Dynamics in quantum Ising chain driven by inhomogeneous transverse magnetization

We study the dynamics caused by transport of transverse magnetization in one dimensional transverse Ising chain at zero temperature. We observe that a class of initial states having product structure in fermionic momentum-space and satisfying certain criteria, produce spatial variation in transverse magnetization. Starting from such a state, we obtain the transverse magnetization analytically and then observe its dynamics in presence of a homogeneous constant field $\Gamma$. In contradiction with general expectation, whatever be the strength of the field, the magnetization of the system does not become homogeneous even after infinite time. At each site, the dynamics is associated with oscillations having two different timescales. The envelope of the larger timescale oscillation decays algebraically with an exponent which is invariant for all such special initial states. The frequency of this oscillation varies differently with external field in ordered and disordered phases. The local magnetization after infinite time also characterizes the quantum phase transition.Comment: 7 pages, 4 figure

Phase Transition in a Long Range Antiferromagnetic Model

We consider an Ising model where longitudinal components of every pair of spins have antiferromagnetic interaction of the same magnitude. When subjected to a transverse magnetic field at zero temperature, the system undergoes a phase transition of second order to an ordered phase and if the temperature is now increased, there is another phase transition to disordered phase. We provide derivation of these features by perturbative treatment up to the second order and argue that the results are non-trivial and not derivable from the known results about related models.Comment: 3 page

Transverse Ising Chain under Periodic Instantaneous Quenches: Dynamical Many-Body Freezing and Emergence of Solitary Oscillation

We study the real-time dynamics of a quantum Ising chain driven periodically by instantaneous quenches of the transverse field (the transverse field varying as rectangular wave symmetric about zero). Two interesting phenomena are reported and analyzed: (1) We observe dynamical many-body freezing or DMF (Phys. Rev. B, vol. 82, 172402, 2010), i.e. strongly non-monotonic freezing of the response (transverse magnetization) with respect to the driving parameters (pulse width and height) resulting from equivocal freezing behavior of all the many-body modes. The freezing occurs due to coherent suppression of dynamics of the many-body modes. For certain combination of the pulse height and period, maximal freezing (freezing peaks) are observed. For those parameter values, a massive collapse of the entire Floquet spectrum occurs. (2) Secondly, we observe emergence of a distinct solitary oscillation with a single frequency, which can be much lower than the driving frequency. This slow oscillation, involving many high-energy modes, dominates the response remarkably in the limit of long observation time. We identify this slow oscillation as the unique survivor of destructive quantum interference between the many-body modes. The oscillation is found to decay algebraically with time to a constant value. All the key features are demonstrated analytically with numerical evaluations for specific results.Comment: Published version (with minor changes and typo corrections

Extreme variability in convergence to structural balance in frustrated dynamical systems

In many complex systems, the dynamical evolution of the different components can result in adaptation of the connections between them. We consider the problem of how a fully connected network of discrete-state dynamical elements which can interact via positive or negative links, approaches structural balance by evolving its links to be consistent with the states of its components. The adaptation process, inspired by Hebb's principle, involves the interaction strengths evolving in accordance with the dynamical states of the elements. We observe that in the presence of stochastic fluctuations in the dynamics of the components, the system can exhibit large dispersion in the time required for converging to the balanced state. This variability is characterized by a bimodal distribution, which points to an intriguing non-trivial problem in the study of evolving energy landscapes.Comment: 6 pages, 4 figure

Nodal line semimetals through boundary

We study the properties of a spinless Hamiltonian which describes topological nodal line semimetals at zero temperature. Our Hamiltonian has some resemblance with the usual Hamiltonian for this type of materials, with that of Weyl semimetals and is some sort of a generalisation of Kitaev Hamiltonian on honeycomb lattice. As the interaction parameter of the Hamiltonian is varied, it shows a transition from a gapless phase to a semimetal phase and then to another semimetal phase. These transitions show up as non-analytic behaviour of ground state energy. We find that one type of the semimetal is characterised by a single closed loop of nodal line, while the other, by two such loops both crossing the boundary of the first Brillouin zone. Surface states with non-zero winding number are found to exist in both of these phases, but over two topologically different regions. We also study the length of nodal line, the area and perimeter of the region over which the winding number is non-zero and the low-frequency electrical conductivity.Comment: 8 pages,10 figure

Multidimensional persistence behaviour in an Ising system

We consider a periodic Ising chain with nearest-neighbour and $r$-th neighbour interaction and quench it from infinite temperature to zero temperature. The persistence probability $P(t)$, measured as the probability that a spin remains unflipped upto time $t$, is studied by computer simulation for suitable values of $r$. We observe that as time progresses, $P(t)$ first decays as $t^{-0.22}$ (-the {\em first} regime), then the $P(t)-t$ curve has a small slope (in log-log scale) for some time (-the {\em second} regime) and at last it decays nearly as $t^{-3/8}$ (-the {\em third} regime). We argue that in the first regime, the persistence behaviour is the usual one for a two-dimensional system, in the second regime it is like that of a non-interacting (`zero-dimensional') system and in the third regime the persistence behaviour is like that of a one dimensional Ising model. We also provide explanations for such behaviour.Comment: 6 pages, 12 figure

A Small World Network of Prime Numbers

According to Goldbach conjecture, any even number can be broken up as the sum of two prime numbers : $n = p + q$. We construct a network where each node is a prime number and corresponding to every even number $n$, we put a link between the component primes $p$ and $q$. In most cases, an even number can be broken up in many ways, and then we chose {\em one} decomposition with a probability $|p - q|^{\alpha}$. Through computation of average shortest distance and clustering coefficient, we conclude that for $\alpha > -1.8$ the network is of small world type and for $\alpha < -1.8$ it is of regular type. We also present a theoretical justification for such behaviour.Comment: 6 pages, 10 figure