We show that for a d-dimensional model in which a quench with a rate tau(-1) takes the system across a (d-m)-dimensional critical surface, the defect density scales as n approximately 1/tau(mnu/(znu+1)), where nu and z are the correlation length and dynamical critical exponents characterizing the critical surface. We explicitly demonstrate that the Kitaev model provides an example of such a scaling with d = 2 and m = nu = z = 1. We also provide the first example of an exact calculation of some multispin correlation functions for a two-dimensional model that can be used to determine the correlation between the defects. We suggest possible experiments to test our theory.

We show that the defect density n, for a slow nonlinear power-law quench with a rate tau(-1) and an exponent alpha>0, which takes the system through a critical point characterized by correlation length and dynamical critical exponents nu and z, scales as n approximately tau(-alphanud/(alphaznu+1)) [n approximately (alphag((alpha-1)/alpha)/tau)(nud/(znu+1))] if the quench takes the system across the critical point at time t=0 [t=t(0) not = 0], where g is a nonuniversal constant and d is the system dimension. These scaling laws constitute the first theoretical results for defect production in nonlinear quenches across quantum critical points and reproduce their well-known counterpart for a linear quench (alpha=1) as a special case. We supplement our results with numerical studies of well-known models and suggest experiments to test our theory.

We study the transport properties of the Dirac fermions with a Fermi velocity v{F} on the surface of a topological insulator across a ferromagnetic strip providing an exchange field J over a region of width d. We show that the conductance of such a junction, in the clean limit and at low temperature, changes from oscillatory to a monotonically decreasing function of d beyond a critical J. This leads to the possible realization of a magnetic switch using these junctions. We also study the conductance of these Dirac fermions across a potential barrier of width d and potential V0 in the presence of such a ferromagnetic strip and show that beyond a critical J, the criteria of conductance maxima changes from chi=eV{0}d/variant Planck's over v{F}=npi to chi=(n+1/2)pi for integer n. We point out that these novel phenomena have no analogs in graphene and suggest experiments which can probe them.

We study the properties of Dirac fermions on the surface of a topological insulator in the presence of crossed electric and magnetic fields. We provide an exact solution to this problem and demonstrate that, in contrast to their counterparts in graphene, these Dirac fermions allow relative tuning of the orbital and Zeeman effects of an applied magnetic field by a crossed electric field along the surface. We also elaborate and extend our earlier results on normal metal-magnetic film-normal metal (NMN) and normal metal-barrier-magnetic film (NBM) junctions of topological insulators [Phys. Rev. Lett. 104, 046403 (2010)]. For NMN junctions, we show that for Dirac fermions with Fermi velocity vF , the transport can be controlled using the exchange field J of a ferromagnetic film over a region of width d. The conductance of such a junction changes from oscillatory to a monotonically decreasing function of d beyond a critical J which leads to the possible realization of magnetic switches using these junctions. For NBM junctions with a potential barrier of width d and potential V0, we find that beyond a critical J , the criteria of conductance maxima changes from χ = eV0d/ vF = nπ to χ = (n + 1/2)π for integer n. Finally, we compute the subgap tunneling conductance of a normal metal-magnetic film-superconductor (NMS) junctions on the surface of a topological insulator and show that the position of the peaks of the zero-bias tunneling conductance can be tuned using the magnetization of the ferromagnetic film. We point out that these phenomena have no analogs in either conventional two-dimensional materials or Dirac electrons in graphene and suggest experiments to test our theory.

We study the non-equilibrium dynamics of ultracold bosons in an optical lattice with a time dependent hopping amplitude J(t) = J0 +δJ cos(ωt) which takes the system from a superfluid phase near the Mott-superfluid transition (J = J0 + δJ) to a Mott phase (J = J0 − δJ) and back through a quantum critical point (J = Jc) and demonstrate dynamic freezing of the boson wavefunction at specific values of ω. At these values, the wavefunction overlap F (defect density P = 1 − F ) approaches unity (zero). We provide a qualitative explanation of the freezing phenomenon, show it's robustness against quantum fluctuations and the presence of a trap, compute residual energy and superfluid order parameter for such dynamics, and suggest experiments to test our theory. PACS numbers: 64.60.Ht, 05.30.Jp, 05.30.Rt Theoretical study of non-equilibrium dynamics in closed quantum systems has seen great progress in recent years [1] mainly due to the possibility of realization of such dynamics using ultracold atom in optical lattices [2,3]. For bosonic atoms, such systems are well described by the Bose-Hubbard model with on-site interaction strength U and nearest neighbor hopping amplitude J [4,5]. Several theoretical studies have been carried out on the quench and ramp dynamics of this model [6][7][8][9][10][11]; some of them have also received support from recent experiments [3]. In contrast, studies on periodically driven closed quantum systems have been undertaken in the past mainly on driven two-level systems [12,13] or on weakly interacting or integrable many-body systems which can be modeled by them [14,15]. Among these, Ref.[15] has predicted freezing of the time-averaged value of the order parameter (magnetization) of an periodically driven one-dimensional (1D) Ising or XY model, when the temporal average is performed over several drive cycles, at specific drive frequencies. Such a freezing occurs in the high frequency regime and exhibits non-monotonic dependence on the drive frequency. However, to the best of our knowledge, the phenomenon of dynamic freezing has never been demonstrated for dynamics involving a single drive cycle and/or for non-integrable quantum systems. Recent studies of periodic dynamics of the Bose-Hubbard model have not addressed this issue [16,17].In this work, we demonstrate, via designing a periodic driving protocol, that the periodically driven Bose-Hubbard model may exhibit dynamic freezing of the boson wavefunction |ψ(t = 0) = |ψ(t = T ) for specific values of the drive frequencies ω = 2π/T . Our driving protocol constitutes a time-dependent hopping amplitude of the bosons J(t) = J 0 + δJ cos(ωt) with J 0 and δJ chosen such that the drive takes the system from a superfluid (SF) (J = J 0 + δJ) to the Mott insulator (MI) state (J = J 0 − δJ) and back through the tip of the Mott lobe where µ = µ tip . We demonstrate, using mean-field theory, that such a freezing phenomenon de-rives from quantum interference of the dynamic phases acquired by the bosons and compute the defect formation probability P = 1−...

The properties of systems near quantum critical points (QCPs) have been studied extensively in recent years [1,2]. A QCP is a point across which the symmetry of the ground state of a quantum system changes in a fundamental way; such a point can be accessed by changing some parameter, say λ , in the Hamiltonian governing the system. The change in the ground state across a QCP is mediated by quantum fluctuations. Unlike conventional thermal critical points, thermal fluctuations do not play a crucial role in such transitions. Similar to its thermal counterparts, the low energy physics near a QCP is associated with a number of critical exponents which characterize the universality class of such a transition. Amongst these exponents, the dynamical critical exponent z provides the signature of the relative scaling of space and time at the transition and has no counterpart in thermal phase transitions. The other exponent which is going to be important for the purpose of this review is the well-known correlation length exponent ν. These exponents are formally defined as follows. As we approach the critical point at λ = λ c , the correlation length diverges as ξ ∼ |λ − λ c | −ν , while the gap between the ground state and first excited state vanishes as ∆ E ∼ ξ −z ∼ |λ − λ c | zν . Exactly at the critical point λ = λ c , the energy of the low-lying excitations vanishes at some wave number k 0 as ω ∼ |k − k 0 | z . The critical exponents are independent of the details of the microscopic Hamiltonian; they depend only on a few parameters such as the dimensionality of the system and the symmetry of the order parameter. These features render the low energy equilibrium physics of a quantum system near a QCP truly universal.In contrast to this well-understood universality of the equilibrium properties of a system near a QCP, relatively few universal features are known in the nonequilibrium behavior of a quantum system. Initial studies in this field aimed at understanding the near-equilibrium finite temperature dynamics near a quantum critical

We study quench dynamics and defect production in the Kitaev and the extended Kitaev models. For the Kitaev model in one dimension, we show that in the limit of slow quench rate, the defect density n ϳ 1 / ͱ , where 1 / is the quench rate. We also compute the defect correlation function by providing an exact calculation of all independent nonzero spin correlation functions of the model. In two dimensions, where the quench dynamics takes the system across a critical line, we elaborate on the results of earlier work ͓K. Sengupta, D. Sen, and S. Mondal, Phys. Rev. Lett. 100, 077204 ͑2008͔͒ to discuss the unconventional scaling of the defect density with the quench rate. In this context, we outline a general proof that for a d-dimensional quantum model, where the quench takes the system through a d − m dimensional gapless ͑critical͒ surface characterized by correlation length exponent and dynamical critical exponent z, the defect density n ϳ 1 / m/͑z+1͒ . We also discuss the variation of the shape and spatial extent of the defect correlation function with both the rate of quench and the model parameters and compute the entropy generated during such a quenching process. Finally, we study the defect scaling law, entropy generation and defect correlation function of the two-dimensional extended Kitaev model.

We study defect production in a quantum system subjected to a nonlinear power-law quench which takes it either through a quantum critical or multicritical point or along a quantum critical line. We elaborate on our earlier work ͓D. Sen, K. Sengupta, and S. Mondal, Phys. Rev. Lett. 101, 016806 ͑2008͔͒ and present a detailed analysis of the scaling of the defect density n with the quench rate and exponent ␣ for each of the abovementioned cases. We also compute the correlation functions for defects generated in nonlinear quenches through a quantum critical point and discuss the dependence of the amplitudes of such correlation functions on the exponent ␣. We discuss several experimental systems where these theoretical predictions can be tested.