Symmetry-breaking Point Research Articles

Two-dimensional fractional [formula omitted]-symmetric cubic–quintic NLS equation: Double-loop symmetry breaking bifurcations, ghost states and dynamics

In this paper, we address the double-loop symmetry breaking bifurcations of optical vortex solitons in the two-dimensional (2D) fractional cubic–quintic nonlinear Schrödinger (FCQNLS) equation with an imprinted partially parity-time (PPT) symmetric complex potential. The continuous branches of asymmetric solitons bifurcate from the fundamental one as the power exceeds some critical value. It is found that the symmetry breaking point is exactly the point at which the fundamental symmetric soliton is destabilized. Intriguingly, the branches of asymmetry solitons (alias ghost states) are existing with complex conjugate propagation constants, which is solely in fractional media. Moreover, the branch of asymmetric solitons crosses the branch of fundamental symmetric ones when it reaches the second critical value, and eventually merges back into it at the third critical power point. The imaginary part of the propagation constant also exhibits a circular shape. And the stability of the fundamental symmetric soliton is restored. In addition, the stability of two branches of asymmetric solitons differs from each other. The reason generating this phenomenon is analyzed in detail. Besides, the branch of dipole solitons for the first excited state is also discussed numerically, which is always stable in the process of symmetry breaking. The stability and dynamics of symmetric (fundamental and the first excited ones) and asymmetric solitons are explored via the linear stability analysis, direct simulation, modulation as well as adiabatic excitation. These results will provide some theoretical basis for the study of spontaneous symmetry breaking phenomena and related experiments in fractional media with PPT-symmetric potentials.

Guide to anomaly-mediated supersymmetry-breaking QCD

We present a careful study of the chiral symmetry breaking minima and the baryonic directions in supersymmetric QCD ($SU({N}_{c})$ with ${N}_{f}$ flavors) perturbed by anomaly mediated supersymmetry breaking (AMSB). For the s-confining case of ${N}_{f}={N}_{c}+1$ and most of the free-magnetic phase (${N}_{f}\ensuremath{\le}1.43{N}_{c}$) we find that naive tree level baryonic runaways are stabilized by loop effects. Runaways are present, however, for the upper end of the free magnetic phase (${N}_{f}\ensuremath{\gtrsim}1.43{N}_{c}$) and into conformal window, signaling the existence of incalculable minima at large field values of $\mathcal{O}(\mathrm{\ensuremath{\Lambda}})$. Nevertheless, the chiral symmetry breaking points are locally stable, and are expected to continuously connect to the vacua of QCD for large SUSY breaking. The case of ${N}_{f}={N}_{c}$ requires particular care due to the inherently strongly coupled nature of the quantum modified moduli space. Due to the incalculability of critical K\"ahler potential terms, the stability of the chiral symmetry breaking point along baryonic directions cannot be determined for ${N}_{f}={N}_{c}$. With the exception of this case, all theories to which AMSB can be applied (${N}_{f}<3{N}_{c}$) possess stable chiral symmetry breaking minima, and all theories with ${N}_{f}\ensuremath{\lesssim}1.43{N}_{c}$ (aside from ${N}_{f}={N}_{c}$) are protected from runaways to incalculable minima.

Dissipation-induced recurrence of non-Hermitian edge burst

In quantum mechanics, the Hermitian Hamiltonian is generally used to describe the ideal closed quantum system, but in reality, the physical system is closely related to the environment, and the open quantum system coupled to the environment can be described by the equivalent non-Hermitian Hamiltonian to a certain extent. Among them, the dissipation intensity is closely related to the dynamic properties of non-Hermitian quantum systems. Therefore, it is of great practical significance to study how dissipation affects particle loss. In this paper, the dynamic law related to dissipation intensity in a one-dimensional non-Hermitian system under open boundary condition is studied, and it is found that dissipation can induce the recurrence of edge burst. After the time-dependent evolution of the particles in the one-dimensional non-Hermitian dissipative lattice system with open boundary condition, there is an edge burst in the system, that is, there is a large probability of particle loss at the edge, and the edge burst disappears after increasing the intracell hopping. It is found that if the dissipation intensity increases or decreases, the edge burst will reappear. This kind of reappearance is different from the original edge burst, which is mainly manifested in the loss probability distribution of particles from the edge distribution to the bulk distribution, which is due to the difference in probability of particle motion direction between the two cases. Under the re-induced edge burst, the particles move leftward and rightward from their initial positions, and rebound from the left after having reached the boundary, forming a more obvious loss probability at the edge and gradually decreasing to the body area. In the original edge burst, the particles only move to the left with a greater probability, and are ‘trapped’ at the edge to completely dissipated, forming a distribution with an independent loss peak at the edge. The movement to the left is due to the non-Hermitian skin effect. The deeper reason for different movement directions is related to parity-time symmetry. Under the parameter near the parity-time symmetry breaking point, the loss probability of the particle is of unilateral distribution, and the loss probability of the particle moving to both sides is of bilateral distribution when it is far away. This is the description of the dissipation-induced edge burst recurrence phenomenon and its characteristics. In addition, this paper also studies the influence of impurity barrier on the probability distribution of particle loss in non-Hermitian dynamics. The results show that placing a small barrier on the non-dissipative <i>A</i>-site can obviously hinder the particle motion, and when the barrier increases to a certain height, its influence on the particle motion tends to be unchanged. And the barrier at the dissipative <i>B</i> lattice has little effect on the dynamics.

General Theory of Josephson Diodes

Motivated by recent progress in the superconductivity nonreciprocal phenomena, we study the general theory of Josephson diodes. The central ingredient for Josephson diodes is the asymmetric proximity process inside the tunneling barrier. From the symmetry breaking point of view, there are two types of Josephson diodes, inversion breaking and time-reversal breaking. For the inversion breaking case, applying voltage bias could effectively tune the proximity process like the voltage-dependent Rashba coupling or electric polarization giving rise to $I_c(V)\neq I_c(-V)$ and $I_{r+}\neq I_{r-}$. For the time-reversal breaking case, the current flow could adjust the internal time-reversal breaking field like magnetism or time-reversal breaking electron-electron pairing, which leads to $I_{c+}\neq I_{c-}$. All these results provide a complete understanding and the general principles of realizing Josephson diodes, especially the recently found NbSe$_2$/Nb$_3$Br$_8$/NbSe$_2$ Josephson diodes.

Topological properties and localization transition in a one-dimensional non-Hermitian lattice with a slowly varying potential

We explore the eigenenergy distribution, localization transition, and topological property of a one-dimensional non-Hermitian lattice with a slowly varying potential, which does not possess parity-time ($\mathcal{PT}$) symmetry. We found that, in contrast to $\mathcal{PT}$-symmetric systems whose eigenenergies can stay real until the $\mathcal{PT}$-symmetry breaking point, the eigenenergies in the present system are fully complex and their distribution in the complex plane depends on the parameters. As non-Hermiticity is introduced into the system, also depending on the parameters, the eigenstates may show a crossover behavior from fully extended to fully localized as non-Hermiticity is increased, or there will be a sharp localization transition at finite non-Hermiticity. Nevertheless, this non-Hermitian system is proven to be topologically trivial, thus we conclude that in general non-Hermitian systems, the localization transition is not always of a topological nature.

PT phase transition in open quantum systems with Lindblad dynamics

We investigate parity-time ($\mathcal{PT}$) phase transitions in open quantum systems and discuss a criterion of Liouvillian $\mathcal{PT}$ symmetry proposed recently by Huber \textit{et al}. [J. Huber \textit{et al}., SciPost Phys. $\textbf{9}$, 52 (2020)]. Using the third quantization, which is a general method to solve the Lindblad equation for open quadratic systems, we show, with a proposed criterion of $\mathcal{PT}$ symmetry, that the eigenvalue structure of the Liouvillian clearly changes at the $\mathcal{PT}$ symmetry breaking point for an open 2-spin model with exactly balanced gain and loss if the total spin is large. In particular, in a $\mathcal{PT}$ unbroken phase, some eigenvalues are pure imaginary numbers while in a $\mathcal{PT}$ broken phase, all the eigenvalues are real. From this result, it is analytically shown for an open quantum system including quantum jumps that the dynamics in the long time limit changes from an oscillatory to an overdamped behavior at the proposed $\mathcal{PT}$ symmetry breaking point. Furthermore, we show a direct relation between the criterion of Huber \textit{et al}. of Liouvillian $\mathcal{PT}$ symmetry and the dynamics of the physical quantities for quadratic bosonic systems. Our results support the validity of the proposed criterion of Liouvillian $\mathcal{PT}$ symmetry.

A Kerr polarization controller

Kerr-effect-induced changes of the polarization state of light are well known in pulsed laser systems. An example is nonlinear polarization rotation, which is critical to the operation of many types of mode-locked lasers. Here, we demonstrate that the Kerr effect in a high-finesse Fabry-Pérot resonator can be utilized to control the polarization of a continuous wave laser. It is shown that a linearly-polarized input field is converted into a left- or right-circularly-polarized field, controlled via the optical power. The observations are explained by Kerr-nonlinearity induced symmetry breaking, which splits the resonance frequencies of degenerate modes with opposite polarization handedness in an otherwise symmetric resonator. The all-optical polarization control is demonstrated at threshold powers down to 7 mW. The physical principle of such Kerr effect-based polarization controllers is generic to high-Q Kerr-nonlinear resonators and could also be implemented in photonic integrated circuits. Beyond polarization control, the spontaneous symmetry breaking of polarization states could be used for polarization filters or highly sensitive polarization sensors when operating close to the symmetry-breaking point.

Large Linear Magnetoresistance and Evidence of Degeneracy Lifting of Valence Bands in Rhombohedral Phase of Topological Crystalline Insulator SnTe

A comprehensive magnetotransport study on a p‐type single crystal of SnTe topological crystalline insulator is reported, across the cubic‐to‐rhombohedral (R3m) transition atTs≈64 K. SnTe exhibits a large unsaturated linear magnetoresistance (LMR) reaching a value of 42% at 5 K and 8 T for (100) plane. LMR is found to have a direct dependence on the mobility and a detailed analysis shows that it follows the classical Parish‐Littlewood model of conductivity fluctuations arising from macroscopic inhomogeneities of tellurium interstitial atoms. Profound SdH oscillations having aπBerry phase are observed in the rhombohedral (R3m) phase having significantly lower carrier density ≈5.32 × 1011 cm−2at 2 K, which provides direct evidence of protected topological surface states in theR3mphase. The Hall conductivity shows a transformation from one band to two‐band behavior across the structural transition, thus providing an experimental evidence for the degeneracy lifting of bulk valence bands below the cubic symmetry breaking point which is consistent with recent band structure calculations. The overall results indicate that magnetotransport studies can distinctly probe the surface and bulk sensitive properties of SnTe, and can also track the band‐splitting of degenerate bands across the cubic‐to‐rhombohedral (R3m) transition.

Exact mobility edges and topological phase transition in two-dimensional non-Hermitian quasicrystals

The emergence of the mobility edge (ME) has been recognized as an important characteristic of Anderson localization. The difficulty in understanding the physics of the MEs in three-dimensional (3D) systems from a microscopic image encourages the development of models in lower-dimensional systems that have exact MEs. While most of the previous studies are concerned with one-dimensional (1D) quasiperiodic systems, the analytic results that allow for an accurate understanding of two-dimensional (2D) cases are rare. In this work, we disclose an exactly solvable 2D quasicrystal model with parity-time ( $$\cal{P}\cal{T}$$ ) symmetry displaying exact MEs. In the thermodynamic limit, we unveil that the extended-localized transition point, observed at the $$\cal{P}\cal{T}$$ symmetry breaking point, is topologically characterized by a hidden winding number defined in the dual space. The coupling waveguide platform can be used to realize the 2D non-Hermitian quasicrystal model, and the excitation dynamics can be used to detect the localization features.

Localization transition, spectrum structure, and winding numbers for one-dimensional non-Hermitian quasicrystals

By analyzing the Lyapunov exponent (LE), we develop a rigorous, fundamental scheme for the study of general non-Hermitian quasicrystals with both complex phase factor and non-reciprocal hopping. Specially, the localization-delocalization transition point, $\mathcal{PT}$-symmetry-breaking point and the winding number transition points are determined by LEs of its dual Hermitian model. The analysis was based on Avila's global theory, and we found that winding number is directly related to the acceleration, the slope of the LE, while quantization of acceleration is the crucial ingredient of Avila's global theory. This result applies as well to the models with higher winding, not only the simplest Aubry-Andr\'{e} model. As typical examples, we obtain the analytical phase boundaries of localization transition for non-Hermitian Aubry-Andr\'{e} model in the whole parameter space, and the complete phase diagram is straightforwardly determined. For the non-Hermitian Soukoulis-Economou model, a high winding model, we show how the phase boundaries of localization transition and winding number transitions relate to the LEs of its dual Hermitian model. Moreover, we discover an intriguing feature of robust spectrum, i.e., the spectrum keeps invariant when one changes the complex phase parameter $h$ or non-reciprocal parameter $g$ in the region of $h<|h_c|$ or $g<|g_c|$ if the system is in the extended or localized state, respectively.

Dynamical evolution in a one-dimensional incommensurate lattice with PT symmetry

We investigate the dynamical evolution of a parity-time ($\mathcal{PT}$) symmetric extension of the Aubry-Andr\'{e} (AA) model, which exhibits the coincidence of a localization-delocalization transition point with a $\mathcal{PT}$ symmetry breaking point. One can apply the evolution of the profile of the wave packet and the long-time survival probability to distinguish the localization regimes in the $\mathcal{PT}$ symmetric AA model. The results of the mean displacement show that when the system is in the $\mathcal{PT}$ symmetry unbroken regime, the wave-packet spreading is ballistic, which is different from that in the $\mathcal{PT}$ symmetry broken regime. Furthermore, we discuss the distinctive features of the Loschmidt echo with the post-quench parameter being localized in different $\mathcal{PT}$ symmetric regimes.

Chaotic dynamics of complex trajectory and its quantum signature* *Project partially supported by the National Natural Science Foundation of China (Grant Nos. 12065009, 11804130, and 11805165) and Zhejiang Provincial Nature Science Foundation, China (Grant No. LY20A050001).

We investigate both the quantum and classical dynamics of a non-Hermitian system via a kicked rotor model with symmetry. For the quantum dynamics, both the mean momentum and mean square of momentum exhibit the staircase growth with time when the system parameter is in the neighborhood of the symmetry breaking point. If the system parameter is much larger than the symmetry breaking point, the accelerator mode results in the directed spreading of the wavepackets as well as the ballistic diffusion in momentum space. For the classical dynamics, the non-Hermitian kicking potential leads to the exponentially-fast increase of classical complex trajectories. As a consequence, the imaginary part of the trajectories exponentially diffuses with time, while the real part exhibits the normal diffusion. Our analytical prediction of the exponential diffusion of imaginary momentum and its breakdown time is in good agreement with numerical results. The quantum signature of the chaotic diffusion of the complex trajectories is reflected by the dynamics of the out-of-time-order correlators (OTOC). In the semiclassical regime, the rate of the exponential increase of the OTOC is equal to that of the exponential diffusion of the complex trajectories.

Holographic QFTs on S2\xd7S2, spontaneous symmetry breaking and Efimov saddle points

Holographic CFTs and holographic RG flows on space-time manifolds which are d-dimensional products of spheres are investigated. On the gravity side, this corresponds to Einstein-dilaton gravity on an asymptotically AdSd+1 geometry, foliated by a product of spheres. We focus on holographic theories on S2× S2, we show that the only regular five-dimensional bulk geometries have an IR endpoint where one of the sphere shrinks to zero size, while the other remains finite. In the Z2-symmetric limit, where the two spheres have the same UV radii, we show the existence of a infinite discrete set of regular solutions, satisfying an Efimov-like discrete scaling. The Z2-symmetric solution in which both spheres shrink to zero at the endpoint is singular, whereas the solution with lowest free energy is regular and breaks Z2 symmetry spontaneously. We explain this phenomenon analytically by identifying an unstable mode in the bulk around the would-be Z2-symmetric solution. The space of theories have two branches that are connected by a conifold transition in the bulk, which is regular and correspond to a quantum first order transition. Our results also imply that AdS5 does not admit a regular slicing by S2× S2.

Asymmetric balance in symmetry breaking

Spontaneous symmetry breaking is central to our understanding of physics and explains many natural phenomena, from cosmic scales to subatomic particles. Its use for applications requires devices with a high level of symmetry, but engineered systems are always imperfect. Surprisingly, the impact of such imperfections has barely been studied, and restricted to a single asymmetry. Here, we experimentally study spontaneous symmetry breaking with two controllable asymmetries. We remarkably find that features typical of spontaneous symmetry breaking, while destroyed by one asymmetry, can be restored by introducing a second asymmetry. In essence, asymmetries are found to balance each other. Our study illustrates aspects of the universal unfolding of the pitchfork bifurcation, and provides new insights into a key fundamental process. It also has practical implications, showing that asymmetry can be exploited as an additional degree of freedom. In particular, it would enable sensors based on symmetry breaking or exceptional points to reach divergent sensitivity even in presence of imperfections. Our experimental implementation built around an optical fiber ring additionally constitutes the first observation of the polarization symmetry breaking of passive driven nonlinear resonators.

Non-Hermitian mobility edges in one-dimensional quasicrystals with parity-time symmetry

We investigate localization-delocalization transition in one-dimensional non-Hermitian quasiperiodic lattices with exponential short-range hopping, which possess parity-time ($\mathcal{PT}$) symmetry. The localization transition induced by the non-Hermitian quasiperiodic potential is found to occur at the $\mathcal{PT}$-symmetry-breaking point. Our results also demonstrate the existence of energy dependent mobility edges, which separate the extended states from localized states and are only associated with the real part of eigen-energies. The level statistics and Loschmidt echo dynamics are also studied.

Observation of an exceptional point in a non-Hermitian metasurface

Abstract Exceptional points (EPs), also known as non-Hermitian degeneracies, have been observed in parity-time symmetric metasurfaces as parity-time symmetry breaking points. However, the parity-time symmetry condition puts constraints on the metasurface parameter space, excluding the full examination of unique properties that stem from an EP. Here, we thus design a general non-Hermitian metasurface with a unit cell containing two orthogonally oriented split-ring resonators (SRRs) with overlapping resonance but different scattering rates and radiation efficiencies. Such a design grants us full access to the parameter space around the EP. The parameter space around the EP is first examined by varying the incident radiation frequency and coupling between SRRs. We further demonstrate that the EP is also observable by varying the incident radiation frequency along with the incident angle. Through both methods, we validate the existence of an EP by observing unique level crossing behavior, eigenstate swapping under encirclement, and asymmetric transmission of circularly polarized light.

Nonlinear nonuniform PT -symmetric Bragg grating structures

We explore the consequences of incorporating parity and time reversal ($\mathcal{PT}$) symmetries on the dynamics of nonreciprocal light propagation exhibited by a class of nonuniform periodic structures known as chirped $\mathcal{PT}$-symmetric fiber Bragg gratings (FBGs). The interplay among various grating parameters such as chirping, detuning, nonlinearities, and gain/loss gives rise to unique bi- and multi-stable states in the unbroken as well as broken $\mathcal{PT}$-symmetric regimes. The role of chirping on the steering dynamics of the hysteresis curve is influenced by the type of nonlinearities and the nature of detuning parameter. Also, incident directions of the input light robustly impact the steering dynamics of bistable and multistable states both in the unbroken and broken $\mathcal{PT}$-symmetric regimes. When the light launching direction is reversed, critical stable states are found to occur at very low intensities which opens up a new avenue for an additional way of controlling light with light. We also analyze the phenomenon of unidirectional wave transport and the reflective bi- and multi-stable characteristics at the so-called $\mathcal{PT}$-symmetry breaking point.

Solitons in the two-dimensional fractional Schrödinger equation with radially symmetric PT potential

We investigate the properties of linear and nonlinear fiber models described by the two-dimensional fractional Schrödinger equation with a radially symmetric parity-time (PT) potential. A nonlinear twisted multicore fiber is constructed with alternating amplifying and absorbing cores, which meet the requirements of the PT symmetry. At the PT symmetry breaking point, the threshold level of gain/loss is non-monotonic function of the fiber twisting rate, which can be reduced to nearly zero. The increasing nonlinearity leads to the reduction of the symmetry-breaking threshold. Furthermore, it is shown that the coupling strength between neighboring cores, global energy transport of the linear modes, and the distribution characteristics of soliton solutions of the model depend on the Lévy index, nonlinearity, gain/loss, and the fiber twisting rate.

Symmetry breaking and restoration of symmetric solitons in partially parity-time-symmetric potentials

We address the symmetry-breaking bifurcation of optical solitons in cubic–quintic media with an imprinted partially parity-time-symmetric potential. An increase in soliton power leads to symmetry breaking, which occurs through a pitchfork bifurcation. In contrast to all previously reported symmetry breaking, restoring bifurcation occurs when soliton power exceeds another critical value. Symmetry-breaking region between symmetry-breaking point and restoration point shrinks with the growth of the strength of quintic nonlinearity. While the base branch of solitons loses their stabilities in the symmetry-breaking region, the stabilities of two branches of bifurcated asymmetric solitons respecting mirror symmetry are opposite. Nonreciprocal propagation of unstable asymmetric solitons is also observed. We thus put forward the first example of restoring bifurcation of symmetric solitons in two dimensions.

Topological Phase Transition in non-Hermitian Quasicrystals.

The discovery of topological phases in non-Hermitian open classical and quantum systems challenges our current understanding of topological order. Non-Hermitian systems exhibit unique features with no counterparts in topological Hermitian models, such as failure of the conventional bulk-boundary correspondence and non-Hermitian skin effect. Advances in the understanding of the topological properties of non-Hermitian lattices with translational invariance have been reported in several recent studies; however little is known about non-Hermitian quasicrystals. Here we disclose topological phases in a quasicrystal with parity-time (PT) symmetry, described by a non-Hermitian extension of the Aubry-André-Harper model. It is shown that the metal-insulating phase transition, observed at the PT symmetry breaking point, is of topological nature and can be expressed in terms of a winding number. A photonic realization of a non-Hermitian quasicrystal is also suggested.