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arXiv:0812.3685 · Physical Review B

Adiabatic dynamics of a quantum critical system coupled to an environment: Scaling and kinetic equation approaches

Dario Patanè, Luigi Amico A, Alessandro Silva, Rosario Fazio, Giuseppe E. Santoro

Published Jul 2, 2009• 60 citations
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Abstract

We study the dynamics of open quantum many-body systems driven across a critical point by quenching a parameter of the Hamiltonian at a certain velocity. General scaling laws are derived for the density of excitations and of energy produced during the quench as a function of the quench velocity and the bath temperature. The scaling laws and their regimes of validity are verified for the $XY$ spin chain locally coupled to bosonic baths. A detailed derivation and analysis of the kinetic equations for the problem is presented.

DOI
10.1103/physrevb.80.024302

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We study the dynamics of open quantum many-body systems driven across a critical point by quenching a parameter of the Hamiltonian at a certain velocity. General scaling laws are derived for the density of excitations and of energy produced during the quench as a function of the quench velocity and the bath temperature. The scaling laws and their regimes of validity are verified for the XYXYXY spin chain locally coupled to bosonic baths. A detailed derivation and analysis of the kinetic equations for the problem is presented.
We investigate the loss of adiabaticity when cooling a many-body quantum system from an initial thermal state toward a quantum critical point. The excitation density, which quantifies the degree of adiabaticity of the dynamics, is found to obey scaling laws in the cooling velocity as well as in the initial and final temperatures of the cooling protocol. The scaling laws are universal, governed by the critical exponents of the quantum phase transition. The validity of these statements is shown analytically for a Kitaev quantum wire coupled to Markovian baths and argued to be valid under rather general conditions. Our results establish that quantum critical properties can be probed dynamically at finite temperature, without even varying the control parameter of the quantum phase transition.
We address the out-of-equilibrium dynamics arising from quantum-quench (QQ) protocols (instantaneous changes of the Hamiltonian parameters) in many-body systems within their quantum critical regime and in contact with (homogeneously coupled) thermal baths. We consider two classes of QQ protocols. In one of them, the thermal bath is used to prepare the initial finite-temperature Gibbs state; then, after quenching, the thermal bath is removed and the dynamics of the system is unitary. We also address a more complex QQ protocol where the thermal bath is not removed after quenching, thus the quantum evolution is also driven by the interaction with the bath, which may be described by appropriate master equations for the density matrix of the system, where a further relevant time scale, or inverse decay rate, characterizes the system-bath coupling. Under these QQ protocols, the critical system develops out-of-equilibrium scaling behaviors, which extend those for isolated critical systems, by introducing further scaling variables proportional to the temperature of the thermal bath and the decay rate of the system-bath interactions. These out-of-equilibrium scaling behaviors are checked by analyzing QQ protocols within fermionic Kitaev wires, or equivalently quantum Ising chains, supplemented with a particular modelization of thermal bath that guarantees the asymptotic thermalization within the Lindblad master equation for the dynamics of open systems.
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 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.
It is well known that the dynamics of a quantum system is always nonadiabatic in passage through a quantum critical point and the defect density in the final state following a quench shows a power-law scaling with the rate of quenching. However, we propose here a possible situation where the dynamics of a quantum system in passage across quantum critical regions is adiabatic and the defect density decays exponentially. This is achieved by incorporating additional interactions which lead to quantum critical behavior and gapless phases but do not participate in the time evolution of the system. To illustrate the general argument, we study the defect generation in the quantum critical dynamics of a spin-1/2 anisotropic quantum XY spin chain with three spin interactions and a linearly driven staggered magnetic field.
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.