Translational Symmetry Research Articles

Eigenstate Thermalisation Hypothesis for Translation Invariant Spin Systems

We prove the Eigenstate Thermalisation Hypothesis (ETH) for local observables in a typical translation invariant system of quantum spins with L-body interactions, where L is the number of spins. This mathematically verifies the observation first made by Santos and Rigol (Phys Rev E 82(3):031130, 2010, https://doi.org/10.1103/PhysRevE.82.031130) that the ETH may hold for systems with additional translational symmetries for a naturally restricted class of observables. We also present numerical support for the same phenomenon for Hamiltonians with local interaction.

The spontaneous disorder according to flat bands

A bosonic Creutz–Hubbard ladder is studied by solving the interacting Bose–Hubbard model, with the help of exact diagonalization method. Under certain parameters, a single-particle energy spectrum possessing two flat energy bands is obtained. Due to these flat bands, we identify the spontaneous disorder in the presence of interaction, which breaks the translational symmetry of lattice system. Out of flat bands and taking a flux quanta as , the checkerboard phase associated with Meissner currents can be observed, as well as the regular biased ladder (BL) phase, which particularly presents a novel type of interlaced chiral current. We further figure out a modulated BL phase, whose imbalance on the occupancies between two legs is constant, whereas the density distribution on either leg oscillates periodically, followed by compound currents.

The Process Analysis Method of SAR Target Recognition in Pre-Trained CNN Models.

Recently, attention has been paid to the convolutional neural network (CNN) based synthetic aperture radar (SAR) target recognition method. Because of its advantages of automatic feature extraction and the preservation of translation invariance, the recognition accuracies are stronger than traditional methods. However, similar to other deep learning models, CNN is a "black-box" model, whose working process is vague. It is difficult to locate the decision reasons. Because of this, we focus on the process analysis of a pre-trained CNN model. The role of the processing to feature extraction and final recognition decision is discussed. The discussed components of CNN models are convolution, activation function, and full connection. Here, the convolution processing can be deemed as image filtering. The activation function provides a nonlinear element of processing. Moreover, the fully connected layers can also further extract features. In the experiment, four classical CNN models, i.e., AlexNet, VGG16, GoogLeNet, and ResNet-50, are trained by public MSTAR data, which can realize ten-category SAR target recognition. These pre-trained CNN models are processing objects of the proposed process analysis method. After the analysis, the content of the SAR image target features concerned by these pre-trained CNN models is further clarified. In summary, we provide a paradigm to process the analysis of pre-trained CNN models used for SAR target recognition in this paper. To some degree, the adaptability of these models to SAR images is verified.

Discovery of charge order in a cuprate Mott insulator

Copper oxide superconductors universally exhibit multiple forms of electronically ordered phases that break the native translational symmetry of the CuO2 planes. In underdoped cuprates with correlated metallic ground states, charge/spin stripes and incommensurate charge density waves (CDWs) have been experimentally observed over the years, while early theoretical studies also predicted the emergence of a Coulomb-frustrated 'charge crystal' phase in the very lightly doped, insulating limit of CuO2 planes. Here, we search for signatures of CDW order in very lightly hole-doped cuprates from the 123 family RBa2Cu3O7 - δ (RBCO; R: Y or rare earth), by using resonant X-ray scattering, electron transport, and muon spin rotation measurements to resolve the electronic and magnetic ground states fully. Specifically, Pr is used to substitute Y at the R-site to systematically suppress the superconductivity and access the extremely low hole-doping regime of the cuprate phase diagram without changing the oxygen stoichiometry. X-ray scattering data taken on Pr-doped YBCO thin films reveal an in-plane CDW order that follows the same linear evolution of wave vector versus hole concentration as oxygen-underdoped YBCO but extends all the way to the insulating and magnetically ordered Mott limit. Combined with the recent observation of charge crystal phase on an insulating surface of Bi2Sr2CaCu2O8 + z, our results in RBCO suggest that this electronic symmetry breaking is universally present in very lightly doped CuO2 planes. These findings bridge the gap between the Mott insulating state and the underdoped metallic state and underscore the prominent role that Coulomb-frustrated electronic phase separation plays among all cuprates.

Tensor Network Efficiently Representing Schmidt Decomposition of Quantum Many-Body States.

Efficient methods to access the entanglement of a quantum many-body state, where the complexity generally scales exponentially with the system size N, have long been a concern. Here we propose the Schmidt tensor network state (Schmidt TNS) that efficiently represents the Schmidt decomposition of finite- and even infinite-size quantum states with nontrivial bipartition boundary. The key idea is to represent the Schmidt coefficients (i.e., entanglement spectrum) and transformations in the decomposition to tensor networks (TNs) with linearly scaled complexity versus N. Specifically, the transformations are written as the TNs formed by local unitary tensors, and the Schmidt coefficients are encoded in a positive-definite matrix product state (MPS). Translational invariance can be imposed on the TNs and MPS for the infinite-size cases. The validity of Schmidt TNS is demonstrated by simulating the ground state of the quasi-one-dimensional spin model with geometrical frustration. Our results show that the MPS encoding the Schmidt coefficients is weakly entangled even when the entanglement entropy of the decomposed state is strong. This justifies the efficiency of using MPS to encode the Schmidt coefficients, and promises an exponential speedup on the full-state sampling tasks.

Symmetry of Dirac two-oscillator system, gauge-invariance, and Landau problem

Role of gauge symmetry in the proton spin problem has intricate and unresolved aspects. One of the interesting approaches to gain physical insights is to explore the Landau problem in this context. A detailed study using the group theoretic method to understand the Landau problem establishes the significance of the gauge transformation intimately related with the space translation symmetry. An important implication of this result is that the E(2)-like Wigner’s little group for massless particles could throw more light on the question of gauge symmetry in QED and QCD. A generalized Landau-Zeeman Hamiltonian is proposed in which Dirac two-oscillator system and the symmetry of the group SO(3,2) become important. It is argued that nontrivial topology of pure gauge field holds promise to resolve the unsettled questions.

Uniqueness in phaseless inverse electromagnetic scattering problem with known superposition of incident electric dipoles

In this paper, we consider the uniqueness of the phaseless inverse electromagnetic scattering problems. The phaseless data are the modulus of the tangential component of the far‐field pattern measured on the unit sphere and generated by the scatterer as well as the electric dipoles. Based on the superpositions of two electric dipoles with different positions and polarization vectors as the incident fields, the translation invariance property of the phaseless far‐field pattern corresponding to a single incident wave can be broken. A rigorous argument is given to illustrate that the location, shape, and boundary conditions of the obstacle or the refractive index of the medium can be uniquely determined by the phaseless far‐field data. Different from the existing method, we do not need to introduce an additional reference sphere. Thus our method makes the argument of uniqueness much simpler and clearer.

Time Translation Symmetry Breaking in an Isolated Spin-Orbit-Coupled Fluid of Light.

We study the interplay between intrinsic spin-orbit coupling and nonlinear photon-photon interactions in a nonparaxial, elliptically polarized fluid of light propagating in a bulk Kerr medium. We find that in situations where the nonlinear interactions induce birefringence, i.e., a polarization-dependent nonlinear refractive index, their interplay with spin-orbit coupling results in an interference between the two polarization components of the fluid traveling at different wave vectors, which entails the breaking of translation symmetry along the propagation direction. This phenomenon leads to a Floquet band structure in the Bogoliubov spectrum of the fluid, and to characteristic oscillations of its intensity correlations. We characterize these oscillations in detail and point out their exponential growth at large propagation distances, revealing the presence of parametric resonances.

COHERENCE ASSEMBLY IN STRUCTURES WITH HEAVY METAL CLUSTERS

A crystallographic analysis of two structures, monoclinic Nа8[{Re4(PO)4}(CN)12]⋅18H2O⋅CH3OH (I) and orthorhombic [(C6H5)4P]4 [Ta6I12(CN6)] (II), in which clusters of heavy atoms are significantly rarefied in space, so that their mutual arrangement cannot be explained only in terms of chemical interaction, has been performed. In structure I the crystallographic planes with a high atomic density (“skeletal” planes) are located in the regions with dhkl = 10–5.5 Å and dhkl 3 Å. The planes in which atomic groups [Re4(PO)4] (playing the role of unified bulk objects) are concentrated are in fact selected in the first region. In the second region, ordering is implemented at the level of individual atoms. A crystallographic analysis showed that the structure basis is determined by the sites of heavy Re cations. A striking fact is that there are 1152 subcells and only 32 Re atoms per unit cell in this structure; i.e., only the fraction of 1/144 provides the basis of structure stability. In structure II “skeletal” planes are also absent in the range of dhkl from ∼7 to ∼4 Å. The planes in the range of large dhkl characterize cluster ordering, whereas the planes in the range of small dhkl characterize ordering of separate atoms. The geometry and local symmetry of the cluster group (Та6 octahedron) dictates the basis of translational symmetry—unified sublattice of sites, most of which are free of these atoms. The considered structures demonstrate the key role of heavy atoms in the formation of translational symmetry—the fundamental difference of the crystalline state from other condensed states. The newly formed structure retains partially local symmetry of cores (templates) of atomic groups, bound by strong chemical interactions, including the interactions between heavy and light atoms. The process of formation of a crystal structure from randomly oriented and randomly located templates—coherence assembly—is implemented according to the laws of dynamics of elastic media, where masses of atoms rather than their chemical characteristics are important.

Nonanomalous heat transport in a one-dimensional composite chain.

Translation-invariant low-dimensional systems are known to exhibit anomalous heat transport. However, there are systems, such as the coupled-rotor chain, where translation invariance is satisfied, yet transport remains diffusive. It has been argued that the restoration of normal diffusion occurs due to the impossibility of defining a global stretch variable with a meaningful dynamics. In this Letter, an alternative mechanism is proposed, namely, that the transition to anomalous heat transport can occur at a scale that, under certain circumstances, may diverge to infinity. To illustrate the mechanism, I consider the case of a composite chain that conserves local energy and momentum as well as global stretch, and at the same time obeys, in the continuum limit, Fourier's law of heat transport. It is shown analytically that for vanishing elasticity the stationary temperature profile of the chain is linear; for finite elasticity, the same property holds in the continuum limit.

Smectic pair-density-wave order in EuRbFe4As4.

The pair density wave (PDW) is a superconducting state in which Cooper pairs carry centre-of-mass momentum in equilibrium, leading to the breaking of translational symmetry1-4. Experimental evidence for such a state exists in high magnetic field5-8 and in some materials that feature density-wave orders that explicitly break translational symmetry9-13. However, evidence for a zero-field PDW state that exists independent of other spatially ordered states has so far been elusive. Here we show that such a state exists in the iron pnictide superconductor EuRbFe4As4, a material that features co-existing superconductivity (superconducting transition temperature (Tc) ≈ 37 kelvin) and magnetism (magnetic transition temperature (Tm) ≈ 15 kelvin)14,15. Using spectroscopic imaging scanning tunnelling microscopy (SI-STM) measurements, we show that the superconducting gap at low temperature has long-range, unidirectional spatial modulations with an incommensurate period of about eight unit cells. Upon increasing the temperature above Tm, the modulated superconductor disappears, but a uniform superconducting gap survives to Tc. When an external magnetic field is applied, gap modulations disappear inside the vortex halo. The SI-STM and bulk measurements show the absence of other density-wave orders, indicating that the PDW state is a primary, zero-field superconducting state in this compound. Both four-fold rotational symmetry and translation symmetry are recovered above Tm, indicating that the PDW is a smectic order.

Heat transfer in MHD thin film flow with concentration using lie point symmetry approach

Heat and mass transfer in a viscous film on an unsteady stretching surface in the presence of a variable magnetic field is investigated using Lie symmetry analysis. Combination of translational and scaling Lie point symmetries is used to obtain invariants of the model. The deduced invariants provide a new generalized class of similarity transformations that have not been used before. These transformations restructure the governing problem in a system of eight-parameter nonlinear ordinary differential equations. Analytic series solutions are obtained for the resulting system of equations for different values of these parameters and are illustrated graphically. We found that coefficients of the translational symmetries do not have any effect on the solutions however, coefficients of the scaling symmetries significantly affect the variation of temperature and concentration distribution across the fluid film and help in controlling the heat and mass transfer rates. Also, increasing the value of unsteadiness and magnetic parameter decreases the film thickness and promotes a uniform temperature and concentration across film thickness. Furthermore, we found that at the lower values of Prandtl and Schmidt number, diffusion dominates the heat and mass transfer while at the higher values advection dominates the heat and mass transfer.

Aftershocks and Fluctuating Diffusivity.

The Omori-Utsu law shows the temporal power-law-like decrease of the frequency of earthquake aftershocks and, interestingly, is found in a variety of complex systems/phenomena exhibiting catastrophes. Now, it may be interpreted as a characteristic response of such systems to large events. Here, hierarchical dynamics with the fast and slow degrees of freedom is studied on the basis of the Fokker-Planck theory for the load-state distribution to formulate the law as a relaxation process, in which diffusion coefficient in the space of the load state is treated as a fluctuating slow variable. The evolution equation reduced from the full Fokker-Planck equation and its Green's function are analyzed for the subdynamics governing the load state as the fast degree of freedom. It is shown that the subsystem has the temporal translational invariance in the logarithmic time, not in the conventional time, and consequently the aging phenomenon appears.

Photonic Dirac cone and topological transition in a moving dielectric slab

The moving media theory is applied to a photonic confined structure which is a continuous dielectric slab waveguide with the uniaxial anisotropy and without the discrete translational symmetry. The moving effect not only brings about non-reciprocity to the whole photonic band structure in the co-moving and counter-moving directions, but also leads to the topological transition of local degenerate points within the band diagram. We demonstrate through calculation that the type-II Dirac point can be turned into type-I Dirac point when the uniaxial slab is moving over certain speed. Our results provide a new approach to regulate the topology of degeneracy for two-dimensional photonic bands in the continuous translational symmetry condition.

Anomalous fractal scaling in two-dimensional electric networks

Much of the qualitative nature of physical systems can be predicted from the way it scales with system size. Contrary to the continuum expectation, we observe a profound deviation from logarithmic scaling in the impedance of a two-dimensional LC circuit network. We find this anomalous impedance contribution to sensitively depend on the number of nodes N in a curious erratic manner and experimentally demonstrate its robustness against perturbations from the contact and parasitic impedance of individual components. This impedance anomaly is traced back to a generalized resonance condition reminiscent of Harper’s equation for electronic lattice transport in a magnetic field, even though our circuit network does not involve magnetic translation symmetry. It exhibits an emergent fractal parametric structure of anomalous impedance peaks for different N that cannot be reconciled with a continuum theory and does not correspond to regular waveguide resonant behavior. This anomalous fractal scaling extends to the transport properties of generic systems described by a network Laplacian whenever a resonance frequency scale is simultaneously present.

A Domain-Knowledge-Inspired Music Embedding Space and a Novel Attention Mechanism for Symbolic Music Modeling

Following the success of the transformer architecture in the natural language domain, transformer-like architectures have been widely applied to the domain of symbolic music recently. Symbolic music and text, however, are two different modalities. Symbolic music contains multiple attributes, both absolute attributes (e.g., pitch) and relative attributes (e.g., pitch interval). These relative attributes shape human perception of musical motifs. These important relative attributes, however, are mostly ignored in existing symbolic music modelling methods with the main reason being the lack of a musically-meaningful embedding space where both the absolute and relative embeddings of the symbolic music tokens can be efficiently represented. In this paper, we propose the Fundamental Music Embedding (FME) for symbolic music based on a bias-adjusted sinusoidal encoding within which both the absolute and the relative attributes can be embedded and the fundamental musical properties (e.g., translational invariance) are explicitly preserved. Taking advantage of the proposed FME, we further propose a novel attention mechanism based on the relative index, pitch and onset embeddings (RIPO attention) such that the musical domain knowledge can be fully utilized for symbolic music modelling. Experiment results show that our proposed model: RIPO transformer which utilizes FME and RIPO attention outperforms the state-of-the-art transformers (i.e., music transformer, linear transformer) in a melody completion task. Moreover, using the RIPO transformer in a downstream music generation task, we notice that the notorious degeneration phenomenon no longer exists and the music generated by the RIPO transformer outperforms the music generated by state-of-the-art transformer models in both subjective and objective evaluations. The code of the proposed method is available online: github.com/guozixunnicolas/FundamentalMusicEmbedding.

Phases of the hard-plate lattice gas on a three-dimensional cubic lattice.

We study the phase diagram of a lattice gas of 2×2×1 hard plates on the three-dimensional cubic lattice. Each plate covers an elementary plaquette of the cubic lattice, with the constraint that a site can belong to utmost one plate. We focus on the isotropic system, with equal fugacities for the three orientations of plates. We show, using grand canonical Monte Carlo simulations, that the system undergoes two phase transitions when the density of plates is increased: the first from a disordered fluid phase to a layered phase, and the second from the layered phase to a sublattice-ordered phase. In the layered phase, the system breaks up into disjoint slabs of thickness two along one spontaneously chosen Cartesian direction, corresponding to a twofold (Z_{2}) symmetry breaking of translation symmetry along the layering direction. Plates with normals perpendicular to this layering direction are preferentially contained entirely within these slabs, while plates straddling two adjacent slabs have a lower density, thus breaking the symmetry between the three types of plates. We show that the slabs exhibit two-dimensional power-law columnar order even in the presence of a nonzero density of vacancies. In contrast, interslab correlations of the two-dimensional columnar order parameter decay exponentially with the separation between the slabs. In the sublattice-ordered phase, there is twofold symmetry breaking of lattice translation symmetry along all three Cartesian directions. We present numerical evidence that the disordered to layered transition is continuous and consistent with universality class of the three-dimensional O(3) model with cubic anisotropy, while the layered to sublattice transition is first-order in nature.

Spontaneous layering and power-law order in the three-dimensional fully packed hard-plate lattice gas.

We obtain the phase diagram of fully packed hard plates on the cubic lattice. Each plate covers an elementary plaquette of the cubic lattice and occupies its four vertices, with each vertex of the cubic lattice occupied by exactly one such plate. We consider the general case with fugacities s_{μ} for "μ plates," whose normal is the μ direction (μ=x,y,z). At and close to the isotropic point, we find, consistent with previous work, a phase with long-range sublattice order. When two of the fugacities s_{μ_{1}} and s_{μ_{2}} are comparable, and the third fugacity s_{μ_{3}} is much smaller, we find a spontaneously layered phase. In this phase, the system breaks up into disjoint slabs of width two stacked along the μ_{3} axis. μ_{1} and μ_{2} plates are preferentially contained entirely within these slabs, while plates straddling two successive slabs have a lower density. This corresponds to a twofold breaking of translation symmetry along the μ_{3} axis. In the opposite limit, with μ_{3}≫μ_{1}∼μ_{2}, we find a phase with long-range columnar order, corresponding to simultaneous twofold symmetry breaking of lattice translation symmetry in directions μ_{1} and μ_{2}. The spontaneously layered phases display critical behavior, with power-law decay of correlations in the μ_{1} and μ_{2} directions when the slabs are stacked in the μ_{3} direction, and represent examples of "floating phases" discussed earlier in the context of coupled Luttinger liquids and quasi-two-dimensional classical systems. We ascribe this remarkable behavior to the constrained motion of defects in this phase, and we sketch a coarse-grained effective field theoretical understanding of the stability of power-law order in this unusual three-dimensional floating phase.

OctFormer: Efficient Octree-Based Transformer for Point Cloud Compression with Local Enhancement

Point cloud compression with a higher compression ratio and tiny loss is essential for efficient data transportation. However, previous methods that depend on 3D convolution or frequent multi-head self-attention operations bring huge computations. To address this problem, we propose an octree-based Transformer compression method called OctFormer, which does not rely on the occupancy information of sibling nodes. Our method uses non-overlapped context windows to construct octree node sequences and share the result of a multi-head self-attention operation among a sequence of nodes. Besides, we introduce a locally-enhance module for exploiting the sibling features and a positional encoding generator for enhancing the translation invariance of the octree node sequence. Compared to the previous state-of-the-art works, our method obtains up to 17% Bpp savings compared to the voxel-context-based baseline and saves an overall 99% coding time compared to the attention-based baseline.

Switching Chirality in Arrays of Shape-Reconfigurable Spindle Microparticles.

The giant circular photo-galvanic effect is realized in chiral metals when illuminated by circularly polarized light. However, the structure itself is not switchable nor is the crystal chirality in the adjacent chiral domains. Here spindle-shaped liquid crystalline elastomer microparticles that can switch from prolate to spherical to oblate reversibly upon heating above the nematic to isotropic transition temperature are synthesized. When arranged in a honeycomb lattice, the continuous shape change of the microparticles leads to lattice reconfiguration, from a right-handed chiral state to an achiral one, then to a left-handed chiral state, without breaking the translational symmetry. Accordingly, the sign of rotation of the polarized light passing through the lattices changes as measured by time-domain terahertz spectroscopy. Further, it can locally alter the chirality in the adjacent domains using near-infrared light illumination. The reconfigurable chiral microarrays will allow us to explore non-trivial symmetry-protected transport modes of topological lattices at the light-matter interface. Specifically, the ability to controllably create chiral states at the boundary of the achiral/chiral domains will lead to rich structures emerging from the interplay of symmetry and topology.