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.
Abstract We study the quantum Ising model in the transverse inhomogeneous magnetic field. Such a system can be approached numerically through exact diagonalization and analytically through the renormalization group techniques. Basic insight into its physics, however, can be obtained by adopting the Kibble–Zurek theory of non-equilibrium phase transitions to description of spatially inhomogeneous systems at equilibrium. We employ all these approaches and focus on derivatives of longitudinal and transverse magnetizations, which have extrema near the critical point. We discuss how these extrema can be used for locating the critical point and for verification of the Kibble–Zurek scaling predictions in the spatial quench.
We review quantum phase transitions of spin systems in transverse magnetic fields taking the examples of the spin-1/2 Ising and XY models in a transverse field. Beginning with an overview of quantum phase transitions, we introduce a number of model Hamiltonians. We provide exact solutions in one spatial dimension connecting them to conformal field theoretical studies. We also discuss Kitaev models and some other exactly solvable spin systems. Studies of quantum phase transitions in the presence of quenched randomness and with frustrating interactions are presented in detail. We discuss novel phenomena like Griffiths-McCoy singularities. We then turn to more recent topics like information theoretic measures of the quantum phase transitions in these models such as concurrence, entanglement entropy, quantum discord and quantum fidelity. We then focus on non-equilibrium dynamics of a variety of transverse field systems across quantum critical points and lines. After mentioning rapid quenching studies, we dwell on slow dynamics and discuss the Kibble-Zurek scaling for the defect density following a quench across critical points and its modifications for quenching across critical lines, gapless regions and multicritical points. Topics like the role of different quenching schemes, local quenching, quenching of models with random interactions and quenching of a spin chain coupled to a heat bath are touched upon. The connection between non-equilibrium dynamics and quantum information theoretic measures is presented at some length. We indicate the connection between Kibble-Zurek scaling and adiabatic evolution of a state as well as the application of adiabatic dynamics as a tool of a quantum optimization technique known as quantum annealing. The final section is dedicated to a detailed discussion on recent experimental studies of transverse Ising-like systems.
We analyze the quantum phase transitions taking place in a one-dimensional transverse-field Ising model with long-range couplings that decay algebraically with distance. We are interested in the Kibble-Zurek universal scaling laws emerging in nonequilibrium dynamics and in the potential for the unambiguous observation of such behavior in a realistic experimental setup based on trapped ions. To this end, we determine the phase diagram of the model and the critical exponents characterizing its quantum phase transitions by means of density-matrix renormalization group calculations and finite-size scaling theory, which allows us to obtain good estimates for different range of ferromagnetic and antiferromagnetic interactions. Beyond critical equilibrium properties, we tackle a nonequilibrium scenario in which quantum Kibble-Zurek scaling laws may be retrieved. Here, it is found that the predicted nonequilibrium universal behavior, i.e., the scaling laws as a function of the quench rate and critical exponents, can be observed in systems comprising an experimentally feasible number of spins. Finally, a scheme is introduced to simulate the algebraically decaying couplings accurately by means of a digital quantum simulation with trapped ions. Our results suggest that quantum Kibble-Zurek physics can be explored and observed in state-of-the-art experiments with trapped ions realizing long-range Ising models.
We study the out-of-equilibrium Kibble-Zurek (KZ) dynamics in quantum Ising chains in a transverse field, driven by a time-dependent longitudinal field
h(t)=t/ts (
ts is the time scale of the protocol), across their first-order quantum transitions (FOQTs) at
h=0. The KZ protocol starts at time
ti<0 from the negatively magnetized ground state for
hi=ti/ts<0. Then, the system evolves unitarily up to a time
tf>0, such that the magnetization of the state at time
tf is positive. In finite-size systems, the KZ dynamics develops out-of-equilibrium finite-size scaling (OFSS) behaviors. Their scaling variables depend either exponentially or with a power law on the size, depending on the boundary conditions (BC). The OFSS functions can be computed in effective models restricted to appropriate low-energy (magnetized and/or kink) states. The KZ scaling behavior drastically changes in the thermodynamic limit (TL), defined as the infinite-size limit keeping
t and
ts fixed, which appears substantially unrelated with the OFSS regime, because it involves higher-energy multi-kink states, which are irrelevant in the OFSS limit. The numerical analyses of the KZ dynamics in the TL show the emergence of a quantum spinodal-like scaling behavior at the FOQTs for all considered BC, which is independent of the BC. The longitudinal magnetization changes sign at
h(t)=h∗>0, where
h∗ decreases with increasing
ts, as
h∗∼1/lnts. Moreover, in the large-
ts limit, the time-dependence of the magnetization is described by a universal function of
Ω=t/τs, with
τs=ts/lnts.
There has been much recent interest in the study of out of equilibrium quantum systems due to experimental advances in trapped cold atomic gases that allow systematic studies of these effects.In this work we study a quantum Ising model in the presence of transverse and longitudinal magnetic fields.The model lacks an analytical solution in the most general case and has to be treated numerically.We first investigate the static properties of the model, including the ground state energy and the zero temperature phase diagram, using the Density Matrix Renormalization Group (DMRG) method.We identify ordered and disordered phases separated by a quantum phase transition.We then consider the time evolution of the system following an abrupt quench of the fields.We study the response of the order parameter to the quenches within the ordered and disordered phases as well as quenches between the phases using the time-dependent DMRG method.The results are in qualitative agreement with experiments and the implemented numerical method in this work can be extended to allow for more general interactions.iii I would like to thank my supervisor, Dr. Malcolm Kennett, whose constant support and hope in the most disappointing situations, his helpful guidance, and endless energy were all new and very invaluable to me.I like to thank my