Optimal linear Glauber model

Contrary to the actual nonlinear Glauber model (NLGM), the linear Glauber model (LGM) is exactly solvable, although the detailed balance condition is not generally satisfied. This motivates us to address the issue of writing the transition rate ($w_j$) in a best possible linear form such that the mean squared error in satisfying the detailed balance condition is least. The advantage of this work is that, by studying the LGM analytically, we will be able to anticipate how the kinetic properties of an arbitrary Ising system depend on the temperature and the coupling constants. The analytical expressions for the optimal values of the parameters involved in the linear $w_j$ are obtained using a simple Moore-Penrose pseudoinverse matrix. This approach is quite general, in principle applicable to any system and can reproduce the exact results for one dimensional Ising system. In the continuum limit, we get a linear time-dependent Ginzburg-Landau (TDGL) equation from the Glauber's microscopic model of non-conservative dynamics. We analyze the critical and dynamic properties of the model, and show that most of the important results obtained in different studies can be reproduced by our new mathematical approach. We will also show in this paper that the effect of magnetic field can easily be studied within our approach; in particular, we show that the inverse of relaxation time changes quadratically with (weak) magnetic field and that the fluctuation-dissipation theorem is valid for our model.Comment: 25 pages; final version; appeared in Journal of Statistical Physic

On twin prime distribution and associated biases

We study some new aspects of the twin prime distribution, focusing especially on how the prime pairs are distributed in arithmetic progressions when they are classified according to the residues of their first members. A modified totient function is seen to play a significant role in this study - we analyze this function and use it to construct a new heuristics for the twin prime conjecture. For the twin primes, we also discuss a sieve similar to Eratosthenes' sieve and a formula similar to the Legendre's formula for the prime counting function. We end our work with a discussion on three types of biases in the distribution of twin primes. Where possible, we compare our results with the corresponding results from the distribution of primes.Comment: 18 pages, title slightly modified, abstract and introduction rewritten, a typo corrected in the proof of Theorem 4.4, results remain the sam

Optimal values of bipartite entanglement in a tripartite system

For a general tripartite system in some pure state, an observer possessing any two parts will see them in a mixed state. By the consequence of Hughston-Jozsa-Wootters theorem, each basis set of local measurement on the third part will correspond to a particular decomposition of the bipartite mixed state into a weighted sum of pure states. It is possible to associate an average bipartite entanglement ($\bar{\mathcal{S}}$) with each of these decompositions. The maximum value of $\bar{\mathcal{S}}$ is called the entanglement of assistance ($E_A$) while the minimum value is called the entanglement of formation ($E_F$). An appropriate choice of the basis set of local measurement will correspond to an optimal value of $\bar{\mathcal{S}}$; we find here a generic optimality condition for the choice of the basis set. In the present context, we analyze the tripartite states $W$ and $GHZ$ and show how they are fundamentally different.Comment: 14+ pages, 1 figure; final version; in different context, a part of the work can be found at arXiv:1201.562

Out-of-equilibrium Kondo Effect in a Quantum Dot: Interplay of Magnetic Field and Spin Accumulation

We present a theoretical study of low temperature nonequilibrium transport through an interacting quantum dot in the presence of Zeeman magnetic field and current injection into one of its leads. By using a self-consistent renormalized equation of motion approach, we show that the injection of a spin-polarized current leads to a modulation of the Zeeman splitting of the Kondo peak in the differential conductance. We find that an appropriate amount of spin accumulation in the lead can restore the Kondo peak by compensating the splitting due to magnetic field. By contrast when the injected current is spin-unpolarized, we establish that both Zeeman-split Kondo peaks are equally shifted and the splitting remains unchanged. Our results quantitatively explain the experimental findings reported in KOBAYASHI T. et al., Phys. Rev. Lett. 104, 036804 (2010). These features could be nicely exploited for the control and manipulation of spin in nanoelectronic and spintronic devices.Comment: 6+ pages; 3 figures; final versio

Magnetic plateaus and jumps in a spin-1/2 ladder with alternate Ising-Heisenberg rungs: a field dependent study

We study a frustrated two-leg spin-1/2 ladder with alternate Ising and isotropic Heisenberg rung exchange interactions, whereas, interactions along legs and diagonals are Ising type. The ground-state (GS) of this model has four exotic phases: (i) the stripe-rung ferromagnet (SRFM), (ii) the anisotropic anti-ferromagnet (AAFM), (iii) the Dimer, and (iv) the stripe-leg ferromagnet (SLFM) in absence of any external magnetic field. In this work, the effect of externally applied longitudinal and transverse fields on quantum phases are studied. In both cases, we show that there exist two plateau phases at $1/4$, and $1/2$ of the saturation of magnetization. Due to the strong rung dimer formation, the system opens a finite spin gap for all the phases resulting in zero magnetization plateau in presence of a longitudinal field. The mechanism of plateau formation is analyzed using spin density, quantum fidelity, and quantum concurrence. In the (i) SRFM phase, Ising exchanges are dominant for all spins but the Heisenberg rungs are weak, and therefore, the magnetization shows a continuous transition as a function of transverse field. In the other three phases [(ii)-(iv)], Ising dimer rungs are weak and broken first to reach the plateau at $1/2$ of the saturation magnetization, having a large gap, which is closed by further application of the field. We use the exact diagonalization (ED) and the transfer matrix method (TM) to solve the Hamiltonian.Comment: 12 pages, 7 figure