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Dynamics of Quantum Phase Transitions: Quantum Kibble-Zurek Mechanism

Jacek Dziarmaga

Published Jul 22, 2015• 0 citations
Publisher

Abstract

Ideally, in an adiabatic quantum computation or quantum state preparation, a simple ground state of an initial Hamiltonian is evolved adiabatically to an interesting ground state of a final Hamiltonian. Unfortunately, the simple and the interesting are often different enough to be separated by a quantum phase transition. Due to a vanishing energy gap between the ground state and excited states at the critical point, near the transition the adiabaticity is bound to fail. This failure is quantified by a quantum version of the Kibble-Zurek mechanism (KZM). In these notes I introduce KZM in its general textbook form, based on adiabatic/impulse approximation, and then support the picture by an exact solution of the integrable transverse field quantum Ising chain and an approximate one of the non-integrable Bose-Hubbard model. The last model illustrates typical problems with adiabatic quantum state preparation that are encoutered in atomic quantum simulators.

DOI
10.1142/9789814704090_0007

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Ideally, in an adiabatic quantum computation or quantum state preparation, a simple ground state of an initial Hamiltonian is evolved adiabatically to an interesting ground state of a final Hamiltonian. Unfortunately, the simple and the interesting are often different enough to be separated by a quantum phase transition. Due to a vanishing energy gap between the ground state and excited states at the critical point, near the transition the adiabaticity is bound to fail. This failure is quantified by a quantum version of the Kibble-Zurek mechanism (KZM). In these notes I introduce KZM in its general textbook form, based on adiabatic/impulse approximation, and then support the picture by an exact solution of the integrable transverse field quantum Ising chain and an approximate one of the non-integrable Bose-Hubbard model. The last model illustrates typical problems with adiabatic quantum state preparation that are encoutered in atomic quantum simulators.
The Kibble-Zurek mechanism (KZM) predicts the density of topological defects produced in the dynamical processes of phase transitions in systems ranging from cosmology to condensed matter and quantum materials. The similarity between KZM and the Landau-Zener transition (LZT), which is a standard tool to describe the dynamics of some non-equilibrium physics in contemporary physics, is being extensively exploited. Here we demonstrate the equivalence between KZM in the Ising model and LZT in a superconducting qubit system. We develop a time-resolved approach to study quantum dynamics of LZT with nano-second resolution. By using this technique, we simulate the key features of KZM in the Ising model with LZT, e.g., the boundary between the adiabatic and impulse regions, the freeze-out phenomenon in the impulse region, especially, the scaling law of the excited state population as the square root of the quenching speed. Our results provide the experimental evidence of the close connection between KZM and LZT, two textbook paradigms to study the dynamics of the non-equilibrium phenomena.
Kibble-Zurek theory (KZ) stands out as the most robust theory of defect generation in the dynamics of phase transitions. KZ utilizes the structure of equilibrium states away from the transition point to estimate the excitations due to the transition using adiabatic and impulse approximations. Here we show, the actual nonequilibrium dynamics lead to a qualitatively different scenario from KZ, as far correlations between the defects (rather than their densities) are concerned. For a quantum Ising chain, we show, this gives rise to a Gaussian spatial decay in the domain wall (kinks) correlations, while KZ would predict an exponential fall. We propose a simple but general framework on top of KZ, based on the ``quantum coarsening'' dynamics of local correlators in the supposed impulse regime. We outline how our picture extends to generic interacting situations.
The Kibble-Zurek mechanism (KZM) was originally proposed to describe the evolution and "freezing" of defects in the early universe, but later it was generalized to study other quantum and classical systems driven by a varying parameter. The basic idea behind the KZM is that, as long as the changing rate (velocity) of the parameter is below a certain critical velocity, 𝑣_crit, the system will remain adiabatic (for isolated quantum systems) or quasi-static (for classical systems with a heat bath). \nThe nonequilibrium finite-size scaling (FSS) method based on KZM has been exploited systematically. Through applying the scaling hypothesis, we can extract the critical exponents and study the dynamic properties of the system. \n \nIn the first few chapters of this dissertation, we discuss the applications of KZM in several classical systems: first, we study the dynamics of 2D and 3D Ising model under a varying temperature as well as a varying magnetic field. Secondly, we examine the classical ℤ₂ gauge model, in which we show that KZM also works for topological phase transitions. Moreover, we also investigate the dynamics of other models with topological ordering only at T=0, where KZM cannot be applied. Lastly, we explore the 2D Ising spin glass with bimodal and gaussian couplings. With bimodal couplings, we find dual time scales associated with the order parameter and the energy correspondingly, while in the gaussian case one unique time scale is involved. \n \nThe systems mentioned above are all classical and the dynamics are approached through simulated annealing (SA), in which thermal fluctuations drives systems to explore the energy landscape in finding the ground state. In the last chapter, we explore the efficiency of Quantum Annealing (QA) on a fully-connected spin glass (or Sherington-Kirkpatrick model) with a transverse field. QA is the counterpart of SA, where quantum fluctuations drive the system toward the ground state when the quantum terms are reduced. QA is currently widely explored as a paradigm for quantum computing to solve optimization problems. Here we compare the scaling of the dynamics (with system size) of the fully-connected spin glass through QA versus SA.
The ground state of the one-dimensional Bose-Hubbard model at unit filling undergoes the Mott-superfluid quantum phase transition. It belongs to the Kosterlitz-Thouless universality class with an exponential divergence of the correlation length in place of the usual power law. We present numerical simulations of a linear quench both from the Mott insulator to superfluid and back. The results satisfy the scaling hypothesis that follows from the Kibble-Zurek mechanism (KZM). In the superfluid-to-Mott quenches there is no significant excitation in the superfluid phase despite its gaplessness. Since all critical superfluid ground states are qualitatively similar, the excitation begins to build up only after crossing the critical point when the ground state begins to change fundamentally. The last process falls into the KZM framework.
The Kibble-Zurek mechanism (KZM) captures the key physics of nonequilibrium dynamics in second order phase transitions, and accurately predicts the density of topological defects formed in such processes. However, the central prediction of KZM--i.e., the scaling of the density of defects with the quench rate--still needs further experimental confirmation, particularly for quantum transitions. Here, we perform a quantum simulation of the nonequilibrium dynamics of the Landau-Zener model based on a nine-stage optical interferometer with an overall visibility of 0.975±0.008. The results support the adiabatic-impulse approximation, which is the core of Kibble-Zurek theory. Moreover, the developed high-fidelity multistage optical interferometer can support more complex linear optical quantum simulations.
The Kibble-Zurek mechanism predicts the formation of topological defects and other excitations that quantify how much a quantum system driven across a quantum critical point fails to be adiabatic. We point out that, thanks to the divergent linear susceptibility at the critical point, even a tiny symmetry breaking bias can restore the adiabaticity. The minimal required bias scales like ${\ensuremath{\tau}}_{Q}^{\ensuremath{-}\ensuremath{\beta}\ensuremath{\delta}/(1+z\ensuremath{\nu})}$, where $\ensuremath{\beta}$, $\ensuremath{\delta}$, zzz, $\ensuremath{\nu}$ are the critical exponents and ${\ensuremath{\tau}}_{Q}$ is a quench time. We test this prediction by DMRG simulations of the quantum Ising chain. It is directly applicable to the recent emulation of quantum phase transition dynamics in the Ising chain with ultracold Rydberg atoms.