Translational Symmetry Research Articles

Parent Berry Curvature and the Ideal Anomalous Hall Crystal

We study a model of electrons moving in a parent band of uniform Berry curvature. At sufficiently high parent Berry curvature, we show that strong repulsive interactions generically lead to the formation of an anomalous Hall crystal: a topological state with spontaneously broken continuous translation symmetry. Our results are established via a mapping to a problem of Wigner crystallization in a regular 2D electron gas. Interestingly, we find that a periodic electrostatic potential induces a competing state with opposite Chern number. Our theory offers a unified perspective for understanding several aspects of the recently observed integer and fractional quantum anomalous Hall effects in rhombohedral multilayer graphene and provides a recipe for engineering new topological states. Published by the American Physical Society 2024

Higher-order and fractional discrete time crystals in Floquet-driven Rydberg atoms

Higher-order and fractional discrete time crystals (DTCs) are exotic phases of matter where the discrete time translation symmetry is broken into higher-order and non-integer category. Generation of these unique DTCs has been widely studied theoretically in different systems. However, no current experimental methods can probe these higher-order and fractional DTCs in any quantum many-body systems. We demonstrate an experimental approach to observe higher-order and fractional DTCs in Floquet-driven Rydberg atomic gases. We have discovered multiple n-DTCs with integer values of n = 2, 3, and 4, and others ranging up to 14, along with fractional n-DTCs with n values beyond the integers. The system response can transition between adjacent integer DTCs, during which the fractional DTCs are investigated. Study of higher-order and fractional DTCs expands fundamental knowledge of non-equilibrium dynamics and is promising for discovery of more complex temporal symmetries beyond the single discrete time translation symmetry.

Superconductivity and nematic order in a new titanium-based kagome metal CsTi3Bi5 without charge density wave order

The cascade of correlated topological quantum states in the newly discovered vanadium-based kagome superconductors, AV3Sb5 (A = K, Rb, and Cs), with a Z2 topological band structure has sparked immense interest. Here, we report the discovery of superconductivity and electronic nematic order in high-quality single-crystals of a new titanium-based kagome metal, CsTi3Bi5, that preserves the translation symmetry, in stark contrast to the charge density wave superconductor AV3Sb5. Transport and magnetic susceptibility measurements show superconductivity with an onset superconducting transition temperature Tc of approximately 4.8 K. Using the scanning tunneling microscopy/spectroscopy and Josephson scanning tunneling spectroscopy, we demonstrate that the single crystals of CsTi3Bi5 exhibit two distinct superconducting gaps. Furthermore, the superconducting gaps break the six-fold crystal rotational symmetry down to two-fold. At low energies, we find that the quasiparticle interference patterns exhibit rotational-symmetry-breaking C2 patterns, revealing a nematic ordered normal state with the same nematic direction as in the superconducting state. Our findings uncover a novel superconducting state in CsTi3Bi5 and provide new insights for the intrinsic electron liquid crystal phases in kagome superconductors.

Observation of vortices in a dipolar supersolid.

Supersolids are states of matter that spontaneously break two continuous symmetries: translational invariance owing to the appearance of a crystal structure and phase invariance owing to phase locking of single-particle wavefunctions, responsible for superfluid phenomena. Although originally predicted to be present in solid helium1-5, ultracold quantum gases provided a first platform to observe supersolids6-10, with particular success coming from dipolar atoms8-12. Phase locking in dipolar supersolids has been investigated through, for example, measurements of the phase coherence8-10 and gapless Goldstone modes13, but quantized vortices, a hydrodynamic fingerprint of superfluidity, have not yet been observed. Here, with the prerequisite pieces at our disposal, namely a method to generate vortices in dipolar gases14,15 and supersolids with two-dimensional crystalline order11,16,17, we report on the theoretical investigation and experimental observation of vortices in the supersolid phase (SSP). Our work reveals a fundamental difference in vortex seeding dynamics between unmodulated and modulated quantum fluids. This opens the door to study the hydrodynamic properties of exotic quantum systems with numerous spontaneously broken symmetries, in disparate domains such as quantum crystals and neutron stars18.

Infinitely rugged intra-cage potential energy landscape in metallic glasses caused by many-body interaction

The absence of translational symmetry in glassy materials poses a significant challenge in establishing effective structure-property relationships in real space. Consequently, the potential energy landscape (PEL) in phase space is widely utilized to comprehend the complex phenomena in glasses. The classical PEL features a two-scale profile comprising mega-basins and sub-basins, corresponding to α-relaxations (e.g. glass transition) and β-relaxations (e.g. local cage-breaking atomic rearrangements), respectively. Recent studies, however, reveal that sub-basins are not smooth and contain finer structures, the origins of which remain elusive. Here we probe the smoothness of sub-basin bottoms in glasses' PEL by introducing small intra-cage cyclic loading and then measuring the net changes in atomic-level stresses. Compared to glasses with pair interaction, glasses with many-body interaction exhibit orders-of-magnitude larger and loading-dependent stress changes even before the first cage-breaking event takes place, which reflect much more feature-rich sub-basins. We further demonstrate this stark contrast stems from the spatial distribution of individual atom's constraining force field. Specifically, at vanishing perturbations, many-body interactions disrupt the positive-definite synchrony in energy variations of the perturbed atom and the whole system, causing inherently less confined atomic responses and infinitely rugged sub-basins. The implications of these findings for the selective addition or removal of fine structures in the PEL and the subsequent tuning of glassy materials' responses to external stimuli are also explored.

The generalized quasilinear approximation of two-dimensional Rayleigh–Bénard convection

In the generalized quasilinear approximation (GQLA) (Marston et al., Phys. Rev. Lett., vol. 116, 2016, 214501), a threshold wavenumber ( $k_0$ ) in the direction of translational symmetry segregates the total into large- ( $l$ ) and small-scale ( $h$ ) fields. While the governing equation for the large-scale field is fully nonlinear, that for the small scales is linearized with respect to the large-scale field. In addition, some nonlinear triad interactions are omitted in the GQLA. Herein, the GQLA is applied to two-dimensional planar Rayleigh–Bénard convection (RBC). A scale separation between the large-scale convection rolls and small-scale turbulent fluctuations is typical in RBC. The present work explores the efficacy of GQLA in capturing the scale-by-scale energy transfer processes in RBC. The initial condition for the GQLA simulations was either the statistically stationary state obtained in direct numerical simulation (DNS) or random fluctuations superimposed on the linear conductive temperature profile and $\boldsymbol {u}=0$ . The GQLA simulations can capture the convection rolls for $k_0$ larger than or equal to the dominant wavenumber for thermal driving of the flow ( $k_0 \ge k_{\hat {Q}}$ ). Additionally, the GQLA emulates the fully nonlinear dynamics for $k_0$ larger than or equal to the first harmonic of the convection-roll wavenumber ( $k_0 \ge 2k_{roll}$ ). In the intermediate regime with $2k_{roll} > k_0 \ge k_{\hat {Q}}$ , the dynamics captured in GQLA simulations is different from the DNS. In DNS, two primary energy transfer processes dominate: (i) the energy transfer to/from the convection rolls and (ii) the scale-by-scale inverse kinetic energy and forward thermal energy cascades mediated by the convection rolls. The fully nonlinear dynamics is emulated by GQLA when these energy transfer processes are faithfully reproduced. Utilizing the framework of altering the triad interactions in GQLA, an additional intrusive calculation, including target triad interactions, is performed here to study their influence. This intrusive calculation shows that the convection rolls are not captured in GQLA for $k_0 < k_{\hat {Q}}$ because of the exclusion of the $h \to l \to l$ and $l \to h \to h$ triad interactions in GQLA. The inclusion of these triad interactions in the intrusive calculation yields the convection rolls, and the reproduced dynamics is similar to that of the intermediate GQLA regime with $2k_{roll} > k_0 \ge k_{\hat {Q}}$ .

Beyond the Blackwell order in dichotomies

I establish a translation invariance property of the Blackwell order for dichotomies and show that the norm of the distance from the identity matrix may be interpreted as a measure of informativeness. The better experiment is closer to the fully revealing experiment; this measure extends the Blackwell order, is complete, and prior-independent.

Elastic Wave Scattering off a Single and Double Array of Periodic Defects

The scattering problem of elastic waves impinging on periodic single and double arrays of parallel cylindrical defects is considered for isotropic materials. An analytic expression for the scattering matrix is obtained by means of the Lippmann–Schwinger formalism and analyzed in the long-wavelength approximation. It is proved that, for a specific curve in the space of physical and geometrical parameters, the scattering is dominated by resonances. The shear mode polarized parallel to the cylinders is decoupled from the other two polarization modes due to the translational symmetry along the cylinders. It is found that a relative mass density and relative Lamé coefficients of the scatterers give opposite contributions to the width of resonances in this mode. A relation between the Bloch phase and material parameters is found to obtain a global minimum of the width. The minimal width is shown to vanish in the leading order of the long wavelength limit for the single array. This new effect is not present in similar acoustic and photonic systems. The shear and compression modes in a plane perpendicular to the cylinders are coupled due to the normal traction boundary condition and have different group velocities. For the double array, it is proved that, under certain conditions on physical and geometrical parameters, there exist resonances with the vanishing width, known as Bound States in the Continuum (BSC). Necessary and sufficient conditions for the existence of BSC are found for any number of open diffraction channels. Analytic BSC solutions are obtained. Spectral parameters of BSC are given in terms of the Bloch phase and group velocities of the shear and compression modes.

Polar charge density wave in a superconductor with crystallographic chirality

Symmetry plays an important role in determining the physical properties in condensed matter physics, as the symmetry operations of any physical property must include the symmetry operations of the point group of the crystal. As a consequence, crystallographic polarity and chirality are expected to have an impact on the Cooper pairing in a superconductor. While superconductivity with crystallographic polarity and chirality have both been found in a few crystalline phases separately; however, their coexistence and material realizations have not been studied. Here, by utilizing transport, Raman scattering, and transmission electron microscopy, we unveil a unique realization of superconductivity in single-crystalline Mo3Al2C (superconducting Tc=8 K) with a polar charge-density-wave phase and well-defined crystallographic chirality. We show that the intriguing charge density wave order leads to a noncentrosymmetric-nonpolar to polar transition below T*=155K via breaking both the translational and rotational symmetries. Superconductivity emerges in this polar and chiral crystal structure below Tc=8 K. Our results establish that Mo3Al2C is a superconductor with crystallographic polarity and chirality simultaneously, and motivate future studies of unconventional superconductivity in this category.

On the computation of lattice sums without translational invariance

This paper introduces a new method for the efficient computation of oscillatory multidimensional lattice sums in geometries with boundaries. Such sums are ubiquitous in both pure and applied mathematics and have immediate applications in condensed matter and topological quantum physics. The challenge in their evaluation results from the combination of singular long-range interactions with the loss of translational invariance caused by the boundaries, rendering standard tools like Ewald summation ineffective. Our work shows that these lattice sums can be generated from a generalization of the Riemann zeta function to multidimensional non-periodic lattice sums. We put forth a new representation of this zeta function together with a numerical algorithm that ensures exponential convergence across an extensive range of geometries. Notably, our method’s runtime is influenced only by the complexity of the considered geometries and not by the number of particles, providing the foundation for efficient and precise simulations of macroscopic condensed matter systems. We showcase the practical utility of our method by computing interaction energies in a three-dimensional crystal structure with 3 × 10 23 3\times 10^{23} particles. Our method’s accuracy is demonstrated through extensive numerical experiments. A reference implementation is provided online along with this article.

The giant dipole resonance (GDR) in odd-mass 181–195Pt nuclei

Abstract This manuscript theoretically examines the Giant Dipole Resonance (GDR) in odd-mass 181-195Pt nuclei with the Translational and Galileo Invariant Qusiparticle Phonon Nuclear Model (TGI-QPNM) for the first time. TGI-QPNM includes axially symmetric Woods-Saxon potential, isovector dipole-dipole interaction and restoration forces for spontaneously broken Galilean and Translation symmetries of the nuclear Hamiltonian. Therefore, TGI-QPNM makes eliminating the spurious contributions in the E1 spectrum possible. The obtained results show that the odd-mass 181-195Pt isotopes have a two-peak structure. In 181-187Pt isotopes, while the second peak is higher than the first, in 189-195Pt isotopes, it’s the opposite. The photo-absorption result for the 195Pt is in reasonably good agreement with the experimental data.

Manipulative Single Electric Dipole with Spontaneous Translational Symmetry Breaking in a Two-Dimensional Crystal.

It is widely acknowledged that quantum entities with minimal mass cannot undergo spontaneous symmetry breaking due to strong quantum fluctuations. Here, we report the discovery of a positionally settled single electric dipole that can be manipulated and electrically polarized in a monolayer CoCl2-graphite heterostructure, which demonstrates an unprecedented example of spontaneous lattice-translational-symmetry breaking. Scanning tunneling microscopy and atomic force microscopy show that the solitons are intrinsic paraelectric dipoles driven by synchronous charge-lattice distortion around individual CoCl6 octahedrons. Both the dipole moment and lateral position of the soliton can be manipulated by the electric field exerted from the tip, which offers polarity-switchable and layout-designable electrostatic potential landscapes that determine the band bending configuration. This study exemplifies a brand-new type of local charge-lattice order, appealing for further research on the mechanism underlying the soliton robustness, and the electrically and positionally controllable single dipole supports the feasibility of band tailoring in device applications.

Polarity of reticular faces and the thermodynamic state of the face system of a crystal

A new concept is introduced into crystallography – the polarity of the reticular faces of a crystal. Reticular faces are divided into basic and additional. The basal faces, systematic in a facet system, only increase as the crystal grows. The shape of the crystal is invariant, stable within the limits of its stability and does not affect the state of its equilibrium. Additional faces, on the contrary, shrink and disappear, and therefore are variable extensive parameters of the facet system. They do not form their own system and associate with the basic ones, determining the variation of the facet system. The polar system is nonequilibrium, spontaneously and irreversibly transforms into the final basic facet system and is in quasi-static equilibrium. The habit of a basic crystal of an invariant shape changes during its growth according to the mechanism of translational symmetry.

Topological transport of a classical droplet in a lattice of time

AbstractThouless charge pumps are quantum mechanical devices whose operation relies on topology. They provide the means for transporting quantum matter in space lattices with a single quantum precision. Contrasting space crystals that spontaneously break a continuous spatial translation symmetry and form crystals in space, time crystals have emerged as novel states of matter that organise into time lattices and spontaneously break a discrete time translation symmetry. The utility of Thouless pumps that enable topologically protected quantised transport of electrons and neutral atoms in spatial superlattices leads to the question if corresponding devices exist for time crystals. Here we show that topological pumps can be realised for time solids by transporting droplets of a liquid forward and backward in time lattices and we measure the topological index that characterises such pumping processes. By exploiting a synthetic time dimension, classical time crystals can circumvent the quantum tunnelling that underpins Thouless charge pumps. Our results establish topological pumping through time instead of space and pave the way for applications of time crystals.Key Points Periodically driven fluid droplets can be made to bounce persistently in a gravitational field and may undergo spontaneous time translation symmetry breaking forming classical time crystals. These classical time crystals can be pumped from one time site to another, and this pumping process is characterised by a topological index. Our observations establish a classical temporal counterpart to Thouless pumping of quantum particles in space lattices.

Progresses on Some Open Problems Related to Infinitely Many Symmetries

The quest to reveal the physical essence of the infinitely many symmetries and/or conservation laws that are intrinsic to integrable systems has historically posed a significant challenge at the confluence of physics and mathematics. This scholarly investigation delves into five open problems related to these boundless symmetries within integrable systems by scrutinizing their multi-wave solutions, employing a fresh analytical methodology. For a specified integrable system, there exist various categories of n-wave solutions, such as the n-soliton solutions, multiple breathers, complexitons, and the n-periodic wave solutions (the algebro-geometric solutions with genus n), wherein n denotes an arbitrary integer that can potentially approach infinity. Each subwave comprising the n-wave solution may possess free parameters, including center parameters ci, width parameters (wave number) ki, and periodic parameters (the Riemann parameters) mi. It is evident that these solutions are translation invariant with respect to all these free parameters. We postulate that the entirety of the recognized infinitely many symmetries merely constitute linear combinations of these finite wave parameter translation symmetries. This conjecture appears to hold true for all integrable systems with n-wave solutions. The conjecture intimates that the currently known infinitely many symmetries is not exhaustive, and an indeterminate number of symmetries remain to be discovered. This conjecture further indicates that by imposing an infinite array of symmetry constraints, it becomes feasible to derive exact multi-wave solutions. By considering the renowned Korteweg–de Vries (KdV) equation and the Burgers equation as simple examples, the conjecture is substantiated for the n-soliton solutions. It is unequivocal that any linear combination of the wave parameter translation symmetries retains its status as a symmetry associated with the particular solution. This observation suggests that by introducing a ren-variable and a ren-symmetric derivative, which serve as generalizations of the Grassmann variable and the super derivative, it may be feasible to unify classical integrable systems, supersymmetric integrable systems, and ren-symmetric integrable systems within a cohesive hierarchical framework. Notably, a ren-symmetric integrable Burgers hierarchy is explicitly derived. Both the supersymmetric and the classical integrable hierarchies are encompassed within the ren-symmetric integrable hierarchy. The results of this paper will make further progresses in nonlinear science: to find more infinitely many symmetries, to establish novel methods to solve nonlinear systems via symmetries, to find more novel exact solutions and new physics, and to open novel integrable theories such as the ren-symmetric integrable systems and the possible relations to fractional integrable systems.

Complex Implications of Translational Invariance in Polymer Field Theory

Complex Implications of Translational Invariance in Polymer Field Theory

Highly-enhanced propagation of long-range kinks in heterogeneous media

We investigate a field-theoretical model that describes the interaction between kinks and antikinks and between kinks and other heterogeneous fields and impurities. We show that the long-range kink can tunnel through a barrier created by heterogeneous fields and impurities even when the energy of the center of mass of the kink is less than the height of the energy barrier. We also study the conditions under which the kink can pass freely through a disordered medium. We introduce the concept of “effective translational symmetry”. We compare our results with those from recent papers published in this journal, where the Bogomol’nyi-Prasad-Sommerfield property is discussed.

Rotation Equivariant Proximal Operator for Deep Unfolding Methods in Image Restoration.

The deep unfolding approach has attracted significant attention in computer vision tasks, which well connects conventional image processing modeling manners with more recent deep learning techniques. Specifically, by establishing a direct correspondence between algorithm operators at each implementation step and network modules within each layer, one can rationally construct an almost "white box" network architecture with high interpretability. In this architecture, only the predefined component of the proximal operator, known as a proximal network, needs manual configuration, enabling the network to automatically extract intrinsic image priors in a data-driven manner. In current deep unfolding methods, such a proximal network is generally designed as a CNN architecture, whose necessity has been proven by a recent theory. That is, CNN structure substantially delivers the translational symmetry image prior, which is the most universally possessed structural prior across various types of images. However, standard CNN-based proximal networks have essential limitations in capturing the rotation symmetry prior, another universal structural prior underlying general images. This leaves a large room for further performance improvement in deep unfolding approaches. To address this issue, this study makes efforts to suggest a high-accuracy rotation equivariant proximal network that effectively embeds rotation symmetry priors into the deep unfolding framework. Especially, we deduce, for the first time, the theoretical equivariant error for such a designed proximal network with arbitrary layers under arbitrary rotation degrees. This analysis should be the most refined theoretical conclusion for such error evaluation to date and is also indispensable for supporting the rationale behind such networks with intrinsic interpretability requirements. Through experimental validation on different vision tasks, including blind image super-resolution, medical image reconstruction, and image de-raining, the proposed method is validated to be capable of directly replacing the proximal network in current deep unfolding architecture and readily enhancing their state-of-the-art performance. This indicates its potential usability in general vision tasks.

Two Distinct Charge Orders in Infinite-Layer PrNiO2+δ Revealed by Resonant X-Ray Diffraction

Abstract Research of infinite-layer nickelates has unveiled a broken translation symmetry, which has sparked significant interest in its root, its relationship to superconductivity, and its comparison to charge order in cuprates. In this study, resonant x-ray scattering measurements were performed on thin films of infinite-layer PrNiO2+δ . The results show significant differences in the superlattice reflection at the Ni L 3 absorption edge compared to that at the Pr M 5 resonance in their dependence on energy, temperature, and local symmetry. These differences point to two distinct charge orders, although they share the same in-plane wavevectors. It is suggested that these dissimilarities could be linked to the excess oxygen dopants, given that the resonant reflections were observed in an incompletely reduced PrNiO2+δ film. Furthermore, azimuthal analysis indicates that the oxygen ligands likely play a crucial role in the charge modulation revealed at the Ni L 3 resonance.

Pentagonal Symmetry in Aperiodic Cyclic Multiply Twinned Nano- and Mesodiamonds.

This Review provides the first comprehensive overview of the structure and properties of exotic sp3-crystalline aperiodic cyclic multiply twinned nano- (10-100 nm) and meso- (up to 1 mm) diamond particles (MTPs) exhibiting pentagonal symmetry. It spans their independent experimental discoveries (1963, 1964, 1972, and 1983) and theoretical structural insights (1993) to recent advancements. The Review focuses on high-symmetry MTPs formed by the fusion of multiple cubic diamond fragments through [111] facets. The sp3 diamond lattice of individual fragments offers a vast range of MTP varieties. These particles are shown to be a special case of aperiodic crystalline solids with limited dimensions and rotational symmetry, leading to a breakdown of translational invariance. Detailed mathematical analysis of the MTPs' lattices highlights the crucial role of central cores in determining the symmetry and effective dimensions of these structures. Both structural and kinetic aspects of the formation mechanisms of pentagonal diamond particles are considered, revealing the main role of embryo seeds (low fullerenes, polyhexacyclo[5.5.1.12,6.18,12.03,11.05,9]pentadecane (C15), and polyheptacyclo[5.5.1.12,7.19,14.03,13.06,11]octadecane (C18)) in determining the MTPs' symmetry and structure. The effective dimensions of multiply twinned diamonds are shown to be limited by structural stress caused by the mismatch of perfect pentagonal and tetrahedral dihedral angles. The extraordinary mechanical and electronic properties of multiply twinned diamonds are discussed, highlighting that hexagonal interfaces between cubic diamond fragments may determine exceptional ultrahard and quantum characteristics. The MTP X-ray diffraction spectra reveal clear pentagonal patterns with single (diamond decahedrons or dodecahedrons) or ten (diamond icosahedra) central 5-fold axes. A comparative analysis of experimental structural data and simulations at different theoretical levels demonstrates a perfect correspondence of theoretical models with crystalline lattices.