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arXiv:1912.13015 · Physical review. B./Physical review. B

Scaling of Loschmidt echo in a boundary-driven critical Z3 Potts model

Naveen Nishad, G. J. Sreejith

Published Apr 6, 2020• 1 citation
arXivHTMLPDF

Abstract

Low-frequency perturbations at the boundary of critical quantum chains can be understood in terms of the sequence of boundary conditions imposed by them, as has been previously demonstrated in the Ising and related fermion models. Using extensive numerical simulations, we explore the scaling behavior of the Loschmidt echo under longitudinal field perturbations at the boundary of a critical ${\mathbb{Z}}_{3}$ Potts model. We show that at times much larger than the relaxation time after a boundary quench, the Loschmidt echo has a power-law scaling with time as expected from interpreting the quench as insertion of boundary condition changing operators. Similar scaling is observed as a function of time period under a low-frequency square-wave pulse. We present numerical evidence for one cycle that indicates that under a sinusoidal or triangular pulse, scaling with time period is modified by the Kibble-Zurek mechanism, again similar to the case of the Ising model. Results confirm the validity, beyond the Ising model, of the treatment of the boundary perturbations in terms of the effect on boundary conditions.

DOI
10.1103/physrevb.101.144302

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Low-frequency perturbations at the boundary of critical quantum chains can be understood in terms of the sequence of boundary conditions imposed by them, as has been previously demonstrated in the Ising and related fermion models. Using extensive numerical simulations, we explore the scaling behavior of the Loschmidt echo under longitudinal field perturbations at the boundary of a critical Z3{\mathbb{Z}}_{3}Z3​ Potts model. We show that at times much larger than the relaxation time after a boundary quench, the Loschmidt echo has a power-law scaling with time as expected from interpreting the quench as insertion of boundary condition changing operators. Similar scaling is observed as a function of time period under a low-frequency square-wave pulse. We present numerical evidence for one cycle that indicates that under a sinusoidal or triangular pulse, scaling with time period is modified by the Kibble-Zurek mechanism, again similar to the case of the Ising model. Results confirm the validity, beyond the Ising model, of the treatment of the boundary perturbations in terms of the effect on boundary conditions.
We study the response to sudden local perturbations of highly excited quantum Ising spin chains. The key quantity encoding this response is the overlap between time-dependent wave functions, which we write as a transient Loschmidt Echo. Its asymptotics at long time differences contain crucial information about the structure of the highly excited nonequilibrium environment induced by the quench. We compute the echo perturbatively for a weak local quench but for arbitrarily large global quench, using a cumulant expansion. Our perturbative results suggest that the echo decays exponentially, rather than power law as in the low-energy orthogonality catastrophe, a further example of quench-induced decoherence already found in the case of quenched Luttinger liquids. The emerging decoherence scale is set by the strength of the local potential and the bulk excitation energy.
Abstract We drive the one-dimensional quantum Ising chain in the transverse field from the paramagnetic phase to the critical point and study its free evolution there. We analyze excitation of such a system at the critical point and dynamics of its transverse magnetization and Loschmidt echo during free evolution. We discuss how the system size and quench-induced scaling relations from the Kibble–Zurek theory of non-equilibrium phase transitions are encoded in quasi-periodic time evolution of the transverse magnetization and Loschmidt echo.
Quantum phase transitions are characterized by the universal scaling laws in the critical region surrounding the transitions. This universality is also manifested in the critical real-time dynamics through the quantum Kibble-Zurek mechanism. In recent experiments on a Rydberg atom quantum simulator, the Kibble-Zurek mechanism was used to probe the nature of quantum phase transitions. In this paper, we analyze the caveats associated with this method and develop strategies to improve its accuracy. Focusing on two minimal models—transverse-field Ising and quantum three-state Potts, both in one dimension—we study the effect of boundary conditions, the location of the endpoints, and some subtleties in the definition of the kink operators. In particular, we show that the critical scaling of the most intuitive types of kinks is extremely sensitive to the correct choice of endpoint, while more advanced types of kinks exhibit remarkably robust universal scaling. Furthermore, we show that when kinks are tracked over the entire chain, fixed boundary conditions improve the accuracy of the scaling. Surprisingly, the Kibble-Zurek critical scaling appears to be equally accurate whether the fixed boundary conditions are chosen to be symmetric or antisymmetric. We also show that the density of kinks extracted in the central part of long chains obeys the predicted universal scaling for all types of boundary conditions. Finally, we test our kink definition for the Ising transition on the period-2 phase of the Rydberg model and show that it is more robust against the endpoint than the standard definition.
A critically enhanced decay of the Loschmidt echo is characteristic of sudden quench dynamics near a quantum phase transition. Here, we demonstrate that the decay and revival of the Loschmidt echo follows power-law scaling in the system size and the distance from a critical point with equilibrium critical exponents. We reveal such dynamical scaling laws by analyzing relevant perturbations to the Loschmidt echo cast in a scaling invariant form. We confirm the validity and the generality of the predicted dynamical scaling laws with a diverse range of critical models such as Ising spin models with a short and long range interaction, a finite-component system phase transition, and a topological phase transition. Moreover, using the integrability of systems in the thermodynamic limit, we derive such scaling laws analytically from a microscopic analysis. Our finding promotes the Loschmidt echo to a quantitative non-equilibrium probe of criticality and allows for quantitative predictions on the role of criticality on various physical scenarios where the Loschmidt echo is central to describing non-equilibrium dynamics.
A quantum critical system described at low energy by a conformal field theory (CFT) and subjected to a time-periodic boundary drive displays multiple dynamical regimes, depending on the drive frequency. We compute the behavior of quantities including the entanglement entropy and Loschmidt echo, confirming analytic predictions from field theory by exact numerics on the transverse field Ising model and demonstrate universality by adding nonintegrable perturbations. The dynamics naturally separate into three regimes: a slow-driving limit, which has an interpretation as multiple quantum quenches with amplitude corrections from CFT; a fast-driving limit, in which the system behaves as though subject to a single quantum quench; and a crossover regime displaying heating. The universal Floquet dynamics in all regimes can be understood using a combination of boundary CFT and Kibble-Zurek scaling arguments.
We show that the dynamics of a quantum impurity subject to a stochastic drive on one side and coupled to a quantum critical system on the other display a universal behavior inherited from the quantum critical scaling. Using boundary conformal field theory, we formulate a generic ansatz for the dynamical scaling form of the typical Loschmidt echo and corroborate it with exact numerical calculations in the case of a spin impurity driven by shot noise in a quantum Ising chain. We find that due to rare events the dynamics of the mean echo can follow very different dynamical scaling than the typical echo for certain classes of drives. Our results are insensitive to irrelevant perturbations of the bulk critical model and apply to all the microscopic models in the same universality class.