Symmetry Breaking Phenomena Research Articles

Re-Examination of the Sel’kov Model of Glycolysis and Its Symmetry-Breaking Instability Due to the Impact of Diffusion with Implications for Cancer Imitation Caused by the Warburg Effect

We revisit the seminal model of glycolysis first proposed by Sel’kov more than fifty years ago. We investigate the onset of instabilities in biological systems described by the Sel’kov model in order to determine the conditions of the model parameters that lead to bifurcations. We analyze the glycolysis reaction under the circumstances when the diffusivity of both ATP and ADP reactants are taken into account. We estimate the critical value of the model’s single compact dimensionless parameter, which is responsible for the onset of reaction instability and the system’s symmetry breaking. It appears that it leads to spatial inhomogeneities of reactants, leading to the formation of standing waves instead of a homogeneous distribution of ATP molecules. The consequences of this model and its results are discussed in the context of the Warburg effect, which signifies a transition from oxidative phosphorylation to glycolysis that is correlated with the initiation and progression of cancer. Our analysis may lead to the selection of therapeutic interventions in order to prevent the symmetry-breaking phenomenon described in our work.

Spontaneous symmetry breaking in generative diffusion models*

Abstract Generative diffusion models have recently emerged as a leading approach for generating high-dimensional data. In this paper, we show that the dynamics of these models exhibit a spontaneous symmetry breaking phenomenon that divides the generative dynamics into two distinct phases: (1) a linear steady-state dynamics around a central fixed-point and, (2) an attractor dynamics directed towards the data manifold. These two ‘phases’ are separated by the change in stability of the central fixed-point, with the resulting window of instability being responsible for the diversity of the generated samples. Using both theoretical and empirical evidence, we show that an accurate simulation of the early dynamics does not significantly contribute to the final generation, since early fluctuations are reverted to the central fixed point. To leverage this insight, we propose a Gaussian late initialization scheme, which significantly improves model performance, achieving up to 3× Fréchet inception distance improvements on fast samplers, while also increasing sample diversity (e.g. racial composition of generated CelebA images). Our work offers a new way to understand the generative dynamics of diffusion models that has the potential to bring about higher performance and less biased fast-samplers.

From Geometry of Hamiltonian Dynamics to Topology of Phase Transitions: A Review.

In this review work, we outline a conceptual path that, starting from the numerical investigation of the transition between weak chaos and strong chaos in Hamiltonian systems with many degrees of freedom, comes to highlight how, at the basis of equilibrium phase transitions, there must be major changes in the topology of submanifolds of the phase space of Hamiltonian systems that describe systems that exhibit phase transitions. In fact, the numerical investigation of Hamiltonian flows of a large number of degrees of freedom that undergo a thermodynamic phase transition has revealed peculiar dynamical signatures detected through the energy dependence of the largest Lyapunov exponent, that is, of the degree of chaoticity of the dynamics at the phase transition point. The geometrization of Hamiltonian flows in terms of geodesic flows on suitably defined Riemannian manifolds, used to explain the origin of deterministic chaos, combined with the investigation of the dynamical counterpart of phase transitions unveils peculiar geometrical changes of the mechanical manifolds in correspondence to the peculiar dynamical changes at the phase transition point. Then, it turns out that these peculiar geometrical changes are the effect of deeper topological changes of the configuration space hypersurfaces ∑v=VN-1(v) as well as of the manifolds {Mv=VN-1((-∞,v])}v∈R bounded by the ∑v. In other words, denoting by vc the critical value of the average potential energy density at which the phase transition takes place, the members of the family {∑v}v<vc are not diffeomorphic to those of the family {∑v}v>vc; additionally, the members of the family {Mv}v>vc are not diffeomorphic to those of {Mv}v>vc. The topological theory of the deep origin of phase transitions allows a unifying framework to tackle phase transitions that may or may not be due to a symmetry-breaking phenomenon (that is, with or without an order parameter) and to finite/small N systems.

Spontaneous symmetry breaking and vortices in a tri-core nonlinear fractional waveguide

We introduce a waveguiding system composed of three linearly-coupled fractional waveguides, with a triangular (prismatic) transverse structure. It may be realized as a tri-core nonlinear optical fiber with fractional group-velocity dispersion (GVD), or, possibly, as a system of coupled Gross–Pitaevskii equations for a set of three tunnel-coupled cigar-shaped traps filled by a Bose–Einstein condensate of particles moving by Lévy flights. The analysis is focused on the phenomenon of spontaneous symmetry breaking (SSB) between components of triple solitons, and the formation and stability of vortex modes. In the self-focusing regime, we identify symmetric and asymmetric soliton states, whose structure and stability are determined by the Lévy index of the fractional GVD, the inter-core coupling strength, and the total energy, which determines the system’s nonlinearity. Bifurcation diagrams (of the supercritical type) reveal regions where SSB occurs, identifying the respective symmetric and asymmetric ground-state soliton modes. In agreement with the general principle of the SSB theory, the solitons with broken inter-component symmetry prevail with the increase of the energy in the weakly-coupled system. Three-components vortex solitons (which do not feature SSB) are studied too. Because the fractional GVD breaks the system’s Galilean invariance, we also address mobility of the vortex solitons, by applying a boost to them.

Physics and chemistry from parsimonious representations: image analysis via invariant variational autoencoders

Electron, optical, and scanning probe microscopy methods are generating ever increasing volume of image data containing information on atomic and mesoscale structures and functionalities. This necessitates the development of the machine learning methods for discovery of physical and chemical phenomena from the data, such as manifestations of symmetry breaking phenomena in electron and scanning tunneling microscopy images, or variability of the nanoparticles. Variational autoencoders (VAEs) are emerging as a powerful paradigm for the unsupervised data analysis, allowing to disentangle the factors of variability and discover optimal parsimonious representation. Here, we summarize recent developments in VAEs, covering the basic principles and intuition behind the VAEs. The invariant VAEs are introduced as an approach to accommodate scale and translation invariances present in imaging data and separate known factors of variations from the ones to be discovered. We further describe the opportunities enabled by the control over VAE architecture, including conditional, semi-supervised, and joint VAEs. Several case studies of VAE applications for toy models and experimental datasets in Scanning Transmission Electron Microscopy are discussed, emphasizing the deep connection between VAE and basic physical principles. Python codes and datasets discussed in this article are available at https://github.com/saimani5/VAE-tutorials and can be used by researchers as an application guide when applying these to their own datasets.

Bistable organic electrochemical transistors: enthalpy vs. entropy

Organic electrochemical transistors (OECTs) underpin a range of emerging technologies, from bioelectronics to neuromorphic computing, owing to their unique coupling of electronic and ionic charge carriers. In this context, various OECT systems exhibit significant hysteresis in their transfer curve, which is frequently leveraged to achieve non-volatility. Meanwhile, a general understanding of its physical origin is missing. Here, we introduce a thermodynamic framework that readily explains the emergence of bistable OECT operation via the interplay of enthalpy and entropy. We validate this model through temperature-resolved characterizations, material manipulation, and thermal imaging. Further, we reveal deviations from Boltzmann statistics for the subthreshold swing and reinterpret existing literature. Capitalizing on these findings, we finally demonstrate a single-OECT Schmitt trigger, thus compacting a multi-component circuit into a single device. These insights provide a fundamental advance for OECT physics and its application in non-conventional computing, where symmetry-breaking phenomena are pivotal to unlock new paradigms of information processing.

Separation of Inverse Altermagnetic Spin-Splitting Effect from Inverse Spin Hall Effect in RuO_{2}.

Recently, a significant amount of attention has been attracted toward a third classification of magnetism, altermagnetism, due to the unique physical properties of altermagnetic materials, which are compensated collinear antiferromagnets that host time-reversal symmetry-breaking phenomena like a ferromagnet. In an altermagnetic material, through the nonrelativistic altermagnetic spin-splitting effect (ASSE), a transverse spin current is generated upon charge current injection. However, it is very challenging to experimentally establish the ASSE since it is inevitably mixed with the spin Hall effect due to the relativistic spin-orbit coupling of the material. Additionally, the dependence on the hard-to-probe and hard-to-control Néel vectors makes it even more difficult to observe and establish the ASSE. In this Letter, we utilize the thermal spin injection from the ferrimagnetic insulator yttrium iron garnet and detect an inverse altermagnetic spin-splitting effect (IASSE) in the high-quality epitaxial altermagnetic RuO_{2} thin films. We observe an opposite sign for the spin-to-charge conversion through the IASSE compared to the inverse spin Hall effect (ISHE). The efficiency of the IASSE is approximately 70% of the ISHE in RuO_{2}. Moreover, we demonstrate that the ASSE or IASSE effect is observable only when the Néel vectors are well aligned. By modifying the Néel vector domains via RuO_{2} crystallinity, we study the ASSE or IASSE unequivocally and quantitatively. Our Letter provides significant insight into the spin-splitting effect in altermagnetic materials.

Broken symmetries in quasi-2D charged systems via negative dielectric confinement

We report spontaneous symmetry breaking (SSB) phenomena in symmetrically charged binary particle systems under planar nanoconfinement with negative dielectric constants. The SSB is triggered solely via the dielectric confinement effect, without any external fields. The mechanism of SSB is found to be caused by the strong polarization field enhanced by nanoconfinement, giving rise to charge/field oscillations in the transverse directions. Interestingly, dielectric contrast can even determine the degree of SSB in transverse and longitudinal dimensions, forming charge-separated interfacial liquids and clusters on square lattices. Furthermore, we analytically show that the formed lattice constant is determined by the dielectric mismatch and the length scale of confinement, which is validated via molecular dynamics simulations. The novel broken symmetry mechanism may provide new insights into the study of quasi-2D systems and the design of future nanodevices.

Vortex-shedding and bistability in cylinder-flexible plate assembly in a channel

This study numerically investigates the fluid–structure interaction dynamics of a stationary circular cylinder with an attached flexible splitter plate in a confined fluid domain. We explore how variations in Reynolds number (Re), plate length, offset from the channel centerline, and blockage ratio influence the vortex-shedding and oscillatory behavior of the plates in laminar flow. Results indicate that at low Reynolds numbers (Re = 100), there are no vortex shedding or oscillations, while at higher Reynolds (Re = 200 to 300), the shorter plate exhibits first-mode oscillations and symmetry breaking, whereas the longer plate shows only transient second-mode oscillations at Re = 300 that diminish in amplitude, suggesting that increased plate length reduces vortex-induced forces. Dynamic Mode Decomposition (DMD) analysis confirms consistent primary flow structures across different Reynolds numbers, with initial modes encapsulating the majority of the system’s dynamics. In addition to determining the critical Re for vortex-shedding, vortex-induced oscillations and bistability, the study also examines the critical blockage ratios on the symmetry breaking phenomenon, and also shows that plate stability is influenced by domain asymmetry. These insights enhance understanding of the biomechanics of fish-like robots and inform the design and control of bioinspired flexible structures in fluid flows.

Room temperature stable twist-bend nematic materials without crystallization over 1 year

The twist-bend nematic (NTB) phase of achiral liquid crystals (LCs) manifests a unique self-assembled heliconical structure with nanometer-scale pitch length, mirroring the chiral symmetry-breaking phenomena in nature, thus sparking widespread research interest. However, the ingenious NTB phase is only stable at high temperatures within a very limited temperature interval, often undergoing inevitable crystallization at low temperatures. Herein, room temperature supercooled NTB material systems composed of meticulously designed LC dimer mixtures with varying molecular curvatures and central flexibility were developed, resulting in complete resistance to crystallization even after 1 year of storage. Furthermore, the proposed NTB material systems demonstrated exceptional compatibility with common nematic LCs, facilitating the tailoring of overall physical parameters, particularly to achieve a sufficiently low bend elastic constant with excellent stability. This work represents a paradigmatic advancement forward in realizing stable NTB phase materials with a broad temperature range and resistance to crystallization, thereby tackling the enduring and seemingly insurmountable challenge while providing impetus for further exploration of their applications in soft matter, crystallography, and advanced photonics.

Existence and symmetry breaking results for positive solutions of elliptic Hamiltonian systems

Abstract In this paper, we are interested in positive solutions of $$ \begin{align*}\left\{ \begin{array}{@{}ll} -\Delta u = a(x)v^{p-1}, \quad &amp;\text{ in } \Omega,\\ -\Delta v = b(x)u^{q-1}, \quad &amp;\text{ in } \Omega,\\ u,v&gt;0, \quad &amp;\text{ in } \Omega,\\ u=v=0, \quad &amp;\text{ on } \partial\Omega, \end{array} \right. \end{align*} $$ where $\Omega $ is a bounded annular domain (not necessarily an annulus) in ${\mathbb {R}}^N (N \ge 3)$ and $ a(x), b(x)$ are positive continuous functions. We show the existence of a positive solution for a range of supercritical values of p and q when the problem enjoys certain mild symmetry and monotonicity conditions. We shall also address the symmetry breaking phenomena where the system is fully symmetric. Indeed, as a consequence of our results, we shall show that problem (1) has $\Bigl \lfloor \frac {N}{2} \Bigr \rfloor $ (the floor of $\frac {N}{2}$ ) positive non-radial solutions when $ a(x)=b(x)=1$ and $\Omega $ is an annulus with certain assumptions on the radii. In general, for the radial case where the domain is an annulus, we prove the existence of a non-radial solution provided $$ \begin{align*}(p-1)(q-1)&gt; \Big(1+\frac{2N}{\lambda_H}\Big)^2\left(\frac{q}{p}\right),\end{align*} $$ where $\lambda _H$ is the best constant for the Hardy inequality on $\Omega .$ We remark that the best constant $\lambda _H$ for the Hardy inequality is just the characteristic of the domain, and is independent of the choices of p and $q.$ For this reason, the aforementioned inequality plays a major role to prove the existence and multiplicity of non-radial solutions when the problem is fully symmetric. Our proofs use a variational formulation on appropriate convex subsets for which the lack of compactness is recovered for the supercritical problem.

Instability mechanisms of symmetry breaking in the wakes of two anti-phase pitching foils at zero free-stream velocity

In the present study, we investigate into the symmetry-breaking phenomenon in the wakes produced by two anti-phase pitching foils at zero free-stream velocity. Numerical simulations are performed in a gap ratio range of 0.6≤g≤2.0 and a Reynolds number range of 70≤Re≤200. Six regions are identified in the plane (g, Re), based on distinct symmetry-breaking (or symmetry-preserving) behaviors of the wakes. Floquet stability analysis and dynamic mode decomposition are performed on some selected cases. By means of modal analysis, the behaviors of wakes in different regions are rationalized and the instability mechanisms of symmetry breaking in the unstable regions are unveiled. In addition, the influence of symmetry-breaking instability on propulsive performance is also discussed.

Indications for an alternative breaking of symmetry in fracture-induced electromagnetic emissions recorded prior to the 2023 Mw7.8 and Mw7.5 Turkey Earthquakes

Several evidence has been reported corroborating the view that the MHz band fracture-induced electromagnetic emissions (FEME), also known as fracture-induced electromagnetic radiation (FEMR), which are detected prior to shallow main earthquakes (EQs) of moment magnitude Mw>5.5 with epicenters on land or near coastline, can be studied in analogy to a Z(2) spin system undergoing a second-order phase transition. Specifically, by analyzing MHz FEME/FEMR time series recorded prior to several earthquakes in the past >20 years, information has been extracted about the organization to critical state, the critical state itself, as well as for the step-by-step development of the spontaneous symmetry breaking (SSB) phenomenon. Using the “method of critical fluctuations” (MCF) and a recently proposed method for the wavelet-based detection of scaling behavior in noisy experimental data, we identified for the first time a different way of breaking the symmetry embedded in the MHz FEME/FEMR time series recorded by the hELlenic Seismo-ElectroMagnetics Network (ELSEM-Net) prior to the devastating (Mw=7.8 and Mw=7.5) EQs that occurred in south-east Turkey on 06/02/2023. Particularly, after the achievement of the critical state, indications were found that, although the underlying system started to break the symmetry according to the second-order phase transition mechanism (through SSB), SSB was not completed. Instead, the system seems to have changed symmetry and presented indications that the break of symmetry was finally achieved according to the first-order phase transition mechanism. Specifically, after the incomplete SSB, it was found to present tricritical behavior, followed by indications of triple degeneration of the order parameter fixed-point. This change of behavior observed in the MHz FEME/FEMR is a puzzling phenomenon for several reasons, which are discussed in detail, whereas suggestions are expressed as a first attempt for the possible explanation of the new findings.

Observation of Replica Symmetry Breaking in Standard Mode-Locked Fiber Laser.

We report the first experimental demonstration of the replica symmetry breaking (RSB) phenomenon in a fiber laser system supporting standard mode-locking (SML) regime. Though theoretically predicted, this photonic glassy phase remained experimentally undisclosed so far. We employ an ytterbium-based mode-locked fiber laser with a very rich phase diagram. Two phase transitions are observed separating three different regimes: cw, quasi-mode-locking (QML), and SML. The regimes are intrinsically related to the distinct dynamics of intensity fluctuations in the laser spectra. We set the connection between the RSB glassy phase with frustrated modes and onset of L-shaped intensity distributions in the QML regime, which impact directly the replica overlap measure.

A High Working Temperature Multiferroic Induced by Inverse Temperature Symmetry Breaking.

Molecular-based multiferroic materials that possess ferroelectric and ferroelastic orders simultaneously have attracted tremendous attention for their potential applications in multiple-state memory devices, molecular switches, and information storage systems. However, it is still a great challenge to effectively construct novel molecular-based multiferroic materials with multifunctionalities. Generally, the structure of these materials possess high symmetry at high temperatures, while processing an obvious order-disorder or displacement-type ferroelastic or ferroelectric phase transition triggered by symmetry breaking during the cooling processes. Therefore, these materials can only function below the Curie temperature (Tc), the low of which is a severe impediment to their practical application. Despite great efforts to elevate Tc, designing single-phase crystalline materials that exhibit multiferroic orders above room temperature remains a challenge. Here, an inverse temperature symmetry-breaking phenomenon was achieved in [FPM][Fe3(μ3-O)(μ-O2CH)8] (FPM stands for 3-(3-formylamino-propyl)-3,4,5,6-tetrahydropyrimidin-1-ium, which acts as the counterions and the rotor component in the network), enabling a ferroelastoelectric phase at a temperature higher than Tc (365 K). Upon heating from room temperature, two-step distinct symmetry breaking with the mm2Fm species leads to the coexistence of ferroelasticity and ferroelectricity in the temperature interval of 365-426 K. In the first step, the FPM cations undergo a conformational flip-induced inverse temperature symmetry breaking; in the second step, a typical ordered-disordered motion-induced symmetry breaking phase transition can be observed, and the abnormal inverse temperature symmetry breaking is unprecedented. Except for the multistep ferroelectric and ferroelastic switching, this complex also exhibits fascinating nonlinear optical switching properties. These discoveries not only signify an important step in designing novel molecular-based multiferroic materials with high working temperatures, but also inspire their multifunctional applications such as multistep switches.

Two-fold symmetric superconductivity in the Kagome superconductor RbV3Sb5

The recently discovered kagome superconductors offer a promising platform for investigating intertwined orders and novel states, including topology, superconductivity, charge density waves, and more. The interplay among these orders can spontaneously break rotational symmetry, giving rise to exotic phenomena such as nematicity or even nematic superconductivity. Here we present our findings on the two-fold symmetric superconductivity in thin-flake RbV3Sb5 in response to direction-dependent in-plane magnetic fields, in contrast to the inherent six-fold structural symmetry of the crystal lattice. The two-fold symmetry was evidenced through a combination of magnetoresistance transport experiments, critical magnetic field measurements, and observations of anisotropic superconducting gaps. Additionally, by altering the experimental configuration, we also detected the presence of six-fold symmetric components superimposed on the two-fold symmetry at the boundary between normal and superconducting states. Our results underscore the correlation-driven symmetry-breaking phenomena and emphasize the potential of this correlated kagome family as a promising platform for investigating intertwined orders, including unconventional superconductivity.

Excited-state symmetry breaking is an ultrasensitive tool for probing microscopic electric fields.

Microscopic electric fields are increasingly found to play a pivotal role in catalysis of enzymatic and chemical reactions. Currently, the vibrational Stark effect is the main experimental method used to measure them. Here, we demonstrate how excited-state symmetry breaking can serve as a much more sensitive tool to assess these fields. Using transient infrared spectroscopy on a quadrupolar probe equipped with nitrile groups we demonstrate both its superior sensitivity and that it does not suffer from the notorious hydrogen-bond induced upshift of the C[triple bond, length as m-dash]N stretch frequency. In combination with conventional ground-state infrared absorption, excited-state symmetry breaking can be used to disentangle even weak specific hydrogen bond interactions from general field effects. We showcase this capability with the example of weak C-H hydrogen bonds in polar aprotic solvents. Additionally, we reveal for the first time symmetry breaking driven not by solvent but by the entropy of the pendant side chains of the chromophore. Our findings not only enhance our understanding of symmetry-breaking charge-transfer phenomena but pave the way toward using them in electric field sensing modality.

Exclusion processes on a roundabout traffic model with constrained resources.

Motivated by the vehicular traffic phenomenon at roundabouts, we examine how the limited availability of resources affects the movement of two distinct types of particles on bidirectional lanes connected by two bridges, with each bridge specifically designated for the transportation of one species. To provide a theoretical ground for our findings, we employ a mean-field framework and successfully validate them through dynamic Monte Carlo simulations. Based on the theoretical analysis, we analytically derive various stationary properties, such as the particle densities, phase boundaries, and particle currents, for all the possible symmetric as well as asymmetric phases. The qualitative as well as quantitative behavior of the system is significantly affected by the constraint on the number of resources. The complexity of the phase diagram shows a nonmonotonic behavior with an increasing number of particles in the system. Analytical arguments enable the identification of several critical values for the total number of particles, leading to a qualitative change in the phase diagrams. The interplay of the finite resources and the bidirectional transport yields unanticipated and unusual features such as back-and-forth transition, the presence of two congested phases where particle movement is halted, as well as shock phases induced by boundaries and the bulk of the system. Also, it is found that spontaneous symmetry-breaking phenomena are induced even for very few particles in the system. Moreover, we thoroughly examine the location of shocks by varying the parameters controlling the system's boundaries, providing insights into possible phase transitions.

Selective Encapsulation and Chiral Induction of C60 and C70 Fullerenes by Axially Chiral Porous Aromatic Cages

AbstractChiral induction has been an important topic in chemistry, not only for its relevance in understanding the mysterious phenomenon of spontaneous symmetry breaking in nature but also due to its critical implications in medicine and the chiral industry. The induced chirality of fullerenes by host–guest interactions has been rarely reported, mainly attributed to their chiral resistance from high symmetry and challenges in their accessibility. Herein, we report two new pairs of chiral porous aromatic cages (PAC), R‐PAC‐2, S‐PAC‐2 (with Br substituents) and R‐PAC‐3, S‐PAC‐3 (with CH3 substituents) enantiomers. PAC‐2, rather than PAC‐3, achieves fullerene encapsulation and selective binding of C70 over C60 in fullerene carbon soot. More significantly, the occurrence of chiral induction between R‐PAC‐2, S‐PAC‐2 and fullerenes is confirmed by single‐crystal X‐ray diffraction and the intense CD signal within the absorption region of fullerenes. DFT calculations reveal the contribution of electrostatic effects originating from face‐to‐face arene‐fullerene interactions dominate C70 selectivity and elucidate the substituent effect on fullerene encapsulation. The disturbance from the differential interactions between fullerene and surrounding chiral cages on the intrinsic highly symmetric electronic structure of fullerene could be the primary reason accounting for the induced chirality of fullerene.

Selective Encapsulation and Chiral Induction of C60 and C70 Fullerenes by Axially Chiral Porous Aromatic Cages.

Chiral induction has been an important topic in chemistry, not only for its relevance in understanding the mysterious phenomenon of spontaneous symmetry breaking in nature but also due to its critical implications in medicine and the chiral industry. The induced chirality of fullerenes by host-guest interactions has been rarely reported, mainly attributed to their chiral resistance from high symmetry and challenges in their accessibility. Herein, we report two new pairs of chiral porous aromatic cages (PAC), R-PAC-2, S-PAC-2 (with Br substituents) and R-PAC-3, S-PAC-3 (with CH3 substituents) enantiomers. PAC-2, rather than PAC-3, achieves fullerene encapsulation and selective binding of C70 over C60 in fullerene carbon soot. More significantly, the occurrence of chiral induction between R-PAC-2, S-PAC-2 and fullerenes is confirmed by single-crystal X-ray diffraction and the intense CD signal within the absorption region of fullerenes. DFT calculations reveal the contribution of electrostatic effects originating from face-to-face arene-fullerene interactions dominate C70 selectivity and elucidate the substituent effect on fullerene encapsulation. The disturbance from the differential interactions between fullerene and surrounding chiral cages on the intrinsic highly symmetric electronic structure of fullerene could be the primary reason accounting for the induced chirality of fullerene.