Finite Two-dimensional Lattice Research Articles

The continuous-time quantum walk on some graphs based on the view of quantum probability

In this paper, continuous-time quantum walk is discussed based on the view of quantum probability, i.e. the quantum decomposition of the adjacency matrix A of graph. Regard adjacency matrix A as Hamiltonian which is a real symmetric matrix with elements 0 or 1, so we regard [Formula: see text] as an unbiased evolution operator, which is related to the calculation of probability amplitude. Combining the quantum decomposition and spectral distribution [Formula: see text] of adjacency matrix A, we calculate the probability amplitude reaching each stratum in continuous-time quantum walk on complete bipartite graphs, finite two-dimensional lattices, binary tree, [Formula: see text]-ary tree and [Formula: see text]-fold star power [Formula: see text]. Of course, this method is also suitable for studying some other graphs, such as growing graphs, hypercube graphs and so on, in addition, the applicability of this method is also explained.

Modulating thermal robustness and entanglement distribution in a two-dimensional dissipative spin system using impurities

We consider a finite two-dimensional Heisenberg triangular spin lattice coupled to a dissipative Markovian environment at finite temperature in the presence of an external uniform magnetic field. We show how inserting a magnetic impurity in the spin system can be used to effectively control the dynamics and asymptotic state of the system. A strong impurity, at a border or central site, was found to enhance its entanglement with the other spins and their thermal robustness to the dissipative environment reaching an asymptotic state that is independent of the system initial state. However, it reduces the entanglement among the other spins in the lattice and their thermal robustness and may diminish them depending on its strength and the environment temperature. Moreover, the effect of the impurity increases significantly as the degree of anisotropy of the spin system increases. Therefore, the entanglement distribution over the different sites of the lattice can be modulated by tuning the impurity strength, system anisotropy and environment temperature. The impurity can be used as a switch that turns the entanglement on among specific spins and off among others simultaneously.

Thermal Robustness of Entanglement in a Dissipative Two-Dimensional Spin System in an Inhomogeneous Magnetic Field.

Recently new novel magnetic phases were shown to exist in the asymptotic steady states of spin systems coupled to dissipative environments at zero temperature. Tuning the different system parameters led to quantum phase transitions among those states. We study, here, a finite two-dimensional Heisenberg triangular spin lattice coupled to a dissipative Markovian Lindblad environment at finite temperature. We show how applying an inhomogeneous magnetic field to the system at different degrees of anisotropy may significantly affect the spin states, and the entanglement properties and distribution among the spins in the asymptotic steady state of the system. In particular, applying an inhomogeneous field with an inward (growing) gradient toward the central spin is found to considerably enhance the nearest neighbor entanglement and its robustness against the thermal dissipative decay effect in the completely anisotropic (Ising) system, whereas the beyond nearest neighbor ones vanish entirely. The spins of the system in this case reach different steady states depending on their positions in the lattice. However, the inhomogeneity of the field shows no effect on the entanglement in the completely isotropic (XXX) system, which vanishes asymptotically under any system configuration and the spins relax to a separable (disentangled) steady state with all the spins reaching a common spin state. Interestingly, applying the same field to a partially anisotropic (XYZ) system does not just enhance the nearest neighbor entanglements and their thermal robustness but all the long-range ones as well, while the spins relax asymptotically to very distinguished spin states, which is a sign of a critical behavior taking place at this combination of system anisotropy and field inhomogeneity.

Majorana modes in emergent-wire phases of helical and cycloidal magnet-superconductor hybrids

Noncollinear magnetism opens exciting possibilities to generate topological superconductivity. Here, we focus on helical and cycloidal magnetic textures in magnet-superconductor hybrid structures in a background magnetic field. We demonstrate that this system can enter a topological phase which can be understood as a set of parallel topological wires. We explore and confirm this idea in depth with three different approaches: a continuum model, a tight-binding model based on the magnetic unit cell, and exact diagonalization on a finite two-dimensional lattice. The key signature of this topological state is the presence of Majorana bound states at certain disclination defects in the magnetic texture. Based on the $C_2$ symmetry imposed by the helical or cycloidal texture, we employ the theory of topological crystalline superconductors with rotation invariants to obtain the Majorana parity at disclinations. Furthermore, we consider a 90-degree helimagnet domain wall, which is formed by a string of alternating disclinations. We discuss how the resulting chain of disclination bound states hybridizes into two chiral modes with different velocities. We suggest that hybrid systems of chiral magnets and superconductors are capable of hosting Majorana modes in various spatial configurations with potentially far less nano-engineering than in, e.g., semiconductor wires.

Dynamics and topology of non-Hermitian elastic lattices with non-local feedback control interactions

We investigate non-Hermitian elastic lattices characterized by non-local feedback interactions. In one-dimensional lattices, proportional feedback produces non-reciprocity associated with complex dispersion relations characterized by gain and loss in opposite propagation directions. For non-local controls, such non-reciprocity occurs over multiple frequency bands characterized by opposite non-reciprocal behavior. The dispersion topology is investigated with focus on winding numbers and non-Hermitian skin effect, which manifests itself through bulk modes localized at the boundaries of finite lattices. In two-dimensional lattices, non-reciprocity is associated with directional wave amplification. Moreover, the combination of skin effect in two directions produces modes that are localized at the corners of finite two-dimensional lattices. Our results describe fundamental properties of non-Hermitian elastic lattices, and suggest new possibilities for the design of meta materials with novel functionalities related to selective wave filtering, amplification and localization. The considered non-local lattices also provide a platform for the investigation of topological phases of non-Hermitian systems.

Electronic transport through correlated electron systems with nonhomogeneous charge orderings

The spinless Falicov-Kimball model exhibits outside the particle-hole symmetric point different stable nonhomogeneous charge orderings. These include the well known charge stripes and a variety of orderings with phase separated domains, which can significantly influence the charge transport through the correlated electron system. We show this by investigating a heterostructure, in which the Falicov-Kimball model on a finite two-dimensional lattice is located between two noninteracting semi-infinite leads. We use a combination of nonequilibrium Green's functions techniques with a sign-problem-free Monte Carlo method for finite temperatures or a simulated annealing technique for the ground state to address steady-state transport through the system. We show that different ground-state phases of the central system can lead to simple metallic-like or insulating charge transport characteristics, but also to more complicated current-voltage dependencies reflecting a multi-band character of the transmission function. Interestingly, with increasing temperature, the orderings tend to form transient phases before the system reaches the disordered phase. This leads to nontrivial temperature dependencies of the transmission function and charge current.

Lattice quantum magnetometry

We put forward the idea of lattice quantum magnetometry, i.e. quantum sensing of magnetic fields by a charged (spinless) particle placed on a finite two-dimensional lattice. In particular, we focus on the detection of a locally static transverse magnetic field, either homogeneous or inhomogeneous, by performing ground state measurements. The system turns out to be of interest as quantum magnetometer, since it provides a non-negligible quantum Fisher information (QFI) in a large range of configurations. Moreover, the QFI shows some relevant peaks, determined by the spectral properties of the Hamiltonian, suggesting that certain values of the magnetic fields may be estimated better than the others, depending on the value of other tunable parameters. We also assess the performance of coarse-grained position measurement, showing that it may be employed to realize nearly optimal estimation strategies.

Optical properties of two-dimensional square lattices with smooth and randomly rough surfaces that include dispersive left-handed materials

The study of photonic crystals (PCs) is very importance for the development of new optical technologies. An interest in the investigation of PCs is the search of totally optical control of information in a circuit, with the idea of developing new technological applications that have great advantages over conventional electronic devices in the miniaturization of circuits. In the present work, we show a numerical study of the electromagnetic response of two-dimensional square lattices such as finite photonic structures formed by cylinders embedded in air and air holes in a finite plate composed of metamaterial. We applied a numerical technique known as Integral Equation Method (IEM) to calculate the optical response by calculating reflectance and transmittance as a function of the angle of incidence of finite systems proposed. The calculations were performed by varying the filling fractions and introducing a random roughness on the surfaces of the cylindrical inclusions that form our proposed systems, for the transverse magnetic field (TM) polarization. The results obtained show that the random roughness on the surfaces of the cylindrical inclusions affects their reflective and transmissive properties of two-dimensional square lattices. This is an important result to consider in manufacturing of finite two-dimensional square lattices, despite the existence of a well-developed technology for the manufacture of surfaces. These structures can be used, for example, for the development of filters, mirrors and lenses.

Gutzwiller Monte Carlo approach for a critical dissipative spin model

We use the Gutzwiller Monte Carlo approach to simulate the dissipative XYZ-model in the vicinity of a dissipative phase transition. This approach captures classical spatial correlations together with the full on-site quantum behavior, while neglecting non-local quantum effects. By considering finite two-dimensional lattices of various sizes, we identify a ferromagnetic and two paramagnetic phases, in agreement with earlier studies. The greatly reduced numerical complexity the Gutzwiller Monte Carlo approach facilitates efficient simulation of relatively large lattice sizes. The inclusion of the spatial correlations allows to describe critical behavior which is completely missed by the widely applied Gutzwiller decoupling of the density matrix.

Simulation of braiding anyons using matrix product states

Anyons exist as point like particles in two dimensions and carry braid statistics which enable interactions that are independent of the distance between the particles. Except for a relatively few number of models which are analytically tractable, much of the physics of anyons remain still unexplored. In this paper, we show how U(1)-symmetry can be combined with the previously proposed anyonic Matrix Product States to simulate ground states and dynamics of anyonic systems on a lattice at any rational particle number density. We provide proof of principle by studying itinerant anyons on a one dimensional chain where no natural notion of braiding arises and also on a two-leg ladder where the anyons hop between sites and possibly braid. We compare the result of the ground state energies of Fibonacci anyons against hardcore bosons and spinless fermions. In addition, we report the entanglement entropies of the ground states of interacting Fibonacci anyons on a fully filled two-leg ladder at different interaction strength, identifying gapped or gapless points in the parameter space. As an outlook, our approach can also prove useful in studying the time dynamics of a finite number of nonabelian anyons on a finite two-dimensional lattice.

Geometry-induced fluctuations of olfactory searches in bounded domains

In olfactory search an immobile target emits chemical molecules at constant rate. The molecules are transported by the medium, which is assumed to be turbulent. Considering a searcher able to detect such chemical signals and whose motion follows the infotaxis strategy, we study the statistics of the first-passage time to the target when the searcher moves on a finite two-dimensional lattice of different geometries. Far from the target, where the concentration of chemicals is low, the direction of the searcher's first movement is determined by the geometry of the domain and the topology of the lattice, inducing strong fluctuations on the average search time with respect to the initial position of the searcher. The domain is partitioned in well-defined regions characterized by the direction of the first movement. If the search starts over the interface between two different regions, large fluctuations in the search time are observed.

Autonomous Search for a Diffusive Source in an Unknown Structured Environment

The paper presents a framework for autonomous search for a diffusive emitting source of a tracer (e.g., aerosol, gas) in an environment with an unknown map of randomly placed and shaped obstacles. The measurements of the tracer concentration are sporadic, noisy and without directional information. The search domain is discretised and modelled by a finite two-dimensional lattice. The links in the lattice represent the traversable paths for emitted particles and for the searcher. A missing link in the lattice indicates a blocked path due to an obstacle. The searcher must simultaneously estimate the source parameters, the map of the search domain and its own location within the map. The solution is formulated in the sequential Bayesian framework and implemented as a Rao-Blackwellised particle filter with entropy-reduction motion control. The numerical results demonstrate the concept and its performance.

First-passage properties of molecular spiders

Molecular spiders are synthetic catalytic DNA-based nanoscale walkers. We study the mean first-passage time for abstract models of spiders moving on a finite two-dimensional lattice with various boundary conditions and compare it with the mean first-passage time of spiders moving on a one-dimensional track. We evaluate by how much the slowdown on newly visited sites, owing to catalysis, can improve the mean first-passage time of spiders and show that in one dimension, when both ends of the track are an absorbing boundary, the performance gain is lower than in two dimensions, when the absorbing boundary is a circle; this persists even when the absorbing boundary is a single site.

Effects of confinement on the statistics of encounter times: exact analytical results for random walks in a partitioned lattice

We study the effects of temporarily and permanently confining domains on the statistics of first-passage times in finite lattices in one and two dimensions. We present exact results for the mean and variance of the first-passage time between arbitrary sites in the following: (1) a finite one-dimensional lattice partitioned into temporarily confining domains and (2) a finite two-dimensional lattice with reflecting boundaries for a single random walker and an immobile target. In the one-dimensional case, we also present the full first-passage time distribution via numerical inversion of Laplace transforms.

Opinion dynamics of random-walking agents on a lattice

The opinion dynamics of random-walking agents on finite two-dimensional lattices is studied. In the model, the opinion is continuous, and both the lattice and the opinion can be either periodic or nonperiodic. At each time step, all agents move randomly on the lattice, and update their opinions based on those of neighbors with whom the differences of opinion are not greater than a given threshold. Due to the effect of repeated averaging, opinions first converge locally, and eventually reach steady states. As in other models with bounded confidence, steady states in general are those with one or more opinion groups in which all agents have the same opinion. When both the lattice and the opinion are periodic, however, metastable states can emerge, in which the whole spectrum of location-dependent opinions can coexist. This result shows that, when a set of continuous opinions forms a structure like a circle, unlike the typically used linear opinions, rich dynamic behavior can arise. When there are geographical restrictions in real situations, a complete consensus is rarely reached, and metastable states here might be one of the explanations for these situations, especially when opinions are not linear.

Generation of entanglement in finite spin systems via adiabatic quantum computation

For a finite XY chain and a finite two-dimensional Ising lattice, it is shown that the paramagnetic ground state is adiabatically transformed to the GHZ state in the ferromagnetic phase by slowly turning on the magnetic field. The fidelity between the GHZ state and an adiabatically evolved state shows a feature of the quantum phase transition.

Thermodynamics of layered Heisenberg magnets with arbitrary spin

We present a spin-rotation-invariant Green-function theory of long- and short-range order in the ferro- and antiferromagnetic Heisenberg model with arbitrary spin quantum number $S$ on a stacked square lattice. The thermodynamic quantities (Curie temperature ${T}_{C}$, N\'eel temperature ${T}_{N}$, specific heat ${C}_{V}$, intralayer, and interlayer correlation lengths) are calculated, where the effects of the interlayer coupling and the $S$ dependence are explored. In addition, exact diagonalizations on finite two-dimensional (2D) lattices with $S\ensuremath{\ge}1$ are performed, and a very good agreement between the results of both approaches is found. For the quasi-2D and isotropic three-dimensional magnets, our theory agrees well with available quantum Monte Carlo and high-temperature series-expansion data. Comparing the quasi-2D $S=1/2$ magnets, we obtain the inequalities ${T}_{N}>{T}_{C}$ and, for small enough interlayer couplings, ${T}_{N}<{T}_{C}$. The results for ${C}_{V}$ and the intralayer correlation length are compared to experiments on the quasi-2D antiferromagnets ${\text{Zn}}_{2}\text{VO}{({\text{PO}}_{4})}_{2}$ with $S=1/2$ and ${\text{La}}_{2}{\text{NiO}}_{4}$ with $S=1$, respectively.

The structure of magnetic translation group for a finite system

The symmetry group (magnetic translation group) of a Bloch electron in a finite two-dimensional lattice in a perpendicular magnetic field is discussed. The details of the structure of this group depend on a gauge of a vector potential of a magnetic field. For two gauges, symmetric and Landau, the group elements are presented in terms of generators, and the relation between the group properties and the number of magnetic fluxes through the elementary cell of the lattice is pointed out.

The QWalk simulator of quantum walks

Several research groups are giving special attention to quantum walks recently, because this research area have been used with success in the development of new efficient quantum algorithms. A general simulator of quantum walks is very important for the development of this area, since it allows the researchers to focus on the mathematical and physical aspects of the research instead of deviating the efforts to the implementation of specific numerical simulations. In this paper we present QWalk, a quantum walk simulator for one- and two-dimensional lattices. Finite two-dimensional lattices with generic topologies can be used. Decoherence can be simulated by performing measurements or by breaking links of the lattice. We use examples to explain the usage of the software and to show some recent results of the literature that are easily reproduced by the simulator. Program summary Program title: QWalk Catalogue identifier: AEAX_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEAX_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: GNU General Public Licence No. of lines in distributed program, including test data, etc.: 10 010 No. of bytes in distributed program, including test data, etc.: 172 064 Distribution format: tar.gz Programming language: C Computer: Any computer with a C compiler that accepts ISO C99 complex arithmetic (recent versions of GCC, for instance). Pre-compiled Windows versions are also provided Operating system: The software should run in any operating system with a recent C compiler. Successful tests were performed in Linux and Windows RAM: Less than 10 MB were required for a two-dimensional lattice of size 201 × 201 . About 400 MB, for a two-dimensional lattice of size 1601 × 1601 Classification: 16.5 Nature of problem: Classical simulation of discrete quantum walks in one- and two-dimensional lattices. Solution method: Iterative approach without explicit representation of evolution operator. Restrictions: The available amount of RAM memory imposes a limit on the size of the simulations. Unusual features: The software provides an easy way of simulating decoherence through detectors or random broken links. In the two-dimensional simulations it also allows the definition of permanent broken links, besides calculation of total variation distance (from the uniform and from an approximate stationary distribution) and the choice between two different physical lattices. It also provides an easy way of performing measurements on specific sites of the 2D lattice and the analysis of observation screens. In one-dimensional simulations it allows the choice between three different lattices. Both one- and two-dimensional simulations facilitates the generation of graphics by automatically generating gnuplot scrips. Additional comments: • An earlier version of QWalk was first presented in [1]. • The simulator generates gnuplot scripts that can be used to make graphics of the output data. • Several examples of input files are provided. Running time: The simulation of 100 steps for a two-dimensional lattice of size 201 × 201 took less than 2 seconds on a Pentium IV 2.6 GHz with 512 MB of RAM memory, 512 KB of cache memory and under Linux. It also took about 15 minutes for a lattice of size 1601 × 1601 on the same computer. Optimization option -O2 was used during compilation for these tests.

Calculation of Conduction Spectra in Quantum Dot Composed of Penrose Lattice

Conductance through finite two-dimensional Penrose lattices (PLs) are calculated as a function of gate voltage. To investigate effects of lattice defects and lattice vibrations, three types of PLs are taken into account; (A) PL without lattice defects, (B) PL with lattice defects, and (C) PL with lattice vibrations, which is applied the electric field across the conductor. When energy levels of PL coincides with the chemical potential of the electrodes with increasing gate voltage, electron transfer through PL takes place. However, electron transfer is forbidden at certain states; These states have confined states, which have multiple-degeneracy in their electronic states. In addition, the states are localized due to interference of their wave-functions though the wave-function of each state is extended. We would like to point out that rigidity of confined states with respect to lattice defects and lattice vibrations.