A nonlinear journey from structural phase transitions to quantum annealing.

Motivated by an exact mapping between equilibrium properties of a one-dimensional chain of quantum Ising spins in a transverse field (the transverse field Ising (TFI) model) and a two-dimensional classical array of particles in double-well potentials (the "ϕ4 model") with weak inter-chain coupling, we explore connections between the driven variants of the two systems. We argue that coupling between the fundamental topological solitary waves in the form of kinks between neighboring chains in the classical ϕ4 system is the analog of the competing effect of the transverse field on spin flips in the quantum TFI model. As an example application, we mimic simplified measurement protocols in a closed quantum model system by studying the classical ϕ4 model subjected to periodic perturbations. This reveals memory/loss of memory and coherence/decoherence regimes, whose quantum analogs are essential in annealing phenomena. In particular, we examine regimes where the topological excitations control the thermal equilibration following perturbations. This paves the way for further explorations of the analogy between lower-dimensional linear quantum and higher-dimensional classical nonlinear systems.

The application of the "inverse problem" method for constructing confining potentials that make N-soliton waveforms exact solutions in the Gross-Pitaevskii equation.

In this work, we discuss an application of the "inverse problem" method to find the external trapping potential, which has particular N trapped soliton-like solutions of the Gross-Pitaevskii equation (GPE) also known as the cubic nonlinear Schrödinger equation (NLSE). This inverse method assumes particular forms for the trapped soliton wave function, which then determines the (unique) external (confining) potential. The latter renders these assumed waveforms exact solutions of the GPE (NLSE) for both attractive (g<0) and repulsive (g>0) self-interactions. For both signs of g, we discuss the stability with respect to self-similar deformations and translations. For g<0, a critical mass Mc or equivalently the number of particles for instabilities to arise can often be found analytically. On the other hand, for the case with g>0 corresponding to repulsive self-interactions which is often discussed in the atomic physics realm of Bose-Einstein condensates, the bound solutions are found to be always stable. For g<0, we also determine the critical mass numerically by using linear stability or Bogoliubov-de Gennes analysis, and compare these results with our analytic estimates. Various analytic forms for the trapped N-soliton solutions in one, two, and three spatial dimensions are discussed, including sums of Gaussians or higher-order eigenfunctions of the harmonic oscillator Hamiltonian.

Quantum fluctuations drive nonmonotonic correlations in a qubit lattice.

Fluctuations may induce the degradation of order by overcoming ordering interactions, consequently leading to an increase of entropy. This is particularly evident in magnetic systems characterized by nontrivial, constrained disorder, where thermal or quantum fluctuations can yield counterintuitive forms of ordering. Using the proven efficiency of quantum annealers as programmable spin system simulators, we present a study based on entropy postulates and experiments on a platform of programmable superconducting qubits to show that a low level of uncertainty can promote ordering in a system impacted by both thermal and quantum fluctuations. A set of experiments is proposed on a lattice of interacting qubits arranged in a triangular geometry with precisely controlled disorder, effective temperature, and quantum fluctuations. Our results demonstrate the creation of ordered ferrimagnetic and layered anisotropic disordered phases, displaying characteristics akin to the elegant order-by-disorder phenomenon. Extensive experimental evidence is provided for the role of quantum fluctuations in lowering the total energy of the system by increasing entropy and defect clustering. Our thorough and comprehensive application of an intentionally introduced noise on a quantum platform provides insight into the dynamics of defects and fluctuations in quantum devices, which may help to reduce the cost associated with quantum processing.

Uniform-density Bose-Einstein condensates of the Gross-Pitaevskii equation found by solving the inverse problem for the confining potential.

In this work, we study the existence and stability of constant density (flat-top) solutions to the Gross-Pitaevskii equation (GPE) in confining potentials. These are constructed by using the "inverse problem" approach which corresponds to the identification of confining potentials that make flat-top waveforms exact solutions to the GPE. In the one-dimensional case, the exact solution is the sum of stationary kink and antikink solutions, and in the overlapping region, the density is constant. In higher spatial dimensions, the exact solutions are generalizations of this wave function. In the absence of self-interactions, the confining potential is similar to a smoothed-out finite square well with minima also at the edges. When self-interactions are added, terms proportional to ±gψ^{*}ψ and ±gM with M representing the mass or number of particles in Bose-Einstein condensates get added to the confining potential and total energy, respectively. In the realm of stability analysis, we find (linearly) stable solutions in the case with repulsive self-interactions which also are stable to self-similar deformations. For attractive interactions, however, the minima at the edges of the potential get deeper and a barrier in the center forms as we increase the norm. This leads to instabilities at a critical value of M. Comparing the stability criteria from Derrick's theorem with Bogoliubov-de Gennes (BdG) analysis stability results, we find that both predict stability for repulsive self-interactions and instability at a critical mass M for attractive interactions. However, the numerical analysis gives a much lower critical mass. This is due to the emergence of symmetry-breaking instabilities that were detected by the BdG analysis and violate the symmetry x→-x assumed by Derrick's theorem.

Spectral Filtering Induced by Non-Hermitian Evolution with Balanced Gain and Loss: Enhancing Quantum Chaos.

The dynamical signatures of quantum chaos in an isolated system are captured by the spectral form factor, which exhibits as a function of time a dip, a ramp, and a plateau, with the ramp being governed by the correlations in the level spacing distribution. While decoherence generally suppresses these dynamical signatures, the nonlinear non-Hermitian evolution with balanced gain and loss (BGL) in an energy-dephasing scenario can enhance manifestations of quantum chaos. In the Sachdev-Ye-Kitaev model and random matrix Hamiltonians, BGL increases the span of the ramp, lowering the dip as well as the value of the plateau, providing an experimentally realizable physical mechanism for spectral filtering. The chaos enhancement due to BGL is optimal over a family of filter functions that can be engineered with fluctuating Hamiltonians.

Thermalization in the one-dimensional Salerno model lattice.

The Salerno model constitutes an intriguing interpolation between the integrable Ablowitz-Ladik (AL) model and the more standard (nonintegrable) discrete nonlinear Schrödinger (DNLS) one. The competition of local on-site nonlinearity and nonlinear dispersion governs the thermalization of this model. Here, we investigate the statistical mechanics of the Salerno one-dimensional lattice model in the nonintegrable case and illustrate the thermalization in the Gibbs regime. As the parameter interpolating between the two limits (from DNLS toward AL) is varied, the region in the space of initial energy and norm densities leading to thermalization expands. The thermalization in the non-Gibbs regime heavily depends on the finite system size; we explore this feature via direct numerical computations for different parametric regimes.

Superconductivity in graphene induced by the rotated layer.

Recent discoveries in graphene bilayers have revealed that when one of the layers is rotated by a specific angle, superconductivity emerges. We provide an explanation for this phenomenon. We find that due to the layer rotations, the spinors are modified in such way that a repulsive interaction becomes attractive in certain directions. We also find that due to rotations the nodal points become angle dependent. The spinor in layer $ i=2 $ depends on the twisting angle in contrast to the spinor in layer $i=1$. As a result, the physics in the two layers depends on the twist and is identified with a twisted phase. In order to observe the twist we use an interaction term which changes sign. The change from a repulsive interaction to an attractive one gives rise to a one dimensional charge-density-wave. Due to tunneling between the two layers, the proximity of layer $i=1$ induces superconductivity in the charge-density-wave phase in layer $i=2$. This result is obtained by following a sequence of steps: when layer $2$ is rotated by an angle $\theta$, this rotation is equivalent to a rotation of an angle $-\theta$ of the linear momentum. Due to the discrete lattice, in layer $1$ the Fourier transform conserves the linear momentum $modulo$ the hexagonal reciprocal lattice vector. In layer $2$, due to the rotation, the linear momentum is conserved $modulo$ the {\it Moir\'e} reciprocal lattice vector. Periodicity is achieved at the $magic $ angles obtained from the condition of commensuration of the two lattices.

Surface-wave-assisted nonreciprocity in spatio-temporally modulated metasurfaces.

Emerging photonic functionalities are mostly governed by the fundamental principle of Lorentz reciprocity. Lifting the constraints imposed by this principle could circumvent deleterious effects that limit the performance of photonic systems. Most efforts to date have been limited to waveguide platforms. Here, we propose and experimentally demonstrate a spatio-temporally modulated metasurface capable of complete violation of Lorentz reciprocity by reflecting an incident beam into far-field radiation in forward scattering, but into near-field surface waves in reverse scattering. These observations are shown both in nonreciprocal beam steering and nonreciprocal focusing. We also demonstrate nonreciprocal behavior of propagative-only waves in the frequency- and momentum-domains, and simultaneously in both. We develop a generalized Bloch-Floquet theory which offers physical insights into Lorentz nonreciprocity for arbitrary spatial phase gradients, and its predictions are in excellent agreement with experiments. Our work opens exciting opportunities in applications where free-space nonreciprocal wave propagation is desired.

Spin Transport Based on Exchange Coupling in Doped Organic Polymers.

We develop a spin diffusion theory based on the exchange mechanism among polarons to understand the organic pure spin current. It is demonstrated that the exchange coupling is strong enough to induce spin transport within the organic layer with impurity concentrations higher than 1018 cm-3. By calculating the inverse spin Hall voltage in an organic spin device, we predict that the voltage depends nonmonotonically on the impurity concentration of the organic material. By tuning the doping concentration, one can achieve a maximum inverse spin Hall voltage. Our results not only explain some recent experimental data but also inspire further experimental investigation on pure spin current in organic devices with variable impurity doping.

Speed-of-light pulses in a massless nonlinear Dirac equation.

In this work, we explore a massless nonlinear Dirac equation, i.e., a nonlinear Weyl equation. We study the dynamics of its pulse solutions and find that a localized one-hump initial condition splits into a localized two-hump pulse, while an associated phase structure emerges in suitable components of the spinor field. For times larger than a transient time t_{s} this pulse moves with the speed of light, effectively featuring linear wave dynamics and maintaining its shape (both in two and three dimensions). We show that for the considered nonlinearity, this pulse represents an exact solution of the nonlinear equation. Finally, we briefly comment on the generalization of the results to a broader class of nonlinearities.

Kink-Kink and Kink-Antikink Interactions with Long-Range Tails.

In this Letter, we address the long-range interaction between kinks and antikinks, as well as kinks and kinks, in φ^{2n+4} field theories for n>1. The kink-antikink interaction is generically attractive, while the kink-kink interaction is generically repulsive. We find that the force of interaction decays with the 2n/(n-1)th power of their separation, and we identify the general prefactor for arbitrary n. Importantly, we test the resulting mathematical prediction with detailed numerical simulations of the dynamic field equation, and obtain good agreement between theory and numerics for the cases of n=2 (φ^{8} model), n=3 (φ^{10} model), and n=4 (φ^{12} model).

Influence of the number of orientational domains on avalanche criticality in ferroelastic transitions.

We study the relationship between avalanche criticality and the number of orientational domains in ferroelastic transitions. To this end, we use a general Ginzburg-Landau model appropriate for displacive transitions of the square lattice. The model includes disorder as a quenched distribution of local transition temperatures. We focus on the square-to-rectangle and the square-to-oblique ferroelastic transitions, which have two and four orientational domains, respectively, which in turn determine the corresponding degeneracy of the ground state of the system. The phase transitions are driven by temperature under the assumption of a strict athermal behavior. That is, we assume that thermal fluctuations do not play any role. Numerical results are obtained using a purely relaxational dynamics, and it is shown that both the square-to-rectangle and the square-to-oblique transitions occur intermittently in the form of avalanches. Avalanche sizes and avalanche energies are found to display power-law distributions, which corroborates avalanche criticality. We compare and contrast the dependence of avalanche criticality on the number of orientational domains of the low-symmetry phase. It is found that the critical exponents depend on that number, in agreement with recent experimental results.

Fragile aspects of topological transition in lossy and parity-time symmetric quantum walks.

Quantum walks often provide telling insights about the structure of the system on which they are performed. In [Formula: see text]-symmetric and lossy dimer lattices, the topological properties of the band structure manifest themselves in the quantization of the mean displacement of such a walker. We investigate the fragile aspects of a topological transition in these two dimer models. We find that the transition is sensitive to the initial state of the walker on the Bloch sphere, and the resultant mean displacement has a robust topological component and a quasiclassical component. In [Formula: see text] symmetric dimer lattices, we also show that the transition is smeared by nonlinear effects that become important in the [Formula: see text]-symmetry broken region. By carrying out consistency checks via analytical calculations, tight-binding results, and beam-propagation-method simulations, we show that our predictions are easily testable in today's experimental systems.

Comparing skyrmions and merons in chiral liquid crystals and magnets.

When chiral liquid crystals or magnets are subjected to applied fields or other anisotropic environments, the competition between favored twist and anisotropy leads to the formation of complex defect structures. In some cases, the defects are skyrmions, which have 180^{∘} double twist going outward from the center, and hence can pack together without singularities in the orientational order. In other cases, the defects are merons, which have 90^{∘} double twist going outward from the center; packing such merons requires singularities in the orientational order. In the liquid crystal context, a lattice of merons is equivalent to a blue phase. Here we perform theoretical and computational studies of skyrmions and merons in chiral liquid crystals and magnets. Through these studies, we calculate the phase diagrams for liquid crystals and magnets in terms of dimensionless ratios of energetic parameters. We also predict the range of metastability for liquid crystal skyrmions and show that these skyrmions can move and interact as effective particles. The results show how the properties of skyrmions and merons depend on the vector or tensor nature of the order parameter.

Statistics of twinning in strained ferroelastics.

In this review, we show that the evolution of the microstructure and kinetics of ferroelastic crystals under external shear can be explored by computer simulations of 2D model materials. We find that the nucleation and propagation of twin boundaries in ferroelastics depend sensitively on temperature. In the plastic regime, the evolution of the ferroelastic microstructure under strain deformation maintains a stick-and-slip mechanism in all temperature regimes, whereas the dynamic behavior changes dramatically from power-law statistics at low temperature to a Kohlrausch law at intermediate temperature, and then to a Vogel-Fulcher law at high temperature. In the yield regime, the distribution of jerk energies follows power-law statistics in all temperature regimes for a large range of strain rates. The non-spanning avalanches in the yield regime follow a parabolic temporal profile. The changes of twin pattern and twin boundaries density represent an important step towards domain boundary engineering.

Emergent inequality and self-organized social classes in a network of power and frustration.

We propose a simple agent-based model on a network to conceptualize the allocation of limited wealth among more abundant expectations at the interplay of power, frustration, and initiative. Concepts imported from the statistical physics of frustrated systems in and out of equilibrium allow us to compare subjective measures of frustration and satisfaction to collective measures of fairness in wealth distribution, such as the Lorenz curve and the Gini index. We find that a completely libertarian, law-of-the-jungle setting, where every agent can acquire wealth from or lose wealth to anybody else invariably leads to a complete polarization of the distribution of wealth vs. opportunity. This picture is however dramatically ameliorated when hard constraints are imposed over agents in the form of a limiting network of transactions. There, an out of equilibrium dynamics of the networks, based on a competition between power and frustration in the decision-making of agents, leads to network coevolution. The ratio of power and frustration controls different dynamical regimes separated by kinetic transitions and characterized by drastically different values of equality. It also leads, for proper values of social initiative, to the emergence of three self-organized social classes, lower, middle, and upper class. Their dynamics, which appears mostly controlled by the middle class, drives a cyclical regime of dramatic social changes.

Speed-of-light pulses in the massless nonlinear Dirac equation with a potential.

We consider the massless nonlinear Dirac (NLD) equation in 1+1 dimension with scalar-scalar self-interaction g^{2}/2(Ψ[over ¯]Ψ)^{2} in the presence of three external electromagnetic real potentials V(x), a potential barrier, a constant potential, and a potential well. By solving numerically the NLD equation, we find different scenarios depending on initial conditions, namely, propagation of the initial pulse along one direction, splitting of the initial pulse into two pulses traveling in opposite directions, and focusing of two initial pulses followed by a splitting. For all considered cases, the final waves travel with the speed of light and are solutions of the massless linear Dirac equation. During these processes the charge and the energy are conserved, whereas the momentum is conserved when the solutions possess specific symmetries. For the case of the constant potential, we derive exact analytical solutions of the massless NLD equation that are also solutions of the massless linearized Dirac equation. Decay or growth of the initial pulse is also predicted from the evolution of the charge for the case of a non-zero imaginary part of the potential.

Variational approach to studying solitary waves in the nonlinear Schrödinger equation with complex potentials.

We discuss the behavior of solitary wave solutions of the nonlinear Schrödinger equation (NLSE) as they interact with complex potentials, using a four-parameter variational approximation based on a dissipation functional formulation of the dynamics. We concentrate on spatially periodic potentials with the periods of the real and imaginary part being either the same or different. Our results for the time evolution of the collective coordinates of our variational ansatz are in good agreement with direct numerical simulation of the NLSE. We compare our method with a collective coordinate approach of Kominis and give examples where the two methods give qualitatively different answers. In our variational approach, we are able to give analytic results for the small oscillation frequency of the solitary wave oscillating parameters which agree with the numerical solution of the collective coordinate equations. We also verify that instabilities set in when the slope dp(t)/dv(t) becomes negative when plotted parametrically as a function of time, where p(t) is the momentum of the solitary wave and v(t) the velocity.

Thermodynamics of multicaloric effects in multiferroic materials: application to metamagnetic shape-memory alloys and ferrotoroidics.

We develop a general thermodynamic framework to investigate multicaloric effects in multiferroic materials. This is applied to the study of both magnetostructural and magnetoelectric multiferroics. Landau models with appropriate interplay between the corresponding ferroic properties (order parameters) are proposed for metamagnetic shape-memory and ferrotoroidic materials, which, respectively, belong to the two classes of multiferroics. For each ferroic property, caloric effects are quantified by the isothermal entropy change induced by the application of the corresponding thermodynamically conjugated field. The multicaloric effect is obtained as a function of the two relevant applied fields in each class of multiferroics. It is further shown that multicaloric effects comprise the corresponding contributions from caloric effects associated with each ferroic property and the cross-contribution arising from the interplay between these ferroic properties.This article is part of the themed issue 'Taking the temperature of phase transitions in cool materials'.