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arXiv:1804.03102 · Physical review. E

Dynamic finite-size scaling after a quench at quantum transitions

Andrea Pelissetto C, Davide Rossini, Ettore Vicari

Published May 31, 2018• 44 citations
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Abstract

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.

DOI
10.1103/physreve.97.052148

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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.
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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.
Uncovering and understanding universal dynamics in matter far from equilibrium remains a key challenge. In this work, we identify a so far unrecognized form of universal behavior that emerges after a sudden symmetry-breaking quench at continuous phase transitions. Our key observation is that the order-parameter fluctuations in Ising models exhibit a compelling temporal collapse across a wide range of system sizes and quench strengths, indicative of an emergent single-variable scaling form. This phenomenon can be explained by introducing a so far unknown dynamical critical exponent for the underlying continuous phase transition. We find evidence for a lower critical effective dimension of this universal regime: it is observed in the 2D quantum and 3D and 4D classical Ising models, but not in the 1D quantum or 2D classical cases. Our results suggest that our observed universal far-from-equilibrium scaling may extend beyond the Ising models studied here and could more broadly characterize systems with non-conserved order parameters, opening new avenues for exploring universal dynamics both theoretically and in current experimental platforms.
We address the quantum dynamics of a system composed of a qubit globally coupled to a many-body system characterized by short-range interactions. We employ a dynamic finite-size scaling framework to investigate the out-of-equilibrium dynamics arising from the sudden variation (turning on) of the interaction between the qubit and the many-body system, in particular when the latter is in proximity to a quantum first-order or continuous-phase transition. Although the approach is quite general, we consider ddd-dimensional quantum Ising spin models in the presence of transverse and longitudinal fields as paradigmatic quantum many-body systems. To characterize the out-of-equilibrium dynamics, we focus on a number of quantum-information-oriented properties of the model. Namely, we concentrate on the decoherence features of the qubit, the energy interchange between the qubit and the many-body system during the out-of-equilibrium dynamics, and the work distribution associated with the quench. The scaling behaviors predicted by the dynamic finite-size scaling theory are verified through extensive numerical computations for the one-dimensional Ising model, which reveal a fast convergence to the expected asymptotic behavior with increasing system size.
We investigate the effects of dissipation on the quantum dynamics of many-body systems at quantum transitions, especially considering those of the first order. This issue is studied within the paradigmatic one-dimensional quantum Ising model. We analyze the out-of-equilibrium dynamics arising from quenches of the Hamiltonian parameters and dissipative mechanisms modeled by a Lindblad master equation, with either local or global spin operators acting as dissipative operators. Analogously to what happens at continuous quantum transitions, we observe a regime where the system develops a nontrivial dynamic scaling behavior, which is realized when the dissipation parameter uuu (globally controlling the decay rate of the dissipation within the Lindblad framework) scales as the energy difference $\mathrm{\ensuremath{\Delta}}$ of the lowest levels of the Hamiltonian, i.e., $u\ensuremath{\sim}\mathrm{\ensuremath{\Delta}}$. However, unlike continuous quantum transitions where $\mathrm{\ensuremath{\Delta}}$ is power-law suppressed, at first-order quantum transitions $\mathrm{\ensuremath{\Delta}}$ is exponentially suppressed with increasing the system size (provided the boundary conditions do not favor any particular phase).
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.