Finite-size Lattices Research Articles

Photonic Mpemba effect.

The Mpemba effect (ME) is the counterintuitive phenomenon in statistical physics for which a far-from-equilibrium state can relax toward equilibrium faster than a state closer to equilibrium. This effect has raised great curiosity for a long time and has been studied extensively in many classical and quantum systems. Here, it is shown that the Mpemba effect can be observed in optics as well. Specifically, the process of light diffusion in finite-sized photonic lattices under incoherent (dephasing) dynamics is considered. Rather surprisingly, it is shown that certain highly localized initial light distributions can diffuse faster than initial broadly delocalized distributions. The effect is illustrated by considering the random walk of optical pulses in fiber-based temporal mesh lattices, which should provide an experimentally accessible setup for the demonstration of the Mpemba effect in optics.

Exploring transitions in finite-size Potts model: comparative analysis using Wang–Landau sampling and parallel tempering

Abstract This research provides a examination of transitions within the various-state Potts model in two-dimensional finite-size lattices. Leveraging the Wang–Landau sampling and parallel tempering, we systematically obtain the density of states, facilitating a comprehensive comparative analysis of the results. The determination of the third-order transitions location are achieved through a meticulous examination of the density of states using microcanonical inflection-point analysis. The remarkable alignment between canonical and microcanonical results for higher-order transition locations affirms the universality of these transitions. Our results further illustrate the universality of the robust and microcanonical inflection-point analysis of Wang–Landau sampling.

On the equivalence between n-state spin and vertex models on the square lattice

In this paper we investigate a correspondence among spin and vertex models with the same number of local states on the square lattice with toroidal boundary conditions. We argue that the partition functions of an arbitrary n-state spin model and of a certain specific n-state vertex model coincide for finite lattice sizes. The equivalent vertex model has n3 non-null Boltzmann weights and their relationship with the edge weights of the spin model is explicitly presented. In particular, the Ising model in a magnetic field is mapped to an eight-vertex model whose weights configurations combine both even and odd number of incoming and outcoming arrows at a vertex. We have studied the Yang-Baxter algebra for such mixed eight-vertex model when the weights are invariant under arrows reversing. We find that while the Lax operator lie on the same elliptic curve of the even eight-vertex model the respective R-matrix can not be presented in terms of the difference of two rapidities. We also argue that the spin-vertex equivalence may be used to imbed an integrable spin model in the realm of the quantum inverse scattering framework. As an example, we show how to determine the R-matrix of the 27-vertex model equivalent to a three-state spin model devised by Fateev and Zamolodchikov.

Finite barrier bound state

A boundary mode localized on one side of a finite-size lattice can tunnel to the opposite side which results in unwanted couplings. Conventional wisdom tells that the tunneling probability decays exponentially with the size of the system which thus requires many lattice sites before eventually becoming negligibly small. Here we show that the tunneling probability for some boundary modes can apparently vanish at specific wavevectors. Thus, similar to bound states in the continuum, a boundary mode can be completely trapped within very few lattice sites where the bulk bandgap is not even well-defined. More intriguingly, the number of trapped states equals the number of lattice sites along the normal direction of the boundary. We provide two configurations and validate the existence of this peculiar finite barrier-bound state experimentally in a dielectric photonic crystal at microwave frequencies. Our work offers extreme flexibility in tuning the coupling between localized states and channels as well as a new mechanism that facilitates unprecedented manipulation of light.

The suppression of finite size effect within a few lattice sites

Boundary modes localized on the boundaries of a finite-size lattice experience a finite size effect (FSE) that could result in unwanted couplings, crosstalks and formation of gaps even in topological boundary modes. It is commonly believed that the FSE decays exponentially with the size of the system and thus requires many lattice sites before eventually becoming negligibly small. Here we consider a two-dimensional strip geometry that is periodic along one direction and truncated along the other direction, in which we identify a special type of FSE of some boundary modes that apparently vanishes at some particular wave vectors along the periodic direction. Meanwhile, the number of wave vectors where the FSE vanishes equals the number of lattice sites across the strip. We analytically prove this type of FSE in a simple model and prove this peculiar feature. We also provide a physical system consisting of a plasmonic sphere array where this FSE is present. Our work points to the possibility of almost arbitrarily tunning of the FSE, which facilitates unprecedented manipulation of the coupling strength between modes or channels such as the integration of multiple waveguides and photonic non-abelian braiding.

Coexistence of extended and localized states in finite-sized mosaic Wannier-Stark lattices

Coexistence of extended and localized states in finite-sized mosaic Wannier-Stark lattices

Topological edge states in one-dimensional non-Hermitian Su-Schrieffer-Heeger systems of finite lattice size: Analytical solutions and exceptional points

Topological edge states in one-dimensional non-Hermitian Su-Schrieffer-Heeger systems of finite lattice size: Analytical solutions and exceptional points

Topological Landau–Zener nanophotonic circuits

Topological edge states (TESs), arising from topologically nontrivial phases, provide a powerful toolkit for the architecture design of photonic integrated circuits, since they are highly robust and strongly localized at the boundaries of topological insulators. It is highly desirable to be able to control TES transport in photonic implementations. Enhancing the coupling between the TESs in a finite-size optical lattice is capable of exchanging light energy between the boundaries of a topological lattice, hence facilitating the flexible control of TES transport. However, existing strategies have paid little attention to enhancing the coupling effects between the TESs through the finite-size effect. Here, we establish a bridge linking the interaction between the TESs in a finite-size optical lattice using the Landau–Zener model so as to provide an alternative way to modulate/control the transport of topological modes. We experimentally demonstrate an edge-to-edge topological transport with high efficiency at telecommunication wavelengths in silicon waveguide lattices. Our results may power up various potential applications for integrated topological photonics.

Study on the bandgap and directional wave propagation mechanism of novel auxiliary semicircle rings lattices

In this study, a novel auxiliary semicircle rings lattice is proposed, and the proposed structure consists of connecting ligaments and auxiliary semicircle rings on ligaments. The bandgap and wave propagation characteristics of the proposed structure are analyzed. The generation mechanism of bandgap is carefully studied according to the vibration shape of the structure. Based on the Bloch's theorem and the finite element method, the dynamic model of the structure is solved to explore the wave behavior in the lattice. Then, the effects of different semicircle directions, the number of semicircle rings and the size of semicircle rings on the bandgap width and position are analyzed. In particular, we calculated the group velocity to analyze the influence of the direction of the semicircle rings on the directional frequency related energy flow in the structure. We also simulated the directional wave behavior in a finite-size lattice. It was found that in the proposed structure, the elastic wave with a specific frequency propagated along a certain direction. This study provides an ideological guidance for the topology design of metamaterials with directional wave propagation.

Stabilization of zero-energy skin modes in finite non-Hermitian lattices

The zero energy of a one-dimensional semi-infinite non-Hermitian lattice with nontrivial spectral topology may disappear when we introduce boundaries to the system. While the corresponding zero-energy state can be considered as a quasi-edge state for the finite lattice with a long survival time, any small disruption (noise) in the initial form of the quasi-edge state can significantly shorten the survival time. Here, by tailoring the couplings at one edge we form an exceptional point allowing for a topological phase transition and the stabilization of the quasi-edge state in a finite-size lattice with open edges. Such a small modification in the lattice does not require closing and opening of the band gap and opens the door for experimental realization of such robust zero-energy edge states.

ЗАКОНОМЕРНОСТИ ПЕРЕХОДНОГО РЕЖИМА В ДИССИПАТИВНОЙ КЛЕТОЧНОЙ МОДЕЛИ ЗЕМЛЕТРЯСЕНИЙ

The purpose of this work was to analyze the mechanisms of the growth of drop clusters,leading on a finite-size lattice to a state close to a critical one with a power-law size distribution ofclusters similar to that observed in a seismic process. At the same time, the question of applicabilityof this model to the description of processes in a real geophysical medium remains. Analysis ofthe elements coupling in the one-dimensional OFC model with open boundary conditions allowsan estimation of the variability of the incoming energy to the lattice elements located at differentdistances from the boundaries. The constructed computational model makes it possible toestimate the size of the boundary areas of high average incoming energy variability at differentvalues of the coupling parameter α. It is shown that, as α grows, the boundary region of inhomogeneityexpands. It is shown that there are two different modes of synchronous drop fo rmation,simulating an earthquake. Both mechanisms are determined by the capture of a neighboringelement and the subsequent synchronization of the drops. This process forms a stabledrop of a larger size. The presence of boundary regions with a high gradient of the input energyrate is the main mechanism for the formation of clusters of lattice elements, demonstrating thesimultaneous drop of the accumulated energy. Such a synchronization is achieved due to thehigh mutual variability of energy at each iteration step. The second important mechanism ofcluster growth is typical for the formed clusters that exceed the size of the near-boundary regionof high inhomogeneity of the energy inflow. As the cluster size grows, the capture area ofneighboring elements that are not included in the cluster expands. Accordingly, the probabi litythat the energy of the neighboring element is in the capture area increases. The calculationsshow that the mean time of reaching the given size of the cluster on the lattice at different sp atialdimensions d and at different coupling parameters confirms the presence of two time intervalswith a different mechanism of cluster formation. In this case, the growth of large clusters hasa power-law character, with an exponent determined by the dimension d.

Incommensurate phases in the two-dimensional XY model with Dzyaloshinskii-Moriya interactions.

The two-dimensional XY model with Dzyaloshinskii-Moriya interaction has been studied through extensive Monte Carlo simulations. A hybrid algorithm consisting of single-spin Metropolis and Swendsen-Wang cluster-spin updates has been employed. Single histogram techniques have been used to obtain the thermodynamic variables of interest and finite-size-scaling analysis has led to the phase transition behavior in the thermodynamic limit. Fluctuating boundary conditions have been utilized in order to match the incommensurability between the spin structures and the finite lattice sizes due to the Dzyaloshinskii-Moriya interaction. The effects of the fluctuating boundary conditions have been analyzed in detail in both commensurate and incommensurate cases. The Berezinskii-Kosterlitz-Thouless transition temperature has been obtained as a function of the Dzyaloshinskii-Moriya interaction and the results are in excellent agreement with the exact equationfor the transition line. The spin-spin correlation function critical exponent has been computed as a function of the Dzyaloshinskii-Moriya interaction and temperature. In the incommensurate cases, optimal sizes for the finite lattices and the distribution of the boundary shift angle have been extracted. Analysis of the low temperature configurations and the corresponding vortex-antivortex pairs have also been addressed in some regions of the phase diagram.

Pseudo-phase transitions of Ising and Baxter–Wu models in two-dimensional finite-size lattices

This article offers a detailed analysis of pseudo-phase transitions of Ising and Baxter–Wu models in two-dimensional finite-size lattices. We carry out Wang–Landau sampling to obtain the density of states. Using microcanonical inflection-point analysis with microcanonical entropy, we obtain the order of the pseudo-phase transitions in the models. The microcanonical analysis results of the second-order transition for the Ising model and the first-order transition for the Baxter–Wu model are consistent with the traditional canonical results. In addition, the third-order transitions are found in both models, implying the universality of higher-order phase transitions.

Persistent homology of Z2 gauge theories

Topologically ordered phases of matter display a number of unique characteristics, including ground states that can be interpreted as patterns of closed strings in the case of general ${\mathbb{Z}}_{2}$ string liquids. In this paper, we consider the problem of detecting and distinguishing closed strings in Ising spin configurations sampled from the classical ${\mathbb{Z}}_{2}$ gauge theory. We address this using the framework of persistent homology, which computes the size and frequency of general loop structures in spin configurations via the formation of geometric complexes. Implemented numerically on finite-size lattices, we show that the first Betti number of the Vietoris-Rips complexes achieves a high density at low temperatures in the ${\mathbb{Z}}_{2}$ gauge theory. In addition, it displays a clear signal at the finite-temperature deconfinement transition of the three-dimensional theory. We argue that persistent homology should be capable of interpreting prominent loop structures that occur in a variety of systems, making it a useful tool in theoretical and experimental searches for topological order.

Dipolar Bose-Hubbard model in finite-size real-space cylindrical lattices

Recent experimental progress in magnetic atoms and polar molecules has created the prospect of simulating dipolar Hubbard models with off-site interactions. When applied to real-space cylindrical optical lattices, these anisotropic dipole-dipole interactions acquire a tunable spatially-dependent component while they remain translationally-invariant in the axial direction, creating a sublattice structure in the azimuthal direction. We numerically study how the coexistence of these classes of interactions affects the ground state of hardcore dipolar bosons at half-filling in a finite-size cylindrical optical lattice with octagonal rings. When these two interaction classes cooperate, we find a solid state where the density order is determined by the azimuthal sublattice structure and builds smoothly as the interaction strength increases. For dipole polarisations where the axial interactions are sufficiently repulsive, the repulsion competes with the sublattice structure, significantly increasing entanglement and creating two distinct ordered density patterns. The spatially-varying interactions cause the emergence of these ordered states in small lattices as a function of interaction strength to be staggered according to the azimuthal sublattices.

Selective and tunable excitation of topological non-Hermitian quasi-edge modes

Non-Hermitian lattices under semi-infinite boundary conditions sustain an extensive number of exponentially localized states, dubbed non-Hermitian quasi-edge modes. Quasi-edge states arise rather generally in systems displaying the non-Hermitian skin effect and can be predicted from the non-trivial topology of the energy spectrum under periodic boundary conditions via a bulk-edge correspondence. However, the selective excitation of the system in one among the infinitely many topological quasi-edge states is challenging both from practical and conceptual viewpoints. In fact, in any realistic system with a finite lattice size most of the quasi-edge states collapse and become metastable states. Here we suggest a route toward the selective and tunable excitation of topological quasi-edge states that avoids the collapse problem by emulating semi-infinite lattice boundaries via tailored on-site potentials at the edges of a finite lattice. We illustrate such a strategy by considering a non-Hermitian topological interface obtained by connecting two Hatano–Nelson chains with opposite imaginary gauge fields, which is amenable for a full analytical treatment.

Finite-size and disorder effects on 1D unipartite and bipartite surface lattice resonances.

Optical resonances in bipartite metal nanostructure lattices are more resilient to finite size-effects than equivalent unipartite lattices, but the complexities of their behaviour in non-ideal settings remain relatively unexplored. Here we investigate the quality factor and extinction efficiency of 1D Ag and Au unipartite and bipartite lattices. By modelling finite size lattices over a range of periods we show that the quality factor of Ag bipartite lattices is significantly better than unipartite lattices. This improvement is less pronounced for Au bipartite lattices. We also show that bipartite lattices are dramatically affected by structure size variations at scales that are typically seen in electron beam lithography fabrication in contrast to unipartite lattices, which are not as sensitive.

Microwave photonic crystals, graphene, and honeycomb-kagome billiards with threefold symmetry: Comparison with nonrelativistic and relativistic quantum billiards

We present results for properties related to the band structure of a microwave photonic crystal, that is, a flat resonator containing metallic cylinders arranged on a triangular grid, referred to as the Dirac (microwave) billiard, with a threefold-symmetric shape. Such systems have been used to investigate finite-size graphene sheets. It was shown recently that the eigenmodes of rectangular Dirac billiards are well described by the tight-binding model of a finite-size honeycomb-kagome lattice of corresponding shape. We compare properties of the eigenstates of the Dirac billiard with those of the associated graphene and honeycomb-kagome billiard and relativistic quantum billiard. We outline how the eigenstates of threefold-symmetric systems can be separated according to their transformation properties under rotation by $\frac{2\ensuremath{\pi}}{3}$ into three subspaces, namely singlets, that are rotationally invariant, and doublets that are noninvariant. We reveal for the doublets in graphene and honeycomb-kagome billiards in quasimomentum space a selective excitation of the valley states associated with the two inequivalent Dirac points. For the understanding of symmetry-related features, we extend known results for nonrelativistic quantum billiards and the associated semiclassical approach to relativistic neutrino billiards.

Experimental demonstration of a quantum anomaly induced by borders

Quantum phase transitions, being in charge of the property of matter, are normally induced by varying the Hamiltonian and are characterized by an avoided level crossing between the ground and an excited state. Recently, a quantum phase anomaly induced by a boundary is predicted theoretically as the number of sites $N$ goes to infinity in a one-dimensional Heisenberg spin $XY$ model without an external magnetic field. The infinite-size lattice apparently is inaccessible in experiment, which prevents the experimental observation of the theoretical prediction. In this paper, we find that the finite-size odd-sited lattice can also give the infinite-size lattice's ground-state behavior, and successfully observe such a critical anomaly between two phases having topologically different boundary conditions for the Mott insulator in a finite-size photonic lattice.

Phases fluctuations, self-similarity breaking and anomalous scalings in driven nonequilibrium critical phenomena

We study in detail the dynamic scaling of the three-dimensional (3D) Ising model driven through its critical point on finite-sized lattices. We show explicitly that while finite-size scaling (FSS) at fixed driving rates and finite-time scaling (FTS) on fixed lattice sizes are satisfied for one set of all four observables we measure in their respective scaling regimes, they can be violated for the other set of the observables even in the same regimes. The different behaviors of the two sets of the observables indicate that the usual critical fluctuations can be divided into the so-called phases fluctuations and magnitude fluctuations. The self-similarity of criticality can also be divided into intrinsic and extrinsic self-similarities. The numerical results show that the phases fluctuations lead to the different behaviors while breaking the extrinsic self-similarity gives rise to the violations of the scalings. The set of the observables that violate the scalings is further divided into a primary observable and a secondary observable. Their different leading behaviors enable us to identify four breaking-of-extrinsic-self-similarity exponents for rectifying the violations of either FSS or FTS in either heating or cooling. Crossovers from the extrinsic-self-similarity-breaking-controlled regimes to the usual FSS or FTS regimes are also discussed. In addition, qualitatively different behaviors of the magnitude fluctuations in cooling and in heating and their origin are revealed. Moreover, both FTS and FSS are good down to quite low temperatures in cooling with the extrinsic self-similarity. This indicates that phase ordering can only have an effect at even lower temperatures. Besides, from the quality of curve collapses, we find that the 3D dynamic critical exponent z in heating and in cooling appears identical but the two-dimensional ones different. Our results demonstrate that new exponents are generally required for scaling in the whole driven process near criticality once the lattice size is taken into account. This opens a new door in critical phenomena and suggest that much is yet to be explored in driven nonequilibrium critical phenomena.