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arXiv:0909.5181 · Physical Review B

Quench dynamics near a quantum critical point

Claudia De Grandi, Vladimir Gritsev, Anatoli Polkovnikov

Published Jan 19, 2010• 230 citations
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Abstract

We study the dynamical response of a system to a sudden change of the tuning parameter $\ensuremath{\lambda}$ starting (or ending) at the quantum critical point. In particular, we analyze the scaling of the excitation probability, number of excited quasiparticles, heat and entropy with the quench amplitude, and the system size. We extend the analysis to quenches with arbitrary power law dependence on time of the tuning parameter, showing a close connection between the scaling behavior of these quantities with the singularities of the adiabatic susceptibilities of order $m$ at the quantum critical point, where $m$ is related to the power of the quench. Precisely for sudden quenches, the relevant susceptibility of the second order coincides with the fidelity susceptibility. We discuss the generalization of the scaling laws to the finite-temperature quenches and show that the statistics of the low-energy excitations becomes important. We illustrate the relevance of those results for cold-atom experiments.

DOI
10.1103/physrevb.81.012303

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We study the dynamical response of a system to a sudden change of the tuning parameter \ensuremathλ\ensuremath{\lambda}\ensuremathλ starting (or ending) at the quantum critical point. In particular, we analyze the scaling of the excitation probability, number of excited quasiparticles, heat and entropy with the quench amplitude, and the system size. We extend the analysis to quenches with arbitrary power law dependence on time of the tuning parameter, showing a close connection between the scaling behavior of these quantities with the singularities of the adiabatic susceptibilities of order mmm at the quantum critical point, where mmm is related to the power of the quench. Precisely for sudden quenches, the relevant susceptibility of the second order coincides with the fidelity susceptibility. We discuss the generalization of the scaling laws to the finite-temperature quenches and show that the statistics of the low-energy excitations becomes important. We illustrate the relevance of those results for cold-atom experiments.
We discuss the quench dynamics near a quantum critical point focusing on the sine-Gordon model as a primary example. We suggest a unified approach to sudden and slow quenches, where the tuning parameter $\ensuremath{\lambda}(t)$ changes in time as $\ensuremath{\lambda}(t)\ensuremath{\sim}\ensuremath{\upsilon}{t}^{r}$, based on the adiabatic expansion of the excitation probability in powers of $\ensuremath{\upsilon}$. We show that the universal scaling of the excitation probability can be understood through the singularity of the generalized adiabatic susceptibility ${\ensuremath{\chi}}_{2r+2}(\ensuremath{\lambda})$, which for sudden quenches (r=0)(r=0)(r=0) reduces to the fidelity susceptibility. In turn this class of susceptibilities is expressed through the moments of the connected correlation function of the quench operator. We analyze the excitations created after a sudden quench of the cosine potential using a combined approach of form-factors expansion and conformal perturbation theory for the low-energy and high-energy sector, respectively. We find the general scaling laws for the probability of exciting the system, the density of excited quasiparticles, the entropy and the heat generated after the quench. In the two limits where the sine-Gordon model maps to hard-core bosons and free massive fermions we provide the exact solutions for the quench dynamics and discuss the finite temperature generalizations.
Quantum critical points of many-body systems can be characterized by studying response of the ground-state wave function to the change of the external parameter, encoded in the ground-state fidelity susceptibility. This quantity characterizes the quench dynamics induced by sudden change of the parameter. In this framework, I analyze scaling relations concerning the probability of excitation and the excitation energy, with the quench amplitude of this parameter. These results are illustrated in the case of one-dimensional sine-Gordon model.
We give an overview of the scaling of density of quasi-particles and excess energy (heat) for nearly adiabatic dynamics near quantum critical points (QCPs). In particular we discuss both sudden quenches of small amplitude and slow sweeps across the QCP. We show close connection between universal scaling of these quantities with the scaling behavior of the fidelity susceptibility and its generalizations. In particular we argue that the Kibble-Zurek scaling can be easily understood using this concept. We discuss how these scalings can be derived within the adiabatic perturbation theory and how using this approach slow and fast quenches can be treated within the same framework. We also describe modifications of these scalings for finite temperature quenches and emphasize the important role of statistics of low-energy excitations. In the end we mention some connections between adiabatic dynamics near critical points with dynamics associated with space-time singularities in the metrics, which naturally emerges in such areas as cosmology and string theory.
We give an overview of the scaling of density of quasi-particles and excess energy (heat) for nearly adiabatic dynamics near quantum critical points (QCPs). In particular we discuss both sudden quenches of small amplitude and slow sweeps across the QCP. We show close connection between universal scaling of these quantities with the scaling behavior of the fidelity susceptibility and its generalizations. In particular we argue that the Kibble-Zurek scaling can be easily understood using this concept. We discuss how these scalings can be derived within the adiabatic perturbation theory and how using this approach slow and fast quenches can be treated within the same framework. We also describe modifications of these scalings for finite temperature quenches and emphasize the important role of statistics of low-energy excitations. In the end we mention some connections between adiabatic dynamics near critical points with dynamics associated with space-time singularities in the metrics, which naturally emerges in such areas as cosmology and string theory.
We study the short-time Loschmidt dynamics after a small sudden quench near a quantum critical point. We show that the initial quadratic growth of the Loschmidt rate function is governed by the variance of the quench operator per system size. For a local quench operator, this variance density is exactly equal to the spatial sum of the equal-time connected two-point correlation function in the initial ground state. This relation connects the early-time Loschmidt response to static critical correlations. Assuming power-law correlations at criticality, we classify the finite-size scaling of the short-time coefficient by the scaling dimension of the quench operator. In one dimension, the coefficient is finite, logarithmically enhanced, or algebraically enhanced with system size. We illustrate this operator dependence in the transverse-field Ising chain, where transverse-field and longitudinal-field quenches couple to different critical operators. We also discuss the first correction beyond the quadratic regime using the fourth cumulant of the post-quench Hamiltonian.
We explore the robustness of universal dynamical scaling behavior in a quantum system near criticality with respect to initialization in a large class of states with finite energy. By focusing on a homogeneous XY\mathit{XY}XY quantum spin chain in a transverse field, we characterize the nonequilibrium response under adiabatic and sudden quench processes originating from a pure as well as a mixed excited initial state, and involving either a regular quantum critical or a multicritical point. We find that the critical exponents of the ground-state quantum phase transition can be encoded in the dynamical scaling exponents despite the finite energy of the initial state. In particular, we identify conditions on the initial distribution of quasiparticle excitation that ensure that Kibble-Zurek scaling persists. The emergence of effective thermal equilibrium behavior following a sudden quench toward criticality is also investigated, with focus on the long-time expectation value of the quasiparticle number operator. Despite the integrability of the XY\mathit{XY}XY model, this observable is found to behave thermally in quenches to a regular quantum critical point, provided that the system is initially prepared at sufficiently high temperature. However, a similar thermalization behavior fails to occur in quenches toward a multicritical point. We argue that the observed lack of thermalization originates in this case in the asymmetry of the impulse region that is also responsible for anomalous multicritical dynamical scaling.