Counterdiabatic driving of the quantum Ising model

The system undergoes adiabatic evolution when its population in the instantaneous eigenbasis of its time-dependent Hamiltonian changes only negligibly. Realization of such dynamics requires slow-enough changes of the parameters of the Hamiltonian, a task that can be hard to achieve near quantum critical points. A powerful alternative is provided by the counterdiabatic modification of the Hamiltonian allowing for an arbitrarily quick implementation of the adiabatic dynamics. Such a counterdiabatic driving protocol has been recently proposed for the quantum Ising model [A. del Campo et al., Phys. Rev. Lett. 109, 115703 (2012)]. We derive an exact closed-form expression for all the coefficients of the counterdiabatic Ising Hamiltonian. We also discuss two approximations to the exact counterdiabatic Ising Hamiltonian quantifying their efficiency of the dynamical preparation of the desired ground state. In particular, these studies show how quantum criticality enhances finite-size effects in the counterdiabatic dynamics.Comment: 11 pages, version accepted in JSTA

Fidelity susceptibility of the quantum Ising model in the transverse field: The exact solution

We derive an exact closed-form expression for fidelity susceptibility of the quantum Ising model in the transverse field. We also establish an exact one-to-one correspondence between fidelity susceptibility in the ferromagnetic and paramagnetic phases of this model. Elegant summation formulas are obtained as a by-product of these studies.Comment: 7 pages, 3 figures, minor changes, some references adde

Fidelity approach to quantum phase transitions in quantum Ising model

Fidelity approach to quantum phase transitions uses the overlap between ground states of the system to gain some information about its quantum phases. Such an overlap is called fidelity. We illustrate how this approach works in the one dimensional quantum Ising model in the transverse field. Several closed-form analytical expressions for fidelity are discussed. An example of what insights fidelity provides into the dynamics of quantum phase transitions is carefully described. The role of fidelity in central spin systems is pointed out.Comment: Proceedings of the 50th Karpacz Winter School of Theoretical Physics, Karpacz, Poland, 2-9 March 201

The quantum Ising model: finite sums and hyperbolic functions

We derive exact closed-form expressions for several sums leading to hyperbolic functions and discuss their applicability for studies of finite-size Ising spin chains. We show how they immediately lead to closed-form expressions for both fidelity susceptibility characterizing the quantum critical point and the coefficients of the counterdiabatic Hamiltonian enabling arbitrarily quick adiabatic driving of the system. Our results generalize and extend the sums presented in the popular Gradshteyn and Ryzhik Table of Integrals, Series, and Products.Comment: 7 pages, new title, small updates, published versio

Properties of the one-dimensional Bose-Hubbard model from a high-order perturbative expansion

We employ a high-order perturbative expansion to characterize the ground state of the Mott phase of the one-dimensional Bose-Hubbard model. We compute for different integer filling factors the energy per lattice site, the two-point and density-density correlations, and expectation values of powers of the on-site number operator determining the local atom number fluctuations (variance, skewness, kurtosis). We compare these expansions to numerical simulations of the infinite-size system to determine their range of applicability. We also discuss a new sum rule for the density-density correlations that can be used in both equilibrium and non-equilibrium systems.Comment: 16 pages, published versio

One-half of the Kibble-Zurek quench followed by free evolution

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.Comment: 19 pages, version accepted for publicatio

Spatial Kibble-Zurek mechanism through susceptibilities: the inhomogeneous quantum Ising model case

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 insights 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.Comment: 14 pages, small updates, published versio

Electric field-based quantization of the gauge invariant Proca theory

We consider the gauge invariant version of the Proca theory, where besides the real vector field there is also the real scalar field. We quantize the theory such that the commutator of the scalar field operator and the electric field operator is given by a predefined three-dimensional vector field, say $\mathcal{E}$ up to a global prefactor. This happens when the field operators of the gauge invariant Proca theory satisfy the proper gauge constraint. In particular, we show that $\mathcal{E}$ given by the classical Coulomb field leads to the Coulomb gauge constraint making the vector field operator divergenceless. We also show that physically unreadable gauge constraints can have a strikingly simple $\mathcal{E}$-representation in our formalism. This leads to the discussion of Debye, Yukawa, etc. gauges. In general terms, we explore the mapping between classical vector fields and gauge constraints imposed on the operators of the studied theory

Periodic charge oscillations in the Proca theory

We consider the Proca theory of the real massive vector field. There is a locally conserved 4-current operator in such a theory, which one may use to define the charge operator. Accordingly, there are charged states in which the expectation value of the charge operator is non-zero. We take a close look at the charge operator and study the dynamics of the certain class of charged states. For this purpose, we discuss the mean electric field and 4-current in such states. The mean electric field has the periodically oscillating Coulomb component, whose presence explains the periodic charge oscillations. A complementary insight at such a phenomenon is provided by the mean 4-current, whose discussion leads to the identification of a certain paradox. Last but not least, we show that there is a shock wave propagating in the studied system, which affects analyticity of the mean electric field and 4-current.Comment: 17 pages, improvements in the discussion of the shock wave fron