocol across the first-order transition have been previously determined. Here, we show that such scaling relations persist when the driving protocol is inverted and the transition is approached again by a far-from-equilibrium state. This results in a quasi-universality of the scaling functions, which keep some dependence on the details of the protocol at the inversion time. We explicitly determine such quasi-universal scaling functions by employing an effective two-level description of the many-body system near the transition. We discuss the validity of this approximation and how this relates to the observed scaling regime. Although our results apply to generic systems, we focus on the prototypical example of a 1D transverse field Ising model in the ferromagnetic regime, which we drive across the first-order transitions through a time-dependent longitudinal field.
We present a general dynamic finite-size scaling theory for the quantum dynamics after an abrupt quench, at both continuous and first-order quantum transitions. For continuous transitions, the scaling laws are naturally ruled by the critical exponents and the renormalization-group dimension of the perturbation at the transition. In the case of first-order transitions, it is possible to recover a universal scaling behavior, which is controlled by the size behavior of the energy gap between the lowest-energy levels. We discuss these findings in the framework of the paradigmatic quantum Ising ring, and support the dynamic scaling laws by numerical evidence.
We study the effects of symmetry-breaking perturbations in the out-of-equilibrium quantum dynamics of many-body systems slowly driven by a time-dependent symmetry-preserving parameter, across the quantum critical regime associated with a continuous quantum transition (CQT). For this purpose, we analyze the out-of-equilibrium dynamics arising from generalized Kibble-Zurek (KZ) protocols, within a dynamic renormalization-group framework allowing for finite-size systems. We show that the time dependence of generic observables develops an out-of-equilibrium finite-size scaling (FSS) behavior, arising from the interplay between the timescale ts{t}_{s}ts​ of the parameter variations in the KZ protocol, the size LLL of the system, and the strength hhh of the symmetry-breaking perturbation, in the limit of large ts{t}_{s}ts​ and LLL. Moreover, scaling arguments based on the first-order adiabatic approximation of slow variations in quantum systems allow us to characterize the approach to the adiabatic regimes for some limits of the model parameters (for example, when we take ${t}_{s}\ensuremath{\rightarrow}\ensuremath{\infty}$ before $L\ensuremath{\rightarrow}\ensuremath{\infty}$), predicting asymptotic power-law suppressions of the nonadiabatic behaviors in the adiabatic limits. This out-of-equilibrium FSS is supported by numerical analyses for the paradigmatic quantum Ising chain along generalized KZ protocols, with a time-dependent transverse field crossing its CQT, in the presence of a static longitudinal field breaking the Z2{\mathbb{Z}}_{2}Z2​ symmetry.
We address the out-of-equilibrium dynamics of many-body systems subject to time-dependent round-trip protocols across quantum and classical (thermal) phase transitions. They are realized by slowly changing one relevant parameter www across its critical point wc=0{w}_{c}=0wc​=0, linearly in time with a large timescale ts{t}_{s}ts​, from wi<0{w}_{i}<0wi​<0 to wf>0{w}_{f}>0wf​>0 and then back to wi<0{w}_{i}<0wi​<0, thus entailing multiple passages through the critical point. Analogously to the one-way Kibble-Zurek protocols across a critical point, round-trip protocols develop dynamic scaling behaviors at both classical and quantum transitions put forward within renormalization-group frameworks. The scaling scenario is analyzed within some paradigmatic models undergoing quantum and classical transitions belonging to the two-dimensional Ising universality class, such as one-dimensional quantum Ising models and fermionic wires, and two-dimensional classical Ising models (supplemented with a purely relaxational dynamics). While the dynamic scaling frameworks are similar for classical and quantum systems, substantial differences emerge due to the different nature of their dynamics, which is purely relaxational for classical systems (implying thermalization in the large-time limit at fixed model parameters), and unitary in the case of quantum systems. In particular, when the critical point separates two gapped (short-ranged) phases and the extreme value wf>0{w}_{f}>0wf​>0 is kept fixed in the large-ts{t}_{s}ts​ limit of the round-trip protocol, we observe hysteresislike scenarios in classical systems, while quantum systems do not apparently develop a sufficiently robust scaling limit along the return way, due to the presence of rapidly oscillating relative phases among the relevant quantum states.
Universality and scaling are fundamental concepts in equilibrium continuous phase transitions. Here, we unveil a unique and universal scaling behavior of the critical time in slowly driven dynamical quantum phase transition. Going beyond the analogy with equilibrium phase transition, we find that the critical time exhibits a power-law scaling with quenching rate and the scaling exponent is fully determined by underlining universality class. We explain this unique scaling behavior based on the adiabatic-impulse scenario in the Kibble-Zurek mechanism. This universal scaling behavior is verified to be valid not only in noninteracting single-particle system, but also in many-body interacting system, and not only in Hermitian system, but also in non-Hermitian system. Our study unravels a deep and fundamental relationship between dynamical phase transition and equilibrium phase tranition.